pax_global_header00006660000000000000000000000064145612712000014510gustar00rootroot0000000000000052 comment=789adda169bbff5d2c2eeb56ddf1840f04fc75e6 statistics-release-1.6.3/000077500000000000000000000000001456127120000153275ustar00rootroot00000000000000statistics-release-1.6.3/CONTRIBUTING.md000066400000000000000000000126421456127120000175650ustar00rootroot00000000000000# Contribution Guidelines #### 1. License and Documentation The **statistics** package is distributed under [GNU General Public License (GPL)](https://www.gnu.org/licenses/gpl-3.0.en.html) with few minor exceptions. Some functions are in the Public Domain. If you are submitting a few function, it should be licensed under GPLv3+ with the following header (use appropriate year, name, etc.) as shown below: ``` ## Copyright (C) 2022 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## Octave is free software; you can redistribute it and/or modify it ## under the terms of the GNU General Public License as published by ## the Free Software Foundation; either version 3 of the License, or ## (at your option) any later version. ## ## Octave is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with Octave; see the file COPYING. If not, ## see . ``` New functions should be properly documented with the embedded help file in [Texinfo](https://www.gnu.org/software/texinfo/) format. This part should be placed outside (before) the function's main body and after the License block. The Texinfo manual can be found [here](https://www.gnu.org/software/texinfo/manual/texinfo/) although reading through existing function files should do the trick. ``` ## -*- texinfo -*- ## @deftypefn {Function File} @var{p} = anova1 (@var{x}) ## ## Help file goes here. ## ## @end deftypefn ``` **Note:** the texinfo is not printed on the command window screen as it appears in the source file. Type e.g. `help anova1` to display the help document in `anova1` function in order to review it. #### 2. Demos Although examples of using a function can go into the help documentation, it is always useful to have example embedded as demos, which the user can invoke with the `demo` command. ``` e.g. >> demo anova1 ``` These should go at the end of the file after the function or any other helper functions that might be present in the file. A small demo sample is shown below. ``` %!demo %! x = meshgrid (1:6); %! x = x + normrnd (0, 1, 6, 6); %! anova1 (x, [], 'off'); ``` Note: demos can also contain comments which are printed when a demo is ran and may be very useful for the user to understand the function's usage. Embed comments in the usual manner after the `%!` starting characters in each line. For example: ``` %!demo %! x = [46 64 83 105 123 150 150]; %! c = [0 0 0 0 0 0 1]; %! f = [1 1 1 1 1 1 4]; %! ## Subtract 30.92 from x to simulate a 3 parameter wbl with gamma = 30.92 %! wblplot (x - 30.92, c, f, 0.05); ``` #### 3. BISTs (testing suite) It is also **very important** that function files contain a testing suite that will test for correct output, properly catching errors etc. BISTs should go at the bottom the file using `%!` starting characters. An example of such a testing suite is shown below. ``` %!error chi2gof () %!error chi2gof ([2,3;3,4]) %!error chi2gof ([1,2,3,4], "nbins", 3, "ctrs", [2,3,4]) %!error chi2gof ([1,2,3,4], "expected", [3,2,2], "nbins", 5) %!test %! x = [1 2 1 3 2 4 3 2 4 3 2 2]; %! [h, p, stats] = chi2gof (x); %! assert (h, 0); %! assert (p, NaN); %! assert (stats.chi2stat, 0.1205375022748029, 1e-14); ``` It is best practice to append tests during development, since it will save time debugging and it will certainly help catching marginal errors that would otherwise have quickly come back as bug reports. It also **important** appending tests when fixing a bug or implementing some new functionality into an existing function. Please, add BISTs, they help a lot with the maintenance :wink::innocent: of the statistics package. There is not such as a thing as too many tests in a function. :metal::v: #### 4. Coding style The coding style of GNU Octave should be used. In general, limit the lines at 80 characters long. - Use `LF` (unix) for end of lines, and NOT `CRLF` (windows). - Use `##` for comments. Don't use `%` or `%%` as in Matlab. - Use `!` instead of `~` for logical NOT. ``` a != 0; b(! isnnan (a)) = []; ``` - **Don't use tabs!** Indent the bodies of statement blocks with 2 spaces. - When calling functions, put spaces after commas and before the calling parentheses, as shown below: ``` x = max (sin (y+3), 2); ``` - An exception are matrix or cell constructors. ``` >> a = [sin(x), cos(x)]; >> b = {sin(x), cos)x)}; ``` **Note:** spaces in the above example would result in a parse error! - For an indexing expression, do not put a space after the identifier (this differentiates indexing and function calls nicely). - When indexing, the space after a comma is not necessary if index expressions are simple, but add a space for complex expressions as shown below: ``` A(:,i,j) A([1:i-1; i+1:n] ``` - Always use a specific end-of-block statement (like endif, endswitch) rather than the generic end. - Enclose the if, while, until, and switch conditions in parentheses. ``` if (isvector (a)) s = sum (a); endif ``` Do not do this, however, with the iteration counter portion of a for statement! ``` for i = 1:n b(i) = sum (a(:,1)); end ``` **That's about it all I suppose!** Just keep more or less consistent with what's in the other function files of the statistics packages and you should be fine :smile: statistics-release-1.6.3/COPYING000066400000000000000000001045121456127120000163650ustar00rootroot00000000000000 GNU GENERAL PUBLIC LICENSE Version 3, 29 June 2007 Copyright (C) 2007 Free Software Foundation, Inc. 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But first, please read .statistics-release-1.6.3/DESCRIPTION000066400000000000000000000004641456127120000170410ustar00rootroot00000000000000Name: statistics Version: 1.6.3 Date: 2024-2-9 Author: various authors Maintainer: Andreas Bertsatos Title: Statistics Description: The Statistics package for GNU Octave. Categories: Statistics Depends: octave (>= 7.2.0) License: GPLv3+ Url: https://github.com/gnu-octave/statistics statistics-release-1.6.3/INDEX000066400000000000000000000100101456127120000161110ustar00rootroot00000000000000statistics >> Statistics Clustering cluster clusterdata cmdscale confusionmat cophenet crossval evalclusters inconsistent kmeans knnsearch linkage mahal mhsample optimalleaforder pdist pdist2 procrustes rangesearch slicesample squareform Classification Classes ClassificationKNN ConfusionMatrixChart Clustering Classes CalinskiHarabaszEvaluation ClusterCriterion DaviesBouldinEvaluation GapEvaluation SilhouetteEvaluation CVpartition (legacy class of set partitions for cross-validation, used in crossval) @cvpartition/cvpartition @cvpartition/display @cvpartition/get @cvpartition/repartition @cvpartition/set @cvpartition/test @cvpartition/training Regression Classes RegressionGAM Data Manipulation combnk crosstab datasample fillmissing grp2idx ismissing isoutlier normalise_distribution rmmissing standardizeMissing tabulate Descriptive Statistics cdfcalc cl_multinom geomean grpstats harmmean jackknife mad mean median nanmax nanmin nansum std trimmean var Distribution Fitting betafit betalike binofit binolike bisafit bisalike burrfit burrlike evfit evlike expfit explike gamfit gamlike geofit gevfit_lmom gevfit gevlike gpfit gplike gumbelfit gumbellike hnfit hnlike invgfit invglike logifit logilike loglfit logllike lognfit lognlike nakafit nakalike nbinfit nbinlike normfit normlike poissfit poisslike raylfit rayllike ricefit ricelike unidfit unifit wblfit wbllike Distribution Functions betacdf betainv betapdf betarnd binocdf binoinv binopdf binornd bisacdf bisainv bisapdf bisarnd burrcdf burrinv burrpdf burrrnd bvncdf bvtcdf cauchycdf cauchyinv cauchypdf cauchyrnd chi2cdf chi2inv chi2pdf chi2rnd copulacdf copulapdf copularnd evcdf evinv evpdf evrnd expcdf expinv exppdf exprnd fcdf finv fpdf frnd gamcdf gaminv gampdf gamrnd geocdf geoinv geopdf geornd gevcdf gevinv gevpdf gevrnd gpcdf gpinv gppdf gprnd gumbelcdf gumbelinv gumbelpdf gumbelrnd hncdf hninv hnpdf hnrnd hygecdf hygeinv hygepdf hygernd invgcdf invginv invgpdf invgrnd iwishpdf iwishrnd jsucdf jsupdf laplacecdf laplaceinv laplacepdf laplacernd logicdf logiinv logipdf logirnd loglcdf loglinv loglpdf loglrnd logncdf logninv lognpdf lognrnd mnpdf mnrnd mvncdf mvnpdf mvnrnd mvtcdf mvtpdf mvtrnd mvtcdfqmc nakacdf nakainv nakapdf nakarnd nbincdf nbininv nbinpdf nbinrnd ncfcdf ncfinv ncfpdf ncfrnd nctcdf nctinv nctpdf nctrnd ncx2cdf ncx2inv ncx2pdf ncx2rnd normcdf norminv normpdf normrnd poisscdf poissinv poisspdf poissrnd raylcdf raylinv raylpdf raylrnd ricecdf riceinv ricepdf ricernd tcdf tinv tpdf trnd tlscdf tlsinv tlspdf tlsrnd tricdf triinv tripdf trirnd unidcdf unidinv unidpdf unidrnd unifcdf unifinv unifpdf unifrnd vmcdf vminv vmpdf vmrnd wblcdf wblinv wblpdf wblrnd wienrnd wishpdf wishrnd Distribution Statistics betastat binostat chi2stat evstat expstat fstat gamstat geostat gevstat gpstat hygestat lognstat nbinstat ncfstat nctstat ncx2stat normstat poisstat raylstat ricestat tlsstat tstat unidstat unifstat wblstat Experimental Design fullfact ff2n sigma_pts x2fx Machine Learning hmmestimate hmmgenerate hmmviterbi svmpredict svmtrain Model Fitting fitcknn fitgmdist fitlm fitrgam Hypothesis Testing adtest anova1 anova2 anovan bartlett_test barttest binotest chi2gof chi2test correlation_test fishertest friedman hotelling_t2test hotelling_t2test2 kruskalwallis kstest kstest2 levene_test manova1 mcnemar_test multcompare ranksum regression_ftest regression_ttest runstest sampsizepwr signtest ttest ttest2 vartest vartest2 vartestn ztest ztest2 I/O libsvmread libsvmwrite Plotting boxplot cdfplot confusionchart dendrogram ecdf einstein gscatter histfit hist3 manovacluster normplot ppplot qqplot silhouette violin wblplot Regression canoncorr cholcov dcov logistic_regression mnrfit monotone_smooth pca pcacov pcares plsregress princomp regress regress_gp ridge stepwisefit Transforms logit probit Wrappers cdf icdf pdf random statistics-release-1.6.3/NEWS000066400000000000000000000163431456127120000160350ustar00rootroot00000000000000 Summary of important user-visible changes for statistics 1.6.0: ------------------------------------------------------------------- Important Notice: 1) dependency changed to Octave>=7.2.0 2) various distribution functions have been renamed, deprecated, or modified extensively so that backwards compatibility is broken 3) `mad`, `mean`, `median`, `std`, `var` functions shadow core Octave's respective functions 4) incompatibility with the `nan` package New functions: ============== ** betafit (fully Matlab compatible) ** betalike (fully Matlab compatible) ** binofit (fully Matlab compatible) ** binolike (similar functionality to MATLAB's negloglik) ** bisafit (similar functionality to MATLAB's mle) ** bisalike (similar functionality to MATLAB's negloglik and mlecov) ** burrfit (similar functionality to MATLAB's mle) ** burrlike (similar functionality to MATLAB's negloglik and mlecov) ** einstein (plotting function for the einstein tile) ** geofit (similar functionality to MATLAB's mle) ** gumbelcdf ** gumbelfit (similar functionality to MATLAB's mle) ** gumbelinv ** gumbellike (similar functionality to MATLAB's negloglik and mlecov) ** gumbelpdf ** gumbelrnd ** hncdf ** hnfit (similar functionality to MATLAB's mle) ** hninv ** hnlike (similar functionality to MATLAB's negloglik and mlecov) ** hnpdf ** hnrnd ** invgcdf ** invgfit (similar functionality to MATLAB's mle) ** invginv ** invglike (similar functionality to MATLAB's negloglik and mlecov) ** invgpdf ** invgrnd ** isoutlier (fully Matlab compatible) ** logifit (similar functionality to MATLAB's mle) ** logilike (similar functionality to MATLAB's negloglik and mlecov) ** loglcdf ** loglfit (similar functionality to MATLAB's mle) ** loglinv ** logllike (similar functionality to MATLAB's negloglik and mlecov) ** loglpdf ** loglrnd ** lognfit (fully Matlab compatible) ** lognlike (fully Matlab compatible) ** nakafit (similar functionality to MATLAB's mle) ** nakalike (similar functionality to MATLAB's negloglik and mlecov) ** nbinfit (similar functionality to MATLAB's mle) ** nbinlike (similar functionality to MATLAB's negloglik and mlecov) ** mad (fully Matlab compatible) ** normfit (fully Matlab compatible) ** poissfit (fully Matlab compatible, extra functionality) ** poisslike (similar functionality to MATLAB's negloglik) ** raylfit (fully Matlab compatible, extra functionality) ** rayllike (similar functionality to MATLAB's negloglik) ** ridge (fully Matlab compatible) ** unidfit (similar functionality to MATLAB's mle) ** unifit (similar functionality to MATLAB's mle) ** vminv (quantile function for the von Mises distribution) ** wblfit (fully Matlab compatible) ** wbllike (fully Matlab compatible) Improvements: ============= ** bisacdf: supports "upper" option ** burrcdf: supports "upper" option ** cauchycdf: supports "upper" option ** cdf: update support for all univariate cumulative distribution functions ** gamfit: fixed MATLAB compatibility ** gamlike: fixed MATLAB compatibility ** icdf: update support for all univariate quantile functions ** laplacecdf: supports "upper" option ** logicdf: supports "upper" option ** mcnemar_test: updated functionality ** nakacdf: supports "upper" option ** pcacov: fixed MATLAB compatibility ** pdf: update support for all univariate probability density functions ** plsregress: fixed MATLAB compatibility ** random: update support for all univariate functions ** runstest: fixed MATLAB compatibility ** tabulate: fixed MATLAB compatibility ** tricdf: supports "upper" option ** trimmean: fixed MATLAB compatibility Removed Functions: ================== ** run_test (replaced by runstest) ** stdnormal_cdf (replaced by normcdf) ** stdnormal_inv (replaced by norminv) ** stdnormal_pdf (replaced by normpdf) ** stdnormal_rnd (replaced by normrnd) Renamed Functions: ================== ** bisacdf (replacing bbscdf) ** bisainv (replacing bbsinv) ** bisapdf (replacing bbspdf) ** bisarnd (replacing bbsrnd) ** cauchycdf (replacing cauchy_cdf) ** cauchyinv (replacing cauchy_inv) ** cauchypdf (replacing cauchy_pdf) ** cauchyrnd (replacing cauchy_rnd) ** laplacecdf (replacing laplace_cdf) ** laplaceinv (replacing laplace_inv) ** laplacepdf (replacing laplace_pdf) ** laplacernd (replacing laplace_rnd) ** logicdf (replacing logistic_cdf) ** logiinv (replacing logistic_inv) ** logipdf (replacing logistic_pdf) ** logirnd (replacing logistic_rnd) Summary of important user-visible changes for statistics 1.6.1: ------------------------------------------------------------------- Important Notice: 1) `mad`, `mean`, `median`, `std`, `var` functions shadow core Octave's respective functions prior to Octave v9.1 2) incompatibility with the `nan` package New functions: ============== ** ClassificationKNN (new classdef) ** predict (for ClassificationKNN classdef) ** fitcknn ** fitrgam ** knnsearch (fully Matlab compatible) ** mnrfit ** rangesearch (fully Matlab compatible) ** RegressionGAM (new classdef) ** predict (for RegressionGAM classdef) Improvements: ============= ** anovan: new features ** friedman: bug fixes ** pdist: updated functionality, fully MATLAB compatible ** pdist2: updated functionality, fully MATLAB compatible ** regress_gp: updated functionality with RBF kernel ** ridge: bug fixes Summary of important user-visible changes for statistics 1.6.2: ------------------------------------------------------------------- Important Notice: 1) `mad`, `mean`, `median`, `std`, `var` functions shadow core Octave's respective functions prior to Octave v9.1 2) incompatibility with the `nan` package New functions: ============== ** ricecdf ** riceinv ** ricepdf ** ricernd Improvements: ============= ** changed ClassificationKNN legacy class to classdef ** changed RegressionGAM legacy class to classdef ** update figure handling in BISTs, bug fix when calling 'pkg test statistics' Summary of important user-visible changes for statistics 1.6.3: ------------------------------------------------------------------- Important Notice: 1) `mad`, `mean`, `median`, `std`, `var` functions shadow core Octave's respective functions prior to Octave v9.1 2) incompatibility with the `nan` package New functions: ============== ** ricefit ** ricelike ** ricestat ** tlscdf ** tlsinv ** tlspdf ** tlsrnd ** tlsstat Improvements: ============= ** cdf: add support for Rician and location-scale T distributions ** gevlike: deprecate gradient as a second output argument (undocumented) ** icdf: add support for Rician and location-scale T distributions ** pdf: add support for Rician and location-scale T distributions ** random: add support for Rician and location-scale T distributions ** ricernd: fixed bug that produced erroneous results ** updated input validation and documentation to distribution *stat functions statistics-release-1.6.3/ONEWS-1.1.x000066400000000000000000000051151456127120000167120ustar00rootroot00000000000000Summary of important user-visible changes for statistics 1.1.3: ------------------------------------------------------------------- ** The following functions are new in 1.1.3: copularnd mvtrnd ** The functions mnpdf and mnrnd are now also usable for greater numbers of categories for which the rows do not exactly sum to 1. Summary of important user-visible changes for statistics 1.1.2: ------------------------------------------------------------------- ** The following functions are new in 1.1.2: mnpdf mnrnd ** The package is now dependent on the io package (version 1.0.18 or later) since the functions that it depended of from miscellaneous package have been moved to io. ** The function `kmeans' now accepts the 'emptyaction' property with the 'singleton' value. This allows for the kmeans algorithm to handle empty cluster better. It also throws an error if the user does not request an empty cluster handling, and there is an empty cluster. Plus, the returned items are now a closer match to Matlab. Summary of important user-visible changes for statistics 1.1.1: ------------------------------------------------------------------- ** The following functions are new in 1.1.1: monotone_smooth kmeans jackknife ** Bug fixes on the functions: normalise_distribution combnk repanova ** The following functions were removed since equivalents are now part of GNU octave core: zscore ** boxplot.m now returns a structure with handles to the plot elemenets. Summary of important user-visible changes for statistics 1.1.0: ------------------------------------------------------------------- ** IMPORTANT note about `fstat' shadowing core library function: GNU octave's 3.2 release added a new function `fstat' to return information of a file. Statistics' `fstat' computes F mean and variance. Since MatLab's `fstat' is the equivalent to statistics' `fstat' (not to core's `fstat'), and to avoid problems with the statistics package, `fstat' has been deprecated in octave 3.4 and will be removed in Octave 3.8. In the mean time, please ignore this warning when installing the package. ** The following functions are new in 1.1.0: normalise_distribution repanova combnk ** The following functions were removed since equivalents are now part of GNU octave core: prctile ** The __tbl_delim__ function is now private. ** The function `boxplot' now accepts named arguments. ** Bug fixes on the functions: harmmean nanmax nanmin regress ** Small improvements on help text. statistics-release-1.6.3/ONEWS-1.2.x000066400000000000000000000047501456127120000167170ustar00rootroot00000000000000Summary of important user-visible changes for statistics 1.2.4: ------------------------------------------------------------------- ** Made princomp work with nargout < 2. ** Renamed dendogram to dendrogram. ** Added isempty check to kmeans. ** Transposed output of hist3. ** Converted calculation in hmmviterbi to log space. ** Bug fixes for stepwisefit wishrnd. ** Rewrite of cmdscale for improved compatibility. ** Fix in squareform for improved compatibility. ** New cvpartition class, with methods: display repartition test training ** New sample data file fisheriris.txt for tests ** The following functions are new: cdf crossval dcov pdist2 qrandn randsample signtest ttest ttest2 vartest vartest2 ztest Summary of important user-visible changes for statistics 1.2.3: ------------------------------------------------------------------- ** Made sure that output of nanstd is real. ** Fixed second output of nanmax and nanmin. ** Corrected handle for outliers in boxplot. ** Bug fix and enhanced functionality for mvnrnd. ** The following functions are new: wishrnd iwishrnd wishpdf iwishpdf cmdscale Summary of important user-visible changes for statistics 1.2.2: ------------------------------------------------------------------- ** Fixed documentation of dendogram and hist3 to work with TexInfo 5. Summary of important user-visible changes for statistics 1.2.1: ------------------------------------------------------------------- ** The following functions are new: pcares pcacov runstest stepwisefit hist3 ** dendogram now returns the leaf node numbers and order that the nodes were displayed in. ** New faster implementation of princomp. Summary of important user-visible changes for statistics 1.2.0: ------------------------------------------------------------------- ** The following functions are new: regress_gp dendogram plsregress ** New functions for the generalized extreme value (GEV) distribution: gevcdf gevfit gevfit_lmom gevinv gevlike gevpdf gevrnd gevstat ** The interface of the following functions has been modified: mvnrnd ** `kmeans' has been fixed to deal with clusters that contain only one element. ** `normplot' has been fixed to avoid use of functions that have been removed from Octave core. Also, the plot produced should now display some aesthetic elements and appropriate legends. ** The help text of `mvtrnd' has been improved. ** Package is no longer autoloaded. statistics-release-1.6.3/ONEWS-1.3.0000066400000000000000000000012641456127120000166050ustar00rootroot00000000000000Summary of important user-visible changes for statistics 1.3.0: ------------------------------------------------------------------- ** The following functions are new: bbscdf bbsinv bbspdf bbsrnd binotest burrcdf burrinv burrpdf burrrnd gpcdf gpinv gppdf gprnd grp2idx mahal mvtpdf nakacdf nakainv nakapdf nakarnd pdf tricdf triinv tripdf trirnd violin ** Other functions that have been changed for smaller bugfixes, increased Matlab compatibility, or performance: betastat binostat cdf combnk gevfit hist3 kmeans linkage randsample squareform ttest statistics-release-1.6.3/ONEWS-1.4.x000066400000000000000000000141701456127120000167160ustar00rootroot00000000000000Summary of important user-visible changes for statistics 1.4.3: ------------------------------------------------------------------- New functions: ============== ** anova1 (patch #10127) kruskalwallis ** cluster (patch #10009) ** clusterdata (patch #10012) ** confusionchart (patch #9985) ** confusionmat (patch #9971) ** cophenet (patch #10040) ** datasample (patch #10050) ** evalclusters (patch #10052) ** expfit (patch #10092) explike ** gscatter (patch #10043) ** ismissing (patch #10102) ** inconsistent (patch #10008) ** mhsample.m (patch #10016) ** ncx2pdf (patch #9711) ** optimalleaforder.m (patch #10034) ** pca (patch #10104) ** rmmissing (patch #10102) ** silhouette (patch #9743) ** slicesample (patch #10019) ** wblplot (patch #8579) Improvements: ============= ** anovan.m: use double instead of toascii (bug #60514) ** binocdf: new option "upper" (bug #43721) ** boxplot: better Matlab compatibility; several Matlab-compatible plot options added (OutlierTags, Sample_IDs, BoxWidth, Widths, BoxStyle, Positions, Labels, Colors) and an Octave-specific one (CapWidhts); demos added; texinfo improved (patch #9930) ** auto MPG (carbig) sample dataset added from https://archive.ics.uci.edu/ml/datasets/Auto+MPG (patch #10045) ** crosstab.m: make n-dimensional (patch #10014) ** dendrogram.m: many improvements (patch #10036) ** fitgmdist.m: fix typo in ComponentProportion (bug #59386) ** gevfit: change orientation of results for Matlab compatibility (bug #47369) ** hygepdf: avoid overflow for certain inputs (bug #35827) ** kmeans: efficiency and compatibility tweaks (patch #10042) ** pdist: option for squared Euclidean distance (patch #10051) ** stepwisefit.m: give another option to select predictors (patch #8584) ** tricdf, triinv: fixes (bug #60113) Summary of important user-visible changes for statistics 1.4.2: ------------------------------------------------------------------- ** canoncorr: allow more variables than observations ** fitgmdist: return fitgmdist parameters (Bug #57917) ** gamfit: invert parameter per docs (Bug #57849) ** geoXXX: update docs 'number of failures (X-1)' => 'number of failures (X)' (Bug #57606) ** kolmogorov_smirnov_test.m: update function handle usage from octave6+ (Bug #57351) ** linkage.m: fix octave6+ parse error (Bug #57348) ** unifrnd: changed unifrnd(a,a) to return a 0 rather than NaN (Bug #56342) ** updates for usage of depreciated octave functions Summary of important user-visible changes for statistics 1.4.1: ------------------------------------------------------------------- ** update install scripts for octave 5.0 depreciated functions ** bug fixes to the following functions: pdist2.m: use max in distEucSq (Bug #50377) normpdf: use eps tolerance in tests (Bug #51963) fitgmdist: fix an output bug in fitgmdist t_test: Set tolerance on t_test BISTS (Bug #54557) gpXXXXX: change order of inputs to match matlab (Bug #54009) bartlett_test: df = k-1 (Bug #45894) gppdf: apply scale factor (Bug #54009) gmdistribution: updates for bug #54278, ##54279 wishrnd: Bug #55860 Summary of important user-visible changes for statistics 1.4.0: ------------------------------------------------------------------- ** The following functions are new: canoncorr fitgmdist gmdistribution sigma_pts ** The following functions have been moved from the statistics package but are conditionally installed: mad ** The following functions have been moved from octave to be conditionally installed: BASE cloglog logit prctile probit qqplot table (renamed to crosstab) DISTRIBUTIONS betacdf betainv betapdf betarnd binocdf binoinv binopdf binornd cauchy_cdf cauchy_inv cauchy_pdf cauchy_rnd chi2cdf chi2inv chi2pdf chi2rnd expcdf expinv exppdf exprnd fcdf finv fpdf frnd gamcdf gaminv gampdf gamrnd geocdf geoinv geopdf geornd hygecdf hygeinv hygepdf hygernd kolmogorov_smirnov_cdf laplace_cdf laplace_inv laplace_pdf laplace_rnd logistic_cdf logistic_inv logistic_pdf logistic_rnd logncdf logninv lognpdf lognrnd nbincdf nbininv nbinpdf nbinrnd normcdf norminv normpdf normrnd poisscdf poissinv poisspdf poissrnd stdnormal_cdf stdnormal_inv stdnormal_pdf stdnormal_rnd tcdf tinv tpdf trnd unidcdf unidinv unidpdf unidrnd unifcdf unifinv unifpdf unifrnd wblcdf wblinv wblpdf wblrnd wienrnd MODELS logistic_regression TESTS anova bartlett_test chisquare_test_homogeneity chisquare_test_independence cor_test f_test_regression hotelling_test hotelling_test_2 kolmogorov_smirnov_test kolmogorov_smirnov_test_2 kruskal_wallis_test manova mcnemar_test prop_test_2 run_test sign_test t_test t_test_2 t_test_regression u_test var_test welch_test wilcoxon_test z_test z_test_2 ** Functions marked with known test failures: grp2idx: bug #51928 gevfir_lmom: bug #31070 ** Other functions that have been changed for smaller bugfixes, increased Matlab compatibility, or performance: dcov: returned dcov instead of dcor. added demo. violin: can be used with subplots. violin quality improved. princomp: Fix expected values of tsquare in unit tests fitgmdist: test number inputs to function hist3: fix removal of rows with NaN values ** added the packages test data to install statistics-release-1.6.3/ONEWS-1.5.x000066400000000000000000000300501456127120000167120ustar00rootroot00000000000000Summary of important user-visible changes for statistics 1.5.0: ------------------------------------------------------------------- Important Notice: 1) dependency change to Octave>=6.1.0 2) `mean` shadows core Octave's respective function 3) removed dependency on `io` package 4) incompatibility with the `nan` package New functions: ============== ** anova2 (fully Matlab compatible) ** bvncdf ** cdfcalc, cdfplot ** chi2gof (fully Matlab compatible. bug #46764) ** chi2test (bug #58838) ** cholcov (fully Matlab compatible) ** ecdf (fully Matlab compatible) ** evfit (fully Matlab compatible) ** evlike (fully Matlab compatible) ** fitlm (mostly Matlab compatible) ** fillmissing (patch #10102) ** friedman (fully Matlab compatible) ** grpstats (complementary to manova1) ** kruskalwallis (fully Matlab compatible) ** kstest (fully Matlab compatible) ** kstest2 (fully Matlab compatible. bug #56572) ** libsvmread, libsvmwrite (I/O functions for LIBSVM data files) ** manova1 (fully Matlab compatible) ** manovacluster (fully Matlab compatible) ** mean (fully Matlab compatible, it shadows mean from core Octave) ** multcompare (fully Matlab compatible) ** mvtcdfqmc ** ranksum (fully Matlab compatible. bug #42079) ** standardizeMissing (patch #10102) ** svmpredict, svmtrain (wrappers for LIBSVM 3.25) ** tiedrank (complementary to ranksum) ** x2fx (missing function: bug #48146) Improvements: ============= ** anova1: added extra feature for performing Welch's ANOVA (PR #15) ** anovan: mostly Matlab compatible, extra features. (patch #10123, PR #1-42) ** binopdf: implement high accuracy Loader algorithm for m>=10 (bug #34362) ** cdf: extended to include all available distributions ** crosstab: can handle char arrays, fixed ordering of groups ** gaminv: fixed accuracy for small 1st argument (bug #56453) ** geomean: fully Matlab compatible. (patch #59410) ** gevlike: fully Matlab compatible. ACOV output is the inverse Fisher Inf Mat ** grp2idx (fully Matlab compatible, indexes in order of appeearance) ** harmmean: fully Matlab compatible. ** hygepdf: added optional parameter "vectorexpand" to facilitate vectorization of other hyge functions. Allows different inputs lengths for x and t,m,n parameters, with broadcast expanded output (bug #34363) ** hygecdf: improved vectorization for non-scalar inputs. hygeinv hygernd ** ismissing: corrects handling of n-D arrays, NaN indicators, and improves matlab compatibility for different data types. (patch #10102) ** kmeans: improved help file, evaluate efficiency (bug #8959) ** laplace_cdf: allow for parameters mu and scale (bug #58688) laplace_inv laplace_pdf logistic_cdf logistic_inv logistic_pdf ** logistic_regression: fixed incorrect results (bug #60348) ** mvncdf: improved performance and accuracy (bug #44130) ** normplot: fixed ploting error (bug #62394), updated features ** pdf: extended to include all available distributions ** pdist: updated the 'cosine' metric to be more efficient (bug #62495) ** rmmissing: corrects cellstr array handling and improves matlab compatibility for different data types. (patch #10102) ** signtest: fix erroneous results, fully Matlab compatible (bug #49961) ** ttest2: can handle NaN values as missing data (bug #58697) ** violin: fix parsing color vector affecting Octave>=6.1.0 (bug #62805) ** wblplot: fixed coding style and help texinfo. (patch #8579) Removed Functions: ================== ** anova (replaced by anova1) ** caseread, casewrite (do not belong here) ** chisquare_test_homogeneity (replaced by chi2test) ** chisquare_test_independence (replaced by chi2test) ** kolmogorov_smirnov_test (replaced by kstest) ** kolmogorov_smirnov_test_2 (replaced by kstest2) ** kruskal_wallis_test (replaced by kruskalwallis) ** manova (replaced by manova1) ** repanova (replaced by anova2) ** sign_test (replaced by updated signtest) ** tblread, tblwrite (belong to `io` package when tables are implemented) ** t_test, t_test_2 (deprecated: use ttest & ttest2) ** wilcoxon_test (replaced by ranksum) Available Data Sets: ==================== ** acetylene Chemical reaction data with correlated predictors ** arrhythmia Cardiac arrhythmia data from the UCI machine learning repository ** carbig Measurements of cars, 1970–1982 ** carsmall Subset of carbig. Measurements of cars, 1970, 1976, 1982 ** cereal Breakfast cereal ingredients ** examgrades Exam grades on a scale of 0–100 ** fisheriris Fisher's 1936 iris data ** hald Heat of cement vs. mix of ingredients ** heart_scale.dat Used for SVM testing ** kmeansdata Four-dimensional clustered data ** mileage Mileage data for three car models from two factories ** morse Recognition of Morse code distinctions by non-coders ** popcorn Popcorn yield by popper type and brand ** stockreturns Simulated stock returns ** weather Daily high temperatures in the same month in two consecutive years Summary of important user-visible changes for statistics 1.5.1: ------------------------------------------------------------------- Important Notice: 1) `mean` shadows core Octave's respective function 2) incompatibility with the `nan` package New functions: ============== ** barttest (fully Matlab compatible) ** evcdf (fully Matlab compatible) ** evinv (fully Matlab compatible) ** evpdf (fully Matlab compatible) ** evrnd (fully Matlab compatible) ** evstat (fully Matlab compatible) ** gpfit (fully Matlab compatible) ** gplike (fully Matlab compatible) ** gpstat (fully Matlab compatible) ** levene_test (options for testtypes, handling NaNs and GROUPS like anova1) ** ncfcdf (fully Matlab compatible) ** ncfinv (fully Matlab compatible) ** ncfpdf (fully Matlab compatible) ** ncfrnd (fully Matlab compatible) ** ncfstat (fully Matlab compatible) ** nctcdf (fully Matlab compatible) ** nctinv (fully Matlab compatible) ** nctpdf (fully Matlab compatible) ** nctrnd (fully Matlab compatible) ** nctstat (fully Matlab compatible) ** ncx2cdf (fully Matlab compatible) ** ncx2inv (fully Matlab compatible) ** ncx2rnd (fully Matlab compatible) ** ncx2stat (fully Matlab compatible) ** normlike (fully Matlab compatible) ** sampsizepwr (fully Matlab compatible with extra functionality) ** vartestn (fully Matlab compatible) Improvements: ============= ** bartlett_test: improved functionality, hanlding NaNs and GROUPS like anova1 ** chi2cdf: added "upper" option and confidence bounds ** chi2test: improved functionality, handles multi-way tables ** crosstab: returns chi-square and p-value for multiway tables ** evfit: fixed bug that caused an error when x is a row vector ** fcdf: added "upper" option and confidence bounds ** gamcdf: added "upper" option and confidence bounds ** gpcdf: added "upper" option and confidence bounds ** mvnpdf: fixed MATLAB compatibility ** mvnrnd: fixed MATLAB compatibility ** ncx2pdf: reimplemented to be fully MATLAB compatible ** normcdf: added "upper" option and confidence bounds ** tcdf: added "upper" option ** vartest: fixed MATLAB compatibility ** vartest2: fixed MATLAB compatibility ** ztest: fixed MATLAB compatibility Removed Functions: ================== ** cloglog ** nanmean (replaced by mean) ** kolmogorov_smirnov_cdf (unused by new kstest, kstest2 functions) ** u_test (replaced by ranksum) ** var_test (replaced by vartest) ** z_test (replaced by ztest) Summary of important user-visible changes for statistics 1.5.2: ------------------------------------------------------------------- Important Notice: 1) `mean`, `median`, `std`, and `var` functions shadow core Octave's respective functions 2) incompatibility with the `nan` package New functions: ============== ** median (fully Matlab compatible) ** std (fully Matlab compatible) ** var (fully Matlab compatible) Improvements: ============= ** mean: fixed MATLAB compatibility ** multcompare: fixed erroneous results for Welch ANOVA, updated features ** tcdf: fixed erroneous results ** ttest: added support for NaN values and matrix inputs ** ttest2: added support for matrices and multiple t-tests Removed Functions: ================== ** nanmedian (replaced by median) ** nanstd (replaced by std) ** nanvar (replaced by var) Summary of important user-visible changes for statistics 1.5.3: ------------------------------------------------------------------- Important Notice: 1) `mean`, `median`, `std`, and `var` functions shadow core Octave's respective functions 2) incompatibility with the `nan` package New functions: ============== ** adtest (fully Matlab compatible) ** hotelling_t2test (new functionality, replacing old hotelling_test) ** hotelling_t2test2 (new functionality, replacing old hotelling_test_2) ** regression_ftest (new functionality, replacing old f_test_regression) ** regression_ttest (replacing old t_test_regression) ** vmcdf (von Mises cummulative distribution function) Improvements: ============= ** betacdf: added "upper" option ** binocdf: added "upper" option ** expcdf: added "upper" option and confidence bounds ** geocdf: added "upper" option ** gevcdf: added "upper" option ** hygecdf: added "upper" option ** laplace_cdf: updated functionality ** laplace_inv: updated functionality ** laplace_pdf: updated functionality ** laplace_rnd: updated functionality ** logistic_cdf: updated functionality ** logistic_inv: updated functionality ** logistic_pdf: updated functionality ** logistic_rnd: updated functionality ** logncdf: added "upper" option and confidence bounds ** mean: fixed MATLAB compatibility ** median: fixed MATLAB compatibility ** multcompare: print PostHoc Test table ** nbincdf: added "upper" option ** poisscdf: added "upper" option ** raylcdf: added "upper" option ** std: fixed MATLAB compatibility ** unidcdf: added "upper" option ** unifcdf: added "upper" option ** var: fixed MATLAB compatibility ** vmpdf: updated functionality ** vmrnd: updated functionality ** wblcdf: added "upper" option and confidence bounds Removed Functions: ================== ** anderson_darling_cdf (replaced by adtest) ** anderson_darling_test (replaced by adtest) ** hotelling_test (replaced by hotelling_t2test) ** hotelling_test_2 (replaced by hotelling_t2test2) ** f_test_regression (replaced by regression_ftest) ** t_test_regression (replaced by regression_ttest) Summary of important user-visible changes for statistics 1.5.4: ------------------------------------------------------------------- Important Notice: 1) `mean`, `median`, `std`, and `var` functions shadow core Octave's respective functions 2) incompatibility with the `nan` package New functions: ============== ** bvtcdf ** correlation_test (new functionality, replacing old cor_test) ** icdf (wrapper for all available *inv distribution functions) ** fishertest (fully Matlab compatible) ** procrustes (fully Matlab compatible) ** ztest2 (new functionality, replacing old prop_test_2) Improvements: ============= ** cdf: updated wrapper for all available *cdf distribution functions ** dcov: handles missing values and multivariate samples ** geomean: fixed MATLAB compatibility ** harmmean: fixed MATLAB compatibility ** mean: fixed MATLAB compatibility ** median: fixed MATLAB compatibility ** mvtcdf: improved speed, fixed Matlab compatibility ** pdf: updated wrapper for all available *pdf distribution functions ** random: updated wrapper for all available *rnd distribution functions ** regression_ttest: new functionality ** std: fixed MATLAB compatibility ** var: fixed MATLAB compatibility Removed Functions: ================== ** cor_test (replaced by correlation_test) ** prop_test_2 (replaced by ztest2) statistics-release-1.6.3/README.md000066400000000000000000000112251456127120000166070ustar00rootroot00000000000000# statistics This is the official repository for the Statistics package for GNU Octave. **Content:** 1. About 2. Install statistics 3. Provide feedback 4. Contribute ## 1. About The **statistics** package is a collection of functions for statistical analysis. As with GNU Octave, the **statistics** package aims to be mostly compatible with MATLAB's equivalent Statistics and Machine Learning Toolbox. However, this is not always applicable of even possible. Hence, identical (in name) functions do not necessarily share the same functionality or behavior. Nevertheless, they produce consistent and correct results, unless there is a bug: see [Murphy's Law](https://en.wikipedia.org/wiki/Murphy's_law) :smile:. As of 10.6.2022, the developemnt of the **statistics** package was moved from [SourceForge](https://octave.sourceforge.io/statistics/) and [Mercurial](https://en.wikipedia.org/wiki/Mercurial) to [GitHub](https://github.com/gnu-octave/statistics) and [Git](https://en.wikipedia.org/wiki/Git). Given the opportunity of this transition, the package has been redesigned, as compared to the its previous point [release 1.4.3](https://octave.sourceforge.io/download.php?package=statistics-1.4.3.tar.gz) at SourceForge, with the aim to keep its structure simplified and easier to maintain. To this end, two major decisions have been made: - Keep a single dependency to the last two major point releases of GNU Octave. - Deprecate old functions once their fully Matlab compatible equivalents are implemented. You can find its documentation at [https://gnu-octave.github.io/statistics/](https://gnu-octave.github.io/statistics/). ## 2. Install statistics To install the latest release (1.6.3) you need Octave (>=7.2.0) installed on your system. Install it by typing: `pkg install -forge statistics` You can automatically download and install the latest development version of the **statistics** package found [here](https://github.com/gnu-octave/statistics/archive/refs/heads/main.zip) by typing: `pkg install "https://github.com/gnu-octave/statistics/archive/refs/heads/main.zip"` If you need to install a specific release, for example `1.4.2`, type: `pkg install "https://github.com/gnu-octave/statistics/archive/refs/tags/release-1.4.2.tar.gz"` After installation, type: - `pkg load statistics` to load the **statistics** package. - `news statistics` to review all the user visible changes since last version. - `pkg test statistics` to run a test suite for all 389 [^1] functions currently available and ensure that they work properly on your system. [^1]: Several functions are still missing from the statistics package, but you are welcome to [contribute](https://github.com/gnu-octave/statistics/blob/main/CONTRIBUTING.md)! ## 3. Provide feedback You are encouraged to provide feedback regarding possible bugs, missing features[^2], discrepancies or incompatibilities with Matlab functions. You may open an [issue](https://github.com/gnu-octave/statistics/issues) to open a discussion to your particular case. **Please, do NOT use the issue tracker for requesting help.** Use the [discourse group](https://octave.discourse.group/c/help/6) for requesting help with using functions and programming in Octave. Please, make sure that when reporting a bug you provide as much information as possible for other users to be able to replicate it. Use [markdown tips](https://docs.github.com/en/get-started/writing-on-github/getting-started-with-writing-and-formatting-on-github/basic-writing-and-formatting-syntax) to make your post clear and easy to read and understand your issue. [^2]: Don't open an issue just for requesting a missing function! Implement it yourself and make an invaluable contribution :innocent: ## 4. Contribute The **statistics** package is **open source**! Everyone is welcome to contribute. If you find a bug and fix it, just [clone](https://github.com/gnu-octave/statistics.git) this repo with `git clone https://github.com/gnu-octave/statistics.git`, make your changes and add a [pull](https://github.com/gnu-octave/statistics/pulls) request. Alternatively, you may open an issue and add a git-patch file, which will be patched by the maintainer. Make sure you follow the coding style already used in the **statistics** package (similar to GNU Octave). For a summary of the coding style rules used in the package see [Contribute](https://github.com/gnu-octave/statistics/blob/main/CONTRIBUTING.md). Contributing is not only about fixing bugs or implementing new functions. Improving the texinfo of the functions help files or adding BISTs and demos at the end of the function files is also important. Fixing a typo in the help file is still of value though. So don't hesitate to contribute! :+1: statistics-release-1.6.3/doc/000077500000000000000000000000001456127120000160745ustar00rootroot00000000000000statistics-release-1.6.3/doc/statistics.png000066400000000000000000000006421456127120000207760ustar00rootroot00000000000000‰PNG  IHDRddÿ€tIMEß Ó@Ÿ…tEXtCommentCreated with GIMPW1IDATxÚíÜAƒ Ph¼ÿ•é¢«&mé(Âû«vaŒO%æRJ’º<À‚ ,X`Á‚ ,°`ÁºQ¶c›åüöw‘ˆ±^:9§¥ž³†°`Á‚K`Á‚kvgØ„6­³a…6­Ûï3Óz&+fžaw<­² <,X뾤œËp™ +§ÈåËî†j–ÔMJï>u<±À·¬ÕÞŽÛHw^à›¼†~êpÕeû Zw7„ ,é]ù÷úaå߈3 Ã,X\àÏè¢cßìíü^X±oöömªY°`Á‚%°`ÁÒî´O¤3¬Êö¬t)G´J†!,X°`ÁX°`Á‚K`Á‚uy¦ý®CÄ’Ä9¿ë`ªY°` ,X°`Á‚%°`Áº>Odœ1ë'ø£#IEND®B`‚statistics-release-1.6.3/docs/000077500000000000000000000000001456127120000162575ustar00rootroot00000000000000statistics-release-1.6.3/docs/@cvpartition_cvpartition.html000066400000000000000000000174011456127120000242340ustar00rootroot00000000000000 Statistics: @cvpartition/cvpartition

Function Reference: @cvpartition/cvpartition

statistics: C = cvpartition (X, [partition_type, [k]])

Create a partition object for cross validation.

X may be a positive integer, interpreted as the number of values n to partition, or a vector of length n containing class designations for the elements, in which case the partitioning types KFold and HoldOut attempt to ensure each partition represents the classes proportionately.

partition_type must be one of the following:

‘KFold’

Divide set into k equal-size subsets (this is the default, with k=10).

‘HoldOut’

Divide set into two subsets, "training" and "validation". If k is a fraction, that is the fraction of values put in the validation subset; if it is a positive integer, that is the number of values in the validation subset (by default k=0.1).

‘LeaveOut’

Leave-one-out partition (each element is placed in its own subset).

‘resubstitution’

Training and validation subsets that both contain all the original elements.

‘Given’

Subset indices are as given in X.

The following fields are defined for the ‘cvpartition’ class:

‘classes’

Class designations for the elements.

‘inds’

Subset indices for the elements.

‘n_classes’

Number of different classes.

‘NumObservations’

n, number of elements in data set.

‘NumTestSets’

Number of testing subsets.

‘TestSize’

Number of elements in (each) testing subset.

‘TrainSize’

Number of elements in (each) training subset.

‘Type’

Partition type.

See also: crossval, @cvpartition/display

Source Code: @cvpartition/cvpartition

Example: 1

  

 ## Partition with Fisher iris dataset (n = 150)
 ## Stratified by species
 load fisheriris
 y = species;
 ## 10-fold cross-validation partition
 c = cvpartition (species, 'KFold', 10)
 ## leave-10-out partition
 c1 = cvpartition (species, 'HoldOut', 10)
 idx1 = test (c, 2);
 idx2 = training (c, 2);
 ## another leave-10-out partition
 c2 = repartition (c1)

K-fold cross validation partition
          N: 150
NumTestSets: 10
  TrainSize: 135  135  135  135  135  135  135  135  135  135
   TestSize: 15  15  15  15  15  15  15  15  15  15
HoldOut cross validation partition
          N: 150
NumTestSets: 1
  TrainSize: 140
   TestSize: 10
HoldOut cross validation partition
          N: 150
NumTestSets: 1
  TrainSize: 140
   TestSize: 10
statistics-release-1.6.3/docs/@cvpartition_display.html000066400000000000000000000076111456127120000233410ustar00rootroot00000000000000 Statistics: @cvpartition/display

Function Reference: @cvpartition/display

statistics: display (C)

Display a ‘cvpartition’ object, C.

See also: @cvpartition/cvpartition

Source Code: @cvpartition/display

statistics-release-1.6.3/docs/@cvpartition_get.html000066400000000000000000000076521456127120000224600ustar00rootroot00000000000000 Statistics: @cvpartition/get

Function Reference: @cvpartition/get

statistics: s = get (C, f)

Get a field, f, from a ‘cvpartition’ object, C.

See also: @cvpartition/cvpartition

Source Code: @cvpartition/get

statistics-release-1.6.3/docs/@cvpartition_repartition.html000066400000000000000000000101711456127120000242270ustar00rootroot00000000000000 Statistics: @cvpartition/repartition

Function Reference: @cvpartition/repartition

statistics: Cnew = repartition (C)

Return a new cvpartition object.

C should be a ‘cvpartition’ object. Cnew will use the same partition_type as C but redo any randomization performed (currently, only the HoldOut type uses randomization).

See also: @cvpartition/cvpartition

Source Code: @cvpartition/repartition

statistics-release-1.6.3/docs/@cvpartition_set.html000066400000000000000000000076741456127120000225000ustar00rootroot00000000000000 Statistics: @cvpartition/set

Function Reference: @cvpartition/set

statistics: Cnew = set (C, field, value)

Set field, in a ‘cvpartition’ object, C.

See also: @cvpartition/cvpartition

Source Code: @cvpartition/set

statistics-release-1.6.3/docs/@cvpartition_test.html000066400000000000000000000100761456127120000226520ustar00rootroot00000000000000 Statistics: @cvpartition/test

Function Reference: @cvpartition/test

statistics: inds = test (C, [i])

Return logical vector for testing-subset indices from a ‘cvpartition’ object, C. i is the fold index (default is 1).

See also: @cvpartition/cvpartition, @cvpartition/training

Source Code: @cvpartition/test

statistics-release-1.6.3/docs/@cvpartition_training.html000066400000000000000000000101131456127120000234760ustar00rootroot00000000000000 Statistics: @cvpartition/training

Function Reference: @cvpartition/training

statistics: inds = training (C, [i])

Return logical vector for training-subset indices from a ‘cvpartition’ object, C. i is the fold index (default is 1).

See also: @cvpartition/cvpartition, @cvpartition/test

Source Code: @cvpartition/training

statistics-release-1.6.3/docs/CalinskiHarabaszEvaluation.html000066400000000000000000000163611456127120000244150ustar00rootroot00000000000000 Statistics: CalinskiHarabaszEvaluation

Class Definition: CalinskiHarabaszEvaluation

statistics: obj = evalclusters (x, clust, CalinskiHarabasz)
statistics: obj = evalclusters (…, Name, Value)

A Calinski-Harabasz object to evaluate clustering solutions.

A CalinskiHarabaszEvaluation object is a ClusterCriterion object used to evaluate clustering solutions using the Calinski-Harabasz criterion.

The Calinski-Harabasz index is based on the ratio between SSb and SSw. SSb is the overall variance between clusters, that is the variance of the distances between the centroids. SSw is the overall variance within clusters, that is the sum of the variances of the distances between each datapoint and its centroid.

The best solution according to the Calinski-Harabasz criterion is the one that scores the highest value.

See also: evalclusters, ClusterCriterion, DaviesBouldinEvaluation, GapEvaluation, SilhouetteEvaluation

Source Code: CalinskiHarabaszEvaluation

Method: addK

CalinskiHarabaszEvaluation: obj = addK (obj, K)

Add a new cluster array to inspect the CalinskiHarabaszEvaluation object.

Method: compact

CalinskiHarabaszEvaluation: obj = compact (obj)

Return a compact CalinskiHarabaszEvaluation object (not implemented yet).

Method: plot

CalinskiHarabaszEvaluation: plot (obj)
CalinskiHarabaszEvaluation: h = plot (obj)

Plot the evaluation results.

Plot the CriterionValues against InspectedK from the CalinskiHarabaszEvaluation, obj, to the current plot. It can also return a handle to the current plot.

statistics-release-1.6.3/docs/ClassificationKNN.html000066400000000000000000000452131456127120000224540ustar00rootroot00000000000000 Statistics: ClassificationKNN

Class Definition: ClassificationKNN

statistics: obj = ClassificationKNN (X, Y)
statistics: obj = ClassificationKNN (…, name, value)

Create a ClassificationKNN class object containing a k-Nearest Neighbor classification model.

obj = ClassificationKNN (X, Y) returns a ClassificationKNN object, with X as the predictor data and Y containing the class labels of observations in X.

  • X must be a N×P numeric matrix of input data where rows correspond to observations and columns correspond to features or variables. X will be used to train the kNN model.
  • Y is N×1 matrix or cell matrix containing the class labels of corresponding predictor data in X. Y can contain any type of categorical data. Y must have same numbers of Rows as X.

obj = ClassificationKNN (…, name, value) returns a ClassificationKNN object with parameters specified by Name-Value pair arguments. Type help fitcknn for more info.

A ClassificationKNN object, obj, stores the labelled training data and various parameters for the k-Nearest Neighbor classification model, which can be accessed in the following fields:

FieldDescription
obj.XUnstandardized predictor data, specified as a numeric matrix. Each column of X represents one predictor (variable), and each row represents one observation.
obj.YClass labels, specified as a logical or numeric vector, or cell array of character vectors. Each value in Y is the observed class label for the corresponding row in X.
obj.NumObservationsNumber of observations used in training the ClassificationKNN model, specified as a positive integer scalar. This number can be less than the number of rows in the training data because rows containing NaN values are not part of the fit.
obj.RowsUsedRows of the original training data used in fitting the ClassificationKNN model, specified as a numerical vector. If you want to use this vector for indexing the training data in X, you have to convert it to a logical vector, i.e X = obj.X(logical (obj.RowsUsed), :);
obj.StandardizeA boolean flag indicating whether the data in X have been standardized prior to training.
obj.SigmaPredictor standard deviations, specified as a numeric vector of the same length as the columns in X. If the predictor variables have not been standardized, then "obj.Sigma" is empty.
obj.MuPredictor means, specified as a numeric vector of the same length as the columns in X. If the predictor variables have not been standardized, then "obj.Mu" is empty.
obj.NumPredictorsThe number of predictors (variables) in X.
obj.PredictorNamesPredictor variable names, specified as a cell array of character vectors. The variable names are in the same order in which they appear in the training data X.
obj.ResponseNameResponse variable name, specified as a character vector.
obj.ClassNamesNames of the classes in the training data Y with duplicates removed, specified as a cell array of character vectors.
obj.BreakTiesTie-breaking algorithm used by predict when multiple classes have the same smallest cost, specified as one of the following character arrays: "smallest" (default), which favors the class with the smallest index among the tied groups, i.e. the one that appears first in the training labelled data. "nearest", which favors the class with the nearest neighbor among the tied groups, i.e. the class with the closest member point according to the distance metric used. "nearest", which randomly picks one class among the tied groups.
obj.PriorPrior probabilities for each class, specified as a numeric vector. The order of the elements in Prior corresponds to the order of the classes in ClassNames.
obj.CostCost of the misclassification of a point, specified as a square matrix. Cost(i,j) is the cost of classifying a point into class j if its true class is i (that is, the rows correspond to the true class and the columns correspond to the predicted class). The order of the rows and columns in Cost corresponds to the order of the classes in ClassNames. The number of rows and columns in Cost is the number of unique classes in the response. By default, Cost(i,j) = 1 if i != j, and Cost(i,j) = 0 if i = j. In other words, the cost is 0 for correct classification and 1 for incorrect classification.
obj.NumNeighborsNumber of nearest neighbors in X used to classify each point during prediction, specified as a positive integer value.
obj.DistanceDistance metric, specified as a character vector. The allowable distance metric names depend on the choice of the neighbor-searcher method. See the available distance metrics in knnseaarch for more info.
obj.DistanceWeightDistance weighting function, specified as a function handle, which accepts a matrix of nonnegative distances, and returns a matrix the same size containing nonnegative distance weights.
obj.DistParameterParameter for the distance metric, specified either as a positive definite covariance matrix (when the distance metric is "mahalanobis", or a positive scalar as the Minkowski distance exponent (when the distance metric is "minkowski", or a vector of positive scale values with length equal to the number of columns of X (when the distance metric is "seuclidean". For any other distance metric, the value of DistParameter is empty.
obj.NSMethodNearest neighbor search method, specified as either "kdtree", which creates and uses a Kd-tree to find nearest neighbors, or "exhaustive", which uses the exhaustive search algorithm by computing the distance values from all points in X to find nearest neighbors.
obj.IncludeTiesA boolean flag indicating whether prediction includes all the neighbors whose distance values are equal to the k^th smallest distance. If IncludeTies is true, prediction includes all of these neighbors. Otherwise, prediction uses exactly k neighbors.
obj.BucketSizeMaximum number of data points in the leaf node of the Kd-tree, specified as positive integer value. This argument is meaningful only when NSMethod is "kdtree".

See also: fitcknn, knnsearch, rangesearch, pdist2

Source Code: ClassificationKNN

Method: predict

ClassificationKNN: label = predict (obj, XC)
ClassificationKNN: [label, score, cost] = predict (obj, XC)

Classify new data points into categories using the kNN algorithm from a k-Nearest Neighbor classification model.

label = predict (obj, XC) returns the matrix of labels predicted for the corresponding instances in XC, using the predictor data in obj.X and corresponding labels, obj.Y, stored in the k-Nearest Neighbor classification model, obj.

XC must be an M×P numeric matrix with the same number of features P as the corresponding predictors of the kNN model in obj.

[label, score, cost] = predict (obj, XC) also returns score, which contains the predicted class scores or posterior probabilities for each instance of the corresponding unique classes, and cost, which is a matrix containing the expected cost of the classifications.

See also: fitcknn, ClassificationKNN

Example: 1

  

 ## Create a k-nearest neighbor classifier for Fisher's iris data with k = 5.
 ## Evaluate some model predictions on new data.

 load fisheriris
 x = meas;
 y = species;
 xc = [min(x); mean(x); max(x)];
 obj = fitcknn (x, y, "NumNeighbors", 5, "Standardize", 1);
 [label, score, cost] = predict (obj, xc)

label =
{
  [1,1] = versicolor
  [2,1] = versicolor
  [3,1] = virginica
}

score =

   0.4000   0.6000        0
        0   1.0000        0
        0        0   1.0000

cost =

   0.6000   0.4000   1.0000
   1.0000        0   1.0000
   1.0000   1.0000        0

Example: 2

  

 ## Train a k-nearest neighbor classifier for k = 10
 ## and plot the decision boundaries.

 load fisheriris
 idx = ! strcmp (species, "setosa");
 X = meas(idx,3:4);
 Y = cast (strcmpi (species(idx), "virginica"), "double");
 obj = fitcknn (X, Y, "Standardize", 1, "NumNeighbors", 10, "NSMethod", "exhaustive")
 x1 = [min(X(:,1)):0.03:max(X(:,1))];
 x2 = [min(X(:,2)):0.02:max(X(:,2))];
 [x1G, x2G] = meshgrid (x1, x2);
 XGrid = [x1G(:), x2G(:)];
 pred = predict (obj, XGrid);
 gidx = logical (str2num (cell2mat (pred)));

 figure
 scatter (XGrid(gidx,1), XGrid(gidx,2), "markerfacecolor", "magenta");
 hold on
 scatter (XGrid(!gidx,1), XGrid(!gidx,2), "markerfacecolor", "red");
 plot (X(Y == 0, 1), X(Y == 0, 2), "ko", X(Y == 1, 1), X(Y == 1, 2), "kx");
 xlabel ("Petal length (cm)");
 ylabel ("Petal width (cm)");
 title ("5-Nearest Neighbor Classifier Decision Boundary");
 legend ({"Versicolor Region", "Virginica Region", ...
         "Sampled Versicolor", "Sampled Virginica"}, ...
         "location", "northwest")
 axis tight
 hold off

obj =

  ClassificationKNN object with properties:

            BreakTies: smallest
           BucketSize: [1x1 double]
           ClassNames: [2x1 cell]
                 Cost: [2x2 double]
        DistParameter: [0x0 double]
             Distance: euclidean
       DistanceWeight: [1x1 function_handle]
          IncludeTies: 0
                   Mu: [1x2 double]
             NSMethod: exhaustive
         NumNeighbors: [1x1 double]
      NumObservations: [1x1 double]
        NumPredictors: [1x1 double]
       PredictorNames: [1x2 cell]
                Prior: [2x1 double]
         ResponseName: Y
             RowsUsed: [100x1 double]
                Sigma: [1x2 double]
          Standardize: 1
                    X: [100x2 double]
                    Y: [100x1 double]
plotted figure

statistics-release-1.6.3/docs/ClusterCriterion.html000066400000000000000000000204511456127120000224470ustar00rootroot00000000000000 Statistics: ClusterCriterion

Class Definition: ClusterCriterion

statistics: obj = ClusterCriterion (x, clust, criterion)

A clustering evaluation object as created by evalclusters.

ClusterCriterion is a superclass for clustering evaluation objects as created by evalclusters.

List of public properties:

ClusteringFunction

a valid clustering funtion name or function handle. It can be empty if the clustering solutions are passed as an input matric.

CriterionName

a valid criterion name to evaluate the clustering solutions.

CriterionValues

a vector of values as generated by the evaluation criterion for each clustering solution.

InspectedK

the list of proposed cluster numbers.

Missing

a logical vector of missing observations. When there are NaN values in the data matrix, the corresponding observation is excluded.

NumObservations

the number of non-missing observations in the data matrix.

OptimalK

the optimal number of clusters.

OptimalY

the clustering solution corresponding to OptimalK.

X

the data matrix.

List of public methods:

addK

add a list of numbers of clusters to evaluate.

compact

return a compact clustering evaluation object. Not implemented

plot

plot the clustering evaluation values against the corresponding number of clusters.

See also: evalclusters, CalinskiHarabaszEvaluation, DaviesBouldinEvaluation, GapEvaluation, SilhouetteEvaluation

Source Code: ClusterCriterion

Method: addK

ClusterCriterion: obj = addK (obj, K)

Add a new cluster array to inspect the ClusterCriterion object.

Method: compact

ClusterCriterion: obj = compact (obj)

Return a compact ClusterCriterion object (not implemented yet).

Method: plot

ClusterCriterion: plot (obj)
ClusterCriterion: h = plot (obj)

Plot the evaluation results.

Plot the CriterionValues against InspectedK from the ClusterCriterion, obj, to the current plot. It can also return a handle to the current plot.

statistics-release-1.6.3/docs/ConfusionMatrixChart.html000066400000000000000000000202441456127120000232610ustar00rootroot00000000000000 Statistics: ConfusionMatrixChart

Class Definition: ConfusionMatrixChart

statistics: cmc = ConfusionMatrixChart ()

Create object cmc, a Confusion Matrix Chart object.

"DiagonalColor"

The color of the patches on the diagonal, default is [0.0, 0.4471, 0.7412].

"OffDiagonalColor"

The color of the patches off the diagonal, default is [0.851, 0.3255, 0.098].

"GridVisible"

Available values: on (default), off.

"Normalization"

Available values: absolute (default), column-normalized, row-normalized, total-normalized.

"ColumnSummary"

Available values: off (default), absolute, column-normalized,total-normalized.

"RowSummary"

Available values: off (default), absolute, row-normalized, total-normalized.

MATLAB compatibility – the not implemented properties are: FontColor, PositionConstraint, InnerPosition, Layout.

See also: confusionchart

Source Code: ConfusionMatrixChart

Method: disp

ConfusionMatrixChart: disp (cmc, order)

Display the properties of the ConfusionMatrixChart object cmc.

Method: sortClasses

ConfusionMatrixChart: sortClasses (cmc, order)

Sort the classes of the ConfusionMatrixChart object cmc according to order.

Valid values for order can be an array or cell array including the same class labels as cm, or a value like "auto", "ascending-diagonal", "descending-diagonal" and "cluster".

Example: 1

  

 ## Create a simple ConfusionMatrixChart Object

 cm = ConfusionMatrixChart (gca, [1 2; 1 2], {"A","B"}, {"XLabel","LABEL A"})
 NormalizedValues = cm.NormalizedValues
 ClassLabels = cm.ClassLabels

cm =

ConfusionMatrixChart with properties:

	NormalizedValues: [ 2x2 double ]
	ClassLabels: { 1x2 cell }


NormalizedValues =

   1   2
   1   2

ClassLabels =
{
  [1,1] = A
  [1,2] = B
}
plotted figure

statistics-release-1.6.3/docs/DaviesBouldinEvaluation.html000066400000000000000000000161671456127120000237400ustar00rootroot00000000000000 Statistics: DaviesBouldinEvaluation

Class Definition: DaviesBouldinEvaluation

Function File: obj = evalclusters (x, clust, DaviesBouldin)
Function File: obj = evalclusters (…, Name, Value)

A Davies-Bouldin object to evaluate clustering solutions.

A DaviesBouldinEvaluation object is a ClusterCriterion object used to evaluate clustering solutions using the Davies-Bouldin criterion.

The Davies-Bouldin criterion is based on the ratio between the distances between clusters and within clusters, that is the distances between the centroids and the distances between each datapoint and its centroid.

The best solution according to the Davies-Bouldin criterion is the one that scores the lowest value.

See also: evalclusters, ClusterCriterion, CalinskiHarabaszEvaluation, GapEvaluation, SilhouetteEvaluation

Source Code: DaviesBouldinEvaluation

Method: addK

DaviesBouldinEvaluation: obj = addK (obj, K)

Add a new cluster array to inspect the DaviesBouldinEvaluation object.

Method: compact

DaviesBouldinEvaluation: obj = compact (obj)

Return a compact DaviesBouldinEvaluation object (not implemented yet).

Method: plot

DaviesBouldinEvaluation: plot (obj)
DaviesBouldinEvaluation: h = plot (obj)

Plot the evaluation results.

Plot the CriterionValues against InspectedK from the DaviesBouldinEvaluation ClusterCriterion, obj, to the current plot. It can also return a handle to the current plot.

statistics-release-1.6.3/docs/GapEvaluation.html000066400000000000000000000207711456127120000217130ustar00rootroot00000000000000 Statistics: GapEvaluation

Class Definition: GapEvaluation

statistics: obj = evalclusters (x, clust, gap)
statistics: obj = evalclusters (…, Name, Value)

A gap object to evaluate clustering solutions.

A GapEvaluation object is a ClusterCriterion object used to evaluate clustering solutions using the gap criterion, which is a mathematical formalization of the elbow method.

List of public properties specific to SilhouetteEvaluation:

B

the number of reference datasets to generate.

Distance

a valid distance metric name, or a function handle as accepted by the pdist function.

ExpectedLogW

a vector of the expected values for the logarithm of the within clusters dispersion.

LogW

a vector of the values of the logarithm of the within clusters dispersion.

ReferenceDistribution

a valid name for the reference distribution, namely: PCA (default) or uniform.

SE

a vector of the standard error of the expected values for the logarithm of the within clusters dispersion.

SearchMethod

a valid name for the search method to use: globalMaxSE (default) or firstMaxSE.

StdLogW

a vector of the standard deviation of the expected values for the logarithm of the within clusters dispersion.

The best solution according to the gap criterion depends on the chosen search method. When the search method is globalMaxSE, the chosen gap value is the smaller one which is inside a standard error from the max gap value; when the search method is firstMaxSE, the chosen gap value is the first one which is inside a standard error from the next gap value.

See also: evalclusters, ClusterCriterion, CalinskiHarabaszEvaluation, DaviesBouldinEvaluation, SilhouetteEvaluation

Source Code: GapEvaluation

Method: addK

GapEvaluation: obj = addK (obj, K)

Add a new cluster array to inspect the GapEvaluation object.

Method: compact

GapEvaluation: obj = compact (obj)

Return a compact GapEvaluation object (not implemented yet).

Method: plot

ClusterCriterion: plot (obj)
ClusterCriterion: h = plot (obj)

Plot the evaluation results.

Plot the CriterionValues against InspectedK from the GapEvaluation ClusterCriterion, obj, and show the standard deviation to the current plot. It can also return a handle to the current plot.

statistics-release-1.6.3/docs/RegressionGAM.html000066400000000000000000000433621456127120000216220ustar00rootroot00000000000000 Statistics: RegressionGAM

Class Definition: RegressionGAM

statistics: obj = RegressionGAM (X, Y)
statistics: obj = RegressionGAM (…, name, value)

Create a RegressionGAM class object containing a Generalised Additive Model (GAM) for regression.

A RegressionGAM class object can store the predictors and response data along with various parameters for the GAM model. It is recommended to use the fitrgam function to create a RegressionGAM object.

obj = RegressionGAM (X, Y) returns an object of class RegressionGAM, with matrix X containing the predictor data and vector Y containing the continuous response data.

  • X must be a N×P numeric matrix of input data where rows correspond to observations and columns correspond to features or variables. X will be used to train the GAM model.
  • Y must be N×1 numeric vector containing the response data corresponding to the predictor data in X. Y must have same number of rows as X.

obj = RegressionGAM (…, name, value) returns an object of class RegressionGAM with additional properties specified by Name-Value pair arguments listed below.

NameValue
"predictors"Predictor Variable names, specified as a row vector cell of strings with the same length as the columns in X. If omitted, the program will generate default variable names (x1, x2, ..., xn) for each column in X.
"responsename"Response Variable Name, specified as a string. If omitted, the default value is "Y".
"formula"a model specification given as a string in the form "Y ~ terms" where Y represents the reponse variable and terms the predictor variables. The formula can be used to specify a subset of variables for training model. For example: "Y ~ x1 + x2 + x3 + x4 + x1:x2 + x2:x3" specifies four linear terms for the first four columns of for predictor data, and x1:x2 and x2:x3 specify the two interaction terms for 1st-2nd and 3rd-4th columns respectively. Only these terms will be used for training the model, but X must have at least as many columns as referenced in the formula. If Predictor Variable names have been defined, then the terms in the formula must reference to those. When "formula" is specified, all terms used for training the model are referenced in the IntMatrix field of the obj class object as a matrix containing the column indexes for each term including both the predictors and the interactions used.
"interactions"a logical matrix, a positive integer scalar, or the string "all" for defining the interactions between predictor variables. When given a logical matrix, it must have the same number of columns as X and each row corresponds to a different interaction term combining the predictors indexed as true. Each interaction term is appended as a column vector after the available predictor column in X. When "all" is defined, then all possible combinations of interactions are appended in X before training. At the moment, parsing a positive integer has the same effect as the "all" option. When "interactions" is specified, only the interaction terms appended to X are referenced in the IntMatrix field of the obj class object.
"knots"a scalar or a row vector with the same columns as X. It defines the knots for fitting a polynomial when training the GAM. As a scalar, it is expanded to a row vector. The default value is 5, hence expanded to ones (1, columns (X)) * 5. You can parse a row vector with different number of knots for each predictor variable to be fitted with, although not recommended.
"order"a scalar or a row vector with the same columns as X. It defines the order of the polynomial when training the GAM. As a scalar, it is expanded to a row vector. The default values is 3, hence expanded to ones (1, columns (X)) * 3. You can parse a row vector with different number of polynomial order for each predictor variable to be fitted with, although not recommended.
"dof"a scalar or a row vector with the same columns as X. It defines the degrees of freedom for fitting a polynomial when training the GAM. As a scalar, it is expanded to a row vector. The default value is 8, hence expanded to ones (1, columns (X)) * 8. You can parse a row vector with different degrees of freedom for each predictor variable to be fitted with, although not recommended.
"tol"a positive scalar to set the tolerance for covergence during training. By defaul, it is set to 1e-3.

You can parse either a "formula" or an "interactions" optional parameter. Parsing both parameters will result an error. Accordingly, you can only pass up to two parameters among "knots", "order", and "dof" to define the required polynomial for training the GAM model.

See also: fitrgam, regress, regress_gp

Source Code: RegressionGAM

Method: predict

RegressionGAM: yFit = predict (obj, Xfit)
RegressionGAM: yFit = predict (…, Name, Value)
RegressionGAM: [yFit, ySD, yInt] = predict (…)

Predict new data points using generalized additive model regression object.

yFit = predict (obj, Xfit returns a vector of predicted responses, yFit, for the predictor data in matrix Xfit based on the Generalized Additive Model in obj. Xfit must have the same number of features/variables as the training data in obj.

  • obj must be a RegressionGAM class object.

[yFit, ySD, yInt] = predict (obj, Xfit also returns the standard deviations, ySD, and prediction intervals, yInt, of the response variable yFit, evaluated at each observation in the predictor data Xfit.

yFit = predict (…, Name, Value) returns the aforementioned results with additional properties specified by Name-Value pair arguments listed below.

NameValue
"alpha"significance level of the prediction intervals yInt, specified as scalar in range [0,1]. The default value is 0.05, which corresponds to 95% prediction intervals.
"includeinteractions"a boolean flag to include interactions to predict new values based on Xfit. By default, "includeinteractions" is true when the GAM model in obj contains a obj.Formula or obj.Interactions fields. Otherwise, is set to false. If set to true when no interactions are present in the trained model, it will result to an error. If set to false when using a model that includes interactions, the predictions will be made on the basic model without any interaction terms. This way you can make predictions from the same GAM model without having to retrain it.

See also: fitrgam, RegressionGAM

Example: 1

  

 ## Train a RegressionGAM Model for synthetic values
 f1 = @(x) cos (3 * x);
 f2 = @(x) x .^ 3;
 x1 = 2 * rand (50, 1) - 1;
 x2 = 2 * rand (50, 1) - 1;
 y = f1(x1) + f2(x2);
 y = y + y .* 0.2 .* rand (50,1);
 X = [x1, x2];
 a = fitrgam (X, y, "tol", 1e-3)

a =

  RegressionGAM object with properties:

            BaseModel: [1x1 struct]
                  DoF: [1x2 double]
              Formula: [0x0 double]
            IntMatrix: [0x0 double]
         Interactions: [0x0 double]
                Knots: [1x2 double]
            ModelwInt: [0x0 double]
      NumObservations: [1x1 double]
        NumPredictors: [1x1 double]
                Order: [1x2 double]
       PredictorNames: [1x2 cell]
         ResponseName: Y
             RowsUsed: [50x1 double]
                  Tol: [1x1 double]
                    X: [50x2 double]
                    Y: [50x1 double]

Example: 2

  

 ## Declare two different functions
 f1 = @(x) cos (3 * x);
 f2 = @(x) x .^ 3;

 ## Generate 80 samples for f1 and f2
 x = [-4*pi:0.1*pi:4*pi-0.1*pi]';
 X1 = f1 (x);
 X2 = f2 (x);

 ## Create a synthetic response by adding noise
 rand ("seed", 3);
 Ytrue = X1 + X2;
 Y = Ytrue + Ytrue .* 0.2 .* rand (80,1);

 ## Assemble predictor data
 X = [X1, X2];

 ## Train the GAM and test on the same data
 a = fitrgam (X, Y, "order", [5, 5]);
 [ypred, ySDsd, yInt] = predict (a, X);

 ## Plot the results
 figure
 [sortedY, indY] = sort (Ytrue);
 plot (sortedY, "r-");
 xlim ([0, 80]);
 hold on
 plot (ypred(indY), "g+")
 plot (yInt(indY,1), "k:")
 plot (yInt(indY,2), "k:")
 xlabel ("Predictor samples");
 ylabel ("Response");
 title ("actual vs predicted values for function f1(x) = cos (3x) ");
 legend ({"Theoretical Response", "Predicted Response", "Prediction Intervals"});

 ## Use 30% Holdout partitioning for training and testing data
 C = cvpartition (80, "HoldOut", 0.3);
 [ypred, ySDsd, yInt] = predict (a, X(test(C),:));

 ## Plot the results
 figure
 [sortedY, indY] = sort (Ytrue(test(C)));
 plot (sortedY, 'r-');
 xlim ([0, sum(test(C))]);
 hold on
 plot (ypred(indY), "g+")
 plot (yInt(indY,1),'k:')
 plot (yInt(indY,2),'k:')
 xlabel ("Predictor samples");
 ylabel ("Response");
 title ("actual vs predicted values for function f1(x) = cos (3x) ");
 legend ({"Theoretical Response", "Predicted Response", "Prediction Intervals"});
plotted figure

plotted figure

statistics-release-1.6.3/docs/SilhouetteEvaluation.html000066400000000000000000000167771456127120000233440ustar00rootroot00000000000000 Statistics: SilhouetteEvaluation

Class Definition: SilhouetteEvaluation

Function File: obj = evalclusters (x, clust, silhouette)
Function File: obj = evalclusters (…, Name, Value)

A silhouette object to evaluate clustering solutions.

A SilhouetteEvaluation object is a ClusterCriterion object used to evaluate clustering solutions using the silhouette criterion.

List of public properties specific to SilhouetteEvaluation:

Distance

a valid distance metric name, or a function handle or a numeric array as generated by the pdist function.

ClusterPriors

a valid name for the evaluation of silhouette values: empirical (default) or equal.

ClusterSilhouettes

a cell array with the silhouette values of each data point for each cluster number.

The best solution according to the silhouette criterion is the one that scores the highest average silhouette value.

See also: evalclusters, ClusterCriterion, CalinskiHarabaszEvaluation, DaviesBouldinEvaluation, GapEvaluation

Source Code: SilhouetteEvaluation

Method: addK

SilhouetteEvaluation: obj = addK (obj, K)

Add a new cluster array to inspect the SilhouetteEvaluation object.

Method: compact

SilhouetteEvaluation: obj = compact (obj)

Return a compact SilhouetteEvaluation object (not implemented yet).

Method: plot

SilhouetteEvaluation: plot (obj)
SilhouetteEvaluation: h = plot (obj)

Plot the evaluation results.

Plot the CriterionValues against InspectedK from the SilhouetteEvaluation ClusterCriterion, obj, to the current plot. It can also return a handle to the current plot.

statistics-release-1.6.3/docs/adtest.html000066400000000000000000000205661456127120000204420ustar00rootroot00000000000000 Statistics: adtest

Function Reference: adtest

statistics: h = adtest (x)
statistics: h = adtest (x, Name, Value)
statistics: [h, pval] = adtest (…)
statistics: [h, pval, adstat, cv] = adtest (…)

Anderson-Darling goodness-of-fit hypothesis test.

h = adtest (x) returns a test decision for the null hypothesis that the data in vector x is from a population with a normal distribution, using the Anderson-Darling test. The alternative hypothesis is that x is not from a population with a normal distribution. The result h is 1 if the test rejects the null hypothesis at the 5% significance level, or 0 otherwise.

h = adtest (x, Name, Value) returns a test decision for the Anderson-Darling test with additional options specified by one or more Name-Value pair arguments. For example, you can specify a null distribution other than normal, or select an alternative method for calculating the p-value, such as a Monte Carlo simulation.

The following parameters can be parsed as Name-Value pair arguments.

NameDescription
"Distribution"The distribution being tested for. It tests whether x could have come from the specified distribution. There are two choise available for parsing distribution parameters:
  • One of the following char strings: "norm", "exp", "ev", "logn", "weibull", for defining either the ’normal’, ’exponential’, ’extreme value’, lognormal, or ’Weibull’ distribution family, accordingly. In this case, x is tested against a composite hypothesis for the specified distribution family and the required distribution parameters are estimated from the data in x. The default is "norm".
  • A cell array defining a distribution in which the first cell contains a char string with the distribution name, as mentioned above, and the consecutive cells containing all specified parameters of the null distribution. In this case, x is tested against a simple hypothesis.
"Alpha"Significance level alpha for the test. Any scalar numeric value between 0 and 1. The default is 0.05 corresponding to the 5% significance level.
"MCTol"Monte-Carlo standard error for the p-value, pval, value. which must be a positive scalar value. In this case, an approximation for the p-value is computed directly, using Monte-Carlo simulations.
"Asymptotic"Method for calculating the p-value of the Anderson-Darling test, which can be either true or false logical value. If you specify ’true’, adtest estimates the p-value using the limiting distribution of the Anderson-Darling test statistic. If you specify ’false’, adtest calculates the p-value based on an analytical formula. For sample sizes greater than 120, the limiting distribution estimate is likely to be more accurate than the small sample size approximation method.
  • If you specify a distribution family with unknown parameters for the distribution Name-Value pair (i.e. composite distribution hypothesis test), the "Asymptotic" option must be false.
  • If you use MCTol to calculate the p-value using a Monte Carlo simulation, the "Asymptotic" option must be false.

[h, pval] = adtest (…) also returns the p-value, pval, of the Anderson-Darling test, using any of the input arguments from the previous syntaxes.

[h, pval, adstat, cv] = adtest (…) also returns the test statistic, adstat, and the critical value, cv, for the Anderson-Darling test.

The Anderson-Darling test statistic belongs to the family of Quadratic Empirical Distribution Function statistics, which are based on the weighted sum of the difference [Fn(×)-F(×)]^2 over the ordered sample values X1 < X2 < ... < Xn, where F is the hypothesized continuous distribution and Fn is the empirical CDF based on the data sample with n sample points.

See also: kstest

Source Code: adtest

statistics-release-1.6.3/docs/anova1.html000066400000000000000000000307411456127120000203370ustar00rootroot00000000000000 Statistics: anova1

Function Reference: anova1

statistics: p = anova1 (x)
statistics: p = anova1 (x, group)
statistics: p = anova1 (x, group, displayopt)
statistics: p = anova1 (x, group, displayopt, vartype)
statistics: [p, atab] = anova1 (x, …)
statistics: [p, atab, stats] = anova1 (x, …)

Perform a one-way analysis of variance (ANOVA) for comparing the means of two or more groups of data under the null hypothesis that the groups are drawn from distributions with the same mean. For planned contrasts and/or diagnostic plots, use anovan instead.

anova1 can take up to three input arguments:

  • x contains the data and it can either be a vector or matrix. If x is a matrix, then each column is treated as a separate group. If x is a vector, then the group argument is mandatory.
  • group contains the names for each group. If x is a matrix, then group can either be a cell array of strings of a character array, with one row per column of x. If you want to omit this argument, enter an empty array ([]). If x is a vector, then group must be a vector of the same length, or a string array or cell array of strings with one row for each element of x. x values corresponding to the same value of group are placed in the same group.
  • displayopt is an optional parameter for displaying the groups contained in the data in a boxplot. If omitted, it is ’on’ by default. If group names are defined in group, these are used to identify the groups in the boxplot. Use ’off’ to omit displaying this figure.
  • vartype is an optional parameter to used to indicate whether the groups can be assumed to come from populations with equal variance. When vartype is "equal" the variances are assumed to be equal (this is the default). When vartype is "unequal" the population variances are not assumed to be equal and Welch’s ANOVA test is used instead.

anova1 can return up to three output arguments:

  • p is the p-value of the null hypothesis that all group means are equal.
  • atab is a cell array containing the results in a standard ANOVA table.
  • stats is a structure containing statistics useful for performing a multiple comparison of means with the MULTCOMPARE function.

If anova1 is called without any output arguments, then it prints the results in a one-way ANOVA table to the standard output. It is also printed when displayopt is ’on’.

Examples:

 
 x = meshgrid (1:6);
 x = x + normrnd (0, 1, 6, 6);
 anova1 (x, [], 'off');
 [p, atab] = anova1(x);
 
 x = ones (50, 4) .* [-2, 0, 1, 5];
 x = x + normrnd (0, 2, 50, 4);
 groups = {"A", "B", "C", "D"};
 anova1 (x, groups);

See also: anova2, anovan, multcompare

Source Code: anova1

Example: 1

  

 x = meshgrid (1:6);
 randn ("seed", 15);    # for reproducibility
 x = x + normrnd (0, 1, 6, 6);
 anova1 (x, [], 'off');


                      ANOVA Table

Source        SS      df        MS       F      Prob>F
------------------------------------------------------
Groups     96.8124     5    19.3625    31.76    0.0000
Error      18.2881    30     0.6096
Total     115.1004    35

Example: 2

  

 x = meshgrid (1:6);
 randn ("seed", 15);    # for reproducibility
 x = x + normrnd (0, 1, 6, 6);
 [p, atab] = anova1(x);


                      ANOVA Table

Source        SS      df        MS       F      Prob>F
------------------------------------------------------
Groups     96.8124     5    19.3625    31.76    0.0000
Error      18.2881    30     0.6096
Total     115.1004    35
plotted figure

Example: 3

  

 x = ones (50, 4) .* [-2, 0, 1, 5];
 randn ("seed", 13);    # for reproducibility
 x = x + normrnd (0, 2, 50, 4);
 groups = {"A", "B", "C", "D"};
 anova1 (x, groups);


                      ANOVA Table

Source        SS      df        MS       F      Prob>F
------------------------------------------------------
Groups   1392.4448     3   464.1483   105.41    0.0000
Error     863.0487   196     4.4033
Total    2255.4935   199
plotted figure

Example: 4

  

 y = [54 87 45; 23 98 39; 45 64 51; 54 77 49; 45 89 50; 47 NaN 55];
 g = [1  2  3 ; 1  2  3 ; 1  2  3 ; 1  2  3 ; 1  2  3 ; 1  2  3 ];
 anova1 (y(:), g(:), "on", "unequal");


           Welch's ANOVA Table

Source        F     df     dfe     Prob>F
-----------------------------------------
Groups     15.52     2    7.58     0.0021
plotted figure

statistics-release-1.6.3/docs/anova2.html000066400000000000000000000260641456127120000203430ustar00rootroot00000000000000 Statistics: anova2

Function Reference: anova2

statistics: p = anova2 (x, reps)
statistics: p = anova2 (x, reps, displayopt)
statistics: p = anova2 (x, reps, displayopt, model)
statistics: [p, atab] = anova2 (…)
statistics: [p, atab, stats] = anova2 (…)

Performs two-way factorial (crossed) or a nested analysis of variance (ANOVA) for balanced designs. For unbalanced factorial designs, diagnostic plots and/or planned contrasts, use anovan instead.

anova2 requires two input arguments with an optional third and fourth:

  • x contains the data and it must be a matrix of at least two columns and two rows.
  • reps is the number of replicates for each combination of factor groups.
  • displayopt is an optional parameter for displaying the ANOVA table, when it is ’on’ (default) and suppressing the display when it is ’off’.
  • model is an optional parameter to specify the model type as either:
    • "interaction" or "full" (default): compute both main effects and their interaction
    • "linear": compute both main effects without an interaction. When reps > 1 the test is suitable for a balanced randomized block design. When reps == 1, the test becomes a One-way Repeated Measures (RM)-ANOVA with Greenhouse-Geisser correction to the column factor degrees of freedom to make the test robust to violations of sphericity
    • "nested": treat the row factor as nested within columns. Note that the row factor is considered a random factor in the calculation of the statistics.

anova2 returns up to three output arguments:

  • p is the p-value of the null hypothesis that all group means are equal.
  • atab is a cell array containing the results in a standard ANOVA table.
  • stats is a structure containing statistics useful for performing a multiple comparison of means with the MULTCOMPARE function.

If anova2 is called without any output arguments, then it prints the results in a one-way ANOVA table to the standard output as if displayopt is ’on’.

Examples:

 
 load popcorn;
 anova2 (popcorn, 3);
 
 [p, anovatab, stats] = anova2 (popcorn, 3, "off");
 disp (p);

See also: anova1, anovan, multcompare

Source Code: anova2

Example: 1

  


 # Factorial (Crossed) Two-way ANOVA with Interaction

 popcorn = [5.5, 4.5, 3.5; 5.5, 4.5, 4.0; 6.0, 4.0, 3.0; ...
            6.5, 5.0, 4.0; 7.0, 5.5, 5.0; 7.0, 5.0, 4.5];

 [p, atab, stats] = anova2(popcorn, 3, "on");


                      ANOVA Table

Source             SS      df        MS       F      Prob>F
-----------------------------------------------------------
Columns         15.7500     2     7.8750    56.70    0.0000
Rows             4.5000     1     4.5000    32.40    0.0001
Interaction      0.0833     2     0.0417     0.30    0.7462
Error            1.6667    12     0.1389
Total           22.0000    17

Example: 2

  


 # One-way Repeated Measures ANOVA (Rows are a crossed random factor)

 data = [54, 43, 78, 111;
         23, 34, 37, 41;
         45, 65, 99, 78;
         31, 33, 36, 35;
         15, 25, 30, 26];

 [p, atab, stats] = anova2 (data, 1, "on", "linear");


                      ANOVA Table

Source             SS      df        MS       F      Prob>F
-----------------------------------------------------------
Columns       2174.9500     3   724.9833     3.62    0.0873
Rows          8371.7000     4  2092.9250    10.45    0.0007
Error         2404.3000    12   200.3583
Total        12950.9500    19

Note: Greenhouse-Geisser's correction was applied to the
degrees of freedom for the Column factor: F(1.74,6.95)

Example: 3

  


 # Balanced Nested One-way ANOVA (Rows are a nested random factor)

 data = [4.5924 7.3809 21.322; -0.5488 9.2085 25.0426; ...
         6.1605 13.1147 22.66; 2.3374 15.2654 24.1283; ...
         5.1873 12.4188 16.5927; 3.3579 14.3951 10.2129; ...
         6.3092 8.5986 9.8934; 3.2831 3.4945 10.0203];

 [p, atab, stats] = anova2 (data, 4, "on", "nested");


                      ANOVA Table

Source             SS      df        MS       F      Prob>F
-----------------------------------------------------------
Columns        745.3603     2   372.6802     4.02    0.1416
Rows           278.0185     3    92.6728     9.26    0.0006
Error          180.1804    18    10.0100
Total         1203.5592    23

Note: Rows are a random factor nested within the columns.
The Column F statistic uses the Row MS instead of the MSE.
statistics-release-1.6.3/docs/anovan.html000066400000000000000000001360621456127120000204370ustar00rootroot00000000000000 Statistics: anovan

Function Reference: anovan

statistics: p = anovan (Y, GROUP)
statistics: p = anovan (Y, GROUP, name, value)
statistics: [p, atab] = anovan (…)
statistics: [p, atab, stats] = anovan (…)
statistics: [p, atab, stats, terms] = anovan (…)

Perform a multi (N)-way analysis of (co)variance (ANOVA or ANCOVA) to evaluate the effect of one or more categorical or continuous predictors (i.e. independent variables) on a continuous outcome (i.e. dependent variable). The algorithms used make anovan suitable for balanced or unbalanced factorial (crossed) designs. By default, anovan treats all factors as fixed. Examples of function usage can be found by entering the command demo anovan. A bootstrap resampling variant of this function, bootlm, is available in the statistics-resampling package and has similar usage.

Data is a single vector Y with groups specified by a corresponding matrix or cell array of group labels GROUP, where each column of GROUP has the same number of rows as Y. For example, if Y = [1.1;1.2]; GROUP = [1,2,1; 1,5,2]; then observation 1.1 was measured under conditions 1,2,1 and observation 1.2 was measured under conditions 1,5,2. If the GROUP provided is empty, then the linear model is fit with just the intercept (no predictors).

anovan can take a number of optional parameters as name-value pairs.

[…] = anovan (Y, GROUP, "continuous", continuous)

  • continuous is a vector of indices indicating which of the columns (i.e. factors) in GROUP should be treated as continuous predictors rather than as categorical predictors. The relationship between continuous predictors and the outcome should be linear.

[…] = anovan (Y, GROUP, "random", random)

  • random is a vector of indices indicating which of the columns (i.e. factors) in GROUP should be treated as random effects rather than fixed effects. Octave anovan provides only basic support for random effects. Specifically, since all F-statistics in anovan are calculated using the mean-squared error (MSE), any interaction terms containing a random effect are dropped from the model term definitions and their associated variance is pooled with the residual, unexplained variance making up the MSE. In effect, the model then fitted equates to a linear mixed model with random intercept(s). Variable names for random factors are appended with a ’ symbol.

[…] = anovan (Y, GROUP, "model", modeltype)

  • modeltype can specified as one of the following:
    • "linear" (default) : compute N main effects with no interactions.
    • "interaction" : compute N effects and N×(N-1) two-factor interactions
    • "full" : compute the N main effects and interactions at all levels
    • a scalar integer : representing the maximum interaction order
    • a matrix of term definitions : each row is a term and each column is a factor
     
     -- Example:
 A two-way ANOVA with interaction would be: [1 0; 0 1; 1 1]

[…] = anovan (Y, GROUP, "sstype", sstype)

  • sstype can specified as one of the following:
    • 1 : Type I sequential sums-of-squares.
    • 2 or "h" : Type II partially sequential (or hierarchical) sums-of-squares
    • 3 (default) : Type III partial, constrained or marginal sums-of-squares

[…] = anovan (Y, GROUP, "varnames", varnames)

  • varnames must be a cell array of strings with each element containing a factor name for each column of GROUP. By default (if not parsed as optional argument), varnames are "X1","X2","X3", etc.

[…] = anovan (Y, GROUP, "alpha", alpha)

  • alpha must be a scalar value between 0 and 1 requesting 100×(1-alpha)% confidence bounds for the regression coefficients returned in stats.coeffs (default 0.05 for 95% confidence).

[…] = anovan (Y, GROUP, "display", dispopt)

  • dispopt can be either "on" (default) or "off" and controls the display of the model formula, table of model parameters, the ANOVA table and the diagnostic plots. The F-statistic and p-values are formatted in APA-style. To avoid p-hacking, the table of model parameters is only displayed if we set planned contrasts (see below).

[…] = anovan (Y, GROUP, "contrasts", contrasts)

  • contrasts can be specified as one of the following:
    • A string corresponding to one of the built-in contrasts listed below:
      • "simple" or "anova" (default): Simple (ANOVA) contrast coding. (The first level appearing in the GROUP column is the reference level)
      • "poly": Polynomial contrast coding for trend analysis.
      • "helmert": Helmert contrast coding: the difference between each level with the mean of the subsequent levels.
      • "effect": Deviation effect coding. (The first level appearing in the GROUP column is omitted).
      • "sdif" or "sdiff": Successive differences contrast coding: the difference between each level with the previous level.
      • "treatment": Treatment contrast (or dummy) coding. (The first level appearing in the GROUP column is the reference level). These contrasts are not compatible with sstype = 3.
    • A matrix containing a custom contrast coding scheme (i.e. the generalized inverse of contrast weights). Rows in the contrast matrices correspond to factor levels in the order that they first appear in the GROUP column. The matrix must contain the same number of columns as there are the number of factor levels minus one.

    If the anovan model contains more than one factor and a built-in contrast coding scheme was specified, then those contrasts are applied to all factors. To specify different contrasts for different factors in the model, contrasts should be a cell array with the same number of cells as there are columns in GROUP. Each cell should define contrasts for the respective column in GROUP by one of the methods described above. If cells are left empty, then the default contrasts are applied. Contrasts for cells corresponding to continuous factors are ignored.

[…] = anovan (Y, GROUP, "weights", weights)

  • weights is an optional vector of weights to be used when fitting the linear model. Weighted least squares (WLS) is used with weights (that is, minimizing sum (weights * residuals .^ 2)); otherwise ordinary least squares (OLS) is used (default is empty for OLS).

anovan can return up to four output arguments:

p = anovan (…) returns a vector of p-values, one for each term.

[p, atab] = anovan (…) returns a cell array containing the ANOVA table.

[p, atab, stats] = anovan (…) returns a structure containing additional statistics, including degrees of freedom and effect sizes for each term in the linear model, the design matrix, the variance-covariance matrix, (weighted) model residuals, and the mean squared error. The columns of stats.coeffs (from left-to-right) report the model coefficients, standard errors, lower and upper 100×(1-alpha)% confidence interval bounds, t-statistics, and p-values relating to the contrasts. The number appended to each term name in stats.coeffnames corresponds to the column number in the relevant contrast matrix for that factor. The stats structure can be used as input for multcompare.

[p, atab, stats, terms] = anovan (…) returns the model term definitions.

See also: anova1, anova2, multcompare, fitlm

Source Code: anovan

Example: 1

  


 # Two-sample unpaired test on independent samples (equivalent to Student's
 # t-test). Note that the absolute value of t-statistic can be obtained by
 # taking the square root of the reported F statistic. In this example,
 # t = sqrt (1.44) = 1.20.

 score = [54 23 45 54 45 43 34 65 77 46 65]';
 gender = {"male" "male" "male" "male" "male" "female" "female" "female" ...
           "female" "female" "female"}';

 [P, ATAB, STATS] = anovan (score, gender, "display", "on", "varnames", "gender");


MODEL FORMULA (based on Wilkinson's notation):

Y ~ 1 + gender

ANOVA TABLE (Type III sums-of-squares):

Source                   Sum Sq.    d.f.    Mean Sq.  R Sq.            F  Prob>F
--------------------------------------------------------------------------------
gender                    318.11       1      318.11  0.138         1.44    .261 
Error                     1992.8       9      221.42
Total                     2310.9      10
plotted figure

Example: 2

  


 # Two-sample paired test on dependent or matched samples equivalent to a
 # paired t-test. As for the first example, the t-statistic can be obtained by
 # taking the square root of the reported F statistic. Note that the interaction
 # between treatment x subject was dropped from the full model by assigning
 # subject as a random factor (').

 score = [4.5 5.6; 3.7 6.4; 5.3 6.4; 5.4 6.0; 3.9 5.7]';
 treatment = {"before" "after"; "before" "after"; "before" "after";
              "before" "after"; "before" "after"}';
 subject = {"GS" "GS"; "JM" "JM"; "HM" "HM"; "JW" "JW"; "PS" "PS"}';

 [P, ATAB, STATS] = anovan (score(:), {treatment(:), subject(:)}, ...
                            "model", "full", "random", 2, "sstype", 2, ...
                            "varnames", {"treatment", "subject"}, ...
                            "display", "on");


MODEL FORMULA (based on Wilkinson's notation):

Y ~ 1 + treatment + (1|subject)

ANOVA TABLE (Type II sums-of-squares):

Source                   Sum Sq.    d.f.    Mean Sq.  R Sq.            F  Prob>F
--------------------------------------------------------------------------------
treatment                  5.329       1       5.329  0.801        16.08    .016 
subject'                   1.674       4      0.4185  0.558         1.26    .413 
Error                      1.326       4      0.3315
Total                      8.329       9
plotted figure

Example: 3

  


 # One-way ANOVA on the data from a study on the strength of structural beams,
 # in Hogg and Ledolter (1987) Engineering Statistics. New York: MacMillan

 strength = [82 86 79 83 84 85 86 87 74 82 ...
            78 75 76 77 79 79 77 78 82 79]';
 alloy = {"st","st","st","st","st","st","st","st", ...
          "al1","al1","al1","al1","al1","al1", ...
          "al2","al2","al2","al2","al2","al2"}';

 [P, ATAB, STATS] = anovan (strength, alloy, "display", "on", ...
                            "varnames", "alloy");


MODEL FORMULA (based on Wilkinson's notation):

Y ~ 1 + alloy

ANOVA TABLE (Type III sums-of-squares):

Source                   Sum Sq.    d.f.    Mean Sq.  R Sq.            F  Prob>F
--------------------------------------------------------------------------------
alloy                      184.8       2        92.4  0.644        15.40   <.001 
Error                        102      17           6
Total                      286.8      19
plotted figure

Example: 4

  


 # One-way repeated measures ANOVA on the data from a study on the number of
 # words recalled by 10 subjects for three time condtions, in Loftus & Masson
 # (1994) Psychon Bull Rev. 1(4):476-490, Table 2. Note that the interaction
 # between seconds x subject was dropped from the full model by assigning
 # subject as a random factor (').

 words = [10 13 13; 6 8 8; 11 14 14; 22 23 25; 16 18 20; ...
          15 17 17; 1 1 4; 12 15 17;  9 12 12;  8 9 12];
 seconds = [1 2 5; 1 2 5; 1 2 5; 1 2 5; 1 2 5; ...
            1 2 5; 1 2 5; 1 2 5; 1 2 5; 1 2 5;];
 subject = [ 1  1  1;  2  2  2;  3  3  3;  4  4  4;  5  5  5; ...
             6  6  6;  7  7  7;  8  8  8;  9  9  9; 10 10 10];

 [P, ATAB, STATS] = anovan (words(:), {seconds(:), subject(:)}, ...
                            "model", "full", "random", 2, "sstype", 2, ...
                            "display", "on", "varnames", {"seconds", "subject"});


MODEL FORMULA (based on Wilkinson's notation):

Y ~ 1 + seconds + (1|subject)

ANOVA TABLE (Type II sums-of-squares):

Source                   Sum Sq.    d.f.    Mean Sq.  R Sq.            F  Prob>F
--------------------------------------------------------------------------------
seconds                   52.267       2      26.133  0.825        42.51   <.001 
subject'                  942.53       9      104.73  0.988       170.34   <.001 
Error                     11.067      18     0.61481
Total                     1005.9      29
plotted figure

Example: 5

  


 # Balanced two-way ANOVA with interaction on the data from a study of popcorn
 # brands and popper types, in Hogg and Ledolter (1987) Engineering Statistics.
 # New York: MacMillan

 popcorn = [5.5, 4.5, 3.5; 5.5, 4.5, 4.0; 6.0, 4.0, 3.0; ...
            6.5, 5.0, 4.0; 7.0, 5.5, 5.0; 7.0, 5.0, 4.5];
 brands = {"Gourmet", "National", "Generic"; ...
           "Gourmet", "National", "Generic"; ...
           "Gourmet", "National", "Generic"; ...
           "Gourmet", "National", "Generic"; ...
           "Gourmet", "National", "Generic"; ...
           "Gourmet", "National", "Generic"};
 popper = {"oil", "oil", "oil"; "oil", "oil", "oil"; "oil", "oil", "oil"; ...
           "air", "air", "air"; "air", "air", "air"; "air", "air", "air"};

 [P, ATAB, STATS] = anovan (popcorn(:), {brands(:), popper(:)}, ...
                            "display", "on", "model", "full", ...
                            "varnames", {"brands", "popper"});


MODEL FORMULA (based on Wilkinson's notation):

Y ~ 1 + brands + popper + brands:popper

ANOVA TABLE (Type III sums-of-squares):

Source                   Sum Sq.    d.f.    Mean Sq.  R Sq.            F  Prob>F
--------------------------------------------------------------------------------
brands                     15.75       2       7.875  0.904        56.70   <.001 
popper                       4.5       1         4.5  0.730        32.40   <.001 
brands*popper           0.083333       2    0.041667  0.048         0.30    .746 
Error                     1.6667      12     0.13889
Total                         22      17
plotted figure

Example: 6

  


 # Unbalanced two-way ANOVA (2x2) on the data from a study on the effects of
 # gender and having a college degree on salaries of company employees,
 # in Maxwell, Delaney and Kelly (2018): Chapter 7, Table 15

 salary = [24 26 25 24 27 24 27 23 15 17 20 16, ...
           25 29 27 19 18 21 20 21 22 19]';
 gender = {"f" "f" "f" "f" "f" "f" "f" "f" "f" "f" "f" "f"...
           "m" "m" "m" "m" "m" "m" "m" "m" "m" "m"}';
 degree = [1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 0 0 0 0 0 0 0]';

 [P, ATAB, STATS] = anovan (salary, {gender, degree}, "model", "full", ...
                            "sstype", 3, "display", "on", "varnames", ...
                            {"gender", "degree"});


MODEL FORMULA (based on Wilkinson's notation):

Y ~ 1 + gender + degree + gender:degree

ANOVA TABLE (Type III sums-of-squares):

Source                   Sum Sq.    d.f.    Mean Sq.  R Sq.            F  Prob>F
--------------------------------------------------------------------------------
gender                    29.371       1      29.371  0.370        10.57    .004 
degree                    264.34       1      264.34  0.841        95.16   <.001 
gender*degree             1.1748       1      1.1748  0.023         0.42    .524 
Error                         50      18      2.7778
Total                     323.86      21
plotted figure

Example: 7

  


 # Unbalanced two-way ANOVA (3x2) on the data from a study of the effect of
 # adding sugar and/or milk on the tendency of coffee to make people babble,
 # in from Navarro (2019): 16.10

 sugar = {"real" "fake" "fake" "real" "real" "real" "none" "none" "none" ...
          "fake" "fake" "fake" "real" "real" "real" "none" "none" "fake"}';
 milk = {"yes" "no" "no" "yes" "yes" "no" "yes" "yes" "yes" ...
         "no" "no" "yes" "no" "no" "no" "no" "no" "yes"}';
 babble = [4.6 4.4 3.9 5.6 5.1 5.5 3.9 3.5 3.7...
           5.6 4.7 5.9 6.0 5.4 6.6 5.8 5.3 5.7]';

 [P, ATAB, STATS] = anovan (babble, {sugar, milk}, "model", "full",  ...
                            "sstype", 3, "display", "on", ...
                            "varnames", {"sugar", "milk"});


MODEL FORMULA (based on Wilkinson's notation):

Y ~ 1 + sugar + milk + sugar:milk

ANOVA TABLE (Type III sums-of-squares):

Source                   Sum Sq.    d.f.    Mean Sq.  R Sq.            F  Prob>F
--------------------------------------------------------------------------------
sugar                     2.1318       2      1.0659  0.403         4.04    .045 
milk                      1.0041       1      1.0041  0.241         3.81    .075 
sugar*milk                5.9439       2      2.9719  0.653        11.28    .002 
Error                     3.1625      12     0.26354
Total                      13.62      17
plotted figure

Example: 8

  


 # Unbalanced three-way ANOVA (3x2x2) on the data from a study of the effects
 # of three different drugs, biofeedback and diet on patient blood pressure,
 # adapted* from Maxwell, Delaney and Kelly (2018): Chapter 8, Table 12
 # * Missing values introduced to make the sample sizes unequal to test the
 #   calculation of different types of sums-of-squares

 drug = {"X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" ...
         "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X";
         "Y" "Y" "Y" "Y" "Y" "Y" "Y" "Y" "Y" "Y" "Y" "Y" ...
         "Y" "Y" "Y" "Y" "Y" "Y" "Y" "Y" "Y" "Y" "Y" "Y";
         "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" ...
         "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z"};
 feedback = [1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0;
             1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0;
             1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0];
 diet = [0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1 1;
         0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1 1;
         0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1 1];
 BP = [170 175 165 180 160 158 161 173 157 152 181 190 ...
       173 194 197 190 176 198 164 190 169 164 176 175;
       186 194 201 215 219 209 164 166 159 182 187 174 ...
       189 194 217 206 199 195 171 173 196 199 180 NaN;
       180 187 199 170 204 194 162 184 183 156 180 173 ...
       202 228 190 206 224 204 205 199 170 160 NaN NaN];

 [P, ATAB, STATS] = anovan (BP(:), {drug(:), feedback(:), diet(:)}, ...
                                    "model", "full", "sstype", 3, ...
                                    "display", "on", ...
                                    "varnames", {"drug", "feedback", "diet"});


MODEL FORMULA (based on Wilkinson's notation):

Y ~ 1 + drug + feedback + diet + drug:feedback + drug:diet + feedback:diet + drug:feedback:diet

ANOVA TABLE (Type III sums-of-squares):

Source                   Sum Sq.    d.f.    Mean Sq.  R Sq.            F  Prob>F
--------------------------------------------------------------------------------
drug                      3430.9       2      1715.4  0.275        10.79   <.001 
feedback                  1833.7       1      1833.7  0.168        11.53    .001 
diet                      5080.5       1      5080.5  0.359        31.94   <.001 
drug*feedback             382.08       2      191.04  0.040         1.20    .308 
drug*diet                 963.04       2      481.52  0.096         3.03    .056 
feedback*diet             44.452       1      44.452  0.005         0.28    .599 
drug*feedback*diet        814.35       2      407.17  0.082         2.56    .086 
Error                     9065.8      57      159.05
Total                      22190      68
plotted figure

Example: 9

  


 # Balanced three-way ANOVA (2x2x2) with one of the factors being a blocking
 # factor. The data is from a randomized block design study on the effects
 # of antioxidant treatment on glutathione-S-transferase (GST) levels in
 # different mouse strains, from Festing (2014), ILAR Journal, 55(3):427-476.
 # Note that all interactions involving block were dropped from the full model
 # by assigning block as a random factor (').

 measurement = [444 614 423 625 408  856 447 719 ...
                764 831 586 782 609 1002 606 766]';
 strain= {"NIH","NIH","BALB/C","BALB/C","A/J","A/J","129/Ola","129/Ola", ...
          "NIH","NIH","BALB/C","BALB/C","A/J","A/J","129/Ola","129/Ola"}';
 treatment={"C" "T" "C" "T" "C" "T" "C" "T" "C" "T" "C" "T" "C" "T" "C" "T"}';
 block = [1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2]';

 [P, ATAB, STATS] = anovan (measurement/10, {strain, treatment, block}, ...
                            "sstype", 2, "model", "full", "random", 3, ...
                            "display", "on", ...
                            "varnames", {"strain", "treatment", "block"});


MODEL FORMULA (based on Wilkinson's notation):

Y ~ 1 + strain + treatment + (1|block) + strain:treatment

ANOVA TABLE (Type II sums-of-squares):

Source                   Sum Sq.    d.f.    Mean Sq.  R Sq.            F  Prob>F
--------------------------------------------------------------------------------
strain                    286.13       3      95.377  0.580         3.23    .091 
treatment                 2275.3       1      2275.3  0.917        76.94   <.001 
block'                    1242.6       1      1242.6  0.857        42.02   <.001 
strain*treatment           495.9       3       165.3  0.706         5.59    .028 
Error                     207.01       7      29.573
Total                     4506.9      15
plotted figure

Example: 10

  


 # One-way ANCOVA on data from a study of the additive effects of species
 # and temperature on chirpy pulses of crickets, from Stitch, The Worst Stats
 # Text eveR

 pulse = [67.9 65.1 77.3 78.7 79.4 80.4 85.8 86.6 87.5 89.1 ...
          98.6 100.8 99.3 101.7 44.3 47.2 47.6 49.6 50.3 51.8 ...
          60 58.5 58.9 60.7 69.8 70.9 76.2 76.1 77 77.7 84.7]';
 temp = [20.8 20.8 24 24 24 24 26.2 26.2 26.2 26.2 28.4 ...
         29 30.4 30.4 17.2 18.3 18.3 18.3 18.9 18.9 20.4 ...
         21 21 22.1 23.5 24.2 25.9 26.5 26.5 26.5 28.6]';
 species = {"ex" "ex" "ex" "ex" "ex" "ex" "ex" "ex" "ex" "ex" "ex" ...
            "ex" "ex" "ex" "niv" "niv" "niv" "niv" "niv" "niv" "niv" ...
            "niv" "niv" "niv" "niv" "niv" "niv" "niv" "niv" "niv" "niv"};

 [P, ATAB, STATS] = anovan (pulse, {species, temp}, "model", "linear", ...
                           "continuous", 2, "sstype", "h", "display", "on", ...
                           "varnames", {"species", "temp"});


MODEL FORMULA (based on Wilkinson's notation):

Y ~ 1 + species + temp

ANOVA TABLE (Type II sums-of-squares):

Source                   Sum Sq.    d.f.    Mean Sq.  R Sq.            F  Prob>F
--------------------------------------------------------------------------------
species                      598       1         598  0.870       187.40   <.001 
temp                      4376.1       1      4376.1  0.980      1371.35   <.001 
Error                      89.35      28      3.1911
Total                     8582.2      30
plotted figure

Example: 11

  


 # Factorial ANCOVA on data from a study of the effects of treatment and
 # exercise on stress reduction score after adjusting for age. Data from R
 # datarium package).

 score = [95.6 82.2 97.2 96.4 81.4 83.6 89.4 83.8 83.3 85.7 ...
          97.2 78.2 78.9 91.8 86.9 84.1 88.6 89.8 87.3 85.4 ...
          81.8 65.8 68.1 70.0 69.9 75.1 72.3 70.9 71.5 72.5 ...
          84.9 96.1 94.6 82.5 90.7 87.0 86.8 93.3 87.6 92.4 ...
          100. 80.5 92.9 84.0 88.4 91.1 85.7 91.3 92.3 87.9 ...
          91.7 88.6 75.8 75.7 75.3 82.4 80.1 86.0 81.8 82.5]';
 treatment = {"yes" "yes" "yes" "yes" "yes" "yes" "yes" "yes" "yes" "yes" ...
              "yes" "yes" "yes" "yes" "yes" "yes" "yes" "yes" "yes" "yes" ...
              "yes" "yes" "yes" "yes" "yes" "yes" "yes" "yes" "yes" "yes" ...
              "no"  "no"  "no"  "no"  "no"  "no"  "no"  "no"  "no"  "no"  ...
              "no"  "no"  "no"  "no"  "no"  "no"  "no"  "no"  "no"  "no"  ...
              "no"  "no"  "no"  "no"  "no"  "no"  "no"  "no"  "no"  "no"}';
 exercise = {"lo"  "lo"  "lo"  "lo"  "lo"  "lo"  "lo"  "lo"  "lo"  "lo"  ...
             "mid" "mid" "mid" "mid" "mid" "mid" "mid" "mid" "mid" "mid" ...
             "hi"  "hi"  "hi"  "hi"  "hi"  "hi"  "hi"  "hi"  "hi"  "hi"  ...
             "lo"  "lo"  "lo"  "lo"  "lo"  "lo"  "lo"  "lo"  "lo"  "lo"  ...
             "mid" "mid" "mid" "mid" "mid" "mid" "mid" "mid" "mid" "mid" ...
             "hi"  "hi"  "hi"  "hi"  "hi"  "hi"  "hi"  "hi"  "hi"  "hi"}';
 age = [59 65 70 66 61 65 57 61 58 55 62 61 60 59 55 57 60 63 62 57 ...
        58 56 57 59 59 60 55 53 55 58 68 62 61 54 59 63 60 67 60 67 ...
        75 54 57 62 65 60 58 61 65 57 56 58 58 58 52 53 60 62 61 61]';

 [P, ATAB, STATS] = anovan (score, {treatment, exercise, age}, ...
                            "model", [1 0 0; 0 1 0; 0 0 1; 1 1 0], ...
                            "continuous", 3, "sstype", "h", "display", "on", ...
                            "varnames", {"treatment", "exercise", "age"});


MODEL FORMULA (based on Wilkinson's notation):

Y ~ 1 + treatment + exercise + age + treatment:exercise

ANOVA TABLE (Type II sums-of-squares):

Source                   Sum Sq.    d.f.    Mean Sq.  R Sq.            F  Prob>F
--------------------------------------------------------------------------------
treatment                  275.7       1       275.7  0.173        11.10    .002 
exercise                  1034.6       2      517.32  0.440        20.82   <.001 
age                       226.36       1      226.36  0.147         9.11    .004 
treatment*exercise        220.94       2      110.47  0.144         4.45    .016 
Error                     1316.9      53      24.848
Total                     3888.3      59
plotted figure

Example: 12

  


 # Unbalanced one-way ANOVA with custom, orthogonal contrasts. The statistics
 # relating to the contrasts are shown in the table of model parameters, and
 # can be retrieved from the STATS.coeffs output.

 dv =  [ 8.706 10.362 11.552  6.941 10.983 10.092  6.421 14.943 15.931 ...
        22.968 18.590 16.567 15.944 21.637 14.492 17.965 18.851 22.891 ...
        22.028 16.884 17.252 18.325 25.435 19.141 21.238 22.196 18.038 ...
        22.628 31.163 26.053 24.419 32.145 28.966 30.207 29.142 33.212 ...
        25.694 ]';
 g = [1 1 1 1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 3 ...
      4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5]';
 C = [ 0.4001601  0.3333333  0.5  0.0
       0.4001601  0.3333333 -0.5  0.0
       0.4001601 -0.6666667  0.0  0.0
      -0.6002401  0.0000000  0.0  0.5
      -0.6002401  0.0000000  0.0 -0.5];

 [P,ATAB, STATS] = anovan (dv, g, "contrasts", C, "varnames", "score", ...
                          "alpha", 0.05, "display", "on");


MODEL FORMULA (based on Wilkinson's notation):

Y ~ 1 + score

MODEL PARAMETERS (contrasts for the fixed effects)

Parameter               Estimate        SE  Lower.CI  Upper.CI        t Prob>|t|
--------------------------------------------------------------------------------
(Intercept)                 19.4     0.483      18.4      20.4    40.16    <.001 
score_1                    -9.33     0.969     -11.3     -7.36    -9.62    <.001 
score_2                       -5      1.31     -7.66     -2.34    -3.82    <.001 
score_3                       -8      1.64     -11.3     -4.66    -4.87    <.001 
score_4                       -8      1.45       -11     -5.04    -5.51    <.001 

ANOVA TABLE (Type III sums-of-squares):

Source                   Sum Sq.    d.f.    Mean Sq.  R Sq.            F  Prob>F
--------------------------------------------------------------------------------
score                     1561.3       4      390.33  0.855        47.10   <.001 
Error                     265.17      32      8.2866
Total                     1826.5      36
plotted figure

Example: 13

  


 # One-way ANOVA with the linear model fit by weighted least squares to
 # account for heteroskedasticity. In this example, the variance appears
 # proportional to the outcome, so weights have been estimated by initially
 # fitting the model without weights and regressing the absolute residuals on
 # the fitted values. Although this data could have been analysed by Welch's
 # ANOVA test, the approach here can generalize to ANOVA models with more than
 # one factor.

 g = [1, 1, 1, 1, 1, 1, 1, 1, ...
      2, 2, 2, 2, 2, 2, 2, 2, ...
      3, 3, 3, 3, 3, 3, 3, 3]';
 y = [13, 16, 16,  7, 11,  5,  1,  9, ...
      10, 25, 66, 43, 47, 56,  6, 39, ...
      11, 39, 26, 35, 25, 14, 24, 17]';

 [P,ATAB,STATS] = anovan(y, g, "display", "off");
 fitted = STATS.X * STATS.coeffs(:,1); # fitted values
 b = polyfit (fitted, abs (STATS.resid), 1);
 v = polyval (b, fitted);  # Variance as a function of the fitted values
 figure("Name", "Regression of the absolute residuals on the fitted values");
 plot (fitted, abs (STATS.resid),'ob');hold on; plot(fitted,v,'-r'); hold off;
 xlabel("Fitted values"); ylabel("Absolute residuals");

 [P,ATAB,STATS] = anovan (y, g, "weights", v.^-1);


MODEL FORMULA (based on Wilkinson's notation):

Y ~ 1 + X1

ANOVA TABLE (Type III sums-of-squares):

Source                   Sum Sq.    d.f.    Mean Sq.  R Sq.            F  Prob>F
--------------------------------------------------------------------------------
X1                        2237.9       2      1118.9  0.522        11.49   <.001 
Error                     2045.5      21      97.405
Total                     6905.6      23
plotted figure

plotted figure

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@/qôèQ×Û¢¢"‹Å2~üxÕy¡»‹ŒŒÜ¸q£ˆÜ¿¿²²Òl6Ϙ1C¯ç0þƒRè´ÊÊʨ¨¨²²²ÿºõðáɉ‰111qqqëÖ­khhpmjii pý÷Óëõþþþª/q»rà˨Kʦ¬¬ìí·ß>|ø©S§è}„ç•SUUuùòeçë¾}û®^½º¹¹¹´´TõÊÐ]Ð8j5sæ̹sç8pàÏ:d·ÛÓÓÓèY³fÑh¬ŠHdddÿþýU/Ï‘ç•c2™>ù䓬¬¬””«Õš››9vìXÕ+CwA×¢•ë1:§OŸ~v«Ãá¸uëVXXØSŸìGŽ)"µµµÎ?ÿ+V¬hmm]±b…ˆÌ˜1cùò媗…ç®K*¾Æ󲩨¨p8+W®|2 ;;{æÌ™ª‡çÈóÊéׯ_^^ÞÖ­[“““^ýõ?þ¸OŸ>ªW†î‚ƱkX,›ÍúÔ¸óÞ×u:.33333Su¾è.4VŽÓøñãËËËU§ õ´”MjjjjjªêLѽhü…3nܸýû÷«NÝ×8v ç×=5,"\D4N¸‡#ŽT:t¨óEMMˆœ;w®°°°OŸ>›7o¾råJYYÙ•+W¶lÙâïï_RRráÂgð¬Y³víÚ-"‹-Úµk×›o¾©}:À=4ŽT ‘ÆÆFqž}~÷ÝwgÏž "o½õÖ´iÓDäúõëíïÍÃé€öqÖ€b:Îõ:33sõêÕzýÓŸi-‹ˆ´µµµ¿+§ÚGã@¥¦¦¦ææf1"âêùnß¾]YYY[[[UUå<ã¬eoN´Æ€J·oßv¾xñÅEÄf³íܹóøñãÕÕÕÎñW_}5::úÚµkîÍÃé€öÑ8PéôéÓ"%"+W®üî»ïBBBRRR¦OŸÑ¿Ù¸q£–ÎÏÃé€öÑ8PÆjµîÝ»WDRRR ÃÇøáÉÏÏ5jÔ“‘­­­îÍÃé€qW55,Ëš5k‚ƒƒ/^,"ÕÕÕv»ý…^ˆŒŒ|2òÑ£GgÏžíp‡NtˆÆÀs÷øñcë>|øóÏ? Statistics: bartlett_test

Function Reference: bartlett_test

statistics: h = bartlett_test (x)
statistics: h = bartlett_test (x, group)
statistics: h = bartlett_test (x, alpha)
statistics: h = bartlett_test (x, group, alpha)
statistics: [h, pval] = bartlett_test (…)
statistics: [h, pval, chisq] = bartlett_test (…)
statistics: [h, pval, chisq, df] = bartlett_test (…)

Perform a Bartlett test for the homogeneity of variances.

Under the null hypothesis of equal variances, the test statistic chisq approximately follows a chi-square distribution with df degrees of freedom.

The p-value (1 minus the CDF of this distribution at chisq) is returned in pval. h = 1 if the null hypothesis is rejected at the significance level of alpha. Otherwise h = 0.

Input Arguments:

  • x contains the data and it can either be a vector or matrix. If x is a matrix, then each column is treated as a separate group. If x is a vector, then the group argument is mandatory. NaN values are omitted.
  • group contains the names for each group. If x is a vector, then group must be a vector of the same length, or a string array or cell array of strings with one row for each element of x. x values corresponding to the same value of group are placed in the same group. If x is a matrix, then group can either be a cell array of strings of a character array, with one row per column of x in the same way it is used in anova1 function. If x is a matrix, then group can be omitted either by entering an empty array ([]) or by parsing only alpha as a second argument (if required to change its default value).
  • alpha is the statistical significance value at which the null hypothesis is rejected. Its default value is 0.05 and it can be parsed either as a second argument (when group is omitted) or as a third argument.

See also: levene_test, vartest2, vartestn

Source Code: bartlett_test

statistics-release-1.6.3/docs/barttest.html000066400000000000000000000116561456127120000210060ustar00rootroot00000000000000 Statistics: barttest

Function Reference: barttest

statistics: ndim = barttest (x)
statistics: ndim = barttest (x, alpha)
statistics: [ndim, pval] = barttest (x, alpha)
statistics: [ndim, pval, chisq] = barttest (x, alpha)

Bartlett’s test of sphericity for correlation.

It compares an observed correlation matrix to the identity matrix in order to check if there is a certain redundancy between the variables that we can summarize with a few number of factors. A statistically significant test shows that the variables (columns) in x are correlated, thus it makes sense to perform some dimensionality reduction of the data in x.

ndim = barttest (x, alpha) returns the number of dimensions necessary to explain the nonrandom variation in the data matrix x at the alpha significance level. alpha is an optional input argument and, when not provided, it is 0.05 by default.

[ndim, pval, chisq] = barttest (…) also returns the significance values pval for the hypothesis test for each dimension as well as the associated chi^2 values in chisq

Source Code: barttest

statistics-release-1.6.3/docs/betacdf.html000066400000000000000000000144141456127120000205410ustar00rootroot00000000000000 Statistics: betacdf

Function Reference: betacdf

statistics: p = betacdf (x, a, b)
statistics: p = betacdf (x, a, b, "upper")

Beta cumulative distribution function (CDF).

For each element of x, compute the cumulative distribution function of the Beta distribution with shape parameters a and b. The size of p is the common size of x, a, and b. A scalar input functions as a constant matrix of the same size as the other inputs.

p = betacdf (x, a, b, "upper") computes the upper tail probability of the Beta distribution with parameters a and b, at the values in x.

Further information about the Beta distribution can be found at https://en.wikipedia.org/wiki/Beta_distribution

See also: betainv, betapdf, betarnd, betafit, betalike, betastat

Source Code: betacdf

Example: 1

  

 ## Plot various CDFs from the Beta distribution
 x = 0:0.005:1;
 p1 = betacdf (x, 0.5, 0.5);
 p2 = betacdf (x, 5, 1);
 p3 = betacdf (x, 1, 3);
 p4 = betacdf (x, 2, 2);
 p5 = betacdf (x, 2, 5);
 plot (x, p1, "-b", x, p2, "-g", x, p3, "-r", x, p4, "-c", x, p5, "-m")
 grid on
 legend ({"α = β = 0.5", "α = 5, β = 1", "α = 1, β = 3", ...
          "α = 2, β = 2", "α = 2, β = 5"}, "location", "northwest")
 title ("Beta CDF")
 xlabel ("values in x")
 ylabel ("probability")
plotted figure

statistics-release-1.6.3/docs/betafit.html000066400000000000000000000205301456127120000205630ustar00rootroot00000000000000 Statistics: betafit

Function Reference: betafit

statistics: paramhat = betafit (x)
statistics: [paramhat, paramci] = betafit (x)
statistics: [paramhat, paramci] = betafit (x, alpha)
statistics: [paramhat, paramci] = betafit (x, alpha, options)

Estimate parameters and confidence intervals for the Beta distribution.

paramhat = betafit (x) returns the maximum likelihood estimates of the parameters of the Beta distribution given the data in vector x. paramhat([1, 2]) corresponds to the α and β shape parameters, respectively. Missing values, NaNs, are ignored.

[paramhat, paramci] = betafit (x) returns the 95% confidence intervals for the parameter estimates.

[…] = betafit (x, alpha) also returns the 100 * (1 - alpha) percent confidence intervals of the estimated parameter. By default, the optional argument alpha is 0.05 corresponding to 95% confidence intervals.

[paramhat, paramci] = nbinfit (x, alpha, options) specifies control parameters for the iterative algorithm used to compute ML estimates with the fminsearch function. options is a structure with the following fields and their default values:

  • options.Display = "off"
  • options.MaxFunEvals = 400
  • options.MaxIter = 200
  • options.TolX = 1e-6

The Beta distribution is defined on the open interval (0,1). However, betafit can also compute the unbounded beta likelihood function for data that include exact zeros or ones. In such cases, zeros and ones are treated as if they were values that have been left-censored at sqrt (realmin) or right-censored at 1 - eps/2, respectively.

Further information about the Beta distribution can be found at https://en.wikipedia.org/wiki/Beta_distribution

See also: betacdf, betainv, betapdf, betarnd, betalike, betastat

Source Code: betafit

Example: 1

  

 ## Sample 2 populations from different Beta distibutions
 randg ("seed", 1);   # for reproducibility
 r1 = betarnd (2, 5, 500, 1);
 randg ("seed", 2);   # for reproducibility
 r2 = betarnd (2, 2, 500, 1);
 r = [r1, r2];

 ## Plot them normalized and fix their colors
 hist (r, 12, 15);
 h = findobj (gca, "Type", "patch");
 set (h(1), "facecolor", "c");
 set (h(2), "facecolor", "g");
 hold on

 ## Estimate their shape parameters
 a_b_A = betafit (r(:,1));
 a_b_B = betafit (r(:,2));

 ## Plot their estimated PDFs
 x = [min(r(:)):0.01:max(r(:))];
 y = betapdf (x, a_b_A(1), a_b_A(2));
 plot (x, y, "-pr");
 y = betapdf (x, a_b_B(1), a_b_B(2));
 plot (x, y, "-sg");
 ylim ([0, 4])
 legend ({"Normalized HIST of sample 1 with α=2 and β=5", ...
          "Normalized HIST of sample 2 with α=2 and β=2", ...
          sprintf("PDF for sample 1 with estimated α=%0.2f and β=%0.2f", ...
                  a_b_A(1), a_b_A(2)), ...
          sprintf("PDF for sample 2 with estimated α=%0.2f and β=%0.2f", ...
                  a_b_B(1), a_b_B(2))})
 title ("Two population samples from different Beta distibutions")
 hold off
plotted figure

statistics-release-1.6.3/docs/betainv.html000066400000000000000000000136171456127120000206050ustar00rootroot00000000000000 Statistics: betainv

Function Reference: betainv

statistics: x = betainv (p, a, b)

Inverse of the Beta distribution (iCDF).

For each element of p, compute the quantile (the inverse of the CDF) of the Beta distribution with shape parameters a and b. The size of x is the common size of x, a, and b. A scalar input functions as a constant matrix of the same size as the other inputs.

Further information about the Beta distribution can be found at https://en.wikipedia.org/wiki/Beta_distribution

See also: betacdf, betapdf, betarnd, betafit, betalike, betastat

Source Code: betainv

Example: 1

  

 ## Plot various iCDFs from the Beta distribution
 p = 0.001:0.001:0.999;
 x1 = betainv (p, 0.5, 0.5);
 x2 = betainv (p, 5, 1);
 x3 = betainv (p, 1, 3);
 x4 = betainv (p, 2, 2);
 x5 = betainv (p, 2, 5);
 plot (p, x1, "-b", p, x2, "-g", p, x3, "-r", p, x4, "-c", p, x5, "-m")
 grid on
 legend ({"α = β = 0.5", "α = 5, β = 1", "α = 1, β = 3", ...
          "α = 2, β = 2", "α = 2, β = 5"}, "location", "southeast")
 title ("Beta iCDF")
 xlabel ("probability")
 ylabel ("values in x")
plotted figure

statistics-release-1.6.3/docs/betalike.html000066400000000000000000000126241456127120000207320ustar00rootroot00000000000000 Statistics: betalike

Function Reference: betalike

statistics: nlogL = betalike (params, x)
statistics: [nlogL, avar] = betalike (params, x)

Negative log-likelihood for the Beta distribution.

nlogL = betalike (params, x) returns the negative log likelihood of the data in x corresponding to the Beta distribution with (1) shape parameter α and (2) shape parameter β given in the two-element vector params. Both parameters must be positive real numbers and the data in the range [0,1]. Out of range parameters or data return NaN.

[nlogL, avar] = betalike (params, x) returns the inverse of Fisher’s information matrix, avar. If the input parameter values in params are the maximum likelihood estimates, the diagonal elements of params are their asymptotic variances.

The Beta distribution is defined on the open interval (0,1). However, betafit can also compute the unbounded beta likelihood function for data that include exact zeros or ones. In such cases, zeros and ones are treated as if they were values that have been left-censored at sqrt (realmin) or right-censored at 1 - eps/2, respectively.

Further information about the Beta distribution can be found at https://en.wikipedia.org/wiki/Beta_distribution

See also: betacdf, betainv, betapdf, betarnd, betafit, betastat

Source Code: betalike

statistics-release-1.6.3/docs/betapdf.html000066400000000000000000000136271456127120000205630ustar00rootroot00000000000000 Statistics: betapdf

Function Reference: betapdf

statistics: y = betapdf (x, a, b)

Beta probability density function (PDF).

For each element of x, compute the probability density function (PDF) of the Beta distribution with shape parameters a and b. The size of y is the common size of x, a, and b. A scalar input functions as a constant matrix of the same size as the other inputs.

Further information about the Beta distribution can be found at https://en.wikipedia.org/wiki/Beta_distribution

See also: betacdf, betainv, betarnd, betafit, betalike, betastat

Source Code: betapdf

Example: 1

  

 ## Plot various PDFs from the Beta distribution
 x = 0.001:0.001:0.999;
 y1 = betapdf (x, 0.5, 0.5);
 y2 = betapdf (x, 5, 1);
 y3 = betapdf (x, 1, 3);
 y4 = betapdf (x, 2, 2);
 y5 = betapdf (x, 2, 5);
 plot (x, y1, "-b", x, y2, "-g", x, y3, "-r", x, y4, "-c", x, y5, "-m")
 grid on
 ylim ([0, 2.5])
 legend ({"α = β = 0.5", "α = 5, β = 1", "α = 1, β = 3", ...
          "α = 2, β = 2", "α = 2, β = 5"}, "location", "north")
 title ("Beta PDF")
 xlabel ("values in x")
 ylabel ("density")
plotted figure

statistics-release-1.6.3/docs/betarnd.html000066400000000000000000000122321456127120000205640ustar00rootroot00000000000000 Statistics: betarnd

Function Reference: betarnd

statistics: r = betarnd (a, b)
statistics: r = betarnd (a, b, rows)
statistics: r = betarnd (a, b, rows, cols, …)
statistics: r = betarnd (a, b, [sz])

Random arrays from the Beta distribution.

r = betarnd (a, b) returns an array of random numbers chosen from the Beta distribution with shape parameters a and b. The size of r is the common size of a and b. A scalar input functions as a constant matrix of the same size as the other inputs.

When called with a single size argument, betarnd returns a square matrix with the dimension specified. When called with more than one scalar argument, the first two arguments are taken as the number of rows and columns and any further arguments specify additional matrix dimensions. The size may also be specified with a row vector of dimensions, sz.

Further information about the Beta distribution can be found at https://en.wikipedia.org/wiki/Beta_distribution

See also: betacdf, betainv, betapdf, betafit, betalike, betastat

Source Code: betarnd

statistics-release-1.6.3/docs/betastat.html000066400000000000000000000107521456127120000207610ustar00rootroot00000000000000 Statistics: betastat

Function Reference: betastat

statistics: [m, v] = betastat (a, b)

Compute statistics of the Beta distribution.

[m, v] = betastat (a, b) returns the mean and variance of the Beta distribution with shape parameters a and b.

The size of m (mean) and v (variance) is the common size of the input arguments. A scalar input functions as a constant matrix of the same size as the other inputs.

Further information about the Beta distribution can be found at https://en.wikipedia.org/wiki/Beta_distribution

See also: betacdf, betainv, betapdf, betarnd, betafit, betalike

Source Code: betastat

statistics-release-1.6.3/docs/binocdf.html000066400000000000000000000145461456127120000205630ustar00rootroot00000000000000 Statistics: binocdf

Function Reference: binocdf

statistics: p = binocdf (x, n, ps)
statistics: p = binocdf (x, n, ps, "upper")

Binomial cumulative distribution function (CDF).

For each element of x, compute the cumulative distribution function (CDF) of the binomial distribution with parameters n and ps, where n is the number of trials and ps is the probability of success. The size of p is the common size of x, n, and ps. A scalar input functions as a constant matrix of the same size as the other inputs.

p = binocdf (x, n, ps, "upper") computes the upper tail probability of the binomial distribution with parameters n and ps, at the values in x.

Further information about the binomial distribution can be found at https://en.wikipedia.org/wiki/Binomial_distribution

See also: binoinv, binopdf, binornd, binofit, binolike, binostat, binotest

Source Code: binocdf

Example: 1

  

 ## Plot various CDFs from the binomial distribution
 x = 0:40;
 p1 = binocdf (x, 20, 0.5);
 p2 = binocdf (x, 20, 0.7);
 p3 = binocdf (x, 40, 0.5);
 plot (x, p1, "*b", x, p2, "*g", x, p3, "*r")
 grid on
 legend ({"n = 20, ps = 0.5", "n = 20, ps = 0.7", ...
          "n = 40, ps = 0.5"}, "location", "southeast")
 title ("Binomial CDF")
 xlabel ("values in x (number of successes)")
 ylabel ("probability")
plotted figure

statistics-release-1.6.3/docs/binofit.html000066400000000000000000000175631456127120000206130ustar00rootroot00000000000000 Statistics: binofit

Function Reference: binofit

statistics: pshat = binofit (x, n)
statistics: [pshat, psci] = binofit (x, n)
statistics: [pshat, psci] = binofit (x, n, alpha)

Estimate parameter and confidence intervals for the binomial distribution.

pshat = binofit (x, n) returns the maximum likelihood estimate (MLE) of the probability of success for the binomial distribution. x and n are scalars containing the number of successes and the number of trials, respectively. If x and n are vectors, binofit returns a vector of estimates whose i-th element is the parameter estimate for x(i) and n(i). A scalar value for x or n is expanded to the same size as the other input.

[pshat, psci] = binofit (x, n, alpha) also returns the 100 * (1 - alpha) percent confidence intervals of the estimated parameter. By default, the optional argument alpha is 0.05 corresponding to 95% confidence intervals.

binofit treats a vector x as a collection of measurements from separate samples, and returns a vector of estimates. If you want to treat x as a single sample and compute a single parameter estimate and confidence interval, use binofit (sum (x), sum (n)) when n is a vector, and binofit (sum (x), n * length (x)) when n is a scalar.

Further information about the binomial distribution can be found at https://en.wikipedia.org/wiki/Binomial_distribution

See also: binocdf, binoinv, binopdf, binornd, binolike, binostat

Source Code: binofit

Example: 1

  

 ## Sample 2 populations from different binomial distibutions
 rand ("seed", 1);    # for reproducibility
 r1 = binornd (50, 0.15, 1000, 1);
 rand ("seed", 2);    # for reproducibility
 r2 = binornd (100, 0.5, 1000, 1);
 r = [r1, r2];

 ## Plot them normalized and fix their colors
 hist (r, 23, 0.35);
 h = findobj (gca, "Type", "patch");
 set (h(1), "facecolor", "c");
 set (h(2), "facecolor", "g");
 hold on

 ## Estimate their probability of success
 pshatA = binofit (r(:,1), 50);
 pshatB = binofit (r(:,2), 100);

 ## Plot their estimated PDFs
 x = [min(r(:,1)):max(r(:,1))];
 y = binopdf (x, 50, mean (pshatA));
 plot (x, y, "-pg");
 x = [min(r(:,2)):max(r(:,2))];
 y = binopdf (x, 100, mean (pshatB));
 plot (x, y, "-sc");
 ylim ([0, 0.2])
 legend ({"Normalized HIST of sample 1 with ps=0.15", ...
          "Normalized HIST of sample 2 with ps=0.50", ...
          sprintf("PDF for sample 1 with estimated ps=%0.2f", ...
                  mean (pshatA)), ...
          sprintf("PDF for sample 2 with estimated ps=%0.2f", ...
                  mean (pshatB))})
 title ("Two population samples from different binomial distibutions")
 hold off
plotted figure

statistics-release-1.6.3/docs/binoinv.html000066400000000000000000000137641456127120000206240ustar00rootroot00000000000000 Statistics: binoinv

Function Reference: binoinv

statistics: x = binoinv (p, n, ps)

Inverse of the Binomial cumulative distribution function (iCDF).

For each element of p, compute the quantile (the inverse of the CDF) of the binomial distribution with parameters n and ps, where n is the number of trials and ps is the probability of success. The size of x is the common size of p, n, and ps. A scalar input functions as a constant matrix of the same size as the other inputs.

Further information about the binomial distribution can be found at https://en.wikipedia.org/wiki/Binomial_distribution

See also: binocdf, binopdf, binornd, binofit, binolike, binostat, binotest

Source Code: binoinv

Example: 1

  

 ## Plot various iCDFs from the binomial distribution
 p = 0.001:0.001:0.999;
 x1 = binoinv (p, 20, 0.5);
 x2 = binoinv (p, 20, 0.7);
 x3 = binoinv (p, 40, 0.5);
 plot (p, x1, "-b", p, x2, "-g", p, x3, "-r")
 grid on
 legend ({"n = 20, ps = 0.5", "n = 20, ps = 0.7", ...
          "n = 40, ps = 0.5"}, "location", "southeast")
 title ("Binomial iCDF")
 xlabel ("probability")
 ylabel ("values in x (number of successes)")
plotted figure

statistics-release-1.6.3/docs/binolike.html000066400000000000000000000122131456127120000207400ustar00rootroot00000000000000 Statistics: binolike

Function Reference: binolike

statistics: nlogL = binolike (params, x)
statistics: [nlogL, acov] = binolike (params, x)

Negative log-likelihood for the binomial distribution.

nlogL = binolike (params, x) returns the negative log likelihood of the binomial distribution with (1) parameter n and (2) parameter ps, given in the two-element vector params, where n is the number of trials and ps is the probability of success, given the number of successes in x. Unlike binofit, which handles each element in x independently, binolike returns the negative log likelihood of the entire vector x.

[nlogL, acov] = binolike (params, x) also returns the inverse of Fisher’s information matrix, acov. If the input parameter values in params are the maximum likelihood estimates, the diagonal elements of params are their asymptotic variances.

Further information about the binomial distribution can be found at https://en.wikipedia.org/wiki/Binomial_distribution

See also: binocdf, binoinv, binopdf, binornd, binofit, binostat

Source Code: binolike

statistics-release-1.6.3/docs/binopdf.html000066400000000000000000000143141456127120000205710ustar00rootroot00000000000000 Statistics: binopdf

Function Reference: binopdf

statistics: y = binopdf (x, n, ps)

Binomial probability density function (PDF).

For each element of x, compute the probability density function (PDF) of the binomial distribution with parameters n and ps, where n is the number of trials and ps is the probability of success. The size of y is the common size of x, n, and ps. A scalar input functions as a constant matrix of the same size as the other inputs.

Matlab incompatibility: Octave’s binopdf does not allow complex input values. Matlab 2021b returns values for complex inputs despite the documentation indicates integer and real value inputs are required.

Further information about the binomial distribution can be found at https://en.wikipedia.org/wiki/Binomial_distribution

See also: binocdf, binoinv, binornd, binofit, binolike, binostat, binotest

Source Code: binopdf

Example: 1

  

 ## Plot various PDFs from the binomial distribution
 x = 0:40;
 y1 = binopdf (x, 20, 0.5);
 y2 = binopdf (x, 20, 0.7);
 y3 = binopdf (x, 40, 0.5);
 plot (x, y1, "*b", x, y2, "*g", x, y3, "*r")
 grid on
 ylim ([0, 0.25])
 legend ({"n = 20, ps = 0.5", "n = 20, ps = 0.7", ...
          "n = 40, ps = 0.5"}, "location", "northeast")
 title ("Binomial PDF")
 xlabel ("values in x (number of successes)")
 ylabel ("density")
plotted figure

statistics-release-1.6.3/docs/binornd.html000066400000000000000000000124561456127120000206100ustar00rootroot00000000000000 Statistics: binornd

Function Reference: binornd

statistics: r = binornd (n, ps)
statistics: r = binornd (n, ps, rows)
statistics: r = binornd (n, ps, rows, cols, …)
statistics: r = binornd (n, ps, [sz])

Random arrays from the Binomial distribution.

r = binornd (n, ps) returns a matrix of random samples from the binomial distribution with parameters n and ps, where n is the number of trials and ps is the probability of success. The size of r is the common size of n and ps. A scalar input functions as a constant matrix of the same size as the other inputs.

When called with a single size argument, binornd returns a square matrix with the dimension specified. When called with more than one scalar argument, the first two arguments are taken as the number of rows and columns and any further arguments specify additional matrix dimensions. The size may also be specified with a row vector of dimensions, sz.

Further information about the binomial distribution can be found at https://en.wikipedia.org/wiki/Binomial_distribution

See also: binocdf, binoinv, binopdf, binofit, binolike, binostat, binotest

Source Code: binornd

statistics-release-1.6.3/docs/binostat.html000066400000000000000000000112011456127120000207630ustar00rootroot00000000000000 Statistics: binostat

Function Reference: binostat

statistics: [m, v] = binostat (n, ps)

Compute statistics of the binomial distribution.

[m, v] = binostat (n, ps) returns the mean and variance of the binomial distribution with parameters n and ps, where n is the number of trials and ps is the probability of success.

The size of m (mean) and v (variance) is the common size of the input arguments. A scalar input functions as a constant matrix of the same size as the other inputs.

Further information about the binomial distribution can be found at https://en.wikipedia.org/wiki/Binomial_distribution

See also: binocdf, binoinv, binopdf, binornd, binofit, binolike, binotest

Source Code: binostat

statistics-release-1.6.3/docs/binotest.html000066400000000000000000000170721456127120000210030ustar00rootroot00000000000000 Statistics: binotest

Function Reference: binotest

statistics: [h, pval, ci] = binotest (pos, N, p0)
statistics: [h, pval, ci] = binotest (pos, N, p0, Name, Value)

Test for probability p of a binomial sample

Perform a test of the null hypothesis p == p0 for a sample of size N with pos positive results.

Name-Value pair arguments can be used to set various options. "alpha" can be used to specify the significance level of the test (the default value is 0.05). The option "tail", can be used to select the desired alternative hypotheses. If the value is "both" (default) the null is tested against the two-sided alternative p != p0. The value of pval is determined by adding the probabilities of all event less or equally likely than the observed number pos of positive events. If the value of "tail" is "right" the one-sided alternative p > p0 is considered. Similarly for "left", the one-sided alternative p < p0 is considered.

If h is 0 the null hypothesis is accepted, if it is 1 the null hypothesis is rejected. The p-value of the test is returned in pval. A 100(1-alpha)% confidence interval is returned in ci.

Source Code: binotest

Example: 1

  

 % flip a coin 1000 times, showing 475 heads
 % Hypothesis: coin is fair, i.e. p=1/2
 [h,p_val,ci] = binotest(475,1000,0.5)
 % Result: h = 0 : null hypothesis not rejected, coin could be fair
 %         P value 0.12, i.e. hypothesis not rejected for alpha up to 12%
 %         0.444 <= p <= 0.506 with 95% confidence

h = 0
p_val = 0.1212
ci =

   0.4437   0.5065

Example: 2

  

 % flip a coin 100 times, showing 65 heads
 % Hypothesis: coin shows less than 50% heads, i.e. p<=1/2
 [h,p_val,ci] = binotest(65,100,0.5,'tail','left','alpha',0.01)
 % Result: h = 1 : null hypothesis is rejected, i.e. coin shows more heads than tails
 %         P value 0.0018, i.e. hypothesis not rejected for alpha up to 0.18%
 %         0 <= p <= 0.76 with 99% confidence

h = 1
p_val = 1.7588e-03
ci =

        0   0.7580
statistics-release-1.6.3/docs/bisacdf.html000066400000000000000000000176611456127120000205530ustar00rootroot00000000000000 Statistics: bisacdf

Function Reference: bisacdf

statistics: p = bisacdf (x, beta, gamma)
statistics: p = bisacdf (x, beta, gamma, "upper")

Birnbaum-Saunders cumulative distribution function (CDF).

For each element of x, compute the cumulative distribution function (CDF) of the Birnbaum-Saunders distribution with scale parameter beta and shape parameter gamma. The size of p is the common size of x, beta and gamma. A scalar input functions as a constant matrix of the same size as the other inputs.

p = bisacdf (x, beta, gamma, "upper") computes the upper tail probability of the Birnbaum-Saunders distribution with parameters beta and gamma, at the values in x.

Further information about the Birnbaum-Saunders distribution can be found at https://en.wikipedia.org/wiki/Birnbaum%E2%80%93Saunders_distribution

See also: bisainv, bisapdf, bisarnd, bisafit, bisalike, bisastat

Source Code: bisacdf

Example: 1

  

 ## Plot various CDFs from the Birnbaum-Saunders distribution
 x = 0.01:0.01:4;
 p1 = bisacdf (x, 1, 0.5);
 p2 = bisacdf (x, 1, 1);
 p3 = bisacdf (x, 1, 2);
 p4 = bisacdf (x, 1, 5);
 p5 = bisacdf (x, 1, 10);
 plot (x, p1, "-b", x, p2, "-g", x, p3, "-r", x, p4, "-c", x, p5, "-m")
 grid on
 legend ({"β = 1, γ = 0.5", "β = 1, γ = 1", "β = 1, γ = 2", ...
          "β = 1, γ = 5", "β = 1, γ = 10"}, "location", "southeast")
 title ("Birnbaum-Saunders CDF")
 xlabel ("values in x")
 ylabel ("probability")
plotted figure

Example: 2

  

 ## Plot various CDFs from the Birnbaum-Saunders distribution
 x = 0.01:0.01:6;
 p1 = bisacdf (x, 1, 0.3);
 p2 = bisacdf (x, 2, 0.3);
 p3 = bisacdf (x, 1, 0.5);
 p4 = bisacdf (x, 3, 0.5);
 p5 = bisacdf (x, 5, 0.5);
 plot (x, p1, "-b", x, p2, "-g", x, p3, "-r", x, p4, "-c", x, p5, "-m")
 grid on
 legend ({"β = 1, γ = 0.3", "β = 2, γ = 0.3", "β = 1, γ = 0.5", ...
          "β = 3, γ = 0.5", "β = 5, γ = 0.5"}, "location", "southeast")
 title ("Birnbaum-Saunders CDF")
 xlabel ("values in x")
 ylabel ("probability")
plotted figure

statistics-release-1.6.3/docs/bisafit.html000066400000000000000000000231031456127120000205650ustar00rootroot00000000000000 Statistics: bisafit

Function Reference: bisafit

statistics: paramhat = bisafit (x)
statistics: [paramhat, paramci] = bisafit (x)
statistics: [paramhat, paramci] = bisafit (x, alpha)
statistics: […] = bisafit (x, alpha, censor)
statistics: […] = bisafit (x, alpha, censor, freq)
statistics: […] = bisafit (x, alpha, censor, freq, options)

Estimate mean and confidence intervals for the Birnbaum-Saunders distribution.

muhat = bisafit (x) returns the maximum likelihood estimates of the parameters of the Birnbaum-Saunders distribution given the data in x. paramhat(1) is the scale parameter, beta, and paramhat(2) is the shape parameter, gamma.

[paramhat, paramci] = bisafit (x) returns the 95% confidence intervals for the parameter estimates.

[…] = bisafit (x, alpha) also returns the 100 * (1 - alpha) percent confidence intervals for the parameter estimates. By default, the optional argument alpha is 0.05 corresponding to 95% confidence intervals. Pass in [] for alpha to use the default values.

[…] = bisafit (x, alpha, censor) accepts a boolean vector, censor, of the same size as x with 1s for observations that are right-censored and 0s for observations that are observed exactly. By default, or if left empty, censor = zeros (size (x)).

[…] = bisafit (x, alpha, censor, freq) accepts a frequency vector, freq, of the same size as x. freq typically contains integer frequencies for the corresponding elements in x, but it can contain any non-integer non-negative values. By default, or if left empty, freq = ones (size (x)).

[…] = bisafit (…, options) specifies control parameters for the iterative algorithm used to compute ML estimates with the fminsearch function. options is a structure with the following fields and their default values:

  • options.Display = "off"
  • options.MaxFunEvals = 400
  • options.MaxIter = 200
  • options.TolX = 1e-6

Further information about the Birnbaum-Saunders distribution can be found at https://en.wikipedia.org/wiki/Birnbaum%E2%80%93Saunders_distribution

See also: bisacdf, bisainv, bisapdf, bisarnd, bisalike, bisastat

Source Code: bisafit

Example: 1

  

 ## Sample 3 populations from different Birnbaum-Saunders distibutions
 rand ("seed", 5);    # for reproducibility
 r1 = bisarnd (1, 0.5, 2000, 1);
 rand ("seed", 2);    # for reproducibility
 r2 = bisarnd (2, 0.3, 2000, 1);
 rand ("seed", 7);    # for reproducibility
 r3 = bisarnd (4, 0.5, 2000, 1);
 r = [r1, r2, r3];

 ## Plot them normalized and fix their colors
 hist (r, 80, 4.2);
 h = findobj (gca, "Type", "patch");
 set (h(1), "facecolor", "c");
 set (h(2), "facecolor", "g");
 set (h(3), "facecolor", "r");
 ylim ([0, 1.1]);
 xlim ([0, 8]);
 hold on

 ## Estimate their α and β parameters
 beta_gammaA = bisafit (r(:,1));
 beta_gammaB = bisafit (r(:,2));
 beta_gammaC = bisafit (r(:,3));

 ## Plot their estimated PDFs
 x = [0:0.1:8];
 y = bisapdf (x, beta_gammaA(1), beta_gammaA(2));
 plot (x, y, "-pr");
 y = bisapdf (x, beta_gammaB(1), beta_gammaB(2));
 plot (x, y, "-sg");
 y = bisapdf (x, beta_gammaC(1), beta_gammaC(2));
 plot (x, y, "-^c");
 hold off
 legend ({"Normalized HIST of sample 1 with β=1 and γ=0.5", ...
          "Normalized HIST of sample 2 with β=2 and γ=0.3", ...
          "Normalized HIST of sample 3 with β=4 and γ=0.5", ...
          sprintf("PDF for sample 1 with estimated β=%0.2f and γ=%0.2f", ...
                  beta_gammaA(1), beta_gammaA(2)), ...
          sprintf("PDF for sample 2 with estimated β=%0.2f and γ=%0.2f", ...
                  beta_gammaB(1), beta_gammaB(2)), ...
          sprintf("PDF for sample 3 with estimated β=%0.2f and γ=%0.2f", ...
                  beta_gammaC(1), beta_gammaC(2))})
 title ("Three population samples from different Birnbaum-Saunders distibutions")
 hold off
plotted figure

statistics-release-1.6.3/docs/bisainv.html000066400000000000000000000171071456127120000206060ustar00rootroot00000000000000 Statistics: bisainv

Function Reference: bisainv

statistics: x = bisainv (p, beta, gamma)

Inverse of the Birnbaum-Saunders cumulative distribution function (iCDF).

For each element of p, compute the quantile (the inverse of the CDF) of the Birnbaum-Saunders distribution with scale parameter beta and shape parameter gamma. The size of x is the common size of p, beta, and gamma. A scalar input functions as a constant matrix of the same size as the other inputs.

Further information about the Birnbaum-Saunders distribution can be found at https://en.wikipedia.org/wiki/Birnbaum%E2%80%93Saunders_distribution

See also: bisainv, bisapdf, bisarnd, bisafit, bisalike, bisastat

Source Code: bisainv

Example: 1

  

 ## Plot various iCDFs from the Birnbaum-Saunders distribution
 p = 0.001:0.001:0.999;
 x1 = bisainv (p, 1, 0.5);
 x2 = bisainv (p, 1, 1);
 x3 = bisainv (p, 1, 2);
 x4 = bisainv (p, 1, 5);
 x5 = bisainv (p, 1, 10);
 plot (p, x1, "-b", p, x2, "-g", p, x3, "-r", p, x4, "-c", p, x5, "-m")
 grid on
 ylim ([0, 10])
 legend ({"β = 1, γ = 0.5", "β = 1, γ = 1", "β = 1, γ = 2", ...
          "β = 1, γ = 5", "β = 1, γ = 10"}, "location", "northwest")
 title ("Birnbaum-Saunders iCDF")
 xlabel ("probability")
 ylabel ("values in x")
plotted figure

Example: 2

  

 ## Plot various iCDFs from the Birnbaum-Saunders distribution
 p = 0.001:0.001:0.999;
 x1 = bisainv (p, 1, 0.3);
 x2 = bisainv (p, 2, 0.3);
 x3 = bisainv (p, 1, 0.5);
 x4 = bisainv (p, 3, 0.5);
 x5 = bisainv (p, 5, 0.5);
 plot (p, x1, "-b", p, x2, "-g", p, x3, "-r", p, x4, "-c", p, x5, "-m")
 grid on
 ylim ([0, 10])
 legend ({"β = 1, γ = 0.3", "β = 2, γ = 0.3", "β = 1, γ = 0.5", ...
          "β = 3, γ = 0.5", "β = 5, γ = 0.5"}, "location", "northwest")
 title ("Birnbaum-Saunders iCDF")
 xlabel ("probability")
 ylabel ("values in x")
plotted figure

statistics-release-1.6.3/docs/bisalike.html000066400000000000000000000137501456127120000207360ustar00rootroot00000000000000 Statistics: bisalike

Function Reference: bisalike

statistics: nlogL = bisalike (params, x)
statistics: [nlogL, acov] = bisalike (params, x)
statistics: […] = bisalike (params, x, censor)
statistics: […] = bisalike (params, x, censor, freq)

Negative log-likelihood for the Birnbaum-Saunders distribution.

nlogL = bisalike (params, x) returns the negative log likelihood of the data in x corresponding to the Birnbaum-Saunders distribution with (1) scale parameter beta and (2) shape parameter gamma given in the two-element vector params.

[nlogL, acov] = bisalike (params, x) also returns the inverse of Fisher’s information matrix, acov. If the input parameter values in params are the maximum likelihood estimates, the diagonal elements of params are their asymptotic variances.

[…] = bisalike (params, x, censor) accepts a boolean vector, censor, of the same size as x with 1s for observations that are right-censored and 0s for observations that are observed exactly. By default, or if left empty, censor = zeros (size (x)).

[…] = bisalike (params, x, censor, freq) accepts a frequency vector, freq, of the same size as x. freq typically contains integer frequencies for the corresponding elements in x, but it can contain any non-integer non-negative values. By default, or if left empty, freq = ones (size (x)).

Further information about the Birnbaum-Saunders distribution can be found at https://en.wikipedia.org/wiki/Birnbaum%E2%80%93Saunders_distribution

See also: bisacdf, bisainv, bisapdf, bisarnd, bisafit, bisastat

Source Code: bisalike

statistics-release-1.6.3/docs/bisapdf.html000066400000000000000000000170361456127120000205640ustar00rootroot00000000000000 Statistics: bisapdf

Function Reference: bisapdf

statistics: y = bisapdf (x, beta, gamma)

Birnbaum-Saunders probability density function (PDF).

For each element of x, compute the probability density function (PDF) of the Birnbaum-Saunders distribution with scale parameter beta and shape parameter gamma. The size of y is the common size of x, beta, and gamma. A scalar input functions as a constant matrix of the same size as the other inputs.

Further information about the Birnbaum-Saunders distribution can be found at https://en.wikipedia.org/wiki/Birnbaum%E2%80%93Saunders_distribution

See also: bisacdf, bisapdf, bisarnd, bisafit, bisalike, bisastat

Source Code: bisapdf

Example: 1

  

 ## Plot various PDFs from the Birnbaum-Saunders distribution
 x = 0.01:0.01:4;
 y1 = bisapdf (x, 1, 0.5);
 y2 = bisapdf (x, 1, 1);
 y3 = bisapdf (x, 1, 2);
 y4 = bisapdf (x, 1, 5);
 y5 = bisapdf (x, 1, 10);
 plot (x, y1, "-b", x, y2, "-g", x, y3, "-r", x, y4, "-c", x, y5, "-m")
 grid on
 ylim ([0, 1.5])
 legend ({"β = 1 ,γ = 0.5", "β = 1, γ = 1", "β = 1, γ = 2", ...
          "β = 1, γ = 5", "β = 1, γ = 10"}, "location", "northeast")
 title ("Birnbaum-Saunders PDF")
 xlabel ("values in x")
 ylabel ("density")
plotted figure

Example: 2

  

 ## Plot various PDFs from the Birnbaum-Saunders distribution
 x = 0.01:0.01:6;
 y1 = bisapdf (x, 1, 0.3);
 y2 = bisapdf (x, 2, 0.3);
 y3 = bisapdf (x, 1, 0.5);
 y4 = bisapdf (x, 3, 0.5);
 y5 = bisapdf (x, 5, 0.5);
 plot (x, y1, "-b", x, y2, "-g", x, y3, "-r", x, y4, "-c", x, y5, "-m")
 grid on
 ylim ([0, 1.5])
 legend ({"β = 1, γ = 0.3", "β = 2, γ = 0.3", "β = 1, γ = 0.5", ...
          "β = 3, γ = 0.5", "β = 5, γ = 0.5"}, "location", "northeast")
 title ("Birnbaum-Saunders CDF")
 xlabel ("values in x")
 ylabel ("density")
plotted figure

statistics-release-1.6.3/docs/bisarnd.html000066400000000000000000000124161456127120000205730ustar00rootroot00000000000000 Statistics: bisarnd

Function Reference: bisarnd

statistics: r = bisarnd (beta, gamma)
statistics: r = bisarnd (beta, gamma, rows)
statistics: r = bisarnd (beta, gamma, rows, cols, …)
statistics: r = bisarnd (beta, gamma, [sz])

Random arrays from the Birnbaum-Saunders distribution.

r = bisarnd (beta, gamma) returns an array of random numbers chosen from the Birnbaum-Saunders distribution with scale parameter beta and shape parameter gamma. The size of r is the common size of beta and gamma. A scalar input functions as a constant matrix of the same size as the other inputs.

When called with a single size argument, bisarnd returns a square matrix with the dimension specified. When called with more than one scalar argument, the first two arguments are taken as the number of rows and columns and any further arguments specify additional matrix dimensions. The size may also be specified with a row vector of dimensions, sz.

Further information about the Birnbaum-Saunders distribution can be found at https://en.wikipedia.org/wiki/Birnbaum%E2%80%93Saunders_distribution

See also: bisacdf, bisainv, bisapdf, bisafit, bisalike, bisastat

Source Code: bisarnd

statistics-release-1.6.3/docs/boxplot.html000066400000000000000000000417241456127120000206440ustar00rootroot00000000000000 Statistics: boxplot

Function Reference: boxplot

statistics: s = boxplot (data)
statistics: s = boxplot (data, group)
statistics: s = boxplot (data, notched, symbol, orientation, whisker, …)
statistics: s = boxplot (data, group, notched, symbol, orientation, whisker, …)
statistics: s = boxplot (data, options)
statistics: s = boxplot (data, group, options, …)
statistics: […, h] = boxplot (data, …)

Produce a box plot.

A box plot is a graphical display that simultaneously describes several important features of a data set, such as center, spread, departure from symmetry, and identification of observations that lie unusually far from the bulk of the data.

Input arguments (case-insensitive) recognized by boxplot are:

  • data is a matrix with one column for each data set, or a cell vector with one cell for each data set. Each cell must contain a numerical row or column vector (NaN and NA are ignored) and not a nested vector of cells.
  • notched = 1 produces a notched-box plot. Notches represent a robust estimate of the uncertainty about the median.

    notched = 0 (default) produces a rectangular box plot.

    notched within the interval (0,1) produces a notch of the specified depth. Notched values outside (0,1) are amusing if not exactly impractical.

  • symbol sets the symbol for the outlier values. The default symbol for points that lie outside 3 times the interquartile range is ’o’; the default symbol for points between 1.5 and 3 times the interquartile range is ’+’.
    Alternative symbol settings:

    symbol = ’.’: points between 1.5 and 3 times the IQR are marked with ’.’ and points outside 3 times IQR with ’o’.

    symbol = [’x’,’*’]: points between 1.5 and 3 times the IQR are marked with ’x’ and points outside 3 times IQR with ’*’.

  • orientation = 0 makes the boxes horizontally.
    orientation = 1 plots the boxes vertically (default). Alternatively, orientation can be passed as a string, e.g., ’vertical’ or ’horizontal’.
  • whisker defines the length of the whiskers as a function of the IQR (default = 1.5). If whisker = 0 then boxplot displays all data values outside the box using the plotting symbol for points that lie outside 3 times the IQR.
  • group may be passed as an optional argument only in the second position after data. group contains a numerical vector defining separate categories, each plotted in a different box, for each set of DATA values that share the same group value or values. With the formalism (data, group), both must be vectors of the same length.
  • options are additional paired arguments passed with the formalism (Name, Value) that provide extra functionality as listed below. options can be passed at any order after the initial arguments and are case-insensitive.
    ’Notch’’on’Notched by 0.25 of the boxes width.
    ’off’Produces a straight box.
    scalarProportional width of the notch.
    ’Symbol’’.’Defines only outliers between 1.5 and 3 IQR.
    [’x’,’*’]2nd character defines outliers > 3 IQR
    ’Orientation’’vertical’Default value, can also be defined with numerical 1.
    ’horizontal’Can also be defined with numerical 0.
    ’Whisker’scalarMultiplier of IQR (default is 1.5).
    ’OutlierTags’’on’ or 1Plot the vector index of the outlier value next to its point.
    ’off’ or 0No tags are plotted (default value).
    ’Sample_IDs’’cell’A cell vector with one cell for each data set containing a nested cell vector with each sample’s ID (should be a string). If this option is passed, then all outliers are tagged with their respective sample’s ID string instead of their vector’s index.
    ’BoxWidth’’proportional’Create boxes with their width proportional to the number of samples in their respective dataset (default value).
    ’fixed’Make all boxes with equal width.
    ’Widths’scalarScaling factor for box widths (default value is 0.4).
    ’CapWidths’scalarScaling factor for whisker cap widths (default value is 1, which results to ’Widths’/8 halflength)
    ’BoxStyle’’outline’Draw boxes as outlines (default value).
    ’filled’Fill boxes with a color (outlines are still plotted).
    ’Positions’vectorNumerical vector that defines the position of each data set. It must have the same length as the number of groups in a desired manner. This vector merely defines the points along the group axis, which by default is [1:number of groups].
    ’Labels’cellA cell vector of strings containing the names of each group. By default each group is labeled numerically according to its order in the data set
    ’Colors’character string or Nx3 numerical matrixIf just one character or 1x3 vector of RGB values, specify the fill color of all boxes when BoxStyle = ’filled’. If a character string or Nx3 matrix is entered, box #1’s fill color corrresponds to the first character or first matrix row, and the next boxes’ fill colors corresponds to the next characters or rows. If the char string or Nx3 array is exhausted the color selection wraps around.

Supplemental arguments not described above (…) are concatenated and passed to the plot() function.

The returned matrix s has one column for each data set as follows:

1Minimum
21st quartile
32nd quartile (median)
43rd quartile
5Maximum
6Lower confidence limit for median
7Upper confidence limit for median

The returned structure h contains handles to the plot elements, allowing customization of the visualization using set/get functions.

Example

 
 title ("Grade 3 heights");
 axis ([0,3]);
 set(gca (), "xtick", [1 2], "xticklabel", {"girls", "boys"});
 boxplot ({randn(10,1)*5+140, randn(13,1)*8+135});

Source Code: boxplot

Example: 1

  

 axis ([0, 3]);
 randn ("seed", 1);    # for reproducibility
 girls = randn (10, 1) * 5 + 140;
 randn ("seed", 2);    # for reproducibility
 boys = randn (13, 1) * 8 + 135;
 boxplot ({girls, boys});
 set (gca (), "xtick", [1 2], "xticklabel", {"girls", "boys"})
 title ("Grade 3 heights");
plotted figure

Example: 2

  

 randn ("seed", 7);    # for reproducibility
 A = randn (10, 1) * 5 + 140;
 randn ("seed", 8);    # for reproducibility
 B = randn (25, 1) * 8 + 135;
 randn ("seed", 9);    # for reproducibility
 C = randn (20, 1) * 6 + 165;
 data = [A; B; C];
 groups = [(ones (10, 1)); (ones (25, 1) * 2); (ones (20, 1) * 3)];
 labels = {"Team A", "Team B", "Team C"};
 pos = [2, 1, 3];
 boxplot (data, groups, "Notch", "on", "Labels", labels, "Positions", pos, ...
          "OutlierTags", "on", "BoxStyle", "filled");
 title ("Example of Group splitting with paired vectors");
plotted figure

Example: 3

  

 randn ("seed", 1);    # for reproducibility
 data = randn (100, 9);
 boxplot (data, "notch", "on", "boxstyle", "filled", ...
          "colors", "ygcwkmb", "whisker", 1.2);
 title ("Example of different colors specified with characters");
plotted figure

Example: 4

  

 randn ("seed", 5);    # for reproducibility
 data = randn (100, 13);
 colors = [0.7 0.7 0.7; ...
           0.0 0.4 0.9; ...
           0.7 0.4 0.3; ...
           0.7 0.1 0.7; ...
           0.8 0.7 0.4; ...
           0.1 0.8 0.5; ...
           0.9 0.9 0.2];
 boxplot (data, "notch", "on", "boxstyle", "filled", ...
          "colors", colors, "whisker", 1.3, "boxwidth", "proportional");
 title ("Example of different colors specified as RGB values");
plotted figure

statistics-release-1.6.3/docs/burrcdf.html000066400000000000000000000150731456127120000206020ustar00rootroot00000000000000 Statistics: burrcdf

Function Reference: burrcdf

statistics: p = burrcdf (x, lambda, c, k)
statistics: p = burrcdf (x, lambda, c, k, "upper")

Burr type XII cumulative distribution function (CDF).

For each element of x, compute the cumulative distribution function (CDF) of the Burr type XII distribution with scale parameter lambda, first shape parameter c, and second shape parameter k. The size of p is the common size of x, lambda, c, and k. A scalar input functions as a constant matrix of the same size as the other inputs.

p = burrcdf (x, beta, gamma, "upper") computes the upper tail probability of the Birnbaum-Saunders distribution with parameters beta and gamma, at the values in x.

Further information about the Burr distribution can be found at https://en.wikipedia.org/wiki/Burr_distribution

See also: burrinv, burrpdf, burrrnd, burrfit, burrlike

Source Code: burrcdf

Example: 1

  

 ## Plot various CDFs from the Burr type XII distribution
 x = 0.001:0.001:5;
 p1 = burrcdf (x, 1, 1, 1);
 p2 = burrcdf (x, 1, 1, 2);
 p3 = burrcdf (x, 1, 1, 3);
 p4 = burrcdf (x, 1, 2, 1);
 p5 = burrcdf (x, 1, 3, 1);
 p6 = burrcdf (x, 1, 0.5, 2);
 plot (x, p1, "-b", x, p2, "-g", x, p3, "-r", ...
       x, p4, "-c", x, p5, "-m", x, p6, "-k")
 grid on
 legend ({"λ = 1, c = 1, k = 1", "λ = 1, c = 1, k = 2", ...
          "λ = 1, c = 1, k = 3", "λ = 1, c = 2, k = 1", ...
          "λ = 1, c = 3, k = 1", "λ = 1, c = 0.5, k = 2"}, ...
         "location", "southeast")
 title ("Burr type XII CDF")
 xlabel ("values in x")
 ylabel ("probability")
plotted figure

statistics-release-1.6.3/docs/burrfit.html000066400000000000000000000233241456127120000206260ustar00rootroot00000000000000 Statistics: burrfit

Function Reference: burrfit

statistics: paramhat = burrfit (x)
statistics: [paramhat, paramci] = burrfit (x)
statistics: [paramhat, paramci] = burrfit (x, alpha)
statistics: […] = burrfit (x, alpha, censor)
statistics: […] = burrfit (x, alpha, censor, freq)
statistics: […] = burrfit (x, alpha, censor, freq, options)

Estimate mean and confidence intervals for the Burr type XII distribution.

muhat = burrfit (x) returns the maximum likelihood estimates of the parameters of the Burr type XII distribution given the data in x. paramhat(1) is the scale parameter, lambda, paramhat(2) is the first shape parameter, c, and paramhat(3) is the second shape parameter, k

[paramhat, paramci] = burrfit (x) returns the 95% confidence intervals for the parameter estimates.

[…] = burrfit (x, alpha) also returns the 100 * (1 - alpha) percent confidence intervals for the parameter estimates. By default, the optional argument alpha is 0.05 corresponding to 95% confidence intervals. Pass in [] for alpha to use the default values.

[…] = burrfit (x, alpha, censor) accepts a boolean vector, censor, of the same size as x with 1s for observations that are right-censored and 0s for observations that are observed exactly. By default, or if left empty, censor = zeros (size (x)).

[…] = burrfit (x, alpha, censor, freq) accepts a frequency vector, freq, of the same size as x. freq typically contains integer frequencies for the corresponding elements in x, but it can contain any non-integer non-negative values. By default, or if left empty, freq = ones (size (x)).

[…] = burrfit (…, options) specifies control parameters for the iterative algorithm used to compute ML estimates with the fminsearch function. options is a structure with the following fields and their default values:

  • options.Display = "off"
  • options.MaxFunEvals = 1000
  • options.MaxIter = 500
  • options.TolX = 1e-6

Further information about the Burr type XII distribution can be found at https://en.wikipedia.org/wiki/Burr_distribution

See also: burrcdf, burrinv, burrpdf, burrrnd, burrlike, burrstat

Source Code: burrfit

Example: 1

  

 ## Sample 3 populations from different Burr type XII distibutions
 rand ("seed", 4);    # for reproducibility
 r1 = burrrnd (3.5, 2, 2.5, 10000, 1);
 rand ("seed", 2);    # for reproducibility
 r2 = burrrnd (1, 3, 1, 10000, 1);
 rand ("seed", 9);    # for reproducibility
 r3 = burrrnd (0.5, 2, 3, 10000, 1);
 r = [r1, r2, r3];

 ## Plot them normalized and fix their colors
 hist (r, [0.1:0.2:20], [18, 5, 3]);
 h = findobj (gca, "Type", "patch");
 set (h(1), "facecolor", "c");
 set (h(2), "facecolor", "g");
 set (h(3), "facecolor", "r");
 ylim ([0, 3]);
 xlim ([0, 5]);
 hold on

 ## Estimate their α and β parameters
 lambda_c_kA = burrfit (r(:,1));
 lambda_c_kB = burrfit (r(:,2));
 lambda_c_kC = burrfit (r(:,3));

 ## Plot their estimated PDFs
 x = [0.01:0.15:15];
 y = burrpdf (x, lambda_c_kA(1), lambda_c_kA(2), lambda_c_kA(3));
 plot (x, y, "-pr");
 y = burrpdf (x, lambda_c_kB(1), lambda_c_kB(2), lambda_c_kB(3));
 plot (x, y, "-sg");
 y = burrpdf (x, lambda_c_kC(1), lambda_c_kC(2), lambda_c_kC(3));
 plot (x, y, "-^c");
 hold off
 legend ({"Normalized HIST of sample 1 with λ=3.5, c=2, and k=2.5", ...
          "Normalized HIST of sample 2 with λ=1, c=3, and k=1", ...
          "Normalized HIST of sample 3 with λ=0.5, c=2, and k=3", ...
  sprintf("PDF for sample 1 with estimated λ=%0.2f, c=%0.2f, and k=%0.2f", ...
          lambda_c_kA(1), lambda_c_kA(2), lambda_c_kA(3)), ...
  sprintf("PDF for sample 2 with estimated λ=%0.2f, c=%0.2f, and k=%0.2f", ...
          lambda_c_kB(1), lambda_c_kB(2), lambda_c_kB(3)), ...
  sprintf("PDF for sample 3 with estimated λ=%0.2f, c=%0.2f, and k=%0.2f", ...
          lambda_c_kC(1), lambda_c_kC(2), lambda_c_kC(3))})
 title ("Three population samples from different Burr type XII distibutions")
 hold off
plotted figure

statistics-release-1.6.3/docs/burrinv.html000066400000000000000000000142511456127120000206370ustar00rootroot00000000000000 Statistics: burrinv

Function Reference: burrinv

statistics: x = burrinv (p, lambda, c, k)

Inverse of the Burr type XII cumulative distribution function (iCDF).

For each element of p, compute the quantile (the inverse of the CDF) of the Burr type XII distribution with scale parameter lambda, first shape parameter c, and second shape parameter k. The size of x is the common size of p, lambda, c, and k. A scalar input functions as a constant matrix of the same size as the other inputs.

Further information about the Burr distribution can be found at https://en.wikipedia.org/wiki/Burr_distribution

See also: burrcdf, burrpdf, burrrnd, burrfit, burrlike

Source Code: burrinv

Example: 1

  

 ## Plot various iCDFs from the Burr type XII distribution
 p = 0.001:0.001:0.999;
 x1 = burrinv (p, 1, 1, 1);
 x2 = burrinv (p, 1, 1, 2);
 x3 = burrinv (p, 1, 1, 3);
 x4 = burrinv (p, 1, 2, 1);
 x5 = burrinv (p, 1, 3, 1);
 x6 = burrinv (p, 1, 0.5, 2);
 plot (p, x1, "-b", p, x2, "-g", p, x3, "-r", ...
       p, x4, "-c", p, x5, "-m", p, x6, "-k")
 grid on
 ylim ([0, 5])
 legend ({"λ = 1, c = 1, k = 1", "λ = 1, c = 1, k = 2", ...
          "λ = 1, c = 1, k = 3", "λ = 1, c = 2, k = 1", ...
          "λ = 1, c = 3, k = 1", "λ = 1, c = 0.5, k = 2"}, ...
         "location", "northwest")
 title ("Burr type XII iCDF")
 xlabel ("probability")
 ylabel ("values in x")
plotted figure

statistics-release-1.6.3/docs/burrlike.html000066400000000000000000000137411456127120000207720ustar00rootroot00000000000000 Statistics: burrlike

Function Reference: burrlike

statistics: nlogL = burrlike (params, x)
statistics: [nlogL, acov] = burrlike (params, x)
statistics: […] = burrlike (params, x, censor)
statistics: […] = burrlike (params, x, censor, freq)

Negative log-likelihood for the Burr type XII distribution.

nlogL = burrlike (params, x) returns the negative log likelihood of the data in x corresponding to the Burr type XII distribution with (1) scale parameter lambda, (2) first shape parameter c, and (3) second shape parameter k given in the three-element vector params.

[nlogL, acov] = burrlike (params, x) also returns the inverse of Fisher’s information matrix, acov. If the input parameter values in params are the maximum likelihood estimates, the diagonal elements of acov are their asymptotic variances.

[…] = burrlike (params, x, censor) accepts a boolean vector, censor, of the same size as x with 1s for observations that are right-censored and 0s for observations that are observed exactly. By default, or if left empty, censor = zeros (size (x)).

[…] = burrlike (params, x, censor, freq) accepts a frequency vector, freq, of the same size as x. freq typically contains integer frequencies for the corresponding elements in x, but it can contain any non-integer non-negative values. By default, or if left empty, freq = ones (size (x)).

Further information about the Burr type XII distribution can be found at https://en.wikipedia.org/wiki/Burr_distribution

See also: burrcdf, burrinv, burrpdf, burrrnd, burrfit, burrstat

Source Code: burrlike

statistics-release-1.6.3/docs/burrpdf.html000066400000000000000000000142221456127120000206120ustar00rootroot00000000000000 Statistics: burrpdf

Function Reference: burrpdf

statistics: y = burrpdf (x, lambda, c, k)

Burr type XII probability density function (PDF).

For each element of x, compute the probability density function (PDF) of the Burr type XII distribution with with scale parameter lambda, first shape parameter c, and second shape parameter k. The size of y is the common size of x, lambda, c, and k. A scalar input functions as a constant matrix of the same size as the other inputs.

Further information about the Burr distribution can be found at https://en.wikipedia.org/wiki/Burr_distribution

See also: burrcdf, burrinv, burrrnd, burrfit, burrlike

Source Code: burrpdf

Example: 1

  

 ## Plot various PDFs from the Burr type XII distribution
 x = 0.001:0.001:3;
 y1 = burrpdf (x, 1, 1, 1);
 y2 = burrpdf (x, 1, 1, 2);
 y3 = burrpdf (x, 1, 1, 3);
 y4 = burrpdf (x, 1, 2, 1);
 y5 = burrpdf (x, 1, 3, 1);
 y6 = burrpdf (x, 1, 0.5, 2);
 plot (x, y1, "-b", x, y2, "-g", x, y3, "-r", ...
       x, y4, "-c", x, y5, "-m", x, y6, "-k")
 grid on
 ylim ([0, 2])
 legend ({"λ = 1, c = 1, k = 1", "λ = 1, c = 1, k = 2", ...
          "λ = 1, c = 1, k = 3", "λ = 1, c = 2, k = 1", ...
          "λ = 1, c = 3, k = 1", "λ = 1, c = 0.5, k = 2"}, ...
         "location", "northeast")
 title ("Burr type XII PDF")
 xlabel ("values in x")
 ylabel ("density")
plotted figure

statistics-release-1.6.3/docs/burrrnd.html000066400000000000000000000125161456127120000206300ustar00rootroot00000000000000 Statistics: burrrnd

Function Reference: burrrnd

statistics: r = burrrnd (lambda, c, k)
statistics: r = burrrnd (lambda, c, k, rows)
statistics: r = burrrnd (lambda, c, k, rows, cols, …)
statistics: r = burrrnd (lambda, c, k, [sz])

Random arrays from the Burr type XII distribution.

r = burrrnd (lambda, c, k) returns an array of random numbers chosen from the Burr type XII distribution with scale parameter lambda, first shape parameter c, and second shape parameterc, and k. The size of r is the common size of lambda, c, and k. LAMBDA scalar input functions as a constant matrix of the same size as the other inputs.

When called with a single size argument, burrrnd returns a square matrix with the dimension specified. When called with more than one scalar argument, the first two arguments are taken as the number of rows and columns and any further arguments specify additional matrix dimensions. The size may also be specified with a row vector of dimensions, sz.

Further information about the Burr distribution can be found at https://en.wikipedia.org/wiki/Burr_distribution

See also: burrcdf, burrinv, burrpdf, burrfit, burrlike

Source Code: burrrnd

statistics-release-1.6.3/docs/bvncdf.html000066400000000000000000000134251456127120000204140ustar00rootroot00000000000000 Statistics: bvncdf

Function Reference: bvncdf

statistics: p = bvncdf (x, mu, sigma)
statistics: p = bvncdf (x, [], sigma)

Bivariate normal cumulative distribution function (CDF).

p = bvncdf (x, mu, sigma) will compute the bivariate normal cumulative distribution function of x given a mean parameter mu and a scale parameter sigma.

  • x must be an N×2 matrix with each variable as a column vector.
  • mu can be either a scalar (common mean) or a two-element row vector (each element corresponds to a variable). If empty, a zero mean is assumed.
  • sigma can be a scalar (common variance) or a 2×2 covariance matrix, which must be positive definite.

See also: mvncdf

Source Code: bvncdf

Example: 1

  

 mu = [1, -1];
 sigma = [0.9, 0.4; 0.4, 0.3];
 [X1, X2] = meshgrid (linspace (-1, 3, 25)', linspace (-3, 1, 25)');
 x = [X1(:), X2(:)];
 p = bvncdf (x, mu, sigma);
 Z = reshape (p, 25, 25);
 surf (X1, X2, Z);
 title ("Bivariate Normal Distribution");
 ylabel "X1"
 xlabel "X2"
plotted figure

statistics-release-1.6.3/docs/bvtcdf.html000066400000000000000000000106141456127120000204170ustar00rootroot00000000000000 Statistics: bvtcdf

Function Reference: bvtcdf

statistics: p = bvtcdf (x, rho, df)
statistics: p = bvtcdf (x, rho, df, Tol)

Bivariate Student’s t cumulative distribution function (CDF).

p = bvtcdf (x, rho, df) will compute the bivariate student’s t cumulative distribution function of x, which must be an N×2 matrix, given a correlation coefficient rho, which must be a scalar, and df degrees of freedom, which can be a scalar or a vector of positive numbers commensurate with x.

Tol is the tolerance for numerical integration and by default Tol = 1e-8.

See also: mvtcdf

Source Code: bvtcdf

statistics-release-1.6.3/docs/canoncorr.html000066400000000000000000000105741456127120000211400ustar00rootroot00000000000000 Statistics: canoncorr

Function Reference: canoncorr

statistics: [A, B, r, U, V] = canoncorr (X, Y)

Canonical correlation analysis.

Given X (size k*m) and Y (k*n), returns projection matrices of canonical coefficients A (size m*d, where d is the smallest of m, n, d) and B (size m*d); the canonical correlations r (1*d, arranged in decreasing order); the canonical variables U, V (both k*d, with orthonormal columns); and stats, a structure containing results from Bartlett’s chi-square and Rao’s F tests of significance.

See also: princomp

Source Code: canoncorr

statistics-release-1.6.3/docs/cauchycdf.html000066400000000000000000000143311456127120000211000ustar00rootroot00000000000000 Statistics: cauchycdf

Function Reference: cauchycdf

statistics: p = cauchycdf (x, x0, gamma)
statistics: p = cauchycdf (x, x0, gamma, "upper")

Cauchy cumulative distribution function (CDF).

For each element of x, compute the cumulative distribution function (CDF) of the Cauchy distribution with location parameter x0 and scale parameter gamma. The size of p is the common size of x, x0, and gamma. A scalar input functions as a constant matrix of the same size as the other inputs.

p = cauchycdf (x, x0, gamma, "upper") computes the upper tail probability of the Cauchy distribution with parameters x0 and gamma, at the values in x.

Further information about the Cauchy distribution can be found at https://en.wikipedia.org/wiki/Cauchy_distribution

See also: cauchyinv, cauchypdf, cauchyrnd

Source Code: cauchycdf

Example: 1

  

 ## Plot various CDFs from the Cauchy distribution
 x = -5:0.01:5;
 p1 = cauchycdf (x, 0, 0.5);
 p2 = cauchycdf (x, 0, 1);
 p3 = cauchycdf (x, 0, 2);
 p4 = cauchycdf (x, -2, 1);
 plot (x, p1, "-b", x, p2, "-g", x, p3, "-r", x, p4, "-c")
 grid on
 xlim ([-5, 5])
 legend ({"x0 = 0, γ = 0.5", "x0 = 0, γ = 1", ...
          "x0 = 0, γ = 2", "x0 = -2, γ = 1"}, "location", "southeast")
 title ("Cauchy CDF")
 xlabel ("values in x")
 ylabel ("probability")
plotted figure

statistics-release-1.6.3/docs/cauchyinv.html000066400000000000000000000135251456127120000211440ustar00rootroot00000000000000 Statistics: cauchyinv

Function Reference: cauchyinv

statistics: x = cauchyinv (p, x0, gamma)

Inverse of the Cauchy cumulative distribution function (iCDF).

For each element of p, compute the quantile (the inverse of the CDF) of the Cauchy distribution with location parameter x0 and scale parameter gamma. The size of x is the common size of p, x0, and gamma. A scalar input functions as a constant matrix of the same size as the other inputs.

Further information about the Cauchy distribution can be found at https://en.wikipedia.org/wiki/Cauchy_distribution

See also: cauchycdf, cauchypdf, cauchyrnd

Source Code: cauchyinv

Example: 1

  

 ## Plot various iCDFs from the Cauchy distribution
 p = 0.001:0.001:0.999;
 x1 = cauchyinv (p, 0, 0.5);
 x2 = cauchyinv (p, 0, 1);
 x3 = cauchyinv (p, 0, 2);
 x4 = cauchyinv (p, -2, 1);
 plot (p, x1, "-b", p, x2, "-g", p, x3, "-r", p, x4, "-c")
 grid on
 ylim ([-5, 5])
 legend ({"x0 = 0, γ = 0.5", "x0 = 0, γ = 1", ...
          "x0 = 0, γ = 2", "x0 = -2, γ = 1"}, "location", "northwest")
 title ("Cauchy iCDF")
 xlabel ("probability")
 ylabel ("values in x")
plotted figure

statistics-release-1.6.3/docs/cauchypdf.html000066400000000000000000000135051456127120000211170ustar00rootroot00000000000000 Statistics: cauchypdf

Function Reference: cauchypdf

statistics: y = cauchypdf (x, x0, gamma)

Cauchy probability density function (PDF).

For each element of x, compute the probability density function (PDF) of the Cauchy distribution with location parameter x0 and scale parameter gamma. The size of y is the common size of x, x0, and gamma. A scalar input functions as a constant matrix of the same size as the other inputs.

Further information about the Cauchy distribution can be found at https://en.wikipedia.org/wiki/Cauchy_distribution

See also: cauchycdf, cauchypdf, cauchyrnd

Source Code: cauchypdf

Example: 1

  

 ## Plot various PDFs from the Cauchy distribution
 x = -5:0.01:5;
 y1 = cauchypdf (x, 0, 0.5);
 y2 = cauchypdf (x, 0, 1);
 y3 = cauchypdf (x, 0, 2);
 y4 = cauchypdf (x, -2, 1);
 plot (x, y1, "-b", x, y2, "-g", x, y3, "-r", x, y4, "-c")
 grid on
 xlim ([-5, 5])
 ylim ([0, 0.7])
 legend ({"x0 = 0, γ = 0.5", "x0 = 0, γ = 1", ...
          "x0 = 0, γ = 2", "x0 = -2, γ = 1"}, "location", "northeast")
 title ("Cauchy PDF")
 xlabel ("values in x")
 ylabel ("density")
plotted figure

statistics-release-1.6.3/docs/cauchyrnd.html000066400000000000000000000122011456127120000211210ustar00rootroot00000000000000 Statistics: cauchyrnd

Function Reference: cauchyrnd

statistics: r = cauchyrnd (x0, gamma)
statistics: r = cauchyrnd (x0, gamma, rows)
statistics: r = cauchyrnd (x0, gamma, rows, cols, …)
statistics: r = cauchyrnd (x0, gamma, [sz])

Random arrays from the Cauchy distribution.

r = cauchyrnd (x0, gamma) returns an array of random numbers chosen from the Cauchy distribution with location parameter x0 and scale parameter gamma. The size of r is the common size of x0 and gamma. A scalar input functions as a constant matrix of the same size as the other inputs.

When called with a single size argument, cauchyrnd returns a square matrix with the dimension specified. When called with more than one scalar argument, the first two arguments are taken as the number of rows and columns and any further arguments specify additional matrix dimensions. The size may also be specified with a row vector of dimensions, sz.

Further information about the Cauchy distribution can be found at https://en.wikipedia.org/wiki/Cauchy_distribution

See also: cauchycdf, cauchyinv, cauchypdf

Source Code: cauchyrnd

statistics-release-1.6.3/docs/cdf.html000066400000000000000000000335631456127120000177130ustar00rootroot00000000000000 Statistics: cdf

Function Reference: cdf

statistics: p = cdf (name, x, A)
statistics: p = cdf (name, x, A, B)
statistics: p = cdf (name, x, A, B, C)
statistics: p = cdf (…, "upper")

Return the CDF of a univariate distribution evaluated at x.

cdf is a wrapper for the univariate cumulative distribution functions available in the statistics package. See the corresponding functions’ help to learn the signification of the parameters after x.

p = cdf (name, x, A) returns the CDF for the one-parameter distribution family specified by name and the distribution parameter A, evaluated at the values in x.

p = cdf (name, x, A, B) returns the CDF for the two-parameter distribution family specified by name and the distribution parameters A and B, evaluated at the values in x.

p = cdf (name, x, A, B, C) returns the CDF for the three-parameter distribution family specified by name and the distribution parameters A, B, and C, evaluated at the values in x.

p = cdf (…, "upper") returns the complement of the CDF using an algorithm that more accurately computes the extreme upper-tail probabilities. "upper" can follow any of the input arguments in the previous syntaxes.

name must be a char string of the name or the abbreviation of the desired cumulative distribution function as listed in the followng table. The last column shows the number of required parameters that should be parsed after x to the desired CDF. The optional input argument "upper" does not count in the required number of parameters.

Distribution NameAbbreviationInput Parameters
"Beta""beta"2
"Binomial""bino"2
"Birnbaum-Saunders""bisa"2
"Burr""burr"3
"Cauchy""cauchy"2
"Chi-squared""chi2"1
"Extreme Value""ev"2
"Exponential""exp"1
"F-Distribution""f"2
"Gamma""gam"2
"Geometric""geo"1
"Generalized Extreme Value""gev"3
"Generalized Pareto""gp"3
"Gumbel""gumbel"2
"Half-normal""hn"2
"Hypergeometric""hyge"3
"Inverse Gaussian""invg"2
"Laplace""laplace"2
"Logistic""logi"2
"Log-Logistic""logl"2
"Lognormal""logn"2
"Nakagami""naka"2
"Negative Binomial""nbin"2
"Noncentral F-Distribution""ncf"3
"Noncentral Student T""nct"2
"Noncentral Chi-Squared""ncx2"2
"Normal""norm"2
"Poisson""poiss"1
"Rayleigh""rayl"1
"Rician""rice"2
"Student T""t"1
"location-scale T""tls"3
"Triangular""tri"3
"Discrete Uniform""unid"1
"Uniform""unif"2
"Von Mises""vm"2
"Weibull""wbl"2

See also: icdf, pdf, cdf, betacdf, binocdf, bisacdf, burrcdf, cauchycdf, chi2cdf, evcdf, expcdf, fcdf, gamcdf, geocdf, gevcdf, gpcdf, gumbelcdf, hncdf, hygecdf, invgcdf, laplacecdf, logicdf, loglcdf, logncdf, nakacdf, nbincdf, ncfcdf, nctcdf, ncx2cdf, normcdf, poisscdf, raylcdf, ricecdf, tcdf, tricdf, unidcdf, unifcdf, vmcdf, wblcdf

Source Code: cdf

statistics-release-1.6.3/docs/cdfcalc.html000066400000000000000000000111561456127120000205300ustar00rootroot00000000000000 Statistics: cdfcalc

Function Reference: cdfcalc

statistics: [yCDF, xCDF, n, emsg, eid] = cdfcalc (x)

Calculate an empirical cumulative distribution function.

[yCDF, xCDF] = cdfcalc (x) calculates an empirical cumulative distribution function (CDF) of the observations in the data sample vector x. x may be a row or column vector, and represents a random sample of observations from some underlying distribution. On return xCDF is the set of x values at which the CDF increases. At XCDF(i), the function increases from YCDF(i) to YCDF(i+1).

[yCDF, xCDF, n] = cdfcalc (x) also returns n, the sample size.

[yCDF, xCDF, n, emsg, eid] = cdfcalc (x) also returns an error message and error id if x is not a vector or if it contains no values other than NaN.

See also: cdfplot

Source Code: cdfcalc

statistics-release-1.6.3/docs/cdfplot.html000066400000000000000000000144411456127120000206040ustar00rootroot00000000000000 Statistics: cdfplot

Function Reference: cdfplot

statistics: hCDF = cdfplot (x)
statistics: [hCDF, stats] = cdfplot (x)

Display an empirical cumulative distribution function.

hCDF = cdfplot (x) plots an empirical cumulative distribution function (CDF) of the observations in the data sample vector x. x may be a row or column vector, and represents a random sample of observations from some underlying distribution.

cdfplot plots F(x), the empirical (or sample) CDF versus the observations in x. The empirical CDF, F(x), is defined as follows:

F(x) = (Number of observations <= x) / (Total number of observations)

for all values in the sample vector x. NaNs are ignored. hCDF is the handle of the empirical CDF curve (a handle hraphics ’line’ object).

[hCDF, stats] = cdfplot (x) also returns a structure with the following fields as a statistical summary.

STATS.minminimum value of x
STATS.maxmaximum value of x
STATS.meansample mean of x
STATS.mediansample median (50th percentile) of x
STATS.stdsample standard deviation of x

See also: qqplot, cdfcalc

Source Code: cdfplot

Example: 1

  

 x = randn(100,1);
 cdfplot (x);
plotted figure

statistics-release-1.6.3/docs/chi2cdf.html000066400000000000000000000145631456127120000204600ustar00rootroot00000000000000 Statistics: chi2cdf

Function Reference: chi2cdf

statistics: p = chi2cdf (x, df)
statistics: p = chi2cdf (x, df, "upper")

Chi-squared cumulative distribution function (CDF).

For each element of x, compute the cumulative distribution function (CDF) of the chi-squared distribution with df degrees of freedom. The chi-squared density function with df degrees of freedom is the same as a gamma density function with parameters df/2 and 2.

The size of p is the common size of x and df. A scalar input functions as a constant matrix of the same size as the other input.

p = chi2cdf (x, df, "upper") computes the upper tail probability of the chi-squared distribution with df degrees of freedom, at the values in x.

Further information about the chi-squared distribution can be found at https://en.wikipedia.org/wiki/Chi-squared_distribution

See also: chi2inv, chi2pdf, chi2rnd, chi2stat

Source Code: chi2cdf

Example: 1

  

 ## Plot various CDFs from the chi-squared distribution
 x = 0:0.01:8;
 p1 = chi2cdf (x, 1);
 p2 = chi2cdf (x, 2);
 p3 = chi2cdf (x, 3);
 p4 = chi2cdf (x, 4);
 p5 = chi2cdf (x, 6);
 p6 = chi2cdf (x, 9);
 plot (x, p1, "-b", x, p2, "-g", x, p3, "-r", ...
       x, p4, "-c", x, p5, "-m", x, p6, "-y")
 grid on
 xlim ([0, 8])
 legend ({"df = 1", "df = 2", "df = 3", ...
          "df = 4", "df = 6", "df = 9"}, "location", "southeast")
 title ("Chi-squared CDF")
 xlabel ("values in x")
 ylabel ("probability")
plotted figure

statistics-release-1.6.3/docs/chi2gof.html000066400000000000000000000343501456127120000204730ustar00rootroot00000000000000 Statistics: chi2gof

Function Reference: chi2gof

statistics: h = chi2gof (x)
statistics: [h, p] = chi2gof (x)
statistics: [p, h, stats] = chi2gof (x)
statistics: […] = chi2gof (x, Name, Value, …)

Chi-square goodness-of-fit test.

chi2gof performs a chi-square goodness-of-fit test for discrete or continuous distributions. The test is performed by grouping the data into bins, calculating the observed and expected counts for those bins, and computing the chi-square test statistic $$ \chi ^ 2 = \sum_{i=1}^N \left (O_i - E_i \right) ^ 2 / E_i $$ where O is the observed counts and E is the expected counts. This test statistic has an approximate chi-square distribution when the counts are sufficiently large.

Bins in either tail with an expected count less than 5 are pooled with neighboring bins until the count in each extreme bin is at least 5. If bins remain in the interior with counts less than 5, chi2gof displays a warning. In that case, you should use fewer bins, or provide bin centers or binedges, to increase the expected counts in all bins.

h = chi2gof (x) performs a chi-square goodness-of-fit test that the data in the vector X are a random sample from a normal distribution with mean and variance estimated from x. The result is h = 0 if the null hypothesis (that x is a random sample from a normal distribution) cannot be rejected at the 5% significance level, or h = 1 if the nullhypothesis can be rejected at the 5% level. chi2gof uses by default 10 bins ("nbins"), and compares the test statistic to a chi-square distribution with nbins - 3 degrees of freedom, to take into account that two parameters were estimated.

[h, p] = chi2gof (x) also returns the p-value p, which is the probability of observing the given result, or one more extreme, by chance if the null hypothesis is true. If there are not enough degrees of freedom to carry out the test, p is NaN.

[h, p, stats] = chi2gof (x) also returns a stats structure with the following fields:

"chi2stat"Chi-square statistic
"df"Degrees of freedom
"binedges"Vector of bin binedges after pooling
"O"Observed count in each bin
"E"Expected count in each bin

[…] = chi2gof (x, Name, Value, …) specifies optional Name/Value pair arguments chosen from the following list.

NameValue
"nbins"The number of bins to use. Default is 10.
"binctrs"A vector of bin centers.
"binedges"A vector of bin binedges.
"cdf"A fully specified cumulative distribution function or a function handle provided in a cell array whose first element is a function handle, and all later elements are its parameter values. The function must take x values as its first argument, and other parameters as later arguments.
"expected"A vector with one element per bin specifying the expected counts for each bin.
"nparams"The number of estimated parameters; used to adjust the degrees of freedom to be nbins - 1 - nparams, where nbins is the number of bins.
"emin"The minimum allowed expected value for a bin; any bin in either tail having an expected value less than this amount is pooled with a neighboring bin. Use the value 0 to prevent pooling. Default is 5.
"frequency"A vector of the same length as x containing the frequency of the corresponding x values.
"alpha"An alpha value such that the hypothesis is rejected if p < alpha. Default is alpha = 0.05.

You should specify either "cdf" or "expected" parameters, but not both. If your "cdf" input contains extra parameters, these are accounted for automatically and there is no need to specify "nparams". If your "expected" input depends on estimated parameters, you should use the "nparams" parameter to ensure that the degrees of freedom for the test is correct.

Source Code: chi2gof

Example: 1

  

 x = normrnd (50, 5, 100, 1);
 [h, p, stats] = chi2gof (x)
 [h, p, stats] = chi2gof (x, "cdf", @(x)normcdf (x, mean(x), std(x)))
 [h, p, stats] = chi2gof (x, "cdf", {@normcdf, mean(x), std(x)})

h = 0
p = 0.5464
stats =

  scalar structure containing the fields:

    chi2stat = 4.0212
    df = 5
    edges =

       38.399   42.726   44.890   47.053   49.217   51.380   53.544   55.708   60.035

    O =

        9    7    9   22   17   14   10   12

    E =

        6.8588    8.2228   13.0313   16.8721   17.8471   15.4236   10.8899   10.8544


h = 0
p = 0.5464
stats =

  scalar structure containing the fields:

    chi2stat = 4.0212
    df = 5
    edges =

       38.399   42.726   44.890   47.053   49.217   51.380   53.544   55.708   60.035

    O =

        9    7    9   22   17   14   10   12

    E =

        6.8588    8.2228   13.0313   16.8721   17.8471   15.4236   10.8899   10.8544


h = 0
p = 0.5464
stats =

  scalar structure containing the fields:

    chi2stat = 4.0212
    df = 5
    edges =

       38.399   42.726   44.890   47.053   49.217   51.380   53.544   55.708   60.035

    O =

        9    7    9   22   17   14   10   12

    E =

        6.8588    8.2228   13.0313   16.8721   17.8471   15.4236   10.8899   10.8544

Example: 2

  

 x = rand (100,1 );
 n = length (x);
 binedges = linspace (0, 1, 11);
 expectedCounts = n * diff (binedges);
 [h, p, stats] = chi2gof (x, "binedges", binedges, "expected", expectedCounts)

h = 0
p = 0.9835
stats =

  scalar structure containing the fields:

    chi2stat = 2.4000
    df = 9
    edges =

     Columns 1 through 8:

       4.2756e-03   1.0230e-01   2.0032e-01   2.9835e-01   3.9637e-01   4.9439e-01   5.9242e-01   6.9044e-01

     Columns 9 through 11:

       7.8847e-01   8.8649e-01   9.8451e-01

    O =

       10    9   12    9    8   10   11    8   10   13

    E =

       10   10   10   10   10   10   10   10   10   10

Example: 3

  

 bins = 0:5;
 obsCounts = [6 16 10 12 4 2];
 n = sum(obsCounts);
 lambdaHat = sum(bins.*obsCounts) / n;
 expCounts = n * poisspdf(bins,lambdaHat);
 [h, p, stats] = chi2gof (bins, "binctrs", bins, "frequency", obsCounts, ...
                          "expected", expCounts, "nparams",1)

h = 0
p = 0.4654
stats =

  scalar structure containing the fields:

    chi2stat = 2.5550
    df = 3
    edges =

       4.9407e-324    8.3333e-01    1.6667e+00    2.5000e+00    3.3333e+00    5.0000e+00

    O =

        6   16   10   12    6

    E =

        7.0429   13.8041   13.5280    8.8383    6.0284
statistics-release-1.6.3/docs/chi2inv.html000066400000000000000000000135501456127120000205130ustar00rootroot00000000000000 Statistics: chi2inv

Function Reference: chi2inv

statistics: x = chi2inv (p, df)

Inverse of the chi-squared cumulative distribution function (iCDF).

For each element of p, compute the quantile (the inverse of the CDF) of the chi-squared distribution with df degrees of freedom. The size of x is the common size of p and df. A scalar input functions as a constant matrix of the same size as the other inputs.

Further information about the chi-squared distribution can be found at https://en.wikipedia.org/wiki/Chi-squared_distribution

See also: chi2cdf, chi2pdf, chi2rnd, chi2stat

Source Code: chi2inv

Example: 1

  

 ## Plot various iCDFs from the chi-squared distribution
 p = 0.001:0.001:0.999;
 x1 = chi2inv (p, 1);
 x2 = chi2inv (p, 2);
 x3 = chi2inv (p, 3);
 x4 = chi2inv (p, 4);
 x5 = chi2inv (p, 6);
 x6 = chi2inv (p, 9);
 plot (p, x1, "-b", p, x2, "-g", p, x3, "-r", ...
       p, x4, "-c", p, x5, "-m", p, x6, "-y")
 grid on
 ylim ([0, 8])
 legend ({"df = 1", "df = 2", "df = 3", ...
          "df = 4", "df = 6", "df = 9"}, "location", "northwest")
 title ("Chi-squared iCDF")
 xlabel ("probability")
 ylabel ("values in x")
plotted figure

statistics-release-1.6.3/docs/chi2pdf.html000066400000000000000000000135271456127120000204740ustar00rootroot00000000000000 Statistics: chi2pdf

Function Reference: chi2pdf

statistics: y = chi2pdf (x, df)

Chi-squared probability density function (PDF).

For each element of x, compute the probability density function (PDF) of the chi-squared distribution with df degrees of freedom. The size of y is the common size of x and df. A scalar input functions as a constant matrix of the same size as the other inputs.

Further information about the chi-squared distribution can be found at https://en.wikipedia.org/wiki/Chi-squared_distribution

See also: chi2cdf, chi2pdf, chi2rnd, chi2stat

Source Code: chi2pdf

Example: 1

  

 ## Plot various PDFs from the chi-squared distribution
 x = 0:0.01:8;
 y1 = chi2pdf (x, 1);
 y2 = chi2pdf (x, 2);
 y3 = chi2pdf (x, 3);
 y4 = chi2pdf (x, 4);
 y5 = chi2pdf (x, 6);
 y6 = chi2pdf (x, 9);
 plot (x, y1, "-b", x, y2, "-g", x, y3, "-r", ...
       x, y4, "-c", x, y5, "-m", x, y6, "-y")
 grid on
 xlim ([0, 8])
 ylim ([0, 0.5])
 legend ({"df = 1", "df = 2", "df = 3", ...
          "df = 4", "df = 6", "df = 9"}, "location", "northeast")
 title ("Chi-squared PDF")
 xlabel ("values in x")
 ylabel ("density")
plotted figure

statistics-release-1.6.3/docs/chi2rnd.html000066400000000000000000000116611456127120000205030ustar00rootroot00000000000000 Statistics: chi2rnd

Function Reference: chi2rnd

statistics: r = chi2rnd (df)
statistics: r = chi2rnd (df, rows)
statistics: r = chi2rnd (df, rows, cols, …)
statistics: r = chi2rnd (df, [sz])

Random arrays from the chi-squared distribution.

r = chi2rnd (df) returns an array of random numbers chosen from the chi-squared distribution with df degrees of freedom. The size of r is the size of df.

When called with a single size argument, chi2rnd returns a square matrix with the dimension specified. When called with more than one scalar argument, the first two arguments are taken as the number of rows and columns and any further arguments specify additional matrix dimensions. The size may also be specified with a row vector of dimensions, sz.

Further information about the chi-squared distribution can be found at https://en.wikipedia.org/wiki/Chi-squared_distribution

See also: chi2cdf, chi2inv, chi2pdf, chi2stat

Source Code: chi2rnd

statistics-release-1.6.3/docs/chi2stat.html000066400000000000000000000104731456127120000206730ustar00rootroot00000000000000 Statistics: chi2stat

Function Reference: chi2stat

statistics: [m, v] = chi2stat (df)

Compute statistics of the chi-squared distribution.

[m, v] = chi2stat (df) returns the mean and variance of the chi-squared distribution with df degrees of freedom.

The size of m (mean) and v (variance) is the same size of the input argument.

Further information about the chi-squared distribution can be found at https://en.wikipedia.org/wiki/Chi-squared_distribution

See also: chi2cdf, chi2inv, chi2pdf, chi2rnd

Source Code: chi2stat

statistics-release-1.6.3/docs/chi2test.html000066400000000000000000000210171456127120000206730ustar00rootroot00000000000000 Statistics: chi2test

Function Reference: chi2test

statistics: pval = chi2test (x)
statistics: [pval, chisq] = chi2test (x)
statistics: [pval, chisq, dF] = chi2test (x)
statistics: [pval, chisq, dF, E] = chi2test (x)
statistics: […] = chi2test (x, name, value)

Perform a chi-squared test (for independence or homogeneity).

For 2-way contingency tables, chi2test performs and a chi-squared test for independence or homogeneity, according to the sampling scheme and related question. Independence means that the the two variables forming the 2-way table are not associated, hence you cannot predict from one another. Homogeneity refers to the concept of similarity, hence they all come from the same distribution.

Both tests are computationally identical and will produce the same result. Nevertheless, they anwser to different questions. Consider two variables, one for gender and another for smoking. To test independence (whether gender and smoking is associated), we would randomly sample from the general population and break them down into categories in the table. To test homogeneity (whether men and women share the same smoking habits), we would sample individuals from within each gender, and then measure their smoking habits (e.g. smokers vs non-smokers).

When chi2test is called without any output arguments, it will print the result in the terminal including p-value, chi^2 statistic, and degrees of freedom. Otherwise it can return the following output arguments:

pvalthe p-value of the relevant test.
chisqthe chi^2 statistic of the relevant test.
dFthe degrees of freedom of the relevant test.
Ethe EXPECTED values of the original contigency table.

Unlike MATLAB, in GNU Octave chi2test also supports 3-way tables, which involve three categorical variables (each in a different dimension of x. In its simplest form, […] = chi2test (x) will will test for mutual independence among the three variables. Alternatively, when called in the form […] = chi2test (x, name, value), it can perform the following tests:

namevalueDescription
"mutual"[]Mutual independence. All variables are independent from each other, (A, B, C). Value must be an empty matrix.
"joint"scalarJoint independence. Two variables are jointly independent of the third, (AB, C). The scalar value corresponds to the dimension of the independent variable (i.e. 3 for C).
"marginal"scalarMarginal independence. Two variables are independent if you ignore the third, (A, C). The scalar value corresponds to the dimension of the variable to be ignored (i.e. 2 for B).
"conditional"scalarConditional independence. Two variables are independent given the third, (AC, BC). The scalar value corresponds to the dimension of the variable that forms the conditional dependence (i.e. 3 for C).
"homogeneous"[]Homogeneous associations. Conditional (partial) odds-ratios are not related on the value of the third, (AB, AC, BC). Value must be an empty matrix.

When testing for homogeneous associations in 3-way tables, the iterative proportional fitting procedure is used. For small samples it is better to use the Cochran-Mantel-Haenszel Test. K-way tables for k > 3 are supported only for testing mutual independence. Similar to 2-way tables, no optional parameters are required for k > 3 multi-way tables.

chi2test produces a warning if any cell of a 2x2 table has an expected frequency less than 5 or if more than 20% of the cells in larger 2-way tables have expected frequencies less than 5 or any cell with expected frequency less than 1. In such cases, use fishertest.

See also: crosstab, fishertest, mcnemar_test

Source Code: chi2test

statistics-release-1.6.3/docs/cholcov.html000066400000000000000000000151041456127120000206030ustar00rootroot00000000000000 Statistics: cholcov

Function Reference: cholcov

statistics: T = cholcov (sigma)
statistics: [T, p = cholcov (sigma)
statistics: […] = cholcov (sigma, flag)

Cholesky-like decomposition for covariance matrix.

T = cholcov (sigma) computes matrix T such that sigma = T’ T. sigma must be square, symmetric, and positive semi-definite.

If sigma is positive definite, then T is the square, upper triangular Cholesky factor. If sigma is not positive definite, T is computed with an eigenvalue decomposition of sigma, but in this case T is not necessarily triangular or square. Any eigenvectors whose corresponding eigenvalue is close to zero (within a tolerance) are omitted. If any remaining eigenvalues are negative, T is empty.

The tolerance is calculated as 10 * eps (max (abs (diag (sigma)))).

[T, p = cholcov (sigma) returns in p the number of negative eigenvalues of sigma. If p > 0, then T is empty, whereas if p = 0, sigma) is positive semi-definite.

If sigma is not square and symmetric, P is NaN and T is empty.

[T, p = cholcov (sigma, 0) returns p = 0 if sigma is positive definite, in which case T is the Cholesky factor. If sigma is not positive definite, p is a positive integer and T is empty.

[…] = cholcov (sigma, 1) is equivalent to […] = cholcov (sigma).

See also: chov

Source Code: cholcov

Example: 1

  

 C1 = [2, 1, 1, 2; 1, 2, 1, 2; 1, 1, 2, 2; 2, 2, 2, 3]
 T = cholcov (C1)
 C2 = T'*T

C1 =

   2   1   1   2
   1   2   1   2
   1   1   2   2
   2   2   2   3

T =

  -0.1247  -0.6365   0.7612        0
   0.8069  -0.5114  -0.2955        0
   1.1547   1.1547   1.1547   1.7321

C2 =

   2   1   1   2
   1   2   1   2
   1   1   2   2
   2   2   2   3
statistics-release-1.6.3/docs/cl_multinom.html000066400000000000000000000204101456127120000214640ustar00rootroot00000000000000 Statistics: cl_multinom

Function Reference: cl_multinom

statistics: CL = cl_multinom (X, N, b)
statistics: CL = cl_multinom (X, N, b, method)

Confidence level of multinomial portions.

cl_multinom returns confidence level of multinomial parameters estimated as p = X / sum(X) with predefined confidence interval b. Finite population is also considered.

This function calculates the level of confidence at which the samples represent the true distribution given that there is a predefined tolerance (confidence interval). This is the upside down case of the typical excercises at which we want to get the confidence interval given the confidence level (and the estimated parameters of the underlying distribution). But once we accept (lets say at elections) that we have a standard predefined maximal acceptable error rate (e.g. b=0.02 ) in the estimation and we just want to know that how sure we can be that the measured proportions are the same as in the entire population (ie. the expected value and mean of the samples are roughly the same) we need to use this function.

Arguments

VariableTypeDescription
Xint vectorsample frequencies bins.
Nint scalarPopulation size that was sampled by X. If N < sum (X), infinite number assumed.
breal vectorconfidence interval. If vector, it should be the size of X containing confence interval for each cells. If scalar, each cell will have the same value of b unless it is zero or -1. If value is 0, b = 0.02 is assumed which is standard choice at elections otherwise it is calculated in a way that one sample in a cell alteration defines the confidence interval.
methodstringAn optional argument for defining the calculation method. Available choices are "bromaghin" (default), "cochran", and agresti_cull.

Note! The agresti_cull method is not exactly the solution at reference given below but an adjustment of the solutions above.

Returns

Confidence level.

Example

CL = cl_multinom ([27; 43; 19; 11], 10000, 0.05) returns 0.69 confidence level.

References

  1. "bromaghin" calculation type (default) is based on the article:

    Jeffrey F. Bromaghin, "Sample Size Determination for Interval Estimation of Multinomial Probabilities", The American Statistician vol 47, 1993, pp 203-206.

  2. "cochran" calculation type is based on article:

    Robert T. Tortora, "A Note on Sample Size Estimation for Multinomial Populations", The American Statistician, , Vol 32. 1978, pp 100-102.

  3. "agresti_cull" calculation type is based on article:

    A. Agresti and B.A. Coull, "Approximate is better than ’exact’ for interval estimation of binomial portions", The American Statistician, Vol. 52, 1998, pp 119-126

Source Code: cl_multinom

Example: 1

  

 CL = cl_multinom ([27; 43; 19; 11], 10000, 0.05)

CL = 0.6923
statistics-release-1.6.3/docs/cluster.html000066400000000000000000000124731456127120000206350ustar00rootroot00000000000000 Statistics: cluster

Function Reference: cluster

statistics: T = cluster (Z, "Cutoff", C)
statistics: T = cluster (Z, "Cutoff", C, "Depth", D)
statistics: T = cluster (Z, "Cutoff", C, "Criterion", criterion)
statistics: T = cluster (Z, "MaxClust", N)

Define clusters from an agglomerative hierarchical cluster tree.

Given a hierarchical cluster tree Z generated by the linkage function, cluster defines clusters, using a threshold value C to identify new clusters (’Cutoff’) or according to a maximum number of desired clusters N (’MaxClust’).

criterion is used to choose the criterion for defining clusters, which can be either "inconsistent" (default) or "distance". When using "inconsistent", cluster compares the threshold value C to the inconsistency coefficient of each link; when using "distance", cluster compares the threshold value C to the height of each link. D is the depth used to evaluate the inconsistency coefficient, its default value is 2.

cluster uses "distance" as a criterion for defining new clusters when it is used the ’MaxClust’ method.

See also: clusterdata, dendrogram, inconsistent, kmeans, linkage, pdist

Source Code: cluster

statistics-release-1.6.3/docs/clusterdata.html000066400000000000000000000141401456127120000214600ustar00rootroot00000000000000 Statistics: clusterdata

Function Reference: clusterdata

statistics: T = clusterdata (X, cutoff)
statistics: T = clusterdata (X, Name, Value)

Wrapper function for linkage and cluster.

If cutoff is used, then clusterdata calls linkage and cluster with default value, using cutoff as a threshold value for cluster. If cutoff is an integer and greater or equal to 2, then cutoff is interpreted as the maximum number of cluster desired and the "MaxClust" option is used for cluster.

If cutoff is not used, then clusterdata expects a list of pair arguments. Then you must specify either the "Cutoff" or "MaxClust" option for cluster. The method and metric used by linkage, are defined through the "linkage" and "distance" arguments.

See also: cluster, dendrogram, inconsistent, kmeans, linkage, pdist

Source Code: clusterdata

Example: 1

  

 randn ("seed", 1)  # for reproducibility
 r1 = randn (10, 2) * 0.25 + 1;
 randn ("seed", 5)  # for reproducibility
 r2 = randn (20, 2) * 0.5 - 1;
 X = [r1; r2];

 wnl = warning ("off", "Octave:linkage_savemem", "local");
 T = clusterdata (X, "linkage", "ward", "MaxClust", 2);
 scatter (X(:,1), X(:,2), 36, T, "filled");
plotted figure

statistics-release-1.6.3/docs/cmdscale.html000066400000000000000000000155511456127120000207270ustar00rootroot00000000000000 Statistics: cmdscale

Function Reference: cmdscale

statistics: Y = cmdscale (D)
statistics: [Y, e] = cmdscale (D)

Classical multidimensional scaling of a matrix.

Takes an n by n distance (or difference, similarity, or dissimilarity) matrix D. Returns Y, a matrix of n points with coordinates in p dimensional space which approximate those distances (or differences, similarities, or dissimilarities). Also returns the eigenvalues e of B = -1/2 * J * (D.^2) * J, where J = eye(n) - ones(n,n)/n. p, the number of columns of Y, is equal to the number of positive real eigenvalues of B.

D can be a full or sparse matrix or a vector of length n*(n-1)/2 containing the upper triangular elements (like the output of the pdist function). It must be symmetric with non-negative entries whose values are further restricted by the type of matrix being represented:

* If D is either a distance, dissimilarity, or difference matrix, then it must have zero entries along the main diagonal. In this case the points Y equal or approximate the distances given by D.

* If D is a similarity matrix, the elements must all be less than or equal to one, with ones along the the main diagonal. In this case the points Y equal or approximate the distances given by D = sqrt(ones(n,n)-D).

D is a Euclidean matrix if and only if B is positive semi-definite. When this is the case, then Y is an exact representation of the distances given in D. If D is non-Euclidean, Y only approximates the distance given in D. The approximation used by cmdscale minimizes the statistical loss function known as strain.

The returned Y is an n by p matrix showing possible coordinates of the points in p dimensional space (p < n). The columns are correspond to the positive eigenvalues of B in descending order. A translation, rotation, or reflection of the coordinates given by Y will satisfy the same distance matrix up to the limits of machine precision.

For any k <= p, if the largest k positive eigenvalues of B are significantly greater in absolute magnitude than its other eigenvalues, the first k columns of Y provide a k-dimensional reduction of Y which approximates the distances given by D. The optional return e can be used to consider various values of k, or to evaluate the accuracy of specific dimension reductions (e.g., k = 2).

Reference: Ingwer Borg and Patrick J.F. Groenen (2005), Modern Multidimensional Scaling, Second Edition, Springer, ISBN: 978-0-387-25150-9 (Print) 978-0-387-28981-6 (Online)

See also: pdist

Source Code: cmdscale

statistics-release-1.6.3/docs/combnk.html000066400000000000000000000112261456127120000204200ustar00rootroot00000000000000 Statistics: combnk

Function Reference: combnk

statistics: c = combnk (data, k)

Return all combinations of k elements in data.

Source Code: combnk

Example: 1

  

 c = combnk (1:5, 2);
 disp ("All pairs of integers between 1 and 5:");
 disp (c);

All pairs of integers between 1 and 5:
   1   2
   1   3
   1   4
   1   5
   2   3
   2   4
   2   5
   3   4
   3   5
   4   5
statistics-release-1.6.3/docs/confusionchart.html000066400000000000000000000270211456127120000221740ustar00rootroot00000000000000 Statistics: confusionchart

Function Reference: confusionchart

statistics: confusionchart (trueLabels, predictedLabels)
statistics: confusionchart (m)
statistics: confusionchart (m, classLabels)
statistics: confusionchart (parent, …)
statistics: confusionchart (…, prop, val, …)
statistics: cm = confusionchart (…)

Display a chart of a confusion matrix.

The two vectors of values trueLabels and predictedLabels, which are used to compute the confusion matrix, must be defined with the same format as the inputs of confusionmat. Otherwise a confusion matrix m as computed by confusionmat can be given.

classLabels is an array of labels, i.e. the list of the class names.

If the first argument is a handle to a figure or to a uipanel, then the confusion matrix chart is displayed inside that object.

Optional property/value pairs are passed directly to the underlying objects, e.g. "xlabel", "ylabel", "title", "fontname", "fontsize" etc.

The optional return value cm is a ConfusionMatrixChart object. Specific properties of a ConfusionMatrixChart object are:

  • "DiagonalColor" The color of the patches on the diagonal, default is [0.0, 0.4471, 0.7412].
  • "OffDiagonalColor" The color of the patches off the diagonal, default is [0.851, 0.3255, 0.098].
  • "GridVisible" Available values: on (default), off.
  • "Normalization" Available values: absolute (default), column-normalized, row-normalized, total-normalized.
  • "ColumnSummary" Available values: off (default), absolute, column-normalized,total-normalized.
  • "RowSummary" Available values: off (default), absolute, row-normalized, total-normalized.

Run demo confusionchart to see some examples.

See also: confusionmat, sortClasses

Source Code: confusionchart

Example: 1

  

 ## Setting the chart properties
 Yt = [8 5 6 8 5 3 1 6 4 2 5 3 1 4]';
 Yp = [8 5 6 8 5 2 3 4 4 5 5 7 2 6]';
 confusionchart (Yt, Yp, "Title", ...
   "Demonstration with summaries","Normalization",...
   "absolute","ColumnSummary", "column-normalized","RowSummary",...
   "row-normalized")

ans =

ConfusionMatrixChart with properties:

	NormalizedValues: [ 8x8 double ]
	ClassLabels: { 8x1 cell }
plotted figure

Example: 2

  

 ## Cellstr as inputs
 Yt = {"Positive", "Positive", "Positive", "Negative", "Negative"};
 Yp = {"Positive", "Positive", "Negative", "Negative", "Negative"};
 m = confusionmat (Yt, Yp);
 confusionchart (m, {"Positive", "Negative"});
plotted figure

Example: 3

  

 ## Editing the object properties
 Yt = {"Positive", "Positive", "Positive", "Negative", "Negative"};
 Yp = {"Positive", "Positive", "Negative", "Negative", "Negative"};
 cm = confusionchart (Yt, Yp);
 cm.Title = "This is an example with a green diagonal";
 cm.DiagonalColor = [0.4660, 0.6740, 0.1880];
plotted figure

Example: 5

  

 ## Sorting classes
 Yt = [8 5 6 8 5 3 1 6 4 2 5 3 1 4]';
 Yp = [8 5 6 8 5 2 3 4 4 5 5 7 2 6]';
 cm = confusionchart (Yt, Yp, "Title", ...
   "Classes are sorted in ascending order");
 cm = confusionchart (Yt, Yp, "Title", ...
   "Classes are sorted according to clusters");
 sortClasses (cm, "cluster");
plotted figure

plotted figure

statistics-release-1.6.3/docs/confusionmat.html000066400000000000000000000123001456127120000216460ustar00rootroot00000000000000 Statistics: confusionmat

Function Reference: confusionmat

statistics: C = confusionmat (group, grouphat)
statistics: C = confusionmat (group, grouphat, "Order", grouporder)
statistics: [C, order] = confusionmat (group, grouphat)

Compute a confusion matrix for classification problems

confusionmat returns the confusion matrix C for the group of actual values group and the group of predicted values grouphat. The row indices of the confusion matrix represent actual values, while the column indices represent predicted values. The indices are the same for both actual and predicted values, so the confusion matrix is a square matrix. Each element of the matrix represents the number of matches between a given actual value (row index) and a given predicted value (column index), hence correct matches lie on the main diagonal of the matrix. The order of the rows and columns is returned in order.

group and grouphat must have the same number of observations and the same data type. Valid data types are numeric vectors, logical vectors, character arrays, string arrays (not implemented yet), cell arrays of strings.

The order of the rows and columns can be specified by setting the grouporder variable. The data type of grouporder must be the same of group and grouphat.

MATLAB compatibility: Octave misses string arrays and categorical vectors.

See also: crosstab

Source Code: confusionmat

statistics-release-1.6.3/docs/cophenet.html000066400000000000000000000137131456127120000207570ustar00rootroot00000000000000 Statistics: cophenet

Function Reference: cophenet

statistics: [c, d] = cophenet (Z, y)

Compute the cophenetic correlation coefficient.

The cophenetic correlation coefficient C of a hierarchical cluster tree Z is the linear correlation coefficient between the cophenetic distances d and the euclidean distances y. $$ c = \frac {\sum_{i < j}(Y_{ij}-{\bar {y}})(Z_{ij}-{\bar{z}})} {\sqrt{\sum_{i < j}(Y_{ij}-{\bar {y}})^2(Z_{ij}-{\bar{z}})^2}} $$

It is a measure of the similarity between the distance of the leaves, as seen in the tree, and the distance of the original data points, which were used to build the tree. When this similarity is greater, that is the coefficient is closer to 1, the tree renders an accurate representation of the distances between the original data points.

Z is a hierarchical cluster tree, as the output of linkage. y is a vector of euclidean distances, as the output of pdist.

The optional output d is a vector of cophenetic distances, in the same lower triangular format as y. The cophenetic distance between two data points is the height of the lowest common node of the tree.

See also: cluster, dendrogram, inconsistent, linkage, pdist, squareform

Source Code: cophenet

Example: 1

  

 randn ("seed", 5)  # for reproducibility
 X = randn (10,2);
 y = pdist (X);
 Z = linkage (y, "average");
 cophenet (Z, y)

ans = 0.8025
statistics-release-1.6.3/docs/copulacdf.html000066400000000000000000000144561456127120000211170ustar00rootroot00000000000000 Statistics: copulacdf

Function Reference: copulacdf

statistics: p = copulacdf (family, x, theta)
statistics: p = copulacdf ('t', x, theta, df)

Copula family cumulative distribution functions (CDF).

Arguments

  • family is the copula family name. Currently, family can be 'Gaussian' for the Gaussian family, 't' for the Student’s t family, 'Clayton' for the Clayton family, 'Gumbel' for the Gumbel-Hougaard family, 'Frank' for the Frank family, 'AMH' for the Ali-Mikhail-Haq family, or 'FGM' for the Farlie-Gumbel-Morgenstern family.
  • x is the support where each row corresponds to an observation.
  • theta is the parameter of the copula. For the Gaussian and Student’s t copula, theta must be a correlation matrix. For bivariate copulas theta can also be a correlation coefficient. For the Clayton family, the Gumbel-Hougaard family, the Frank family, and the Ali-Mikhail-Haq family, theta must be a vector with the same number of elements as observations in x or be scalar. For the Farlie-Gumbel-Morgenstern family, theta must be a matrix of coefficients for the Farlie-Gumbel-Morgenstern polynomial where each row corresponds to one set of coefficients for an observation in x. A single row is expanded. The coefficients are in binary order.
  • df is the degrees of freedom for the Student’s t family. df must be a vector with the same number of elements as observations in x or be scalar.

Return values

  • p is the cumulative distribution of the copula at each row of x and corresponding parameter theta.

Examples

 
 
 x = [0.2:0.2:0.6; 0.2:0.2:0.6];
 theta = [1; 2];
 p = copulacdf ("Clayton", x, theta)
 
 
 x = [0.2:0.2:0.6; 0.2:0.1:0.4];
 theta = [0.2, 0.1, 0.1, 0.05];
 p = copulacdf ("FGM", x, theta)

References

  1. Roger B. Nelsen. An Introduction to Copulas. Springer, New York, second edition, 2006.

See also: copulapdf, copularnd

Source Code: copulacdf

statistics-release-1.6.3/docs/copulapdf.html000066400000000000000000000131261456127120000211250ustar00rootroot00000000000000 Statistics: copulapdf

Function Reference: copulapdf

statistics: y = copulapdf (family, x, theta)

Copula family probability density functions (PDF).

Arguments

  • family is the copula family name. Currently, family can be 'Clayton' for the Clayton family, 'Gumbel' for the Gumbel-Hougaard family, 'Frank' for the Frank family, or 'AMH' for the Ali-Mikhail-Haq family.
  • x is the support where each row corresponds to an observation.
  • theta is the parameter of the copula. The elements of theta must be greater than or equal to -1 for the Clayton family, greater than or equal to 1 for the Gumbel-Hougaard family, arbitrary for the Frank family, and greater than or equal to -1 and lower than 1 for the Ali-Mikhail-Haq family. Moreover, theta must be non-negative for dimensions greater than 2. theta must be a column vector with the same number of rows as x or be scalar.

Return values

  • y is the probability density of the copula at each row of x and corresponding parameter theta.

Examples

 
 
 x = [0.2:0.2:0.6; 0.2:0.2:0.6];
 theta = [1; 2];
 y = copulapdf ("Clayton", x, theta)
 
 
 y = copulapdf ("Gumbel", x, 2)

References

  1. Roger B. Nelsen. An Introduction to Copulas. Springer, New York, second edition, 2006.

See also: copulacdf, copularnd

Source Code: copulapdf

statistics-release-1.6.3/docs/copularnd.html000066400000000000000000000140401456127120000211330ustar00rootroot00000000000000 Statistics: copularnd

Function Reference: copularnd

Function File: r = copularnd (family, theta, n)
Function File: r = copularnd (family, theta, n, d)
Function File: r = copularnd ('t', theta, df, n)

Random arrays from the copula family distributions.

Arguments

  • family is the copula family name. Currently, family can be 'Gaussian' for the Gaussian family, 't' for the Student’s t family, or 'Clayton' for the Clayton family.
  • theta is the parameter of the copula. For the Gaussian and Student’s t copula, theta must be a correlation matrix. For bivariate copulas theta can also be a correlation coefficient. For the Clayton family, theta must be a vector with the same number of elements as samples to be generated or be scalar.
  • df is the degrees of freedom for the Student’s t family. df must be a vector with the same number of elements as samples to be generated or be scalar.
  • n is the number of rows of the matrix to be generated. n must be a non-negative integer and corresponds to the number of samples to be generated.
  • d is the number of columns of the matrix to be generated. d must be a positive integer and corresponds to the dimension of the copula.

Return values

  • r is a matrix of random samples from the copula with n samples of distribution dimension d.

Examples

 
 
 theta = 0.5;
 r = copularnd ("Gaussian", theta);
 
 
 theta = 0.5;
 df = 2;
 r = copularnd ("t", theta, df);
 
 
 theta = 0.5;
 n = 2;
 r = copularnd ("Clayton", theta, n);

References

  1. Roger B. Nelsen. An Introduction to Copulas. Springer, New York, second edition, 2006.

Source Code: copularnd

statistics-release-1.6.3/docs/correlation_test.html000066400000000000000000000167651456127120000225440ustar00rootroot00000000000000 Statistics: correlation_test

Function Reference: correlation_test

statistics: h = correlation_test (x, y)
statistics: [h, pval] = correlation_test (y, x)
statistics: [h, pval, stats] = correlation_test (y, x)
statistics: […] = correlation_test (y, x, Name, Value)

Perform a correlation coefficient test whether two samples x and y come from uncorrelated populations.

h = correlation_test (y, x) tests the null hypothesis that the two samples x and y come from uncorrelated populations. The result is h = 0 if the null hypothesis cannot be rejected at the 5% significance level, or h = 1 if the null hypothesis can be rejected at the 5% level. y and x must be vectors of equal length with finite real numbers.

The p-value of the test is returned in pval. stats is a structure with the following fields:

FieldValue
methodthe type of correlation coefficient used for the test
dfthe degrees of freedom (where applicable)
corrcoefthe correlation coefficient
statthe test’s statistic
distthe respective distribution for the test
altthe alternative hypothesis for the test

[…] = correlation_test (…, name, value) specifies one or more of the following name/value pairs:

NameValue
"alpha"the significance level. Default is 0.05.
"tail"a string specifying the alternative hypothesis
"both"corrcoef is not 0 (two-tailed, default)
"left"corrcoef is less than 0 (left-tailed)
"right"corrcoef is greater than 0 (right-tailed)
"method"a string specifying the correlation coefficient used for the test
"pearson"Pearson’s product moment correlation (Default)
"kendall"Kendall’s rank correlation tau
"spearman"Spearman’s rank correlation rho

See also: regression_ftest, regression_ttest

Source Code: correlation_test

statistics-release-1.6.3/docs/crosstab.html000066400000000000000000000105651456127120000207740ustar00rootroot00000000000000 Statistics: crosstab

Function Reference: crosstab

statistics: t = crosstab (x1, x2)
statistics: t = crosstab (x1, …, xn)
statistics: [t, chisq, p, labels] = crosstab (…)

Create a cross-tabulation (contingency table) t from data vectors.

The inputs x1, x2, ... xn must be vectors of equal length with a data type of numeric, logical, char, or string (cell array).

As additional return values crosstab returns the chi-square statistics chisq, its p-value p and a cell array labels, containing the labels of each input argument.

See also: grp2idx, tabulate

Source Code: crosstab

statistics-release-1.6.3/docs/crossval.html000066400000000000000000000146151456127120000210100ustar00rootroot00000000000000 Statistics: crossval

Function Reference: crossval

statistics: results = crossval (f, X, y)
statistics: results = crossval (f, X, y, name, value)

Perform cross validation on given data.

f should be a function that takes 4 inputs xtrain, ytrain, xtest, ytest, fits a model based on xtrain, ytrain, applies the fitted model to xtest, and returns a goodness of fit measure based on comparing the predicted and actual ytest. crossval returns an array containing the values returned by f for every cross-validation fold or resampling applied to the given data.

X should be an n by m matrix of predictor values

y should be an n by 1 vector of predicand values

Optional arguments may include name-value pairs as follows:

"KFold"

Divide set into k equal-size subsets, using each one successively for validation.

"HoldOut"

Divide set into two subsets, training and validation. If the value k is a fraction, that is the fraction of values put in the validation subset (by default k=0.1); if it is a positive integer, that is the number of values in the validation subset.

"LeaveOut"

Leave-one-out partition (each element is placed in its own subset). The value is ignored.

"Partition"

The value should be a cvpartition object.

"Given"

The value should be an n by 1 vector specifying in which partition to put each element.

"stratify"

The value should be an n by 1 vector containing class designations for the elements, in which case the "KFold" and "HoldOut" partitionings attempt to ensure each partition represents the classes proportionately.

"mcreps"

The value should be a positive integer specifying the number of times to resample based on different partitionings. Currently only works with the partition type "HoldOut".

Only one of "KFold", "HoldOut", "LeaveOut", "Given", "Partition" should be specified. If none is specified, the default is "KFold" with k = 10.

See also: cvpartition

Source Code: crossval

statistics-release-1.6.3/docs/datasample.html000066400000000000000000000121651456127120000212650ustar00rootroot00000000000000 Statistics: datasample

Function Reference: datasample

statistics: y = datasample (data, k)
statistics: y = datasample (data, k, dim)
statistics: y = datasample (…, Name, Value)
statistics: [y idcs] = datasample (…)

Randomly sample data.

Return k observations randomly sampled from data. data can be a vector or a matrix of any data. When data is a matrix or a n-dimensional array, the samples are the subarrays of size n - 1, taken along the dimension dim. The default value for dim is 1, that is the row vectors when sampling a matrix.

Output y is the returned sampled data. Optional output idcs is the vector of the indices to build y from data.

Additional options are set through pairs of parameter name and value. Available parameters are:

Replace

a logical value that can be true (default) or false: when set to true, datasample returns data sampled with replacement.

Weigths

a vector of positive numbers that sets the probability of each element. It must have the same size as data along dimension dim.

See also: rand, randi, randperm, randsample

Source Code: datasample

statistics-release-1.6.3/docs/dcov.html000066400000000000000000000161021456127120000201000ustar00rootroot00000000000000 Statistics: dcov

Function Reference: dcov

statistics: [dCor, dCov, dVarX, dVarY] = dcov (x, y)

Distance correlation, covariance and correlation statistics.

It returns the distance correlation (dCor) and the distance covariance (dCov) between x and y, the distance variance of x in (dVarX) and the distance variance of y in (dVarY).

x and y must have the same number of observations (rows) but they can have different number of dimensions (columns). Rows with missing values (NaN) in either x or y are omitted.

The Brownian covariance is the same as the distance covariance:

$$ cov_W (X, Y) = dCov(X, Y) $$

and thus Brownian correlation is the same as distance correlation.

See also: corr, cov

Source Code: dcov

Example: 1

  

 base=@(x) (x- min(x))./(max(x)-min(x));
 N = 5e2;
 x = randn (N,1); x = base (x);
 z = randn (N,1); z = base (z);
 # Linear relations
 cy = [1 0.55 0.3 0 -0.3 -0.55 -1];
 ly = x .* cy;
 ly(:,[1:3 5:end]) = base (ly(:,[1:3 5:end]));
 # Correlated Gaussian
 cz = 1 - abs (cy);
 gy = base ( ly + cz.*z);
 # Shapes
 sx      = repmat (x,1,7);
 sy      = zeros (size (ly));
 v       = 2 * rand (size(x,1),2) - 1;
 sx(:,1) = v(:,1); sy(:,1) = cos(2*pi*sx(:,1)) + 0.5*v(:,2).*exp(-sx(:,1).^2/0.5);
 R       =@(d) [cosd(d) sind(d); -sind(d) cosd(d)];
 tmp     = R(35) * v.';
 sx(:,2) = tmp(1,:); sy(:,2) = tmp(2,:);
 tmp     = R(45) * v.';
 sx(:,3) = tmp(1,:); sy(:,3) = tmp(2,:);
 sx(:,4) = v(:,1); sy(:,4) = sx(:,4).^2 + 0.5*v(:,2);
 sx(:,5) = v(:,1); sy(:,5) = 3*sign(v(:,2)).*(sx(:,5)).^2  + v(:,2);
 sx(:,6) = cos (2*pi*v(:,1)) + 0.5*(x-0.5);
 sy(:,6) = sin (2*pi*v(:,1)) + 0.5*(z-0.5);
 sx(:,7) = x + sign(v(:,1)); sy(:,7) = z + sign(v(:,2));
 sy      = base (sy);
 sx      = base (sx);
 # scaled shape
 sc  = 1/3;
 ssy = (sy-0.5) * sc + 0.5;
 n = size (ly,2);
 ym = 1.2;
 xm = 0.5;
 fmt={'horizontalalignment','center'};
 ff = "% .2f";
 figure (1)
 for i=1:n
   subplot(4,n,i);
   plot (x, gy(:,i), '.b');
   axis tight
   axis off
   text (xm,ym,sprintf (ff, dcov (x,gy(:,i))),fmt{:})

   subplot(4,n,i+n);
   plot (x, ly(:,i), '.b');
   axis tight
   axis off
   text (xm,ym,sprintf (ff, dcov (x,ly(:,i))),fmt{:})

   subplot(4,n,i+2*n);
   plot (sx(:,i), sy(:,i), '.b');
   axis tight
   axis off
   text (xm,ym,sprintf (ff, dcov (sx(:,i),sy(:,i))),fmt{:})
   v = axis ();

   subplot(4,n,i+3*n);
   plot (sx(:,i), ssy(:,i), '.b');
   axis (v)
   axis off
   text (xm,ym,sprintf (ff, dcov (sx(:,i),ssy(:,i))),fmt{:})
 endfor
plotted figure

statistics-release-1.6.3/docs/dendrogram.html000066400000000000000000000311741456127120000212750ustar00rootroot00000000000000 Statistics: dendrogram

Function Reference: dendrogram

statistics: dendrogram (tree)
statistics: dendrogram (tree, p)
statistics: dendrogram (tree, prop, val)
statistics: dendrogram (tree, p, prop, val )
statistics: h = dendrogram (…)
statistics: [h, t, perm] = dendrogram (…)

Plot a dendrogram of a hierarchical binary cluster tree.

Given tree, a hierarchical binary cluster tree as the output of linkage, plot a dendrogram of the tree. The number of leaves shown by the dendrogram plot is limited to p. The default value for p is 30. Set p to 0 to plot all leaves.

The optional outputs are h, t and perm:

  • h is a handle to the lines of the plot.
  • t is the vector with the numbers assigned to each leaf. Each element of t is a leaf of tree and its value is the number shown in the plot. When the dendrogram plot is collapsed, that is when the number of shown leaves p is inferior to the total number of leaves, a single leaf of the plot can represent more than one leaf of tree: in that case multiple elements of t share the same value, that is the same leaf of the plot. When the dendrogram plot is not collapsed, each leaf of the plot is the leaf of tree with the same number.
  • perm is the vector list of the leaves as ordered as in the plot.

Additional input properties can be specified by pairs of properties and values. Known properties are:

  • "Reorder" Reorder the leaves of the dendrogram plot using a numerical vector of size n, the number of leaves. When p is smaller than n, the reordering cannot break the p groups of leaves.
  • "Orientation" Change the orientation of the plot. Available values: top (default), bottom, left, right.
  • "CheckCrossing" Check if the lines of a reordered dendrogram cross each other. Available values: true (default), false.
  • "ColorThreshold" Not implemented.
  • "Labels" Use a char, string or cellstr array of size n to set the label for each leaf; the label is dispayed only for nodes with just one leaf.

See also: cluster, clusterdata, cophenet, inconsistent, linkage, pdist

Source Code: dendrogram

Example: 1

  

 ## simple dendrogram
 y = [4, 5; 2, 6; 3, 7; 8, 9; 1, 10];
 y(:,3) = 1:5;
 dendrogram (y);
 title ("simple dendrogram");
plotted figure

Example: 2

  

 ## another simple dendrogram
 v = 2 * rand (30, 1) - 1;
 d = abs (bsxfun (@minus, v(:, 1), v(:, 1)'));
 y = linkage (squareform (d, "tovector"));
 dendrogram (y);
 title ("another simple dendrogram");
plotted figure

Example: 3

  

 ## collapsed tree, find all the leaves of node 5
 X = randn (60, 2);
 D = pdist (X);
 y = linkage (D, "average");
 subplot (2, 1, 1);
 title ("original tree");
 dendrogram (y, 0);
 subplot (2, 1, 2);
 title ("collapsed tree");
 [~, t] = dendrogram (y, 20);
 find(t == 5)

ans = 5
plotted figure

Example: 4

  

 ## optimal leaf order
 X = randn (30, 2);
 D = pdist (X);
 y = linkage (D, "average");
 order = optimalleaforder (y, D);
 subplot (2, 1, 1);
 title ("original leaf order");
 dendrogram (y);
 subplot (2, 1, 2);
 title ("optimal leaf order");
 dendrogram (y, "Reorder", order);
plotted figure

Example: 5

  

 ## horizontal orientation and labels
 X = randn (8, 2);
 D = pdist (X);
 L = ["Snow White"; "Doc"; "Grumpy"; "Happy"; "Sleepy"; "Bashful"; ...
      "Sneezy"; "Dopey"];
 y = linkage (D, "average");
 dendrogram (y, "Orientation", "left", "Labels", L);
 title ("horizontal orientation and labels");
plotted figure

statistics-release-1.6.3/docs/ecdf.html000066400000000000000000000225141456127120000200520ustar00rootroot00000000000000 Statistics: ecdf

Function Reference: ecdf

statistics: [f, x] = ecdf (y)
statistics: [f, x, flo, fup] = ecdf (y)
statistics: ecdf (…)
statistics: ecdf (ax, …)
statistics: […] = ecdf (y, name, value, …)
statistics: […] = ecdf (ax, y, name, value, …)

Empirical (Kaplan-Meier) cumulative distribution function.

[f, x] = ecdf (y) calculates the Kaplan-Meier estimate of the cumulative distribution function (cdf), also known as the empirical cdf. y is a vector of data values. f is a vector of values of the empirical cdf evaluated at x.

[f, x, flo, fup] = ecdf (y) also returns lower and upper confidence bounds for the cdf. These bounds are calculated using Greenwood’s formula, and are not simultaneous confidence bounds.

ecdf (…) without output arguments produces a plot of the empirical cdf.

ecdf (ax, …) plots into existing axes ax.

[…] = ecdf (y, name, value, …) specifies additional parameter name/value pairs chosen from the following:

namevalue
"censoring"A boolean vector of the same size as Y that is 1 for observations that are right-censored and 0 for observations that are observed exactly. Default is all observations observed exactly.
"frequency"A vector of the same size as Y containing non-negative integer counts. The jth element of this vector gives the number of times the jth element of Y was observed. Default is 1 observation per Y element.
"alpha"A value alpha between 0 and 1 specifying the significance level. Default is 0.05 for 5% significance.
"function"The type of function returned as the F output argument, chosen from "cdf" (the default), "survivor", or "cumulative hazard".
"bounds"Either "on" to include bounds or "off" (the default) to omit them. Used only for plotting.

Type demo ecdf to see examples of usage.

See also: cdfplot, ecdfhist

Source Code: ecdf

Example: 1

  

 y = exprnd (10, 50, 1);    ## random failure times are exponential(10)
 d = exprnd (20, 50, 1);    ## drop-out times are exponential(20)
 t = min (y, d);            ## we observe the minimum of these times
 censored = (y > d);        ## we also observe whether the subject failed

 ## Calculate and plot the empirical cdf and confidence bounds
 [f, x, flo, fup] = ecdf (t, "censoring", censored);
 stairs (x, f);
 hold on;
 stairs (x, flo, "r:"); stairs (x, fup, "r:");

 ## Superimpose a plot of the known true cdf
 xx = 0:.1:max (t); yy = 1 - exp (-xx / 10); plot (xx, yy, "g-");
 hold off;
plotted figure

Example: 2

  

 R = wblrnd (100, 2, 100, 1);
 ecdf (R, "Function", "survivor", "Alpha", 0.01, "Bounds", "on");
 hold on
 x = 1:1:250;
 wblsurv = 1 - cdf ("weibull", x, 100, 2);
 plot (x, wblsurv, "g-", "LineWidth", 2)
 legend ("Empirical survivor function", "Lower confidence bound", ...
         "Upper confidence bound", "Weibull survivor function", ...
         "Location", "northeast");
 hold off
plotted figure

statistics-release-1.6.3/docs/einstein.html000066400000000000000000000212201456127120000207600ustar00rootroot00000000000000 Statistics: einstein

Function Reference: einstein

statistics: einstein ()
statistics: tiles = einstein (a, b)
statistics: [tiles, rhat] = einstein (a, b)
statistics: [tiles, rhat, that] = einstein (a, b)
statistics: [tiles, rhat, that, shat] = einstein (a, b)
statistics: [tiles, rhat, that, shat, phat] = einstein (a, b)
statistics: [tiles, rhat, that, shat, phat, fhat] = einstein (a, b)

Plots the tiling of the basic clusters of einstein tiles.

Scalars a and b define the shape of the einstein tile. See Smith et al (2023) for details: https://arxiv.org/abs/2303.10798

  • tiles is a structure containing the coordinates of the einstein tiles that are tiled on the plot. Each field contains the tile coordinates of the corresponding clusters.
    • tiles.rhat contains the reflected einstein tiles
    • tiles.that contains the three-hat shells
    • tiles.shat contains the single-hat clusters
    • tiles.phat contains the paired-hat clusters
    • tiles.fhat contains the fylfot clusters
  • rhat contains the coordinates of the first reflected tile
  • that contains the coordinates of the first three-hat shell
  • shat contains the coordinates of the first single-hat cluster
  • phat contains the coordinates of the first paired-hat cluster
  • fhat contains the coordinates of the first fylfot cluster

Source Code: einstein

Example: 1

  

 einstein (0.4, 0.6)
plotted figure

Example: 2

  

 einstein (0.2, 0.5)
plotted figure

Example: 3

  

 einstein (0.6, 0.1)
plotted figure

statistics-release-1.6.3/docs/evalclusters.html000066400000000000000000000226261456127120000216710ustar00rootroot00000000000000 Statistics: evalclusters

Function Reference: evalclusters

statistics: eva = evalclusters (x, clust, criterion)
statistics: eva = evalclusters (…, Name, Value)

Create a clustering evaluation object to find the optimal number of clusters.

evalclusters creates a clustering evaluation object to evaluate the optimal number of clusters for data x, using criterion criterion. The input data x is a matrix with n observations of p variables. The evaluation criterion criterion is one of the following:

CalinskiHarabasz

to create a CalinskiHarabaszEvaluation object.

DaviesBouldin

to create a DaviesBouldinEvaluation object.

gap

to create a GapEvaluation object.

silhouette

to create a SilhouetteEvaluation object.

The clustering algorithm clust is one of the following:

kmeans

to cluster the data using kmeans with EmptyAction set to singleton and Replicates set to 5.

linkage

to cluster the data using clusterdata with linkage set to Ward.

gmdistribution

to cluster the data using fitgmdist with SharedCov set to true and Replicates set to 5.

If the criterion is CalinskiHarabasz, DaviesBouldin, or silhouette, clust can also be a function handle to a function of the form c = clust(x, k), where x is the input data, k the number of clusters to evaluate and c the clustering result. The clustering result can be either an array of size n with k different integer values, or a matrix of size n by k with a likelihood value assigned to each one of the n observations for each one of the k clusters. In the latter case, each observation is assigned to the cluster with the higher value. If the criterion is CalinskiHarabasz, DaviesBouldin, or silhouette, clust can also be a matrix of size n by k, where k is the number of proposed clustering solutions, so that each column of clust is a clustering solution.

In addition to the obligatory x, clust and criterion inputs there is a number of optional arguments, specified as pairs of Name and Value options. The known Name arguments are:

KList

a vector of positive integer numbers, that is the cluster sizes to evaluate. This option is necessary, unless clust is a matrix of proposed clustering solutions.

Distance

a distance metric as accepted by the chosen clust. It can be the name of the distance metric as a string or a function handle. When criterion is silhouette, it can be a vector as created by function pdist. Valid distance metric strings are: sqEuclidean (default), Euclidean, cityblock, cosine, correlation, Hamming, Jaccard. Only used by silhouette and gap evaluation.

ClusterPriors

the prior probabilities of each cluster, which can be either empirical (default), or equal. When empirical the silhouette value is the average of the silhouette values of all points; when equal the silhouette value is the average of the average silhouette value of each cluster. Only used by silhouette evaluation.

B

the number of reference datasets generated from the reference distribution. Only used by gap evaluation.

ReferenceDistribution

the reference distribution used to create the reference data. It can be PCA (default) for a distribution based on the principal components of X, or uniform for a uniform distribution based on the range of the observed data. PCA is currently not implemented. Only used by gap evaluation.

SearchMethod

the method for selecting the optimal value with a gap evaluation. It can be either globalMaxSE (default) for selecting the smallest number of clusters which is inside the standard error of the maximum gap value, or firstMaxSE for selecting the first number of clusters which is inside the standard error of the following cluster number. Only used by gap evaluation.

Output eva is a clustering evaluation object.

See also: CalinskiHarabaszEvaluation, DaviesBouldinEvaluation, GapEvaluation, SilhouetteEvaluation

Source Code: evalclusters

statistics-release-1.6.3/docs/evcdf.html000066400000000000000000000177771456127120000202570ustar00rootroot00000000000000 Statistics: evcdf

Function Reference: evcdf

statistics: p = evcdf (x)
statistics: p = evcdf (x, mu)
statistics: p = evcdf (x, mu, sigma)
statistics: p = evcdf (…, "upper")
statistics: [p, plo, pup] = evcdf (x, mu, sigma, pcov)
statistics: [p, plo, pup] = evcdf (x, mu, sigma, pcov, alpha)
statistics: [p, plo, pup] = evcdf (…, "upper")

Extreme value cumulative distribution function (CDF).

For each element of x, compute the cumulative distribution function (CDF) of the extreme value distribution (also known as the Gumbel or the type I generalized extreme value distribution) at the values in x with location parameter mu and scale parameter sigma. The size of p is the common size of x, mu and sigma. A scalar input functions as a constant matrix of the same size as the other inputs.

Default values are mu = 0 and sigma = 1.

When called with three output arguments, i.e. [p, plo, pup], evcdf computes the confidence bounds for p when the input parameters mu and sigma are estimates. In such case, pcov, a 2×2 matrix containing the covariance matrix of the estimated parameters, is necessary. Optionally, alpha, which has a default value of 0.05, specifies the 100 * (1 - alpha) percent confidence bounds. plo and pup are arrays of the same size as p containing the lower and upper confidence bounds.

[…] = evcdf (…, "upper") computes the upper tail probability of the extreme value distribution with parameters x0 and gamma, at the values in x.

The Gumbel distribution is used to model the distribution of the maximum (or the minimum) of a number of samples of various distributions. This version is suitable for modeling minima. For modeling maxima, use the alternative Gumbel CDF, gumbelcdf.

Further information about the Gumbel distribution can be found at https://en.wikipedia.org/wiki/Gumbel_distribution

See also: evinv, evpdf, evrnd, evfit, evlike, evstat, gumbelcdf

Source Code: evcdf

Example: 1

  

 ## Plot various CDFs from the extreme value distribution
 x = -10:0.01:10;
 p1 = evcdf (x, 0.5, 2);
 p2 = evcdf (x, 1.0, 2);
 p3 = evcdf (x, 1.5, 3);
 p4 = evcdf (x, 3.0, 4);
 plot (x, p1, "-b", x, p2, "-g", x, p3, "-r", x, p4, "-c")
 grid on
 legend ({"μ = 0.5, σ = 2", "μ = 1.0, σ = 2", ...
          "μ = 1.5, σ = 3", "μ = 3.0, σ = 4"}, "location", "southeast")
 title ("Extreme value CDF")
 xlabel ("values in x")
 ylabel ("probability")
plotted figure

statistics-release-1.6.3/docs/evfit.html000066400000000000000000000232061456127120000202650ustar00rootroot00000000000000 Statistics: evfit

Function Reference: evfit

statistics: paramhat = evfit (x)
statistics: [paramhat, paramci] = evfit (x)
statistics: [paramhat, paramci] = evfit (x, alpha)
statistics: […] = evfit (x, alpha, censor)
statistics: […] = evfit (x, alpha, censor, freq)
statistics: […] = evfit (x, alpha, censor, freq, options)

Estimate parameters and confidence intervals for the extreme value distribution.

paramhat = evfit (x) returns the maximum likelihood estimates of the parameters of the extreme value distribution (also known as the Gumbel or the type I generalized extreme value distribution) given the data in x. paramhat(1) is the location parameter, mu, and paramhat(2) is the scale parameter, sigma.

[paramhat, paramci] = evfit (x) returns the 95% confidence intervals for the parameter estimates.

[…] = evfit (x, alpha) also returns the 100 * (1 - alpha) percent confidence intervals for the parameter estimates. By default, the optional argument alpha is 0.05 corresponding to 95% confidence intervals. Pass in [] for alpha to use the default values.

[…] = evfit (x, alpha, censor) accepts a boolean vector, censor, of the same size as x with 1s for observations that are right-censored and 0s for observations that are observed exactly. By default, or if left empty, censor = zeros (size (x)).

[…] = evfit (x, alpha, censor, freq) accepts a frequency vector, freq, of the same size as x. freq typically contains integer frequencies for the corresponding elements in x, but it can contain any non-integer non-negative values. By default, or if left empty, freq = ones (size (x)).

[…] = evfit (…, options) specifies control parameters for the iterative algorithm used to compute the maximum likelihood estimates. options is a structure with the following field and its default value:

  • options.TolX = 1e-6

The Gumbel distribution is used to model the distribution of the maximum (or the minimum) of a number of samples of various distributions. This version is suitable for modeling minima. For modeling maxima, use the alternative Gumbel fitting function, gumbelfit.

Further information about the Gumbel distribution can be found at https://en.wikipedia.org/wiki/Gumbel_distribution

See also: evcdf, evinv, evpdf, evrnd, evlike, evstat, gumbelfit

Source Code: evfit

Example: 1

  

 ## Sample 3 populations from different extreme value distibutions
 rand ("seed", 1);    # for reproducibility
 r1 = evrnd (2, 5, 400, 1);
 rand ("seed", 12);    # for reproducibility
 r2 = evrnd (-5, 3, 400, 1);
 rand ("seed", 13);    # for reproducibility
 r3 = evrnd (14, 8, 400, 1);
 r = [r1, r2, r3];

 ## Plot them normalized and fix their colors
 hist (r, 25, 0.4);
 h = findobj (gca, "Type", "patch");
 set (h(1), "facecolor", "c");
 set (h(2), "facecolor", "g");
 set (h(3), "facecolor", "r");
 ylim ([0, 0.28])
 xlim ([-30, 30]);
 hold on

 ## Estimate their MU and SIGMA parameters
 mu_sigmaA = evfit (r(:,1));
 mu_sigmaB = evfit (r(:,2));
 mu_sigmaC = evfit (r(:,3));

 ## Plot their estimated PDFs
 x = [min(r(:)):max(r(:))];
 y = evpdf (x, mu_sigmaA(1), mu_sigmaA(2));
 plot (x, y, "-pr");
 y = evpdf (x, mu_sigmaB(1), mu_sigmaB(2));
 plot (x, y, "-sg");
 y = evpdf (x, mu_sigmaC(1), mu_sigmaC(2));
 plot (x, y, "-^c");
 legend ({"Normalized HIST of sample 1 with μ=2 and σ=5", ...
          "Normalized HIST of sample 2 with μ=-5 and σ=3", ...
          "Normalized HIST of sample 3 with μ=14 and σ=8", ...
          sprintf("PDF for sample 1 with estimated μ=%0.2f and σ=%0.2f", ...
                  mu_sigmaA(1), mu_sigmaA(2)), ...
          sprintf("PDF for sample 2 with estimated μ=%0.2f and σ=%0.2f", ...
                  mu_sigmaB(1), mu_sigmaB(2)), ...
          sprintf("PDF for sample 3 with estimated μ=%0.2f and σ=%0.2f", ...
                  mu_sigmaC(1), mu_sigmaC(2))})
 title ("Three population samples from different extreme value distibutions")
 hold off
plotted figure

statistics-release-1.6.3/docs/evinv.html000066400000000000000000000170571456127120000203060ustar00rootroot00000000000000 Statistics: evinv

Function Reference: evinv

statistics: x = evinv (p)
statistics: x = evinv (p, mu)
statistics: x = evinv (p, mu, sigma)
statistics: [x, xlo, xup] = evinv (p, mu, sigma, pcov)
statistics: [x, xlo, xup] = evinv (p, mu, sigma, pcov, alpha)

Inverse of the extreme value cumulative distribution function (iCDF).

For each element of p, compute the quantile (the inverse of the CDF) of the extreme value distribution (also known as the Gumbel or the type I generalized extreme value distribution) with location parameter mu and scale parameter sigma. The size of x is the common size of p, mu and sigma. A scalar input functions as a constant matrix of the same size as the other inputs.

Default values are mu = 0 and sigma = 1.

When called with three output arguments, i.e. [x, xlo, xup], evinv computes the confidence bounds for x when the input parameters mu and sigma are estimates. In such case, pcov, a 2×2 matrix containing the covariance matrix of the estimated parameters, is necessary. Optionally, alpha, which has a default value of 0.05, specifies the 100 * (1 - alpha) percent confidence bounds. xlo and xup are arrays of the same size as x containing the lower and upper confidence bounds.

The Gumbel distribution is used to model the distribution of the maximum (or the minimum) of a number of samples of various distributions. This version is suitable for modeling minima. For modeling maxima, use the alternative Gumbel iCDF, gumbelinv.

Further information about the Gumbel distribution can be found at https://en.wikipedia.org/wiki/Gumbel_distribution

See also: evcdf, evpdf, evrnd, evfit, evlike, evstat, gumbelinv

Source Code: evinv

Example: 1

  

 ## Plot various iCDFs from the extreme value distribution
 p = 0.001:0.001:0.999;
 x1 = evinv (p, 0.5, 2);
 x2 = evinv (p, 1.0, 2);
 x3 = evinv (p, 1.5, 3);
 x4 = evinv (p, 3.0, 4);
 plot (p, x1, "-b", p, x2, "-g", p, x3, "-r", p, x4, "-c")
 grid on
 ylim ([-10, 10])
 legend ({"μ = 0.5, σ = 2", "μ = 1.0, σ = 2", ...
          "μ = 1.5, σ = 3", "μ = 3.0, σ = 4"}, "location", "northwest")
 title ("Extreme value iCDF")
 xlabel ("probability")
 ylabel ("values in x")
plotted figure

statistics-release-1.6.3/docs/evlike.html000066400000000000000000000144771456127120000204410ustar00rootroot00000000000000 Statistics: evlike

Function Reference: evlike

statistics: nlogL = evlike (params, x)
statistics: [nlogL, acov] = evlike (params, x)
statistics: […] = evlike (params, x, censor)
statistics: […] = evlike (params, x, censor, freq)

Negative log-likelihood for the extreme value distribution.

nlogL = evlike (params, x) returns the negative log likelihood of the data in x corresponding to the extreme value distribution (also known as the Gumbel or the type I generalized extreme value distribution) with (1) location parameter mu and (2) scale parameter sigma given in the two-element vector params.

[nlogL, acov] = evlike (params, x) also returns the inverse of Fisher’s information matrix, acov. If the input parameter values in params are the maximum likelihood estimates, the diagonal elements of acov are their asymptotic variances.

[…] = evlike (params, x, censor) accepts a boolean vector, censor, of the same size as x with 1s for observations that are right-censored and 0s for observations that are observed exactly. By default, or if left empty, censor = zeros (size (x)).

[…] = evlike (params, x, censor, freq) accepts a frequency vector, freq, of the same size as x. freq typically contains integer frequencies for the corresponding elements in x, but it can contain any non-integer non-negative values. By default, or if left empty, freq = ones (size (x)).

The Gumbel distribution is used to model the distribution of the maximum (or the minimum) of a number of samples of various distributions. This version is suitable for modeling minima. For modeling maxima, use the alternative Gumbel likelihood function, gumbellike.

Further information about the Gumbel distribution can be found at https://en.wikipedia.org/wiki/Gumbel_distribution

See also: evcdf, evinv, evpdf, evrnd, evfit, evstat, gumbellike

Source Code: evlike

statistics-release-1.6.3/docs/evpdf.html000066400000000000000000000150161456127120000202540ustar00rootroot00000000000000 Statistics: evpdf

Function Reference: evpdf

statistics: y = evpdf (x)
statistics: y = evpdf (x, mu)
statistics: y = evpdf (x, mu, sigma)

Extreme value probability density function (PDF).

For each element of x, compute the probability density function (PDF) of the extreme value distribution (also known as the Gumbel or the type I generalized extreme value distribution) with location parameter mu and scale parameter sigma. The size of y is the common size of x, mu and sigma. A scalar input functions as a constant matrix of the same size as the other inputs.

Default values are mu = 0 and sigma = 1.

The Gumbel distribution is used to model the distribution of the maximum (or the minimum) of a number of samples of various distributions. This version is suitable for modeling minima. For modeling maxima, use the alternative Gumbel iCDF, gumbelinv.

Further information about the Gumbel distribution can be found at https://en.wikipedia.org/wiki/Gumbel_distribution

See also: evcdf, evinv, evrnd, evfit, evlike, evstat, gumbelpdf

Source Code: evpdf

Example: 1

  

 ## Plot various PDFs from the Extreme value distribution
 x = -10:0.001:10;
 y1 = evpdf (x, 0.5, 2);
 y2 = evpdf (x, 1.0, 2);
 y3 = evpdf (x, 1.5, 3);
 y4 = evpdf (x, 3.0, 4);
 plot (x, y1, "-b", x, y2, "-g", x, y3, "-r", x, y4, "-c")
 grid on
 ylim ([0, 0.2])
 legend ({"μ = 0.5, σ = 2", "μ = 1.0, σ = 2", ...
          "μ = 1.5, σ = 3", "μ = 3.0, σ = 4"}, "location", "northeast")
 title ("Extreme value PDF")
 xlabel ("values in x")
 ylabel ("density")
plotted figure

statistics-release-1.6.3/docs/evrnd.html000066400000000000000000000130421456127120000202630ustar00rootroot00000000000000 Statistics: evrnd

Function Reference: evrnd

statistics: r = evrnd (mu, sigma)
statistics: r = evrnd (mu, sigma, rows)
statistics: r = evrnd (mu, sigma, rows, cols, …)
statistics: r = evrnd (mu, sigma, [sz])

Random arrays from the extreme value distribution.

r = evrnd (mu, sigma) returns an array of random numbers chosen from the extreme value distribution (also known as the Gumbel or the type I generalized extreme value distribution) with location parameter mu and scale parameter sigma. The size of r is the common size of mu and sigma. A scalar input functions as a constant matrix of the same size as the other inputs.

When called with a single size argument, evrnd returns a square matrix with the dimension specified. When called with more than one scalar argument, the first two arguments are taken as the number of rows and columns and any further arguments specify additional matrix dimensions. The size may also be specified with a row vector of dimensions, sz.

The Gumbel distribution is used to model the distribution of the maximum (or the minimum) of a number of samples of various distributions. This version is suitable for modeling minima. For modeling maxima, use the alternative Gumbel iCDF, gumbelinv.

Further information about the Gumbel distribution can be found at https://en.wikipedia.org/wiki/Gumbel_distribution

See also: evcdf, evinv, evpdf, evfit, evlike, evstat

Source Code: evrnd

statistics-release-1.6.3/docs/evstat.html000066400000000000000000000117041456127120000204560ustar00rootroot00000000000000 Statistics: evstat

Function Reference: evstat

statistics: [m, v] = evstat (mu, sigma)

Compute statistics of the extreme value distribution.

[m, v] = evstat (mu, sigma) returns the mean and variance of the extreme value distribution (also known as the Gumbel or the type I generalized extreme value distribution) with location parameter mu and scale parameter sigma.

The size of m (mean) and v (variance) is the common size of the input arguments. A scalar input functions as a constant matrix of the same size as the other inputs.

The type 1 extreme value distribution is also known as the Gumbel distribution. This version is suitable for modeling minima. The mirror image of this distribution can be used to model maxima by negating x. If y has a Weibull distribution, then x = log (y) has the type 1 extreme value distribution.

Further information about the Gumbel distribution can be found at https://en.wikipedia.org/wiki/Gumbel_distribution

See also: evcdf, evinv, evpdf, evrnd, evfit, evlike

Source Code: evstat

statistics-release-1.6.3/docs/expcdf.html000066400000000000000000000170741456127120000204270ustar00rootroot00000000000000 Statistics: expcdf

Function Reference: expcdf

statistics: p = expcdf (x)
statistics: p = expcdf (x, mu)
statistics: p = expcdf (…, "upper")
statistics: [p, plo, pup] = expcdf (x, mu, pcov)
statistics: [p, plo, pup] = expcdf (x, mu, pcov, alpha)
statistics: [p, plo, pup] = expcdf (…, "upper")

Exponential cumulative distribution function (CDF).

For each element of x, compute the cumulative distribution function (CDF) of the exponential distribution with mean parameter mu. The size of p is the common size of x and mu. A scalar input functions as a constant matrix of the same size as the other inputs.

Default value is mu = 1.

A common alternative parameterization of the exponential distribution is to use the parameter λ defined as the mean number of events in an interval as opposed to the parameter μ, which is the mean wait time for an event to occur. λ and μ are reciprocals, i.e. μ = 1 / λ.

When called with three output arguments, i.e. [p, plo, pup], expcdf computes the confidence bounds for p when the input parameter mu is an estimate. In such case, pcov, a scalar value with the variance of the estimated parameter mu, is necessary. Optionally, alpha, which has a default value of 0.05, specifies the 100 * (1 - alpha) percent confidence bounds. plo and pup are arrays of the same size as p containing the lower and upper confidence bounds.

[…] = expcdf (…, "upper") computes the upper tail probability of the exponential distribution with parameter mu, at the values in x.

Further information about the exponential distribution can be found at https://en.wikipedia.org/wiki/Exponential_distribution

See also: expinv, exppdf, exprnd, expfit, explike, expstat

Source Code: expcdf

Example: 1

  

 ## Plot various CDFs from the exponential distribution
 x = 0:0.01:5;
 p1 = expcdf (x, 2/3);
 p2 = expcdf (x, 1.0);
 p3 = expcdf (x, 2.0);
 plot (x, p1, "-b", x, p2, "-g", x, p3, "-r")
 grid on
 legend ({"μ = 2/3", "μ = 1", "μ = 2"}, "location", "southeast")
 title ("Exponential CDF")
 xlabel ("values in x")
 ylabel ("probability")
plotted figure

statistics-release-1.6.3/docs/expfit.html000066400000000000000000000232251456127120000204500ustar00rootroot00000000000000 Statistics: expfit

Function Reference: expfit

statistics: muhat = expfit (x)
statistics: [muhat, muci] = expfit (x)
statistics: [muhat, muci] = expfit (x, alpha)
statistics: […] = expfit (x, alpha, censor)
statistics: […] = expfit (x, alpha, censor, freq)

Estimate mean and confidence intervals for the exponential distribution.

muhat = expfit (x) returns the maximum likelihood estimate of the mean parameter, muhat, of the exponential distribution given the data in x. x is expected to be a non-negative vector. If x is an array, the mean will be computed for each column of x. If any elements of x are NaN, that vector’s mean will be returned as NaN.

[muhat, muci] = expfit (x) returns the 95% confidence intervals for the parameter estimate. If x is a vector, muci is a two element column vector. If x is an array, each column of data will have a confidence interval returned as a two-row array.

[…] = evfit (x, alpha) also returns the 100 * (1 - alpha) percent confidence intervals for the parameter estimates. By default, the optional argument alpha is 0.05 corresponding to 95% confidence intervals. Pass in [] for alpha to use the default values. Any invalid values for alpha will return NaN for both CI bounds.

[…] = expfit (x, alpha, censor) accepts a logical or numeric array, censor, of the same size as x with 1s for observations that are right-censored and 0s for observations that are observed exactly. Any non-zero elements are regarded as 1s. By default, or if left empty, censor = zeros (size (x)).

[…] = expfit (x, alpha, censor, freq) accepts a frequency array, freq, of the same size as x. freq typically contains integer frequencies for the corresponding elements in x, but it can contain any non-integer non-negative values. By default, or if left empty, freq = ones (size (x)).

Matlab incompatibility: Matlab’s expfit produces unpredictable results for some cases with higher dimensions (specifically 1 x m x n x ... arrays). Octave’s implementation allows for n×D arrays, consistently performing calculations on individual column vectors. Additionally, censor and freq can be used with arrays of any size, whereas Matlab only allows their use when x is a vector.

A common alternative parameterization of the exponential distribution is to use the parameter λ defined as the mean number of events in an interval as opposed to the parameter μ, which is the mean wait time for an event to occur. λ and μ are reciprocals, i.e. μ = 1 / λ.

Further information about the exponential distribution can be found at https://en.wikipedia.org/wiki/Exponential_distribution

See also: expcdf, expinv, explpdf, exprnd, explike, expstat

Source Code: expfit

Example: 1

  

 ## Sample 3 populations from 3 different exponential distibutions
 rande ("seed", 1);   # for reproducibility
 r1 = exprnd (2, 4000, 1);
 rande ("seed", 2);   # for reproducibility
 r2 = exprnd (5, 4000, 1);
 rande ("seed", 3);   # for reproducibility
 r3 = exprnd (12, 4000, 1);
 r = [r1, r2, r3];

 ## Plot them normalized and fix their colors
 hist (r, 48, 0.52);
 h = findobj (gca, "Type", "patch");
 set (h(1), "facecolor", "c");
 set (h(2), "facecolor", "g");
 set (h(3), "facecolor", "r");
 hold on

 ## Estimate their mu parameter
 muhat = expfit (r);

 ## Plot their estimated PDFs
 x = [0:max(r(:))];
 y = exppdf (x, muhat(1));
 plot (x, y, "-pr");
 y = exppdf (x, muhat(2));
 plot (x, y, "-sg");
 y = exppdf (x, muhat(3));
 plot (x, y, "-^c");
 ylim ([0, 0.6])
 xlim ([0, 40])
 legend ({"Normalized HIST of sample 1 with μ=2", ...
          "Normalized HIST of sample 2 with μ=5", ...
          "Normalized HIST of sample 3 with μ=12", ...
          sprintf("PDF for sample 1 with estimated μ=%0.2f", muhat(1)), ...
          sprintf("PDF for sample 2 with estimated μ=%0.2f", muhat(2)), ...
          sprintf("PDF for sample 3 with estimated μ=%0.2f", muhat(3))})
 title ("Three population samples from different exponential distibutions")
 hold off
plotted figure

statistics-release-1.6.3/docs/expinv.html000066400000000000000000000162241456127120000204630ustar00rootroot00000000000000 Statistics: expinv

Function Reference: expinv

statistics: x = expinv (p)
statistics: x = expinv (p, mu)
statistics: [x, xlo, xup] = expinv (p, mu, pcov)
statistics: [x, xlo, xup] = expinv (p, mu, pcov, alpha)

Inverse of the exponential cumulative distribution function (iCDF).

For each element of p, compute the quantile (the inverse of the CDF) of the exponential distribution with mean mu. The size of x is the common size of p and mu. A scalar input functions as a constant matrix of the same size as the other inputs.

Default value is mu = 1.

A common alternative parameterization of the exponential distribution is to use the parameter λ defined as the mean number of events in an interval as opposed to the parameter μ, which is the mean wait time for an event to occur. λ and μ are reciprocals, i.e. μ = 1 / λ.

When called with three output arguments, i.e. [x, xlo, xup], expinv computes the confidence bounds for x when the input parameter mu is an estimate. In such case, pcov, a scalar value with the variance of the estimated parameter mu, is necessary. Optionally, alpha, which has a default value of 0.05, specifies the 100 * (1 - alpha) percent confidence bounds. xlo and xup are arrays of the same size as x containing the lower and upper confidence bounds.

Further information about the exponential distribution can be found at https://en.wikipedia.org/wiki/Exponential_distribution

See also: expcdf, exppdf, exprnd, expfit, explike, expstat

Source Code: expinv

Example: 1

  

 ## Plot various iCDFs from the exponential distribution
 p = 0.001:0.001:0.999;
 x1 = expinv (p, 2/3);
 x2 = expinv (p, 1.0);
 x3 = expinv (p, 2.0);
 plot (p, x1, "-b", p, x2, "-g", p, x3, "-r")
 grid on
 ylim ([0, 5])
 legend ({"μ = 2/3", "μ = 1", "μ = 2"}, "location", "northwest")
 title ("Exponential iCDF")
 xlabel ("probability")
 ylabel ("values in x")
plotted figure

statistics-release-1.6.3/docs/explike.html000066400000000000000000000143301456127120000206070ustar00rootroot00000000000000 Statistics: explike

Function Reference: explike

statistics: nlogL = explike (mu, x)
statistics: [nlogL, avar] = explike (mu, x)
statistics: […] = explike (mu, x, censor)
statistics: […] = explike (mu, x, censor, freq)

Negative log-likelihood for the exponential distribution.

nlogL = explike (mu, x) returns the negative log likelihood of the data in x corresponding to the exponential distribution with mean parameter mu. x must be a vector of non-negative values, otherwise NaN is returned.

[nlogL, avar] = explike (mu, x) also returns the inverse of Fisher’s information matrix, avar. If the input mean parameter, mu, is the maximum likelihood estimate, avar is its asymptotic variance.

[…] = explike (mu, x, censor) accepts a boolean vector, censor, of the same size as x with 1s for observations that are right-censored and 0s for observations that are observed exactly. By default, or if left empty, censor = zeros (size (x)).

[…] = explike (mu, x, censor, freq) accepts a frequency vector, freq, of the same size as x. freq typically contains integer frequencies for the corresponding elements in x, but it can contain any non-integer non-negative values. By default, or if left empty, freq = ones (size (x)).

A common alternative parameterization of the exponential distribution is to use the parameter λ defined as the mean number of events in an interval as opposed to the parameter μ, which is the mean wait time for an event to occur. λ and μ are reciprocals, i.e. μ = 1 / λ.

Further information about the exponential distribution can be found at https://en.wikipedia.org/wiki/Exponential_distribution

See also: expcdf, expinv, exppdf, exprnd, expfit, expstat

Source Code: explike

statistics-release-1.6.3/docs/exppdf.html000066400000000000000000000143101456127120000204320ustar00rootroot00000000000000 Statistics: exppdf

Function Reference: exppdf

statistics: y = exppdf (x)
statistics: y = exppdf (x, mu)

Exponential probability density function (PDF).

For each element of x, compute the probability density function (PDF) of the exponential distribution with mean parameter mu. The size of y is the common size of x and mu. A scalar input functions as a constant matrix of the same size as the other inputs.

Default value for mu = 1.

A common alternative parameterization of the exponential distribution is to use the parameter λ defined as the mean number of events in an interval as opposed to the parameter μ, which is the mean wait time for an event to occur. λ and μ are reciprocals, i.e. μ = 1 / λ.

Further information about the exponential distribution can be found at https://en.wikipedia.org/wiki/Exponential_distribution

See also: expcdf, expinv, exprnd, expfit, explike, expstat

Source Code: exppdf

Example: 1

  

 ## Plot various PDFs from the exponential distribution
 x = 0:0.01:5;
 y1 = exppdf (x, 2/3);
 y2 = exppdf (x, 1.0);
 y3 = exppdf (x, 2.0);
 plot (x, y1, "-b", x, y2, "-g", x, y3, "-r")
 grid on
 ylim ([0, 1.5])
 legend ({"μ = 2/3", "μ = 1", "μ = 2"}, "location", "northeast")
 title ("Exponential PDF")
 xlabel ("values in x")
 ylabel ("density")
plotted figure

statistics-release-1.6.3/docs/exprnd.html000066400000000000000000000125121456127120000204460ustar00rootroot00000000000000 Statistics: exprnd

Function Reference: exprnd

statistics: r = exprnd (mu)
statistics: r = exprnd (mu, rows)
statistics: r = exprnd (mu, rows, cols, …)
statistics: r = exprnd (mu, [sz])

Random arrays from the exponential distribution.

r = exprnd (mu) returns an array of random numbers chosen from the exponential distribution with mean parameter mu. The size of r is the size of mu.

When called with a single size argument, exprnd returns a square matrix with the dimension specified. When called with more than one scalar argument, the first two arguments are taken as the number of rows and columns and any further arguments specify additional matrix dimensions. The size may also be specified with a row vector of dimensions, sz.

A common alternative parameterization of the exponential distribution is to use the parameter λ defined as the mean number of events in an interval as opposed to the parameter μ, which is the mean wait time for an event to occur. λ and μ are reciprocals, i.e. μ = 1 / λ.

Further information about the exponential distribution can be found at https://en.wikipedia.org/wiki/Exponential_distribution

See also: expcdf, expinv, exppdf, expfit, explike, expstat

Source Code: exprnd

statistics-release-1.6.3/docs/expstat.html000066400000000000000000000113301456127120000206330ustar00rootroot00000000000000 Statistics: expstat

Function Reference: expstat

statistics: [m, v] = expstat (mu)

Compute statistics of the exponential distribution.

[m, v] = expstat (mu) returns the mean and variance of the exponential distribution with mean parameter mu.

The size of m (mean) and v (variance) is the same size of the input argument.

A common alternative parameterization of the exponential distribution is to use the parameter λ defined as the mean number of events in an interval as opposed to the parameter μ, which is the mean wait time for an event to occur. λ and μ are reciprocals, i.e. μ = 1 / λ.

Further information about the exponential distribution can be found at https://en.wikipedia.org/wiki/Exponential_distribution

See also: expcdf, expinv, exppdf, exprnd, expfit, explike

Source Code: expstat

statistics-release-1.6.3/docs/fcdf.html000066400000000000000000000143431456127120000200540ustar00rootroot00000000000000 Statistics: fcdf

Function Reference: fcdf

statistics: p = fcdf (x, df1, df2)
statistics: p = fcdf (x, df1, df2, "upper")

F-cumulative distribution function (CDF).

For each element of x, compute the cumulative distribution function (CDF) of the F-distribution with df1 and df2 degrees of freedom. The size of p is the common size of x, df1, and df2. A scalar input functions as a constant matrix of the same size as the other inputs.

p = fcdf (x, df1, df2, "upper") computes the upper tail probability of the F-distribution with df1 and df2 degrees of freedom, at the values in x.

Further information about the F-distribution can be found at https://en.wikipedia.org/wiki/F-distribution

See also: finv, fpdf, frnd, fstat

Source Code: fcdf

Example: 1

  

 ## Plot various CDFs from the F distribution
 x = 0.01:0.01:4;
 p1 = fcdf (x, 1, 2);
 p2 = fcdf (x, 2, 1);
 p3 = fcdf (x, 5, 2);
 p4 = fcdf (x, 10, 1);
 p5 = fcdf (x, 100, 100);
 plot (x, p1, "-b", x, p2, "-g", x, p3, "-r", x, p4, "-c", x, p5, "-m")
 grid on
 legend ({"df1 = 1, df2 = 2", "df1 = 2, df2 = 1", ...
          "df1 = 5, df2 = 2", "df1 = 10, df2 = 1", ...
          "df1 = 100, df2 = 100"}, "location", "southeast")
 title ("F CDF")
 xlabel ("values in x")
 ylabel ("probability")
plotted figure

statistics-release-1.6.3/docs/ff2n.html000066400000000000000000000077661456127120000200200ustar00rootroot00000000000000 Statistics: ff2n

Function Reference: ff2n

statistics: dFF2 = ff2n (n)

Two-level full factorial design.

dFF2 = ff2n (n) gives factor settings dFF2 for a two-level full factorial design with n factors. dFF2 is m-by-n, where m is the number of treatments in the full-factorial design. Each row of dFF2 corresponds to a single treatment. Each column contains the settings for a single factor, with values of 0 and 1 for the two levels.

@seealso {fullfact

Source Code: ff2n

statistics-release-1.6.3/docs/fillmissing.html000066400000000000000000000307441456127120000214750ustar00rootroot00000000000000 Statistics: fillmissing

Function Reference: fillmissing

statistics: B = fillmissing (A, 'constant', v)
statistics: B = fillmissing (A, method)
statistics: B = fillmissing (A, move_method, window_size)
statistics: B = fillmissing (A, fill_function, window_size)
statistics: B = fillmissing (…, dim)
statistics: B = fillmissing (…, PropertyName, PropertyValue)
statistics: [B, idx] = fillmissing (…)

Replace missing entries of array A either with values in v or as determined by other specified methods. ’missing’ values are determined by the data type of A as identified by the function @ref{ismissing}, curently defined as:

  • NaN: single, double
  • " " (white space): char
  • {""} (white space in cell): string cells.

A can be a numeric scalar or array, a character vector or array, or a cell array of character vectors (a.k.a. string cells).

v can be a scalar or an array containing values for replacing the missing values in A with a compatible data type for isertion into A. The shape of v must be a scalar or an array with number of elements in v equal to the number of elements orthoganal to the operating dimension. E.g., if size(A) = [3 5 4], operating along dim = 2 requires v to contain either 1 or 3x4=12 elements.

If requested, the optional output idx will contain a logical array the same shape as A indicating with 1’s which locations in A were filled.

Alternate Input Arguments and Values:

  • method - replace missing values with:
    next
    previous
    nearest

    next, previous, or nearest non-missing value (nearest defaults to next when equidistant as determined by SamplePoints.)

    linear

    linear interpolation of neigboring, non-missing values

    spline

    piecewise cubic spline interpolation of neigboring, non-missing values

    pchip

    ’shape preserving’ piecewise cubic spline interposaliton of neighboring, non-missing values

  • move_method - moving window calculated replacement values:
    movmean
    movmedian

    moving average or median using a window determined by window_size. window_size must be either a positive scalar value or a two element positive vector of sizes [nb, na] measured in the same units as SamplePoints. For scalar values, the window is centered on the missing element and includes all data points within a distance of half of window_size on either side of the window center point. Note that for compatability, when using a scalar value, the backward window limit is inclusive and the forward limit is exclusive. If a two-element window_size vector is specified, the window includes all points within a distance of nb backward and na forward from the current element at the window center (both limits inclusive).

  • fill_function - custom method specified as a function handle. The supplied fill function must accept three inputs in the following order for each missing gap in the data:
    A_values -

    elements of A within the window on either side of the gap as determined by window_size. (Note these elements can include missing values from other nearby gaps.)

    A_locs -

    locations of the reference data, A_values, in terms of the default or specified SamplePoints.

    gap_locs -

    location of the gap data points that need to be filled in terms of the default or specified SamplePoints.

    The supplied function must return a scalar or vector with the same number of elements in gap_locs. The required window_size parameter follows similar rules as for the moving average and median methods described above, with the two exceptions that (1) each gap is processed as a single element, rather than gap elements being processed individually, and (2) the window extended on either side of the gap has inclusive endpoints regardless of how window_size is specified.

  • dim - specify a dimension for vector operation (default = first non-singeton dimension)
  • PropertyName-PropertyValue pairs
    SamplePoints

    PropertyValue is a vector of sample point values representing the sorted and unique x-axis values of the data in A. If unspecified, the default is assumed to be the vector [1 : size (A, dim)]. The values in SamplePoints will affect methods and properties that rely on the effective distance between data points in A, such as interpolants and moving window functions where the window_size specified for moving window functions is measured relative to the SamplePoints.

    EndValues

    Apply a separate handling method for missing values at the front or back of the array. PropertyValue can be:

    • A constant scalar or array with the same shape requirments as v.
    • none - Do not fill end gap values.
    • extrap - Use the same procedure as method to fill the end gap values.
    • Any valid method listed above except for movmean, movmedian, and fill_function. Those methods can only be applied to end gap values with extrap.
    MissingLocations

    PropertyValue must be a logical array the same size as A indicating locations of known missing data with a value of true. (cannot be combined with MaxGap)

    MaxGap

    PropertyValue is a numeric scalar indicating the maximum gap length to fill, and assumes the same distance scale as the sample points. Gap length is calculated by the difference in locations of the sample points on either side of the gap, and gaps larger than MaxGap are ignored by fillmissing. (cannot be combined with MissingLocations)

Compatibility Notes:

  • Numerical and logical inputs for A and v may be specified in any combination. The output will be the same class as A, with the v converted to that data type for filling. Only single and double have defined ’missing’ values, so except for when the missinglocations option specifies the missing value identification of logical and other numeric data types, the output will always be B = A with idx = false(size(A)).
  • All interpolation methods can be individually applied to EndValues.
  • MATLAB’s fill_function method currently has several inconsistencies with the other methods (tested against version 2022a), and Octave’s implementation has chosen the following consistent behavior over compatibility: (1) a column full of missing data is considered part of EndValues, (2) such columns are then excluded from fill_function processing because the moving window is always empty. (3) operation in dimensions higher than 2 perform identically to operations in dims 1 and 2, most notable on vectors.
  • Method "makima" is not yet implemented in interp1, which is used by fillmissing. Attempting to call this method will produce an error until the method is implemented in interp1.

See also: ismissing, rmmissing, standardizeMissing

Source Code: fillmissing

statistics-release-1.6.3/docs/finv.html000066400000000000000000000135471456127120000201210ustar00rootroot00000000000000 Statistics: finv

Function Reference: finv

statistics: x = finv (p, df1, df2)

Inverse of the F-cumulative distribution function (iCDF).

For each element of p, compute the quantile (the inverse of the CDF) of the F-distribution with df1 and df2 degrees of freedom. The size of x is the common size of p, df1, and df2. A scalar input functions as a constant matrix of the same size as the other inputs.

Further information about the F-distribution can be found at https://en.wikipedia.org/wiki/F-distribution

See also: fcdf, fpdf, frnd, fstat

Source Code: finv

Example: 1

  

 ## Plot various iCDFs from the F distribution
 p = 0.001:0.001:0.999;
 x1 = finv (p, 1, 1);
 x2 = finv (p, 2, 1);
 x3 = finv (p, 5, 2);
 x4 = finv (p, 10, 1);
 x5 = finv (p, 100, 100);
 plot (p, x1, "-b", p, x2, "-g", p, x3, "-r", p, x4, "-c", p, x5, "-m")
 grid on
 ylim ([0, 4])
 legend ({"df1 = 1, df2 = 2", "df1 = 2, df2 = 1", ...
          "df1 = 5, df2 = 2", "df1 = 10, df2 = 1", ...
          "df1 = 100, df2 = 100"}, "location", "northwest")
 title ("F iCDF")
 xlabel ("probability")
 ylabel ("values in x")
plotted figure

statistics-release-1.6.3/docs/fishertest.html000066400000000000000000000201161456127120000213250ustar00rootroot00000000000000 Statistics: fishertest

Function Reference: fishertest

statistics: h = fishertest (x)
statistics: h = fishertest (x, param1, value1, …)
statistics: [h, pval] = fishertest (…)
statistics: [h, pval, stats] = fishertest (…)

Fisher’s exact test.

h = fishertest (x) performs Fisher’s exact test on a 2×2 contingency table given in matrix x. This is a test of the hypothesis that there are no non-random associations between the two 2-level categorical variables in x. fishertest returns the result of the tested hypothsis in h. h = 0 indicates that the null hypothesis (of no association) cannot be rejected at the 5% significance level. h = 1 indicates that the null hypothesis can be rejected at the 5% level. x must contain only non-negative integers. Use the crostab function to generate the contingency table from samples of two categorical variables. Fisher’s exact test is not suitable when all integers in x are very large. Use can use the Chi-square test in this case.

[h, pval] = fishertest (x) returns the p-value in pval. That is the probability of observing the given result, or one more extreme, by chance if the null hypothesis is true. Small values of pval cast doubt on the validity of the null hypothesis.

[p, pval, stats] = fishertest (…) returns the structure stats with the following fields:

OddsRatio– the odds ratio
ConfidenceInterval– the asymptotic confidence interval for the odds ratio. If any of the four entries in the contingency table x is zero, the confidence interval will not be computed, and [-Inf Inf] will be displayed.

[…] = fishertest (…, name, value, …) specifies one or more of the following name/value pairs:

NameValue
"alpha"the significance level. Default is 0.05.
"tail"a string specifying the alternative hypothesis
"both"odds ratio not equal to 1, indicating association between two variables (two-tailed test, default)
"left"odds ratio greater than 1 (right-tailed test)
"right"odds ratio is less than 1 (left-tailed test)

See also: crosstab, chi2test, mcnemar_test, ztest2

Source Code: fishertest

Example: 1

  

 ## A Fisher's exact test example

 x = [3, 1; 1, 3]
 [h, p, stats] = fishertest(x)

x =

   3   1
   1   3

h = 0
p = 0.4857
stats =

  scalar structure containing the fields:

    OddsRatio = 9
    ConfidenceInterval =

         0.3666   220.9270
statistics-release-1.6.3/docs/fitcknn.html000066400000000000000000000364511456127120000206120ustar00rootroot00000000000000 Statistics: fitcknn

Function Reference: fitcknn

statistics: obj = fitcknn (X, Y)
statistics: obj = fitcknn (…, name, value)

Fit a k-Nearest Neighbor classification model.

obj = fitcknn (X, Y) returns a k-Nearest Neighbor classification model, obj, with X being the predictor data, and Y the class labels of observations in X.

  • X must be a N×P numeric matrix of input data where rows correspond to observations and columns correspond to features or variables. X will be used to train the kNN model.
  • Y is N×1 matrix or cell matrix containing the class labels of corresponding predictor data in X. Y can contain any type of categorical data. Y must have same numbers of Rows as X.

obj = fitcknn (…, name, value) returns a k-Nearest Neighbor classification model with additional options specified by Name-Value pair arguments listed below.

NameValue
"PredictorNames"A cell array of character vectors specifying the predictor variable names. The variable names are assumed to be in the same order as they appear in the training data X.
"ResponseName"A character vector specifying the name of the response variable.
"ClassNames"A cell array of character vectors specifying the names of the classes in the training data Y.
"BreakTies"Tie-breaking algorithm used by predict when multiple classes have the same smallest cost. By default, ties occur when multiple classes have the same number of nearest points among the k nearest neighbors. The available options are specified by the following character arrays:
"smallest"This is the default and it favors the class with the smallest index among the tied groups, i.e. the one that appears first in the training labelled data.
"nearest"This favors the class with the nearest neighbor among the tied groups, i.e. the class with the closest member point according to the distance metric used.
"nearest"This randomly picks one class among the tied groups.
"BucketSize"The maximum number of data points in the leaf node of the Kd-tree and it must be a positive integer. By default, it is 50. This argument is meaningful only when the selected search method is "kdtree".
"Cost"A N×R numeric matrix containing misclassification cost for the corresponding instances in X where R is the number of unique categories in Y. If an instance is correctly classified into its category the cost is calculated to be 1, If not then 0. cost matrix can be altered use obj.cost = somecost. default value cost = ones(rows(X),numel(unique(Y))).
"Prior"A numeric vector specifying the prior probabilities for each class. The order of the elements in Prior corresponds to the order of the classes in ClassNames.
"NumNeighbors"A positive integer value specifying the number of nearest neighbors to be found in the kNN search. By default, it is 1.
"Exponent"A positive scalar (usually an integer) specifying the Minkowski distance exponent. This argument is only valid when the selected distance metric is "minkowski". By default it is 2.
"Scale"A nonnegative numeric vector specifying the scale parameters for the standardized Euclidean distance. The vector length must be equal to the number of columns in X. This argument is only valid when the selected distance metric is "seuclidean", in which case each coordinate of X is scaled by the corresponding element of "scale", as is each query point in Y. By default, the scale parameter is the standard deviation of each coordinate in X. If a variable in X is constant, i.e. zero variance, this value is forced to 1 to avoid division by zero. This is the equivalent of this variable not being standardized.
"Cov"A square matrix with the same number of columns as X specifying the covariance matrix for computing the mahalanobis distance. This must be a positive definite matrix matching. This argument is only valid when the selected distance metric is "mahalanobis".
"Distance"is the distance metric used by knnsearch as specified below:
"euclidean"Euclidean distance.
"seuclidean"standardized Euclidean distance. Each coordinate difference between the rows in X and the query matrix Y is scaled by dividing by the corresponding element of the standard deviation computed from X. To specify a different scaling, use the "Scale" name-value argument.
"cityblock"City block distance.
"chebychev"Chebychev distance (maximum coordinate difference).
"minkowski"Minkowski distance. The default exponent is 2. To specify a different exponent, use the "P" name-value argument.
"mahalanobis"Mahalanobis distance, computed using a positive definite covariance matrix. To change the value of the covariance matrix, use the "Cov" name-value argument.
"cosine"Cosine distance.
"correlation"One minus the sample linear correlation between observations (treated as sequences of values).
"spearman"One minus the sample Spearman’s rank correlation between observations (treated as sequences of values).
"hamming"Hamming distance, which is the percentage of coordinates that differ.
"jaccard"One minus the Jaccard coefficient, which is the percentage of nonzero coordinates that differ.
@distfunCustom distance function handle. A distance function of the form function D2 = distfun (XI, YI), where XI is a 1×P vector containing a single observation in P-dimensional space, YI is an N×P matrix containing an arbitrary number of observations in the same P-dimensional space, and D2 is an N×P vector of distances, where (D2k) is the distance between observations XI and (YIk,:).
"DistanceWeight"A distance weighting function, specified either as a function handle, which accepts a matrix of nonnegative distances and returns a matrix the same size containing nonnegative distance weights, or one of the following values: "equal", which corresponds to no weighting; "inverse", which corresponds to a weight equal to 1/distance; "squaredinverse", which corresponds to a weight equal to 1/distance^2.
"IncludeTies"A boolean flag to indicate if the returned values should contain the indices that have same distance as the K^th neighbor. When false, knnsearch chooses the observation with the smallest index among the observations that have the same distance from a query point. When true, knnsearch includes all nearest neighbors whose distances are equal to the K^th smallest distance in the output arguments. To specify K, use the "K" name-value pair argument.
"NSMethod"is the nearest neighbor search method used by knnsearch as specified below.
"kdtree"Creates and uses a Kd-tree to find nearest neighbors. "kdtree" is the default value when the number of columns in X is less than or equal to 10, X is not sparse, and the distance metric is "euclidean", "cityblock", "manhattan", "chebychev", or "minkowski". Otherwise, the default value is "exhaustive". This argument is only valid when the distance metric is one of the four aforementioned metrics.
"exhaustive"Uses the exhaustive search algorithm by computing the distance values from all the points in X to each point in Y.

See also: ClassificationKNN, knnsearch, rangesearch, pdist2

Source Code: fitcknn

statistics-release-1.6.3/docs/fitgmdist.html000066400000000000000000000235671456127120000211540ustar00rootroot00000000000000 Statistics: fitgmdist

Function Reference: fitgmdist

statistics: GMdist = fitgmdist (data, k, param1, value1, …)

Fit a Gaussian mixture model with k components to data. Each row of data is a data sample. Each column is a variable.

Optional parameters are:

  • "start": Initialization conditions. Possible values are:
    • "randSample" (default) Takes means uniformly from rows of data.
    • "plus" Use k-means++ to initialize means.
    • "cluster" Performs an initial clustering with 10% of the data.
    • vector A vector whose length is the number of rows in data, and whose values are 1 to k specify the components each row is initially allocated to. The mean, variance, and weight of each component is calculated from that.
    • structure A structure with fields mu, Sigma and ComponentProportion.

    For "randSample", "plus", and "cluster", the initial variance of each component is the variance of the entire data sample.

  • "Replicates": Number of random restarts to perform.
  • "RegularizationValue" or "Regularize": A small number added to the diagonal entries of the covariance to prevent singular covariances.
  • "SharedCovariance" or "SharedCov" (logical). True if all components must share the same variance, to reduce the number of free parameters
  • "CovarianceType" or "CovType" (string). Possible values are:
    • "full" (default) Allow arbitrary covariance matrices.
    • "diagonal" Force covariances to be diagonal, to reduce the number of free parameters.
  • "Options": A structure with all of the following fields:
    • MaxIter Maximum number of EM iterations (default 100).
    • TolFun Threshold increase in likelihood to terminate EM (default 1e-6).
    • Display Possible values are:
      • "off" (default): Display nothing.
      • "final": Display the total number of iterations and likelihood once the execution completes.
      • "iter": Display the number of iteration and likelihood after each iteration.
  • "Weight": A column vector or N×2 matrix. The first column consists of non-negative weights given to the samples. If these are all integers, this is equivalent to specifying weight(i) copies of row i of data, but potentially faster. If a row of data is used to represent samples that are similar but not identical, then the second column of weight indicates the variance of those original samples. Specifically, in the EM algorithm, the contribution of row i towards the variance is set to at least weight(i,2), to prevent spurious components with zero variance.

See also: gmdistribution, kmeans

Source Code: fitgmdist

Example: 1

  

 ## Generate a two-cluster problem
 C1 = randn (100, 2) + 2;
 C2 = randn (100, 2) - 2;
 data = [C1; C2];

 ## Perform clustering
 GMModel = fitgmdist (data, 2);

 ## Plot the result
 figure
 [heights, bins] = hist3([C1; C2]);
 [xx, yy] = meshgrid(bins{1}, bins{2});
 bbins = [xx(:), yy(:)];
 contour (reshape (GMModel.pdf (bbins), size (heights)));
plotted figure

Example: 2

  

 Angle_Theta = [ 30 + 10 * randn(1, 10),  60 + 10 * randn(1, 10) ]';
 nbOrientations = 2;
 initial_orientations = [38.0; 18.0];
 initial_weights = ones (1, nbOrientations) / nbOrientations;
 initial_Sigma = 10 * ones (1, 1, nbOrientations);
 start = struct ("mu", initial_orientations, "Sigma", initial_Sigma, ...
                 "ComponentProportion", initial_weights);
 GMModel_Theta = fitgmdist (Angle_Theta, nbOrientations, "Start", start , ...
                            "RegularizationValue", 0.0001)

Gaussian mixture distribution with 2 components in 1 dimension(s)
Clust 1: weight 0.70508
	Mean: 52.5568 
	Variance:156.17
Clust 2: weight 0.29492
	Mean: 20.0915 
	Variance:1.8459
AIC=164.862 BIC=169.841 NLogL=77.431 Iter=5 Cged=1 Reg=0.0001
statistics-release-1.6.3/docs/fitlm.html000066400000000000000000000335511456127120000202670ustar00rootroot00000000000000 Statistics: fitlm

Function Reference: fitlm

statistics: tab = fitlm (X, y)
statistics: tab = fitlm (X, y, name, value)
statistics: tab = fitlm (X, y, modelspec)
statistics: tab = fitlm (X, y, modelspec, name, value)
statistics: [tab] = fitlm (…)
statistics: [tab, stats] = fitlm (…)
statistics: [tab, stats] = fitlm (…)

Regress the continuous outcome (i.e. dependent variable) y on continuous or categorical predictors (i.e. independent variables) X by minimizing the sum-of-squared residuals. Unless requested otherwise, fitlm prints the model formula, the regression coefficients (i.e. parameters/contrasts) and an ANOVA table. Note that unlike anovan, fitlm treats all factors as continuous by default. A bootstrap resampling variant of this function, bootlm, is available in the statistics-resampling package and has similar usage.

X must be a column major matrix or cell array consisting of the predictors. A constant term (intercept) should not be included in X - it is automatically added to the model. y must be a column vector corresponding to the outcome variable. modelspec can specified as one of the following:

  • "constant" : model contains only a constant (intercept) term.
  • "linear" (default) : model contains an intercept and linear term for each predictor.
  • "interactions" : model contains an intercept, linear term for each predictor and all products of pairs of distinct predictors.
  • "full" : model contains an intercept, linear term for each predictor and all combinations of the predictors.
  • a matrix of term definitions : an t-by-(N+1) matrix specifying terms in a model, where t is the number of terms, N is the number of predictor variables, and +1 accounts for the outcome variable. The outcome variable is the last column in the terms matrix and must be a column of zeros. An intercept must be specified in the first row of the terms matrix and must be a row of zeros.

fitlm can take a number of optional parameters as name-value pairs.

[…] = fitlm (..., "CategoricalVars", categorical)

  • categorical is a vector of indices indicating which of the columns (i.e. variables) in X should be treated as categorical predictors rather than as continuous predictors.

fitlm also accepts optional anovan parameters as name-value pairs (except for the "model" parameter). The accepted parameter names from anovan and their default values in fitlm are:

  • CONTRASTS : "treatment"
  • SSTYPE: 2
  • ALPHA: 0.05
  • DISPLAY: "on"
  • WEIGHTS: [] (empty)
  • RANDOM: [] (empty)
  • CONTINUOUS: [1:N]
  • VARNAMES: [] (empty)

Type ’help anovan’ to find out more about what these options do.

fitlm can return up to two output arguments:

[tab] = fitlm (…) returns a cell array containing a table of model parameters

[tab, stats] = fitlm (…) returns a structure containing additional statistics, including degrees of freedom and effect sizes for each term in the linear model, the design matrix, the variance-covariance matrix, (weighted) model residuals, and the mean squared error. The columns of stats.coeffs (from left-to-right) report the model coefficients, standard errors, lower and upper 100*(1-alpha)% confidence interval bounds, t-statistics, and p-values relating to the contrasts. The number appended to each term name in stats.coeffnames corresponds to the column number in the relevant contrast matrix for that factor. The stats structure can be used as input for multcompare. Note that if the model contains a continuous variable and you wish to use the STATS output as input to multcompare, then the model needs to be refit with the "contrast" parameter set to a sum-to-zero contrast coding scheme, e.g."simple".

See also: anovan, multcompare

Source Code: fitlm

Example: 1

  

 y =  [ 8.706 10.362 11.552  6.941 10.983 10.092  6.421 14.943 15.931 ...
        22.968 18.590 16.567 15.944 21.637 14.492 17.965 18.851 22.891 ...
        22.028 16.884 17.252 18.325 25.435 19.141 21.238 22.196 18.038 ...
        22.628 31.163 26.053 24.419 32.145 28.966 30.207 29.142 33.212 ...
        25.694 ]';
 X = [1 1 1 1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5]';

 [TAB,STATS] = fitlm (X,y,"linear","CategoricalVars",1,"display","on");


MODEL FORMULA (based on Wilkinson's notation):

Y ~ 1 + X1

MODEL PARAMETERS (contrasts for the fixed effects)

Parameter               Estimate        SE  Lower.CI  Upper.CI        t Prob>|t|
--------------------------------------------------------------------------------
(Intercept)                   10      1.02      7.93      12.1     9.83    <.001 
X1_1                           8      1.64      4.66      11.3     4.87    <.001 
X1_2                           9      1.44      6.07      11.9     6.25    <.001 
X1_3                          11      1.49      7.97        14     7.38    <.001 
X1_4                          19       1.4      16.2      21.8    13.58    <.001 

ANOVA TABLE (Type II sums-of-squares):

Source                   Sum Sq.    d.f.    Mean Sq.  R Sq.            F  Prob>F
--------------------------------------------------------------------------------
X1                        1561.3       4      390.33  0.855        47.10   <.001 
Error                     265.17      32      8.2866
Total                     1826.5      36
plotted figure

Example: 2

  

 popcorn = [5.5, 4.5, 3.5; 5.5, 4.5, 4.0; 6.0, 4.0, 3.0; ...
            6.5, 5.0, 4.0; 7.0, 5.5, 5.0; 7.0, 5.0, 4.5];
 brands = {'Gourmet', 'National', 'Generic'; ...
           'Gourmet', 'National', 'Generic'; ...
           'Gourmet', 'National', 'Generic'; ...
           'Gourmet', 'National', 'Generic'; ...
           'Gourmet', 'National', 'Generic'; ...
           'Gourmet', 'National', 'Generic'};
 popper = {'oil', 'oil', 'oil'; 'oil', 'oil', 'oil'; 'oil', 'oil', 'oil'; ...
           'air', 'air', 'air'; 'air', 'air', 'air'; 'air', 'air', 'air'};

 [TAB, STATS] = fitlm ({brands(:),popper(:)},popcorn(:),"interactions",...
                          "CategoricalVars",[1,2],"display","on");


MODEL FORMULA (based on Wilkinson's notation):

Y ~ 1 + X1 + X2 + X1:X2

MODEL PARAMETERS (contrasts for the fixed effects)

Parameter               Estimate        SE  Lower.CI  Upper.CI        t Prob>|t|
--------------------------------------------------------------------------------
(Intercept)                 5.67     0.215       5.2      6.14    26.34    <.001 
X1_1                       -1.33     0.304        -2     -0.67    -4.38    <.001 
X1_2                       -2.17     0.304     -2.83      -1.5    -7.12    <.001 
X2_1                        1.17     0.304     0.504      1.83     3.83     .002 
X1:X2_1                   -0.333      0.43     -1.27     0.604    -0.77     .454 
X1:X2_2                   -0.167      0.43      -1.1     0.771    -0.39     .705 

ANOVA TABLE (Type II sums-of-squares):

Source                   Sum Sq.    d.f.    Mean Sq.  R Sq.            F  Prob>F
--------------------------------------------------------------------------------
X1                         15.75       2       7.875  0.904        56.70   <.001 
X2                           4.5       1         4.5  0.730        32.40   <.001 
X1*X2                   0.083333       2    0.041667  0.048         0.30    .746 
Error                     1.6667      12     0.13889
Total                         22      17
plotted figure

statistics-release-1.6.3/docs/fitrgam.html000066400000000000000000000271111456127120000206000ustar00rootroot00000000000000 Statistics: fitrgam

Function Reference: fitrgam

statistics: obj = fitrgam (X, Y)
statistics: obj = fitrgam (X, Y, name, value)

Fit a Generalised Additive Model (GAM) for regression.

obj = fitrgam (X, Y) returns an object of class RegressionGAM, with matrix X containing the predictor data and vector Y containing the continuous response data.

  • X must be a N×P numeric matrix of input data where rows correspond to observations and columns correspond to features or variables. X will be used to train the GAM model.
  • Y must be N×1 numeric vector containing the response data corresponding to the predictor data in X. Y must have same number of rows as X.

obj = fitrgam (…, name, value) returns an object of class RegressionGAM with additional properties specified by Name-Value pair arguments listed below.

NameValue
"predictors"Predictor Variable names, specified as a row vector cell of strings with the same length as the columns in X. If omitted, the program will generate default variable names (x1, x2, ..., xn) for each column in X.
"responsename"Response Variable Name, specified as a string. If omitted, the default value is "Y".
"formula"a model specification given as a string in the form "Y ~ terms" where Y represents the reponse variable and terms the predictor variables. The formula can be used to specify a subset of variables for training model. For example: "Y ~ x1 + x2 + x3 + x4 + x1:x2 + x2:x3" specifies four linear terms for the first four columns of for predictor data, and x1:x2 and x2:x3 specify the two interaction terms for 1st-2nd and 3rd-4th columns respectively. Only these terms will be used for training the model, but X must have at least as many columns as referenced in the formula. If Predictor Variable names have been defined, then the terms in the formula must reference to those. When "formula" is specified, all terms used for training the model are referenced in the IntMatrix field of the obj class object as a matrix containing the column indexes for each term including both the predictors and the interactions used.
"interactions"a logical matrix, a positive integer scalar, or the string "all" for defining the interactions between predictor variables. When given a logical matrix, it must have the same number of columns as X and each row corresponds to a different interaction term combining the predictors indexed as true. Each interaction term is appended as a column vector after the available predictor column in X. When "all" is defined, then all possible combinations of interactions are appended in X before training. At the moment, parsing a positive integer has the same effect as the "all" option. When "interactions" is specified, only the interaction terms appended to X are referenced in the IntMatrix field of the obj class object.
"knots"a scalar or a row vector with the same columns as X. It defines the knots for fitting a polynomial when training the GAM. As a scalar, it is expanded to a row vector. The default value is 5, hence expanded to ones (1, columns (X)) * 5. You can parse a row vector with different number of knots for each predictor variable to be fitted with, although not recommended.
"order"a scalar or a row vector with the same columns as X. It defines the order of the polynomial when training the GAM. As a scalar, it is expanded to a row vector. The default values is 3, hence expanded to ones (1, columns (X)) * 3. You can parse a row vector with different number of polynomial order for each predictor variable to be fitted with, although not recommended.
"dof"a scalar or a row vector with the same columns as X. It defines the degrees of freedom for fitting a polynomial when training the GAM. As a scalar, it is expanded to a row vector. The default value is 8, hence expanded to ones (1, columns (X)) * 8. You can parse a row vector with different degrees of freedom for each predictor variable to be fitted with, although not recommended.
"tol"a positive scalar to set the tolerance for covergence during training. By defaul, it is set to 1e-3.

You can parse either a "formula" or an "interactions" optional parameter. Parsing both parameters will result an error. Accordingly, you can only pass up to two parameters among "knots", "order", and "dof" to define the required polynomial for training the GAM model.

See also: RegressionGAM, regress, regress_gp

Source Code: fitrgam

Example: 1

  

 # Train a RegressionGAM Model for synthetic values

 f1 = @(x) cos (3 *x);
 f2 = @(x) x .^ 3;

 # generate x1 and x2 for f1 and f2
 x1 = 2 * rand (50, 1) - 1;
 x2 = 2 * rand (50, 1) - 1;

 # calculate y
 y = f1(x1) + f2(x2);

 # add noise
 y = y + y .* 0.2 .* rand (50,1);
 X = [x1, x2];

 # create an object
 a = fitrgam (X, y, "tol", 1e-3)

a =

  RegressionGAM object with properties:

            BaseModel: [1x1 struct]
                  DoF: [1x2 double]
              Formula: [0x0 double]
            IntMatrix: [0x0 double]
         Interactions: [0x0 double]
                Knots: [1x2 double]
            ModelwInt: [0x0 double]
      NumObservations: [1x1 double]
        NumPredictors: [1x1 double]
                Order: [1x2 double]
       PredictorNames: [1x2 cell]
         ResponseName: Y
             RowsUsed: [50x1 double]
                  Tol: [1x1 double]
                    X: [50x2 double]
                    Y: [50x1 double]
statistics-release-1.6.3/docs/fpdf.html000066400000000000000000000135121456127120000200660ustar00rootroot00000000000000 Statistics: fpdf

Function Reference: fpdf

statistics: y = fpdf (x, df1, df2)

F-probability density function (PDF).

For each element of x, compute the probability density function (PDF) of the F-distribution with df1 and df2 degrees of freedom. The size of y is the common size of x, df1, and df2. A scalar input functions as a constant matrix of the same size as the other inputs.

Further information about the F-distribution can be found at https://en.wikipedia.org/wiki/F-distribution

See also: fcdf, finv, frnd, fstat

Source Code: fpdf

Example: 1

  

 ## Plot various PDFs from the F distribution
 x = 0.01:0.01:4;
 y1 = fpdf (x, 1, 1);
 y2 = fpdf (x, 2, 1);
 y3 = fpdf (x, 5, 2);
 y4 = fpdf (x, 10, 1);
 y5 = fpdf (x, 100, 100);
 plot (x, y1, "-b", x, y2, "-g", x, y3, "-r", x, y4, "-c", x, y5, "-m")
 grid on
 ylim ([0, 2.5])
 legend ({"df1 = 1, df2 = 2", "df1 = 2, df2 = 1", ...
          "df1 = 5, df2 = 2", "df1 = 10, df2 = 1", ...
          "df1 = 100, df2 = 100"}, "location", "northeast")
 title ("F PDF")
 xlabel ("values in x")
 ylabel ("density")
plotted figure

statistics-release-1.6.3/docs/friedman.html000066400000000000000000000176521456127120000207450ustar00rootroot00000000000000 Statistics: friedman

Function Reference: friedman

statistics: p = friedman (x)
statistics: p = friedman (x, reps)
statistics: p = friedman (x, reps, displayopt)
statistics: [p, atab] = friedman (…)
statistics: [p, atab, stats] = friedman (…)

Performs the nonparametric Friedman’s test to compare column effects in a two-way layout. friedman tests the null hypothesis that the column effects are all the same against the alternative that they are not all the same.

friedman requires one up to three input arguments:

  • x contains the data and it must be a matrix of at least two columns and two rows.
  • reps is the number of replicates for each combination of factor groups. If not provided, no replicates are assumed.
  • displayopt is an optional parameter for displaying the Friedman’s ANOVA table, when it is ’on’ (default) and suppressing the display when it is ’off’.

friedman returns up to three output arguments:

  • p is the p-value of the null hypothesis that all group means are equal.
  • atab is a cell array containing the results in a Friedman’s ANOVA table.
  • stats is a structure containing statistics useful for performing a multiple comparison of medians with the MULTCOMPARE function.

If friedman is called without any output arguments, then it prints the results in a one-way ANOVA table to the standard output as if displayopt is ’on’.

Examples:

 
 load popcorn;
 friedman (popcorn, 3);
 
 [p, anovatab, stats] = friedman (popcorn, 3, "off");
 disp (p);

See also: anova2, kruskalwallis, multcompare

Source Code: friedman

Example: 1

  

 load popcorn;
 friedman (popcorn, 3);

              Friedman's ANOVA Table
Source            SS      df        MS    Chi-sq    Prob>Chi-sq
---------------------------------------------------------------
Columns         99.7500     2    49.8750    13.76    0.0010
Interaction      0.0833     2     0.0417  
Error           16.1667    12     1.3472
Total          116.0000    17

Example: 2

  

 load popcorn;
 [p, atab] = friedman (popcorn, 3, "off");
 disp (p);

1.0289e-03
statistics-release-1.6.3/docs/frnd.html000066400000000000000000000121131456127120000200740ustar00rootroot00000000000000 Statistics: frnd

Function Reference: frnd

statistics: r = frnd (df1, df2)
statistics: r = frnd (df1, df2, rows)
statistics: r = frnd (df1, df2, rows, cols, …)
statistics: r = frnd (df1, df2, [sz])

Random arrays from the F-distribution.

r = frnd (df1, df2) returns an array of random numbers chosen from the F-distribution with df1 and df2 degrees of freedom. The size of r is the common size of df1 and df2. A scalar input functions as a constant matrix of the same size as the other inputs.

When called with a single size argument, frnd returns a square matrix with the dimension specified. When called with more than one scalar argument, the first two arguments are taken as the number of rows and columns and any further arguments specify additional matrix dimensions. The size may also be specified with a row vector of dimensions, sz.

Further information about the F-distribution can be found at https://en.wikipedia.org/wiki/F-distribution

See also: fcdf, finv, fpdf, fstat

Source Code: frnd

statistics-release-1.6.3/docs/fstat.html000066400000000000000000000106261456127120000202730ustar00rootroot00000000000000 Statistics: fstat

Function Reference: fstat

statistics: [m, v] = fstat (df1, df2)

Compute statistics of the F-distribution.

[m, v] = fstat (df1, df2) returns the mean and variance of the F-distribution with df1 and df2 degrees of freedom.

The size of m (mean) and v (variance) is the common size of the input arguments. A scalar input functions as a constant matrix of the same size as the other inputs.

Further information about the F-distribution can be found at https://en.wikipedia.org/wiki/F-distribution

See also: fcdf, finv, fpdf, frnd

Source Code: fstat

statistics-release-1.6.3/docs/fullfact.html000066400000000000000000000141461456127120000207530ustar00rootroot00000000000000 Statistics: fullfact

Function Reference: fullfact

statistics: A = fullfact (N)

Full factorial design.

If N is a scalar, return the full factorial design with N binary choices, 0 and 1.

If N is a vector, return the full factorial design with ordinal choices 1 through n_i for each factor i.

Values in N must be positive integers.

@seealso {ff2n

Source Code: fullfact

Example: 1

  

 ## Full factorial design with 3 binary variables
 fullfact (3)

ans =

   0   0   0
   0   0   1
   0   1   0
   0   1   1
   1   0   0
   1   0   1
   1   1   0
   1   1   1

Example: 2

  

 ## Full factorial design with 3 ordinal variables
 fullfact ([2, 3, 4])

ans =

   1   1   1
   1   1   2
   1   1   3
   1   1   4
   1   2   1
   1   2   2
   1   2   3
   1   2   4
   1   3   1
   1   3   2
   1   3   3
   1   3   4
   2   1   1
   2   1   2
   2   1   3
   2   1   4
   2   2   1
   2   2   2
   2   2   3
   2   2   4
   2   3   1
   2   3   2
   2   3   3
   2   3   4
statistics-release-1.6.3/docs/gamcdf.html000066400000000000000000000177731456127120000204050ustar00rootroot00000000000000 Statistics: gamcdf

Function Reference: gamcdf

statistics: p = gamcdf (x, k)
statistics: p = gamcdf (x, k, theta)
statistics: p = gamcdf (…, "upper")
statistics: [p, plo, pup] = evcdf (x, k, theta, pcov)
statistics: [p, plo, pup] = evcdf (x, k, theta, pcov, alpha)
statistics: [p, plo, pup] = evcdf (…, "upper")

Gamma cumulative distribution function (CDF).

For each element of x, compute the cumulative distribution function (CDF) of the Gamma distribution with shape parameter k and scale parameter theta. When called with only one parameter, then theta defaults to 1. The size of p is the common size of x, k, and theta. A scalar input functions as a constant matrix of the same size as the other inputs.

When called with three output arguments, i.e. [p, plo, pup], gamcdf computes the confidence bounds for p when the input parameters k and theta are estimates. In such case, pcov, a 2×2 matrix containing the covariance matrix of the estimated parameters, is necessary. Optionally, alpha, which has a default value of 0.05, specifies the 100 * (1 - alpha) percent confidence bounds. plo and pup are arrays of the same size as p containing the lower and upper confidence bounds.

[…] = gamcdf (…, "upper") computes the upper tail probability of the Gamma distribution with parameters k and theta, at the values in x.

There are two equivalent parameterizations in common use:

  1. With a shape parameter k and a scale parameter θ, which is used by gamcdf.
  2. With a shape parameter α = k and an inverse scale parameter β = 1 / θ, called a rate parameter.

Further information about the Gamma distribution can be found at https://en.wikipedia.org/wiki/Gamma_distribution

See also: gaminv, gampdf, gamrnd, gamfit, gamlike, gamstat

Source Code: gamcdf

Example: 1

  

 ## Plot various CDFs from the Gamma distribution
 x = 0:0.01:20;
 p1 = gamcdf (x, 1, 2);
 p2 = gamcdf (x, 2, 2);
 p3 = gamcdf (x, 3, 2);
 p4 = gamcdf (x, 5, 1);
 p5 = gamcdf (x, 9, 0.5);
 p6 = gamcdf (x, 7.5, 1);
 p7 = gamcdf (x, 0.5, 1);
 plot (x, p1, "-r", x, p2, "-g", x, p3, "-y", x, p4, "-m", ...
       x, p5, "-k", x, p6, "-b", x, p7, "-c")
 grid on
 legend ({"α = 1, θ = 2", "α = 2, θ = 2", "α = 3, θ = 2", ...
          "α = 5, θ = 1", "α = 9, θ = 0.5", "α = 7.5, θ = 1", ...
          "α = 0.5, θ = 1"}, "location", "southeast")
 title ("Gamma CDF")
 xlabel ("values in x")
 ylabel ("probability")
plotted figure

statistics-release-1.6.3/docs/gamfit.html000066400000000000000000000233031456127120000204150ustar00rootroot00000000000000 Statistics: gamfit

Function Reference: gamfit

statistics: paramhat = gamfit (x)
statistics: [paramhat, paramci] = gamfit (x)
statistics: [paramhat, paramci] = gamfit (x, alpha)
statistics: […] = gamfit (x, alpha, censor)
statistics: […] = gamfit (x, alpha, censor, freq)
statistics: […] = gamfit (x, alpha, censor, freq, options)

Estimate parameters and confidence intervals for the Gamma distribution.

paramhat = gamfit (x) returns the maximum likelihood estimates of the parameters of the Gamma distribution given the data in x. paramhat(1) is the shape parameter, k, and paramhat(2) is the scale parameter, theta.

[paramhat, paramci] = gamfit (x) returns the 95% confidence intervals for the parameter estimates.

[…] = gamfit (x, alpha) also returns the 100 * (1 - alpha) percent confidence intervals for the parameter estimates. By default, the optional argument alpha is 0.05 corresponding to 95% confidence intervals. Pass in [] for alpha to use the default values.

[…] = gamfit (x, alpha, censor) accepts a boolean vector, censor, of the same size as x with 1s for observations that are right-censored and 0s for observations that are observed exactly. By default, or if left empty, censor = zeros (size (x)).

[…] = gamfit (x, alpha, censor, freq) accepts a frequency vector, freq, of the same size as x. freq typically contains integer frequencies for the corresponding elements in x, but it can contain any non-integer non-negative values. By default, or if left empty, freq = ones (size (x)).

[…] = gamfit (…, options) specifies control parameters for the iterative algorithm used to compute the maximum likelihood estimates. options is a structure with the following field and its default value:

  • options.Display = "off"
  • options.MaxFunEvals = 1000
  • options.MaxIter = 500
  • options.TolX = 1e-6

There are two equivalent parameterizations in common use:

  1. With a shape parameter k and a scale parameter θ, which is used by gamcdf.
  2. With a shape parameter α = k and an inverse scale parameter β = 1 / θ, called a rate parameter.

Further information about the Gamma distribution can be found at https://en.wikipedia.org/wiki/Gamma_distribution

See also: gamcdf, gampdf, gaminv, gamrnd, gamlike

Source Code: gamfit

Example: 1

  

 ## Sample 3 populations from different Gamma distibutions
 randg ("seed", 5);    # for reproducibility
 r1 = gamrnd (1, 2, 2000, 1);
 randg ("seed", 2);    # for reproducibility
 r2 = gamrnd (2, 2, 2000, 1);
 randg ("seed", 7);    # for reproducibility
 r3 = gamrnd (7.5, 1, 2000, 1);
 r = [r1, r2, r3];

 ## Plot them normalized and fix their colors
 hist (r, 75, 4);
 h = findobj (gca, "Type", "patch");
 set (h(1), "facecolor", "c");
 set (h(2), "facecolor", "g");
 set (h(3), "facecolor", "r");
 ylim ([0, 0.62]);
 xlim ([0, 12]);
 hold on

 ## Estimate their α and β parameters
 k_thetaA = gamfit (r(:,1));
 k_thetaB = gamfit (r(:,2));
 k_thetaC = gamfit (r(:,3));

 ## Plot their estimated PDFs
 x = [0.01,0.1:0.2:18];
 y = gampdf (x, k_thetaA(1), k_thetaA(2));
 plot (x, y, "-pr");
 y = gampdf (x, k_thetaB(1), k_thetaB(2));
 plot (x, y, "-sg");
 y = gampdf (x, k_thetaC(1), k_thetaC(2));
 plot (x, y, "-^c");
 hold off
 legend ({"Normalized HIST of sample 1 with k=1 and θ=2", ...
          "Normalized HIST of sample 2 with k=2 and θ=2", ...
          "Normalized HIST of sample 3 with k=7.5 and θ=1", ...
          sprintf("PDF for sample 1 with estimated k=%0.2f and θ=%0.2f", ...
                  k_thetaA(1), k_thetaA(2)), ...
          sprintf("PDF for sample 2 with estimated k=%0.2f and θ=%0.2f", ...
                  k_thetaB(1), k_thetaB(2)), ...
          sprintf("PDF for sample 3 with estimated k=%0.2f and θ=%0.2f", ...
                  k_thetaC(1), k_thetaC(2))})
 title ("Three population samples from different Gamma distibutions")
 hold off
plotted figure

statistics-release-1.6.3/docs/gaminv.html000066400000000000000000000146341456127120000204360ustar00rootroot00000000000000 Statistics: gaminv

Function Reference: gaminv

statistics: x = gaminv (p, k, theta)

Inverse of the Gamma cumulative distribution function (iCDF).

For each element of p, compute the quantile (the inverse of the CDF) of the Gamma distribution with shape parameter k and scale parameter theta. The size of x is the common size of p, k, and theta. A scalar input functions as a constant matrix of the same size as the other inputs.

There are two equivalent parameterizations in common use:

  1. With a shape parameter k and a scale parameter θ, which is used by gaminv.
  2. With a shape parameter α = k and an inverse scale parameter β = 1 / θ, called a rate parameter.

Further information about the Gamma distribution can be found at https://en.wikipedia.org/wiki/Gamma_distribution

See also: gamcdf, gampdf, gamrnd, gamfit, gamlike, gamstat

Source Code: gaminv

Example: 1

  

 ## Plot various iCDFs from the Gamma distribution
 p = 0.001:0.001:0.999;
 x1 = gaminv (p, 1, 2);
 x2 = gaminv (p, 2, 2);
 x3 = gaminv (p, 3, 2);
 x4 = gaminv (p, 5, 1);
 x5 = gaminv (p, 9, 0.5);
 x6 = gaminv (p, 7.5, 1);
 x7 = gaminv (p, 0.5, 1);
 plot (p, x1, "-r", p, x2, "-g", p, x3, "-y", p, x4, "-m", ...
       p, x5, "-k", p, x6, "-b", p, x7, "-c")
 ylim ([0, 20])
 grid on
 legend ({"α = 1, θ = 2", "α = 2, θ = 2", "α = 3, θ = 2", ...
          "α = 5, θ = 1", "α = 9, θ = 0.5", "α = 7.5, θ = 1", ...
          "α = 0.5, θ = 1"}, "location", "northwest")
 title ("Gamma iCDF")
 xlabel ("probability")
 ylabel ("x")
plotted figure

statistics-release-1.6.3/docs/gamlike.html000066400000000000000000000143151456127120000205620ustar00rootroot00000000000000 Statistics: gamlike

Function Reference: gamlike

statistics: nlogL = gamlike (params, x)
statistics: [nlogL, acov] = gamlike (params, x)
statistics: […] = gamlike (params, x, censor)
statistics: […] = gamlike (params, x, censor, freq)

Negative log-likelihood for the Gamma distribution.

nlogL = gamlike (params, x) returns the negative log likelihood of the data in x corresponding to the Gamma distribution with (1) shape parameter k and (2) scale parameter theta given in the two-element vector params.

[nlogL, acov] = gamlike (params, x) also returns the inverse of Fisher’s information matrix, acov. If the input parameter values in params are the maximum likelihood estimates, the diagonal elements of acov are their asymptotic variances.

[…] = gamlike (params, x, censor) accepts a boolean vector, censor, of the same size as x with 1s for observations that are right-censored and 0s for observations that are observed exactly. By default, or if left empty, censor = zeros (size (x)).

[…] = gamlike (params, x, censor, freq) accepts a frequency vector, freq, of the same size as x. freq typically contains integer frequencies for the corresponding elements in x, but it can contain any non-integer non-negative values. By default, or if left empty, freq = ones (size (x)).

There are two equivalent parameterizations in common use:

  1. With a shape parameter k and a scale parameter θ, which is used by gamcdf.
  2. With a shape parameter α = k and an inverse scale parameter β = 1 / θ, called a rate parameter.

Further information about the Gamma distribution can be found at https://en.wikipedia.org/wiki/Gamma_distribution

See also: gamcdf, gampdf, gaminv, gamrnd, gamfit

Source Code: gamlike

statistics-release-1.6.3/docs/gampdf.html000066400000000000000000000146041456127120000204100ustar00rootroot00000000000000 Statistics: gampdf

Function Reference: gampdf

statistics: y = gampdf (x, k, theta)

Gamma probability density function (PDF).

For each element of x, compute the probability density function (PDF) of the Gamma distribution with shape parameter k and scale parameter theta. The size of y is the common size of x, k and theta. A scalar input functions as a constant matrix of the same size as the other inputs.

There are two equivalent parameterizations in common use:

  1. With a shape parameter k and a scale parameter θ, which is used by gampdf.
  2. With a shape parameter α = k and an inverse scale parameter β = 1 / θ, called a rate parameter.

Further information about the Gamma distribution can be found at https://en.wikipedia.org/wiki/Gamma_distribution

See also: gamcdf, gaminv, gamrnd, gamfit, gamlike, gamstat

Source Code: gampdf

Example: 1

  

 ## Plot various PDFs from the Gamma distribution
 x = 0:0.01:20;
 y1 = gampdf (x, 1, 2);
 y2 = gampdf (x, 2, 2);
 y3 = gampdf (x, 3, 2);
 y4 = gampdf (x, 5, 1);
 y5 = gampdf (x, 9, 0.5);
 y6 = gampdf (x, 7.5, 1);
 y7 = gampdf (x, 0.5, 1);
 plot (x, y1, "-r", x, y2, "-g", x, y3, "-y", x, y4, "-m", ...
       x, y5, "-k", x, y6, "-b", x, y7, "-c")
 grid on
 ylim ([0,0.5])
 legend ({"α = 1, θ = 2", "α = 2, θ = 2", "α = 3, θ = 2", ...
          "α = 5, θ = 1", "α = 9, θ = 0.5", "α = 7.5, θ = 1", ...
          "α = 0.5, θ = 1"}, "location", "northeast")
 title ("Gamma PDF")
 xlabel ("values in x")
 ylabel ("density")
plotted figure

statistics-release-1.6.3/docs/gamrnd.html000066400000000000000000000130141456127120000204140ustar00rootroot00000000000000 Statistics: gamrnd

Function Reference: gamrnd

statistics: r = gamrnd (k, theta)
statistics: r = gamrnd (k, theta, rows)
statistics: r = gamrnd (k, theta, rows, cols, …)
statistics: r = gamrnd (k, theta, [sz])

Random arrays from the Gamma distribution.

r = gamrnd (k, theta) returns an array of random numbers chosen from the Gamma distribution with shape parameter k and scale parameter theta. The size of r is the common size of k and theta. A scalar input functions as a constant matrix of the same size as the other inputs.

When called with a single size argument, gamrnd returns a square matrix with the dimension specified. When called with more than one scalar argument, the first two arguments are taken as the number of rows and columns and any further arguments specify additional matrix dimensions. The size may also be specified with a row vector of dimensions, sz.

There are two equivalent parameterizations in common use:

  1. With a shape parameter k and a scale parameter θ, which is used by gamrnd.
  2. With a shape parameter α = k and an inverse scale parameter β = 1 / θ, called a rate parameter.

Further information about the Gamma distribution can be found at https://en.wikipedia.org/wiki/Gamma_distribution

See also: gamcdf, gaminv, gampdf, gamfit, gamlike, gamstat

Source Code: gamrnd

statistics-release-1.6.3/docs/gamstat.html000066400000000000000000000115241456127120000206100ustar00rootroot00000000000000 Statistics: gamstat

Function Reference: gamstat

statistics: [m, v] = gamstat (k, theta)

Compute statistics of the Gamma distribution.

[m, v] = gamstat (k, theta) returns the mean and variance of the Gamma distribution with with shape parameter k and scale parameter theta.

The size of m (mean) and v (variance) is the common size of the input arguments. A scalar input functions as a constant matrix of the same size as the other inputs.

There are two equivalent parameterizations in common use:

  1. With a shape parameter k and a scale parameter θ, which is used by gamrnd.
  2. With a shape parameter α = k and an inverse scale parameter β = 1 / θ, called a rate parameter.

Further information about the Gamma distribution can be found at https://en.wikipedia.org/wiki/Gamma_distribution

See also: gamcdf, gaminv, gampdf, gamrnd, gamfit, gamlike

Source Code: gamstat

statistics-release-1.6.3/docs/geocdf.html000066400000000000000000000143401456127120000203760ustar00rootroot00000000000000 Statistics: geocdf

Function Reference: geocdf

statistics: p = geocdf (x, ps)
statistics: p = geocdf (x, ps, "upper")

Geometric cumulative distribution function (CDF).

For each element of x, compute the cumulative distribution function (CDF) of the geometric distribution with probability of success parameter ps. The size of p is the common size of x and ps. A scalar input functions as a constant matrix of the same size as the other inputs.

p = geocdf (x, ps, "upper") computes the upper tail probability of the geometric distribution with parameter ps, at the values in x.

The geometric distribution models the number of failures (x) of a Bernoulli trial with probability ps before the first success.

Further information about the geometric distribution can be found at https://en.wikipedia.org/wiki/Geometric_distribution

See also: geoinv, geopdf, geornd, geofit, geostat

Source Code: geocdf

Example: 1

  

 ## Plot various CDFs from the geometric distribution
 x = 0:10;
 p1 = geocdf (x, 0.2);
 p2 = geocdf (x, 0.5);
 p3 = geocdf (x, 0.7);
 plot (x, p1, "*b", x, p2, "*g", x, p3, "*r")
 grid on
 xlim ([0, 10])
 legend ({"ps = 0.2", "ps = 0.5", "ps = 0.7"}, "location", "southeast")
 title ("Geometric CDF")
 xlabel ("values in x (number of failures)")
 ylabel ("probability")
plotted figure

statistics-release-1.6.3/docs/geofit.html000066400000000000000000000171411456127120000204260ustar00rootroot00000000000000 Statistics: geofit

Function Reference: geofit

statistics: pshat = geofit (x)
statistics: [pshat, psci] = geofit (x)
statistics: [pshat, psci] = geofit (x, alpha)
statistics: [pshat, psci] = geofit (x, alpha, freq)

Estimate parameter and confidence intervals for the geometric distribution.

pshat = geofit (x) returns the maximum likelihood estimate (MLE) of the probability of success for the geometric distribution. x must be a vector.

[pshat, psci] = geofit (x, alpha) also returns the 100 * (1 - alpha) percent confidence intervals of the estimated parameter. By default, the optional argument alpha is 0.05 corresponding to 95% confidence intervals. Pass in [] for alpha to use the default values.

[…] = geofit (x, alpha, freq) accepts a frequency vector, freq, of the same size as x. freq typically contains integer frequencies for the corresponding elements in x, but it can contain any non-integer non-negative values. By default, or if left empty, freq = ones (size (x)).

The geometric distribution models the number of failures (x) of a Bernoulli trial with probability ps before the first success.

Further information about the geometric distribution can be found at https://en.wikipedia.org/wiki/Geometric_distribution

See also: geocdf, geoinv, geopdf, geornd, geostat

Source Code: geofit

Example: 1

  

 ## Sample 2 populations from different geometric distibutions
 rande ("seed", 1);    # for reproducibility
 r1 = geornd (0.15, 1000, 1);
 rande ("seed", 2);    # for reproducibility
 r2 = geornd (0.5, 1000, 1);
 r = [r1, r2];

 ## Plot them normalized and fix their colors
 hist (r, 0:0.5:20.5, 1);
 h = findobj (gca, "Type", "patch");
 set (h(1), "facecolor", "c");
 set (h(2), "facecolor", "g");
 hold on

 ## Estimate their probability of success
 pshatA = geofit (r(:,1));
 pshatB = geofit (r(:,2));

 ## Plot their estimated PDFs
 x = [0:15];
 y = geopdf (x, pshatA);
 plot (x, y, "-pg");
 y = geopdf (x, pshatB);
 plot (x, y, "-sc");
 xlim ([0, 15])
 ylim ([0, 0.6])
 legend ({"Normalized HIST of sample 1 with ps=0.15", ...
          "Normalized HIST of sample 2 with ps=0.50", ...
          sprintf("PDF for sample 1 with estimated ps=%0.2f", ...
                  mean (pshatA)), ...
          sprintf("PDF for sample 2 with estimated ps=%0.2f", ...
                  mean (pshatB))})
 title ("Two population samples from different geometric distibutions")
 hold off
plotted figure

statistics-release-1.6.3/docs/geoinv.html000066400000000000000000000136351456127120000204440ustar00rootroot00000000000000 Statistics: geoinv

Function Reference: geoinv

statistics: x = geoinv (p, ps)

Inverse of the geometric cumulative distribution function (iCDF).

For each element of p, compute the quantile (the inverse of the CDF) of the geometric distribution with probability of success parameter ps. The size of x is the common size of p and ps. A scalar input functions as a constant matrix of the same size as the other inputs.

The geometric distribution models the number of failures (p) of a Bernoulli trial with probability ps before the first success.

Further information about the geometric distribution can be found at https://en.wikipedia.org/wiki/Geometric_distribution

See also: geocdf, geopdf, geornd, geofit, geostat

Source Code: geoinv

Example: 1

  

 ## Plot various iCDFs from the geometric distribution
 p = 0.001:0.001:0.999;
 x1 = geoinv (p, 0.2);
 x2 = geoinv (p, 0.5);
 x3 = geoinv (p, 0.7);
 plot (p, x1, "-b", p, x2, "-g", p, x3, "-r")
 grid on
 ylim ([0, 10])
 legend ({"ps = 0.2", "ps = 0.5", "ps = 0.7"}, "location", "northwest")
 title ("Geometric iCDF")
 xlabel ("probability")
 ylabel ("values in x (number of failures)")
plotted figure

statistics-release-1.6.3/docs/geomean.html000066400000000000000000000143621456127120000205660ustar00rootroot00000000000000 Statistics: geomean

Function Reference: geomean

statistics: m = geomean (x)
statistics: m = geomean (x, "all")
statistics: m = geomean (x, dim)
statistics: m = geomean (x, vecdim)
statistics: m = geomean (…, nanflag)

Compute the geometric mean of x.

  • If x is a vector, then geomean(x) returns the geometric mean of the elements in x defined as $$ {\rm geomean}(x) = \left( \prod_{i=1}^N x_i \right)^\frac{1}{N} = exp \left({1\over N} \sum_{i=1}^N log x_i \right) $$ where N is the length of the x vector.
  • If x is a matrix, then geomean(x) returns a row vector with the geometric mean of each columns in x.
  • If x is a multidimensional array, then geomean(x) operates along the first nonsingleton dimension of x.
  • x must not contain any negative or complex values.

geomean(x, "all") returns the geometric mean of all the elements in x. If x contains any 0, then the returned value is 0.

geomean(x, dim) returns the geometric mean along the operating dimension dim of x. Calculating the harmonic mean of any subarray containing any 0 will return 0.

geomean(x, vecdim) returns the geometric mean over the dimensions specified in the vector vecdim. For example, if x is a 2-by-3-by-4 array, then geomean(x, [1 2]) returns a 1-by-1-by-4 array. Each element of the output array is the geometric mean of the elements on the corresponding page of x. If vecdim indexes all dimensions of x, then it is equivalent to geomean (x, "all"). Any dimension in vecdim greater than ndims (x) is ignored.

geomean(…, nanflag) specifies whether to exclude NaN values from the calculation, using any of the input argument combinations in previous syntaxes. By default, geomean includes NaN values in the calculation (nanflag has the value "includenan"). To exclude NaN values, set the value of nanflag to "omitnan".

See also: harmmean, mean

Source Code: geomean

statistics-release-1.6.3/docs/geopdf.html000066400000000000000000000135701456127120000204170ustar00rootroot00000000000000 Statistics: geopdf

Function Reference: geopdf

statistics: y = geopdf (x, ps)

Geometric probability density function (PDF).

For each element of x, compute the probability density function (PDF) of the geometric distribution with probability of success parameter ps. The size of y is the common size of x and ps. A scalar input functions as a constant matrix of the same size as the other inputs.

The geometric distribution models the number of failures (x) of a Bernoulli trial with probability ps before the first success.

Further information about the geometric distribution can be found at https://en.wikipedia.org/wiki/Geometric_distribution

See also: geocdf, geoinv, geornd, geofit, geostat

Source Code: geopdf

Example: 1

  

 ## Plot various PDFs from the geometric distribution
 x = 0:10;
 y1 = geopdf (x, 0.2);
 y2 = geopdf (x, 0.5);
 y3 = geopdf (x, 0.7);
 plot (x, y1, "*b", x, y2, "*g", x, y3, "*r")
 grid on
 ylim ([0, 0.8])
 legend ({"ps = 0.2", "ps = 0.5", "ps = 0.7"}, "location", "northeast")
 title ("Geometric PDF")
 xlabel ("values in x (number of failures)")
 ylabel ("density")
plotted figure

statistics-release-1.6.3/docs/geornd.html000066400000000000000000000121601456127120000204230ustar00rootroot00000000000000 Statistics: geornd

Function Reference: geornd

statistics: r = geornd (ps)
statistics: r = geornd (ps, rows)
statistics: r = geornd (ps, rows, cols, …)
statistics: r = geornd (ps, [sz])

Random arrays from the geometric distribution.

r = geornd (ps) returns an array of random numbers chosen from the Birnbaum-Saunders distribution with probability of success parameter ps. The size of r is the size of ps.

When called with a single size argument, geornd returns a square matrix with the dimension specified. When called with more than one scalar argument, the first two arguments are taken as the number of rows and columns and any further arguments specify additional matrix dimensions. The size may also be specified with a row vector of dimensions, sz.

The geometric distribution models the number of failures (x) of a Bernoulli trial with probability ps before the first success.

Further information about the geometric distribution can be found at https://en.wikipedia.org/wiki/Geometric_distribution

See also: geocdf, geoinv, geopdf, geofit, geostat

Source Code: geornd

statistics-release-1.6.3/docs/geostat.html000066400000000000000000000105271456127120000206200ustar00rootroot00000000000000 Statistics: geostat

Function Reference: geostat

statistics: [m, v] = geostat (ps)

Compute statistics of the geometric distribution.

[m, v] = geostat (ps) returns the mean and variance of the geometric distribution with probability of success parameter ps.

The size of m (mean) and v (variance) is the same size of the input argument.

Further information about the geometric distribution can be found at https://en.wikipedia.org/wiki/Geometric_distribution

See also: geocdf, geoinv, geopdf, geornd, geofit

Source Code: geostat

statistics-release-1.6.3/docs/gevcdf.html000066400000000000000000000177021456127120000204120ustar00rootroot00000000000000 Statistics: gevcdf

Function Reference: gevcdf

statistics: p = gevcdf (x, k, sigma, mu)
statistics: p = gevcdf (x, k, sigma, mu, "upper")

Generalized extreme value (GEV) cumulative distribution function (CDF).

For each element of x, compute the cumulative distribution function (CDF) of the GEV distribution with shape parameter k, scale parameter sigma, and location parameter mu. The size of p is the common size of x, k, sigma, and mu. A scalar input functions as a constant matrix of the same size as the other inputs.

[…] = gevcdf (x, k, sigma, mu, "upper") computes the upper tail probability of the GEV distribution with parameters k, sigma, and mu, at the values in x.

When k < 0, the GEV is the type III extreme value distribution. When k > 0, the GEV distribution is the type II, or Frechet, extreme value distribution. If W has a Weibull distribution as computed by the wblcdf function, then -W has a type III extreme value distribution and 1/W has a type II extreme value distribution. In the limit as k approaches 0, the GEV is the mirror image of the type I extreme value distribution as computed by the evcdf function.

The mean of the GEV distribution is not finite when k >= 1, and the variance is not finite when k >= 1/2. The GEV distribution has positive density only for values of x such that k * (x - mu) / sigma > -1.

Further information about the generalized extreme value distribution can be found at https://en.wikipedia.org/wiki/Generalized_extreme_value_distribution

References

  1. Rolf-Dieter Reiss and Michael Thomas. Statistical Analysis of Extreme Values with Applications to Insurance, Finance, Hydrology and Other Fields. Chapter 1, pages 16-17, Springer, 2007.

See also: gevinv, gevpdf, gevrnd, gevfit, gevlike, gevstat

Source Code: gevcdf

Example: 1

  

 ## Plot various CDFs from the generalized extreme value distribution
 x = -1:0.001:10;
 p1 = gevcdf (x, 1, 1, 1);
 p2 = gevcdf (x, 0.5, 1, 1);
 p3 = gevcdf (x, 1, 1, 5);
 p4 = gevcdf (x, 1, 2, 5);
 p5 = gevcdf (x, 1, 5, 5);
 p6 = gevcdf (x, 1, 0.5, 5);
 plot (x, p1, "-b", x, p2, "-g", x, p3, "-r", ...
       x, p4, "-c", x, p5, "-m", x, p6, "-k")
 grid on
 xlim ([-1, 10])
 legend ({"ξ = 1, σ = 1, μ = 1", "ξ = 0.5, σ = 1, μ = 1", ...
          "ξ = 1, σ = 1, μ = 5", "ξ = 1, σ = 2, μ = 5", ...
          "ξ = 1, σ = 5, μ = 5", "ξ = 1, σ = 0.5, μ = 5"}, ...
         "location", "southeast")
 title ("Generalized extreme value CDF")
 xlabel ("values in x")
 ylabel ("probability")
plotted figure

statistics-release-1.6.3/docs/gevfit.html000066400000000000000000000227751456127120000204460ustar00rootroot00000000000000 Statistics: gevfit

Function Reference: gevfit

statistics: paramhat = gevfit (x)
statistics: [paramhat, paramci] = gevfit (x)
statistics: [paramhat, paramci] = gevfit (x, alpha)
statistics: […] = gevfit (x, alpha, options)

Estimate parameters and confidence intervals for the generalized extreme value (GEV) distribution.

Arguments

paramhat = gevfit (x) returns the maximum likelihood estimates of the parameters of the GEV distribution given the data in x. paramhat(1) is the shape parameter, k, and paramhat(2) is the scale parameter, sigma, and paramhat(3) is the location parameter, mu.

[paramhat, paramci] = gevfit (x) returns the 95% confidence intervals for the parameter estimates.

[…] = gevfit (x, alpha) also returns the 100 * (1 - alpha) percent confidence intervals for the parameter estimates. By default, the optional argument alpha is 0.05 corresponding to 95% confidence intervals. Pass in [] for alpha to use the default values.

[…] = gevfit (…, options) specifies control parameters for the iterative algorithm used to compute the maximum likelihood estimates. options is a structure with the following field and its default value:

  • options.Display = "off"
  • options.MaxFunEvals = 1000
  • options.MaxIter = 500
  • options.TolX = 1e-6

When k < 0, the GEV is the type III extreme value distribution. When k > 0, the GEV distribution is the type II, or Frechet, extreme value distribution. If W has a Weibull distribution as computed by the wblcdf function, then -W has a type III extreme value distribution and 1/W has a type II extreme value distribution. In the limit as k approaches 0, the GEV is the mirror image of the type I extreme value distribution as computed by the evcdf function.

The mean of the GEV distribution is not finite when k >= 1, and the variance is not finite when k >= 1/2. The GEV distribution has positive density only for values of x such that k * (x - mu) / sigma > -1.

Further information about the generalized extreme value distribution can be found at https://en.wikipedia.org/wiki/Generalized_extreme_value_distribution

References

  1. Rolf-Dieter Reiss and Michael Thomas. Statistical Analysis of Extreme Values with Applications to Insurance, Finance, Hydrology and Other Fields. Chapter 1, pages 16-17, Springer, 2007.

See also: gevcdf, gevinv, gevpdf, gevrnd, gevlike, gevstat

Source Code: gevfit

Example: 1

  

 ## Sample 2 populations from 2 different exponential distibutions
 rand ("seed", 1);   # for reproducibility
 r1 = gevrnd (-0.5, 1, 2, 5000, 1);
 rand ("seed", 2);   # for reproducibility
 r2 = gevrnd (0, 1, -4, 5000, 1);
 r = [r1, r2];

 ## Plot them normalized and fix their colors
 hist (r, 50, 5);
 h = findobj (gca, "Type", "patch");
 set (h(1), "facecolor", "c");
 set (h(2), "facecolor", "g");
 hold on

 ## Estimate their k, sigma, and mu parameters
 k_sigma_muA = gevfit (r(:,1));
 k_sigma_muB = gevfit (r(:,2));

 ## Plot their estimated PDFs
 x = [-10:0.5:20];
 y = gevpdf (x, k_sigma_muA(1), k_sigma_muA(2), k_sigma_muA(3));
 plot (x, y, "-pr");
 y = gevpdf (x, k_sigma_muB(1), k_sigma_muB(2), k_sigma_muB(3));
 plot (x, y, "-sg");
 ylim ([0, 0.7])
 xlim ([-7, 5])
 legend ({"Normalized HIST of sample 1 with ξ=-0.5, σ=1, μ=2", ...
          "Normalized HIST of sample 2 with ξ=0, σ=1, μ=-4",
     sprintf("PDF for sample 1 with estimated ξ=%0.2f, σ=%0.2f, μ=%0.2f", ...
                 k_sigma_muA(1), k_sigma_muA(2), k_sigma_muA(3)), ...
     sprintf("PDF for sample 3 with estimated ξ=%0.2f, σ=%0.2f, μ=%0.2f", ...
                 k_sigma_muB(1), k_sigma_muB(2), k_sigma_muB(3))})
 title ("Two population samples from different exponential distibutions")
 hold off
plotted figure

statistics-release-1.6.3/docs/gevfit_lmom.html000066400000000000000000000115531456127120000214620ustar00rootroot00000000000000 Statistics: gevfit_lmom

Function Reference: gevfit_lmom

statistics: [paramhat, paramci] = gevfit_lmom (data)

Find an estimator (paramhat) of the generalized extreme value (GEV) distribution fitting data using the method of L-moments.

Arguments

  • data is the vector of given values.

Return values

  • parmhat is the 3-parameter maximum-likelihood parameter vector [k; sigma; mu], where k is the shape parameter of the GEV distribution, sigma is the scale parameter of the GEV distribution, and mu is the location parameter of the GEV distribution.
  • paramci has the approximate 95% confidence intervals of the parameter values (currently not implemented).

Examples

 
 
 data = gevrnd (0.1, 1, 0, 100, 1);
 [pfit, pci] = gevfit_lmom (data);
 p1 = gevcdf (data,pfit(1),pfit(2),pfit(3));
 [f, x] = ecdf (data);
 plot(data, p1, 's', x, f)

See also: gevfit

Source Code: gevfit_lmom

statistics-release-1.6.3/docs/gevinv.html000066400000000000000000000170351456127120000204510ustar00rootroot00000000000000 Statistics: gevinv

Function Reference: gevinv

statistics: x = gevinv (p, k, sigma, mu)

Inverse of the generalized extreme value (GEV) cumulative distribution function (iCDF).

For each element of p, compute the quantile (the inverse of the CDF) of the GEV distribution with shape parameter k, scale parameter sigma, and location parameter mu. The size of p is the common size of x, k, sigma, and mu. A scalar input functions as a constant matrix of the same size as the other inputs.

When k < 0, the GEV is the type III extreme value distribution. When k > 0, the GEV distribution is the type II, or Frechet, extreme value distribution. If W has a Weibull distribution as computed by the wblcdf function, then -W has a type III extreme value distribution and 1/W has a type II extreme value distribution. In the limit as k approaches 0, the GEV is the mirror image of the type I extreme value distribution as computed by the evcdf function.

The mean of the GEV distribution is not finite when k >= 1, and the variance is not finite when k >= 1/2. The GEV distribution has positive density only for values of x such that k * (x - mu) / sigma > -1.

Further information about the generalized extreme value distribution can be found at https://en.wikipedia.org/wiki/Generalized_extreme_value_distribution

References

  1. Rolf-Dieter Reiss and Michael Thomas. Statistical Analysis of Extreme Values with Applications to Insurance, Finance, Hydrology and Other Fields. Chapter 1, pages 16-17, Springer, 2007.

See also: gevcdf, gevpdf, gevrnd, gevfit, gevlike, gevstat

Source Code: gevinv

Example: 1

  

 ## Plot various iCDFs from the generalized extreme value distribution
 p = 0.001:0.001:0.999;
 x1 = gevinv (p, 1, 1, 1);
 x2 = gevinv (p, 0.5, 1, 1);
 x3 = gevinv (p, 1, 1, 5);
 x4 = gevinv (p, 1, 2, 5);
 x5 = gevinv (p, 1, 5, 5);
 x6 = gevinv (p, 1, 0.5, 5);
 plot (p, x1, "-b", p, x2, "-g", p, x3, "-r", ...
       p, x4, "-c", p, x5, "-m", p, x6, "-k")
 grid on
 ylim ([-1, 10])
 legend ({"ξ = 1, σ = 1, μ = 1", "ξ = 0.5, σ = 1, μ = 1", ...
          "ξ = 1, σ = 1, μ = 5", "ξ = 1, σ = 2, μ = 5", ...
          "ξ = 1, σ = 5, μ = 5", "ξ = 1, σ = 0.5, μ = 5"}, ...
         "location", "northwest")
 title ("Generalized extreme value iCDF")
 xlabel ("probability")
 ylabel ("values in x")
plotted figure

statistics-release-1.6.3/docs/gevlike.html000066400000000000000000000143231456127120000205760ustar00rootroot00000000000000 Statistics: gevlike

Function Reference: gevlike

statistics: nlogL = gevlike (params, x)
statistics: [nlogL, acov] = gevlike (params, x)

Negative log-likelihood for the generalized extreme value (GEV) distribution.

nlogL = gevlike (params, x) returns the negative log likelihood of the data in x corresponding to the GEV distribution with (1) shape parameter k, (2) scale parameter sigma, and (3) location parameter mu given in the three-element vector params.

[nlogL, acov] = gevlike (params, x) also returns the inverse of Fisher’s information matrix, acov. If the input parameter values in params are the maximum likelihood estimates, the diagonal elements of acov are their asymptotic variances.

When k < 0, the GEV is the type III extreme value distribution. When k > 0, the GEV distribution is the type II, or Frechet, extreme value distribution. If W has a Weibull distribution as computed by the wblcdf function, then -W has a type III extreme value distribution and 1/W has a type II extreme value distribution. In the limit as k approaches 0, the GEV is the mirror image of the type I extreme value distribution as computed by the evcdf function.

The mean of the GEV distribution is not finite when k >= 1, and the variance is not finite when k >= 1/2. The GEV distribution has positive density only for values of x such that k * (x - mu) / sigma > -1.

Further information about the generalized extreme value distribution can be found at https://en.wikipedia.org/wiki/Generalized_extreme_value_distribution

References

  1. Rolf-Dieter Reiss and Michael Thomas. Statistical Analysis of Extreme Values with Applications to Insurance, Finance, Hydrology and Other Fields. Chapter 1, pages 16-17, Springer, 2007.

See also: gevcdf, gevinv, gevpdf, gevrnd, gevfit, gevstat

Source Code: gevlike

statistics-release-1.6.3/docs/gevpdf.html000066400000000000000000000170161456127120000204250ustar00rootroot00000000000000 Statistics: gevpdf

Function Reference: gevpdf

statistics: y = gevpdf (x, k, sigma, mu)

Generalized extreme value (GEV) probability density function (PDF).

For each element of x, compute the probability density function (PDF) of the GEV distribution with shape parameter k, scale parameter sigma, and location parameter mu. The size of y is the common size of x, k, sigma, and mu. A scalar input functions as a constant matrix of the same size as the other inputs.

When k < 0, the GEV is the type III extreme value distribution. When k > 0, the GEV distribution is the type II, or Frechet, extreme value distribution. If W has a Weibull distribution as computed by the wblcdf function, then -W has a type III extreme value distribution and 1/W has a type II extreme value distribution. In the limit as k approaches 0, the GEV is the mirror image of the type I extreme value distribution as computed by the evcdf function.

The mean of the GEV distribution is not finite when k >= 1, and the variance is not finite when k >= 1/2. The GEV distribution has positive density only for values of x such that k * (x - mu) / sigma > -1.

Further information about the generalized extreme value distribution can be found at https://en.wikipedia.org/wiki/Generalized_extreme_value_distribution

References

  1. Rolf-Dieter Reiss and Michael Thomas. Statistical Analysis of Extreme Values with Applications to Insurance, Finance, Hydrology and Other Fields. Chapter 1, pages 16-17, Springer, 2007.

See also: gevcdf, gevinv, gevrnd, gevfit, gevlike, gevstat

Source Code: gevpdf

Example: 1

  

 ## Plot various PDFs from the generalized extreme value distribution
 x = -1:0.001:10;
 y1 = gevpdf (x, 1, 1, 1);
 y2 = gevpdf (x, 0.5, 1, 1);
 y3 = gevpdf (x, 1, 1, 5);
 y4 = gevpdf (x, 1, 2, 5);
 y5 = gevpdf (x, 1, 5, 5);
 y6 = gevpdf (x, 1, 0.5, 5);
 plot (x, y1, "-b", x, y2, "-g", x, y3, "-r", ...
       x, y4, "-c", x, y5, "-m", x, y6, "-k")
 grid on
 xlim ([-1, 10])
 ylim ([0, 1.1])
 legend ({"ξ = 1, σ = 1, μ = 1", "ξ = 0.5, σ = 1, μ = 1", ...
          "ξ = 1, σ = 1, μ = 5", "ξ = 1, σ = 2, μ = 5", ...
          "ξ = 1, σ = 5, μ = 5", "ξ = 1, σ = 0.5, μ = 5"}, ...
         "location", "northeast")
 title ("Generalized extreme value PDF")
 xlabel ("values in x")
 ylabel ("density")
plotted figure

statistics-release-1.6.3/docs/gevrnd.html000066400000000000000000000152011456127120000204310ustar00rootroot00000000000000 Statistics: gevrnd

Function Reference: gevrnd

statistics: r = gevrnd (k, sigma, mu)
statistics: r = gevrnd (k, sigma, mu, rows)
statistics: r = gevrnd (k, sigma, mu, rows, cols, …)
statistics: r = gevrnd (k, sigma, mu, [sz])

Random arrays from the generalized extreme value (GEV) distribution.

r = gevrnd (k, sigma, mu returns an array of random numbers chosen from the GEV distribution with shape parameter k, scale parameter sigma, and location parameter mu. The size of r is the common size of k, sigma, and mu. A scalar input functions as a constant matrix of the same size as the other inputs.

When called with a single size argument, gevrnd returns a square matrix with the dimension specified. When called with more than one scalar argument, the first two arguments are taken as the number of rows and columns and any further arguments specify additional matrix dimensions. The size may also be specified with a row vector of dimensions, sz.

When k < 0, the GEV is the type III extreme value distribution. When k > 0, the GEV distribution is the type II, or Frechet, extreme value distribution. If W has a Weibull distribution as computed by the wblcdf function, then -W has a type III extreme value distribution and 1/W has a type II extreme value distribution. In the limit as k approaches 0, the GEV is the mirror image of the type I extreme value distribution as computed by the evcdf function.

The mean of the GEV distribution is not finite when k >= 1, and the variance is not finite when k >= 1/2. The GEV distribution has positive density only for values of x such that k * (x - mu) / sigma > -1.

Further information about the generalized extreme value distribution can be found at https://en.wikipedia.org/wiki/Generalized_extreme_value_distribution

References

  1. Rolf-Dieter Reiss and Michael Thomas. Statistical Analysis of Extreme Values with Applications to Insurance, Finance, Hydrology and Other Fields. Chapter 1, pages 16-17, Springer, 2007.

See also: gevcdf, gevinv, gevpdf, gevfit, gevlike, gevstat

Source Code: gevrnd

statistics-release-1.6.3/docs/gevstat.html000066400000000000000000000124331456127120000206250ustar00rootroot00000000000000 Statistics: gevstat

Function Reference: gevstat

statistics: [m, v] = gevstat (k, sigma, mu)

Compute statistics of the generalized extreme value distribution.

[m, v] = gevstat (k, sigma, mu) returns the mean and variance of the generalized extreme value distribution with shape parameter k, scale parameter sigma, and location parameter mu.

The size of m (mean) and v (variance) is the common size of the input arguments. A scalar input functions as a constant matrix of the same size as the other inputs.

The mean of the GEV distribution is not finite when k >= 1, and the variance is not finite when k >= 1/2. The GEV distribution has positive density only for values of x such that k * (x - mu) / sigma > -1.

Further information about the generalized extreme value distribution can be found at https://en.wikipedia.org/wiki/Generalized_extreme_value_distribution

References

  1. Rolf-Dieter Reiss and Michael Thomas. Statistical Analysis of Extreme Values with Applications to Insurance, Finance, Hydrology and Other Fields. Chapter 1, pages 16-17, Springer, 2007.

See also: gevcdf, gevinv, gevpdf, gevrnd, gevfit, gevlike

Source Code: gevstat

statistics-release-1.6.3/docs/gpcdf.html000066400000000000000000000166221456127120000202370ustar00rootroot00000000000000 Statistics: gpcdf

Function Reference: gpcdf

statistics: p = gpcdf (x, k, sigma, mu)
statistics: p = gpcdf (x, k, sigma, mu, "upper")

Generalized Pareto cumulative distribution function (CDF).

For each element of x, compute the cumulative distribution function (CDF) of the generalized Pareto distribution with shape parameter k, scale parameter sigma, and location parameter mu. The size of p is the common size of x, k, sigma, and mu. A scalar input functions as a constant matrix of the same size as the other inputs.

[…] = gpcdf(x, k, sigma, mu, "upper") computes the upper tail probability of the generalized Pareto distribution with parameters k, sigma, and mu, at the values in x.

When k = 0 and mu = 0, the Generalized Pareto CDF is equivalent to the exponential distribution. When k > 0 and mu = k / k the Generalized Pareto is equivalent to the Pareto distribution. The mean of the Generalized Pareto is not finite when k >= 1 and the variance is not finite when k >= 1/2. When k >= 0, the Generalized Pareto has positive density for x > mu, or, when mu < 0, for 0 <= (x - mu) / sigma <= -1 / k.

Further information about the generalized Pareto distribution can be found at https://en.wikipedia.org/wiki/Generalized_Pareto_distribution

See also: gpinv, gppdf, gprnd, gpfit, gplike, gpstat

Source Code: gpcdf

Example: 1

  

 ## Plot various CDFs from the generalized Pareto distribution
 x = 0:0.001:5;
 p1 = gpcdf (x, 1, 1, 0);
 p2 = gpcdf (x, 5, 1, 0);
 p3 = gpcdf (x, 20, 1, 0);
 p4 = gpcdf (x, 1, 2, 0);
 p5 = gpcdf (x, 5, 2, 0);
 p6 = gpcdf (x, 20, 2, 0);
 plot (x, p1, "-b", x, p2, "-g", x, p3, "-r", ...
       x, p4, "-c", x, p5, "-m", x, p6, "-k")
 grid on
 xlim ([0, 5])
 legend ({"ξ = 1, σ = 1, μ = 0", "ξ = 5, σ = 1, μ = 0", ...
          "ξ = 20, σ = 1, μ = 0", "ξ = 1, σ = 2, μ = 0", ...
          "ξ = 5, σ = 2, μ = 0", "ξ = 20, σ = 2, μ = 0"}, ...
         "location", "northwest")
 title ("Generalized Pareto CDF")
 xlabel ("values in x")
 ylabel ("probability")
plotted figure

statistics-release-1.6.3/docs/gpfit.html000066400000000000000000000224371456127120000202660ustar00rootroot00000000000000 Statistics: gpfit

Function Reference: gpfit

statistics: paramhat = gpfit (x)
statistics: [paramhat, paramci] = gpfit (x)
statistics: [paramhat, paramci] = gpfit (x, alpha)
statistics: […] = gpfit (x, alpha, options)

Estimate parameters and confidence intervals for the generalized Pareto distribution.

paramhat = gpfit (x) returns the maximum likelihood estimates of the parameters of the generalized Pareto distribution given the data in x. paramhat(1) is the shape parameter, k, and paramhat(2) is the scale parameter, sigma. Other functions for the generalized Pareto, such as gpcdf, allow a location parameter, mu. However, gpfit does not estimate a location parameter, and it must be assumed known, and subtracted from x before calling gpfit.

[paramhat, paramci] = gpfit (x) returns the 95% confidence intervals for the parameter estimates.

[…] = gpfit (x, alpha) also returns the 100 * (1 - alpha) percent confidence intervals for the parameter estimates. By default, the optional argument alpha is 0.05 corresponding to 95% confidence intervals. Pass in [] for alpha to use the default values.

[…] = gpfit (x, alpha, options) specifies control parameters for the iterative algorithm used to compute ML estimates with the fminsearch function. options is a structure with the following fields and their default values:

  • options.Display = "off"
  • options.MaxFunEvals = 400
  • options.MaxIter = 200
  • options.TolBnd = 1e-6
  • options.TolFun = 1e-6
  • options.TolX = 1e-6

When k = 0 and mu = 0, the Generalized Pareto CDF is equivalent to the exponential distribution. When k > 0 and mu = k / k the Generalized Pareto is equivalent to the Pareto distribution. The mean of the Generalized Pareto is not finite when k >= 1 and the variance is not finite when k >= 1/2. When k >= 0, the Generalized Pareto has positive density for x > mu, or, when mu < 0, for 0 <= (x - mu) / sigma <= -1 / k.

Further information about the generalized Pareto distribution can be found at https://en.wikipedia.org/wiki/Generalized_Pareto_distribution

See also: gpcdf, gpinv, gppdf, gprnd, gplike, gpstat

Source Code: gpfit

Example: 1

  

 ## Sample 2 populations from different generalized Pareto distibutions
 ## Assume location parameter is known
 mu = 0;
 rand ("seed", 5);    # for reproducibility
 r1 = gprnd (1, 2, mu, 20000, 1);
 rand ("seed", 2);    # for reproducibility
 r2 = gprnd (3, 1, mu, 20000, 1);
 r = [r1, r2];

 ## Plot them normalized and fix their colors
 hist (r, [0.1:0.2:100], 5);
 h = findobj (gca, "Type", "patch");
 set (h(1), "facecolor", "r");
 set (h(2), "facecolor", "c");
 ylim ([0, 1]);
 xlim ([0, 5]);
 hold on

 ## Estimate their α and β parameters
 k_sigmaA = gpfit (r(:,1));
 k_sigmaB = gpfit (r(:,2));

 ## Plot their estimated PDFs
 x = [0.01, 0.1:0.2:18];
 y = gppdf (x, k_sigmaA(1), k_sigmaA(2), mu);
 plot (x, y, "-pc");
 y = gppdf (x, k_sigmaB(1), k_sigmaB(2), mu);
 plot (x, y, "-sr");
 hold off
 legend ({"Normalized HIST of sample 1 with k=1 and σ=2", ...
          "Normalized HIST of sample 2 with k=2 and σ=2", ...
          sprintf("PDF for sample 1 with estimated k=%0.2f and σ=%0.2f", ...
                  k_sigmaA(1), k_sigmaA(2)), ...
          sprintf("PDF for sample 3 with estimated k=%0.2f and σ=%0.2f", ...
                  k_sigmaB(1), k_sigmaB(2))})
 title ("Three population samples from different generalized Pareto distibutions")
 text (2, 0.7, "Known location parameter μ = 0")
 hold off
plotted figure

statistics-release-1.6.3/docs/gpinv.html000066400000000000000000000157421456127120000203010ustar00rootroot00000000000000 Statistics: gpinv

Function Reference: gpinv

statistics: x = gpinv (p, k, sigma, mu)

Inverse of the generalized Pareto cumulative distribution function (iCDF).

For each element of p, compute the quantile (the inverse of the CDF) of the generalized Pareto distribution with shape parameter k, scale parameter sigma, and location parameter mu. The size of x is the common size of p, k, sigma, and mu. A scalar input functions as a constant matrix of the same size as the other inputs.

When k = 0 and mu = 0, the Generalized Pareto CDF is equivalent to the exponential distribution. When k > 0 and mu = k / k the Generalized Pareto is equivalent to the Pareto distribution. The mean of the Generalized Pareto is not finite when k >= 1 and the variance is not finite when k >= 1/2. When k >= 0, the Generalized Pareto has positive density for x > mu, or, when mu < 0, for 0 <= (x - mu) / sigma <= -1 / k.

Further information about the generalized Pareto distribution can be found at https://en.wikipedia.org/wiki/Generalized_Pareto_distribution

See also: gpcdf, gppdf, gprnd, gpfit, gplike, gpstat

Source Code: gpinv

Example: 1

  

 ## Plot various iCDFs from the generalized Pareto distribution
 p = 0.001:0.001:0.999;
 x1 = gpinv (p, 1, 1, 0);
 x2 = gpinv (p, 5, 1, 0);
 x3 = gpinv (p, 20, 1, 0);
 x4 = gpinv (p, 1, 2, 0);
 x5 = gpinv (p, 5, 2, 0);
 x6 = gpinv (p, 20, 2, 0);
 plot (p, x1, "-b", p, x2, "-g", p, x3, "-r", ...
       p, x4, "-c", p, x5, "-m", p, x6, "-k")
 grid on
 ylim ([0, 5])
 legend ({"ξ = 1, σ = 1, μ = 0", "ξ = 5, σ = 1, μ = 0", ...
          "ξ = 20, σ = 1, μ = 0", "ξ = 1, σ = 2, μ = 0", ...
          "ξ = 5, σ = 2, μ = 0", "ξ = 20, σ = 2, μ = 0"}, ...
         "location", "southeast")
 title ("Generalized Pareto iCDF")
 xlabel ("probability")
 ylabel ("values in x")
plotted figure

statistics-release-1.6.3/docs/gplike.html000066400000000000000000000136071456127120000204270ustar00rootroot00000000000000 Statistics: gplike

Function Reference: gplike

statistics: nlogL = gplike (params, x)
statistics: [nlogL, acov] = gplike (params, x)

Negative log-likelihood for the generalized Pareto distribution.

nlogL = gplike (params, x) returns the negative log-likelihood of the data in x corresponding to the generalized Pareto distribution with (1) shape parameter k and (2) scale parameter sigma given in the two-element vector params. gplike does not allow a location parameter and it must be assumed known, and subtracted from x before calling gplike.

[nlogL, acov] = gplike (params, x) returns the inverse of Fisher’s information matrix, acov. If the input parameter values in params are the maximum likelihood estimates, the diagonal elements of acov are their asymptotic variances. acov is based on the observed Fisher’s information, not the expected information.

When k = 0 and mu = 0, the Generalized Pareto CDF is equivalent to the exponential distribution. When k > 0 and mu = k / k the Generalized Pareto is equivalent to the Pareto distribution. The mean of the Generalized Pareto is not finite when k >= 1 and the variance is not finite when k >= 1/2. When k >= 0, the Generalized Pareto has positive density for x > mu, or, when mu < 0, for 0 <= (x - mu) / sigma <= -1 / k.

Further information about the generalized Pareto distribution can be found at https://en.wikipedia.org/wiki/Generalized_Pareto_distribution

See also: gpcdf, gpinv, gppdf, gprnd, gpfit, gpstat

Source Code: gplike

statistics-release-1.6.3/docs/gppdf.html000066400000000000000000000157201456127120000202520ustar00rootroot00000000000000 Statistics: gppdf

Function Reference: gppdf

statistics: y = gppdf (x, k, sigma, mu)

Generalized Pareto probability density function (PDF).

For each element of x, compute the probability density function (PDF) of the generalized Pareto distribution with shape parameter k, scale parameter sigma, and location parameter mu. The size of y is the common size of p, k, sigma, and mu. A scalar input functions as a constant matrix of the same size as the other inputs.

When k = 0 and mu = 0, the Generalized Pareto CDF is equivalent to the exponential distribution. When k > 0 and mu = k / k the Generalized Pareto is equivalent to the Pareto distribution. The mean of the Generalized Pareto is not finite when k >= 1 and the variance is not finite when k >= 1/2. When k >= 0, the Generalized Pareto has positive density for x > mu, or, when mu < 0, for 0 <= (x - mu) / sigma <= -1 / k.

Further information about the generalized Pareto distribution can be found at https://en.wikipedia.org/wiki/Generalized_Pareto_distribution

See also: gpcdf, gpinv, gprnd, gpfit, gplike, gpstat

Source Code: gppdf

Example: 1

  

 ## Plot various PDFs from the generalized Pareto distribution
 x = 0:0.001:5;
 y1 = gppdf (x, 1, 1, 0);
 y2 = gppdf (x, 5, 1, 0);
 y3 = gppdf (x, 20, 1, 0);
 y4 = gppdf (x, 1, 2, 0);
 y5 = gppdf (x, 5, 2, 0);
 y6 = gppdf (x, 20, 2, 0);
 plot (x, y1, "-b", x, y2, "-g", x, y3, "-r", ...
       x, y4, "-c", x, y5, "-m", x, y6, "-k")
 grid on
 xlim ([0, 5])
 ylim ([0, 1])
 legend ({"ξ = 1, σ = 1, μ = 0", "ξ = 5, σ = 1, μ = 0", ...
          "ξ = 20, σ = 1, μ = 0", "ξ = 1, σ = 2, μ = 0", ...
          "ξ = 5, σ = 2, μ = 0", "ξ = 20, σ = 2, μ = 0"}, ...
         "location", "northeast")
 title ("Generalized Pareto PDF")
 xlabel ("values in x")
 ylabel ("density")
plotted figure

statistics-release-1.6.3/docs/gprnd.html000066400000000000000000000141371456127120000202650ustar00rootroot00000000000000 Statistics: gprnd

Function Reference: gprnd

statistics: r = gprnd (k, sigma, mu)
statistics: r = gprnd (k, sigma, mu, rows)
statistics: r = gprnd (k, sigma, mu, rows, cols, …)
statistics: r = gprnd (k, sigma, mu, [sz])

Random arrays from the generalized Pareto distribution.

r = gprnd (k, sigma, mu) returns an array of random numbers chosen from the generalized Pareto distribution with shape parameter k, scale parameter sigma, and location parameter mu. The size of r is the common size of k, sigma, and mu. A scalar input functions as a constant matrix of the same size as the other inputs.

When called with a single size argument, gprnd returns a square matrix with the dimension specified. When called with more than one scalar argument, the first two arguments are taken as the number of rows and columns and any further arguments specify additional matrix dimensions. The size may also be specified with a row vector of dimensions, sz.

When k = 0 and mu = 0, the Generalized Pareto CDF is equivalent to the exponential distribution. When k > 0 and mu = k / k the Generalized Pareto is equivalent to the Pareto distribution. The mean of the Generalized Pareto is not finite when k >= 1 and the variance is not finite when k >= 1/2. When k >= 0, the Generalized Pareto has positive density for x > mu, or, when mu < 0, for 0 <= (x - mu) / sigma <= -1 / k.

Further information about the generalized Pareto distribution can be found at https://en.wikipedia.org/wiki/Generalized_Pareto_distribution

See also: gpcdf, gpinv, gppdf, gpfit, gplike, gpstat

Source Code: gprnd

statistics-release-1.6.3/docs/gpstat.html000066400000000000000000000126051456127120000204530ustar00rootroot00000000000000 Statistics: gpstat

Function Reference: gpstat

statistics: [m, v] = gpstat (k, sigma, mu)

Compute statistics of the generalized Pareto distribution.

[m, v] = gpstat (k, sigma, mu) returns the mean and variance of the generalized Pareto distribution with shape parameter k, scale parameter sigma, and location parameter mu.

The size of m (mean) and v (variance) is the common size of the input arguments. A scalar input functions as a constant matrix of the same size as the other inputs.

When k = 0 and mu = 0, the generalized Pareto distribution is equivalent to the exponential distribution. When k > 0 and mu = sigma / k, the generalized Pareto distribution is equivalent to the Pareto distribution. The mean of the generalized Pareto distribution is not finite when k >= 1, and the variance is not finite when k >= 1/2. When k >= 0, the generalized Pareto distribution has positive density for x > mu, or, when k < 0, for 0 <= (x - mu) / sigma <= -1 / k.

Further information about the generalized Pareto distribution can be found at https://en.wikipedia.org/wiki/Generalized_Pareto_distribution

See also: gpcdf, gpinv, gppdf, gprnd, gpfit, gplike

Source Code: gpstat

statistics-release-1.6.3/docs/grp2idx.html000066400000000000000000000101551456127120000205260ustar00rootroot00000000000000 Statistics: grp2idx

Function Reference: grp2idx

statistics: [g, gn, gl] = grp2idx (s)

Get index for group variables.

For variable s, returns the indices g, into the variable groups gn and gl. The first has a string representation of the groups while the later has its actual values. The group indices are allocated in order of appearance in s.

NaNs and empty strings in s appear as NaN in g and are not present on either gn and gl.

See also: grpstats

Source Code: grp2idx

statistics-release-1.6.3/docs/grpstats.html000066400000000000000000000240541456127120000210210ustar00rootroot00000000000000 Statistics: grpstats

Function Reference: grpstats

statistics: mean = grpstats (x)
statistics: mean = grpstats (x, group)
statistics: [a, b, …] = grpstats (x, group, whichstats)
statistics: [a, b, …] = grpstats (x, group, whichstats, alpha)

Summary statistics by group. grpstats computes groupwise summary statistics, for data in a matrix x. grpstats treats NaNs as missing values, and removes them.

means = grpstats (x, group), when X is a matrix of observations, returns the means of each column of x by group. group is a grouping variable defined as a categorical variable, numeric, string array, or cell array of strings. group can be [] or omitted to compute the mean of the entire sample without grouping.

[a, b, …] = grpstats (x, group, whichstats), for a numeric matrix X, returns the statistics specified by whichstats, as separate arrays a, b, …. whichstats can be a single function name, or a cell array containing multiple function names. The number of output arguments must match the number of function names in whichstats. Names in whichstats can be chosen from among the following:

"mean"mean
"median"median
"sem"standard error of the mean
"std"standard deviation
"var"variance
"min"minimum value
"max"maximum value
"range"maximum - minimum
"numel"count, or number of elements
"meanci"95% confidence interval for the mean
"predci"95% prediction interval for a new observation
"gname"group name

[…] = grpstats (x, group, whichstats, alpha) specifies the confidence level as 100(1-ALPHA)% for the "meanci" and "predci" options. Default value for alpha is 0.05.

Examples:

 
 load carsmall;
 [m,p,g] = grpstats (Weight, Model_Year, {"mean", "predci", "gname"})
 n = length(m);
 errorbar((1:n)',m,p(:,2)-m)
 set (gca, "xtick", 1:n, "xticklabel", g);
 title ("95% prediction intervals for mean weight by year")

See also: grp2idx

Source Code: grpstats

Example: 1

  

 load carsmall;
 [m,p,g] = grpstats (Weight, Model_Year, {"mean", "predci", "gname"})
 n = length(m);
 errorbar((1:n)',m,p(:,2)-m);
 set (gca, "xtick", 1:n, "xticklabel", g);
 title ("95% prediction intervals for mean weight by year");

m =

   3441.3
   3078.7
   2453.5

p =

   1847.1   5035.6
   1457.0   4700.4
   1754.1   3153.0

g =
{
  [1,1] = 70
  [2,1] = 76
  [3,1] = 82
}
plotted figure

Example: 2

  

 load carsmall;
 [m,p,g] = grpstats ([Acceleration,Weight/1000],Cylinders, ...
                     {"mean", "meanci", "gname"}, 0.05)
 [c,r] = size (m);
 errorbar((1:c)'.*ones(c,r),m,p(:,[(1:r)])-m);
 set (gca, "xtick", 1:c, "xticklabel", g);
 title ("95% prediction intervals for mean weight by year");

m =

   11.6406    3.9703
   16.6706    2.3726
   16.4765    3.1255

p =

   11.1762    3.8899   12.1050    4.0506
   16.1385    2.2998   17.2027    2.4454
   16.1622    3.0518   16.7907    3.1992

g =
{
  [1,1] = 8
  [2,1] = 4
  [3,1] = 6
}
plotted figure

statistics-release-1.6.3/docs/gscatter.html000066400000000000000000000157271456127120000207750ustar00rootroot00000000000000 Statistics: gscatter

Function Reference: gscatter

statistics: gscatter (x, y, g)
statistics: gscatter (x, y, g, clr, sym, siz)
statistics: gscatter (…, doleg, xnam, ynam)
statistics: h = gscatter (…)

Draw a scatter plot with grouped data.

gscatter is a utility function to draw a scatter plot of x and y, according to the groups defined by g. Input x and y are numeric vectors of the same size, while g is either a vector of the same size as x or a character matrix with the same number of rows as the size of x. As a vector g can be numeric, logical, a character array, a string array (not implemented), a cell string or cell array.

A number of optional inputs change the appearance of the plot:

  • "clr" defines the color for each group; if not enough colors are defined by "clr", gscatter cycles through the specified colors. Colors can be defined as named colors, as rgb triplets or as indices for the current colormap. The default value is a different color for each group, according to the current colormap.
  • "sym" is a char array of symbols for each group; if not enough symbols are defined by "sym", gscatter cycles through the specified symbols.
  • "siz" is a numeric array of sizes for each group; if not enough sizes are defined by "siz", gscatter cycles through the specified sizes.
  • "doleg" is a boolean value to show the legend; it can be either on (default) or off.
  • "xnam" is a character array, the name for the x axis.
  • "ynam" is a character array, the name for the y axis.

Output h is an array of graphics handles to the line object of each group.

See also: scatter

Source Code: gscatter

Example: 1

  

 load fisheriris;
 X = meas(:,3:4);
 cidcs = kmeans (X, 3, "Replicates", 5);
 gscatter (X(:,1), X(:,2), cidcs, [.75 .75 0; 0 .75 .75; .75 0 .75], "os^");
 title ("Fisher's iris data");
plotted figure

statistics-release-1.6.3/docs/gumbelcdf.html000066400000000000000000000202531456127120000210770ustar00rootroot00000000000000 Statistics: gumbelcdf

Function Reference: gumbelcdf

statistics: p = gumbelcdf (x)
statistics: p = gumbelcdf (x, mu)
statistics: p = gumbelcdf (x, mu, beta)
statistics: p = gumbelcdf (…, "upper")
statistics: [p, plo, pup] = gumbelcdf (x, mu, beta, pcov)
statistics: [p, plo, pup] = gumbelcdf (x, mu, beta, pcov, alpha)
statistics: [p, plo, pup] = gumbelcdf (…, "upper")

Gumbel cumulative distribution function (CDF).

For each element of x, compute the cumulative distribution function (CDF) of the Gumbel distribution (also known as the extreme value or the type I generalized extreme value distribution) with location parameter mu and scale parameter beta. The size of p is the common size of x, mu and beta. A scalar input functions as a constant matrix of the same size as the other inputs.

Default values are mu = 0 and beta = 1.

When called with three output arguments, i.e. [p, plo, pup], gumbelcdf computes the confidence bounds for p when the input parameters mu and beta are estimates. In such case, pcov, a 2×2 matrix containing the covariance matrix of the estimated parameters, is necessary. Optionally, alpha, which has a default value of 0.05, specifies the 100 * (1 - alpha) percent confidence bounds. plo and pup are arrays of the same size as p containing the lower and upper confidence bounds.

[…] = gumbelcdf (…, "upper") computes the upper tail probability of the Gumbel distribution with parameters mu and beta, at the values in x.

The Gumbel distribution is used to model the distribution of the maximum (or the minimum) of a number of samples of various distributions. This version is suitable for modeling maxima. For modeling minima, use the alternative extreme value CDF, evcdf.

[…] = gumbelcdf (…, "upper") computes the upper tail probability of the extreme value (Gumbel) distribution.

Further information about the Gumbel distribution can be found at https://en.wikipedia.org/wiki/Gumbel_distribution

See also: gumbelinv, gumbelpdf, gumbelrnd, gumbelfit, gumbellike, gumbelstat, evcdf

Source Code: gumbelcdf

Example: 1

  

 ## Plot various CDFs from the Gumbel distribution
 x = -5:0.01:20;
 p1 = gumbelcdf (x, 0.5, 2);
 p2 = gumbelcdf (x, 1.0, 2);
 p3 = gumbelcdf (x, 1.5, 3);
 p4 = gumbelcdf (x, 3.0, 4);
 plot (x, p1, "-b", x, p2, "-g", x, p3, "-r", x, p4, "-c")
 grid on
 legend ({"μ = 0.5, β = 2", "μ = 1.0, β = 2", ...
          "μ = 1.5, β = 3", "μ = 3.0, β = 4"}, "location", "southeast")
 title ("Gumbel CDF")
 xlabel ("values in x")
 ylabel ("probability")
plotted figure

statistics-release-1.6.3/docs/gumbelfit.html000066400000000000000000000233071456127120000211300ustar00rootroot00000000000000 Statistics: gumbelfit

Function Reference: gumbelfit

statistics: paramhat = gumbelfit (x)
statistics: [paramhat, paramci] = gumbelfit (x)
statistics: [paramhat, paramci] = gumbelfit (x, alpha)
statistics: […] = gumbelfit (x, alpha, censor)
statistics: […] = gumbelfit (x, alpha, censor, freq)
statistics: […] = gumbelfit (x, alpha, censor, freq, options)

Estimate parameters and confidence intervals for Gumbel distribution.

paramhat = gumbelfit (x) returns the maximum likelihood estimates of the parameters of the Gumbel distribution (also known as the extreme value or the type I generalized extreme value distribution) given in x. paramhat(1) is the location parameter, mu, and paramhat(2) is the scale parameter, beta.

[paramhat, paramci] = gumbelfit (x) returns the 95% confidence intervals for the parameter estimates.

[…] = gumbelfit (x, alpha) also returns the 100 * (1 - alpha) percent confidence intervals for the parameter estimates. By default, the optional argument alpha is 0.05 corresponding to 95% confidence intervals. Pass in [] for alpha to use the default values.

[…] = gumbelfit (x, alpha, censor) accepts a boolean vector, censor, of the same size as x with 1s for observations that are right-censored and 0s for observations that are observed exactly. By default, or if left empty, censor = zeros (size (x)).

[…] = gumbelfit (x, alpha, censor, freq) accepts a frequency vector, freq, of the same size as x. freq typically contains integer frequencies for the corresponding elements in x, but it can contain any non-integer non-negative values. By default, or if left empty, freq = ones (size (x)).

[…] = gumbelfit (…, options) specifies control parameters for the iterative algorithm used to compute the maximum likelihood estimates. options is a structure with the following field and its default value:

  • options.TolX = 1e-6

The Gumbel distribution is used to model the distribution of the maximum (or the minimum) of a number of samples of various distributions. This version is suitable for modeling maxima. For modeling minima, use the alternative extreme value fitting function, evfit.

Further information about the Gumbel distribution can be found at https://en.wikipedia.org/wiki/Gumbel_distribution

See also: gumbelcdf, gumbelinv, gumbelpdf, gumbelrnd, gumbellike, gumbelstat, evfit

Source Code: gumbelfit

Example: 1

  

 ## Sample 3 populations from different Gumbel distibutions
 rand ("seed", 1);    # for reproducibility
 r1 = gumbelrnd (2, 5, 400, 1);
 rand ("seed", 11);    # for reproducibility
 r2 = gumbelrnd (-5, 3, 400, 1);
 rand ("seed", 16);    # for reproducibility
 r3 = gumbelrnd (14, 8, 400, 1);
 r = [r1, r2, r3];

 ## Plot them normalized and fix their colors
 hist (r, 25, 0.32);
 h = findobj (gca, "Type", "patch");
 set (h(1), "facecolor", "c");
 set (h(2), "facecolor", "g");
 set (h(3), "facecolor", "r");
 ylim ([0, 0.28])
 xlim ([-11, 50]);
 hold on

 ## Estimate their MU and BETA parameters
 mu_betaA = gumbelfit (r(:,1));
 mu_betaB = gumbelfit (r(:,2));
 mu_betaC = gumbelfit (r(:,3));

 ## Plot their estimated PDFs
 x = [min(r(:)):max(r(:))];
 y = gumbelpdf (x, mu_betaA(1), mu_betaA(2));
 plot (x, y, "-pr");
 y = gumbelpdf (x, mu_betaB(1), mu_betaB(2));
 plot (x, y, "-sg");
 y = gumbelpdf (x, mu_betaC(1), mu_betaC(2));
 plot (x, y, "-^c");
 legend ({"Normalized HIST of sample 1 with μ=2 and β=5", ...
          "Normalized HIST of sample 2 with μ=-5 and β=3", ...
          "Normalized HIST of sample 3 with μ=14 and β=8", ...
          sprintf("PDF for sample 1 with estimated μ=%0.2f and β=%0.2f", ...
                  mu_betaA(1), mu_betaA(2)), ...
          sprintf("PDF for sample 2 with estimated μ=%0.2f and β=%0.2f", ...
                  mu_betaB(1), mu_betaB(2)), ...
          sprintf("PDF for sample 3 with estimated μ=%0.2f and β=%0.2f", ...
                  mu_betaC(1), mu_betaC(2))})
 title ("Three population samples from different Gumbel distibutions")
 hold off
plotted figure

statistics-release-1.6.3/docs/gumbelinv.html000066400000000000000000000171311456127120000211400ustar00rootroot00000000000000 Statistics: gumbelinv

Function Reference: gumbelinv

statistics: x = gumbelinv (p)
statistics: x = gumbelinv (p, mu)
statistics: x = gumbelinv (p, mu, beta)
statistics: [x, xlo, xup] = gumbelinv (p, mu, beta, pcov)
statistics: [x, xlo, xup] = gumbelinv (p, mu, beta, pcov, alpha)

Inverse of the Gumbel cumulative distribution function (iCDF).

For each element of p, compute the quantile (the inverse of the CDF) of the Gumbel distribution (also known as the extreme value or the type I generalized extreme value distribution) with location parameter mu and scale parameter beta. The size of x is the common size of p, mu and beta. A scalar input functions as a constant matrix of the same size as the other inputs.

Default values are mu = 0 and beta = 1.

When called with three output arguments, i.e. [x, xlo, xup], gumbelinv computes the confidence bounds for x when the input parameters mu and beta are estimates. In such case, pcov, a 2×2 matrix containing the covariance matrix of the estimated parameters, is necessary. Optionally, alpha, which has a default value of 0.05, specifies the 100 * (1 - alpha) percent confidence bounds. xlo and xup are arrays of the same size as x containing the lower and upper confidence bounds.

The Gumbel distribution is used to model the distribution of the maximum (or the minimum) of a number of samples of various distributions. This version is suitable for modeling maxima. For modeling minima, use the alternative extreme value iCDF, evinv.

Further information about the Gumbel distribution can be found at https://en.wikipedia.org/wiki/Gumbel_distribution

See also: gumbelcdf, gumbelpdf, gumbelrnd, gumbelfit, gumbellike, gumbelstat, evinv

Source Code: gumbelinv

Example: 1

  

 ## Plot various iCDFs from the Gumbel distribution
 p = 0.001:0.001:0.999;
 x1 = gumbelinv (p, 0.5, 2);
 x2 = gumbelinv (p, 1.0, 2);
 x3 = gumbelinv (p, 1.5, 3);
 x4 = gumbelinv (p, 3.0, 4);
 plot (p, x1, "-b", p, x2, "-g", p, x3, "-r", p, x4, "-c")
 grid on
 ylim ([-5, 20])
 legend ({"μ = 0.5, β = 2", "μ = 1.0, β = 2", ...
          "μ = 1.5, β = 3", "μ = 3.0, β = 4"}, "location", "northwest")
 title ("Gumbel iCDF")
 xlabel ("probability")
 ylabel ("values in x")
plotted figure

statistics-release-1.6.3/docs/gumbellike.html000066400000000000000000000145711456127120000212750ustar00rootroot00000000000000 Statistics: gumbellike

Function Reference: gumbellike

statistics: nlogL = gumbellike (params, x)
statistics: [nlogL, avar] = gumbellike (params, x)
statistics: […] = gumbellike (params, x, censor)
statistics: […] = gumbellike (params, x, censor, freq)

Negative log-likelihood for the extreme value distribution.

nlogL = gumbellike (params, x) returns the negative log likelihood of the data in x corresponding to the Gumbel distribution (also known as the extreme value or the type I generalized extreme value distribution) with (1) location parameter mu and (2) scale parameter beta given in the two-element vector params.

[nlogL, acov] = gumbellike (params, x) also returns the inverse of Fisher’s information matrix, acov. If the input parameter values in params are the maximum likelihood estimates, the diagonal elements of acov are their asymptotic variances.

[…] = gumbellike (params, x, censor) accepts a boolean vector, censor, of the same size as x with 1s for observations that are right-censored and 0s for observations that are observed exactly. By default, or if left empty, censor = zeros (size (x)).

[…] = gumbellike (params, x, censor, freq) accepts a frequency vector, freq, of the same size as x. freq typically contains integer frequencies for the corresponding elements in x, but it can contain any non-integer non-negative values. By default, or if left empty, freq = ones (size (x)).

The Gumbel distribution is used to model the distribution of the maximum (or the minimum) of a number of samples of various distributions. This version is suitable for modeling maxima. For modeling minima, use the alternative extreme value likelihood function, evlike.

Further information about the Gumbel distribution can be found at https://en.wikipedia.org/wiki/Gumbel_distribution

See also: gumbelcdf, gumbelinv, gumbelpdf, gumbelrnd, gumbelfit, gumbelstat, evlike

Source Code: gumbellike

statistics-release-1.6.3/docs/gumbelpdf.html000066400000000000000000000150751456127120000211220ustar00rootroot00000000000000 Statistics: gumbelpdf

Function Reference: gumbelpdf

statistics: y = gumbelpdf (x)
statistics: y = gumbelpdf (x, mu)
statistics: y = gumbelpdf (x, mu, beta)

Gumbel probability density function (PDF).

For each element of x, compute the probability density function (PDF) of the Gumbel distribution (also known as the extreme value or the type I generalized extreme value distribution) with location parameter mu and scale parameter beta. The size of y is the common size of x, mu and beta. A scalar input functions as a constant matrix of the same size as the other inputs.

Default values are mu = 0 and beta = 1.

The Gumbel distribution is used to model the distribution of the maximum (or the minimum) of a number of samples of various distributions. This version is suitable for modeling maxima. For modeling minima, use the alternative extreme value iCDF, evpdf.

Further information about the Gumbel distribution can be found at https://en.wikipedia.org/wiki/Gumbel_distribution

See also: gumbelcdf, gumbelinv, gumbelrnd, gumbelfit, gumbellike, gumbelstat, evpdf

Source Code: gumbelpdf

Example: 1

  

 ## Plot various PDFs from the Extreme value distribution
 x = -5:0.001:20;
 y1 = gumbelpdf (x, 0.5, 2);
 y2 = gumbelpdf (x, 1.0, 2);
 y3 = gumbelpdf (x, 1.5, 3);
 y4 = gumbelpdf (x, 3.0, 4);
 plot (x, y1, "-b", x, y2, "-g", x, y3, "-r", x, y4, "-c")
 grid on
 ylim ([0, 0.2])
 legend ({"μ = 0.5, β = 2", "μ = 1.0, β = 2", ...
          "μ = 1.5, β = 3", "μ = 3.0, β = 4"}, "location", "northeast")
 title ("Extreme value PDF")
 xlabel ("values in x")
 ylabel ("density")
plotted figure

statistics-release-1.6.3/docs/gumbelrnd.html000066400000000000000000000131621456127120000211270ustar00rootroot00000000000000 Statistics: gumbelrnd

Function Reference: gumbelrnd

statistics: r = gumbelrnd (mu, beta)
statistics: r = gumbelrnd (mu, beta, rows)
statistics: r = gumbelrnd (mu, beta, rows, cols, …)
statistics: r = gumbelrnd (mu, beta, [sz])

Random arrays from the Gumbel distribution.

r = gumbelrnd (mu, beta) returns an array of random numbers chosen from the Gumbel distribution (also known as the extreme value or the type I generalized extreme value distribution) with location parameter mu and scale parameter beta. The size of r is the common size of mu and beta. A scalar input functions as a constant matrix of the same size as the other inputs.

When called with a single size argument, gumbelrnd returns a square matrix with the dimension specified. When called with more than one scalar argument, the first two arguments are taken as the number of rows and columns and any further arguments specify additional matrix dimensions. The size may also be specified with a row vector of dimensions, sz.

The Gumbel distribution is used to model the distribution of the maximum (or the minimum) of a number of samples of various distributions. This version is suitable for modeling maxima. For modeling minima, use the alternative extreme value iCDF, evinv.

Further information about the Gumbel distribution can be found at https://en.wikipedia.org/wiki/Gumbel_distribution

See also: gumbelcdf, gumbelinv, gumbelpdf, gumbelfit, gumbellike, gumbelstat, evrnd

Source Code: gumbelrnd

statistics-release-1.6.3/docs/harmmean.html000066400000000000000000000143001456127120000207330ustar00rootroot00000000000000 Statistics: harmmean

Function Reference: harmmean

statistics: m = harmmean (x)
statistics: m = harmmean (x, "all")
statistics: m = harmmean (x, dim)
statistics: m = harmmean (x, vecdim)
statistics: m = harmmean (…, nanflag)

Compute the harmonic mean of x.

  • If x is a vector, then harmmean(x) returns the harmonic mean of the elements in x defined as $$ {\rm harmmean}(x) = \frac{N}{\sum_{i=1}^N \frac{1}{x_i}} $$ where N is the length of the x vector.
  • If x is a matrix, then harmmean(x) returns a row vector with the harmonic mean of each columns in x.
  • If x is a multidimensional array, then harmmean(x) operates along the first nonsingleton dimension of x.
  • x must not contain any negative or complex values.

harmmean(x, "all") returns the harmonic mean of all the elements in x. If x contains any 0, then the returned value is 0.

harmmean(x, dim) returns the harmonic mean along the operating dimension dim of x. Calculating the harmonic mean of any subarray containing any 0 will return 0.

harmmean(x, vecdim) returns the harmonic mean over the dimensions specified in the vector vecdim. For example, if x is a 2-by-3-by-4 array, then harmmean(x, [1 2]) returns a 1-by-1-by-4 array. Each element of the output array is the harmonic mean of the elements on the corresponding page of x. If vecdim indexes all dimensions of x, then it is equivalent to harmmean (x, "all"). Any dimension in vecdim greater than ndims (x) is ignored.

harmmean(…, nanflag) specifies whether to exclude NaN values from the calculation, using any of the input argument combinations in previous syntaxes. By default, harmmean includes NaN values in the calculation (nanflag has the value "includenan"). To exclude NaN values, set the value of nanflag to "omitnan".

See also: geomean, mean

Source Code: harmmean

statistics-release-1.6.3/docs/hist3.html000066400000000000000000000166771456127120000202200ustar00rootroot00000000000000 Statistics: hist3

Function Reference: hist3

statistics: hist3 (X)
statistics: hist3 (X, nbins)
statistics: hist3 (X, "Nbins", nbins)
statistics: hist3 (X, centers)
statistics: hist3 (X, "Ctrs", centers)
statistics: hist3 (X, "Edges", edges)
statistics: [N, C] = hist3 (…)
statistics: hist3 (…, prop, val, …)
statistics: hist3 (hax, …)

Produce bivariate (2D) histogram counts or plots.

The elements to produce the histogram are taken from the Nx2 matrix X. Any row with NaN values are ignored. The actual bins can be configured in 3 different: number, centers, or edges of the bins:

Number of bins (default)

Produces equally spaced bins between the minimum and maximum values of X. Defined as a 2 element vector, nbins, one for each dimension. Defaults to [10 10].

Center of bins

Defined as a cell array of 2 monotonically increasing vectors, centers. The width of each bin is determined from the adjacent values in the vector with the initial and final bin, extending to Infinity.

Edge of bins

Defined as a cell array of 2 monotonically increasing vectors, edges. N(i,j) contains the number of elements in X for which:

  • edges{1}(i) <= X(:,1) < edges{1}(i+1)
  • edges{2}(j) <= X(:,2) < edges{2}(j+1)

The consequence of this definition is that values outside the initial and final edge values are ignored, and that the final bin only contains the number of elements exactly equal to the final edge.

The return values, N and C, are the bin counts and centers respectively. These are specially useful to produce intensity maps:

 
 [counts, centers] = hist3 (data);
 imagesc (centers{1}, centers{2}, counts)

If there is no output argument, or if the axes graphics handle hax is defined, the function will plot a 3 dimensional bar graph. Any extra property/value pairs are passed directly to the underlying surface object.

See also: hist, histc, lookup, mesh

Source Code: hist3

Example: 1

  

 X = [
    1    1
    1    1
    1   10
    1   10
    5    5
    5    5
    5    5
    5    5
    5    5
    7    3
    7    3
    7    3
   10   10
   10   10];
 hist3 (X)
plotted figure

statistics-release-1.6.3/docs/histfit.html000066400000000000000000000124551456127120000206260ustar00rootroot00000000000000 Statistics: histfit

Function Reference: histfit

statistics: histfit (x, nbins)
statistics: h = histfit (x, nbins)

Plot histogram with superimposed fitted normal density.

histfit (x, nbins) plots a histogram of the values in the vector x using nbins bars in the histogram. With one input argument, nbins is set to the square root of the number of elements in x.

h = histfit (x, nbins) returns the bins and fitted line handles of the plot in h.

Example

 
 histfit (randn (100, 1))

See also: bar, hist, pareto

Source Code: histfit

Example: 1

  

 histfit (randn (100, 1))
plotted figure

statistics-release-1.6.3/docs/hmmestimate.html000066400000000000000000000214471456127120000214720ustar00rootroot00000000000000 Statistics: hmmestimate

Function Reference: hmmestimate

statistics: [transprobest, outprobest] = hmmestimate (sequence, states)
statistics: […] = hmmestimate (…, "statenames", statenames)
statistics: […] = hmmestimate (…, "symbols", symbols)
statistics: […] = hmmestimate (…, "pseudotransitions", pseudotransitions)
statistics: […] = hmmestimate (…, "pseudoemissions", pseudoemissions)

Estimation of a hidden Markov model for a given sequence.

Estimate the matrix of transition probabilities and the matrix of output probabilities of a given sequence of outputs and states generated by a hidden Markov model. The model assumes that the generation starts in state 1 at step 0 but does not include step 0 in the generated states and sequence.

Arguments

  • sequence is a vector of a sequence of given outputs. The outputs must be integers ranging from 1 to the number of outputs of the hidden Markov model.
  • states is a vector of the same length as sequence of given states. The states must be integers ranging from 1 to the number of states of the hidden Markov model.

Return values

  • transprobest is the matrix of the estimated transition probabilities of the states. transprobest(i, j) is the estimated probability of a transition to state j given state i.
  • outprobest is the matrix of the estimated output probabilities. outprobest(i, j) is the estimated probability of generating output j given state i.

If 'symbols' is specified, then sequence is expected to be a sequence of the elements of symbols instead of integers. symbols can be a cell array.

If 'statenames' is specified, then states is expected to be a sequence of the elements of statenames instead of integers. statenames can be a cell array.

If 'pseudotransitions' is specified then the integer matrix pseudotransitions is used as an initial number of counted transitions. pseudotransitions(i, j) is the initial number of counted transitions from state i to state j. transprobest will have the same size as pseudotransitions. Use this if you have transitions that are very unlikely to occur.

If 'pseudoemissions' is specified then the integer matrix pseudoemissions is used as an initial number of counted outputs. pseudoemissions(i, j) is the initial number of counted outputs j given state i. If 'pseudoemissions' is also specified then the number of rows of pseudoemissions must be the same as the number of rows of pseudotransitions. outprobest will have the same size as pseudoemissions. Use this if you have outputs or states that are very unlikely to occur.

Examples

 
 
 transprob = [0.8, 0.2; 0.4, 0.6];
 outprob = [0.2, 0.4, 0.4; 0.7, 0.2, 0.1];
 [sequence, states] = hmmgenerate (25, transprob, outprob);
 [transprobest, outprobest] = hmmestimate (sequence, states)
 
 
 symbols = {"A", "B", "C"};
 statenames = {"One", "Two"};
 [sequence, states] = hmmgenerate (25, transprob, outprob, ...
                                   "symbols", symbols, ...
                                   "statenames", statenames);
 [transprobest, outprobest] = hmmestimate (sequence, states, ...
                                   "symbols', symbols, ...
                                   "statenames', statenames)
 
 
 pseudotransitions = [8, 2; 4, 6];
 pseudoemissions = [2, 4, 4; 7, 2, 1];
 [sequence, states] = hmmgenerate (25, transprob, outprob);
 [transprobest, outprobest] = hmmestimate (sequence, states, ...
                              "pseudotransitions", pseudotransitions, ...
                              "pseudoemissions", pseudoemissions)

References

  1. Wendy L. Martinez and Angel R. Martinez. Computational Statistics Handbook with MATLAB. Appendix E, pages 547-557, Chapman & Hall/CRC, 2001.
  2. Lawrence R. Rabiner. A Tutorial on Hidden Markov Models and Selected Applications in Speech Recognition. Proceedings of the IEEE, 77(2), pages 257-286, February 1989.

Source Code: hmmestimate

statistics-release-1.6.3/docs/hmmgenerate.html000066400000000000000000000155601456127120000214500ustar00rootroot00000000000000 Statistics: hmmgenerate

Function Reference: hmmgenerate

statistics: [sequence, states] = hmmgenerate (len, transprob, outprob)
statistics: […] = hmmgenerate (…, "symbols", symbols)
statistics: […] = hmmgenerate (…, "statenames", statenames)

Output sequence and hidden states of a hidden Markov model.

Generate an output sequence and hidden states of a hidden Markov model. The model starts in state 1 at step 0 but will not include step 0 in the generated states and sequence.

Arguments

  • len is the number of steps to generate. sequence and states will have len entries each.
  • transprob is the matrix of transition probabilities of the states. transprob(i, j) is the probability of a transition to state j given state i.
  • outprob is the matrix of output probabilities. outprob(i, j) is the probability of generating output j given state i.

Return values

  • sequence is a vector of length len of the generated outputs. The outputs are integers ranging from 1 to columns (outprob).
  • states is a vector of length len of the generated hidden states. The states are integers ranging from 1 to columns (transprob).

If "symbols" is specified, then the elements of symbols are used for the output sequence instead of integers ranging from 1 to columns (outprob). symbols can be a cell array.

If "statenames" is specified, then the elements of statenames are used for the states instead of integers ranging from 1 to columns (transprob). statenames can be a cell array.

Examples

 
 
 transprob = [0.8, 0.2; 0.4, 0.6];
 outprob = [0.2, 0.4, 0.4; 0.7, 0.2, 0.1];
 [sequence, states] = hmmgenerate (25, transprob, outprob)
 
 
 symbols = {"A", "B", "C"};
 statenames = {"One", "Two"};
 [sequence, states] = hmmgenerate (25, transprob, outprob, ...
                                   "symbols", symbols, ...
                                   "statenames", statenames)

References

  1. Wendy L. Martinez and Angel R. Martinez. Computational Statistics Handbook with MATLAB. Appendix E, pages 547-557, Chapman & Hall/CRC, 2001.
  2. Lawrence R. Rabiner. A Tutorial on Hidden Markov Models and Selected Applications in Speech Recognition. Proceedings of the IEEE, 77(2), pages 257-286, February 1989.

Source Code: hmmgenerate

statistics-release-1.6.3/docs/hmmviterbi.html000066400000000000000000000156611456127120000213240ustar00rootroot00000000000000 Statistics: hmmviterbi

Function Reference: hmmviterbi

statistics: vpath = hmmviterbi (sequence, transprob, outprob)
statistics: vpath = hmmviterbi (…, "symbols", symbols)
statistics: vpath = hmmviterbi (…, "statenames", statenames)

Viterbi path of a hidden Markov model.

Use the Viterbi algorithm to find the Viterbi path of a hidden Markov model given a sequence of outputs. The model assumes that the generation starts in state 1 at step 0 but does not include step 0 in the generated states and sequence.

Arguments

  • sequence is the vector of length len of given outputs. The outputs must be integers ranging from 1 to columns (outprob).
  • transprob is the matrix of transition probabilities of the states. transprob(i, j) is the probability of a transition to state j given state i.
  • outprob is the matrix of output probabilities. outprob(i, j) is the probability of generating output j given state i.

Return values

  • vpath is the vector of the same length as sequence of the estimated hidden states. The states are integers ranging from 1 to columns (transprob).

If "symbols" is specified, then sequence is expected to be a sequence of the elements of symbols instead of integers ranging from 1 to columns (outprob). symbols can be a cell array.

If "statenames" is specified, then the elements of statenames are used for the states in vpath instead of integers ranging from 1 to columns (transprob). statenames can be a cell array.

Examples

 
 
 transprob = [0.8, 0.2; 0.4, 0.6];
 outprob = [0.2, 0.4, 0.4; 0.7, 0.2, 0.1];
 [sequence, states] = hmmgenerate (25, transprob, outprob);
 vpath = hmmviterbi (sequence, transprob, outprob);
 
 
 symbols = {"A", "B", "C"};
 statenames = {"One", "Two"};
 [sequence, states] = hmmgenerate (25, transprob, outprob, ...
                      "symbols", symbols, "statenames", statenames);
 vpath = hmmviterbi (sequence, transprob, outprob, ...
         "symbols", symbols, "statenames", statenames);

References

  1. Wendy L. Martinez and Angel R. Martinez. Computational Statistics Handbook with MATLAB. Appendix E, pages 547-557, Chapman & Hall/CRC, 2001.
  2. Lawrence R. Rabiner. A Tutorial on Hidden Markov Models and Selected Applications in Speech Recognition. Proceedings of the IEEE, 77(2), pages 257-286, February 1989.

Source Code: hmmviterbi

statistics-release-1.6.3/docs/hncdf.html000066400000000000000000000173451456127120000202410ustar00rootroot00000000000000 Statistics: hncdf

Function Reference: hncdf

statistics: p = hncdf (x, mu, sigma)
statistics: p = hncdf (x, mu, sigma, "upper")

Half-normal cumulative distribution function (CDF).

For each element of x, compute the cumulative distribution function (CDF) of the half-normal distribution with location parameter mu and scale parameter sigma. The size of p is the common size of x, mu and sigma. A scalar input functions as a constant matrix of the same size as the other inputs.

[…] = hncdf (x, mu, sigma, "upper") computes the upper tail probability of the half-normal distribution with parameters mu and sigma, at the values in x.

The half-normal CDF is only defined for x >= mu.

Further information about the half-normal distribution can be found at https://en.wikipedia.org/wiki/Half-normal_distribution

See also: hninv, hnpdf, hnrnd, hnfit, hnlike

Source Code: hncdf

Example: 1

  

 ## Plot various CDFs from the half-normal distribution
 x = 0:0.001:10;
 p1 = hncdf (x, 0, 1);
 p2 = hncdf (x, 0, 2);
 p3 = hncdf (x, 0, 3);
 p4 = hncdf (x, 0, 5);
 plot (x, p1, "-b", x, p2, "-g", x, p3, "-r", x, p4, "-c")
 grid on
 xlim ([0, 10])
 legend ({"μ = 0, σ = 1", "μ = 0, σ = 2", ...
          "μ = 0, σ = 3", "μ = 0, σ = 5"}, "location", "southeast")
 title ("Half-normal CDF")
 xlabel ("values in x")
 ylabel ("probability")
plotted figure

Example: 2

  

 ## Plot half-normal against normal cumulative distribution function
 x = -5:0.001:5;
 p1 = hncdf (x, 0, 1);
 p2 = normcdf (x);
 plot (x, p1, "-b", x, p2, "-g")
 grid on
 xlim ([-5, 5])
 legend ({"half-normal with μ = 0, σ = 1", ...
          "standart normal (μ = 0, σ = 1)"}, "location", "southeast")
 title ("Half-normal against standard normal CDF")
 xlabel ("values in x")
 ylabel ("probability")
plotted figure

statistics-release-1.6.3/docs/hnfit.html000066400000000000000000000172731456127120000202670ustar00rootroot00000000000000 Statistics: hnfit

Function Reference: hnfit

statistics: paramhat = hnfit (x, mu)
statistics: [paramhat, paramci] = hnfit (x, mu)
statistics: [paramhat, paramci] = hnfit (x, mu, alpha)

Estimate parameters and confidence intervals for the half-normal distribution.

paramhat = hnfit (x) returns the maximum likelihood estimates of the parameters of the half-normal distribution given the data in vector x. paramhat(1) is the location parameter, mu, and paramhat(2) is the scale parameter, sigma. Although mu is returned in the estimated paramhat, hnfit does not estimate the location parameter mu, and it must be assumed to be known.

[paramhat, paramci] = hnfit (x, mu) returns the 95% confidence intervals for the estimated scale parameter sigma. The first colummn of paramci includes the location parameter mu without any confidence bounds.

[…] = hnfit (x, alpha) also returns the 100 * (1 - alpha) percent confidence intervals of the estimated scale parameter. By default, the optional argument alpha is 0.05 corresponding to 95% confidence intervals.

The half-normal CDF is only defined for x >= mu.

Further information about the half-normal distribution can be found at https://en.wikipedia.org/wiki/Half-normal_distribution

See also: hncdf, hninv, hnpdf, hnrnd, hnlike

Source Code: hnfit

Example: 1

  

 ## Sample 2 populations from different half-normal distibutions
 rand ("seed", 1);   # for reproducibility
 r1 = hnrnd (0, 5, 5000, 1);
 rand ("seed", 2);   # for reproducibility
 r2 = hnrnd (0, 2, 5000, 1);
 r = [r1, r2];

 ## Plot them normalized and fix their colors
 hist (r, [0.5:20], 1);
 h = findobj (gca, "Type", "patch");
 set (h(1), "facecolor", "c");
 set (h(2), "facecolor", "g");
 hold on

 ## Estimate their shape parameters
 mu_sigmaA = hnfit (r(:,1), 0);
 mu_sigmaB = hnfit (r(:,2), 0);

 ## Plot their estimated PDFs
 x = [0:0.2:10];
 y = hnpdf (x, mu_sigmaA(1), mu_sigmaA(2));
 plot (x, y, "-pr");
 y = hnpdf (x, mu_sigmaB(1), mu_sigmaB(2));
 plot (x, y, "-sg");
 xlim ([0, 10])
 ylim ([0, 0.5])
 legend ({"Normalized HIST of sample 1 with μ=0 and σ=5", ...
          "Normalized HIST of sample 2 with μ=0 and σ=2", ...
          sprintf("PDF for sample 1 with estimated μ=%0.2f and σ=%0.2f", ...
                  mu_sigmaA(1), mu_sigmaA(2)), ...
          sprintf("PDF for sample 2 with estimated μ=%0.2f and σ=%0.2f", ...
                  mu_sigmaB(1), mu_sigmaB(2))})
 title ("Two population samples from different half-normal distibutions")
 hold off
plotted figure

statistics-release-1.6.3/docs/hninv.html000066400000000000000000000135601456127120000202740ustar00rootroot00000000000000 Statistics: hninv

Function Reference: hninv

statistics: x = hninv (p, mu, sigma)

Inverse of the half-normal cumulative distribution function (iCDF).

For each element of p, compute the quantile (the inverse of the CDF) of the half-normal distribution with location parameter mu and scale parameter sigma. The size of x is the common size of p, mu, and sigma. A scalar input functions as a constant matrix of the same size as the other inputs.

Further information about the half-normal distribution can be found at https://en.wikipedia.org/wiki/Half-normal_distribution

See also: hncdf, hnpdf, hnrnd, hnfit, hnlike

Source Code: hninv

Example: 1

  

 ## Plot various iCDFs from the half-normal distribution
 p = 0.001:0.001:0.999;
 x1 = hninv (p, 0, 1);
 x2 = hninv (p, 0, 2);
 x3 = hninv (p, 0, 3);
 x4 = hninv (p, 0, 5);
 plot (p, x1, "-b", p, x2, "-g", p, x3, "-r", p, x4, "-c")
 grid on
 ylim ([0, 10])
 legend ({"μ = 0, σ = 1", "μ = 0, σ = 2", ...
          "μ = 0, σ = 3", "μ = 0, σ = 5"}, "location", "northwest")
 title ("Half-normal iCDF")
 xlabel ("probability")
 ylabel ("x")
plotted figure

statistics-release-1.6.3/docs/hnlike.html000066400000000000000000000116411456127120000204220ustar00rootroot00000000000000 Statistics: hnlike

Function Reference: hnlike

statistics: nlogL = hnlike (params, x)
statistics: [nlogL, acov] = hnlike (params, x)

Negative log-likelihood for the half-normal distribution.

nlogL = hnlike (params, x) returns the negative log likelihood of the data in x corresponding to the half-normal distribution with (1) location parameter mu and (2) scale parameter sigma given in the two-element vector params.

[nlogL, acov] = hnlike (params, x) returns the inverse of Fisher’s information matrix, acov. If the input parameter values in params are the maximum likelihood estimates, the diagonal elements of params are their asymptotic variances.

The half-normal CDF is only defined for x >= mu.

Further information about the half-normal distribution can be found at https://en.wikipedia.org/wiki/Half-normal_distribution

See also: hncdf, hninv, hnpdf, hnrnd, hnfit

Source Code: hnlike

statistics-release-1.6.3/docs/hnpdf.html000066400000000000000000000165401456127120000202520ustar00rootroot00000000000000 Statistics: hnpdf

Function Reference: hnpdf

statistics: y = hnpdf (x, mu, sigma)

Half-normal probability density function (PDF).

For each element of x, compute the probability density function (PDF) of the half-normal distribution with location parameter mu and scale parameter sigma. The size of y is the common size of x, mu, and sigma. A scalar input functions as a constant matrix of the same size as the other inputs.

The half-normal CDF is only defined for x >= mu.

Further information about the half-normal distribution can be found at https://en.wikipedia.org/wiki/Half-normal_distribution

See also: hncdf, hninv, hnrnd, hnfit, hnlike

Source Code: hnpdf

Example: 1

  

 ## Plot various PDFs from the half-normal distribution
 x = 0:0.001:10;
 y1 = hnpdf (x, 0, 1);
 y2 = hnpdf (x, 0, 2);
 y3 = hnpdf (x, 0, 3);
 y4 = hnpdf (x, 0, 5);
 plot (x, y1, "-b", x, y2, "-g", x, y3, "-r", x, y4, "-c")
 grid on
 xlim ([0, 10])
 ylim ([0, 0.9])
 legend ({"μ = 0, σ = 1", "μ = 0, σ = 2", ...
          "μ = 0, σ = 3", "μ = 0, σ = 5"}, "location", "northeast")
 title ("Half-normal PDF")
 xlabel ("values in x")
 ylabel ("density")
plotted figure

Example: 2

  

 ## Plot half-normal against normal probability density function
 x = -5:0.001:5;
 y1 = hnpdf (x, 0, 1);
 y2 = normpdf (x);
 plot (x, y1, "-b", x, y2, "-g")
 grid on
 xlim ([-5, 5])
 ylim ([0, 0.9])
 legend ({"half-normal with μ = 0, σ = 1", ...
          "standart normal (μ = 0, σ = 1)"}, "location", "northeast")
 title ("Half-normal against standard normal PDF")
 xlabel ("values in x")
 ylabel ("density")
plotted figure

statistics-release-1.6.3/docs/hnrnd.html000066400000000000000000000122421456127120000202570ustar00rootroot00000000000000 Statistics: hnrnd

Function Reference: hnrnd

statistics: r = hnrnd (mu, sigma)
statistics: r = hnrnd (mu, sigma, rows)
statistics: r = hnrnd (mu, sigma, rows, cols, …)
statistics: r = hnrnd (mu, sigma, [sz])

Random arrays from the half-normal distribution.

r = hnrnd (mu, sigma) returns an array of random numbers chosen from the half-normal distribution with location parameter mu and scale parameter sigma. The size of r is the common size of mu and sigma. A scalar input functions as a constant matrix of the same size as the other inputs.

When called with a single size argument, hnrnd returns a square matrix with the dimension specified. When called with more than one scalar argument, the first two arguments are taken as the number of rows and columns and any further arguments specify additional matrix dimensions. The size may also be specified with a row vector of dimensions, sz.

Further information about the half-normal distribution can be found at https://en.wikipedia.org/wiki/Half-normal_distribution

See also: hncdf, hninv, hnpdf, hnfit, hnlike

Source Code: hnrnd

statistics-release-1.6.3/docs/hotelling_t2test.html000066400000000000000000000137021456127120000224420ustar00rootroot00000000000000 Statistics: hotelling_t2test

Function Reference: hotelling_t2test

statistics: [h, pval, stats] = hotelling_t2test (x)
statistics: […] = hotelling_t2test (x, m)
statistics: […] = hotelling_t2test (x, y)
statistics: […] = hotelling_t2test (x, m, Name, Value)
statistics: […] = hotelling_t2test (x, y, Name, Value)

Compute Hotelling’s T^2 ("T-squared") test for a single sample or two dependent samples (paired-samples).

For a sample x from a multivariate normal distribution with unknown mean and covariance matrix, test the null hypothesis that mean (x) == m.

For two dependent samples x and y from a multivariate normal distributions with unknown means and covariance matrices, test the null hypothesis that mean (x - y) == 0.

hotelling_t2test treats NaNs as missing values, and ignores the corresponding rows.

Name-Value pair arguments can be used to set statistical significance. "alpha" can be used to specify the significance level of the test (the default value is 0.05).

If h is 1 the null hypothesis is rejected, meaning that the tested sample does not come from a multivariate distribution with mean m, or in case of two dependent samples that they do not come from the same multivariate distribution. If h is 0, then the null hypothesis cannot be rejected and it can be assumed that it holds true.

The p-value of the test is returned in pval.

stats is a structure containing the value of the Hotelling’s T^2 test statistic in the field "Tsq", and the degrees of freedom of the F distribution in the fields "df1" and "df2". Under the null hypothesis, (n-p) T^2 / (p(n-1)) has an F distribution with p and n-p degrees of freedom, where n and p are the numbers of samples and variables, respectively.

See also: hotelling_t2test2

Source Code: hotelling_t2test

statistics-release-1.6.3/docs/hotelling_t2test2.html000066400000000000000000000126731456127120000225320ustar00rootroot00000000000000 Statistics: hotelling_t2test2

Function Reference: hotelling_t2test2

statistics: [h, pval, stats] = hotelling_t2test2 (x, y)
statistics: […] = hotelling_t2test2 (x, y, Name, Value)

Compute Hotelling’s T^2 ("T-squared") test for two independent samples.

For two samples x from multivariate normal distributions with the same number of variables (columns), unknown means and unknown equal covariance matrices, test the null hypothesis mean (x) == mean (y).

hotelling_t2test2 treats NaNs as missing values, and ignores the corresponding rows for each sample independently.

Name-Value pair arguments can be used to set statistical significance. "alpha" can be used to specify the significance level of the test (the default value is 0.05).

If h is 1 the null hypothesis is rejected, meaning that the tested samples do not come from the same multivariate distribution. If h is 0, then the null hypothesis cannot be rejected and it can be assumed that both samples come from the same multivariate distribution.

The p-value of the test is returned in pval.

stats is a structure containing the value of the Hotelling’s T^2 test statistic in the field "Tsq", and the degrees of freedom of the F distribution in the fields "df1" and "df2". Under the null hypothesis, $$ {(n_x+n_y-p-1) T^2 \over p(n_x+n_y-2)} $$ has an F distribution with p and n_×+n_y-p-1 degrees of freedom, where n_× and n_y are the sample sizes and p is the number of variables.

See also: hotelling_t2test

Source Code: hotelling_t2test2

statistics-release-1.6.3/docs/hygecdf.html000066400000000000000000000153351456127120000205650ustar00rootroot00000000000000 Statistics: hygecdf

Function Reference: hygecdf

statistics: p = hygecdf (x, m, k, n)
statistics: p = hygecdf (x, m, k, n, "upper")

Hypergeometric cumulative distribution function (CDF).

For each element of x, compute the cumulative distribution function (CDF) of the hypergeometric distribution with parameters m, k, and n. The size of p is the common size of x, m, k, and n. A scalar input functions as a constant matrix of the same size as the other inputs.

This is the cumulative probability of obtaining not more than x marked items when randomly drawing a sample of size n without replacement from a population of total size m containing k marked items. The parameters m, k, and n must be positive integers with k and n not greater than m.

[…] = hygecdf (x, m, k, n, "upper") computes the upper tail probability of the hypergeometric distribution with parameters m, k, and n, at the values in x.

Further information about the hypergeometric distribution can be found at https://en.wikipedia.org/wiki/Hypergeometric_distribution

See also: hygeinv, hygepdf, hygernd, hygestat

Source Code: hygecdf

Example: 1

  

 ## Plot various CDFs from the hypergeometric distribution
 x = 0:60;
 p1 = hygecdf (x, 500, 50, 100);
 p2 = hygecdf (x, 500, 60, 200);
 p3 = hygecdf (x, 500, 70, 300);
 plot (x, p1, "*b", x, p2, "*g", x, p3, "*r")
 grid on
 xlim ([0, 60])
 legend ({"m = 500, k = 50, n = 100", "m = 500, k = 60, n = 200", ...
          "m = 500, k = 70, n = 300"}, "location", "southeast")
 title ("Hypergeometric CDF")
 xlabel ("values in x (number of successes)")
 ylabel ("probability")
plotted figure

statistics-release-1.6.3/docs/hygeinv.html000066400000000000000000000145001456127120000206160ustar00rootroot00000000000000 Statistics: hygeinv

Function Reference: hygeinv

statistics: x = hygeinv (p, m, k, n)

Inverse of the hypergeometric cumulative distribution function (iCDF).

For each element of p, compute the quantile (the inverse of the CDF) of the hypergeometric distribution with parameters m, k, and n. The size of x is the common size of p, m, k, and n. A scalar input functions as a constant matrix of the same size as the other inputs.

This is the number of drawn marked items x given a probability p, when randomly drawing a sample of size n without replacement from a population of total size m containing k marked items. The parameters m, k, and n must be positive integers with k and n not greater than m.

Further information about the hypergeometric distribution can be found at https://en.wikipedia.org/wiki/Hypergeometric_distribution

See also: hygeinv, hygepdf, hygernd, hygestat

Source Code: hygeinv

Example: 1

  

 ## Plot various iCDFs from the hypergeometric distribution
 p = 0.001:0.001:0.999;
 x1 = hygeinv (p, 500, 50, 100);
 x2 = hygeinv (p, 500, 60, 200);
 x3 = hygeinv (p, 500, 70, 300);
 plot (p, x1, "-b", p, x2, "-g", p, x3, "-r")
 grid on
 ylim ([0, 60])
 legend ({"m = 500, k = 50, n = 100", "m = 500, k = 60, n = 200", ...
          "m = 500, k = 70, n = 300"}, "location", "northwest")
 title ("Hypergeometric iCDF")
 xlabel ("probability")
 ylabel ("values in p (number of successes)")
plotted figure

statistics-release-1.6.3/docs/hygepdf.html000066400000000000000000000157241456127120000206040ustar00rootroot00000000000000 Statistics: hygepdf

Function Reference: hygepdf

statistics: y = hygepdf (x, m, k, n)
statistics: y = hygepdf (…, "vectorexpand")

Hypergeometric probability density function (PDF).

For each element of x, compute the probability density function (PDF) of the hypergeometric distribution with parameters m, k, and n. The size of y is the common size of x, m, k, and n. A scalar input functions as a constant matrix of the same size as the other inputs.

This is the probability of obtaining x marked items when randomly drawing a sample of size n without replacement from a population of total size m containing k marked items. The parameters m, k, and n must be positive integers with k and n not greater than m.

If the optional parameter vectorexpand is provided, x may be an array with size different from parameters m, k, and n (which must still be of a common size or scalar). Each element of x will be evaluated against each set of parameters m, k, and n in columnwise order. The output y will be an array of size r x s, where r = numel (m), and s = numel (x).

Further information about the hypergeometric distribution can be found at https://en.wikipedia.org/wiki/Hypergeometric_distribution

See also: hygecdf, hygeinv, hygernd, hygestat

Source Code: hygepdf

Example: 1

  

 ## Plot various PDFs from the hypergeometric distribution
 x = 0:60;
 y1 = hygepdf (x, 500, 50, 100);
 y2 = hygepdf (x, 500, 60, 200);
 y3 = hygepdf (x, 500, 70, 300);
 plot (x, y1, "*b", x, y2, "*g", x, y3, "*r")
 grid on
 xlim ([0, 60])
 ylim ([0, 0.18])
 legend ({"m = 500, k = 50, μ = 100", "m = 500, k = 60, μ = 200", ...
          "m = 500, k = 70, μ = 300"}, "location", "northeast")
 title ("Hypergeometric PDF")
 xlabel ("values in x (number of successes)")
 ylabel ("density")
plotted figure

statistics-release-1.6.3/docs/hygernd.html000066400000000000000000000125751456127120000206170ustar00rootroot00000000000000 Statistics: hygernd

Function Reference: hygernd

statistics: r = hygernd (m, k, n)
statistics: r = hygernd (m, k, n, rows)
statistics: r = hygernd (m, k, n, rows, cols, …)
statistics: r = hygernd (m, k, n, [sz])

Random arrays from the hypergeometric distribution.

r = hygernd ((m, k, n returns an array of random numbers chosen from the hypergeometric distribution with parameters m, k, and n. The size of r is the common size of m, k, and n. A scalar input functions as a constant matrix of the same size as the other inputs.

The parameters m, k, and n must be positive integers with k and n not greater than m.

When called with a single size argument, hygernd returns a square matrix with the dimension specified. When called with more than one scalar argument, the first two arguments are taken as the number of rows and columns and any further arguments specify additional matrix dimensions. The size may also be specified with a row vector of dimensions, sz.

Further information about the hypergeometric distribution can be found at https://en.wikipedia.org/wiki/Hypergeometric_distribution

See also: hygecdf, hygeinv, hygepdf, hygestat

Source Code: hygernd

statistics-release-1.6.3/docs/hygestat.html000066400000000000000000000117161456127120000210030ustar00rootroot00000000000000 Statistics: hygestat

Function Reference: hygestat

statistics: [mn, v] = hygestat (m, k, n)

Compute statistics of the hypergeometric distribution.

[mn, v] = hygestat (m, k, n) returns the mean and variance of the hypergeometric distribution parameters m, k, and n.

  • m is the total size of the population of the hypergeometric distribution. The elements of m must be positive natural numbers.
  • k is the number of marked items of the hypergeometric distribution. The elements of k must be natural numbers.
  • n is the size of the drawn sample of the hypergeometric distribution. The elements of n must be positive natural numbers.

The size of mn (mean) and v (variance) is the common size of the input arguments. A scalar input functions as a constant matrix of the same size as the other inputs.

Further information about the hypergeometric distribution can be found at https://en.wikipedia.org/wiki/Hypergeometric_distribution

See also: hygecdf, hygeinv, hygepdf, hygernd

Source Code: hygestat

statistics-release-1.6.3/docs/icdf.html000066400000000000000000000326111456127120000200550ustar00rootroot00000000000000 Statistics: icdf

Function Reference: icdf

statistics: x = icdf (name, p, A)
statistics: x = icdf (name, p, A, B)
statistics: x = icdf (name, p, A, B, C)

Return the inverse CDF of a univariate distribution evaluated at p.

icdf is a wrapper for the univariate quantile distribution functions (iCDF) available in the statistics package. See the corresponding functions’ help to learn the signification of the parameters after p.

x = icdf (name, p, A) returns the iCDF for the one-parameter distribution family specified by name and the distribution parameter A, evaluated at the values in p.

x = icdf (name, p, A, B) returns the iCDF for the two-parameter distribution family specified by name and the distribution parameters A and B, evaluated at the values in p.

x = icdf (name, p, A, B, C) returns the iCDF for the three-parameter distribution family specified by name and the distribution parameters A, B, and C, evaluated at the values in p.

name must be a char string of the name or the abbreviation of the desired quantile distribution function as listed in the followng table. The last column shows the number of required parameters that should be parsed after x to the desired iCDF.

Distribution NameAbbreviationInput Parameters
"Beta""beta"2
"Binomial""bino"2
"Birnbaum-Saunders""bisa"2
"Burr""burr"3
"Cauchy""cauchy"2
"Chi-squared""chi2"1
"Extreme Value""ev"2
"Exponential""exp"1
"F-Distribution""f"2
"Gamma""gam"2
"Geometric""geo"1
"Generalized Extreme Value""gev"3
"Generalized Pareto""gp"3
"Gumbel""gumbel"2
"Half-normal""hn"2
"Hypergeometric""hyge"3
"Inverse Gaussian""invg"2
"Laplace""laplace"2
"Logistic""logi"2
"Log-Logistic""logl"2
"Lognormal""logn"2
"Nakagami""naka"2
"Negative Binomial""nbin"2
"Noncentral F-Distribution""ncf"3
"Noncentral Student T""nct"2
"Noncentral Chi-Squared""ncx2"2
"Normal""norm"2
"Poisson""poiss"1
"Rayleigh""rayl"1
"Rician""rice"2
"Student T""t"1
"location-scale T""tls"3
"Triangular""tri"3
"Discrete Uniform""unid"1
"Uniform""unif"2
"Von Mises""vm"2
"Weibull""wbl"2

See also: icdf, pdf, random, betainv, binoinv, bisainv, burrinv, cauchyinv, chi2inv, evinv, expinv, finv, gaminv, geoinv, gevinv, gpinv, gumbelinv, hninv, hygeinv, invginv, laplaceinv, logiinv, loglinv, logninv, nakainv, nbininv, ncfinv, nctinv, ncx2inv, norminv, poissinv, raylinv, riceinv, tinv, triinv, unidinv, unifinv, vminv, wblinv

Source Code: icdf

statistics-release-1.6.3/docs/inconsistent.html000066400000000000000000000116021456127120000216650ustar00rootroot00000000000000 Statistics: inconsistent

Function Reference: inconsistent

statistics: Y = inconsistent (Z)
statistics: Y = inconsistent (Z, d)

Compute the inconsistency coefficient for each link of a hierarchical cluster tree.

Given a hierarchical cluster tree Z generated by the linkage function, inconsistent computes the inconsistency coefficient for each link of the tree, using all the links down to the d-th level below that link.

The default depth d is 2, which means that only two levels are considered: the level of the computed link and the level below that.

Each row of Y corresponds to the row of same index of Z. The columns of Y are respectively: the mean of the heights of the links used for the calculation, the standard deviation of the heights of those links, the number of links used, the inconsistency coefficient.

Reference Jain, A., and R. Dubes. Algorithms for Clustering Data. Upper Saddle River, NJ: Prentice-Hall, 1988.

See also: cluster, clusterdata, dendrogram, linkage, pdist, squareform

Source Code: inconsistent

statistics-release-1.6.3/docs/index.html000066400000000000000000003547121456127120000202700ustar00rootroot00000000000000 Statistics
statistics

statistics

1.6.3 2024-2-9

The Statistics package for GNU Octave.

Select Category:

Clustering

cluster Define clusters from an agglomerative hierarchical cluster tree.
clusterdata Wrapper function for 'linkage' and 'cluster'.
cmdscale Classical multidimensional scaling of a matrix.
confusionmat Compute a confusion matrix for classification problems
cophenet Compute the cophenetic correlation coefficient.
crossval Perform cross validation on given data.
evalclusters Create a clustering evaluation object to find the optimal number of clusters.
inconsistent Compute the inconsistency coefficient for each link of a hierarchical cluster tree.
kmeans Perform a K-means clustering of the NxD matrix DATA.
knnsearch Find k-nearest neighbors from input data.
linkage Produce a hierarchical clustering dendrogram.
mahal Mahalanobis' D-square distance.
mhsample Draws NSAMPLES samples from a target stationary distribution PDF using Metropolis-Hastings algorithm.
optimalleaforder Compute the optimal leaf ordering of a hierarchical binary cluster tree.
pdist Return the distance between any two rows in X.
pdist2 Compute pairwise distance between two sets of vectors.
procrustes Procrustes Analysis.
rangesearch Find all neighbors within specified distance from input data.
slicesample Draws NSAMPLES samples from a target stationary distribution PDF using slice sampling of Radford M.
squareform Interchange between distance matrix and distance vector formats.

Classification Classes

ClassificationKNN Create a ClassificationKNN class object containing a k-Nearest Neighbor classification model.
ConfusionMatrixChart Create object CMC, a Confusion Matrix Chart object.

Clustering Classes

CalinskiHarabaszEvaluation A Calinski-Harabasz object to evaluate clustering solutions.
ClusterCriterion A clustering evaluation object as created by 'evalclusters'.
DaviesBouldinEvaluation A Davies-Bouldin object to evaluate clustering solutions.
GapEvaluation A gap object to evaluate clustering solutions.
SilhouetteEvaluation A silhouette object to evaluate clustering solutions.

CVpartition (legacy class of set partitions for cross-validation, used in crossval)

@cvpartition/cvpartition Create a partition object for cross validation.
@cvpartition/display Display a 'cvpartition' object, C.
@cvpartition/get Get a field, F, from a 'cvpartition' object, C.
@cvpartition/repartition Return a new cvpartition object.
@cvpartition/set Set FIELD, in a 'cvpartition' object, C.
@cvpartition/test Return logical vector for testing-subset indices from a 'cvpartition' object, C.
@cvpartition/training Return logical vector for training-subset indices from a 'cvpartition' object, C.

Regression Classes

RegressionGAM Create a RegressionGAM class object containing a Generalised Additive Model (GAM) for regression.

Data Manipulation

combnk Return all combinations of K elements in DATA.
crosstab Create a cross-tabulation (contingency table) T from data vectors.
datasample Randomly sample data.
fillmissing Replace missing entries of array A either with values in V or as determined by other specified methods. 'missing' values are determined by the data type of A as identified by the function ismissing, curently defined as:
grp2idx Get index for group variables.
ismissing Find missing data in a numeric or string array.
isoutlier Find outliers in data
normalise_distribution Transform a set of data so as to be N(0,1) distributed according to an idea by van Albada and Robinson.
rmmissing Remove missing or incomplete data from an array.
standardizeMissing Replace data values specified by INDICATOR in A by the standard 'missing' data value for that data type.
tabulate Calculate a frequency table.

Descriptive Statistics

cdfcalc Calculate an empirical cumulative distribution function.
cl_multinom Confidence level of multinomial portions.
geomean Compute the geometric mean of X.
grpstats Summary statistics by group. 'grpstats' computes groupwise summary statistics, for data in a matrix X. 'grpstats' treats NaNs as missing values, and removes them.
harmmean Compute the harmonic mean of X.
jackknife Compute jackknife estimates of a parameter taking one or more given samples as parameters.
mad Compute the mean or median absolute deviation (MAD).
mean Compute the mean of the elements of X.
median Compute the median value of the elements of X.
nanmax Find the maximal element while ignoring NaN values.
nanmin Find the minimal element while ignoring NaN values.
nansum Compute the sum while ignoring NaN values.
std Compute the standard deviation of the elements of X.
trimmean Compute the trimmed mean.
var Compute the variance of the elements of X.

Distribution Fitting

betafit Estimate parameters and confidence intervals for the Beta distribution.
betalike Negative log-likelihood for the Beta distribution.
binofit Estimate parameter and confidence intervals for the binomial distribution.
binolike Negative log-likelihood for the binomial distribution.
bisafit Estimate mean and confidence intervals for the Birnbaum-Saunders distribution.
bisalike Negative log-likelihood for the Birnbaum-Saunders distribution.
burrfit Estimate mean and confidence intervals for the Burr type XII distribution.
burrlike Negative log-likelihood for the Burr type XII distribution.
evfit Estimate parameters and confidence intervals for the extreme value distribution.
evlike Negative log-likelihood for the extreme value distribution.
expfit Estimate mean and confidence intervals for the exponential distribution.
explike Negative log-likelihood for the exponential distribution.
gamfit Estimate parameters and confidence intervals for the Gamma distribution.
gamlike Negative log-likelihood for the Gamma distribution.
geofit Estimate parameter and confidence intervals for the geometric distribution.
gevfit_lmom Find an estimator (PARAMHAT) of the generalized extreme value (GEV) distribution fitting DATA using the method of L-moments.
gevfit Estimate parameters and confidence intervals for the generalized extreme value (GEV) distribution.
gevlike Negative log-likelihood for the generalized extreme value (GEV) distribution.
gpfit Estimate parameters and confidence intervals for the generalized Pareto distribution.
gplike Negative log-likelihood for the generalized Pareto distribution.
gumbelfit Estimate parameters and confidence intervals for Gumbel distribution.
gumbellike Negative log-likelihood for the extreme value distribution.
hnfit Estimate parameters and confidence intervals for the half-normal distribution.
hnlike Negative log-likelihood for the half-normal distribution.
invgfit Estimate mean and confidence intervals for the inverse Gaussian distribution.
invglike Negative log-likelihood for the inverse Gaussian distribution.
logifit Estimate mean and confidence intervals for the logistic distribution.
logilike Negative log-likelihood for the logistic distribution.
loglfit Estimate mean and confidence intervals for the log-logistic distribution.
logllike Negative log-likelihood for the log-logistic distribution.
lognfit Estimate parameters and confidence intervals for the log-normal distribution.
lognlike Negative log-likelihood for the log-normal distribution.
nakafit Estimate mean and confidence intervals for the Nakagami distribution.
nakalike Negative log-likelihood for the Nakagami distribution.
nbinfit Estimate parameter and confidence intervals for the negative binomial distribution.
nbinlike Negative log-likelihood for the negative binomial distribution.
normfit Estimate parameters and confidence intervals for the normal distribution.
normlike Negative log-likelihood for the normal distribution.
poissfit Estimate parameter and confidence intervals for the Poisson distribution.
poisslike Negative log-likelihood for the Poisson distribution.
raylfit Estimate parameter and confidence intervals for the Rayleigh distribution.
rayllike Negative log-likelihood for the Rayleigh distribution.
ricefit Estimate parameters and confidence intervals for the Gamma distribution.
ricelike Negative log-likelihood for the Rician distribution.
unidfit Estimate parameter and confidence intervals for the discrete uniform distribution.
unifit Estimate parameter and confidence intervals for the continuous uniform distribution.
wblfit Estimate mean and confidence intervals for the Weibull distribution.
wbllike Negative log-likelihood for the Weibull distribution.

Distribution Functions

betacdf Beta cumulative distribution function (CDF).
betainv Inverse of the Beta distribution (iCDF).
betapdf Beta probability density function (PDF).
betarnd Random arrays from the Beta distribution.
binocdf Binomial cumulative distribution function (CDF).
binoinv Inverse of the Binomial cumulative distribution function (iCDF).
binopdf Binomial probability density function (PDF).
binornd Random arrays from the Binomial distribution.
bisacdf Birnbaum-Saunders cumulative distribution function (CDF).
bisainv Inverse of the Birnbaum-Saunders cumulative distribution function (iCDF).
bisapdf Birnbaum-Saunders probability density function (PDF).
bisarnd Random arrays from the Birnbaum-Saunders distribution.
burrcdf Burr type XII cumulative distribution function (CDF).
burrinv Inverse of the Burr type XII cumulative distribution function (iCDF).
burrpdf Burr type XII probability density function (PDF).
burrrnd Random arrays from the Burr type XII distribution.
bvncdf Bivariate normal cumulative distribution function (CDF).
bvtcdf Bivariate Student's t cumulative distribution function (CDF).
cauchycdf Cauchy cumulative distribution function (CDF).
cauchyinv Inverse of the Cauchy cumulative distribution function (iCDF).
cauchypdf Cauchy probability density function (PDF).
cauchyrnd Random arrays from the Cauchy distribution.
chi2cdf Chi-squared cumulative distribution function (CDF).
chi2inv Inverse of the chi-squared cumulative distribution function (iCDF).
chi2pdf Chi-squared probability density function (PDF).
chi2rnd Random arrays from the chi-squared distribution.
copulacdf Copula family cumulative distribution functions (CDF).
copulapdf Copula family probability density functions (PDF).
copularnd Random arrays from the copula family distributions.
evcdf Extreme value cumulative distribution function (CDF).
evinv Inverse of the extreme value cumulative distribution function (iCDF).
evpdf Extreme value probability density function (PDF).
evrnd Random arrays from the extreme value distribution.
expcdf Exponential cumulative distribution function (CDF).
expinv Inverse of the exponential cumulative distribution function (iCDF).
exppdf Exponential probability density function (PDF).
exprnd Random arrays from the exponential distribution.
fcdf F-cumulative distribution function (CDF).
finv Inverse of the F-cumulative distribution function (iCDF).
fpdf F-probability density function (PDF).
frnd Random arrays from the F-distribution.
gamcdf Gamma cumulative distribution function (CDF).
gaminv Inverse of the Gamma cumulative distribution function (iCDF).
gampdf Gamma probability density function (PDF).
gamrnd Random arrays from the Gamma distribution.
geocdf Geometric cumulative distribution function (CDF).
geoinv Inverse of the geometric cumulative distribution function (iCDF).
geopdf Geometric probability density function (PDF).
geornd Random arrays from the geometric distribution.
gevcdf Generalized extreme value (GEV) cumulative distribution function (CDF).
gevinv Inverse of the generalized extreme value (GEV) cumulative distribution function (iCDF).
gevpdf Generalized extreme value (GEV) probability density function (PDF).
gevrnd Random arrays from the generalized extreme value (GEV) distribution.
gpcdf Generalized Pareto cumulative distribution function (CDF).
gpinv Inverse of the generalized Pareto cumulative distribution function (iCDF).
gppdf Generalized Pareto probability density function (PDF).
gprnd Random arrays from the generalized Pareto distribution.
gumbelcdf Gumbel cumulative distribution function (CDF).
gumbelinv Inverse of the Gumbel cumulative distribution function (iCDF).
gumbelpdf Gumbel probability density function (PDF).
gumbelrnd Random arrays from the Gumbel distribution.
hncdf Half-normal cumulative distribution function (CDF).
hninv Inverse of the half-normal cumulative distribution function (iCDF).
hnpdf Half-normal probability density function (PDF).
hnrnd Random arrays from the half-normal distribution.
hygecdf Hypergeometric cumulative distribution function (CDF).
hygeinv Inverse of the hypergeometric cumulative distribution function (iCDF).
hygepdf Hypergeometric probability density function (PDF).
hygernd Random arrays from the hypergeometric distribution.
invgcdf Inverse Gaussian cumulative distribution function (CDF).
invginv Inverse of the inverse Gaussian cumulative distribution function (iCDF).
invgpdf Inverse Gaussian probability density function (PDF).
invgrnd Random arrays from the inverse Gaussian distribution.
iwishpdf Compute the probability density function of the inverse Wishart distribution.
iwishrnd Return a random matrix sampled from the inverse Wishart distribution with given parameters.
jsucdf Johnson SU cumulative distribution function (CDF).
jsupdf Johnson SU probability density function (PDF).
laplacecdf Laplace cumulative distribution function (CDF).
laplaceinv Inverse of the Laplace cumulative distribution function (iCDF).
laplacepdf Laplace probability density function (PDF).
laplacernd Random arrays from the Laplace distribution.
logicdf Logistic cumulative distribution function (CDF).
logiinv Inverse of the logistic cumulative distribution function (iCDF).
logipdf Logistic probability density function (PDF).
logirnd Random arrays from the logistic distribution.
loglcdf Log-logistic cumulative distribution function (CDF).
loglinv Inverse of the log-logistic cumulative distribution function (iCDF).
loglpdf Log-logistic probability density function (PDF).
loglrnd Random arrays from the log-logistic distribution.
logncdf Log-normal cumulative distribution function (CDF).
logninv Inverse of the log-normal cumulative distribution function (iCDF).
lognpdf Lognormal probability density function (PDF).
lognrnd Random arrays from the lognormal distribution.
mnpdf Multinomial probability density function (PDF).
mnrnd Random arrays from the multinomial distribution.
mvncdf Multivariate normal cumulative distribution function (CDF).
mvnpdf Multivariate normal probability density function (PDF).
mvnrnd Random vectors from the multivariate normal distribution.
mvtcdf Multivariate Student's t cumulative distribution function (CDF).
mvtpdf Multivariate Student's t probability density function (PDF).
mvtrnd Random vectors from the multivariate Student's t distribution.
mvtcdfqmc Quasi-Monte-Carlo computation of the multivariate Student's T CDF.
nakacdf Nakagami cumulative distribution function (CDF).
nakainv Inverse of the Nakagami cumulative distribution function (iCDF).
nakapdf Nakagami probability density function (PDF).
nakarnd Random arrays from the Nakagami distribution.
nbincdf Negative binomial cumulative distribution function (CDF).
nbininv Inverse of the negative binomial cumulative distribution function (iCDF).
nbinpdf Negative binomial probability density function (PDF).
nbinrnd Random arrays from the negative binomial distribution.
ncfcdf Noncentral F-cumulative distribution function (CDF).
ncfinv Inverse of the noncentral F-cumulative distribution function (iCDF).
ncfpdf Noncentral F-probability density function (PDF).
ncfrnd Random arrays from the noncentral F-distribution.
nctcdf Noncentral t-cumulative distribution function (CDF).
nctinv Inverse of the non-central t-cumulative distribution function (iCDF).
nctpdf Noncentral t-probability density function (PDF).
nctrnd Random arrays from the noncentral t-distribution.
ncx2cdf Noncentral chi-squared cumulative distribution function (CDF).
ncx2inv Inverse of the noncentral chi-squared cumulative distribution function (iCDF).
ncx2pdf Noncentral chi-squared probability distribution function (PDF).
ncx2rnd Random arrays from the noncentral chi-squared distribution.
normcdf Normal cumulative distribution function (CDF).
norminv Inverse of the normal cumulative distribution function (iCDF).
normpdf Normal probability density function (PDF).
normrnd Random arrays from the normal distribution.
poisscdf Poisson cumulative distribution function (CDF).
poissinv Inverse of the Poisson cumulative distribution function (iCDF).
poisspdf Poisson probability density function (PDF).
poissrnd Random arrays from the Poisson distribution.
raylcdf Rayleigh cumulative distribution function (CDF).
raylinv Inverse of the Rayleigh cumulative distribution function (iCDF).
raylpdf Rayleigh probability density function (PDF).
raylrnd Random arrays from the Rayleigh distribution.
ricecdf Rician cumulative distribution function (CDF).
riceinv Inverse of the Rician distribution (iCDF).
ricepdf Rician probability density function (PDF).
ricernd Random arrays from the Rician distribution.
tcdf Student's T cumulative distribution function (CDF).
tinv Inverse of the Student's T cumulative distribution function (iCDF).
tpdf Student's T probability density function (PDF).
trnd Random arrays from the Student's T distribution.
tlscdf Location-scale Student's T cumulative distribution function (CDF).
tlsinv Inverse of the location-scale Student's T cumulative distribution function (iCDF).
tlspdf Location-scale Student's T probability density function (PDF).
tlsrnd Random arrays from the location-scale Student's T distribution.
tricdf Triangular cumulative distribution function (CDF).
triinv Inverse of the triangular cumulative distribution function (iCDF).
tripdf Triangular probability density function (PDF).
trirnd Random arrays from the triangular distribution.
unidcdf Discrete uniform cumulative distribution function (CDF).
unidinv Inverse of the discrete uniform cumulative distribution function (iCDF).
unidpdf Discrete uniform probability density function (PDF).
unidrnd Random arrays from the discrete uniform distribution.
unifcdf Continuous uniform cumulative distribution function (CDF).
unifinv Inverse of the continuous uniform cumulative distribution function (iCDF).
unifpdf Continuous uniform probability density function (PDF).
unifrnd Random arrays from the continuous uniform distribution.
vmcdf Von Mises probability density function (PDF).
vminv Inverse of the von Mises cumulative distribution function (iCDF).
vmpdf Von Mises probability density function (PDF).
vmrnd Random arrays from the von Mises distribution.
wblcdf Weibull cumulative distribution function (CDF).
wblinv Inverse of the Weibull cumulative distribution function (iCDF).
wblpdf Weibull probability density function (PDF).
wblrnd Random arrays from the Weibull distribution.
wienrnd Return a simulated realization of the D-dimensional Wiener Process on the interval [0, T].
wishpdf Compute the probability density function of the Wishart distribution
wishrnd Return a random matrix sampled from the Wishart distribution with given parameters

Distribution Statistics

betastat Compute statistics of the Beta distribution.
binostat Compute statistics of the binomial distribution.
chi2stat Compute statistics of the chi-squared distribution.
evstat Compute statistics of the extreme value distribution.
expstat Compute statistics of the exponential distribution.
fstat Compute statistics of the F-distribution.
gamstat Compute statistics of the Gamma distribution.
geostat Compute statistics of the geometric distribution.
gevstat Compute statistics of the generalized extreme value distribution.
gpstat Compute statistics of the generalized Pareto distribution.
hygestat Compute statistics of the hypergeometric distribution.
lognstat Compute statistics of the log-normal distribution.
nbinstat Compute statistics of the negative binomial distribution.
ncfstat Compute statistics for the noncentral F-distribution.
nctstat Compute statistics for the noncentral t-distribution.
ncx2stat Compute statistics for the noncentral chi-squared distribution.
normstat Compute statistics of the normal distribution.
poisstat Compute statistics of the Poisson distribution.
raylstat Compute statistics of the Rayleigh distribution.
ricestat Compute statistics of the Rician distribution.
tlsstat Compute statistics of the location-scale Student's T distribution.
tstat Compute statistics of the Student's T distribution.
unidstat Compute statistics of the discrete uniform cumulative distribution.
unifstat Compute statistics of the continuous uniform cumulative distribution.
wblstat Compute statistics of the Weibull distribution.

Experimental Design

fullfact Full factorial design.
ff2n Two-level full factorial design.
sigma_pts Calculates 2*N+1 sigma points in N dimensions.
x2fx Convert predictors to design matrix.

Machine Learning

hmmestimate Estimation of a hidden Markov model for a given sequence.
hmmgenerate Output sequence and hidden states of a hidden Markov model.
hmmviterbi Viterbi path of a hidden Markov model.
svmpredict This function predicts new labels from a testing instance matrtix based on an SVM MODEL created with 'svmtrain'.
svmtrain This function trains an SVM MODEL based on known LABELS and their corresponding DATA which comprise an instance matrtix.

Model Fitting

fitcknn Fit a k-Nearest Neighbor classification model.
fitgmdist Fit a Gaussian mixture model with K components to DATA.
fitlm Regress the continuous outcome (i.e. dependent variable) Y on continuous or categorical predictors (i.e. independent variables) X by minimizing the sum-of-squared residuals.
fitrgam Fit a Generalised Additive Model (GAM) for regression.

Hypothesis Testing

adtest Anderson-Darling goodness-of-fit hypothesis test.
anova1 Perform a one-way analysis of variance (ANOVA) for comparing the means of two or more groups of data under the null hypothesis that the groups are drawn from distributions with the same mean.
anova2 Performs two-way factorial (crossed) or a nested analysis of variance (ANOVA) for balanced designs.
anovan Perform a multi (N)-way analysis of (co)variance (ANOVA or ANCOVA) to evaluate the effect of one or more categorical or continuous predictors (i.e. independent variables) on a continuous outcome (i.e. dependent variable).
bartlett_test Perform a Bartlett test for the homogeneity of variances.
barttest Bartlett's test of sphericity for correlation.
binotest Test for probability P of a binomial sample
chi2gof Chi-square goodness-of-fit test.
chi2test Perform a chi-squared test (for independence or homogeneity).
correlation_test Perform a correlation coefficient test whether two samples X and Y come from uncorrelated populations.
fishertest Fisher's exact test.
friedman Performs the nonparametric Friedman's test to compare column effects in a two-way layout. friedman tests the null hypothesis that the column effects are all the same against the alternative that they are not all the same.
hotelling_t2test Compute Hotelling's T^2 ("T-squared") test for a single sample or two dependent samples (paired-samples).
hotelling_t2test2 Compute Hotelling's T^2 ("T-squared") test for two independent samples.
kruskalwallis Perform a Kruskal-Wallis test, the non-parametric alternative of a one-way analysis of variance (ANOVA), for comparing the means of two or more groups of data under the null hypothesis that the groups are drawn from the same population, ...
kstest Single sample Kolmogorov-Smirnov (K-S) goodness-of-fit hypothesis test.
kstest2 Two-sample Kolmogorov-Smirnov goodness-of-fit hypothesis test.
levene_test Perform a Levene's test for the homogeneity of variances.
manova1 One-way multivariate analysis of variance (MANOVA).
mcnemar_test Perform a McNemar's test on paired nominal data.
multcompare Perform posthoc multiple comparison tests or p-value adjustments to control the family-wise error rate (FWER) or false discovery rate (FDR).
ranksum Wilcoxon rank sum test for equal medians.
regression_ftest F-test for General Linear Regression Analysis
regression_ttest Perform a linear regression t-test.
runstest Run test for randomness in the vector X.
sampsizepwr Sample size and power calculation for hypothesis test.
signtest Test for median.
ttest Test for mean of a normal sample with unknown variance.
ttest2 Perform a t-test to compare the means of two groups of data under the null hypothesis that the groups are drawn from distributions with the same mean.
vartest One-sample test of variance.
vartest2 Two-sample F test for equal variances.
vartestn Test for equal variances across multiple groups.
ztest One-sample Z-test.
ztest2 Two proportions Z-test.

I/O

libsvmread This function reads the labels and the corresponding instance_matrix from a LIBSVM data file and stores them in LABELS and DATA respectively.
libsvmwrite This function saves the labels and the corresponding instance_matrix in a file specified by FILENAME.

Plotting

boxplot Produce a box plot.
cdfplot Display an empirical cumulative distribution function.
confusionchart Display a chart of a confusion matrix.
dendrogram Plot a dendrogram of a hierarchical binary cluster tree.
ecdf Empirical (Kaplan-Meier) cumulative distribution function.
einstein Plots the tiling of the basic clusters of einstein tiles.
gscatter Draw a scatter plot with grouped data.
histfit Plot histogram with superimposed fitted normal density.
hist3 Produce bivariate (2D) histogram counts or plots.
manovacluster Cluster group means using manova1 output.
normplot Produce normal probability plot of the data in X.
ppplot Perform a PP-plot (probability plot).
qqplot Perform a QQ-plot (quantile plot).
silhouette Compute the silhouette values of clustered data and show them on a plot.
violin Produce a Violin plot of the data X.
wblplot Plot a column vector DATA on a Weibull probability plot using rank regression.

Regression

canoncorr Canonical correlation analysis.
cholcov Cholesky-like decomposition for covariance matrix.
dcov Distance correlation, covariance and correlation statistics.
logistic_regression Perform ordinal logistic regression.
mnrfit Perform logistic regression for binomial responses or multiple ordinal responses.
monotone_smooth Produce a smooth monotone increasing approximation to a sampled functional dependence.
pca Performs a principal component analysis on a data matrix.
pcacov Perform principal component analysis on covariance matrix
pcares Calculate residuals from principal component analysis.
plsregress Calculate partial least squares regression using SIMPLS algorithm.
princomp Performs a principal component analysis on a NxP data matrix X.
regress Multiple Linear Regression using Least Squares Fit of Y on X with the model 'y = X * beta + e'.
regress_gp Regression using Gaussian Processes.
ridge Ridge regression.
stepwisefit Linear regression with stepwise variable selection.

Transforms

logit Compute the logit for each value of P
probit Probit transformation

Wrappers

cdf Return the CDF of a univariate distribution evaluated at X.
icdf Return the inverse CDF of a univariate distribution evaluated at P.
pdf Return the PDF of a univariate distribution evaluated at X.
random Random arrays from a given one-, two-, or three-parameter distribution.
statistics-release-1.6.3/docs/invgcdf.html000066400000000000000000000150241456127120000205670ustar00rootroot00000000000000 Statistics: invgcdf

Function Reference: invgcdf

statistics: p = invgcdf (x, mu, lambda)
statistics: p = invgcdf (x, mu, lambda, "upper")

Inverse Gaussian cumulative distribution function (CDF).

For each element of x, compute the cumulative distribution function (CDF) of the inverse Gaussian distribution with scale parameter mu and shape parameter lambda. The size of p is the common size of x, mu and lambda. A scalar input functions as a constant matrix of the same size as the other inputs.

p = invgcdf (x, mu, lambda, "upper") computes the upper tail probability of the inverse Gaussian distribution with parameters mu and lambda, at the values in x.

The inverse Gaussian CDF is only defined for mu > 0 and lambda > 0.

Further information about the inverse Gaussian distribution can be found at https://en.wikipedia.org/wiki/Inverse_Gaussian_distribution

See also: invginv, invgpdf, invgrnd, invgfit, invglike

Source Code: invgcdf

Example: 1

  

 ## Plot various CDFs from the inverse Gaussian distribution
 x = 0:0.001:3;
 p1 = invgcdf (x, 1, 0.2);
 p2 = invgcdf (x, 1, 1);
 p3 = invgcdf (x, 1, 3);
 p4 = invgcdf (x, 3, 0.2);
 p5 = invgcdf (x, 3, 1);
 plot (x, p1, "-b", x, p2, "-g", x, p3, "-r", x, p4, "-c", x, p5, "-y")
 grid on
 xlim ([0, 3])
 legend ({"μ = 1, σ = 0.2", "μ = 1, σ = 1", "μ = 1, σ = 3", ...
          "μ = 3, σ = 0.2", "μ = 3, σ = 1"}, "location", "southeast")
 title ("Inverse Gaussian CDF")
 xlabel ("values in x")
 ylabel ("probability")
plotted figure

statistics-release-1.6.3/docs/invgfit.html000066400000000000000000000231131456127120000206130ustar00rootroot00000000000000 Statistics: invgfit

Function Reference: invgfit

statistics: paramhat = invgfit (x)
statistics: [paramhat, paramci] = invgfit (x)
statistics: [paramhat, paramci] = invgfit (x, alpha)
statistics: […] = invgfit (x, alpha, censor)
statistics: […] = invgfit (x, alpha, censor, freq)
statistics: […] = invgfit (x, alpha, censor, freq, options)

Estimate mean and confidence intervals for the inverse Gaussian distribution.

mu0 = invgfit (x) returns the maximum likelihood estimates of the parameters of the inverse Gaussian distribution given the data in x. paramhat(1) is the scale parameter, mu, and paramhat(2) is the shape parameter, lambda.

[paramhat, paramci] = invgfit (x) returns the 95% confidence intervals for the parameter estimates.

[…] = invgfit (x, alpha) also returns the 100 * (1 - alpha) percent confidence intervals for the parameter estimates. By default, the optional argument alpha is 0.05 corresponding to 95% confidence intervals. Pass in [] for alpha to use the default values.

[…] = invgfit (x, alpha, censor) accepts a boolean vector, censor, of the same size as x with 1s for observations that are right-censored and 0s for observations that are observed exactly. By default, or if left empty, censor = zeros (size (x)).

[…] = invgfit (x, alpha, censor, freq) accepts a frequency vector, freq, of the same size as x. freq typically contains integer frequencies for the corresponding elements in x, but it can contain any non-integer non-negative values. By default, or if left empty, freq = ones (size (x)).

[…] = invgfit (…, options) specifies control parameters for the iterative algorithm used to compute ML estimates with the fminsearch function. options is a structure with the following fields and their default values:

  • options.Display = "off"
  • options.MaxFunEvals = 400
  • options.MaxIter = 200
  • options.TolX = 1e-6

Further information about the inverse Gaussian distribution can be found at https://en.wikipedia.org/wiki/Inverse_Gaussian_distribution

See also: invgcdf, invginv, invgpdf, invgrnd, invglike

Source Code: invgfit

Example: 1

  

 ## Sample 3 populations from different inverse Gaussian distibutions
 rand ("seed", 5); randn ("seed", 5);   # for reproducibility
 r1 = invgrnd (1, 0.2, 2000, 1);
 rand ("seed", 2); randn ("seed", 2);   # for reproducibility
 r2 = invgrnd (1, 3, 2000, 1);
 rand ("seed", 7); randn ("seed", 7);   # for reproducibility
 r3 = invgrnd (3, 1, 2000, 1);
 r = [r1, r2, r3];

 ## Plot them normalized and fix their colors
 hist (r, [0.1:0.1:3.2], 9);
 h = findobj (gca, "Type", "patch");
 set (h(1), "facecolor", "c");
 set (h(2), "facecolor", "g");
 set (h(3), "facecolor", "r");
 ylim ([0, 3]);
 xlim ([0, 3]);
 hold on

 ## Estimate their MU and LAMBDA parameters
 mu_lambdaA = invgfit (r(:,1));
 mu_lambdaB = invgfit (r(:,2));
 mu_lambdaC = invgfit (r(:,3));

 ## Plot their estimated PDFs
 x = [0:0.1:3];
 y = invgpdf (x, mu_lambdaA(1), mu_lambdaA(2));
 plot (x, y, "-pr");
 y = invgpdf (x, mu_lambdaB(1), mu_lambdaB(2));
 plot (x, y, "-sg");
 y = invgpdf (x, mu_lambdaC(1), mu_lambdaC(2));
 plot (x, y, "-^c");
 hold off
 legend ({"Normalized HIST of sample 1 with μ=1 and λ=0.5", ...
          "Normalized HIST of sample 2 with μ=2 and λ=0.3", ...
          "Normalized HIST of sample 3 with μ=4 and λ=0.5", ...
          sprintf("PDF for sample 1 with estimated μ=%0.2f and λ=%0.2f", ...
                  mu_lambdaA(1), mu_lambdaA(2)), ...
          sprintf("PDF for sample 2 with estimated μ=%0.2f and λ=%0.2f", ...
                  mu_lambdaB(1), mu_lambdaB(2)), ...
          sprintf("PDF for sample 3 with estimated μ=%0.2f and λ=%0.2f", ...
                  mu_lambdaC(1), mu_lambdaC(2))})
 title ("Three population samples from different inverse Gaussian distibutions")
 hold off
plotted figure

statistics-release-1.6.3/docs/invginv.html000066400000000000000000000141761456127120000206360ustar00rootroot00000000000000 Statistics: invginv

Function Reference: invginv

statistics: x = invginv (p, mu, lambda)

Inverse of the inverse Gaussian cumulative distribution function (iCDF).

For each element of p, compute the quantile (the inverse of the CDF) of the inverse Gaussian distribution with scale parameter mu and shape parameter lambda. The size of x is the common size of p, mu, and lambda. A scalar input functions as a constant matrix of the same size as the other inputs.

The inverse Gaussian CDF is only defined for mu > 0 and lambda > 0.

Further information about the inverse Gaussian distribution can be found at https://en.wikipedia.org/wiki/Inverse_Gaussian_distribution

See also: invgcdf, invgpdf, invgrnd, invgfit, invglike

Source Code: invginv

Example: 1

  

 ## Plot various iCDFs from the inverse Gaussian distribution
 p = 0.001:0.001:0.999;
 x1 = invginv (p, 1, 0.2);
 x2 = invginv (p, 1, 1);
 x3 = invginv (p, 1, 3);
 x4 = invginv (p, 3, 0.2);
 x5 = invginv (p, 3, 1);
 plot (p, x1, "-b", p, x2, "-g", p, x3, "-r", p, x4, "-c", p, x5, "-y")
 grid on
 ylim ([0, 3])
 legend ({"μ = 1, σ = 0.2", "μ = 1, σ = 1", "μ = 1, σ = 3", ...
          "μ = 3, σ = 0.2", "μ = 3, σ = 1"}, "location", "northwest")
 title ("Inverse Gaussian iCDF")
 xlabel ("probability")
 ylabel ("x")
plotted figure

statistics-release-1.6.3/docs/invglike.html000066400000000000000000000137071456127120000207650ustar00rootroot00000000000000 Statistics: invglike

Function Reference: invglike

statistics: nlogL = invglike (params, x)
statistics: [nlogL, acov] = invglike (params, x)
statistics: […] = invglike (params, x, censor)
statistics: […] = invglike (params, x, censor, freq)

Negative log-likelihood for the inverse Gaussian distribution.

nlogL = invglike (params, x) returns the negative log likelihood of the data in x corresponding to the inverse Gaussian distribution with (1) scale parameter mu and (2) shape parameter lambda given in the two-element vector params.

[nlogL, acov] = invglike (params, x) also returns the inverse of Fisher’s information matrix, acov. If the input parameter values in params are the maximum likelihood estimates, the diagonal elements of params are their asymptotic variances.

[…] = invglike (params, x, censor) accepts a boolean vector, censor, of the same size as x with 1s for observations that are right-censored and 0s for observations that are observed exactly. By default, or if left empty, censor = zeros (size (x)).

[…] = invglike (params, x, censor, freq) accepts a frequency vector, freq, of the same size as x. freq typically contains integer frequencies for the corresponding elements in x, but it can contain any non-integer non-negative values. By default, or if left empty, freq = ones (size (x)).

Further information about the inverse Gaussian distribution can be found at https://en.wikipedia.org/wiki/Inverse_Gaussian_distribution

See also: invgcdf, invginv, invgpdf, invgrnd, invgfit

Source Code: invglike

statistics-release-1.6.3/docs/invgpdf.html000066400000000000000000000141661456127120000206120ustar00rootroot00000000000000 Statistics: invgpdf

Function Reference: invgpdf

statistics: y = invgpdf (x, mu, lambda)

Inverse Gaussian probability density function (PDF).

For each element of x, compute the probability density function (PDF) of the inverse Gaussian distribution with scale parameter mu and shape parameter lambda. The size of y is the common size of x, mu, and lambda. A scalar input functions as a constant matrix of the same size as the other inputs.

The inverse Gaussian CDF is only defined for mu > 0 and lambda > 0.

Further information about the inverse Gaussian distribution can be found at https://en.wikipedia.org/wiki/Inverse_Gaussian_distribution

See also: invgcdf, invginv, invgrnd, invgfit, invglike

Source Code: invgpdf

Example: 1

  

 ## Plot various PDFs from the inverse Gaussian distribution
 x = 0:0.001:3;
 y1 = invgpdf (x, 1, 0.2);
 y2 = invgpdf (x, 1, 1);
 y3 = invgpdf (x, 1, 3);
 y4 = invgpdf (x, 3, 0.2);
 y5 = invgpdf (x, 3, 1);
 plot (x, y1, "-b", x, y2, "-g", x, y3, "-r", x, y4, "-c", x, y5, "-y")
 grid on
 xlim ([0, 3])
 ylim ([0, 3])
 legend ({"μ = 1, σ = 0.2", "μ = 1, σ = 1", "μ = 1, σ = 3", ...
          "μ = 3, σ = 0.2", "μ = 3, σ = 1"}, "location", "northeast")
 title ("Inverse Gaussian PDF")
 xlabel ("values in x")
 ylabel ("density")
plotted figure

statistics-release-1.6.3/docs/invgrnd.html000066400000000000000000000125561456127120000206250ustar00rootroot00000000000000 Statistics: invgrnd

Function Reference: invgrnd

statistics: r = invgrnd (mu, lambda)
statistics: r = invgrnd (mu, lambda, rows)
statistics: r = invgrnd (mu, lambda, rows, cols, …)
statistics: r = invgrnd (mu, lambda, [sz])

Random arrays from the inverse Gaussian distribution.

r = invgrnd (mu, lambda) returns an array of random numbers chosen from the inverse Gaussian distribution with location parameter mu and scale parameter lambda. The size of r is the common size of mu and lambda. A scalar input functions as a constant matrix of the same size as the other inputs.

When called with a single size argument, invgrnd returns a square matrix with the dimension specified. When called with more than one scalar argument, the first two arguments are taken as the number of rows and columns and any further arguments specify additional matrix dimensions. The size may also be specified with a row vector of dimensions, sz.

The inverse Gaussian CDF is only defined for mu > 0 and lambda > 0.

Further information about the inverse Gaussian distribution can be found at https://en.wikipedia.org/wiki/Inverse_Gaussian_distribution

See also: invgcdf, invginv, invgpdf, invgfit, invglike

Source Code: invgrnd

statistics-release-1.6.3/docs/ismissing.html000066400000000000000000000122201456127120000211470ustar00rootroot00000000000000 Statistics: ismissing

Function Reference: ismissing

statistics: TF = ismissing (A)
statistics: TF = ismissing (A, indicator)

Find missing data in a numeric or string array.

Given an input numeric data array, char array, or array of cell strings A, ismissing returns a logical array TF with the same dimensions as A, where true values match missing values in the input data.

The optional input indicator is an array of values that represent missing values in the input data. The values which represent missing data by default depend on the data type of A:

  • NaN: single, double.
  • ' ' (white space): char.
  • {''}: string cells.

Note: logical and numeric data types may be used in any combination for A and indicator. A and the indicator values will be compared as type double, and the output will have the same class as A. Data types other than those specified above have no defined ’missing’ value. As such, the TF output for those inputs will always be false(size(A)). The exception to this is that indicator can be specified for logical and numeric inputs to designate values that will register as ’missing’.

See also: fillmissing, rmmissing, standardizeMissing

Source Code: ismissing

statistics-release-1.6.3/docs/isoutlier.html000066400000000000000000000415221456127120000211700ustar00rootroot00000000000000 Statistics: isoutlier

Function Reference: isoutlier

statistics: TF = isoutlier (x)
statistics: TF = isoutlier (x, method)
statistics: TF = isoutlier (x, "percentiles", threshold)
statistics: TF = isoutlier (x, movmethod, window)
statistics: TF = isoutlier (…, dim)
statistics: TF = isoutlier (…, Name, Value)
statistics: [TF, L, U, C] = isoutlier (…)

Find outliers in data

isoutlier (x) returns a logical array whose elements are true when an outlier is detected in the corresponding element of x. isoutlier treats NaNs as missing values and removes them.

  • If x is a matrix, then isoutlier operates on each column of x separately.
  • If x is a multidimensional array, then isoutlier operates along the first dimension of x whose size does not equal 1.

By default, an outlier is a value that is more than three scaled median absolute deviations (MAD) from the median. The scaled median is defined as c*median(abs(A-median(A))), where c=-1/(sqrt(2)*erfcinv(3/2)).

isoutlier (x, method) specifies a method for detecting outliers. The following methods are available:

MethodDescription
"median"Outliers are defined as elements more than three scaled MAD from the median.
"mean"Outliers are defined as elements more than three standard deviations from the mean.
"quartiles"Outliers are defined as elements more than 1.5 interquartile ranges above the upper quartile (75 percent) or below the lower quartile (25 percent). This method is useful when the data in x is not normally distributed.
"grubbs"Outliers are detected using Grubbs’ test for outliers, which removes one outlier per iteration based on hypothesis testing. This method assumes that the data in x is normally distributed.
"gesd"Outliers are detected using the generalized extreme Studentized deviate test for outliers. This iterative method is similar to "grubbs", but can perform better when there are multiple outliers masking each other.

isoutlier (x, "percentiles", threshold) detects outliers based on a percentile thresholds, specified as a two-element row vector whose elements are in the interval [0, 100]. The first element indicates the lower percentile threshold, and the second element indicates the upper percentile threshold. The first element of threshold must be less than the second element.

isoutlier (x, movmethod, window) specifies a moving method for detecting outliers. The following methods are available:

MethodDescription
"movmedian"Outliers are defined as elements more than three local scaled MAD from the local median over a window length specified by window.
"movmean"Outliers are defined as elements more than three local standard deviations from the from the local mean over a window length specified by window.

window must be a positive integer scalar or a two-element vector of positive integers. When window is a scalar, if it is an odd number, the window is centered about the current element and contains window - 1 neighboring elements. If even, then the window is centered about the current and previous elements. When window is a two-element vector of positive integers [nb, na], the window contains the current element, nb elements before the current element, and na elements after the current element. When "SamplePoints" are also specified, window can take any real positive values (either as a scalar or a two-element vector) and in this case, the windows are computed relative to the sample points.

dim specifies the operating dimension and it must be a positive integer scalar. If not specified, then, by default, isoutlier operates along the first non-singleton dimension of x.

The following optional parameters can be specified as Name/Value paired arguments.

  • "SamplePoints" can be specified as a vector of sample points with equal length as the operating dimension. The sample points represent the x-axis location of the data and must be sorted and contain unique elements. Sample points do not need to be uniformly sampled. By default, the vector is [1, 2, 3, …, n], where n = size (x, dim). You can use unequally spaced "SamplePoints" to define a variable-length window for one of the moving methods available.
  • "ThresholdFactor" can be specified as a nonnegative scalar. For methods "median" and "movmedian", the detection threshold factor replaces the number of scaled MAD, which is 3 by default. For methods "mean" and "movmean", the detection threshold factor replaces the number of standard deviations, which is 3 by default. For methods "grubbs" and "gesd", the detection threshold factor ranges from 0 to 1, specifying the critical alpha-value of the respective test, and it is 0.05 by default. For the "quartiles" method, the detection threshold factor replaces the number of interquartile ranges, which is 1.5 by default. "ThresholdFactor" is not supported for the "quartiles" method.
  • "MaxNumOutliers" is only relevant to the "gesd" method and it must be a positive integer scalar specifying the maximum number of outliers returned by the "gesd" method. By default, it is the integer nearest to the 10% of the number of elements along the operating dimension in x. The "gesd" method assumes the nonoutlier input data is sampled from an approximate normal distribution. When the data is not sampled in this way, the number of returned outliers might exceed the MaxNumOutliers value.

[TF, L, U, C] = isoutlier (…) returns up to 4 output arguments as described below.

  • TF is the outlier indicator with the same size a x.
  • L is the lower threshold used by the outlier detection method. If method is used for outlier detection, then L has the same size as x in all dimensions except for the operating dimension where the length is 1. If movmethod is used, then L has the same size as x.
  • U is the upper threshold used by the outlier detection method. If method is used for outlier detection, then U has the same size as x in all dimensions except for the operating dimension where the length is 1. If movmethod is used, then U has the same size as x.
  • C is the center value used by the outlier detection method. If method is used for outlier detection, then C has the same size as x in all dimensions except for the operating dimension where the length is 1. If movmethod is used, then C has the same size as x. For "median", "movmedian", "mean", and "movmean" methods, C is computed by taking into acount the outlier values. For "grubbs" and "gesd" methods, C is computed by excluding the outliers. For the "percentiles" method, C is the average between U and L thresholds.

See also: filloutliers, rmoutliers, ismissing

Source Code: isoutlier

Example: 1

  

 A = [57 59 60 100 59 58 57 58 300 61 62 60 62 58 57];
 TF = isoutlier (A, "mean")

TF =

  0  0  0  0  0  0  0  0  1  0  0  0  0  0  0

Example: 2

  

 ## Use a moving detection method to detect local outliers in a sine wave

 x = -2*pi:0.1:2*pi;
 A = sin(x);
 A(47) = 0;
 time = datenum (2023,1,1,0,0,0) + (1/24)*[0:length(x)-1] - 730485;
 TF = isoutlier (A, "movmedian", 5*(1/24), "SamplePoints", time);
 plot (time, A)
 hold on
 plot (time(TF), A(TF), "x")
 datetick ('x', 20, 'keepticks')
 legend ("Original Data", "Outlier Data")
plotted figure

Example: 3

  

 ## Locate an outlier in a vector of data and visualize the outlier

 x = 1:10;
 A = [60 59 49 49 58 100 61 57 48 58];
 [TF, L, U, C] = isoutlier (A);
 plot (x, A);
 hold on
 plot (x(TF), A(TF), "x");
 xlim ([1,10]);
 line ([1,10], [L, L], "Linestyle", ":");
 text (1.1, L-2, "Lower Threshold");
 line ([1,10], [U, U], "Linestyle", ":");
 text (1.1, U-2, "Upper Threshold");
 line ([1,10], [C, C], "Linestyle", ":");
 text (1.1, C-3, "Center Value");
 legend ("Original Data", "Outlier Data");
plotted figure

statistics-release-1.6.3/docs/iwishpdf.html000066400000000000000000000107401456127120000207640ustar00rootroot00000000000000 Statistics: iwishpdf

Function Reference: iwishpdf

statistics: y = iwishpdf (W, Tau, df, log_y=false)

Compute the probability density function of the inverse Wishart distribution.

Inputs: A p x p matrix W where to find the PDF and the p x p positive definite scale matrix Tau and scalar degrees of freedom parameter df characterizing the inverse Wishart distribution. (For the density to be finite, need df > (p - 1).) If the flag log_y is set, return the log probability density – this helps avoid underflow when the numerical value of the density is very small.

Output: y is the probability density of Wishart(Sigma, df) at W.

See also: iwishrnd, wishpdf, wishrnd

Source Code: iwishpdf

statistics-release-1.6.3/docs/iwishrnd.html000066400000000000000000000121041456127120000207720ustar00rootroot00000000000000 Statistics: iwishrnd

Function Reference: iwishrnd

statistics: [W, DI] = iwishrnd (Tau, df, DI, n=1)

Return a random matrix sampled from the inverse Wishart distribution with given parameters.

Inputs: the p× p positive definite matrix Tau and scalar degrees of freedom parameter df (and optionally the transposed Cholesky factor DI of Sigma = inv(Tau)).

df can be non-integer as long as df > d

Output: a random p× p matrix W from the inverse Wishart(Tau, df) distribution. (inv(W) is from the Wishart(inv(Tau), df) distribution.) If n > 1, then W is p x p x n and holds n such random matrices. (Optionally, the transposed Cholesky factor DI of Sigma is also returned.)

Averaged across many samples, the mean of W should approach Tau / (df - p - 1).

References

  1. Yu-Cheng Ku and Peter Bloomfield (2010), Generating Random Wishart Matrices with Fractional Degrees of Freedom in OX, http://www.gwu.edu/~forcpgm/YuChengKu-030510final-WishartYu-ChengKu.pdf

See also: iwishpdf, wishpdf, wishrnd

Source Code: iwishrnd

statistics-release-1.6.3/docs/jackknife.html000066400000000000000000000216361456127120000211020ustar00rootroot00000000000000 Statistics: jackknife

Function Reference: jackknife

statistics: jackstat = jackknife (E, x)
statistics: jackstat = jackknife (E, x, …)

Compute jackknife estimates of a parameter taking one or more given samples as parameters.

In particular, E is the estimator to be jackknifed as a function name, handle, or inline function, and x is the sample for which the estimate is to be taken. The i-th entry of jackstat will contain the value of the estimator on the sample x with its i-th row omitted.

 
 
 jackstat (i) = E(x(1 : i - 1, i + 1 : length(x)))

Depending on the number of samples to be used, the estimator must have the appropriate form:

  • If only one sample is used, then the estimator need not be concerned with cell arrays, for example jackknifing the standard deviation of a sample can be performed with jackstat = jackknife (@std, rand (100, 1)).
  • If, however, more than one sample is to be used, the samples must all be of equal size, and the estimator must address them as elements of a cell-array, in which they are aggregated in their order of appearance:
 
 
 jackstat = jackknife (@(x) std(x{1})/var(x{2}),
 rand (100, 1), randn (100, 1))

If all goes well, a theoretical value P for the parameter is already known, n is the sample size,

t = n * E(x) - (n - 1) * mean(jackstat)

and

v = sumsq(n * E(x) - (n - 1) * jackstat - t) / (n * (n - 1))

then

(t-P)/sqrt(v) should follow a t-distribution with n-1 degrees of freedom.

Jackknifing is a well known method to reduce bias. Further details can be found in:

References

  1. Rupert G. Miller. The jackknife - a review. Biometrika (1974), 61(1):1-15. doi:10.1093/biomet/61.1.1
  2. Rupert G. Miller. Jackknifing Variances. Ann. Math. Statist. (1968), Volume 39, Number 2, 567-582. doi:10.1214/aoms/1177698418

Source Code: jackknife

Example: 1

  

 for k = 1:1000
   rand ("seed", k);  # for reproducibility
   x = rand (10, 1);
   s(k) = std (x);
   jackstat = jackknife (@std, x);
   j(k) = 10 * std (x) - 9 * mean (jackstat);
 endfor
 figure();
 hist ([s', j'], 0:sqrt(1/12)/10:2*sqrt(1/12))
plotted figure

Example: 2

  

 for k = 1:1000
   randn ("seed", k); # for reproducibility
   x = randn (1, 50);
   rand ("seed", k);  # for reproducibility
   y = rand (1, 50);
   jackstat = jackknife (@(x) std(x{1})/std(x{2}), y, x);
   j(k) = 50 * std (y) / std (x) - 49 * mean (jackstat);
   v(k) = sumsq ((50 * std (y) / std (x) - 49 * jackstat) - j(k)) / (50 * 49);
 endfor
 t = (j - sqrt (1 / 12)) ./ sqrt (v);
 figure();
 plot (sort (tcdf (t, 49)), ...
       "-;Almost linear mapping indicates good fit with t-distribution.;")
plotted figure

statistics-release-1.6.3/docs/jsucdf.html000066400000000000000000000105601456127120000204250ustar00rootroot00000000000000 Statistics: jsucdf

Function Reference: jsucdf

statistics: p = jsucdf (x)
statistics: p = jsucdf (x, alpha1)
statistics: p = jsucdf (x, alpha1, alpha2)

Johnson SU cumulative distribution function (CDF).

For each element of x, return the cumulative distribution functions (CDF) at x of the Johnson SU distribution with shape parameters alpha1 and alpha2. The size of p is the common size of the input arguments x, alpha1, and alpha2. A scalar input functions as a constant matrix of the same size as the other

Default values are alpha1 = 1, alpha2 = 1.

See also: jsupdf

Source Code: jsucdf

statistics-release-1.6.3/docs/jsupdf.html000066400000000000000000000105501456127120000204410ustar00rootroot00000000000000 Statistics: jsupdf

Function Reference: jsupdf

statistics: y = jsupdf (x)
statistics: y = jsupdf (x, alpha1)
statistics: y = jsupdf (x, alpha1, alpha2)

Johnson SU probability density function (PDF).

For each element of x, compute the probability density function (PDF) at x of the Johnson SU distribution with shape parameters alpha1 and alpha2. The size of p is the common size of the input arguments x, alpha1, and alpha2. A scalar input functions as a constant matrix of the same size as the other

Default values are alpha1 = 1, alpha2 = 1.

See also: jsucdf

Source Code: jsupdf

statistics-release-1.6.3/docs/kmeans.html000066400000000000000000000527451456127120000204400ustar00rootroot00000000000000 Statistics: kmeans

Function Reference: kmeans

statistics: idx = kmeans (data, k)
statistics: [idx, centers] = kmeans (data, k)
statistics: [idx, centers, sumd] = kmeans (data, k)
statistics: [idx, centers, sumd, dist] = kmeans (data, k)
statistics: […] = kmeans (data, k, param1, value1, …)
statistics: […] = kmeans (data, [], "start", start, …)

Perform a k-means clustering of the N×D matrix data.

If parameter "start" is specified, then k may be empty in which case k is set to the number of rows of start.

The outputs are:

idxAn N×1 vector whose i-th element is the class to which row i of data is assigned.
centersA K×D array whose i-th row is the centroid of cluster i.
sumdA k×1 vector whose i-th entry is the sum of the distances from samples in cluster i to centroid i.
distAn N×k matrix whose ij-th element is the distance from sample i to centroid j.

The following parameters may be placed in any order. Each parameter must be followed by its value, as in Name-Value pairs.

NameDescription
"Start"The initialization method for the centroids.
ValueDescription
"plus"The k-means++ algorithm. (Default)
"sample"A subset of k rows from data, sampled uniformly without replacement.
"cluster"Perform a pilot clustering on 10% of the rows of data.
"uniform"Each component of each centroid is drawn uniformly from the interval between the maximum and minimum values of that component within data. This performs poorly and is implemented only for Matlab compatibility.
numeric matrixA k×D matrix of centroid starting locations. The rows correspond to seeds.
numeric arrayA k×D×r array of centroid starting locations. The third dimension invokes replication of the clustering routine. Page r contains the set of seeds for replicate r. kmeans infers the number of replicates (specified by the "Replicates" Name-Value pair argument) from the size of the third dimension.
NameDescription
"Distance"The distance measure used for partitioning and calculating centroids.
ValueDescription
"sqeuclidean"The squared Euclidean distance. i.e. the sum of the squares of the differences between corresponding components. In this case, the centroid is the arithmetic mean of all samples in its cluster. This is the only distance for which this algorithm is truly "k-means".
"cityblock"The sum metric, or L1 distance, i.e. the sum of the absolute differences between corresponding components. In this case, the centroid is the median of all samples in its cluster. This gives the k-medians algorithm.
"cosine"One minus the cosine of the included angle between points (treated as vectors). Each centroid is the mean of the points in that cluster, after normalizing those points to unit Euclidean length.
"correlation"One minus the sample correlation between points (treated as sequences of values). Each centroid is the component-wise mean of the points in that cluster, after centering and normalizing those points to zero mean and unit standard deviation.
"hamming"The number of components in which the sample and the centroid differ. In this case, the centroid is the median of all samples in its cluster. Unlike Matlab, Octave allows non-logical data.
NameDescription
"EmptyAction"What to do when a centroid is not the closest to any data sample.
ValueDescription
"error"Throw an error.
"singleton"(Default) Select the row of data that has the highest error and use that as the new centroid.
"drop"Remove the centroid, and continue computation with one fewer centroid. The dimensions of the outputs centroids and d are unchanged, with values for omitted centroids replaced by NaN.
NameDescription
"Display"Display a text summary.
ValueDescription
"off"(Default) Display no summary.
"final"Display a summary for each clustering operation.
"iter"Display a summary for each iteration of a clustering operation.
NameValue
"Replicates"A positive integer specifying the number of independent clusterings to perform. The output values are the values for the best clustering, i.e., the one with the smallest value of sumd. If Start is numeric, then Replicates defaults to (and must equal) the size of the third dimension of Start. Otherwise it defaults to 1.
"MaxIter"The maximum number of iterations to perform for each replicate. If the maximum change of any centroid is less than 0.001, then the replicate terminates even if MaxIter iterations have no occurred. The default is 100.

Example:

[~,c] = kmeans (rand(10, 3), 2, "emptyaction", "singleton");

See also: linkage

Source Code: kmeans

Example: 1

  

 ## Generate a two-cluster problem
 randn ("seed", 31)  # for reproducibility
 C1 = randn (100, 2) + 1;
 randn ("seed", 32)  # for reproducibility
 C2 = randn (100, 2) - 1;
 data = [C1; C2];

 ## Perform clustering
 rand ("seed", 1)  # for reproducibility
 [idx, centers] = kmeans (data, 2);

 ## Plot the result
 figure;
 plot (data (idx==1, 1), data (idx==1, 2), "ro");
 hold on;
 plot (data (idx==2, 1), data (idx==2, 2), "bs");
 plot (centers (:, 1), centers (:, 2), "kv", "markersize", 10);
 hold off;
plotted figure

Example: 2

  

 ## Cluster data using k-means clustering, then plot the cluster regions
 ## Load Fisher's iris data set and use the petal lengths and widths as
 ## predictors

 load fisheriris
 X = meas(:,3:4);

 figure;
 plot (X(:,1), X(:,2), "k*", "MarkerSize", 5);
 title ("Fisher's Iris Data");
 xlabel ("Petal Lengths (cm)");
 ylabel ("Petal Widths (cm)");

 ## Cluster the data. Specify k = 3 clusters
 rand ("seed", 1)  # for reproducibility
 [idx, C] = kmeans (X, 3);
 x1 = min (X(:,1)):0.01:max (X(:,1));
 x2 = min (X(:,2)):0.01:max (X(:,2));
 [x1G, x2G] = meshgrid (x1, x2);
 XGrid = [x1G(:), x2G(:)];

 idx2Region = kmeans (XGrid, 3, "MaxIter", 1, "Start", C);
 figure;
 gscatter (XGrid(:,1), XGrid(:,2), idx2Region, ...
           [0, 0.75, 0.75; 0.75, 0, 0.75; 0.75, 0.75, 0], "..");
 hold on;
 plot (X(:,1), X(:,2), "k*", "MarkerSize", 5);
 title ("Fisher's Iris Data");
 xlabel ("Petal Lengths (cm)");
 ylabel ("Petal Widths (cm)");
 legend ("Region 1", "Region 2", "Region 3", "Data", "Location", "SouthEast");
 hold off

warning: kmeans: failed to converge in 1 iterations
warning: called from
    kmeans at line 442 column 7
    build_DEMOS at line 94 column 11
    function_texi2html at line 112 column 14
    package_texi2html at line 290 column 9
plotted figure

plotted figure

Example: 3

  

 ## Partition Data into Two Clusters

 randn ("seed", 1)  # for reproducibility
 r1 = randn (100, 2) * 0.75 + ones (100, 2);
 randn ("seed", 2)  # for reproducibility
 r2 = randn (100, 2) * 0.5 - ones (100, 2);
 X = [r1; r2];

 figure;
 plot (X(:,1), X(:,2), ".");
 title ("Randomly Generated Data");
 rand ("seed", 1)  # for reproducibility
 [idx, C] = kmeans (X, 2, "Distance", "cityblock", ...
                          "Replicates", 5, "Display", "final");
 figure;
 plot (X(idx==1,1), X(idx==1,2), "r.", "MarkerSize", 12);
 hold on
 plot(X(idx==2,1), X(idx==2,2), "b.", "MarkerSize", 12);
 plot (C(:,1), C(:,2), "kx", "MarkerSize", 15, "LineWidth", 3);
 legend ("Cluster 1", "Cluster 2", "Centroids", "Location", "NorthWest");
 title ("Cluster Assignments and Centroids");
 hold off

Replicate 1, 5 iterations, total sum of distances = 197.416.
Replicate 2, 1 iterations, total sum of distances = 253.651.
Replicate 3, 1 iterations, total sum of distances = 401.899.
Replicate 4, 1 iterations, total sum of distances = 426.406.
Replicate 5, 1 iterations, total sum of distances = 663.780.
Best total sum of distances = 197.416
plotted figure

plotted figure

Example: 4

  

 ## Assign New Data to Existing Clusters

 ## Generate a training data set using three distributions
 randn ("seed", 5)  # for reproducibility
 r1 = randn (100, 2) * 0.75 + ones (100, 2);
 randn ("seed", 7)  # for reproducibility
 r2 = randn (100, 2) * 0.5 - ones (100, 2);
 randn ("seed", 9)  # for reproducibility
 r3 = randn (100, 2) * 0.75;
 X = [r1; r2; r3];

 ## Partition the training data into three clusters by using kmeans

 rand ("seed", 1)  # for reproducibility
 [idx, C] = kmeans (X, 3);

 ## Plot the clusters and the cluster centroids

 figure
 gscatter (X(:,1), X(:,2), idx, "bgm", "***");
 hold on
 plot (C(:,1), C(:,2), "kx");
 legend ("Cluster 1", "Cluster 2", "Cluster 3", "Cluster Centroid")

 ## Generate a test data set
 randn ("seed", 25)  # for reproducibility
 r1 = randn (100, 2) * 0.75 + ones (100, 2);
 randn ("seed", 27)  # for reproducibility
 r2 = randn (100, 2) * 0.5 - ones (100, 2);
 randn ("seed", 29)  # for reproducibility
 r3 = randn (100, 2) * 0.75;
 Xtest = [r1; r2; r3];

 ## Classify the test data set using the existing clusters
 ## Find the nearest centroid from each test data point by using pdist2

 D = pdist2 (C, Xtest, "euclidean");
 [group, ~] = find (D == min (D));

 ## Plot the test data and label the test data using idx_test with gscatter

 gscatter (Xtest(:,1), Xtest(:,2), group, "bgm", "ooo");
 legend ("Cluster 1", "Cluster 2", "Cluster 3", "Cluster Centroid", ...
         "Data classified to Cluster 1", "Data classified to Cluster 2", ...
         "Data classified to Cluster 3", "Location", "NorthWest");
 title ("Assign New Data to Existing Clusters");
plotted figure

statistics-release-1.6.3/docs/knnsearch.html000066400000000000000000000412441456127120000211260ustar00rootroot00000000000000 Statistics: knnsearch

Function Reference: knnsearch

statistics: idx = knnsearch (X, Y)
statistics: [idx, D] = knnsearch (X, Y)
statistics: […] = knnsearch (…, name, value)

Find k-nearest neighbors from input data.

idx = knnsearch (X, Y) finds K nearest neighbors in X for Y. It returns idx which contains indices of K nearest neighbors of each row of Y, If not specified, K = 1. X must be an N×P numeric matrix of input data, where rows correspond to observations and columns correspond to features or variables. Y is an M×P numeric matrix with query points, which must have the same numbers of column as X.

[idx, D] = knnsearch (X, Y) also returns the the distances, D, which correspond to the K nearest neighbour in X for each Y

Additional parameters can be specified by Name-Value pair arguments.

NameValue
"K"is the number of nearest neighbors to be found in the kNN search. It must be a positive integer value and by default it is 1.
"P"is the Minkowski distance exponent and it must be a positive scalar. This argument is only valid when the selected distance metric is "minkowski". By default it is 2.
"Scale"is the scale parameter for the standardized Euclidean distance and it must be a nonnegative numeric vector of equal length to the number of columns in X. This argument is only valid when the selected distance metric is "seuclidean", in which case each coordinate of X is scaled by the corresponding element of "scale", as is each query point in Y. By default, the scale parameter is the standard deviation of each coordinate in X.
"Cov"is the covariance matrix for computing the mahalanobis distance and it must be a positive definite matrix matching the the number of columns in X. This argument is only valid when the selected distance metric is "mahalanobis".
"BucketSize"is the maximum number of data points in the leaf node of the Kd-tree and it must be a positive integer. This argument is only valid when the selected search method is "kdtree".
"SortIndices"is a boolean flag to sort the returned indices in ascending order by distance and it is true by default. When the selected search method is "exhaustive" or the "IncludeTies" flag is true, knnsearch always sorts the returned indices.
"Distance"is the distance metric used by knnsearch as specified below:
"euclidean"Euclidean distance.
"seuclidean"standardized Euclidean distance. Each coordinate difference between the rows in X and the query matrix Y is scaled by dividing by the corresponding element of the standard deviation computed from X. To specify a different scaling, use the "Scale" name-value argument.
"cityblock"City block distance.
"chebychev"Chebychev distance (maximum coordinate difference).
"minkowski"Minkowski distance. The default exponent is 2. To specify a different exponent, use the "P" name-value argument.
"mahalanobis"Mahalanobis distance, computed using a positive definite covariance matrix. To change the value of the covariance matrix, use the "Cov" name-value argument.
"cosine"Cosine distance.
"correlation"One minus the sample linear correlation between observations (treated as sequences of values).
"spearman"One minus the sample Spearman’s rank correlation between observations (treated as sequences of values).
"hamming"Hamming distance, which is the percentage of coordinates that differ.
"jaccard"One minus the Jaccard coefficient, which is the percentage of nonzero coordinates that differ.
@distfunCustom distance function handle. A distance function of the form function D2 = distfun (XI, YI), where XI is a 1×P vector containing a single observation in P-dimensional space, YI is an N×P matrix containing an arbitrary number of observations in the same P-dimensional space, and D2 is an N×P vector of distances, where (D2k) is the distance between observations XI and (YIk,:).
"NSMethod"is the nearest neighbor search method used by knnsearch as specified below.
"kdtree"Creates and uses a Kd-tree to find nearest neighbors. "kdtree" is the default value when the number of columns in X is less than or equal to 10, X is not sparse, and the distance metric is "euclidean", "cityblock", "manhattan", "chebychev", or "minkowski". Otherwise, the default value is "exhaustive". This argument is only valid when the distance metric is one of the four aforementioned metrics.
"exhaustive"Uses the exhaustive search algorithm by computing the distance values from all the points in X to each point in Y.
"IncludeTies"is a boolean flag to indicate if the returned values should contain the indices that have same distance as the K^th neighbor. When false, knnsearch chooses the observation with the smallest index among the observations that have the same distance from a query point. When true, knnsearch includes all nearest neighbors whose distances are equal to the K^th smallest distance in the output arguments. To specify K, use the "K" name-value pair argument.

See also: rangesearch, pdist2, fitcknn

Source Code: knnsearch

Example: 1

  

 ## find 10 nearest neighbour of a point using different distance metrics
 ## and compare the results by plotting
 load fisheriris
 X = meas(:,3:4);
 Y = species;
 point = [5, 1.45];

 ## calculate 10 nearest-neighbours by minkowski distance
 [id, d] = knnsearch (X, point, "K", 10);

 ## calculate 10 nearest-neighbours by minkowski distance
 [idm, dm] = knnsearch (X, point, "K", 10, "distance", "minkowski", "p", 5);

 ## calculate 10 nearest-neighbours by chebychev distance
 [idc, dc] = knnsearch (X, point, "K", 10, "distance", "chebychev");

 ## plotting the results
 gscatter (X(:,1), X(:,2), species, [.75 .75 0; 0 .75 .75; .75 0 .75], ".", 20);
 title ("Fisher's Iris Data - Nearest Neighbors with different types of distance metrics");
 xlabel("Petal length (cm)");
 ylabel("Petal width (cm)");

 line (point(1), point(2), "marker", "X", "color", "k", ...
       "linewidth", 2, "displayname", "query point")
 line (X(id,1), X(id,2), "color", [0.5 0.5 0.5], "marker", "o", ...
       "linestyle", "none", "markersize", 10, "displayname", "eulcidean")
 line (X(idm,1), X(idm,2), "color", [0.5 0.5 0.5], "marker", "d", ...
       "linestyle", "none", "markersize", 10, "displayname", "Minkowski")
 line (X(idc,1), X(idc,2), "color", [0.5 0.5 0.5], "marker", "p", ...
       "linestyle", "none", "markersize", 10, "displayname", "chebychev")
 xlim ([4.5 5.5]);
 ylim ([1 2]);
 axis square;
plotted figure

Example: 2

  

 ## knnsearch on iris dataset using kdtree method
 load fisheriris
 X = meas(:,3:4);
 gscatter (X(:,1), X(:,2), species, [.75 .75 0; 0 .75 .75; .75 0 .75], ".", 20);
 title ("Fisher's iris dataset : Nearest Neighbors with kdtree search");

 ## new point to be predicted
 point = [5 1.45];

 line (point(1), point(2), "marker", "X", "color", "k", ...
       "linewidth", 2, "displayname", "query point")

 ## knnsearch using kdtree method
 [idx, d] = knnsearch (X, point, "K", 10, "NSMethod", "kdtree");

 ## plotting predicted neighbours
 line (X(idx,1), X(idx,2), "color", [0.5 0.5 0.5], "marker", "o", ...
       "linestyle", "none", "markersize", 10, ...
       "displayname", "nearest neighbour")
 xlim ([4 6])
 ylim ([1 3])
 axis square
 ## details of predicted labels
 tabulate (species(idx))

 ctr = point - d(end);
 diameter = 2 * d(end);
 ##  Draw a circle around the 10 nearest neighbors.
 h = rectangle ("position", [ctr, diameter, diameter], "curvature", [1 1]);

 ## here only 8 neighbours are plotted instead of 10 since the dataset
 ## contains duplicate values

       Value    Count    Percent
   virginica        2      20.00%
  versicolor        8      80.00%
plotted figure

statistics-release-1.6.3/docs/kruskalwallis.html000066400000000000000000000246651456127120000220520ustar00rootroot00000000000000 Statistics: kruskalwallis

Function Reference: kruskalwallis

statistics: p = kruskalwallis (x)
statistics: p = kruskalwallis (x, group)
statistics: p = kruskalwallis (x, group, displayopt)
statistics: [p, tbl] = kruskalwallis (x, …)
statistics: [p, tbl, stats] = kruskalwallis (x, …)

Perform a Kruskal-Wallis test, the non-parametric alternative of a one-way analysis of variance (ANOVA), for comparing the means of two or more groups of data under the null hypothesis that the groups are drawn from the same population, i.e. the group means are equal.

kruskalwallis can take up to three input arguments:

  • x contains the data and it can either be a vector or matrix. If x is a matrix, then each column is treated as a separate group. If x is a vector, then the group argument is mandatory.
  • group contains the names for each group. If x is a matrix, then group can either be a cell array of strings of a character array, with one row per column of x. If you want to omit this argument, enter an empty array ([]). If x is a vector, then group must be a vector of the same lenth, or a string array or cell array of strings with one row for each element of x. x values corresponding to the same value of group are placed in the same group.
  • displayopt is an optional parameter for displaying the groups contained in the data in a boxplot. If omitted, it is ’on’ by default. If group names are defined in group, these are used to identify the groups in the boxplot. Use ’off’ to omit displaying this figure.

kruskalwallis can return up to three output arguments:

  • p is the p-value of the null hypothesis that all group means are equal.
  • tbl is a cell array containing the results in a standard ANOVA table.
  • stats is a structure containing statistics useful for performing a multiple comparison of means with the MULTCOMPARE function.

If kruskalwallis is called without any output arguments, then it prints the results in a one-way ANOVA table to the standard output. It is also printed when displayopt is ’on’.

Examples:

 
 x = meshgrid (1:6);
 x = x + normrnd (0, 1, 6, 6);
 [p, atab] = kruskalwallis(x);
 
 x = ones (50, 4) .* [-2, 0, 1, 5];
 x = x + normrnd (0, 2, 50, 4);
 group = {"A", "B", "C", "D"};
 kruskalwallis (x, group);

Source Code: kruskalwallis

Example: 1

  

 x = meshgrid (1:6);
 x = x + normrnd (0, 1, 6, 6);
 kruskalwallis (x, [], 'off');

              Kruskal-Wallis ANOVA Table
Source        SS      df      MS      Chi-sq  Prob>Chi-sq
---------------------------------------------------------
Columns    3139.67     5     627.93    28.29  3.20103e-05
Error       745.33    30      24.84
Total      3885.00    35

Example: 2

  

 x = meshgrid (1:6);
 x = x + normrnd (0, 1, 6, 6);
 [p, atab] = kruskalwallis(x);

              Kruskal-Wallis ANOVA Table
Source        SS      df      MS      Chi-sq  Prob>Chi-sq
---------------------------------------------------------
Columns    3232.33     5     646.47    29.12  2.19627e-05
Error       652.67    30      21.76
Total      3885.00    35
plotted figure

Example: 3

  

 x = ones (30, 4) .* [-2, 0, 1, 5];
 x = x + normrnd (0, 2, 30, 4);
 group = {"A", "B", "C", "D"};
 kruskalwallis (x, group);

              Kruskal-Wallis ANOVA Table
Source        SS      df      MS      Chi-sq  Prob>Chi-sq
---------------------------------------------------------
Columns   93937.00     3   31312.33    77.63  1.11022e-16
Error     50053.00   116     431.49
Total    143990.00   119
plotted figure

statistics-release-1.6.3/docs/kstest.html000066400000000000000000000200361456127120000204630ustar00rootroot00000000000000 Statistics: kstest

Function Reference: kstest

statistics: h = kstest (x)
statistics: h = kstest (x, name, value)
statistics: [h, p] = kstest (…)
statistics: [h, p, ksstat, cv] = kstest (…)

Single sample Kolmogorov-Smirnov (K-S) goodness-of-fit hypothesis test.

h = kstest (x) performs a Kolmogorov-Smirnov (K-S) test to determine if a random sample x could have come from a standard normal distribution. h indicates the results of the null hypothesis test.

  • h = 0 => Do not reject the null hypothesis at the 5% significance
  • h = 1 => Reject the null hypothesis at the 5% significance

x is a vector representing a random sample from some unknown distribution with a cumulative distribution function F(X). Missing values declared as NaNs in x are ignored.

h = kstest (x, name, value) returns a test decision for a single-sample K-S test with additional options specified by one or more name-value pair arguments as shown below.

"alpha"A value alpha between 0 and 1 specifying the significance level. Default is 0.05 for 5% significance.
"CDF"CDF is the c.d.f. under the null hypothesis. It can be specified either as a function handle or a a function name of an existing cdf function or as a two-column matrix. If not provided, the default is the standard normal, N(0,1).
"tail"A string indicating the type of test:
"unequal""F(X) not equal to CDF(X)" (two-sided) (Default)
"larger""F(X) > CDF(X)" (one-sided)
"smaller""CDF(X) < F(X)" (one-sided)

Let S(X) be the empirical c.d.f. estimated from the sample vector x, F(X) be the corresponding true (but unknown) population c.d.f., and CDF be the known input c.d.f. specified under the null hypothesis. For tail = "unequal", "larger", and "smaller", the test statistics are max|S(X) - CDF(X)|, max[S(X) - CDF(X)], and max[CDF(X) - S(X)], respectively.

[h, p] = kstest (…) also returns the asymptotic p-value p.

[h, p, ksstat] = kstest (…) returns the K-S test statistic ksstat defined above for the test type indicated by the "tail" option

In the matrix version of CDF, column 1 contains the x-axis data and column 2 the corresponding y-axis c.d.f data. Since the K-S test statistic will occur at one of the observations in x, the calculation is most efficient when CDF is only specified at the observations in x. When column 1 of CDF represents x-axis points independent of x, CDF is linearly interpolated at the observations found in the vector x. In this case, the interval along the x-axis (the column 1 spread of CDF) must span the observations in x for successful interpolation.

The decision to reject the null hypothesis is based on comparing the p-value p with the "alpha" value, not by comparing the statistic ksstat with the critical value cv. cv is computed separately using an approximate formula or by interpolation using Miller’s approximation table. The formula and table cover the range 0.01 <= "alpha" <= 0.2 for two-sided tests and 0.005 <= "alpha" <= 0.1 for one-sided tests. CV is returned as NaN if "alpha" is outside this range. Since CV is approximate, a comparison of ksstat with cv may occasionally lead to a different conclusion than a comparison of p with "alpha".

See also: kstest2, cdfplot

Source Code: kstest

statistics-release-1.6.3/docs/kstest2.html000066400000000000000000000150541456127120000205510ustar00rootroot00000000000000 Statistics: kstest2

Function Reference: kstest2

statistics: h = kstest2 (x1, x2)
statistics: h = kstest2 (x1, x2, name, value)
statistics: [h, p] = kstest2 (…)
statistics: [h, p, ks2stat] = kstest2 (…)

Two-sample Kolmogorov-Smirnov goodness-of-fit hypothesis test.

h = kstest2 (x1, x2) returns a test decision for the null hypothesis that the data in vectors x1 and x2 are from the same continuous distribution, using the two-sample Kolmogorov-Smirnov test. The alternative hypothesis is that x1 and x2 are from different continuous distributions. The result h is 1 if the test rejects the null hypothesis at the 5% significance level, and 0 otherwise.

h = kstest2 (x1, x2, name, value) returns a test decision for a two-sample Kolmogorov-Smirnov test with additional options specified by one or more name-value pair arguments as shown below.

"alpha"A value alpha between 0 and 1 specifying the significance level. Default is 0.05 for 5% significance.
"tail"A string indicating the type of test:
"unequal""F(X1) not equal to F(X2)" (two-sided) [Default]
"larger""F(X1) > F(X2)" (one-sided)
"smaller""F(X1) < F(X2)" (one-sided)

The two-sided test uses the maximum absolute difference between the cdfs of the distributions of the two data vectors. The test statistic is D* = max(|F1(x) - F2(x)|), where F1(x) is the proportion of x1 values less or equal to x and F2(x) is the proportion of x2 values less than or equal to x. The one-sided test uses the actual value of the difference between the cdfs of the distributions of the two data vectors rather than the absolute value. The test statistic is D* = max(F1(x) - F2(x)) or D* = max(F2(x) - F1(x)) for tail = "larger" or "smaller", respectively.

[h, p] = kstest2 (…) also returns the asymptotic p-value p.

[h, p, ks2stat] = kstest2 (…) also returns the Kolmogorov-Smirnov test statistic ks2stat defined above for the test type indicated by tail.

See also: kstest, cdfplot

Source Code: kstest2

statistics-release-1.6.3/docs/laplacecdf.html000066400000000000000000000146431456127120000212330ustar00rootroot00000000000000 Statistics: laplacecdf

Function Reference: laplacecdf

statistics: p = laplacecdf (x, mu, beta)
statistics: p = laplacecdf (x, mu, beta, "upper")

Laplace cumulative distribution function (CDF).

For each element of x, compute the cumulative distribution function (CDF) of the Laplace distribution with location parameter mu and scale parameter (i.e. "diversity") beta. The size of p is the common size of x, mu, and beta. A scalar input functions as a constant matrix of the same size as the other inputs.

Both parameters must be reals and beta > 0. For beta <= 0, NaN is returned.

p = laplacecdf (x, mu, beta, "upper") computes the upper tail probability of the Laplace distribution with parameters mu and beta, at the values in x.

Further information about the Laplace distribution can be found at https://en.wikipedia.org/wiki/Laplace_distribution

See also: laplaceinv, laplacepdf, laplacernd

Source Code: laplacecdf

Example: 1

  

 ## Plot various CDFs from the Laplace distribution
 x = -10:0.01:10;
 p1 = laplacecdf (x, 0, 1);
 p2 = laplacecdf (x, 0, 2);
 p3 = laplacecdf (x, 0, 4);
 p4 = laplacecdf (x, -5, 4);
 plot (x, p1, "-b", x, p2, "-g", x, p3, "-r", x, p4, "-c")
 grid on
 xlim ([-10, 10])
 legend ({"μ = 0, β = 1", "μ = 0, β = 2", ...
          "μ = 0, β = 4", "μ = -5, β = 4"}, "location", "southeast")
 title ("Laplace CDF")
 xlabel ("values in x")
 ylabel ("probability")
plotted figure

statistics-release-1.6.3/docs/laplaceinv.html000066400000000000000000000140311456127120000212620ustar00rootroot00000000000000 Statistics: laplaceinv

Function Reference: laplaceinv

statistics: x = laplaceinv (p, mu, beta)

Inverse of the Laplace cumulative distribution function (iCDF).

For each element of p, compute the quantile (the inverse of the CDF) of the Laplace distribution with location parameter mu and scale parameter (i.e. "diversity") beta. The size of x is the common size of p, mu, and beta. A scalar input functions as a constant matrix of the same size as the other inputs.

Both parameters must be reals and beta > 0. For beta <= 0, NaN is returned.

Further information about the Laplace distribution can be found at https://en.wikipedia.org/wiki/Laplace_distribution

See also: laplaceinv, laplacepdf, laplacernd

Source Code: laplaceinv

Example: 1

  

 ## Plot various iCDFs from the Laplace distribution
 p = 0.001:0.001:0.999;
 x1 = cauchyinv (p, 0, 1);
 x2 = cauchyinv (p, 0, 2);
 x3 = cauchyinv (p, 0, 4);
 x4 = cauchyinv (p, -5, 4);
 plot (p, x1, "-b", p, x2, "-g", p, x3, "-r", p, x4, "-c")
 grid on
 ylim ([-10, 10])
 legend ({"μ = 0, β = 1", "μ = 0, β = 2", ...
          "μ = 0, β = 4", "μ = -5, β = 4"}, "location", "northwest")
 title ("Laplace iCDF")
 xlabel ("probability")
 ylabel ("values in x")
plotted figure

statistics-release-1.6.3/docs/laplacepdf.html000066400000000000000000000140171456127120000212430ustar00rootroot00000000000000 Statistics: laplacepdf

Function Reference: laplacepdf

statistics: y = laplacepdf (x, mu, beta)

Laplace probability density function (PDF).

For each element of x, compute the probability density function (PDF) of the Laplace distribution with location parameter mu and scale parameter (i.e. "diversity") beta. The size of y is the common size of x, mu, and beta. A scalar input functions as a constant matrix of the same size as the other inputs.

Both parameters must be reals and beta > 0. For beta <= 0, NaN is returned.

Further information about the Laplace distribution can be found at https://en.wikipedia.org/wiki/Laplace_distribution

See also: laplacecdf, laplacepdf, laplacernd

Source Code: laplacepdf

Example: 1

  

 ## Plot various PDFs from the Laplace distribution
 x = -10:0.01:10;
 y1 = laplacepdf (x, 0, 1);
 y2 = laplacepdf (x, 0, 2);
 y3 = laplacepdf (x, 0, 4);
 y4 = laplacepdf (x, -5, 4);
 plot (x, y1, "-b", x, y2, "-g", x, y3, "-r", x, y4, "-c")
 grid on
 xlim ([-10, 10])
 ylim ([0, 0.6])
 legend ({"μ = 0, β = 1", "μ = 0, β = 2", ...
          "μ = 0, β = 4", "μ = -5, β = 4"}, "location", "northeast")
 title ("Laplace PDF")
 xlabel ("values in x")
 ylabel ("density")
plotted figure

statistics-release-1.6.3/docs/laplacernd.html000066400000000000000000000124501456127120000212540ustar00rootroot00000000000000 Statistics: laplacernd

Function Reference: laplacernd

statistics: r = laplacernd (mu, beta)
statistics: r = laplacernd (mu, beta, rows)
statistics: r = laplacernd (mu, beta, rows, cols, …)
statistics: r = laplacernd (mu, beta, [sz])

Random arrays from the Laplace distribution.

r = laplacernd (mu, beta) returns an array of random numbers chosen from the Laplace distribution with location parameter mu and scale parameter beta. The size of r is the common size of mu and beta. A scalar input functions as a constant matrix of the same size as the other inputs.

Both parameters must be reals and beta > 0. For beta <= 0, NaN is returned.

When called with a single size argument, laplacernd returns a square matrix with the dimension specified. When called with more than one scalar argument, the first two arguments are taken as the number of rows and columns and any further arguments specify additional matrix dimensions. The size may also be specified with a row vector of dimensions, sz.

Further information about the Laplace distribution can be found at https://en.wikipedia.org/wiki/Laplace_distribution

See also: laplacecdf, laplaceinv, laplacernd

Source Code: laplacernd

statistics-release-1.6.3/docs/levene_test.html000066400000000000000000000160201456127120000214610ustar00rootroot00000000000000 Statistics: levene_test

Function Reference: levene_test

statistics: h = levene_test (x)
statistics: h = levene_test (x, group)
statistics: h = levene_test (x, alpha)
statistics: h = levene_test (x, testtype)
statistics: h = levene_test (x, group, alpha)
statistics: h = levene_test (x, group, testtype)
statistics: h = levene_test (x, group, alpha, testtype)
statistics: [h, pval] = levene_test (…)
statistics: [h, pval, W] = levene_test (…)
statistics: [h, pval, W, df] = levene_test (…)

Perform a Levene’s test for the homogeneity of variances.

Under the null hypothesis of equal variances, the test statistic W approximately follows an F distribution with df degrees of freedom being a vector ([k-1, N-k]).

The p-value (1 minus the CDF of this distribution at W) is returned in pval. h = 1 if the null hypothesis is rejected at the significance level of alpha. Otherwise h = 0.

Input Arguments:

  • x contains the data and it can either be a vector or matrix. If x is a matrix, then each column is treated as a separate group. If x is a vector, then the group argument is mandatory. NaN values are omitted.
  • group contains the names for each group. If x is a vector, then group must be a vector of the same length, or a string array or cell array of strings with one row for each element of x. x values corresponding to the same value of group are placed in the same group. If x is a matrix, then group can either be a cell array of strings of a character array, with one row per column of x in the same way it is used in anova1 function. If x is a matrix, then group can be omitted either by entering an empty array ([]) or by parsing only alpha as a second argument (if required to change its default value).
  • alpha is the statistical significance value at which the null hypothesis is rejected. Its default value is 0.05 and it can be parsed either as a second argument (when group is omitted) or as a third argument.
  • testtype is a string determining the type of Levene’s test. By default it is set to "absolute", but the user can also parse "quadratic" in order to perform Levene’s Quadratic test for equal variances or "median" in order to to perform the Brown-Forsythe’s test. These options determine how the Z_ij values are computed. If an invalid name is parsed for testtype, then the Levene’s Absolute test is performed.

See also: bartlett_test, vartest2, vartestn

Source Code: levene_test

statistics-release-1.6.3/docs/libsvmread.html000066400000000000000000000075051456127120000213040ustar00rootroot00000000000000 Statistics: libsvmread

Function Reference: libsvmread

statistics: [labels, data] = libsvmread (filename)

This function reads the labels and the corresponding instance_matrix from a LIBSVM data file and stores them in labels and data respectively. These can then be used as inputs to svmtrain or svmpredict function.

Source Code: libsvmread

statistics-release-1.6.3/docs/libsvmwrite.html000066400000000000000000000074401456127120000215210ustar00rootroot00000000000000 Statistics: libsvmwrite

Function Reference: libsvmwrite

statistics: libsvmwrite (filename, labels, data)

This function saves the labels and the corresponding instance_matrix in a file specified by filename. data must be a sparse matrix. Both labels, data must be of double type.

Source Code: libsvmwrite

statistics-release-1.6.3/docs/linkage.html000066400000000000000000000163221456127120000205630ustar00rootroot00000000000000 Statistics: linkage

Function Reference: linkage

statistics: y = linkage (d)
statistics: y = linkage (d, method)
statistics: y = linkage (x)
statistics: y = linkage (x, method)
statistics: y = linkage (x, method, metric)
statistics: y = linkage (x, method, arglist)

Produce a hierarchical clustering dendrogram.

d is the dissimilarity matrix relative to n observations, formatted as a (n-1)×n/2x1 vector as produced by pdist. Alternatively, x contains data formatted for input to pdist, metric is a metric for pdist and arglist is a cell array containing arguments that are passed to pdist.

linkage starts by putting each observation into a singleton cluster and numbering those from 1 to n. Then it merges two clusters, chosen according to method, to create a new cluster numbered n+1, and so on until all observations are grouped into a single cluster numbered 2(n-1). Row k of the (m-1)x3 output matrix relates to cluster n+k: the first two columns are the numbers of the two component clusters and column 3 contains their distance.

method defines the way the distance between two clusters is computed and how they are recomputed when two clusters are merged:

‘"single" (default)’

Distance between two clusters is the minimum distance between two elements belonging each to one cluster. Produces a cluster tree known as minimum spanning tree.

‘"complete"’

Furthest distance between two elements belonging each to one cluster.

‘"average"’

Unweighted pair group method with averaging (UPGMA). The mean distance between all pair of elements each belonging to one cluster.

‘"weighted"’

Weighted pair group method with averaging (WPGMA). When two clusters A and B are joined together, the new distance to a cluster C is the mean between distances A-C and B-C.

‘"centroid"’

Unweighted Pair-Group Method using Centroids (UPGMC). Assumes Euclidean metric. The distance between cluster centroids, each centroid being the center of mass of a cluster.

‘"median"’

Weighted pair-group method using centroids (WPGMC). Assumes Euclidean metric. Distance between cluster centroids. When two clusters are joined together, the new centroid is the midpoint between the joined centroids.

‘"ward"’

Ward’s sum of squared deviations about the group mean (ESS). Also known as minimum variance or inner squared distance. Assumes Euclidean metric. How much the moment of inertia of the merged cluster exceeds the sum of those of the individual clusters.

Reference Ward, J. H. Hierarchical Grouping to Optimize an Objective Function J. Am. Statist. Assoc. 1963, 58, 236-244, http://iv.slis.indiana.edu/sw/data/ward.pdf.

See also: pdist, squareform

Source Code: linkage

statistics-release-1.6.3/docs/logicdf.html000066400000000000000000000146711456127120000205650ustar00rootroot00000000000000 Statistics: logicdf

Function Reference: logicdf

statistics: p = logicdf (x, mu, s)
statistics: p = logicdf (x, mu, s, "upper")

Logistic cumulative distribution function (CDF).

For each element of x, compute the cumulative distribution function (CDF) of the logistic distribution with location parameter mu and scale parameter s. The size of p is the common size of x, mu, and s. A scalar input functions as a constant matrix of the same size as the other inputs.

Both parameters must be reals and s > 0. For s <= 0, NaN is returned.

p = logicdf (x, mu, s, "upper") computes the upper tail probability of the logistic distribution with parameters mu and s, at the values in x.

Further information about the logistic distribution can be found at https://en.wikipedia.org/wiki/Logistic_distribution

See also: logiinv, logipdf, logirnd, logifit, logilike, logistat

Source Code: logicdf

Example: 1

  

 ## Plot various CDFs from the logistic distribution
 x = -5:0.01:20;
 p1 = logicdf (x, 5, 2);
 p2 = logicdf (x, 9, 3);
 p3 = logicdf (x, 9, 4);
 p4 = logicdf (x, 6, 2);
 p5 = logicdf (x, 2, 1);
 plot (x, p1, "-b", x, p2, "-g", x, p3, "-r", x, p4, "-c", x, p5, "-m")
 grid on
 legend ({"μ = 5, s = 2", "μ = 9, s = 3", "μ = 9, s = 4", ...
          "μ = 6, s = 2", "μ = 2, s = 1"}, "location", "southeast")
 title ("Logistic CDF")
 xlabel ("values in x")
 ylabel ("probability")
plotted figure

statistics-release-1.6.3/docs/logifit.html000066400000000000000000000225761456127120000206160ustar00rootroot00000000000000 Statistics: logifit

Function Reference: logifit

statistics: paramhat = logifit (x)
statistics: [paramhat, paramci] = logifit (x)
statistics: [paramhat, paramci] = logifit (x, alpha)
statistics: […] = logifit (x, alpha, censor)
statistics: […] = logifit (x, alpha, censor, freq)
statistics: […] = logifit (x, alpha, censor, freq, options)

Estimate mean and confidence intervals for the logistic distribution.

mu0 = logifit (x) returns the maximum likelihood estimates of the parameters of the logistic distribution given the data in x. paramhat(1) is the scale parameter, mu, and paramhat(2) is the shape parameter, s.

[paramhat, paramci] = logifit (x) returns the 95% confidence intervals for the parameter estimates.

[…] = logifit (x, alpha) also returns the 100 * (1 - alpha) percent confidence intervals for the parameter estimates. By default, the optional argument alpha is 0.05 corresponding to 95% confidence intervals. Pass in [] for alpha to use the default values.

[…] = logifit (x, alpha, censor) accepts a boolean vector, censor, of the same size as x with 1s for observations that are right-censored and 0s for observations that are observed exactly. By default, or if left empty, censor = zeros (size (x)).

[…] = logifit (x, alpha, censor, freq) accepts a frequency vector, freq, of the same size as x. freq typically contains integer frequencies for the corresponding elements in x, but it can contain any non-integer non-negative values. By default, or if left empty, freq = ones (size (x)).

[…] = logifit (…, options) specifies control parameters for the iterative algorithm used to compute ML estimates with the fminsearch function. options is a structure with the following fields and their default values:

  • options.Display = "off"
  • options.MaxFunEvals = 400
  • options.MaxIter = 200
  • options.TolX = 1e-6

Further information about the logistic distribution can be found at https://en.wikipedia.org/wiki/Logistic_distribution

See also: logicdf, logiinv, logipdf, logirnd, logilike

Source Code: logifit

Example: 1

  

 ## Sample 3 populations from different logistic distibutions
 rand ("seed", 5)  # for reproducibility
 r1 = logirnd (2, 1, 2000, 1);
 rand ("seed", 2)   # for reproducibility
 r2 = logirnd (5, 2, 2000, 1);
 rand ("seed", 7)   # for reproducibility
 r3 = logirnd (9, 4, 2000, 1);
 r = [r1, r2, r3];

 ## Plot them normalized and fix their colors
 hist (r, [-6:20], 1);
 h = findobj (gca, "Type", "patch");
 set (h(1), "facecolor", "c");
 set (h(2), "facecolor", "g");
 set (h(3), "facecolor", "r");
 ylim ([0, 0.3]);
 xlim ([-5, 20]);
 hold on

 ## Estimate their MU and LAMBDA parameters
 mu_sA = logifit (r(:,1));
 mu_sB = logifit (r(:,2));
 mu_sC = logifit (r(:,3));

 ## Plot their estimated PDFs
 x = [-5:0.5:20];
 y = logipdf (x, mu_sA(1), mu_sA(2));
 plot (x, y, "-pr");
 y = logipdf (x, mu_sB(1), mu_sB(2));
 plot (x, y, "-sg");
 y = logipdf (x, mu_sC(1), mu_sC(2));
 plot (x, y, "-^c");
 hold off
 legend ({"Normalized HIST of sample 1 with μ=1 and s=0.5", ...
          "Normalized HIST of sample 2 with μ=2 and s=0.3", ...
          "Normalized HIST of sample 3 with μ=4 and s=0.5", ...
          sprintf("PDF for sample 1 with estimated μ=%0.2f and s=%0.2f", ...
                  mu_sA(1), mu_sA(2)), ...
          sprintf("PDF for sample 2 with estimated μ=%0.2f and s=%0.2f", ...
                  mu_sB(1), mu_sB(2)), ...
          sprintf("PDF for sample 3 with estimated μ=%0.2f and s=%0.2f", ...
                  mu_sC(1), mu_sC(2))})
 title ("Three population samples from different logistic distibutions")
 hold off
plotted figure

statistics-release-1.6.3/docs/logiinv.html000066400000000000000000000140701456127120000206160ustar00rootroot00000000000000 Statistics: logiinv

Function Reference: logiinv

statistics: x = logiinv (p, mu, s)

Inverse of the logistic cumulative distribution function (iCDF).

For each element of p, compute the quantile (the inverse of the CDF) of the logistic distribution with location parameter mu and scale parameter s. The size of p is the common size of x, mu, and s. A scalar input functions as a constant matrix of the same size as the other inputs.

Both parameters must be reals and s > 0. For s <= 0, NaN is returned.

Further information about the logistic distribution can be found at https://en.wikipedia.org/wiki/Logistic_distribution

See also: logicdf, logipdf, logirnd, logifit, logilike, logistat

Source Code: logiinv

Example: 1

  

 ## Plot various iCDFs from the logistic distribution
 p = 0.001:0.001:0.999;
 x1 = logiinv (p, 5, 2);
 x2 = logiinv (p, 9, 3);
 x3 = logiinv (p, 9, 4);
 x4 = logiinv (p, 6, 2);
 x5 = logiinv (p, 2, 1);
 plot (p, x1, "-b", p, x2, "-g", p, x3, "-r", p, x4, "-c", p, x5, "-m")
 grid on
 legend ({"μ = 5, s = 2", "μ = 9, s = 3", "μ = 9, s = 4", ...
          "μ = 6, s = 2", "μ = 2, s = 1"}, "location", "southeast")
 title ("Logistic iCDF")
 xlabel ("probability")
 ylabel ("x")
plotted figure

statistics-release-1.6.3/docs/logilike.html000066400000000000000000000136351456127120000207540ustar00rootroot00000000000000 Statistics: logilike

Function Reference: logilike

statistics: nlogL = logilike (params, x)
statistics: [nlogL, acov] = logilike (params, x)
statistics: […] = logilike (params, x, censor)
statistics: […] = logilike (params, x, censor, freq)

Negative log-likelihood for the logistic distribution.

nlogL = logilike (params, x) returns the negative log likelihood of the data in x corresponding to the logistic distribution with (1) location parameter mu and (2) scale parameter s given in the two-element vector params.

[nlogL, acov] = logilike (params, x) also returns the inverse of Fisher’s information matrix, acov. If the input parameter values in params are the maximum likelihood estimates, the diagonal elements of params are their asymptotic variances.

[…] = logilike (params, x, censor) accepts a boolean vector, censor, of the same size as x with 1s for observations that are right-censored and 0s for observations that are observed exactly. By default, or if left empty, censor = zeros (size (x)).

[…] = logilike (params, x, censor, freq) accepts a frequency vector, freq, of the same size as x. freq typically contains integer frequencies for the corresponding elements in x, but it can contain any non-integer non-negative values. By default, or if left empty, freq = ones (size (x)).

Further information about the logistic distribution can be found at https://en.wikipedia.org/wiki/Logistic_distribution

See also: logicdf, logiinv, logipdf, logirnd, logifit

Source Code: logilike

statistics-release-1.6.3/docs/logipdf.html000066400000000000000000000140631456127120000205750ustar00rootroot00000000000000 Statistics: logipdf

Function Reference: logipdf

statistics: y = logipdf (x, mu, s)

Logistic probability density function (PDF).

For each element of x, compute the probability density function (PDF) of the logistic distribution with location parameter mu and scale parameter s. The size of p is the common size of x, mu, and s. A scalar input functions as a constant matrix of the same size as the other inputs.

Both parameters must be reals and s > 0. For s <= 0, NaN is returned.

Further information about the logistic distribution can be found at https://en.wikipedia.org/wiki/Logistic_distribution

See also: logicdf, logiinv, logirnd, logifit, logilike, logistat

Source Code: logipdf

Example: 1

  

 ## Plot various PDFs from the logistic distribution
 x = -5:0.01:20;
 y1 = logipdf (x, 5, 2);
 y2 = logipdf (x, 9, 3);
 y3 = logipdf (x, 9, 4);
 y4 = logipdf (x, 6, 2);
 y5 = logipdf (x, 2, 1);
 plot (x, y1, "-b", x, y2, "-g", x, y3, "-r", x, y4, "-c", x, y5, "-m")
 grid on
 ylim ([0, 0.3])
 legend ({"μ = 5, s = 2", "μ = 9, s = 3", "μ = 9, s = 4", ...
          "μ = 6, s = 2", "μ = 2, s = 1"}, "location", "northeast")
 title ("Logistic PDF")
 xlabel ("values in x")
 ylabel ("density")
plotted figure

statistics-release-1.6.3/docs/logirnd.html000066400000000000000000000124371456127120000206120ustar00rootroot00000000000000 Statistics: logirnd

Function Reference: logirnd

statistics: r = logirnd (mu, s)
statistics: r = logirnd (mu, s, rows)
statistics: r = logirnd (mu, s, rows, cols, …)
statistics: r = logirnd (mu, s, [sz])

Random arrays from the logistic distribution.

r = logirnd (mu, s) returns an array of random numbers chosen from the logistic distribution with location parameter mu and scale parameter s. The size of r is the common size of mu and s. A scalar input functions as a constant matrix of the same size as the other inputs.

Both parameters must be reals and s > 0. For s <= 0, NaN is returned.

When called with a single size argument, logirnd returns a square matrix with the dimension specified. When called with more than one scalar argument, the first two arguments are taken as the number of rows and columns and any further arguments specify additional matrix dimensions. The size may also be specified with a row vector of dimensions, sz.

Further information about the logistic distribution can be found at https://en.wikipedia.org/wiki/Logistic_distribution

See also: logcdf, logiinv, logipdf, logifit, logilike, logistat

Source Code: logirnd

statistics-release-1.6.3/docs/logistic_regression.html000066400000000000000000000152561456127120000232330ustar00rootroot00000000000000 Statistics: logistic_regression

Function Reference: logistic_regression

statistics: [intercept, slope, dev, dl, d2l, P, stats] = logistic_regression (y, x, print, intercept, slope)

Perform ordinal logistic regression.

Suppose y takes values in k ordered categories, and let P_i (x) be the cumulative probability that y falls in one of the first i categories given the covariate x. Then

 
 [intercept, slope] = logistic_regression (y, x)

fits the model

 
 logit (P_i (x)) = x * slope + intercept_i,   i = 1 … k-1

The number of ordinal categories, k, is taken to be the number of distinct values of round (y). If k equals 2, y is binary and the model is ordinary logistic regression. The matrix x is assumed to have full column rank.

Given y only, intercept = logistic_regression (y) fits the model with baseline logit odds only.

The full form is

 
 
 [intercept, slope, dev, dl, d2l, P, stats]
    = logistic_regression (y, x, print, intercept, slope)

in which all output arguments and all input arguments except y are optional.

Setting print to 1 requests summary information about the fitted model to be displayed. Setting print to 2 requests information about convergence at each iteration. Other values request no information to be displayed. The input arguments intercept and slope give initial estimates for intercept and slope.

The returned value dev holds minus twice the log-likelihood.

The returned values dl and d2l are the vector of first and the matrix of second derivatives of the log-likelihood with respect to intercept and slope.

P holds estimates for the conditional distribution of y given x.

stats returns a structure that contains the following fields:

  • "intercept": intercept coefficients
  • "slope": slope coefficients
  • "coeff": regression coefficients (intercepts and slops)
  • "covb": estimated covariance matrix for coefficients (coeff)
  • "coeffcorr": correlation matrix for coeff
  • "se": standard errors of the coeff
  • "z": z statistics for coeff
  • "pval": p-values for coeff

Source Code: logistic_regression

statistics-release-1.6.3/docs/logit.html000066400000000000000000000074031456127120000202670ustar00rootroot00000000000000 Statistics: logit

Function Reference: logit

statistics: x = logit (p)

Compute the logit for each value of p

The logit is defined as $$ {\rm logit}(p) = \log\Big({p \over 1-p}\Big) $$ See also: probit, logicdf

Source Code: logit

statistics-release-1.6.3/docs/loglcdf.html000066400000000000000000000156351456127120000205710ustar00rootroot00000000000000 Statistics: loglcdf

Function Reference: loglcdf

statistics: p = loglcdf (x, a, b)
statistics: p = loglcdf (x, a, b, "upper")

Log-logistic cumulative distribution function (CDF).

For each element of x, compute the cumulative distribution function (CDF) of the log-logistic distribution with scale parameter a and shape parameter b. The size of p is the common size of x, a, and b. A scalar input functions as a constant matrix of the same size as the other inputs.

Both parameters, a and b, must be positive reals and x is supported in the range [0,inf), otherwise NaN is returned.

p = loglcdf (x, a, b, "upper") computes the upper tail probability of the log-logistic distribution with parameters a and b, at the values in x.

Further information about the log-logistic distribution can be found at https://en.wikipedia.org/wiki/Log-logistic_distribution

MATLAB compatibility: MATLAB uses an alternative parameterization given by the pair μ, s, i.e. mu and s, in analogy with the logistic distribution. Their relation to the a and b parameters is given below:

  • a = exp (mu)
  • b = 1 / s

See also: loglinv, loglpdf, loglrnd, loglfit, logllike

Source Code: loglcdf

Example: 1

  

 ## Plot various CDFs from the log-logistic distribution
 x = 0:0.001:2;
 p1 = loglcdf (x, 1, 0.5);
 p2 = loglcdf (x, 1, 1);
 p3 = loglcdf (x, 1, 2);
 p4 = loglcdf (x, 1, 4);
 p5 = loglcdf (x, 1, 8);
 plot (x, p1, "-b", x, p2, "-g", x, p3, "-r", x, p4, "-c", x, p5, "-m")
 legend ({"β = 0.5", "β = 1", "β = 2", "β = 4", "β = 8"}, ...
         "location", "northwest")
 grid on
 title ("Log-logistic CDF")
 xlabel ("values in x")
 ylabel ("probability")
 text (0.05, 0.64, "α = 1, values of β as shown in legend")
plotted figure

statistics-release-1.6.3/docs/loglfit.html000066400000000000000000000234401456127120000206100ustar00rootroot00000000000000 Statistics: loglfit

Function Reference: loglfit

statistics: paramhat = loglfit (x)
statistics: [paramhat, paramci] = loglfit (x)
statistics: [paramhat, paramci] = loglfit (x, alpha)
statistics: […] = loglfit (x, alpha, censor)
statistics: […] = loglfit (x, alpha, censor, freq)
statistics: […] = loglfit (x, alpha, censor, freq, options)

Estimate mean and confidence intervals for the log-logistic distribution.

mu0 = loglfit (x) returns the maximum likelihood estimates of the parameters of the log-logistic distribution given the data in x. paramhat(1) is the scale parameter, a, and paramhat(2) is the shape parameter, b.

[paramhat, paramci] = loglfit (x) returns the 95% confidence intervals for the parameter estimates.

[…] = loglfit (x, alpha) also returns the 100 * (1 - alpha) percent confidence intervals for the parameter estimates. By default, the optional argument alpha is 0.05 corresponding to 95% confidence intervals. Pass in [] for alpha to use the default values.

[…] = loglfit (x, alpha, censor) accepts a boolean vector, censor, of the same size as x with 1s for observations that are right-censored and 0s for observations that are observed exactly. By default, or if left empty, censor = zeros (size (x)).

[…] = loglfit (x, alpha, censor, freq) accepts a frequency vector, freq, of the same size as x. freq typically contains integer frequencies for the corresponding elements in x, but it can contain any non-integer non-negative values. By default, or if left empty, freq = ones (size (x)).

[…] = loglfit (…, options) specifies control parameters for the iterative algorithm used to compute ML estimates with the fminsearch function. options is a structure with the following fields and their default values:

  • options.Display = "off"
  • options.MaxFunEvals = 400
  • options.MaxIter = 200
  • options.TolX = 1e-6

Further information about the log-logistic distribution can be found at https://en.wikipedia.org/wiki/Log-logistic_distribution

MATLAB compatibility: MATLAB uses an alternative parameterization given by the pair μ, s, i.e. mu and s, in analogy with the logistic distribution. Their relation to the a and b parameters is given below:

  • a = exp (mu)
  • b = 1 / s

See also: loglcdf, loglinv, loglpdf, loglrnd, logllike

Source Code: loglfit

Example: 1

  

 ## Sample 3 populations from different log-logistic distibutions
 rand ("seed", 5)  # for reproducibility
 r1 = loglrnd (1, 1, 2000, 1);
 rand ("seed", 2)   # for reproducibility
 r2 = loglrnd (1, 2, 2000, 1);
 rand ("seed", 7)   # for reproducibility
 r3 = loglrnd (1, 8, 2000, 1);
 r = [r1, r2, r3];

 ## Plot them normalized and fix their colors
 hist (r, [0.05:0.1:2.5], 10);
 h = findobj (gca, "Type", "patch");
 set (h(1), "facecolor", "c");
 set (h(2), "facecolor", "g");
 set (h(3), "facecolor", "r");
 ylim ([0, 3.5]);
 xlim ([0, 2.0]);
 hold on

 ## Estimate their MU and LAMBDA parameters
 a_bA = loglfit (r(:,1));
 a_bB = loglfit (r(:,2));
 a_bC = loglfit (r(:,3));

 ## Plot their estimated PDFs
 x = [0.01:0.1:2.01];
 y = loglpdf (x, a_bA(1), a_bA(2));
 plot (x, y, "-pr");
 y = loglpdf (x, a_bB(1), a_bB(2));
 plot (x, y, "-sg");
 y = loglpdf (x, a_bC(1), a_bC(2));
 plot (x, y, "-^c");
 legend ({"Normalized HIST of sample 1 with α=1 and β=1", ...
          "Normalized HIST of sample 2 with α=1 and β=2", ...
          "Normalized HIST of sample 3 with α=1 and β=8", ...
          sprintf("PDF for sample 1 with estimated α=%0.2f and β=%0.2f", ...
                  a_bA(1), a_bA(2)), ...
          sprintf("PDF for sample 2 with estimated α=%0.2f and β=%0.2f", ...
                  a_bB(1), a_bB(2)), ...
          sprintf("PDF for sample 3 with estimated α=%0.2f and β=%0.2f", ...
                  a_bC(1), a_bC(2))})
 title ("Three population samples from different log-logistic distibutions")
 hold off
plotted figure

statistics-release-1.6.3/docs/loglinv.html000066400000000000000000000150521456127120000206220ustar00rootroot00000000000000 Statistics: loglinv

Function Reference: loglinv

statistics: x = loglinv (p, a, b)

Inverse of the log-logistic cumulative distribution function (iCDF).

For each element of p, compute the quantile (the inverse of the CDF) of the log-logistic distribution with scale parameter a and shape parameter b. The size of x is the common size of p, a, and b. A scalar input functions as a constant matrix of the same size as the other inputs.

Both parameters, a and b, must be positive reals and p is supported in the range [0,1], otherwise NaN is returned.

Further information about the log-logistic distribution can be found at https://en.wikipedia.org/wiki/Log-logistic_distribution

MATLAB compatibility: MATLAB uses an alternative parameterization given by the pair μ, s, i.e. mu and s, in analogy with the logistic distribution. Their relation to the a and b parameters is given below:

  • a = exp (mu)
  • b = 1 / s

See also: loglcdf, loglpdf, loglrnd, loglfit, logllike

Source Code: loglinv

Example: 1

  

 ## Plot various iCDFs from the log-logistic distribution
 p = 0.001:0.001:0.999;
 x1 = loglinv (p, 1, 0.5);
 x2 = loglinv (p, 1, 1);
 x3 = loglinv (p, 1, 2);
 x4 = loglinv (p, 1, 4);
 x5 = loglinv (p, 1, 8);
 plot (p, x1, "-b", p, x2, "-g", p, x3, "-r", p, x4, "-c", p, x5, "-m")
 ylim ([0, 20])
 grid on
 legend ({"β = 0.5", "β = 1", "β = 2", "β = 4", "β = 8"}, ...
         "location", "northwest")
 title ("Log-logistic iCDF")
 xlabel ("probability")
 ylabel ("x")
 text (0.03, 12.5, "α = 1, values of β as shown in legend")
plotted figure

statistics-release-1.6.3/docs/logllike.html000066400000000000000000000145011456127120000207500ustar00rootroot00000000000000 Statistics: logllike

Function Reference: logllike

statistics: nlogL = logllike (params, x)
statistics: [nlogL, acov] = logllike (params, x)
statistics: […] = logllike (params, x, censor)
statistics: […] = logllike (params, x, censor, freq)

Negative log-likelihood for the log-logistic distribution.

nlogL = logllike (params, x) returns the negative log likelihood of the data in x corresponding to the log-logistic distribution with (1) scale parameter a and (2) shape parameter b given in the two-element vector params.

[nlogL, acov] = logllike (params, x) also returns the inverse of Fisher’s information matrix, acov. If the input parameter values in params are the maximum likelihood estimates, the diagonal elements of params are their asymptotic variances.

[…] = logllike (params, x, censor) accepts a boolean vector, censor, of the same size as x with 1s for observations that are right-censored and 0s for observations that are observed exactly. By default, or if left empty, censor = zeros (size (x)).

[…] = logllike (params, x, censor, freq) accepts a frequency vector, freq, of the same size as x. freq typically contains integer frequencies for the corresponding elements in x, but it can contain any non-integer non-negative values. By default, or if left empty, freq = ones (size (x)).

Further information about the log-logistic distribution can be found at https://en.wikipedia.org/wiki/Log-logistic_distribution

MATLAB compatibility: MATLAB uses an alternative parameterization given by the pair μ, s, i.e. mu and s, in analogy with the logistic distribution. Their relation to the a and b parameters is given below:

  • a = exp (mu)
  • b = 1 / s

See also: loglcdf, loglinv, loglpdf, loglrnd, loglfit

Source Code: logllike

statistics-release-1.6.3/docs/loglpdf.html000066400000000000000000000150741456127120000206030ustar00rootroot00000000000000 Statistics: loglpdf

Function Reference: loglpdf

statistics: y = loglpdf (x, a, b)

Log-logistic probability density function (PDF).

For each element of x, compute the probability density function (PDF) of the log-logistic distribution with with scale parameter a and shape parameter b. The size of y is the common size of x, a, and b. A scalar input functions as a constant matrix of the same size as the other inputs.

Both parameters, a and b, must be positive reals, otherwise NaN is returned. x is supported in the range [0,Inf), otherwise 0 is returned.

Further information about the log-logistic distribution can be found at https://en.wikipedia.org/wiki/Log-logistic_distribution

MATLAB compatibility: MATLAB uses an alternative parameterization given by the pair μ, s, i.e. mu and s, in analogy with the logistic distribution. Their relation to the a and b parameters is given below:

  • a = exp (mu)
  • b = 1 / s

See also: loglcdf, loglinv, loglrnd, loglfit, logllike

Source Code: loglpdf

Example: 1

  

 ## Plot various PDFs from the log-logistic distribution
 x = 0:0.001:2;
 y1 = loglpdf (x, 1, 0.5);
 y2 = loglpdf (x, 1, 1);
 y3 = loglpdf (x, 1, 2);
 y4 = loglpdf (x, 1, 4);
 y5 = loglpdf (x, 1, 8);
 plot (x, y1, "-b", x, y2, "-g", x, y3, "-r", x, y4, "-c", x, y5, "-m")
 grid on
 ylim ([0,3])
 legend ({"β = 0.5", "β = 1", "β = 2", "β = 4", "β = 8"}, ...
         "location", "northeast")
 title ("Log-logistic PDF")
 xlabel ("values in x")
 ylabel ("density")
 text (0.5, 2.8, "α = 1, values of β as shown in legend")
plotted figure

statistics-release-1.6.3/docs/loglrnd.html000066400000000000000000000132261456127120000206120ustar00rootroot00000000000000 Statistics: loglrnd

Function Reference: loglrnd

statistics: r = loglrnd (a, b)
statistics: r = loglrnd (a, b, rows)
statistics: r = loglrnd (a, b, rows, cols, …)
statistics: r = loglrnd (a, b, [sz])

Random arrays from the log-logistic distribution.

r = loglrnd (a, b) returns an array of random numbers chosen from the log-logistic distribution with scale parameter a and shape parameter b. The size of r is the common size of a and b. A scalar input functions as a constant matrix of the same size as the other inputs.

Both parameters must be positive reals, otherwise NaN is returned.

When called with a single size argument, loglrnd returns a square matrix with the dimension specified. When called with more than one scalar argument, the first two arguments are taken as the number of rows and columns and any further arguments specify additional matrix dimensions. The size may also be specified with a row vector of dimensions, sz.

Further information about the log-logistic distribution can be found at https://en.wikipedia.org/wiki/Log-logistic_distribution

MATLAB compatibility: MATLAB uses an alternative parameterization given by the pair μ, s, i.e. mu and s, in analogy with the logistic distribution. Their relation to the a and b parameters is given below:

  • a = exp (mu)
  • b = 1 / s

See also: loglcdf, loglinv, loglpdf, loglfit, logllike

Source Code: loglrnd

statistics-release-1.6.3/docs/logncdf.html000066400000000000000000000177501456127120000205730ustar00rootroot00000000000000 Statistics: logncdf

Function Reference: logncdf

statistics: p = logncdf (x)
statistics: p = logncdf (x, mu)
statistics: p = logncdf (x, mu, sigma)
statistics: p = logncdf (…, "upper")
statistics: [p, plo, pup] = logncdf (x, mu, sigma, pcov)
statistics: [p, plo, pup] = logncdf (x, mu, sigma, pcov, alpha)
statistics: [p, plo, pup] = logncdf (…, "upper")

Log-normal cumulative distribution function (CDF).

For each element of x, compute the cumulative distribution function (CDF) of the log-normal distribution with mean parameter mu and standard deviation parameter sigma, each corresponding to the associated normal distribution. The size of p is the common size of x, mu, and sigma. A scalar input functions as a constant matrix of the same size as the other inputs.

If a random variable follows this distribution, its logarithm is normally distributed with mean mu and standard deviation sigma.

Default parameter values are mu = 0 and sigma = 1. Both parameters must be reals and sigma > 0. For sigma <= 0, NaN is returned.

When called with three output arguments, i.e. [p, plo, pup], logncdf computes the confidence bounds for p when the input parameters mu and sigma are estimates. In such case, pcov, a 2×2 matrix containing the covariance matrix of the estimated parameters, is necessary. Optionally, alpha, which has a default value of 0.05, specifies the 100 * (1 - alpha) percent confidence bounds. plo and pup are arrays of the same size as p containing the lower and upper confidence bounds.

[…] = logncdf (…, "upper") computes the upper tail probability of the log-normal distribution with parameters mu and sigma, at the values in x.

Further information about the log-normal distribution can be found at https://en.wikipedia.org/wiki/Log-normal_distribution

See also: logninv, lognpdf, lognrnd, lognfit, lognlike, lognstat

Source Code: logncdf

Example: 1

  

 ## Plot various CDFs from the log-normal distribution
 x = 0:0.01:3;
 p1 = logncdf (x, 0, 1);
 p2 = logncdf (x, 0, 0.5);
 p3 = logncdf (x, 0, 0.25);
 plot (x, p1, "-b", x, p2, "-g", x, p3, "-r")
 grid on
 legend ({"μ = 0, σ = 1", "μ = 0, σ = 0.5", "μ = 0, σ = 0.25"}, ...
         "location", "southeast")
 title ("Log-normal CDF")
 xlabel ("values in x")
 ylabel ("probability")
plotted figure

statistics-release-1.6.3/docs/lognfit.html000066400000000000000000000236741456127120000206230ustar00rootroot00000000000000 Statistics: lognfit

Function Reference: lognfit

statistics: paramhat = lognfit (x)
statistics: [paramhat, paramci] = lognfit (x)
statistics: [paramhat, paramci] = lognfit (x, alpha)
statistics: […] = lognfit (x, alpha, censor)
statistics: […] = lognfit (x, alpha, censor, freq)
statistics: […] = lognfit (x, alpha, censor, freq, options)

Estimate parameters and confidence intervals for the log-normal distribution.

paramhat = lognfit (x) returns the maximum likelihood estimates of the parameters of the log-normal distribution given the data in vector x. paramhat([1, 2]) corresponds to the mean and standard deviation, respectively, of the associated normal distribution.

If a random variable follows this distribution, its logarithm is normally distributed with mean mu and standard deviation sigma.

[paramhat, paramci] = lognfit (x) returns the 95% confidence intervals for the parameter estimates.

[…] = lognfit (x, alpha) also returns the 100 * (1 - alpha) percent confidence intervals for the parameter estimates. By default, the optional argument alpha is 0.05 corresponding to 95% confidence intervals. Pass in [] for alpha to use the default values.

[…] = lognfit (x, alpha, censor) accepts a boolean vector, censor, of the same size as x with 1s for observations that are right-censored and 0s for observations that are observed exactly. By default, or if left empty, censor = zeros (size (x)).

[…] = lognfit (x, alpha, censor, freq) accepts a frequency vector, freq, of the same size as x. freq typically contains integer frequencies for the corresponding elements in x, but it can contain any non-integer non-negative values. By default, or if left empty, freq = ones (size (x)).

[…] = lognfit (…, options) specifies control parameters for the iterative algorithm used to compute ML estimates with the fminsearch function. options is a structure with the following fields and their default values:

  • options.Display = "off"
  • options.MaxFunEvals = 400
  • options.MaxIter = 200
  • options.TolX = 1e-6

With no censor, the estimate of the standard deviation, paramhat(2), is the square root of the unbiased estimate of the variance of log (x). With censored data, the maximum likelihood estimate is returned.

Further information about the log-normal distribution can be found at https://en.wikipedia.org/wiki/Log-normal_distribution

See also: logncdf, logninv, lognpdf, lognrnd, lognlike, lognstat

Source Code: lognfit

Example: 1

  

 ## Sample 3 populations from 3 different log-normal distibutions
 randn ("seed", 1);    # for reproducibility
 r1 = lognrnd (0, 0.25, 1000, 1);
 randn ("seed", 2);    # for reproducibility
 r2 = lognrnd (0, 0.5, 1000, 1);
 randn ("seed", 3);    # for reproducibility
 r3 = lognrnd (0, 1, 1000, 1);
 r = [r1, r2, r3];

 ## Plot them normalized and fix their colors
 hist (r, 30, 2);
 h = findobj (gca, "Type", "patch");
 set (h(1), "facecolor", "c");
 set (h(2), "facecolor", "g");
 set (h(3), "facecolor", "r");
 hold on

 ## Estimate their mu and sigma parameters
 mu_sigmaA = lognfit (r(:,1));
 mu_sigmaB = lognfit (r(:,2));
 mu_sigmaC = lognfit (r(:,3));

 ## Plot their estimated PDFs
 x = [0:0.1:6];
 y = lognpdf (x, mu_sigmaA(1), mu_sigmaA(2));
 plot (x, y, "-pr");
 y = lognpdf (x, mu_sigmaB(1), mu_sigmaB(2));
 plot (x, y, "-sg");
 y = lognpdf (x, mu_sigmaC(1), mu_sigmaC(2));
 plot (x, y, "-^c");
 ylim ([0, 2])
 xlim ([0, 6])
 hold off
 legend ({"Normalized HIST of sample 1 with mu=0, σ=0.25", ...
          "Normalized HIST of sample 2 with mu=0, σ=0.5", ...
          "Normalized HIST of sample 3 with mu=0, σ=1", ...
          sprintf("PDF for sample 1 with estimated mu=%0.2f and σ=%0.2f", ...
                  mu_sigmaA(1), mu_sigmaA(2)), ...
          sprintf("PDF for sample 2 with estimated mu=%0.2f and σ=%0.2f", ...
                  mu_sigmaB(1), mu_sigmaB(2)), ...
          sprintf("PDF for sample 3 with estimated mu=%0.2f and σ=%0.2f", ...
                  mu_sigmaC(1), mu_sigmaC(2))}, "location", "northeast")
 title ("Three population samples from different log-normal distibutions")
 hold off
plotted figure

statistics-release-1.6.3/docs/logninv.html000066400000000000000000000150521456127120000206240ustar00rootroot00000000000000 Statistics: logninv

Function Reference: logninv

statistics: x = logninv (p)
statistics: x = logninv (p, mu)
statistics: x = logninv (p, mu, sigma)

Inverse of the log-normal cumulative distribution function (iCDF).

For each element of p, compute the quantile (the inverse of the CDF) of the log-normal distribution with mean parameter mu and standard deviation parameter sigma, each corresponding to the associated normal distribution. The size of x is the common size of p, mu, and sigma. A scalar input functions as a constant matrix of the same size as the other inputs.

If a random variable follows this distribution, its logarithm is normally distributed with mean mu and standard deviation sigma.

Default parameter values are mu = 0 and sigma = 1. Both parameters must be reals and sigma > 0. For sigma <= 0, NaN is returned.

Further information about the log-normal distribution can be found at https://en.wikipedia.org/wiki/Log-normal_distribution

See also: logncdf, lognpdf, lognrnd, lognfit, lognlike, lognstat

Source Code: logninv

Example: 1

  

 ## Plot various iCDFs from the log-normal distribution
 p = 0.001:0.001:0.999;
 x1 = logninv (p, 0, 1);
 x2 = logninv (p, 0, 0.5);
 x3 = logninv (p, 0, 0.25);
 plot (p, x1, "-b", p, x2, "-g", p, x3, "-r")
 grid on
 ylim ([0, 3])
 legend ({"μ = 0, σ = 1", "μ = 0, σ = 0.5", "μ = 0, σ = 0.25"}, ...
         "location", "northwest")
 title ("Log-normal iCDF")
 xlabel ("probability")
 ylabel ("values in x")
plotted figure

statistics-release-1.6.3/docs/lognlike.html000066400000000000000000000146411456127120000207570ustar00rootroot00000000000000 Statistics: lognlike

Function Reference: lognlike

statistics: nlogL = lognlike (params, x)
statistics: [nlogL, avar] = lognlike (params, x)
statistics: […] = lognlike (params, x, censor)
statistics: […] = lognlike (params, x, censor, freq)

Negative log-likelihood for the log-normal distribution.

nlogL = lognlike (params, x) returns the negative log-likelihood of the data in x corresponding to the log-normal distribution with (1) location parameter mu and (2) scale parameter sigma given in the two-element vector params, which correspond to the mean and standard deviation of the associated normal distribution. Missing values, NaNs, are ignored. Negative values of x are treated as missing values.

If a random variable follows this distribution, its logarithm is normally distributed with mean mu and standard deviation sigma.

[nlogL, avar] = lognlike (params, x) returns the inverse of Fisher’s information matrix, avar. If the input parameter values in params are the maximum likelihood estimates, the diagonal elements of avar are their asymptotic variances. avar is based on the observed Fisher’s information, not the expected information.

[…] = lognlike (params, x, censor) accepts a boolean vector, censor, of the same size as x with 1s for observations that are right-censored and 0s for observations that are observed exactly. By default, or if left empty, censor = zeros (size (x)).

[…] = lognlike (params, x, censor, freq) accepts a frequency vector, freq, of the same size as x. freq typically contains integer frequencies for the corresponding elements in x, but it can contain any non-integer non-negative values. By default, or if left empty, freq = ones (size (x)).

Further information about the log-normal distribution can be found at https://en.wikipedia.org/wiki/Log-normal_distribution

See also: logncdf, logninv, lognpdf, lognrnd, lognfit, lognstat

Source Code: lognlike

statistics-release-1.6.3/docs/lognpdf.html000066400000000000000000000150071456127120000206010ustar00rootroot00000000000000 Statistics: lognpdf

Function Reference: lognpdf

statistics: y = lognpdf (x)
statistics: y = lognpdf (x, mu)
statistics: y = lognpdf (x, mu, sigma)

Lognormal probability density function (PDF).

For each element of x, compute the probability density function (PDF) of the lognormal distribution with mean parameter mu and standard deviation parameter sigma, each corresponding to the associated normal distribution. The size of y is the common size of p, mu, and sigma. A scalar input functions as a constant matrix of the same size as the other inputs.

If a random variable follows this distribution, its logarithm is normally distributed with mean mu and standard deviation sigma.

Default parameter values are mu = 0 and sigma = 1. Both parameters must be reals and sigma > 0. For sigma <= 0, NaN is returned.

Further information about the log-normal distribution can be found at https://en.wikipedia.org/wiki/Log-normal_distribution

See also: logncdf, logninv, lognrnd, lognfit, lognlike, lognstat

Source Code: lognpdf

Example: 1

  

 ## Plot various PDFs from the log-normal distribution
 x = 0:0.01:5;
 y1 = lognpdf (x, 0, 1);
 y2 = lognpdf (x, 0, 0.5);
 y3 = lognpdf (x, 0, 0.25);
 plot (x, y1, "-b", x, y2, "-g", x, y3, "-r")
 grid on
 ylim ([0, 2])
 legend ({"μ = 0, σ = 1", "μ = 0, σ = 0.5", "μ = 0, σ = 0.25"}, ...
          "location", "northeast")
 title ("Log-normal PDF")
 xlabel ("values in x")
 ylabel ("density")
plotted figure

statistics-release-1.6.3/docs/lognrnd.html000066400000000000000000000131411456127120000206100ustar00rootroot00000000000000 Statistics: lognrnd

Function Reference: lognrnd

statistics: r = lognrnd (mu, sigma)
statistics: r = lognrnd (mu, sigma, rows)
statistics: r = lognrnd (mu, sigma, rows, cols, …)
statistics: r = lognrnd (mu, sigma, [sz])

Random arrays from the lognormal distribution.

r = lognrnd (mu, sigma) returns an array of random numbers chosen from the lognormal distribution with mean parameter mu and standard deviation parameter sigma, each corresponding to the associated normal distribution. The size of r is the common size of mu, and sigma. A scalar input functions as a constant matrix of the same size as the other inputs. Both parameters must be reals and sigma > 0. For sigma <= 0, NaN is returned.

Both parameters must be reals and sigma > 0. For sigma <= 0, NaN is returned.

When called with a single size argument, lognrnd returns a square matrix with the dimension specified. When called with more than one scalar argument, the first two arguments are taken as the number of rows and columns and any further arguments specify additional matrix dimensions. The size may also be specified with a row vector of dimensions, sz.

Further information about the log-normal distribution can be found at https://en.wikipedia.org/wiki/Log-normal_distribution

See also: logncdf, logninv, lognpdf, lognfit, lognlike, lognstat

Source Code: lognrnd

statistics-release-1.6.3/docs/lognstat.html000066400000000000000000000111551456127120000210030ustar00rootroot00000000000000 Statistics: lognstat

Function Reference: lognstat

statistics: [m, v] = lognstat (mu, sigma)

Compute statistics of the log-normal distribution.

[m, v] = lognstat (mu, sigma) returns the mean and variance of the log-normal distribution with mean parameter mu and standard deviation parameter sigma, each corresponding to the associated normal distribution.

The size of m (mean) and v (variance) is the common size of the input arguments. A scalar input functions as a constant matrix of the same size as the other inputs.

Further information about the log-normal distribution can be found at https://en.wikipedia.org/wiki/Log-normal_distribution

See also: logncdf, logninv, lognpdf, lognrnd, lognfit, lognlike

Source Code: lognstat

statistics-release-1.6.3/docs/mad.html000066400000000000000000000142211456127120000177060ustar00rootroot00000000000000 Statistics: mad

Function Reference: mad

statistics: m = mad (x)
statistics: m = mad (x, flag)
statistics: m = mad (x, flag, "all")
statistics: m = mad (x, flag, dim)
statistics: m = mad (x, flag, vecdim)

Compute the mean or median absolute deviation (MAD).

mad (x) returns the mean absolute deviation of the values in x. mad treats NaNs as missing values and removes them.

  • If x is a vector, then mad returns the mean or median absolute deviation of the values in X.
  • If x is a matrix, then mad returns the mean or median absolute deviation of each column of X.
  • If x is an multidimensional array, then mad (x) operates along the first non-singleton dimension of x.

mad (x, flag) specifies whether to compute the mean absolute deviation (flag = 0, the default) or the median absolute deviation (flag = 1). Passing an empty variable, defaults to 0.

mad (x, flag, "all") returns the MAD of all the elements in x.

The optional variable dim forces mad to operate over the specified dimension, which must be a positive integer-valued number. Specifying any singleton dimension in x, including any dimension exceeding ndims (x), will result in a MAD equal to zeros (size (x)), while non-finite elements are returned as NaNs.

mad (x, flag, vecdim) returns the MAD over the dimensions specified in the vector vecdim. For example, if x is a 2-by-3-by-4 array, then mad (x, [1 2]) returns a 1-by-1-by-4 array. Each element of the output array is the median of the elements on the corresponding page of x. If vecdim indexes all dimensions of x, then it is equivalent to mad (x, "all"). Any dimension in vecdim greater than ndims (x) is ignored.

See also: median, mean, mode

Source Code: mad

statistics-release-1.6.3/docs/mahal.html000066400000000000000000000102441456127120000202300ustar00rootroot00000000000000 Statistics: mahal

Function Reference: mahal

statistics: d = mahal (y, x)

Mahalanobis’ D-square distance.

Return the Mahalanobis’ D-square distance of the points in y from the distribution implied by points x.

Specifically, it uses a Cholesky decomposition to set

 
  answer(i) = (y(i,:) - mean (x)) * inv (A) * (y(i,:)-mean (x))'

where A is the covariance of x.

The data x and y must have the same number of components (columns), but may have a different number of observations (rows).

Source Code: mahal

statistics-release-1.6.3/docs/manova1.html000066400000000000000000000212331456127120000205100ustar00rootroot00000000000000 Statistics: manova1

Function Reference: manova1

statistics: d = manova1 (x, group)
statistics: d = manova1 (x, group, alpha)
statistics: [d, p] = manova1 (…)
statistics: [d, p, stats] = manova1 (…)

One-way multivariate analysis of variance (MANOVA).

d = manova1 (x, group, alpha) performs a one-way MANOVA for comparing the mean vectors of two or more groups of multivariate data.

x is a matrix with each row representing a multivariate observation, and each column representing a variable.

group is a numeric vector, string array, or cell array of strings with the same number of rows as x. x values are in the same group if they correspond to the same value of GROUP.

alpha is the scalar significance level and is 0.05 by default.

d is an estimate of the dimension of the group means. It is the smallest dimension such that a test of the hypothesis that the means lie on a space of that dimension is not rejected. If d = 0 for example, we cannot reject the hypothesis that the means are the same. If d = 1, we reject the hypothesis that the means are the same but we cannot reject the hypothesis that they lie on a line.

[d, p] = manova1 (…) returns P, a vector of p-values for testing the null hypothesis that the mean vectors of the groups lie on various dimensions. P(1) is the p-value for a test of dimension 0, P(2) for dimension 1, etc.

[d, p, stats] = manova1 (…) returns a STATS structure with the following fields:

"W"within-group sum of squares and products matrix
"B"between-group sum of squares and products matrix
"T"total sum of squares and products matrix
"dfW"degrees of freedom for WSSP matrix
"dfB"degrees of freedom for BSSP matrix
"dfT"degrees of freedom for TSSP matrix
"lambda"value of Wilk’s lambda (the test statistic)
"chisq"transformation of lambda to a chi-square distribution
"chisqdf"degrees of freedom for chisq
"eigenval"eigenvalues of (WSSP^-1) * BSSP
"eigenvec"eigenvectors of (WSSP^-1) * BSSP; these are the coefficients for canonical variables, and they are scaled so the within-group variance of C is 1
"canon"canonical variables, equal to XC*eigenvec, where XC is X with columns centered by subtracting their means
"mdist"Mahalanobis distance from each point to its group mean
"gmdist"Mahalanobis distances between each pair of group means
"gnames"Group names

The canonical variables C have the property that C(:,1) is the linear combination of the x columns that has the maximum separation between groups, C(:,2) has the maximum separation subject to it being orthogonal to C(:,1), and so on.

Source Code: manova1

Example: 1

  

 load carbig
 [d,p] = manova1([MPG, Acceleration, Weight, Displacement], Origin)

d = 3
p =

        0
   0.0000
   0.0075
   0.1934
statistics-release-1.6.3/docs/manovacluster.html000066400000000000000000000143301456127120000220310ustar00rootroot00000000000000 Statistics: manovacluster

Function Reference: manovacluster

statistics: manovacluster (stats)
statistics: manovacluster (stats, method)
statistics: h = manovacluster (stats)
statistics: h = manovacluster (stats, method)

Cluster group means using manova1 output.

manovacluster (stats) draws a dendrogram showing the clustering of group means, calculated using the output STATS structure from manova1 and applying the single linkage algorithm. See the dendrogram function for more information about the figure.

manovacluster (stats, method) uses the method algorithm in place of single linkage. The available methods are:

"single"— nearest distance
"complete"— furthest distance
"average"— average distance
"centroid"— center of mass distance
"ward"— inner squared distance

h = manovacluster (…) returns a vector of line handles.

See also: manova1

Source Code: manovacluster

Example: 1

  

 load carbig
 X = [MPG Acceleration Weight Displacement];
 [d, p, stats] = manova1 (X, Origin);
 manovacluster (stats)
plotted figure

statistics-release-1.6.3/docs/mcnemar_test.html000066400000000000000000000137561456127120000216420ustar00rootroot00000000000000 Statistics: mcnemar_test

Function Reference: mcnemar_test

statistics: [h, pval, chisq] = mcnemar_test (x)
statistics: [h, pval, chisq] = mcnemar_test (x, alpha)
statistics: [h, pval, chisq] = mcnemar_test (x, testtype)
statistics: [h, pval, chisq] = mcnemar_test (x, alpha, testtype)

Perform a McNemar’s test on paired nominal data.

@nospell{McNemar’s} test is applied to a 2×2 contingency table x with a dichotomous trait, with matched pairs of subjects, of data cross-classified on the row and column variables to testing the null hypothesis of symmetry of the classification probabilities. More formally, the null hypothesis of marginal homogeneity states that the two marginal probabilities for each outcome are the same.

Under the null, with a sufficiently large number of discordants (x(1,2) + x(2,1) >= 25), the test statistic, chisq, follows a chi-squared distribution with 1 degree of freedom. When the number of discordants is less than 25, then the mid-P exact McNemar test is used.

testtype will force mcnemar_test to apply a particular method for testing the null hypothesis independently of the number of discordants. Valid options for testtype:

  • "asymptotic" Original McNemar test statistic
  • "corrected" Edwards’ version with continuity correction
  • "exact" An exact binomial test
  • "mid-p" The mid-P McNemar test (mid-p binomial test)

The test decision is returned in h, which is 1 when the null hypothesis is rejected (pval < alpha) or 0 otherwise. alpha defines the critical value of statistical significance for the test.

Further information about the McNemar’s test can be found at https://en.wikipedia.org/wiki/McNemar%27s_test

See also: crosstab, chi2test, fishertest

Source Code: mcnemar_test

statistics-release-1.6.3/docs/mean.html000066400000000000000000000155521456127120000200750ustar00rootroot00000000000000 Statistics: mean

Function Reference: mean

statistics: m = mean (x)
statistics: m = mean (x, "all")
statistics: m = mean (x, dim)
statistics: m = mean (x, vecdim)
statistics: m = mean (…, outtype)
statistics: m = mean (…, nanflag)

Compute the mean of the elements of x.

  • If x is a vector, then mean(x) returns the mean of the elements in x defined as $$ {\rm mean}(x) = \bar{x} = {1\over N} \sum_{i=1}^N x_i $$ where N is the length of the x vector.
  • If x is a matrix, then mean(x) returns a row vector with the mean of each columns in x.
  • If x is a multidimensional array, then mean(x) operates along the first nonsingleton dimension of x.

mean (x, dim) returns the mean along the operating dimension dim of x. For dim greater than ndims (x), then m = x.

mean (x, vecdim) returns the mean over the dimensions specified in the vector vecdim. For example, if x is a 2-by-3-by-4 array, then mean (x, [1 2]) returns a 1-by-1-by-4 array. Each element of the output array is the mean of the elements on the corresponding page of x. If vecdim indexes all dimensions of x, then it is equivalent to mean (x, "all"). Any dimension in vecdim greater than ndims (x) is ignored.

mean (x, "all") returns the mean of all the elements in x. The optional flag "all" cannot be used together with dim or vecdim input arguments.

mean (…, outtype) returns the mean with a specified data type, using any of the input arguments in the previous syntaxes. outtype can take the following values:

  • "default" Output is of type double, unless the input is single in which case the output is of type single.
  • "double" Output is of type double.
  • "native". Output is of the same type as the input (class (x)), unless the input is logical in which case the output is of type double or a character array in which case an error is produced.

mean (…, nanflag) specifies whether to exclude NaN values from the calculation, using any of the input argument combinations in previous syntaxes. By default, NaN values are included in the calculation (nanflag has the value "includenan"). To exclude NaN values, set the value of nanflag to "omitnan".

See also: trimmean, median, mad, mode

Source Code: mean

statistics-release-1.6.3/docs/median.html000066400000000000000000000157141456127120000204120ustar00rootroot00000000000000 Statistics: median

Function Reference: median

statistics: m = median (x)
statistics: m = median (x, "all")
statistics: m = median (x, dim)
statistics: m = median (x, vecdim)
statistics: m = median (…, outtype)
statistics: m = median (…, nanflag)

Compute the median value of the elements of x.

When the elements of x are sorted, say s = sort (x), the median is defined as $$ {\rm median} (x) = \cases{s(\lceil N/2\rceil), & $N$ odd; \cr (s(N/2)+s(N/2+1))/2, & $N$ even.} $$ where N is the number of elements of x.

If x is an array, then median (x) operates along the first non-singleton dimension of x.

The optional variable dim forces median to operate over the specified dimension, which must be a positive integer-valued number. Specifying any singleton dimension in x, including any dimension exceeding ndims (x), will result in a median equal to x.

median (x, vecdim) returns the median over the dimensions specified in the vector vecdim. For example, if x is a 2-by-3-by-4 array, then median (x, [1 2]) returns a 1-by-1-by-4 array. Each element of the output array is the median of the elements on the corresponding page of x. If vecdim indexes all dimensions of x, then it is equivalent to median (x, "all"). Any dimension in vecdim greater than ndims (x) is ignored.

median (x, "all") returns the median of all the elements in x. The optional flag "all" cannot be used together with dim or vecdim input arguments.

median (…, outtype) returns the median with a specified data type, using any of the input arguments in the previous syntaxes. outtype can take the following values:

  • "default" Output is of type double, unless the input is single in which case the output is of type single.
  • "double" Output is of type double.
  • "native" Output is of the same type as the input (class (x)), unless the input is logical in which case the output is of type double.

The optional variable nanflag specifies whether to include or exclude NaN values from the calculation using any of the previously specified input argument combinations. The default value for nanflag is "includenan" which keeps NaN values in the calculation. To exclude NaN values set the value of nanflag to "omitnan". The output will still contain NaN values if x consists of all NaN values in the operating dimension.

See also: mean, mad, mode

Source Code: median

statistics-release-1.6.3/docs/mhsample.html000066400000000000000000000271621456127120000207630ustar00rootroot00000000000000 Statistics: mhsample

Function Reference: mhsample

statistics: [smpl, accept] = mhsample (start, nsamples, property, value, …)

Draws nsamples samples from a target stationary distribution pdf using Metropolis-Hastings algorithm.

Inputs:

  • start is a nchain by dim matrix of starting points for each Markov chain. Each row is the starting point of a different chain and each column corresponds to a different dimension.
  • nsamples is the number of samples, the length of each Markov chain.

Some property-value pairs can or must be specified, they are:

(Required) One of:

  • "pdf" pdf: a function handle of the target stationary distribution to be sampled. The function should accept different locations in each row and each column corresponds to a different dimension.

    or

  • "logpdf" logpdf: a function handle of the log of the target stationary distribution to be sampled. The function should accept different locations in each row and each column corresponds to a different dimension.

In case optional argument symmetric is set to false (the default), one of:

  • "proppdf" proppdf: a function handle of the proposal distribution that is sampled from with proprnd to give the next point in the chain. The function should accept two inputs, the random variable and the current location each input should accept different locations in each row and each column corresponds to a different dimension.

    or

  • "logproppdf" logproppdf: the log of "proppdf".

The following input property/pair values may be needed depending on the desired outut:

  • "proprnd" proprnd: (Required) a function handle which generates random numbers from proppdf. The function should accept different locations in each row and each column corresponds to a different dimension corresponding with the current location.
  • "symmetric" symmetric: true or false based on whether proppdf is a symmetric distribution. If true, proppdf (or logproppdf) need not be specified. The default is false.
  • "burnin" burnin the number of points to discard at the beginning, the default is 0.
  • "thin" thin: omits thin-1 of every thin points in the generated Markov chain. The default is 1.
  • "nchain" nchain: the number of Markov chains to generate. The default is 1.

Outputs:

  • smpl: a nsamples x dim x nchain tensor of random values drawn from pdf, where the rows are different random values, the columns correspond to the dimensions of pdf, and the third dimension corresponds to different Markov chains.
  • accept is a vector of the acceptance rate for each chain.

Example : Sampling from a normal distribution

 
 
 start = 1;
 nsamples = 1e3;
 pdf = @(x) exp (-.5 * x .^ 2) / (pi ^ .5 * 2 ^ .5);
 proppdf = @(x,y) 1 / 6;
 proprnd = @(x) 6 * (rand (size (x)) - .5) + x;
 [smpl, accept] = mhsample (start, nsamples, "pdf", pdf, "proppdf", ...
 proppdf, "proprnd", proprnd, "thin", 4);
 histfit (smpl);

See also: rand, slicesample

Source Code: mhsample

Example: 1

  

 ## Define function to sample
 d = 2;
 mu = [-1; 2];
 rand ("seed", 5)  # for reproducibility
 Sigma = rand (d);
 Sigma = (Sigma + Sigma');
 Sigma += eye (d) * abs (eigs (Sigma, 1, "sa")) * 1.1;
 pdf = @(x)(2*pi)^(-d/2)*det(Sigma)^-.5*exp(-.5*sum((x.'-mu).*(Sigma\(x.'-mu)),1));
 ## Inputs
 start = ones (1, 2);
 nsamples = 500;
 sym = true;
 K = 500;
 m = 10;
 rand ("seed", 8)  # for reproducibility
 proprnd = @(x) (rand (size (x)) - .5) * 3 + x;
 [smpl, accept] = mhsample (start, nsamples, "pdf", pdf, "proprnd", proprnd, ...
                            "symmetric", sym, "burnin", K, "thin", m);
 figure;
 hold on;
 plot (smpl(:, 1), smpl(:, 2), 'x');
 [x, y] = meshgrid (linspace (-6, 4), linspace(-3, 7));
 z = reshape (pdf ([x(:), y(:)]), size(x));
 mesh (x, y, z, "facecolor", "None");
 ## Using sample points to find the volume of half a sphere with radius of .5
 f = @(x) ((.25-(x(:,1)+1).^2-(x(:,2)-2).^2).^.5.*(((x(:,1)+1).^2+(x(:,2)-2).^2)<.25)).';
 int = mean (f (smpl) ./ pdf (smpl));
 errest = std (f (smpl) ./ pdf (smpl)) / nsamples ^ .5;
 trueerr = abs (2 / 3 * pi * .25 ^ (3 / 2) - int);
 printf ("Monte Carlo integral estimate int f(x) dx = %f\n", int);
 printf ("Monte Carlo integral error estimate %f\n", errest);
 printf ("The actual error %f\n", trueerr);
 mesh (x, y, reshape (f([x(:), y(:)]), size(x)), "facecolor", "None");

Monte Carlo integral estimate int f(x) dx = 0.242643
Monte Carlo integral error estimate 0.028039
The actual error 0.019156
plotted figure

Example: 2

  

 ## Integrate truncated normal distribution to find normilization constant
 pdf = @(x) exp (-.5*x.^2)/(pi^.5*2^.5);
 nsamples = 1e3;
 rand ("seed", 5)  # for reproducibility
 proprnd = @(x) (rand (size (x)) - .5) * 3 + x;
 [smpl, accept] = mhsample (1, nsamples, "pdf", pdf, "proprnd", proprnd, ...
                            "symmetric", true, "thin", 4);
 f = @(x) exp(-.5 * x .^ 2) .* (x >= -2 & x <= 2);
 x = linspace (-3, 3, 1000);
 area(x, f(x));
 xlabel ('x');
 ylabel ('f(x)');
 int = mean (f (smpl) ./ pdf (smpl));
 errest = std (f (smpl) ./ pdf (smpl)) / nsamples^ .5;
 trueerr = abs (erf (2 ^ .5) * 2 ^ .5 * pi ^ .5 - int);
 printf ("Monte Carlo integral estimate int f(x) dx = %f\n", int);
 printf ("Monte Carlo integral error estimate %f\n", errest);
 printf ("The actual error %f\n", trueerr);

Monte Carlo integral estimate int f(x) dx = 2.346204
Monte Carlo integral error estimate 0.019410
The actual error 0.046372
plotted figure

statistics-release-1.6.3/docs/mnpdf.html000066400000000000000000000131331456127120000202520ustar00rootroot00000000000000 Statistics: mnpdf

Function Reference: mnpdf

statistics: y = mnpdf (x, pk)

Multinomial probability density function (PDF).

Arguments

  • x is vector with a single sample of a multinomial distribution with parameter pk or a matrix of random samples from multinomial distributions. In the latter case, each row of x is a sample from a multinomial distribution with the corresponding row of pk being its parameter.
  • pk is a vector with the probabilities of the categories or a matrix with each row containing the probabilities of a multinomial sample.

Return values

  • y is a vector of probabilites of the random samples x from the multinomial distribution with corresponding parameter pk. The parameter n of the multinomial distribution is the sum of the elements of each row of x. The length of y is the number of columns of x. If a row of pk does not sum to 1, then the corresponding element of y will be NaN.

Examples

 
 
 x = [1, 4, 2];
 pk = [0.2, 0.5, 0.3];
 y = mnpdf (x, pk);
 
 
 x = [1, 4, 2; 1, 0, 9];
 pk = [0.2, 0.5, 0.3; 0.1, 0.1, 0.8];
 y = mnpdf (x, pk);

References

  1. Wendy L. Martinez and Angel R. Martinez. Computational Statistics Handbook with MATLAB. Appendix E, pages 547-557, Chapman & Hall/CRC, 2001.
  2. Merran Evans, Nicholas Hastings and Brian Peacock. Statistical Distributions. pages 134-136, Wiley, New York, third edition, 2000.

See also: mnrnd

Source Code: mnpdf

statistics-release-1.6.3/docs/mnrfit.html000066400000000000000000000153761456127120000204600ustar00rootroot00000000000000 Statistics: mnrfit

Function Reference: mnrfit

statistics: B = mnrfit (X, Y)
statistics: B = mnrfit (X, Y, name, value)
statistics: [B, dev] = mnrrfit (…)
statistics: [B, dev, stats] = mnrfit (…)

Perform logistic regression for binomial responses or multiple ordinal responses.

Note: This function is currently a wrapper for the logistic_regression function. It can only be used for fitting an ordinal logistic model and a nominal model with 2 categories (which is an ordinal case). Hierarchical models as well as nominal model with more than two classes are not currently supported. This function is a work in progress.

B = mnrfit (X, Y) returns a matrix, B, of coefficient estimates for a multinomial logistic regression of the nominal responses in Y on the predictors in X. X is an N×P numeric matrix the observations on predictor variables, where N corresponds to the number of observations and P corresponds to predictor variables. Y contains the response category labels and it either be an N×P categorical or numerical matrix (containing only 1s and 0s) or an N×1 numeric vector with positive integer values, a cell array of character vectors and a logical vector. Y can also be defined as a character matrix with each row corresponding to an observation of X.

B = mnrfit (X, Y, name, value) returns a matrix, B, of coefficient estimates for a multinomial model fit with additional parameterss specified Name-Value pair arguments.

NameValue
"model"Specifies the type of model to fit. Currently, only "ordinal" is fully supported. "nominal" is only supported for 2 classes in Y.
"display"A flag to enable/disable displaying information about the fitted model. Default is "off".

[B, dev, stats] = mnrfit (… also returns the deviance of the fit, dev, and the structure stats for any of the previous input arguments. stats currently only returns values for the fields "beta", same as B, "coeffcorr", the estimated correlation matrix for B, "covd", the estimated covariance matrix for B, and "se", the standard errors of the coefficient estimates B.

See also: logistic_regression

Source Code: mnrfit

statistics-release-1.6.3/docs/mnrnd.html000066400000000000000000000144431456127120000202710ustar00rootroot00000000000000 Statistics: mnrnd

Function Reference: mnrnd

statistics: r = mnrnd (n, pk)
statistics: r = mnrnd (n, pk, s)

Random arrays from the multinomial distribution.

Arguments

  • n is the first parameter of the multinomial distribution. n can be scalar or a vector containing the number of trials of each multinomial sample. The elements of n must be non-negative integers.
  • pk is the second parameter of the multinomial distribution. pk can be a vector with the probabilities of the categories or a matrix with each row containing the probabilities of a multinomial sample. If pk has more than one row and n is non-scalar, then the number of rows of pk must match the number of elements of n.
  • s is the number of multinomial samples to be generated. s must be a non-negative integer. If s is specified, then n must be scalar and pk must be a vector.

Return values

  • r is a matrix of random samples from the multinomial distribution with corresponding parameters n and pk. Each row corresponds to one multinomial sample. The number of columns, therefore, corresponds to the number of columns of pk. If s is not specified, then the number of rows of r is the maximum of the number of elements of n and the number of rows of pk. If a row of pk does not sum to 1, then the corresponding row of r will contain only NaN values.

Examples

 
 
 n = 10;
 pk = [0.2, 0.5, 0.3];
 r = mnrnd (n, pk);
 
 
 n = 10 * ones (3, 1);
 pk = [0.2, 0.5, 0.3];
 r = mnrnd (n, pk);
 
 
 n = (1:2)';
 pk = [0.2, 0.5, 0.3; 0.1, 0.1, 0.8];
 r = mnrnd (n, pk);

References

  1. Wendy L. Martinez and Angel R. Martinez. Computational Statistics Handbook with MATLAB. Appendix E, pages 547-557, Chapman & Hall/CRC, 2001.
  2. Merran Evans, Nicholas Hastings and Brian Peacock. Statistical Distributions. pages 134-136, Wiley, New York, third edition, 2000.

See also: mnpdf

Source Code: mnrnd

statistics-release-1.6.3/docs/monotone_smooth.html000066400000000000000000000132211456127120000223730ustar00rootroot00000000000000 Statistics: monotone_smooth

Function Reference: monotone_smooth

statistics: yy = monotone_smooth (x, y, h)

Produce a smooth monotone increasing approximation to a sampled functional dependence.

A kernel method is used (an Epanechnikov smoothing kernel is applied to y(x); this is integrated to yield the monotone increasing form. See Reference 1 for details.)

Arguments

  • x is a vector of values of the independent variable.
  • y is a vector of values of the dependent variable, of the same size as x. For best performance, it is recommended that the y already be fairly smooth, e.g. by applying a kernel smoothing to the original values if they are noisy.
  • h is the kernel bandwidth to use. If h is not given, a "reasonable" value is computed.

Return values

  • yy is the vector of smooth monotone increasing function values at x.

Examples

 
 
 x = 0:0.1:10;
 y = (x .^ 2) + 3 * randn(size(x)); # typically non-monotonic from the added
 noise
 ys = ([y(1) y(1:(end-1))] + y + [y(2:end) y(end)])/3; # crudely smoothed via
 moving average, but still typically non-monotonic
 yy = monotone_smooth(x, ys); # yy is monotone increasing in x
 plot(x, y, '+', x, ys, x, yy)

References

  1. Holger Dette, Natalie Neumeyer and Kay F. Pilz (2006), A simple nonparametric estimator of a strictly monotone regression function, Bernoulli, 12:469-490
  2. Regine Scheder (2007), R Package ’monoProc’, Version 1.0-6, http://cran.r-project.org/web/packages/monoProc/monoProc.pdf (The implementation here is based on the monoProc function mono.1d)

Source Code: monotone_smooth

statistics-release-1.6.3/docs/multcompare.html000066400000000000000000000657041456127120000215110ustar00rootroot00000000000000 Statistics: multcompare

Function Reference: multcompare

statistics: C = multcompare (STATS)
statistics: C = multcompare (STATS, "name", value)
statistics: [C, M] = multcompare (...)
statistics: [C, M, H] = multcompare (...)
statistics: [C, M, H, GNAMES] = multcompare (...)
statistics: padj = multcompare (p)
statistics: padj = multcompare (p, "ctype", CTYPE)

Perform posthoc multiple comparison tests or p-value adjustments to control the family-wise error rate (FWER) or false discovery rate (FDR).

C = multcompare (STATS) performs a multiple comparison using a STATS structure that is obtained as output from any of the following functions: anova1, anova2, anovan, kruskalwallis, and friedman. The return value C is a matrix with one row per comparison and six columns. Columns 1-2 are the indices of the two samples being compared. Columns 3-5 are a lower bound, estimate, and upper bound for their difference, where the bounds are for 95% confidence intervals. Column 6-8 are the multiplicity adjusted p-values for each individual comparison, the test statistic and the degrees of freedom. All tests by multcompare are two-tailed.

multcompare can take a number of optional parameters as name-value pairs.

[…] = multcompare (STATS, "alpha", ALPHA)

  • ALPHA sets the significance level of null hypothesis significance tests to ALPHA, and the central coverage of two-sided confidence intervals to 100*(1-ALPHA)%. (Default ALPHA is 0.05).

[…] = multcompare (STATS, "ControlGroup", REF)

  • REF is the index of the control group to limit comparisons to. The index must be a positive integer scalar value. For each dimension (d) listed in DIM, multcompare uses STATS.grpnames{d}(idx) as the control group. (Default is empty, i.e. [], for full pairwise comparisons)

[…] = multcompare (STATS, "ctype", CTYPE)

  • CTYPE is the type of comparison test to use. In order of increasing power, the choices are: "bonferroni", "scheffe", "mvt", "holm" (default), "hochberg", "fdr", or "lsd". The first five methods control the family-wise error rate. The "fdr" method controls false discovery rate (by the original Benjamini-Hochberg step-up procedure). The final method, "lsd" (or "none"), makes no attempt to control the Type 1 error rate of multiple comparisons. The coverage of confidence intervals are only corrected for multiple comparisons in the cases where CTYPE is "bonferroni", "scheffe" or "mvt", which control the Type 1 error rate for simultaneous inference.

    The "mvt" method uses the multivariate t distribution to assess the probability or critical value of the maximum statistic across the tests, thereby accounting for correlations among comparisons in the control of the family-wise error rate with simultaneous inference. In the case of pairwise comparisons, it simulates Tukey’s (or the Games-Howell) test, in the case of comparisons with a single control group, it simulates Dunnett’s test. CTYPE values "tukey-kramer" and "hsd" are recognised but set the value of CTYPE and REF to "mvt" and empty respectively. A CTYPE value "dunnett" is recognised but sets the value of CTYPE to "mvt", and if REF is empty, sets REF to 1. Since the algorithm uses a Monte Carlo method (of 1e+06 random samples), you can expect the results to fluctuate slightly with each call to multcompare and the calculations may be slow to complete for a large number of comparisons. If the parallel package is installed and loaded, multcompare will automatically accelerate computations by parallel processing. Note that p-values calculated by the "mvt" are truncated at 1e-06.

[…] = multcompare (STATS, "df", DF)

  • DF is an optional scalar value to set the number of degrees of freedom in the calculation of p-values for the multiple comparison tests. By default, this value is extracted from the STATS structure of the ANOVA test, but setting DF maybe necessary to approximate Satterthwaite correction if anovan was performed using weights.

[…] = multcompare (STATS, "dim", DIM)

  • DIM is a vector specifying the dimension or dimensions over which the estimated marginal means are to be calculated. Used only if STATS comes from anovan. The value [1 3], for example, computes the estimated marginal mean for each combination of the first and third predictor values. The default is to compute over the first dimension (i.e. 1). If the specified dimension is, or includes, a continuous factor then multcompare will return an error.

[…] = multcompare (STATS, "estimate", ESTIMATE)

  • ESTIMATE is a string specifying the estimates to be compared when computing multiple comparisons after anova2; this argument is ignored by anovan and anova1. Accepted values for ESTIMATE are either "column" (default) to compare column means, or "row" to compare row means. If the model type in anova2 was "linear" or "nested" then only "column" is accepted for ESTIMATE since the row factor is assumed to be a random effect.

[…] = multcompare (STATS, "display", DISPLAY)

  • DISPLAY is either "on" (the default): to display a table and graph of the comparisons (e.g. difference between means), their 100*(1-ALPHA)% intervals and multiplicity adjusted p-values in APA style; or "off": to omit the table and graph. On the graph, markers and error bars colored red have multiplicity adjusted p-values < ALPHA, otherwise the markers and error bars are blue.

[…] = multcompare (STATS, "seed", SEED)

  • SEED is a scalar value used to initialize the random number generator so that CTYPE "mvt" produces reproducible results.

[C, M, H, GNAMES] = multcompare (…) returns additional outputs. M is a matrix where columns 1-2 are the estimated marginal means and their standard errors, and columns 3-4 are lower and upper bounds of the confidence intervals for the means; the critical value of the test statistic is scaled by a factor of 2^(-0.5) before multiplying by the standard errors of the group means so that the intervals overlap when the difference in means becomes significant at approximately the level ALPHA. When ALPHA is 0.05, this corresponds to confidence intervals with 83.4% central coverage. H is a handle to the figure containing the graph. GNAMES is a cell array with one row for each group, containing the names of the groups.

padj = multcompare (p) calculates and returns adjusted p-values (padj) using the Holm-step down Bonferroni procedure to control the family-wise error rate.

padj = multcompare (p, "ctype", CTYPE) calculates and returns adjusted p-values (padj) computed using the method CTYPE. In order of increasing power, CTYPE for p-value adjustment can be either "bonferroni", "holm" (default), "hochberg", or "fdr". See above for further information about the CTYPE methods.

See also: anova1, anova2, anovan, kruskalwallis, friedman, fitlm

Source Code: multcompare

Example: 1

  


 ## Demonstration using balanced one-way ANOVA from anova1

 x = ones (50, 4) .* [-2, 0, 1, 5];
 randn ("seed", 1);    # for reproducibility
 x = x + normrnd (0, 2, 50, 4);
 groups = {"A", "B", "C", "D"};
 [p, tbl, stats] = anova1 (x, groups, "off");
 multcompare (stats);


        HOLM Multiple Comparison (Post Hoc) Test for ANOVA1

Group ID  Group ID   LBoundDiff   EstimatedDiff   UBoundDiff   p-value
----------------------------------------------------------------------
    1         2         -2.789        -2.009         -1.228      <.001
    1         3         -2.489        -1.709         -0.928      <.001
    1         4         -7.365        -6.585         -5.804      <.001
    2         3         -0.480         0.300          1.081       .449
    2         4         -5.356        -4.576         -3.795      <.001
    3         4         -5.656        -4.876         -4.095      <.001
plotted figure

Example: 2

  


 ## Demonstration using unbalanced one-way ANOVA example from anovan

 dv =  [ 8.706 10.362 11.552  6.941 10.983 10.092  6.421 14.943 15.931 ...
        22.968 18.590 16.567 15.944 21.637 14.492 17.965 18.851 22.891 ...
        22.028 16.884 17.252 18.325 25.435 19.141 21.238 22.196 18.038 ...
        22.628 31.163 26.053 24.419 32.145 28.966 30.207 29.142 33.212 ...
        25.694 ]';
 g = [1 1 1 1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 3 ...
      4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5]';

 [P,ATAB, STATS] = anovan (dv, g, "varnames", "score", "display", "off");

 [C, M, H, GNAMES] = multcompare (STATS, "dim", 1, "ctype", "holm", ...
                                  "ControlGroup", 1, "display", "on")



        HOLM Multiple Comparison (Post Hoc) Test for ANOVAN

Group ID  Group ID   LBoundDiff   EstimatedDiff   UBoundDiff   p-value
----------------------------------------------------------------------
    1         2        -11.343        -8.000         -4.657      <.001
    1         3        -11.932        -9.000         -6.068      <.001
    1         4        -14.035       -11.000         -7.965      <.001
    1         5        -21.849       -19.000        -16.151      <.001

C =

   1.0000e+00   2.0000e+00  -1.1343e+01  -8.0000e+00  -4.6572e+00   2.8581e-05  -4.8748e+00   3.2000e+01
   1.0000e+00   3.0000e+00  -1.1932e+01  -9.0000e+00  -6.0682e+00   1.0459e-06  -6.2529e+00   3.2000e+01
   1.0000e+00   4.0000e+00  -1.4035e+01  -1.1000e+01  -7.9654e+00   6.3838e-08  -7.3834e+00   3.2000e+01
   1.0000e+00   5.0000e+00  -2.1849e+01  -1.9000e+01  -1.6151e+01   3.1974e-14  -1.3583e+01   3.2000e+01

M =

   10.0000    1.0178    8.5341   11.4659
   18.0000    1.2874   16.1458   19.8542
   19.0000    1.0178   17.5341   20.4659
   21.0001    1.0880   19.4330   22.5673
   29.0001    0.9595   27.6180   30.3822

H = 1
GNAMES =
{
  [1,1] = score=1
  [2,1] = score=2
  [3,1] = score=3
  [4,1] = score=4
  [5,1] = score=5
}
plotted figure

Example: 3

  


 ## Demonstration using factorial ANCOVA example from anovan

 score = [95.6 82.2 97.2 96.4 81.4 83.6 89.4 83.8 83.3 85.7 ...
 97.2 78.2 78.9 91.8 86.9 84.1 88.6 89.8 87.3 85.4 ...
 81.8 65.8 68.1 70.0 69.9 75.1 72.3 70.9 71.5 72.5 ...
 84.9 96.1 94.6 82.5 90.7 87.0 86.8 93.3 87.6 92.4 ...
 100. 80.5 92.9 84.0 88.4 91.1 85.7 91.3 92.3 87.9 ...
 91.7 88.6 75.8 75.7 75.3 82.4 80.1 86.0 81.8 82.5]';
 treatment = {"yes" "yes" "yes" "yes" "yes" "yes" "yes" "yes" "yes" "yes" ...
              "yes" "yes" "yes" "yes" "yes" "yes" "yes" "yes" "yes" "yes" ...
              "yes" "yes" "yes" "yes" "yes" "yes" "yes" "yes" "yes" "yes" ...
              "no"  "no"  "no"  "no"  "no"  "no"  "no"  "no"  "no"  "no"  ...
              "no"  "no"  "no"  "no"  "no"  "no"  "no"  "no"  "no"  "no"  ...
              "no"  "no"  "no"  "no"  "no"  "no"  "no"  "no"  "no"  "no"}';
 exercise = {"lo"  "lo"  "lo"  "lo"  "lo"  "lo"  "lo"  "lo"  "lo"  "lo"  ...
             "mid" "mid" "mid" "mid" "mid" "mid" "mid" "mid" "mid" "mid" ...
             "hi"  "hi"  "hi"  "hi"  "hi"  "hi"  "hi"  "hi"  "hi"  "hi"  ...
             "lo"  "lo"  "lo"  "lo"  "lo"  "lo"  "lo"  "lo"  "lo"  "lo"  ...
             "mid" "mid" "mid" "mid" "mid" "mid" "mid" "mid" "mid" "mid" ...
             "hi"  "hi"  "hi"  "hi"  "hi"  "hi"  "hi"  "hi"  "hi"  "hi"}';
 age = [59 65 70 66 61 65 57 61 58 55 62 61 60 59 55 57 60 63 62 57 ...
 58 56 57 59 59 60 55 53 55 58 68 62 61 54 59 63 60 67 60 67 ...
 75 54 57 62 65 60 58 61 65 57 56 58 58 58 52 53 60 62 61 61]';

 [P, ATAB, STATS] = anovan (score, {treatment, exercise, age}, "model", ...
                            [1 0 0; 0 1 0; 0 0 1; 1 1 0], "continuous", 3, ...
                            "sstype", "h", "display", "off", "contrasts", ...
                            {"simple","poly",""});

 [C, M, H, GNAMES] = multcompare (STATS, "dim", [1 2], "ctype", "holm", ...
                                  "display", "on")



        HOLM Multiple Comparison (Post Hoc) Test for ANOVAN

Group ID  Group ID   LBoundDiff   EstimatedDiff   UBoundDiff   p-value
----------------------------------------------------------------------
    1         2         -4.544        -0.017          4.509      1.000
    1         3          8.963        13.703         18.443      <.001
    1         4         -6.002        -1.529          2.945      1.000
    1         5         -6.174        -1.701          2.771      1.000
    1         6         -0.692         3.957          8.605       .662
    2         3          9.166        13.721         18.276      <.001
    2         4         -6.060        -1.511          3.038      1.000
    2         5         -6.195        -1.684          2.828      1.000
    2         6         -0.533         3.974          8.481       .662
    3         4        -20.018       -15.232        -10.446      <.001
    3         5        -20.112       -15.404        -10.697      <.001
    3         6        -14.228        -9.747         -5.265      <.001
    4         5         -4.650        -0.172          4.305      1.000
    4         6          0.798         5.485         10.172       .204
    5         6          1.035         5.658         10.280       .174

C =

   1.0000e+00   2.0000e+00  -4.5437e+00  -1.7457e-02   4.5087e+00   1.0000e+00  -7.7360e-03   5.3000e+01
   1.0000e+00   3.0000e+00   8.9634e+00   1.3703e+01   1.8443e+01   4.5333e-06   5.7988e+00   5.3000e+01
   1.0000e+00   4.0000e+00  -6.0019e+00  -1.5286e+00   2.9447e+00   1.0000e+00  -6.8538e-01   5.3000e+01
   1.0000e+00   5.0000e+00  -6.1735e+00  -1.7011e+00   2.7714e+00   1.0000e+00  -7.6287e-01   5.3000e+01
   1.0000e+00   6.0000e+00  -6.9212e-01   3.9565e+00   8.6051e+00   6.6201e-01   1.7071e+00   5.3000e+01
   2.0000e+00   3.0000e+00   9.1656e+00   1.3721e+01   1.8276e+01   2.0212e-06   6.0416e+00   5.3000e+01
   2.0000e+00   4.0000e+00  -6.0600e+00  -1.5111e+00   3.0378e+00   1.0000e+00  -6.6630e-01   5.3000e+01
   2.0000e+00   5.0000e+00  -6.1953e+00  -1.6836e+00   2.8281e+00   1.0000e+00  -7.4847e-01   5.3000e+01
   2.0000e+00   6.0000e+00  -5.3340e-01   3.9740e+00   8.4813e+00   6.6201e-01   1.7684e+00   5.3000e+01
   3.0000e+00   4.0000e+00  -2.0018e+01  -1.5232e+01  -1.0446e+01   6.1827e-07  -6.3835e+00   5.3000e+01
   3.0000e+00   5.0000e+00  -2.0112e+01  -1.5404e+01  -1.0697e+01   3.4083e-07  -6.5634e+00   5.3000e+01
   3.0000e+00   6.0000e+00  -1.4228e+01  -9.7468e+00  -5.2654e+00   6.5704e-04  -4.3623e+00   5.3000e+01
   4.0000e+00   5.0000e+00  -4.6499e+00  -1.7249e-01   4.3050e+00   1.0000e+00  -7.7268e-02   5.3000e+01
   4.0000e+00   6.0000e+00   7.9808e-01   5.4851e+00   1.0172e+01   2.0411e-01   2.3473e+00   5.3000e+01
   5.0000e+00   6.0000e+00   1.0354e+00   5.6576e+00   1.0280e+01   1.7402e-01   2.4550e+00   5.3000e+01

M =

   86.9788    1.6031   84.7051   89.2525
   86.9962    1.5774   84.7590   89.2334
   73.2755    1.6514   70.9334   75.6176
   88.5074    1.6166   86.2146   90.8002
   88.6799    1.5948   86.4180   90.9417
   83.0223    1.6130   80.7346   85.3100

H = 1
GNAMES =
{
  [1,1] = X1=yes, X2=lo
  [2,1] = X1=yes, X2=mid
  [3,1] = X1=yes, X2=hi
  [4,1] = X1=no, X2=lo
  [5,1] = X1=no, X2=mid
  [6,1] = X1=no, X2=hi
}
plotted figure

Example: 4

  


 ## Demonstration using one-way ANOVA from anovan, with fit by weighted least
 ## squares to account for heteroskedasticity.

 g = [1, 1, 1, 1, 1, 1, 1, 1, ...
      2, 2, 2, 2, 2, 2, 2, 2, ...
      3, 3, 3, 3, 3, 3, 3, 3]';

 y = [13, 16, 16,  7, 11,  5,  1,  9, ...
      10, 25, 66, 43, 47, 56,  6, 39, ...
      11, 39, 26, 35, 25, 14, 24, 17]';

 [P,ATAB,STATS] = anovan(y, g, "display", "off");
 fitted = STATS.X * STATS.coeffs(:,1); # fitted values
 b = polyfit (fitted, abs (STATS.resid), 1);
 v = polyval (b, fitted);  # Variance as a function of the fitted values
 [P,ATAB,STATS] = anovan (y, g, "weights", v.^-1, "display", "off");
 [C, M] =  multcompare (STATS, "display", "on", "ctype", "mvt")


        MVT Multiple Comparison (Post Hoc) Test for ANOVAN

Group ID  Group ID   LBoundDiff   EstimatedDiff   UBoundDiff   p-value
----------------------------------------------------------------------
    1         2        -42.077       -26.750        -11.423      <.001
    1         3        -26.789       -14.125         -1.461       .027
    2         3         -5.216        12.625         30.466       .199

C =

   1.0000e+00   2.0000e+00  -4.2077e+01  -2.6750e+01  -1.1423e+01   7.5000e-04  -4.3716e+00   2.1000e+01
   1.0000e+00   3.0000e+00  -2.6789e+01  -1.4125e+01  -1.4610e+00   2.7188e-02  -2.7937e+00   2.1000e+01
   2.0000e+00   3.0000e+00  -5.2159e+00   1.2625e+01   3.0466e+01   1.9913e-01   1.7725e+00   2.1000e+01

M =

    9.7500    2.4770    5.3629   14.1371
   36.5000    5.5953   26.5900   46.4100
   23.8750    4.4076   16.0685   31.6815
plotted figure

Example: 5

  


 ## Demonstration of p-value adjustments to control the false discovery rate
 ## Data from Westfall (1997) JASA. 92(437):299-306

 p = [.005708; .023544; .024193; .044895; ...
       .048805; .221227; .395867; .693051; .775755];

 padj = multcompare(p,'ctype','fdr')

padj =

   0.051372
   0.072579
   0.072579
   0.087849
   0.087849
   0.331840
   0.508972
   0.775755
   0.775755
statistics-release-1.6.3/docs/mvncdf.html000066400000000000000000000240001456127120000204160ustar00rootroot00000000000000 Statistics: mvncdf

Function Reference: mvncdf

statistics: p = mvncdf (x)
statistics: p = mvncdf (x, mu, sigma)
statistics: p = mvncdf (x_lo, x_up, mu, sigma)
statistics: p = mvncdf (…, options)
statistics: [p, err] = mvncdf (…)

Multivariate normal cumulative distribution function (CDF).

p = mvncdf (x) returns the cumulative probability of the multivariate normal distribution evaluated at each row of x with zero mean and an identity covariance matrix. The rows of matrix x correspond to observations and its columns to variables. The return argument p is a column vector with the same number of rows as in x.

p = mvncdf (x, mu, sigma) returns cumulative probability of the multivariate normal distribution evaluated at each row of x with mean mu and a covariance matrix sigma. mu can be either a scalar (the same of every variable) or a row vector with the same number of elements as the number of variables in x. sigma covariance matrix may be specified a row vector if it only contains variances along its diagonal and zero covariances of the diagonal. In such a case, the diagonal vector sigma must have the same number of elements as the number of variables (columns) in x. If you only want to specify sigma, you can pass an empty matrix for mu.

The multivariate normal cumulative probability at x is defined as the probability that a random vector V, distributed as multivariate normal, will fall within the semi-infinite rectangle with upper limits defined by x.

  • PrV(1)<=X(1), V(2)<=X(2), ... V(D)<=X(D).

p = mvncdf (x_lo, x_hi, mu, sigma) returns the multivariate normal cumulative probability evaluated over the rectangle (hyper-rectangle for multivariate data in x) with lower and upper limits defined by x_lo and x_hi, respectively.

[p, err] = mvncdf (…) also returns an error estimate err in p.

p = mvncdf (…, options) specifies the structure, which controls specific parameters for the numerical integration used to compute p. The required fieds are:

"TolFun"Maximum absolute error tolerance. Default is 1e-8 for D < 4, or 1e-4 for D >= 4. Note that for bivariate normal cdf, the Octave implementation has a presicion of more than 1e-10.
"MaxFunEvals"Maximum number of integrand evaluations. Default is 1e7 for D > 4.
"Display"Display options. Choices are "off" (default), "iter", which shows the probability and estimated error at each repetition, and "final", which shows the final probability and related error after the integrand has converged successfully.

See also: bvncdf, mvnpdf, mvnrnd

Source Code: mvncdf

Example: 1

  

 mu = [1, -1];
 Sigma = [0.9, 0.4; 0.4, 0.3];
 [X1, X2] = meshgrid (linspace (-1, 3, 25)', linspace (-3, 1, 25)');
 X = [X1(:), X2(:)];
 p = mvncdf (X, mu, Sigma);
 Z = reshape (p, 25, 25);
 surf (X1, X2, Z);
 title ("Bivariate Normal Distribution");
 ylabel "X1"
 xlabel "X2"
plotted figure

Example: 2

  

 mu = [0, 0];
 Sigma = [0.25, 0.3; 0.3, 1];
 p = mvncdf ([0 0], [1 1], mu, Sigma);
 x1 = -3:.2:3;
 x2 = -3:.2:3;
 [X1, X2] = meshgrid (x1, x2);
 X = [X1(:), X2(:)];
 p = mvnpdf (X, mu, Sigma);
 p = reshape (p, length (x2), length (x1));
 contour (x1, x2, p, [0.0001, 0.001, 0.01, 0.05, 0.15, 0.25, 0.35]);
 xlabel ("x");
 ylabel ("p");
 title ("Probability over Rectangular Region");
 line ([0, 0, 1, 1, 0], [1, 0, 0, 1, 1], "Linestyle", "--", "Color", "k");
plotted figure

statistics-release-1.6.3/docs/mvnpdf.html000066400000000000000000000155261456127120000204500ustar00rootroot00000000000000 Statistics: mvnpdf

Function Reference: mvnpdf

statistics: y = mvnpdf (x, mu, sigma)

Multivariate normal probability density function (PDF).

y = mvnpdf (x) returns the probability density of the multivariate normal distribution with zero mean and identity covariance matrix, evaluated at each row of x. Rows of the N-by-D matrix x correspond to observations orpoints, and columns correspond to variables or coordinates. y is an N-by-1 vector.

y = mvnpdf (x, mu) returns the density of the multivariate normal distribution with mean MU and identity covariance matrix, evaluated at each row of x. mu is a 1-by-D vector, or an N-by-D matrix, in which case the density is evaluated for each row of x with the corresponding row of mu. mu can also be a scalar value, which MVNPDF replicates to match the size of x.

y = mvnpdf (x, mu, sigma) returns the density of the multivariate normal distribution with mean mu and covariance sigma, evaluated at each row of x. sigma is a D-by-D matrix, or an D-by-D-by-N array, in which case the density is evaluated for each row of x with the corresponding page of sigma, i.e., mvnpdf computes y(i) using x(i,:) and sigma(:,:,i). If the covariance matrix is diagonal, containing variances along the diagonal and zero covariances off the diagonal, sigma may also be specified as a 1-by-D matrix or a 1-by-D-by-N array, containing just the diagonal. Pass in the empty matrix for mu to use its default value when you want to only specify sigma.

If x is a 1-by-D vector, mvnpdf replicates it to match the leading dimension of mu or the trailing dimension of sigma.

See also: mvncdf, mvnrnd

Source Code: mvnpdf

Example: 1

  

 mu = [1, -1];
 sigma = [0.9, 0.4; 0.4, 0.3];
 [X1, X2] = meshgrid (linspace (-1, 3, 25)', linspace (-3, 1, 25)');
 x = [X1(:), X2(:)];
 p = mvnpdf (x, mu, sigma);
 surf (X1, X2, reshape (p, 25, 25));
plotted figure

statistics-release-1.6.3/docs/mvnrnd.html000066400000000000000000000140771456127120000204620ustar00rootroot00000000000000 Statistics: mvnrnd

Function Reference: mvnrnd

statistics: r = mvnrnd (mu, sigma)
statistics: r = mvnrnd (mu, sigma, n)
statistics: r = mvnrnd (mu, sigma, n, T)
statistics: [r, T] = mvnrnd (…)

Random vectors from the multivariate normal distribution.

r = mvnrnd (mu, sigma) returns an N-by-D matrix r of random vectors chosen from the multivariate normal distribution with mean vector mu and covariance matrix sigma. mu is an N-by-D matrix, and mvnrnd generates each N of r using the corresponding N of mu. sigma is a D-by-D symmetric positive semi-definite matrix, or a D-by-D-by-N array. If sigma is an array, mvnrnd generates each N of r using the corresponding page of sigma, i.e., mvnrnd computes r(i,:) using mu(i,:) and sigma(:,:,i). If the covariance matrix is diagonal, containing variances along the diagonal and zero covariances off the diagonal, sigma may also be specified as a 1-by-D matrix or a 1-by-D-by-N array, containing just the diagonal. If mu is a 1-by-D vector, mvnrnd replicates it to match the trailing dimension of SIGMA.

r = mvnrnd (mu, sigma, n) returns a N-by-D matrix R of random vectors chosen from the multivariate normal distribution with 1-by-D mean vector mu, and D-by-D covariance matrix sigma.

r = mvnrnd (mu, sigma, n, T) supplies the Cholesky factor T of sigma, so that sigma(:,:,J) == T(:,:,J)’*T(:,:,J) if sigma is a 3D array or sigma == T’*T if sigma is a matrix. No error checking is done on T.

[r, T] = mvnrnd (…) returns the Cholesky factor T, so it can be re-used to make later calls more efficient, although there are greater efficiency gains when SIGMA can be specified as a diagonal instead.

See also: mvncdf, mvnpdf

Source Code: mvnrnd

statistics-release-1.6.3/docs/mvtcdf.html000066400000000000000000000210651456127120000204340ustar00rootroot00000000000000 Statistics: mvtcdf

Function Reference: mvtcdf

statistics: p = mvtcdf (x, rho, df)
statistics: p = mvncdf (x_lo, x_up, rho, df)
statistics: p = mvncdf (…, options)
statistics: [p, err] = mvncdf (…)

Multivariate Student’s t cumulative distribution function (CDF).

p = mvtcdf (x, rho, df) returns the cumulative probability of the multivariate student’s t distribution with correlation parameters rho and degrees of freedom df, evaluated at each row of x. The rows of the N×D matrix x correspond to sample observations and its columns correspond to variables or coordinates. The return argument p is a column vector with the same number of rows as in x.

rho is a symmetric, positive definite, D×D correlation matrix. dF is a scalar or a vector with N elements.

Note: mvtcdf computes the CDF for the standard multivariate Student’s t distribution, centered at the origin, with no scale parameters. If rho is a covariance matrix, i.e. diag(rho) is not all ones, mvtcdf rescales rho to transform it to a correlation matrix. mvtcdf does not rescale x, though.

The multivariate Student’s t cumulative probability at x is defined as the probability that a random vector T, distributed as multivariate normal, will fall within the semi-infinite rectangle with upper limits defined by x.

  • PrT(1)<=X(1), T(2)<=X(2), ... T(D)<=X(D).

p = mvtcdf (x_lo, x_hi, rho, df) returns the multivariate Student’s t cumulative probability evaluated over the rectangle (hyper-rectangle for multivariate data in x) with lower and upper limits defined by x_lo and x_hi, respectively.

[p, err] = mvtcdf (…) also returns an error estimate err in p.

p = mvtcdf (…, options) specifies the structure, which controls specific parameters for the numerical integration used to compute p. The required fieds are:

"TolFun"Maximum absolute error tolerance. Default is 1e-8 for D < 4, or 1e-4 for D >= 4.
"MaxFunEvals"Maximum number of integrand evaluations when D >= 4. Default is 1e7. Ignored when D < 4.
"Display"Display options. Choices are "off" (default), "iter", which shows the probability and estimated error at each repetition, and "final", which shows the final probability and related error after the integrand has converged successfully. Ignored when D < 4.

See also: bvtcdf, mvtpdf, mvtrnd, mvtcdfqmc

Source Code: mvtcdf

Example: 1

  

 ## Compute the cdf of a multivariate Student's t distribution with
 ## correlation parameters rho = [1, 0.4; 0.4, 1] and 2 degrees of freedom.

 rho = [1, 0.4; 0.4, 1];
 df = 2;
 [X1, X2] = meshgrid (linspace (-2, 2, 25)', linspace (-2, 2, 25)');
 X = [X1(:), X2(:)];
 p = mvtcdf (X, rho, df);
 surf (X1, X2, reshape (p, 25, 25));
 title ("Bivariate Student's t cummulative distribution function");
plotted figure

statistics-release-1.6.3/docs/mvtcdfqmc.html000066400000000000000000000131051456127120000211310ustar00rootroot00000000000000 Statistics: mvtcdfqmc

Function Reference: mvtcdfqmc

statistics: p = mvtcdfqmc (A, B, Rho, df)
statistics: p = mvtcdfqmc (…, TolFun)
statistics: p = mvtcdfqmc (…, TolFun, MaxFunEvals)
statistics: p = mvtcdfqmc (…, TolFun, MaxFunEvals, Display)
statistics: [p, err] = mvtcdfqmc (…)
statistics: [p, err, FunEvals] = mvtcdfqmc (…)

Quasi-Monte-Carlo computation of the multivariate Student’s T CDF.

The QMC multivariate Student’s t distribution is evaluated between the lower limit A and upper limit B of the hyper-rectangle with a correlation matrix Rho and degrees of freedom df.

"TolFun"— Maximum absolute error tolerance. Default is 1e-4.
"MaxFunEvals"— Maximum number of integrand evaluations. Default is 1e7 for D > 4.
"Display"— Display options. Choices are "off" (default), "iter", which shows the probability and estimated error at each repetition, and "final", which shows the final probability and related error after the integrand has converged successfully.

[p, err, FunEvals] = mvtcdfqmc (…) returns the estimated probability, p, an estimate of the error, err, and the number of iterations until a successful convergence is met, unless the value in MaxFunEvals was reached.

See also: mvtcdf, mvtpdf, mvtrnd

Source Code: mvtcdfqmc

statistics-release-1.6.3/docs/mvtpdf.html000066400000000000000000000146771456127120000204640ustar00rootroot00000000000000 Statistics: mvtpdf

Function Reference: mvtpdf

statistics: y = mvtpdf (x, rho, df)

Multivariate Student’s t probability density function (PDF).

Arguments

  • x are the points at which to find the probability, where each row corresponds to an observation. (N×D matrix)
  • rho is the correlation matrix. (D×D symmetric positive definite matrix)
  • df is the degrees of freedom. (scalar or vector of length N)

The distribution is assumed to be centered (zero mean).

Return values

  • y is the probability density for each row of x. (N×1 vector)

Examples

 
 
 x = [1 2];
 rho = [1.0 0.5; 0.5 1.0];
 df = 4;
 y = mvtpdf (x, rho, df)

References

  1. Michael Roth, On the Multivariate t Distribution, Technical report from Automatic Control at Linkoepings universitet, http://users.isy.liu.se/en/rt/roth/student.pdf

See also: mvtcdf, mvtcdfqmc, mvtrnd

Source Code: mvtpdf

Example: 1

  

 ## Compute the pdf of a multivariate t distribution with correlation
 ## parameters rho = [1 .4; .4 1] and 2 degrees of freedom.

 rho = [1, 0.4; 0.4, 1];
 df = 2;
 [X1, X2] = meshgrid (linspace (-2, 2, 25)', linspace (-2, 2, 25)');
 X = [X1(:), X2(:)];
 y = mvtpdf (X, rho, df);
 surf (X1, X2, reshape (y, 25, 25));
 title ("Bivariate Student's t probability density function");
plotted figure

statistics-release-1.6.3/docs/mvtrnd.html000066400000000000000000000140371456127120000204640ustar00rootroot00000000000000 Statistics: mvtrnd

Function Reference: mvtrnd

statistics: r = mvtrnd (rho, df)
statistics: r = mvtrnd (rho, df, n)

Random vectors from the multivariate Student’s t distribution.

Arguments

  • rho is the matrix of correlation coefficients. If there are any non-unit diagonal elements then rho will be normalized, so that the resulting covariance of the obtained samples r follows: cov (r) = df/(df-2) * rho ./ (sqrt (diag (rho) * diag (rho))). In order to obtain samples distributed according to a standard multivariate student’s t-distribution, rho must be equal to the identity matrix. To generate multivariate student’s t-distribution samples r with arbitrary covariance matrix rho, the following scaling might be used: r = mvtrnd (rho, df, n) * diag (sqrt (diag (rho))).
  • df is the degrees of freedom for the multivariate t-distribution. df must be a vector with the same number of elements as samples to be generated or be scalar.
  • n is the number of rows of the matrix to be generated. n must be a non-negative integer and corresponds to the number of samples to be generated.

Return values

  • r is a matrix of random samples from the multivariate t-distribution with n row samples.

Examples

 
 
 rho = [1, 0.5; 0.5, 1];
 df = 3;
 n = 10;
 r = mvtrnd (rho, df, n);
 
 
 rho = [1, 0.5; 0.5, 1];
 df = [2; 3];
 n = 2;
 r = mvtrnd (rho, df, 2);

References

  1. Wendy L. Martinez and Angel R. Martinez. Computational Statistics Handbook with MATLAB. Appendix E, pages 547-557, Chapman & Hall/CRC, 2001.
  2. Samuel Kotz and Saralees Nadarajah. Multivariate t Distributions and Their Applications. Cambridge University Press, Cambridge, 2004.

See also: mvtcdf, mvtcdfqmc, mvtpdf

Source Code: mvtrnd

statistics-release-1.6.3/docs/nakacdf.html000066400000000000000000000152331456127120000205400ustar00rootroot00000000000000 Statistics: nakacdf

Function Reference: nakacdf

statistics: p = nakacdf (x, mu, omega)
statistics: p = nakacdf (x, mu, omega, "upper")

Nakagami cumulative distribution function (CDF).

For each element of x, compute the cumulative distribution function (CDF) of the Nakagami distribution with shape parameter mu and spread parameter omega. The size of p is the common size of x, mu, and omega. A scalar input functions as a constant matrix of the same size as the other inputs.

Both parameters must be positive reals and mu >= 0.5. For mu < 0.5 or omega <= 0, NaN is returned.

p = nakacdf (x, mu, omega, "upper") computes the upper tail probability of the Nakagami distribution with parameters mu and beta, at the values in x.

Further information about the Nakagami distribution can be found at https://en.wikipedia.org/wiki/Nakagami_distribution

See also: nakainv, nakapdf, nakarnd, nakafit, nakalike

Source Code: nakacdf

Example: 1

  

 ## Plot various CDFs from the Nakagami distribution
 x = 0:0.01:3;
 p1 = nakacdf (x, 0.5, 1);
 p2 = nakacdf (x, 1, 1);
 p3 = nakacdf (x, 1, 2);
 p4 = nakacdf (x, 1, 3);
 p5 = nakacdf (x, 2, 1);
 p6 = nakacdf (x, 2, 2);
 p7 = nakacdf (x, 5, 1);
 plot (x, p1, "-r", x, p2, "-g", x, p3, "-y", x, p4, "-m", ...
       x, p5, "-k", x, p6, "-b", x, p7, "-c")
 grid on
 xlim ([0, 3])
 legend ({"μ = 0.5, ω = 1", "μ = 1, ω = 1", "μ = 1, ω = 2", ...
          "μ = 1, ω = 3", "μ = 2, ω = 1", "μ = 2, ω = 2", ...
          "μ = 5, ω = 1"}, "location", "southeast")
 title ("Nakagami CDF")
 xlabel ("values in x")
 ylabel ("probability")
plotted figure

statistics-release-1.6.3/docs/nakafit.html000066400000000000000000000227071456127120000205720ustar00rootroot00000000000000 Statistics: nakafit

Function Reference: nakafit

statistics: paramhat = nakafit (x)
statistics: [paramhat, paramci] = nakafit (x)
statistics: [paramhat, paramci] = nakafit (x, alpha)
statistics: […] = nakafit (x, alpha, censor)
statistics: […] = nakafit (x, alpha, censor, freq)
statistics: […] = nakafit (x, alpha, censor, freq, options)

Estimate mean and confidence intervals for the Nakagami distribution.

mu0 = nakafit (x) returns the maximum likelihood estimates of the parameters of the Nakagami distribution given the data in x. paramhat(1) is the scale parameter, mu, and paramhat(2) is the shape parameter, omega.

[paramhat, paramci] = nakafit (x) returns the 95% confidence intervals for the parameter estimates.

[…] = nakafit (x, alpha) also returns the 100 * (1 - alpha) percent confidence intervals for the parameter estimates. By default, the optional argument alpha is 0.05 corresponding to 95% confidence intervals. Pass in [] for alpha to use the default values.

[…] = nakafit (x, alpha, censor) accepts a boolean vector, censor, of the same size as x with 1s for observations that are right-censored and 0s for observations that are observed exactly. By default, or if left empty, censor = zeros (size (x)).

[…] = nakafit (x, alpha, censor, freq) accepts a frequency vector, freq, of the same size as x. freq typically contains integer frequencies for the corresponding elements in x, but it can contain any non-integer non-negative values. By default, or if left empty, freq = ones (size (x)).

[…] = nakafit (…, options) specifies control parameters for the iterative algorithm used to compute ML estimates with the fminsearch function. options is a structure with the following fields and their default values:

  • options.Display = "off"
  • options.MaxFunEvals = 400
  • options.MaxIter = 200
  • options.TolX = 1e-6

Further information about the Nakagami distribution can be found at https://en.wikipedia.org/wiki/Nakagami_distribution

See also: nakacdf, nakainv, nakapdf, nakarnd, nakalike

Source Code: nakafit

Example: 1

  

 ## Sample 3 populations from different Nakagami distibutions
 randg ("seed", 5)  # for reproducibility
 r1 = nakarnd (0.5, 1, 2000, 1);
 randg ("seed", 2)   # for reproducibility
 r2 = nakarnd (5, 1, 2000, 1);
 randg ("seed", 7)   # for reproducibility
 r3 = nakarnd (2, 2, 2000, 1);
 r = [r1, r2, r3];

 ## Plot them normalized and fix their colors
 hist (r, [0.05:0.1:3.5], 10);
 h = findobj (gca, "Type", "patch");
 set (h(1), "facecolor", "c");
 set (h(2), "facecolor", "g");
 set (h(3), "facecolor", "r");
 ylim ([0, 2.5]);
 xlim ([0, 3.0]);
 hold on

 ## Estimate their MU and LAMBDA parameters
 mu_omegaA = nakafit (r(:,1));
 mu_omegaB = nakafit (r(:,2));
 mu_omegaC = nakafit (r(:,3));

 ## Plot their estimated PDFs
 x = [0.01:0.1:3.01];
 y = nakapdf (x, mu_omegaA(1), mu_omegaA(2));
 plot (x, y, "-pr");
 y = nakapdf (x, mu_omegaB(1), mu_omegaB(2));
 plot (x, y, "-sg");
 y = nakapdf (x, mu_omegaC(1), mu_omegaC(2));
 plot (x, y, "-^c");
 legend ({"Normalized HIST of sample 1 with μ=0.5 and ω=1", ...
          "Normalized HIST of sample 2 with μ=5 and ω=1", ...
          "Normalized HIST of sample 3 with μ=2 and ω=2", ...
          sprintf("PDF for sample 1 with estimated μ=%0.2f and ω=%0.2f", ...
                  mu_omegaA(1), mu_omegaA(2)), ...
          sprintf("PDF for sample 2 with estimated μ=%0.2f and ω=%0.2f", ...
                  mu_omegaB(1), mu_omegaB(2)), ...
          sprintf("PDF for sample 3 with estimated μ=%0.2f and ω=%0.2f", ...
                  mu_omegaC(1), mu_omegaC(2))})
 title ("Three population samples from different Nakagami distibutions")
 hold off
plotted figure

statistics-release-1.6.3/docs/nakainv.html000066400000000000000000000144331456127120000206010ustar00rootroot00000000000000 Statistics: nakainv

Function Reference: nakainv

statistics: x = nakacdf (x, mu, omega)

Inverse of the Nakagami cumulative distribution function (iCDF).

For each element of p, compute the quantile (the inverse of the CDF) of the Nakagami distribution with shape parameter mu and spread parameter omega. The size of x is the common size of x, mu, and omega. A scalar input functions as a constant matrix of the same size as the other inputs.

Both parameters must be positive reals and mu >= 0.5. For mu < 0.5 or omega <= 0, NaN is returned.

Further information about the Nakagami distribution can be found at https://en.wikipedia.org/wiki/Nakagami_distribution

See also: nakacdf, nakapdf, nakarnd, nakafit, nakalike

Source Code: nakainv

Example: 1

  

 ## Plot various iCDFs from the Nakagami distribution
 p = 0.001:0.001:0.999;
 x1 = nakainv (p, 0.5, 1);
 x2 = nakainv (p, 1, 1);
 x3 = nakainv (p, 1, 2);
 x4 = nakainv (p, 1, 3);
 x5 = nakainv (p, 2, 1);
 x6 = nakainv (p, 2, 2);
 x7 = nakainv (p, 5, 1);
 plot (p, x1, "-r", p, x2, "-g", p, x3, "-y", p, x4, "-m", ...
       p, x5, "-k", p, x6, "-b", p, x7, "-c")
 grid on
 ylim ([0, 3])
 legend ({"μ = 0.5, ω = 1", "μ = 1, ω = 1", "μ = 1, ω = 2", ...
          "μ = 1, ω = 3", "μ = 2, ω = 1", "μ = 2, ω = 2", ...
          "μ = 5, ω = 1"}, "location", "northwest")
 title ("Nakagami iCDF")
 xlabel ("probability")
 ylabel ("values in x")
plotted figure

statistics-release-1.6.3/docs/nakalike.html000066400000000000000000000136361456127120000207350ustar00rootroot00000000000000 Statistics: nakalike

Function Reference: nakalike

statistics: nlogL = nakalike (params, x)
statistics: [nlogL, acov] = nakalike (params, x)
statistics: […] = nakalike (params, x, censor)
statistics: […] = nakalike (params, x, censor, freq)

Negative log-likelihood for the Nakagami distribution.

nlogL = nakalike (params, x) returns the negative log likelihood of the data in x corresponding to the Nakagami distribution with (1) scale parameter mu and (2) shape parameter omega given in the two-element vector params.

[nlogL, acov] = nakalike (params, x) also returns the inverse of Fisher’s information matrix, acov. If the input parameter values in params are the maximum likelihood estimates, the diagonal elements of params are their asymptotic variances.

[…] = nakalike (params, x, censor) accepts a boolean vector, censor, of the same size as x with 1s for observations that are right-censored and 0s for observations that are observed exactly. By default, or if left empty, censor = zeros (size (x)).

[…] = nakalike (params, x, censor, freq) accepts a frequency vector, freq, of the same size as x. freq typically contains integer frequencies for the corresponding elements in x, but it can contain any non-integer non-negative values. By default, or if left empty, freq = ones (size (x)).

Further information about the Nakagami distribution can be found at https://en.wikipedia.org/wiki/Nakagami_distribution

See also: nakacdf, nakainv, nakapdf, nakarnd, nakafit

Source Code: nakalike

statistics-release-1.6.3/docs/nakapdf.html000066400000000000000000000144101456127120000205510ustar00rootroot00000000000000 Statistics: nakapdf

Function Reference: nakapdf

statistics: y = nakapdf (x, mu, omega)

Nakagami probability density function (PDF).

For each element of x, compute the probability density function (PDF) of the Nakagami distribution with shape parameter mu and spread parameter omega. The size of y is the common size of x, mu, and omega. A scalar input functions as a constant matrix of the same size as the other inputs.

Both parameters must be positive reals and mu >= 0.5. For mu < 0.5 or omega <= 0, NaN is returned.

Further information about the Nakagami distribution can be found at https://en.wikipedia.org/wiki/Nakagami_distribution

See also: nakacdf, nakapdf, nakarnd, nakafit, nakalike

Source Code: nakapdf

Example: 1

  

 ## Plot various PDFs from the Nakagami distribution
 x = 0:0.01:3;
 y1 = nakapdf (x, 0.5, 1);
 y2 = nakapdf (x, 1, 1);
 y3 = nakapdf (x, 1, 2);
 y4 = nakapdf (x, 1, 3);
 y5 = nakapdf (x, 2, 1);
 y6 = nakapdf (x, 2, 2);
 y7 = nakapdf (x, 5, 1);
 plot (x, y1, "-r", x, y2, "-g", x, y3, "-y", x, y4, "-m", ...
       x, y5, "-k", x, y6, "-b", x, y7, "-c")
 grid on
 xlim ([0, 3])
 ylim ([0, 2])
 legend ({"μ = 0.5, ω = 1", "μ = 1, ω = 1", "μ = 1, ω = 2", ...
          "μ = 1, ω = 3", "μ = 2, ω = 1", "μ = 2, ω = 2", ...
          "μ = 5, ω = 1"}, "location", "northeast")
 title ("Nakagami PDF")
 xlabel ("values in x")
 ylabel ("density")
plotted figure

statistics-release-1.6.3/docs/nakarnd.html000066400000000000000000000124651456127120000205730ustar00rootroot00000000000000 Statistics: nakarnd

Function Reference: nakarnd

statistics: r = nakarnd (mu, omega)
statistics: r = nakarnd (mu, omega, rows)
statistics: r = nakarnd (mu, omega, rows, cols, …)
statistics: r = nakarnd (mu, omega, [sz])

Random arrays from the Nakagami distribution.

r = nakarnd (mu, omega) returns an array of random numbers chosen from the Nakagami distribution with shape parameter mu and spread parameter omega. The size of r is the common size of mu and omega. A scalar input functions as a constant matrix of the same size as the other inputs.

Both parameters must be positive reals and mu >= 0.5. For mu < 0.5 or omega <= 0, NaN is returned.

When called with a single size argument, nakarnd returns a square matrix with the dimension specified. When called with more than one scalar argument, the first two arguments are taken as the number of rows and columns and any further arguments specify additional matrix dimensions. The size may also be specified with a row vector of dimensions, sz.

Further information about the Nakagami distribution can be found at https://en.wikipedia.org/wiki/Nakagami_distribution

See also: nakacdf, nakainv, nakapdf

Source Code: nakarnd

statistics-release-1.6.3/docs/nanmax.html000066400000000000000000000100361456127120000204270ustar00rootroot00000000000000 Statistics: nanmax

Function Reference: nanmax

statistics: [v, idx] = nanmax (X)
statistics: [v, idx] = nanmax (X, Y)

Find the maximal element while ignoring NaN values.

nanmax is identical to the max function except that NaN values are ignored. If all values in a column are NaN, the maximum is returned as NaN rather than [].

See also: max, nansum, nanmin

Source Code: nanmax

statistics-release-1.6.3/docs/nanmin.html000066400000000000000000000100361456127120000204250ustar00rootroot00000000000000 Statistics: nanmin

Function Reference: nanmin

statistics: [v, idx] = nanmin (X)
statistics: [v, idx] = nanmin (X, Y)

Find the minimal element while ignoring NaN values.

nanmin is identical to the min function except that NaN values are ignored. If all values in a column are NaN, the minimum is returned as NaN rather than [].

See also: min, nansum, nanmax

Source Code: nanmin

statistics-release-1.6.3/docs/nansum.html000066400000000000000000000104701456127120000204500ustar00rootroot00000000000000 Statistics: nansum

Function Reference: nansum

statistics: s = nansum (x)
statistics: s = nansum (x, dim)
statistics: s = nansum (…, "native")
statistics: s = nansum (…, "double")
statistics: s = nansum (…, "extra")

Compute the sum while ignoring NaN values.

nansum is identical to the sum function except that NaN values are treated as 0 and so ignored. If all values are NaN, the sum is returned as 0.

See help text of sum for details on the options.

See also: sum, nanmin, nanmax

Source Code: nansum

statistics-release-1.6.3/docs/nbincdf.html000066400000000000000000000175041456127120000205570ustar00rootroot00000000000000 Statistics: nbincdf

Function Reference: nbincdf

statistics: p = nbincdf (x, r, ps)
statistics: p = nbincdf (x, r, ps, "upper")

Negative binomial cumulative distribution function (CDF).

For each element of x, compute the cumulative distribution function (CDF) of the negative binomial distribution with parameters r and ps, where r is the number of successes until the experiment is stopped and ps is the probability of success in each experiment, given the number of failures in x. The size of p is the common size of x, r, and ps. A scalar input functions as a constant matrix of the same size as the other inputs.

The algorithm uses the cumulative sums of the binomial masses.

p = nbincdf (x, r, ps, "upper") computes the upper tail probability of the negative binomial distribution with parameters r and ps, at the values in x.

When r is an integer, the negative binomial distribution is also known as the Pascal distribution and it models the number of failures in x before a specified number of successes is reached in a series of independent, identical trials. Its parameters are the probability of success in a single trial, ps, and the number of successes, r. A special case of the negative binomial distribution, when r = 1, is the geometric distribution, which models the number of failures before the first success.

r can also have non-integer positive values, in which form the negative binomial distribution, also known as the Polya distribution, has no interpretation in terms of repeated trials, but, like the Poisson distribution, it is useful in modeling count data. The negative binomial distribution is more general than the Poisson distribution because it has a variance that is greater than its mean, making it suitable for count data that do not meet the assumptions of the Poisson distribution. In the limit, as r increases to infinity, the negative binomial distribution approaches the Poisson distribution.

Further information about the negative binomial distribution can be found at https://en.wikipedia.org/wiki/Negative_binomial_distribution

See also: nbininv, nbinpdf, nbinrnd, nbinfit, nbinlike, nbinstat

Source Code: nbincdf

Example: 1

  

 ## Plot various CDFs from the negative binomial distribution
 x = 0:50;
 p1 = nbincdf (x, 2, 0.15);
 p2 = nbincdf (x, 5, 0.2);
 p3 = nbincdf (x, 4, 0.4);
 p4 = nbincdf (x, 10, 0.3);
 plot (x, p1, "*r", x, p2, "*g", x, p3, "*k", x, p4, "*m")
 grid on
 xlim ([0, 40])
 legend ({"r = 2, ps = 0.15", "r = 5, ps = 0.2", "r = 4, p = 0.4", ...
          "r = 10, ps = 0.3"}, "location", "southeast")
 title ("Negative binomial CDF")
 xlabel ("values in x (number of failures)")
 ylabel ("probability")
plotted figure

statistics-release-1.6.3/docs/nbinfit.html000066400000000000000000000226101456127120000205770ustar00rootroot00000000000000 Statistics: nbinfit

Function Reference: nbinfit

statistics: paramhat = nbinfit (x)
statistics: [paramhat, paramci] = nbinfit (x)
statistics: [paramhat, paramci] = nbinfit (x, alpha)
statistics: [paramhat, paramci] = nbinfit (x, alpha, options)

Estimate parameter and confidence intervals for the negative binomial distribution.

paramhat = nbinfit (x) returns the maximum likelihood estimates of the parameters of the negative binomial distribution given the data in vector x. paramhat(1) is the number of successes until the experiment is stopped, r, and paramhat(1) is the probability of success in each experiment, ps.

[paramhat, paramci] = nbinfit (x) returns the 95% confidence intervals for the parameter estimates.

[paramhat, paramci] = nbinfit (x, alpha) also returns the 100 * (1 - alpha) percent confidence intervals of the estimated parameter. By default, the optional argument alpha is 0.05 corresponding to 95% confidence intervals.

[paramhat, paramci] = nbinfit (x, alpha, options) specifies control parameters for the iterative algorithm used to compute ML estimates with the fminsearch function. options is a structure with the following fields and their default values:

  • options.Display = "off"
  • options.MaxFunEvals = 400
  • options.MaxIter = 200
  • options.TolX = 1e-6

When r is an integer, the negative binomial distribution is also known as the Pascal distribution and it models the number of failures in x before a specified number of successes is reached in a series of independent, identical trials. Its parameters are the probability of success in a single trial, ps, and the number of successes, r. A special case of the negative binomial distribution, when r = 1, is the geometric distribution, which models the number of failures before the first success.

r can also have non-integer positive values, in which form the negative binomial distribution, also known as the Polya distribution, has no interpretation in terms of repeated trials, but, like the Poisson distribution, it is useful in modeling count data. The negative binomial distribution is more general than the Poisson distribution because it has a variance that is greater than its mean, making it suitable for count data that do not meet the assumptions of the Poisson distribution. In the limit, as r increases to infinity, the negative binomial distribution approaches the Poisson distribution.

Further information about the negative binomial distribution can be found at https://en.wikipedia.org/wiki/Negative_binomial_distribution

See also: nbincdf, nbininv, nbinpdf, nbinrnd, nbinlike, nbinstat

Source Code: nbinfit

Example: 1

  

 ## Sample 2 populations from different negative binomial distibutions
 randp ("seed", 5); randg ("seed", 5);    # for reproducibility
 r1 = nbinrnd (2, 0.15, 5000, 1);
 randp ("seed", 8); randg ("seed", 8);    # for reproducibility
 r2 = nbinrnd (5, 0.2, 5000, 1);
 r = [r1, r2];

 ## Plot them normalized and fix their colors
 hist (r, [0:51], 1);
 h = findobj (gca, "Type", "patch");
 set (h(1), "facecolor", "c");
 set (h(2), "facecolor", "g");
 hold on

 ## Estimate their probability of success
 r_psA = nbinfit (r(:,1));
 r_psB = nbinfit (r(:,2));

 ## Plot their estimated PDFs
 x = [0:40];
 y = nbinpdf (x, r_psA(1), r_psA(2));
 plot (x, y, "-pg");
 x = [min(r(:,2)):max(r(:,2))];
 y = nbinpdf (x, r_psB(1), r_psB(2));
 plot (x, y, "-sc");
 ylim ([0, 0.1])
 xlim ([0, 50])
 legend ({"Normalized HIST of sample 1 with r=2 and ps=0.15", ...
          "Normalized HIST of sample 2 with r=5 and ps=0.2", ...
          sprintf("PDF for sample 1 with estimated r=%0.2f and ps=%0.2f", ...
                  r_psA(1), r_psA(2)), ...
          sprintf("PDF for sample 2 with estimated r=%0.2f and ps=%0.2f", ...
                  r_psB(1), r_psB(2))})
 title ("Two population samples from negative different binomial distibutions")
 hold off
plotted figure

statistics-release-1.6.3/docs/nbininv.html000066400000000000000000000165731456127120000206240ustar00rootroot00000000000000 Statistics: nbininv

Function Reference: nbininv

statistics: x = nbininv (p, r, ps)

Inverse of the negative binomial cumulative distribution function (iCDF).

For each element of p, compute the quantile (the inverse of the CDF) of the negative binomial distribution with parameters r and ps, where r is the number of successes until the experiment is stopped and ps is the probability of success in each experiment, given the probability in p. The size of x is the common size of p, r, and ps. A scalar input functions as a constant matrix of the same size as the other inputs.

When r is an integer, the negative binomial distribution is also known as the Pascal distribution and it models the number of failures in x before a specified number of successes is reached in a series of independent, identical trials. Its parameters are the probability of success in a single trial, ps, and the number of successes, r. A special case of the negative binomial distribution, when r = 1, is the geometric distribution, which models the number of failures before the first success.

r can also have non-integer positive values, in which form the negative binomial distribution, also known as the Polya distribution, has no interpretation in terms of repeated trials, but, like the Poisson distribution, it is useful in modeling count data. The negative binomial distribution is more general than the Poisson distribution because it has a variance that is greater than its mean, making it suitable for count data that do not meet the assumptions of the Poisson distribution. In the limit, as r increases to infinity, the negative binomial distribution approaches the Poisson distribution.

Further information about the negative binomial distribution can be found at https://en.wikipedia.org/wiki/Negative_binomial_distribution

See also: nbininv, nbinpdf, nbinrnd, nbinfit, nbinlike, nbinstat

Source Code: nbininv

Example: 1

  

 ## Plot various iCDFs from the negative binomial distribution
 p = 0.001:0.001:0.999;
 x1 = nbininv (p, 2, 0.15);
 x2 = nbininv (p, 5, 0.2);
 x3 = nbininv (p, 4, 0.4);
 x4 = nbininv (p, 10, 0.3);
 plot (p, x1, "-r", p, x2, "-g", p, x3, "-k", p, x4, "-m")
 grid on
 ylim ([0, 40])
 legend ({"r = 2, ps = 0.15", "r = 5, ps = 0.2", "r = 4, p = 0.4", ...
          "r = 10, ps = 0.3"}, "location", "northwest")
 title ("Negative binomial iCDF")
 xlabel ("probability")
 ylabel ("values in x (number of failures)")
plotted figure

statistics-release-1.6.3/docs/nbinlike.html000066400000000000000000000144021456127120000207410ustar00rootroot00000000000000 Statistics: nbinlike

Function Reference: nbinlike

statistics: nlogL = nbinlike (params, x)
statistics: [nlogL, avar] = nbinlike (params, x)

Negative log-likelihood for the negative binomial distribution.

nlogL = nbinlike (params, x) returns the negative log likelihood of the negative binomial distribution with (1) parameter r and (2) parameter ps, given in the two-element vector params, where r is the number of successes until the experiment is stopped and ps is the probability of success in each experiment, given the number of failures in x.

[nlogL, avar] = nbinlike (params, x) also returns the inverse of Fisher’s information matrix, avar. If the input parameter values in params are the maximum likelihood estimates, the diagonal elements of params are their asymptotic variances.

When r is an integer, the negative binomial distribution is also known as the Pascal distribution and it models the number of failures in x before a specified number of successes is reached in a series of independent, identical trials. Its parameters are the probability of success in a single trial, ps, and the number of successes, r. A special case of the negative binomial distribution, when r = 1, is the geometric distribution, which models the number of failures before the first success.

r can also have non-integer positive values, in which form the negative binomial distribution, also known as the Polya distribution, has no interpretation in terms of repeated trials, but, like the Poisson distribution, it is useful in modeling count data. The negative binomial distribution is more general than the Poisson distribution because it has a variance that is greater than its mean, making it suitable for count data that do not meet the assumptions of the Poisson distribution. In the limit, as r increases to infinity, the negative binomial distribution approaches the Poisson distribution.

Further information about the negative binomial distribution can be found at https://en.wikipedia.org/wiki/Negative_binomial_distribution

See also: nbincdf, nbininv, nbinpdf, nbinrnd, nbinfit, nbinstat

Source Code: nbinlike

statistics-release-1.6.3/docs/nbinpdf.html000066400000000000000000000165761456127120000206040ustar00rootroot00000000000000 Statistics: nbinpdf

Function Reference: nbinpdf

statistics: y = nbinpdf (x, r, ps)

Negative binomial probability density function (PDF).

For each element of x, compute the probability density function (PDF) at x of the negative binomial distribution with parameters r and ps, where r is the number of successes until the experiment is stopped and ps is the probability of success in each experiment, given the number of failures in x. The size of y is the common size of x, r, and ps. A scalar input functions as a constant matrix of the same size as the other inputs.

When r is an integer, the negative binomial distribution is also known as the Pascal distribution and it models the number of failures in x before a specified number of successes is reached in a series of independent, identical trials. Its parameters are the probability of success in a single trial, ps, and the number of successes, r. A special case of the negative binomial distribution, when r = 1, is the geometric distribution, which models the number of failures before the first success.

r can also have non-integer positive values, in which form the negative binomial distribution, also known as the Polya distribution, has no interpretation in terms of repeated trials, but, like the Poisson distribution, it is useful in modeling count data. The negative binomial distribution is more general than the Poisson distribution because it has a variance that is greater than its mean, making it suitable for count data that do not meet the assumptions of the Poisson distribution. In the limit, as r increases to infinity, the negative binomial distribution approaches the Poisson distribution.

Further information about the negative binomial distribution can be found at https://en.wikipedia.org/wiki/Negative_binomial_distribution

See also: nbininv, nbininv, nbinrnd, nbinfit, nbinlike, nbinstat

Source Code: nbinpdf

Example: 1

  

 ## Plot various PDFs from the negative binomial distribution
 x = 0:40;
 y1 = nbinpdf (x, 2, 0.15);
 y2 = nbinpdf (x, 5, 0.2);
 y3 = nbinpdf (x, 4, 0.4);
 y4 = nbinpdf (x, 10, 0.3);
 plot (x, y1, "*r", x, y2, "*g", x, y3, "*k", x, y4, "*m")
 grid on
 xlim ([0, 40])
 ylim ([0, 0.12])
 legend ({"r = 2, ps = 0.15", "r = 5, ps = 0.2", "r = 4, p = 0.4", ...
          "r = 10, ps = 0.3"}, "location", "northeast")
 title ("Negative binomial PDF")
 xlabel ("values in x (number of failures)")
 ylabel ("density")
plotted figure

statistics-release-1.6.3/docs/nbinrnd.html000066400000000000000000000151201456127120000205760ustar00rootroot00000000000000 Statistics: nbinrnd

Function Reference: nbinrnd

statistics: rnd = nbinrnd (r, ps)
statistics: rnd = nbinrnd (r, ps, rows)
statistics: rnd = nbinrnd (r, ps, rows, cols, …)
statistics: rnd = nbinrnd (r, ps, [sz])

Random arrays from the negative binomial distribution.

rnd = nbinrnd (r, ps) returns an array of random numbers chosen from the Laplace distribution with parameters r and ps, where r is the number of successes until the experiment is stopped and ps is the probability of success in each experiment, given the number of failures in x. The size of rnd is the common size of r and ps. A scalar input functions as a constant matrix of the same size as the other inputs.

When called with a single size argument, return a square matrix with the dimension specified. When called with more than one scalar argument the first two arguments are taken as the number of rows and columns and any further arguments specify additional matrix dimensions. The size may also be specified with a vector of dimensions sz.

When r is an integer, the negative binomial distribution is also known as the Pascal distribution and it models the number of failures in x before a specified number of successes is reached in a series of independent, identical trials. Its parameters are the probability of success in a single trial, ps, and the number of successes, r. A special case of the negative binomial distribution, when r = 1, is the geometric distribution, which models the number of failures before the first success.

r can also have non-integer positive values, in which form the negative binomial distribution, also known as the Polya distribution, has no interpretation in terms of repeated trials, but, like the Poisson distribution, it is useful in modeling count data. The negative binomial distribution is more general than the Poisson distribution because it has a variance that is greater than its mean, making it suitable for count data that do not meet the assumptions of the Poisson distribution. In the limit, as r increases to infinity, the negative binomial distribution approaches the Poisson distribution.

Further information about the negative binomial distribution can be found at https://en.wikipedia.org/wiki/Negative_binomial_distribution

See also: nbininv, nbininv, nbinpdf, nbinfit, nbinlike, nbinstat

Source Code: nbinrnd

statistics-release-1.6.3/docs/nbinstat.html000066400000000000000000000113521456127120000207710ustar00rootroot00000000000000 Statistics: nbinstat

Function Reference: nbinstat

statistics: [m, v] = nbinstat (r, ps)

Compute statistics of the negative binomial distribution.

[m, v] = nbinstat (r, ps) returns the mean and variance of the negative binomial distribution with parameters r and ps, where r is the number of successes until the experiment is stopped and ps is the probability of success in each experiment, given the number of failures in x.

The size of m (mean) and v (variance) is the common size of the input arguments. A scalar input functions as a constant matrix of the same size as the other inputs.

Further information about the negative binomial distribution can be found at https://en.wikipedia.org/wiki/Negative_binomial_distribution

See also: nbincdf, nbininv, nbininv, nbinrnd, nbinfit, nbinlike

Source Code: nbinstat

statistics-release-1.6.3/docs/ncfcdf.html000066400000000000000000000176561456127120000204070ustar00rootroot00000000000000 Statistics: ncfcdf

Function Reference: ncfcdf

statistics: p = ncfcdf (x, df1, df2, lambda)
statistics: p = ncfcdf (x, df1, df2, lambda, "upper")

Noncentral F-cumulative distribution function (CDF).

For each element of x, compute the cumulative distribution function (CDF) of the noncentral F-distribution with df1 and df2 degrees of freedom and noncentrality parameter lambda. The size of p is the common size of x, df1, df2, and lambda. A scalar input functions as a constant matrix of the same size as the other inputs.

p = ncfcdf (x, df1, df2, lambda, "upper") computes the upper tail probability of the noncentral F-distribution with parameters df1, df2, and lambda, at the values in x.

Further information about the noncentral F-distribution can be found at https://en.wikipedia.org/wiki/Noncentral_F-distribution

See also: ncfinv, ncfpdf, ncfrnd, ncfstat, fcdf

Source Code: ncfcdf

Example: 1

  

 ## Plot various CDFs from the noncentral F distribution
 x = 0:0.01:5;
 p1 = ncfcdf (x, 2, 5, 1);
 p2 = ncfcdf (x, 2, 5, 2);
 p3 = ncfcdf (x, 5, 10, 1);
 p4 = ncfcdf (x, 10, 20, 10);
 plot (x, p1, "-r", x, p2, "-g", x, p3, "-k", x, p4, "-m")
 grid on
 xlim ([0, 5])
 legend ({"df1 = 2, df2 = 5, λ = 1", "df1 = 2, df2 = 5, λ = 2", ...
          "df1 = 5, df2 = 10, λ = 1", "df1 = 10, df2 = 20, λ = 10"}, ...
         "location", "southeast")
 title ("Noncentral F CDF")
 xlabel ("values in x")
 ylabel ("probability")
plotted figure

Example: 2

  

 ## Compare the noncentral F CDF with LAMBDA = 10 to the F CDF with the
 ## same number of numerator and denominator degrees of freedom (5, 20)

 x = 0.01:0.1:10.01;
 p1 = ncfcdf (x, 5, 20, 10);
 p2 = fcdf (x, 5, 20);
 plot (x, p1, "-", x, p2, "-");
 grid on
 xlim ([0, 10])
 legend ({"Noncentral F(5,20,10)", "F(5,20)"}, "location", "southeast")
 title ("Noncentral F vs F CDFs")
 xlabel ("values in x")
 ylabel ("probability")
plotted figure

statistics-release-1.6.3/docs/ncfinv.html000066400000000000000000000171151456127120000204350ustar00rootroot00000000000000 Statistics: ncfinv

Function Reference: ncfinv

statistics: x = ncfinv (p, df1, df2, lambda)

Inverse of the noncentral F-cumulative distribution function (iCDF).

For each element of p, compute the quantile (the inverse of the CDF) of the noncentral F-distribution with df1 and df2 degrees of freedom and noncentrality parameter lambda. The size of x is the common size of p, df1, df2, and lambda. A scalar input functions as a constant matrix of the same size as the other inputs.

ncfinv uses Newton’s method to converge to the solution.

Further information about the noncentral F-distribution can be found at https://en.wikipedia.org/wiki/Noncentral_F-distribution

See also: ncfcdf, ncfpdf, ncfrnd, ncfstat, finv

Source Code: ncfinv

Example: 1

  

 ## Plot various iCDFs from the noncentral F distribution
 p = 0.001:0.001:0.999;
 x1 = ncfinv (p, 2, 5, 1);
 x2 = ncfinv (p, 2, 5, 2);
 x3 = ncfinv (p, 5, 10, 1);
 x4 = ncfinv (p, 10, 20, 10);
 plot (p, x1, "-r", p, x2, "-g", p, x3, "-k", p, x4, "-m")
 grid on
 ylim ([0, 5])
 legend ({"df1 = 2, df2 = 5, λ = 1", "df1 = 2, df2 = 5, λ = 2", ...
          "df1 = 5, df2 = 10, λ = 1", "df1 = 10, df2 = 20, λ = 10"}, ...
         "location", "northwest")
 title ("Noncentral F iCDF")
 xlabel ("probability")
 ylabel ("values in x")
plotted figure

Example: 2

  

 ## Compare the noncentral F iCDF with LAMBDA = 10 to the F iCDF with the
 ## same number of numerator and denominator degrees of freedom (5, 20)

 p = 0.001:0.001:0.999;
 x1 = ncfinv (p, 5, 20, 10);
 x2 = finv (p, 5, 20);
 plot (p, x1, "-", p, x2, "-");
 grid on
 ylim ([0, 10])
 legend ({"Noncentral F(5,20,10)", "F(5,20)"}, "location", "northwest")
 title ("Noncentral F vs F quantile functions")
 xlabel ("probability")
 ylabel ("values in x")
plotted figure

statistics-release-1.6.3/docs/ncfpdf.html000066400000000000000000000167411456127120000204160ustar00rootroot00000000000000 Statistics: ncfpdf

Function Reference: ncfpdf

statistics: y = ncfpdf (x, df1, df2, lambda)

Noncentral F-probability density function (PDF).

For each element of x, compute the probability density function (PDF) of the noncentral F-distribution with df1 and df2 degrees of freedom and noncentrality parameter lambda. The size of y is the common size of x, df1, df2, and lambda. A scalar input functions as a constant matrix of the same size as the other inputs.

Further information about the noncentral F-distribution can be found at https://en.wikipedia.org/wiki/Noncentral_F-distribution

See also: ncfcdf, ncfinv, ncfrnd, ncfstat, fpdf

Source Code: ncfpdf

Example: 1

  

 ## Plot various PDFs from the noncentral F distribution
 x = 0:0.01:5;
 y1 = ncfpdf (x, 2, 5, 1);
 y2 = ncfpdf (x, 2, 5, 2);
 y3 = ncfpdf (x, 5, 10, 1);
 y4 = ncfpdf (x, 10, 20, 10);
 plot (x, y1, "-r", x, y2, "-g", x, y3, "-k", x, y4, "-m")
 grid on
 xlim ([0, 5])
 ylim ([0, 0.8])
 legend ({"df1 = 2, df2 = 5, λ = 1", "df1 = 2, df2 = 5, λ = 2", ...
          "df1 = 5, df2 = 10, λ = 1", "df1 = 10, df2 = 20, λ = 10"}, ...
         "location", "northeast")
 title ("Noncentral F PDF")
 xlabel ("values in x")
 ylabel ("density")
plotted figure

Example: 2

  

 ## Compare the noncentral F PDF with LAMBDA = 10 to the F PDF with the
 ## same number of numerator and denominator degrees of freedom (5, 20)

 x = 0.01:0.1:10.01;
 y1 = ncfpdf (x, 5, 20, 10);
 y2 = fpdf (x, 5, 20);
 plot (x, y1, "-", x, y2, "-");
 grid on
 xlim ([0, 10])
 ylim ([0, 0.8])
 legend ({"Noncentral F(5,20,10)", "F(5,20)"}, "location", "northeast")
 title ("Noncentral F vs F PDFs")
 xlabel ("values in x")
 ylabel ("density")
plotted figure

statistics-release-1.6.3/docs/ncfrnd.html000066400000000000000000000130241456127120000204170ustar00rootroot00000000000000 Statistics: ncfrnd

Function Reference: ncfrnd

statistics: r = ncfrnd (df1, df2, lambda)
statistics: r = ncfrnd (df1, df2, lambda, rows, cols, …)
statistics: r = ncfrnd (df1, df2, lambda, [sz])

Random arrays from the noncentral F-distribution.

x = ncfrnd (p, df1, df2, lambda) returns an array of random numbers chosen from the noncentral F-distribution with df1 and df2 degrees of freedom and noncentrality parameter lambda. The size of r is the common size of df1, df2, and lambda. A scalar input functions as a constant matrix of the same size as the other input.

ncfrnd generates values using the definition of a noncentral F random variable, as the ratio of a noncentral chi-squared distribution and a (central) chi-squared distribution.

When called with a single size argument, ncfrnd returns a square matrix with the dimension specified. When called with more than one scalar argument, the first two arguments are taken as the number of rows and columns and any further arguments specify additional matrix dimensions. The size may also be specified with a row vector of dimensions, sz.

Further information about the noncentral F-distribution can be found at https://en.wikipedia.org/wiki/Noncentral_F-distribution

See also: ncfcdf, ncfinv, ncfpdf, ncfstat, frnd, ncx2rnd, chi2rnd

Source Code: ncfrnd

statistics-release-1.6.3/docs/ncfstat.html000066400000000000000000000111421456127120000206060ustar00rootroot00000000000000 Statistics: ncfstat

Function Reference: ncfstat

statistics: [m, v] = ncfstat (df1, df1, lambda)

Compute statistics for the noncentral F-distribution.

[m, v] = ncfstat (df1, df1, lambda) returns the mean and variance of the noncentral F-distribution with df1 and df2 degrees of freedom and noncentrality parameter lambda.

The size of m (mean) and v (variance) is the common size of the input arguments. A scalar input functions as a constant matrix of the same size as the other inputs.

Further information about the noncentral F-distribution can be found at https://en.wikipedia.org/wiki/Noncentral_F-distribution

See also: ncfcdf, ncfinv, ncfpdf, ncfrnd, fstat

Source Code: ncfstat

statistics-release-1.6.3/docs/nctcdf.html000066400000000000000000000172611456127120000204150ustar00rootroot00000000000000 Statistics: nctcdf

Function Reference: nctcdf

statistics: p = nctcdf (x, df, mu)
statistics: p = nctcdf (x, df, mu, "upper")

Noncentral t-cumulative distribution function (CDF).

For each element of x, compute the cumulative distribution function (CDF) of the noncentral t-distribution with df degrees of freedom and noncentrality parameter mu. The size of p is the common size of x, df, and mu. A scalar input functions as a constant matrix of the same size as the other inputs.

p = nctcdf (x, df, mu, "upper") computes the upper tail probability of the noncentral t-distribution with parameters df and mu, at the values in x.

Further information about the noncentral t-distribution can be found at https://en.wikipedia.org/wiki/Noncentral_t-distribution

See also: nctinv, nctpdf, nctrnd, nctstat, tcdf

Source Code: nctcdf

Example: 1

  

 ## Plot various CDFs from the noncentral Τ distribution
 x = -5:0.01:5;
 p1 = nctcdf (x, 1, 0);
 p2 = nctcdf (x, 4, 0);
 p3 = nctcdf (x, 1, 2);
 p4 = nctcdf (x, 4, 2);
 plot (x, p1, "-r", x, p2, "-g", x, p3, "-k", x, p4, "-m")
 grid on
 xlim ([-5, 5])
 legend ({"df = 1, μ = 0", "df = 4, μ = 0", ...
          "df = 1, μ = 2", "df = 4, μ = 2"}, "location", "southeast")
 title ("Noncentral Τ CDF")
 xlabel ("values in x")
 ylabel ("probability")
plotted figure

Example: 2

  

 ## Compare the noncentral T CDF with MU = 1 to the T CDF
 ## with the same number of degrees of freedom (10).

 x = -5:0.1:5;
 p1 = nctcdf (x, 10, 1);
 p2 = tcdf (x, 10);
 plot (x, p1, "-", x, p2, "-")
 grid on
 xlim ([-5, 5])
 legend ({"Noncentral T(10,1)", "T(10)"}, "location", "southeast")
 title ("Noncentral T vs T CDFs")
 xlabel ("values in x")
 ylabel ("probability")
plotted figure

statistics-release-1.6.3/docs/nctinv.html000066400000000000000000000166301456127120000204540ustar00rootroot00000000000000 Statistics: nctinv

Function Reference: nctinv

statistics: x = ncx2inv (p, df, mu)

Inverse of the non-central t-cumulative distribution function (iCDF).

For each element of p, compute the quantile (the inverse of the CDF) of the noncentral t-distribution with df degrees of freedom and noncentrality parameter mu. The size of x is the common size of p, df, and mu. A scalar input functions as a constant matrix of the same size as the other inputs.

nctinv uses Newton’s method to converge to the solution.

Further information about the noncentral t-distribution can be found at https://en.wikipedia.org/wiki/Noncentral_t-distribution

See also: nctcdf, nctpdf, nctrnd, nctstat, tinv

Source Code: nctinv

Example: 1

  

 ## Plot various iCDFs from the noncentral T distribution
 p = 0.001:0.001:0.999;
 x1 = nctinv (p, 1, 0);
 x2 = nctinv (p, 4, 0);
 x3 = nctinv (p, 1, 2);
 x4 = nctinv (p, 4, 2);
 plot (p, x1, "-r", p, x2, "-g", p, x3, "-k", p, x4, "-m")
 grid on
 ylim ([-5, 5])
 legend ({"df = 1, μ = 0", "df = 4, μ = 0", ...
          "df = 1, μ = 2", "df = 4, μ = 2"}, "location", "northwest")
 title ("Noncentral T iCDF")
 xlabel ("probability")
 ylabel ("values in x")
plotted figure

Example: 2

  

 ## Compare the noncentral T iCDF with MU = 1 to the T iCDF
 ## with the same number of degrees of freedom (10).

 p = 0.001:0.001:0.999;
 x1 = nctinv (p, 10, 1);
 x2 = tinv (p, 10);
 plot (p, x1, "-", p, x2, "-");
 grid on
 ylim ([-5, 5])
 legend ({"Noncentral T(10,1)", "T(10)"}, "location", "northwest")
 title ("Noncentral T vs T quantile functions")
 xlabel ("probability")
 ylabel ("values in x")
plotted figure

statistics-release-1.6.3/docs/nctpdf.html000066400000000000000000000164531456127120000204340ustar00rootroot00000000000000 Statistics: nctpdf

Function Reference: nctpdf

statistics: y = nctpdf (x, df, mu)

Noncentral t-probability density function (PDF).

For each element of x, compute the probability density function (PDF) of the noncentral t-distribution with df degrees of freedom and noncentrality parameter mu. The size of y is the common size of x, df, and mu. A scalar input functions as a constant matrix of the same size as the other inputs.

Further information about the noncentral t-distribution can be found at https://en.wikipedia.org/wiki/Noncentral_t-distribution

See also: nctcdf, nctinv, nctrnd, nctstat, tpdf

Source Code: nctpdf

Example: 1

  

 ## Plot various PDFs from the noncentral T distribution
 x = -5:0.01:10;
 y1 = nctpdf (x, 1, 0);
 y2 = nctpdf (x, 4, 0);
 y3 = nctpdf (x, 1, 2);
 y4 = nctpdf (x, 4, 2);
 plot (x, y1, "-r", x, y2, "-g", x, y3, "-k", x, y4, "-m")
 grid on
 xlim ([-5, 10])
 ylim ([0, 0.4])
 legend ({"df = 1, μ = 0", "df = 4, μ = 0", ...
          "df = 1, μ = 2", "df = 4, μ = 2"}, "location", "northeast")
 title ("Noncentral T PDF")
 xlabel ("values in x")
 ylabel ("density")
plotted figure

Example: 2

  

 ## Compare the noncentral T PDF with MU = 1 to the T PDF
 ## with the same number of degrees of freedom (10).

 x = -5:0.1:5;
 y1 = nctpdf (x, 10, 1);
 y2 = tpdf (x, 10);
 plot (x, y1, "-", x, y2, "-");
 grid on
 xlim ([-5, 5])
 ylim ([0, 0.4])
 legend ({"Noncentral χ^2(4,2)", "χ^2(4)"}, "location", "northwest")
 title ("Noncentral T vs T PDFs")
 xlabel ("values in x")
 ylabel ("density")
plotted figure

statistics-release-1.6.3/docs/nctrnd.html000066400000000000000000000126261456127120000204440ustar00rootroot00000000000000 Statistics: nctrnd

Function Reference: nctrnd

statistics: r = nctrnd (df, mu)
statistics: r = nctrnd (df, mu, rows, cols, …)
statistics: r = nctrnd (df, mu, [sz])

Random arrays from the noncentral t-distribution.

x = nctrnd (p, df, mu) returns an array of random numbers chosen from the noncentral t-distribution with df degrees of freedom and noncentrality parameter mu. The size of r is the common size of df and mu. A scalar input functions as a constant matrix of the same size as the other input.

nctrnd generates values using the definition of a noncentral t random variable, as the ratio of a normal distribution with non-zero mean and the sqrt of a chi-squared distribution.

When called with a single size argument, nctrnd returns a square matrix with the dimension specified. When called with more than one scalar argument, the first two arguments are taken as the number of rows and columns and any further arguments specify additional matrix dimensions. The size may also be specified with a row vector of dimensions, sz.

Further information about the noncentral t-distribution can be found at https://en.wikipedia.org/wiki/Noncentral_t-distribution

See also: nctcdf, nctinv, nctpdf, nctstat, trnd, normrnd, chi2rnd

Source Code: nctrnd

statistics-release-1.6.3/docs/nctstat.html000066400000000000000000000110371456127120000206270ustar00rootroot00000000000000 Statistics: nctstat

Function Reference: nctstat

statistics: [m, v] = nctstat (df, mu)

Compute statistics for the noncentral t-distribution.

[m, v] = nctstat (df, mu) returns the mean and variance of the noncentral t-distribution with df degrees of freedom and noncentrality parameter mu.

The size of m (mean) and v (variance) is the common size of the input arguments. A scalar input functions as a constant matrix of the same size as the other inputs.

Further information about the noncentral t-distribution can be found at https://en.wikipedia.org/wiki/Noncentral_t-distribution

See also: nctcdf, nctinv, nctpdf, nctrnd, tstat

Source Code: nctstat

statistics-release-1.6.3/docs/ncx2cdf.html000066400000000000000000000176671456127120000205150ustar00rootroot00000000000000 Statistics: ncx2cdf

Function Reference: ncx2cdf

statistics: p = ncx2cdf (x, df, lambda)
statistics: p = ncx2cdf (x, df, lambda, "upper")

Noncentral chi-squared cumulative distribution function (CDF).

For each element of x, compute the cumulative distribution function (CDF) of the noncentral chi-squared distribution with df degrees of freedom and noncentrality parameter lambda. The size of p is the common size of x, df, and lambda. A scalar input functions as a constant matrix of the same size as the other inputs.

p = ncx2cdf (x, df, lambda, "upper") computes the upper tail probability of the noncentral chi-squared distribution with parameters df and lambda, at the values in x.

Further information about the noncentral chi-squared distribution can be found at https://en.wikipedia.org/wiki/Noncentral_chi-squared_distribution

See also: ncx2inv, ncx2pdf, ncx2rnd, ncx2stat, chi2cdf

Source Code: ncx2cdf

Example: 1

  

 ## Plot various CDFs from the noncentral chi-squared distribution
 x = 0:0.1:10;
 p1 = ncx2cdf (x, 2, 1);
 p2 = ncx2cdf (x, 2, 2);
 p3 = ncx2cdf (x, 2, 3);
 p4 = ncx2cdf (x, 4, 1);
 p5 = ncx2cdf (x, 4, 2);
 p6 = ncx2cdf (x, 4, 3);
 plot (x, p1, "-r", x, p2, "-g", x, p3, "-k", ...
       x, p4, "-m", x, p5, "-c", x, p6, "-y")
 grid on
 xlim ([0, 10])
 legend ({"df = 2, λ = 1", "df = 2, λ = 2", ...
          "df = 2, λ = 3", "df = 4, λ = 1", ...
          "df = 4, λ = 2", "df = 4, λ = 3"}, "location", "southeast")
 title ("Noncentral chi-squared CDF")
 xlabel ("values in x")
 ylabel ("probability")
plotted figure

Example: 2

  

 ## Compare the noncentral chi-squared CDF with LAMBDA = 2 to the
 ## chi-squared CDF with the same number of degrees of freedom (4).

 x = 0:0.1:10;
 p1 = ncx2cdf (x, 4, 2);
 p2 = chi2cdf (x, 4);
 plot (x, p1, "-", x, p2, "-")
 grid on
 xlim ([0, 10])
 legend ({"Noncentral χ^2(4,2)", "χ^2(4)"}, "location", "northwest")
 title ("Noncentral chi-squared vs chi-squared CDFs")
 xlabel ("values in x")
 ylabel ("probability")
plotted figure

statistics-release-1.6.3/docs/ncx2inv.html000066400000000000000000000172131456127120000205400ustar00rootroot00000000000000 Statistics: ncx2inv

Function Reference: ncx2inv

statistics: x = ncx2inv (p, df, lambda)

Inverse of the noncentral chi-squared cumulative distribution function (iCDF).

For each element of p, compute the quantile (the inverse of the CDF) of the noncentral chi-squared distribution with df degrees of freedom and noncentrality parameter mu. The size of x is the common size of p, df, and mu. A scalar input functions as a constant matrix of the same size as the other inputs.

ncx2inv uses Newton’s method to converge to the solution.

Further information about the noncentral chi-squared distribution can be found at https://en.wikipedia.org/wiki/Noncentral_chi-squared_distribution

See also: ncx2cdf, ncx2pdf, ncx2rnd, ncx2stat, chi2inv

Source Code: ncx2inv

Example: 1

  

 ## Plot various iCDFs from the noncentral chi-squared distribution
 p = 0.001:0.001:0.999;
 x1 = ncx2inv (p, 2, 1);
 x2 = ncx2inv (p, 2, 2);
 x3 = ncx2inv (p, 2, 3);
 x4 = ncx2inv (p, 4, 1);
 x5 = ncx2inv (p, 4, 2);
 x6 = ncx2inv (p, 4, 3);
 plot (p, x1, "-r", p, x2, "-g", p, x3, "-k", ...
       p, x4, "-m", p, x5, "-c", p, x6, "-y")
 grid on
 ylim ([0, 10])
 legend ({"df = 2, λ = 1", "df = 2, λ = 2", ...
          "df = 2, λ = 3", "df = 4, λ = 1", ...
          "df = 4, λ = 2", "df = 4, λ = 3"}, "location", "northwest")
 title ("Noncentral chi-squared iCDF")
 xlabel ("probability")
 ylabel ("values in x")
plotted figure

Example: 2

  

 ## Compare the noncentral chi-squared CDF with LAMBDA = 2 to the
 ## chi-squared CDF with the same number of degrees of freedom (4).

 p = 0.001:0.001:0.999;
 x1 = ncx2inv (p, 4, 2);
 x2 = chi2inv (p, 4);
 plot (p, x1, "-", p, x2, "-");
 grid on
 ylim ([0, 10])
 legend ({"Noncentral χ^2(4,2)", "χ^2(4)"}, "location", "northwest")
 title ("Noncentral chi-squared vs chi-squared quantile functions")
 xlabel ("probability")
 ylabel ("values in x")
plotted figure

statistics-release-1.6.3/docs/ncx2pdf.html000066400000000000000000000170451456127120000205200ustar00rootroot00000000000000 Statistics: ncx2pdf

Function Reference: ncx2pdf

statistics: y = ncx2pdf (x, df, lambda)

Noncentral chi-squared probability distribution function (PDF).

For each element of x, compute the probability density function (PDF) of the noncentral chi-squared distribution with df degrees of freedom and noncentrality parameter lambda. The size of y is the common size of x, df, and lambda. A scalar input functions as a constant matrix of the same size as the other inputs.

Further information about the noncentral chi-squared distribution can be found at https://en.wikipedia.org/wiki/Noncentral_chi-squared_distribution

See also: ncx2cdf, ncx2inv, ncx2rnd, ncx2stat, chi2pdf

Source Code: ncx2pdf

Example: 1

  

 ## Plot various PDFs from the noncentral chi-squared distribution
 x = 0:0.1:10;
 y1 = ncx2pdf (x, 2, 1);
 y2 = ncx2pdf (x, 2, 2);
 y3 = ncx2pdf (x, 2, 3);
 y4 = ncx2pdf (x, 4, 1);
 y5 = ncx2pdf (x, 4, 2);
 y6 = ncx2pdf (x, 4, 3);
 plot (x, y1, "-r", x, y2, "-g", x, y3, "-k", ...
       x, y4, "-m", x, y5, "-c", x, y6, "-y")
 grid on
 xlim ([0, 10])
 ylim ([0, 0.32])
 legend ({"df = 2, λ = 1", "df = 2, λ = 2", ...
          "df = 2, λ = 3", "df = 4, λ = 1", ...
          "df = 4, λ = 2", "df = 4, λ = 3"}, "location", "northeast")
 title ("Noncentral chi-squared PDF")
 xlabel ("values in x")
 ylabel ("density")
plotted figure

Example: 2

  

 ## Compare the noncentral chi-squared PDF with LAMBDA = 2 to the
 ## chi-squared PDF with the same number of degrees of freedom (4).

 x = 0:0.1:10;
 y1 = ncx2pdf (x, 4, 2);
 y2 = chi2pdf (x, 4);
 plot (x, y1, "-", x, y2, "-");
 grid on
 xlim ([0, 10])
 ylim ([0, 0.32])
 legend ({"Noncentral T(10,1)", "T(10)"}, "location", "northwest")
 title ("Noncentral chi-squared vs chi-squared PDFs")
 xlabel ("values in x")
 ylabel ("density")
plotted figure

statistics-release-1.6.3/docs/ncx2rnd.html000066400000000000000000000121631456127120000205260ustar00rootroot00000000000000 Statistics: ncx2rnd

Function Reference: ncx2rnd

statistics: r = ncx2rnd (df, lambda)
statistics: r = ncx2rnd (df, lambda, rows, cols, …)
statistics: r = ncx2rnd (df, lambda, [sz])

Random arrays from the noncentral chi-squared distribution.

r = ncx2rnd (df, lambda) returns an array of random numbers chosen from the noncentral chi-squared distribution with df degrees of freedom and noncentrality parameter lambda. The size of r is the common size of df and lambda. A scalar input functions as a constant matrix of the same size as the other input.

When called with a single size argument, ncx2rnd returns a square matrix with the dimension specified. When called with more than one scalar argument, the first two arguments are taken as the number of rows and columns and any further arguments specify additional matrix dimensions. The size may also be specified with a row vector of dimensions, sz.

Further information about the noncentral chi-squared distribution can be found at https://en.wikipedia.org/wiki/Noncentral_chi-squared_distribution

See also: ncx2cdf, ncx2inv, ncx2pdf, ncx2stat

Source Code: ncx2rnd

statistics-release-1.6.3/docs/ncx2stat.html000066400000000000000000000110411456127120000207100ustar00rootroot00000000000000 Statistics: ncx2stat

Function Reference: ncx2stat

statistics: [m, v] = ncx2stat (df, lambda)

Compute statistics for the noncentral chi-squared distribution.

[m, v] = ncx2stat (df, lambda) returns the mean and variance of the noncentral chi-squared distribution with df degrees of freedom and noncentrality parameter lambda.

The size of m (mean) and v (variance) is the common size of the input arguments. A scalar input functions as a constant matrix of the same size as the other inputs.

Further information about the noncentral chi-squared distribution can be found at https://en.wikipedia.org/wiki/Noncentral_chi-squared_distribution

See also: ncx2cdf, ncx2inv, ncx2pdf, ncx2rnd

Source Code: ncx2stat

statistics-release-1.6.3/docs/normalise_distribution.html000066400000000000000000000134271456127120000237440ustar00rootroot00000000000000 Statistics: normalise_distribution

Function Reference: normalise_distribution

statistics: NORMALISED = normalise_distribution (DATA)
statistics: NORMALISED = normalise_distribution (DATA, DISTRIBUTION)
statistics: NORMALISED = normalise_distribution (DATA, DISTRIBUTION, DIMENSION)

Transform a set of data so as to be N(0,1) distributed according to an idea by van Albada and Robinson.

This is achieved by first passing it through its own cumulative distribution function (CDF) in order to get a uniform distribution, and then mapping the uniform to a normal distribution.

The data must be passed as a vector or matrix in DATA. If the CDF is unknown, then [] can be passed in DISTRIBUTION, and in this case the empirical CDF will be used. Otherwise, if the CDFs for all data are known, they can be passed in DISTRIBUTION, either in the form of a single function name as a string, or a single function handle, or a cell array consisting of either all function names as strings, or all function handles. In the latter case, the number of CDFs passed must match the number of rows, or columns respectively, to normalise. If the data are passed as a matrix, then the transformation will operate either along the first non-singleton dimension, or along DIMENSION if present.

Notes: The empirical CDF will map any two sets of data having the same size and their ties in the same places after sorting to some permutation of the same normalised data:

 
 normalise_distribution([1 2 2 3 4])
 ⇒ -1.28  0.00  0.00  0.52  1.28

 normalise_distribution([1 10 100 10 1000])
 ⇒ -1.28  0.00  0.52  0.00  1.28

Original source: S.J. van Albada, P.A. Robinson "Transformation of arbitrary distributions to the normal distribution with application to EEG test-retest reliability" Journal of Neuroscience Methods, Volume 161, Issue 2, 15 April 2007, Pages 205-211 ISSN 0165-0270, 10.1016/j.jneumeth.2006.11.004. (http://www.sciencedirect.com/science/article/pii/S0165027006005668)

Source Code: normalise_distribution

statistics-release-1.6.3/docs/normcdf.html000066400000000000000000000174221456127120000206030ustar00rootroot00000000000000 Statistics: normcdf

Function Reference: normcdf

statistics: p = normcdf (x)
statistics: p = normcdf (x, mu)
statistics: p = normcdf (x, mu, sigma)
statistics: p = normcdf (…, "upper")
statistics: [p, plo, pup] = normcdf (x, mu, sigma, pcov)
statistics: [p, plo, pup] = normcdf (x, mu, sigma, pcov, alpha)
statistics: [p, plo, pup] = normcdf (…, "upper")

Normal cumulative distribution function (CDF).

For each element of x, compute the cumulative distribution function (CDF) of the normal distribution with mean mu and standard deviation sigma. The size of p is the common size of x, mu and sigma. A scalar input functions as a constant matrix of the same size as the other inputs.

Default values are mu = 0, sigma = 1.

When called with three output arguments, i.e. [p, plo, pup], normcdf computes the confidence bounds for p when the input parameters mu and sigma are estimates. In such case, pcov, a 2×2 matrix containing the covariance matrix of the estimated parameters, is necessary. Optionally, alpha, which has a default value of 0.05, specifies the 100 * (1 - alpha) percent confidence bounds. plo and pup are arrays of the same size as p containing the lower and upper confidence bounds.

[…] = normcdf (…, "upper") computes the upper tail probability of the normal distribution with parameters mu and sigma, at the values in x. This can be used to compute a right-tailed p-value. To compute a two-tailed p-value, use 2 * normcdf (-abs (x), mu, sigma).

Further information about the normal distribution can be found at https://en.wikipedia.org/wiki/Normal_distribution

See also: norminv, normpdf, normrnd, normfit, normlike, normstat

Source Code: normcdf

Example: 1

  

 ## Plot various CDFs from the normal distribution
 x = -5:0.01:5;
 p1 = normcdf (x, 0, 0.5);
 p2 = normcdf (x, 0, 1);
 p3 = normcdf (x, 0, 2);
 p4 = normcdf (x, -2, 0.8);
 plot (x, p1, "-b", x, p2, "-g", x, p3, "-r", x, p4, "-c")
 grid on
 xlim ([-5, 5])
 legend ({"μ = 0, σ = 0.5", "μ = 0, σ = 1", ...
          "μ = 0, σ = 2", "μ = -2, σ = 0.8"}, "location", "southeast")
 title ("Normal CDF")
 xlabel ("values in x")
 ylabel ("probability")
plotted figure

statistics-release-1.6.3/docs/normfit.html000066400000000000000000000247501456127120000206330ustar00rootroot00000000000000 Statistics: normfit

Function Reference: normfit

statistics: muhat = normfit (x)
statistics: [muhat, sigmahat] = normfit (x)
statistics: [muhat, sigmahat, muci] = normfit (x)
statistics: [muhat, sigmahat, muci, sigmaci] = normfit (x)
statistics: […] = normfit (x, alpha)
statistics: […] = normfit (x, alpha, censor)
statistics: […] = normfit (x, alpha, censor, freq)
statistics: […] = normfit (x, alpha, censor, freq, options)

Estimate parameters and confidence intervals for the normal distribution.

[muhat, sigmahat] = normfit (x) estimates the parameters of the normal distribution given the data in x. muhat is an estimate of the mean, and sigmahat is an estimate of the standard deviation.

[muhat, sigmahat, muci, sigmaci] = normfit (x) returns the 95% confidence intervals for the mean and standard deviation estimates in the arrays muci and sigmaci, respectively.

  • x can be a vector or a matrix. When x is a matrix, the parameter estimates and their confidence intervals are computed for each column. In this case, normfit supports only 2 input arguments, x and alpha. Optional arguments censor, freq, and options can be used only when x is a vector.
  • alpha is a scalar value in the range (0,1) specifying the confidence level for the confidence intervals calculated as 100×(1 – alpha)%. By default, the optional argument alpha is 0.05 corresponding to 95% confidence intervals. Pass in [] for alpha to use the default values.
  • censor is a logical vector of the same length as x specifying whether each value in x is right-censored or not. 1 indicates observations that are right-censored and 0 indicates observations that are fully observed. With censoring, muhat and sigmahat are the maximum likelihood estimates (MLEs). If empty, the default is an array of 0s, meaning that all observations are fully observed.
  • freq is a vector of the same length as x and it typically contains non-negative integer counts of the corresponding elements in x. If empty, the default is an array of 1s, meaning one observation per element of x. To obtain the weighted MLEs for a data set with censoring, specify weights of observations, normalized to the number of observations in x. However, when there is no censored data (default), the returned estimate for standard deviation is not exactly the WMLE. To compute the weighted MLE, multiply the value returned in sigmahat by (SUM (freq) - 1) / SUM (freq). This correction is needed because normfit normally computes sigmahat using an unbiased variance estimator when there is no censored data. When there is censoring in the data, the correction is not needed, since normfit does not use the unbiased variance estimator in that case.
  • options is a structure with the control parameters for fminsearch which is used internally to compute MLEs for censored data. By default, it uses the following options:
    • options.Display = "off"
    • options.MaxFunEvals = 400
    • options.MaxIter = 200
    • options.TolX = 1e-6

See also: normcdf, norminv, normpdf, normrnd, normlike, normstat

Source Code: normfit

Example: 1

  

 ## Sample 3 populations from 3 different normal distibutions
 randn ("seed", 1);    # for reproducibility
 r1 = normrnd (2, 5, 5000, 1);
 randn ("seed", 2);    # for reproducibility
 r2 = normrnd (5, 2, 5000, 1);
 randn ("seed", 3);    # for reproducibility
 r3 = normrnd (9, 4, 5000, 1);
 r = [r1, r2, r3];

 ## Plot them normalized and fix their colors
 hist (r, 15, 0.4);
 h = findobj (gca, "Type", "patch");
 set (h(1), "facecolor", "c");
 set (h(2), "facecolor", "g");
 set (h(3), "facecolor", "r");
 hold on

 ## Estimate their mu and sigma parameters
 [muhat, sigmahat] = normfit (r);

 ## Plot their estimated PDFs
 x = [min(r(:)):max(r(:))];
 y = normpdf (x, muhat(1), sigmahat(1));
 plot (x, y, "-pr");
 y = normpdf (x, muhat(2), sigmahat(2));
 plot (x, y, "-sg");
 y = normpdf (x, muhat(3), sigmahat(3));
 plot (x, y, "-^c");
 ylim ([0, 0.5])
 xlim ([-20, 20])
 hold off
 legend ({"Normalized HIST of sample 1 with mu=2, σ=5", ...
          "Normalized HIST of sample 2 with mu=5, σ=2", ...
          "Normalized HIST of sample 3 with mu=9, σ=4", ...
          sprintf("PDF for sample 1 with estimated mu=%0.2f and σ=%0.2f", ...
                  muhat(1), sigmahat(1)), ...
          sprintf("PDF for sample 2 with estimated mu=%0.2f and σ=%0.2f", ...
                  muhat(2), sigmahat(2)), ...
          sprintf("PDF for sample 3 with estimated mu=%0.2f and σ=%0.2f", ...
                  muhat(3), sigmahat(3))}, "location", "northwest")
 title ("Three population samples from different normal distibutions")
 hold off
plotted figure

statistics-release-1.6.3/docs/norminv.html000066400000000000000000000146441456127120000206460ustar00rootroot00000000000000 Statistics: norminv

Function Reference: norminv

statistics: x = norminv (p)
statistics: x = norminv (p, mu)
statistics: x = norminv (p, mu, sigma)

Inverse of the normal cumulative distribution function (iCDF).

For each element of p, compute the quantile (the inverse of the CDF) of the normal distribution with mean mu and standard deviation sigma. The size of p is the common size of p, mu and sigma. A scalar input functions as a constant matrix of the same size as the other inputs.

Default values are mu = 0, sigma = 1.

The default values correspond to the standard normal distribution and computing its quantile function is also possible with the probit function, which is faster but it does not perform any input validation.

Further information about the normal distribution can be found at https://en.wikipedia.org/wiki/Normal_distribution

See also: norminv, normpdf, normrnd, normfit, normlike, normstat, probit

Source Code: norminv

Example: 1

  

 ## Plot various iCDFs from the normal distribution
 p = 0.001:0.001:0.999;
 x1 = norminv (p, 0, 0.5);
 x2 = norminv (p, 0, 1);
 x3 = norminv (p, 0, 2);
 x4 = norminv (p, -2, 0.8);
 plot (p, x1, "-b", p, x2, "-g", p, x3, "-r", p, x4, "-c")
 grid on
 ylim ([-5, 5])
 legend ({"μ = 0, σ = 0.5", "μ = 0, σ = 1", ...
          "μ = 0, σ = 2", "μ = -2, σ = 0.8"}, "location", "northwest")
 title ("Normal iCDF")
 xlabel ("probability")
 ylabel ("values in x")
plotted figure

statistics-release-1.6.3/docs/normlike.html000066400000000000000000000132151456127120000207670ustar00rootroot00000000000000 Statistics: normlike

Function Reference: normlike

statistics: nlogL = normlike (params, x)
statistics: [nlogL, avar] = normlike (params, x)
statistics: […] = normlike (params, x, censor)
statistics: […] = normlike (params, x, censor, freq)

Negative log-likelihood for the normal distribution.

nlogL = normlike (params, x) returns the negative log-likelihood for the normal distribution, evaluated at parameters params(1) = mean and params(2) = standard deviation, given x. nlogL is a scalar.

[nlogL, avar] = normlike (params, x) returns the inverse of Fisher’s information matrix, avar. If the input parameter values in params are the maximum likelihood estimates, the diagonal elements of avar are their asymptotic variances. avar is based on the observed Fisher’s information, not the expected information.

[…] = normlike (params, x, censor) accepts a boolean vector of the same size as x that is 1 for observations that are right-censored and 0 for observations that are observed exactly.

[…] = normlike (params, x, censor, freq) accepts a frequency vector of the same size as x. freq typically contains integer frequencies for the corresponding elements in x, but it may contain any non-integer non-negative values. Pass in [] for censor to use its default value.

See also: normcdf, norminv, normpdf, normrnd, normfit, normstat

Source Code: normlike

statistics-release-1.6.3/docs/normpdf.html000066400000000000000000000142121456127120000206120ustar00rootroot00000000000000 Statistics: normpdf

Function Reference: normpdf

statistics: y = normpdf (x)
statistics: y = normpdf (x, mu)
statistics: y = normpdf (x, mu, sigma)

Normal probability density function (PDF).

For each element of x, compute the probability density function (PDF) of the normal distribution with mean mu and standard deviation sigma. The size of y is the common size of p, mu and sigma. A scalar input functions as a constant matrix of the same size as the other inputs.

Default values are mu = 0, sigma = 1.

Further information about the normal distribution can be found at https://en.wikipedia.org/wiki/Normal_distribution

See also: norminv, norminv, normrnd, normfit, normlike, normstat

Source Code: normpdf

Example: 1

  

 ## Plot various PDFs from the normal distribution
 x = -5:0.01:5;
 y1 = normpdf (x, 0, 0.5);
 y2 = normpdf (x, 0, 1);
 y3 = normpdf (x, 0, 2);
 y4 = normpdf (x, -2, 0.8);
 plot (x, y1, "-b", x, y2, "-g", x, y3, "-r", x, y4, "-c")
 grid on
 xlim ([-5, 5])
 ylim ([0, 0.9])
 legend ({"μ = 0, σ = 0.5", "μ = 0, σ = 1", ...
          "μ = 0, σ = 2", "μ = -2, σ = 0.8"}, "location", "northeast")
 title ("Normal PDF")
 xlabel ("values in x")
 ylabel ("density")
plotted figure

statistics-release-1.6.3/docs/normplot.html000066400000000000000000000171741456127120000210310ustar00rootroot00000000000000 Statistics: normplot

Function Reference: normplot

Function File: normplot (x)
Function File: normplot (ax, x)
Function File: h = normplot (…)

Produce normal probability plot of the data in x. If x is a matrix, normplot plots the data for each column. NaN values are ignored.

h = normplot (ax, x) takes a handle ax in addition to the data in x and it uses that axes for ploting. You may get this handle of an existing plot with gca/.

The line joing the 1st and 3rd quantile is drawn solid whereas its extensions to both ends are dotted. If the underlying distribution is normal, the points will cluster around the solid part of the line. Other distribution types will introduce curvature in the plot.

See also: cdfplot, wblplot

Source Code: normplot

Example: 1

  

 h = normplot([1:20]);
plotted figure

Example: 2

  

 h = normplot([1:20;5:2:44]');
plotted figure

Example: 3

  

 ax = newplot();
 h = normplot(ax, [1:20]);
 ax = gca;
 h = normplot(ax, [-10:10]);
 set (ax, "xlim", [-11, 21]);
plotted figure

statistics-release-1.6.3/docs/normrnd.html000066400000000000000000000124651456127120000206340ustar00rootroot00000000000000 Statistics: normrnd

Function Reference: normrnd

statistics: r = normrnd (mu, sigma)
statistics: r = normrnd (mu, sigma, rows)
statistics: r = normrnd (mu, sigma, rows, cols, …)
statistics: r = normrnd (mu, sigma, [sz])

Random arrays from the normal distribution.

r = normrnd (mu, sigma) returns an array of random numbers chosen from the normal distribution with mean mu and standard deviation sigma. The size of r is the common size of mu and sigma. A scalar input functions as a constant matrix of the same size as the other inputs. Both parameters must be finite real numbers and sigma > 0, otherwise NaN is returned.

When called with a single size argument, normrnd returns a square matrix with the dimension specified. When called with more than one scalar argument, the first two arguments are taken as the number of rows and columns and any further arguments specify additional matrix dimensions. The size may also be specified with a row vector of dimensions, sz.

Further information about the normal distribution can be found at https://en.wikipedia.org/wiki/Normal_distribution

See also: norminv, norminv, normpdf, normfit, normlike, normstat

Source Code: normrnd

statistics-release-1.6.3/docs/normstat.html000066400000000000000000000110461456127120000210160ustar00rootroot00000000000000 Statistics: normstat

Function Reference: normstat

statistics: [m, v] = normstat (mu, sigma)

Compute statistics of the normal distribution.

[m, v] = normstat (mu, sigma) returns the mean and variance of the normal distribution with non-centrality (distance) parameter mu and scale parameter sigma.

The size of m (mean) and v (variance) is the common size of the input arguments. A scalar input functions as a constant matrix of the same size as the other inputs.

Further information about the normal distribution can be found at https://en.wikipedia.org/wiki/Normal_distribution

See also: norminv, norminv, normpdf, normrnd, normfit, normlike

Source Code: normstat

statistics-release-1.6.3/docs/optimalleaforder.html000066400000000000000000000142521456127120000225020ustar00rootroot00000000000000 Statistics: optimalleaforder

Function Reference: optimalleaforder

statistics: leafOrder = optimalleaforder (tree, D)
statistics: leafOrder = optimalleaforder (…, Name, Value)

Compute the optimal leaf ordering of a hierarchical binary cluster tree.

The optimal leaf ordering of a tree is the ordering which minimizes the sum of the distances between each leaf and its adjacent leaves, without altering the structure of the tree, that is without redefining the clusters of the tree.

Required inputs:

  • tree: a hierarchical cluster tree tree generated by the linkage function.
  • D: a matrix of distances as computed by pdist.

Optional inputs can be the following property/value pairs:

  • property ’Criteria’ at the moment can only have the value ’adjacent’, for minimizing the distances between leaves.
  • property ’Transformation’ can have one of the values ’linear’, ’inverse’ or a handle to a custom function which computes S the similarity matrix.

optimalleaforder’s output leafOrder is the optimal leaf ordering.

Reference Bar-Joseph, Z., Gifford, D.K., and Jaakkola, T.S. Fast optimal leaf ordering for hierarchical clustering. Bioinformatics vol. 17 suppl. 1, 2001.

See also: dendrogram, linkage, pdist

Source Code: optimalleaforder

Example: 1

  

 randn ("seed", 5)  # for reproducibility
 X = randn (10, 2);
 D = pdist (X);
 tree = linkage(D, 'average');
 optimalleaforder (tree, D, 'Transformation', 'linear')

ans =

   10    8    3    5    9    7    2    4    6    1
statistics-release-1.6.3/docs/pca.html000066400000000000000000000176161456127120000177230ustar00rootroot00000000000000 Statistics: pca

Function Reference: pca

statistics: coeff = pca (x)
statistics: coeff = pca (x, Name, Value)
statistics: [coeff, score, latent] = pca (…)
statistics: [coeff, score, latent, tsquared] = pca (…)
statistics: [coeff, score, latent, tsquared, explained, mu] = pca (…)

Performs a principal component analysis on a data matrix.

A principal component analysis of a data matrix of N observations in a D dimensional space returns a D×D transformation matrix, to perform a change of basis on the data. The first component of the new basis is the direction that maximizes the variance of the projected data.

Input argument:

  • x : a N×D data matrix

The following Name, Value pair arguments can be used:

  • "Algorithm" defines the algorithm to use:
    • "svd" (default), for singular value decomposition
    • "eig" for eigenvalue decomposition
  • "Centered" is a boolean indicator for centering the observation data. It is true by default.
  • "Economy" is a boolean indicator for the economy size output. It is true by default. Hence, pca returns only the elements of latent that are not necessarily zero, and the corresponding columns of coeff and score, that is, when N <= D, only the first N - 1.
  • "NumComponents" defines the number of components k to return. If k < p, then only the first k columns of coeff and score are returned.
  • "Rows" defines how to handle missing values:
    • "complete" (default), missing values are removed before computation.
    • "pairwise" (only valid when "Algorithm" is "eig"), the covariance of rows with missing data is computed using the available data, but the covariance matrix could be not positive definite, which triggers the termination of pca.
    • "complete", missing values are not allowed, pca terminates with an error if there are any.
  • "Weights" defines observation weights as a vector of positive values of length N.
  • "VariableWeights" defines variable weights:
    • a vector of positive values of length D.
    • the string "variance" to use the sample variance as weights.

Return values:

  • coeff : the principal component coefficients, a D×D transformation matrix
  • score : the principal component scores, the representation of x in the principal component space
  • latent : the principal component variances, i.e., the eigenvalues of the covariance matrix of x
  • tsquared : Hotelling’s T-squared Statistic for each observation in x
  • explained : the percentage of the variance explained by each principal component
  • mu : the estimated mean of each variable of x, it is zero if the data are not centered

Matlab compatibility note: the alternating least square method ’als’ and associated options ’Coeff0’, ’Score0’, and ’Options’ are not yet implemented

References

  1. Jolliffe, I. T., Principal Component Analysis, 2nd Edition, Springer, 2002

See also: barttest, factoran, pcacov, pcares

Source Code: pca

statistics-release-1.6.3/docs/pcacov.html000066400000000000000000000160111456127120000204170ustar00rootroot00000000000000 Statistics: pcacov

Function Reference: pcacov

statistics: coeff = pcacov (K)
statistics: [coeff, latent] = pcacov (K)
statistics: [coeff, latent, explained] = pcacov (K)

Perform principal component analysis on covariance matrix

coeff = pcacov (K) performs principal component analysis on the square covariance matrix K and returns the principal component coefficients, also known as loadings. The columns are in order of decreasing component variance.

[coeff, latent] = pcacov (K) also returns a vector with the principal component variances, i.e. the eigenvalues of K. latent has a length of size (coeff, 1).

[coeff, latent, explained] = pcacov (K) also returns a vector with the percentage of the total variance explained by each principal component. explained has the same size as latent. The entries in explained range from 0 (none of the variance is explained) to 100 (all of the variance is explained).

pcacov does not standardize K to have unit variances. In order to perform principal component analysis on standardized variables, use the correlation matrix R = K ./ (SD * SD'), where SD = sqrt (diag (K)), in place of K. To perform principal component analysis directly on the data matrix, use pca.

References

  1. Jolliffe, I. T., Principal Component Analysis, 2nd Edition, Springer, 2002

See also: bartlett, factoran, pcares, pca

Source Code: pcacov

Example: 1

  

 x = [ 7    26     6    60;
       1    29    15    52;
      11    56     8    20;
      11    31     8    47;
       7    52     6    33;
      11    55     9    22;
       3    71    17     6;
       1    31    22    44;
       2    54    18    22;
      21    47     4    26;
       1    40    23    34;
      11    66     9    12;
      10    68     8    12
     ];
 Kxx = cov (x);
 [coeff, latent, explained] = pcacov (Kxx)

coeff =

  -0.067800  -0.646018   0.567315   0.506180
  -0.678516  -0.019993  -0.543969   0.493268
   0.029021   0.755310   0.403553   0.515567
   0.730874  -0.108480  -0.468398   0.484416

latent =

   517.7969
    67.4964
    12.4054
     0.2372

explained =

   8.6597e+01
   1.1288e+01
   2.0747e+00
   3.9662e-02
statistics-release-1.6.3/docs/pcares.html000066400000000000000000000152431456127120000204270ustar00rootroot00000000000000 Statistics: pcares

Function Reference: pcares

statistics: residuals = pcares (x, ndim)
statistics: [residuals, reconstructed] = pcares (x, ndim)

Calculate residuals from principal component analysis.

residuals = pcares (x, ndim) returns the residuals obtained by retaining ndim principal components of the N×D matrix x. Rows of x correspond to observations, columns of x correspond to variables. ndim is a scalar and must be less than or equal to D. residuals is a matrix of the same size as x. Use the data matrix, not the covariance matrix, with this function.

[residuals, reconstructed] = pcares (x, ndim) returns the reconstructed observations, i.e. the approximation to x obtained by retaining its first ndim principal components.

pcares does not normalize the columns of x. Use pcares (zscore (x), ndim) in order to perform the principal components analysis based on standardized variables, i.e. based on correlations. Use pcacov in order to perform principal components analysis directly on a covariance or correlation matrix without constructing residuals.

References

  1. Jolliffe, I. T., Principal Component Analysis, 2nd Edition, Springer, 2002

See also: factoran, pcacov, pca

Source Code: pcares

Example: 1

  

 x = [ 7    26     6    60;
       1    29    15    52;
      11    56     8    20;
      11    31     8    47;
       7    52     6    33;
      11    55     9    22;
       3    71    17     6;
       1    31    22    44;
       2    54    18    22;
      21    47     4    26;
       1    40    23    34;
      11    66     9    12;
      10    68     8    12];

 ## As we increase the number of principal components, the norm
 ## of the residuals matrix will decrease
 r1 = pcares (x,1);
 n1 = norm (r1)
 r2 = pcares (x,2);
 n2 = norm (r2)
 r3 = pcares (x,3);
 n3 = norm (r3)
 r4 = pcares (x,4);
 n4 = norm (r4)

n1 = 28.460
n2 = 12.201
n3 = 1.6870
n4 = 4.2168e-14
statistics-release-1.6.3/docs/pdf.html000066400000000000000000000325571456127120000177320ustar00rootroot00000000000000 Statistics: pdf

Function Reference: pdf

statistics: y = pdf (name, x, A)
statistics: y = pdf (name, x, A, B)
statistics: y = pdf (name, x, A, B, C)

Return the PDF of a univariate distribution evaluated at x.

pdf is a wrapper for the univariate cumulative distribution functions available in the statistics package. See the corresponding functions’ help to learn the signification of the parameters after x.

y = pdf (name, x, A) returns the CDF for the one-parameter distribution family specified by name and the distribution parameter A, evaluated at the values in x.

y = pdf (name, x, A, B) returns the CDF for the two-parameter distribution family specified by name and the distribution parameters A and B, evaluated at the values in x.

y = pdf (name, x, A, B, C) returns the CDF for the three-parameter distribution family specified by name and the distribution parameters A, B, and C, evaluated at the values in x.

name must be a char string of the name or the abbreviation of the desired cumulative distribution function as listed in the followng table. The last column shows the number of required parameters that should be parsed after x to the desired PDF.

Distribution NameAbbreviationInput Parameters
"Beta""beta"2
"Binomial""bino"2
"Birnbaum-Saunders""bisa"2
"Burr""burr"3
"Cauchy""cauchy"2
"Chi-squared""chi2"1
"Extreme Value""ev"2
"Exponential""exp"1
"F-Distribution""f"2
"Gamma""gam"2
"Geometric""geo"1
"Generalized Extreme Value""gev"3
"Generalized Pareto""gp"3
"Gumbel""gumbel"2
"Half-normal""hn"2
"Hypergeometric""hyge"3
"Inverse Gaussian""invg"2
"Laplace""laplace"2
"Logistic""logi"2
"Log-Logistic""logl"2
"Lognormal""logn"2
"Nakagami""naka"2
"Negative Binomial""nbin"2
"Noncentral F-Distribution""ncf"3
"Noncentral Student T""nct"2
"Noncentral Chi-Squared""ncx2"2
"Normal""norm"2
"Poisson""poiss"1
"Rayleigh""rayl"1
"Rician""rice"2
"Student T""t"1
"location-scale T""tls"3
"Triangular""tri"3
"Discrete Uniform""unid"1
"Uniform""unif"2
"Von Mises""vm"2
"Weibull""wbl"2

See also: cdf, icdf, random, betapdf, binopdf, bisapdf, burrpdf, cauchypdf, chi2pdf, evpdf, exppdf, fpdf, gampdf, geopdf, gevpdf, gppdf, gumbelpdf, hnpdf, hygepdf, invgpdf, laplacepdf, logipdf, loglpdf, lognpdf, nakapdf, nbinpdf, ncfpdf, nctpdf, ncx2pdf, normpdf, poisspdf, raylpdf, ricepdf, tpdf, tripdf, unidpdf, unifpdf, vmpdf, wblpdf

Source Code: pdf

statistics-release-1.6.3/docs/pdist.html000066400000000000000000000207011456127120000202700ustar00rootroot00000000000000 Statistics: pdist

Function Reference: pdist

statistics: D = pdist (X)
statistics: D = pdist (X, Distance)
statistics: D = pdist (X, Distance, DistParameter)

Return the distance between any two rows in X.

D = pdist (X calculates the euclidean distance between pairs of observations in X. X must be an M×P numeric matrix representing M points in P-dimensional space. This function computes the pairwise distances returned in D as an M×(M-1)/P row vector. Use Z = squareform (D) to convert the row vector D into a an M×M symmetric matrix Z, where Z(i,j) corresponds to the pairwise distance between points i and j.

D = pdist (X, Y, Distance) returns the distance between pairs of observations in X using the metric specified by Distance, which can be any of the following options.

"euclidean"Euclidean distance.
"squaredeuclidean"Squared Euclidean distance.
"seuclidean"standardized Euclidean distance. Each coordinate difference between the rows in X and the query matrix Y is scaled by dividing by the corresponding element of the standard deviation computed from X. A different scaling vector can be specified with the subsequent DistParameter input argument.
"mahalanobis"Mahalanobis distance, computed using a positive definite covariance matrix. A different covariance matrix can be specified with the subsequent DistParameter input argument.
"cityblock"City block distance.
"minkowski"Minkowski distance. The default exponent is 2. A different exponent can be specified with the subsequent DistParameter input argument.
"chebychev"Chebychev distance (maximum coordinate difference).
"cosine"One minus the cosine of the included angle between points (treated as vectors).
"correlation"One minus the sample linear correlation between observations (treated as sequences of values).
"hamming"Hamming distance, which is the percentage of coordinates that differ.
"jaccard"One minus the Jaccard coefficient, which is the percentage of nonzero coordinates that differ.
"spearman"One minus the sample Spearman’s rank correlation between observations (treated as sequences of values).
@distfunCustom distance function handle. A distance function of the form function D2 = distfun (XI, YI), where XI is a 1×P vector containing a single observation in P-dimensional space, YI is an N×P matrix containing an arbitrary number of observations in the same P-dimensional space, and D2 is an N×P vector of distances, where (D2k) is the distance between observations XI and (YIk,:).

D = pdist (X, Y, Distance, DistParameter) returns the distance using the metric specified by Distance and DistParameter. The latter one can only be specified when the selected Distance is "seuclidean", "minkowski", and "mahalanobis".

See also: pdist2, squareform, linkage

Source Code: pdist

statistics-release-1.6.3/docs/pdist2.html000066400000000000000000000240421456127120000203540ustar00rootroot00000000000000 Statistics: pdist2

Function Reference: pdist2

statistics: D = pdist2 (X, Y)
statistics: D = pdist2 (X, Y, Distance)
statistics: D = pdist2 (X, Y, Distance, DistParameter)
statistics: D = pdist2 (…, Name, Value)
statistics: [D, I] = pdist2 (…, Name, Value)

Compute pairwise distance between two sets of vectors.

D = pdist2 (X, Y) calculates the euclidean distance between each pair of observations in X and Y. Let X be an M×P matrix representing M points in P-dimensional space and Y be an N×P matrix representing another set of points in the same space. This function computes the M×N distance matrix D, where D(i,j) is the distance between X(i,:) and Y(j,:).

D = pdist2 (X, Y, Distance) returns the distance between each pair of observations in X and Y using the metric specified by Distance, which can be any of the following options.

"euclidean"Euclidean distance.
"squaredeuclidean"Squared Euclidean distance.
"seuclidean"standardized Euclidean distance. Each coordinate difference between the rows in X and the query matrix Y is scaled by dividing by the corresponding element of the standard deviation computed from X. A different scaling vector can be specified with the subsequent DistParameter input argument.
"mahalanobis"Mahalanobis distance, computed using a positive definite covariance matrix. A different covariance matrix can be specified with the subsequent DistParameter input argument.
"cityblock"City block distance.
"minkowski"Minkowski distance. The default exponent is 2. A different exponent can be specified with the subsequent DistParameter input argument.
"chebychev"Chebychev distance (maximum coordinate difference).
"cosine"One minus the cosine of the included angle between points (treated as vectors).
"correlation"One minus the sample linear correlation between observations (treated as sequences of values).
"hamming"Hamming distance, which is the percentage of coordinates that differ.
"jaccard"One minus the Jaccard coefficient, which is the percentage of nonzero coordinates that differ.
"spearman"One minus the sample Spearman’s rank correlation between observations (treated as sequences of values).
@distfunCustom distance function handle. A distance function of the form function D2 = distfun (XI, YI), where XI is a 1×P vector containing a single observation in P-dimensional space, YI is an N×P matrix containing an arbitrary number of observations in the same P-dimensional space, and D2 is an N×P vector of distances, where (D2k) is the distance between observations XI and (YIk,:).

D = pdist2 (X, Y, Distance, DistParameter) returns the distance using the metric specified by Distance and DistParameter. The latter one can only be specified when the selected Distance is "seuclidean", "minkowski", and "mahalanobis".

D = pdist2 (…, Name, Value) for any previous arguments, modifies the computation using Name-Value parameters.

  • D = pdist2 (X, Y, Distance, "Smallest", K) computes the distance using the metric specified by Distance and returns the K smallest pairwise distances to observations in X for each observation in Y in ascending order.
  • D = pdist2 (X, Y, Distance, DistParameter, "Largest", K) computes the distance using the metric specified by Distance and DistParameter and returns the K largest pairwise distances in descending order.

[D, I] = pdist2 (…, Name, Value) also returns the matrix I, which contains the indices of the observations in X corresponding to the distances in D. You must specify either "Smallest" or "Largest" as an optional Name-Value pair pair argument to compute the second output argument.

See also: pdist, knnsearch, rangesearch

Source Code: pdist2

statistics-release-1.6.3/docs/plsregress.html000066400000000000000000000351321456127120000213420ustar00rootroot00000000000000 Statistics: plsregress

Function Reference: plsregress

statistics: [xload, yload] = plsregress (X, Y)
statistics: [xload, yload] = plsregress (X, Y, NCOMP)
statistics: [xload, yload, xscore, yscore, coef, pctVar, mse, stats] = plsregress (X, Y, NCOMP)
statistics: [xload, yload, xscore, yscore, coef, pctVar, mse, stats] = plsregress (…, Name, Value)

Calculate partial least squares regression using SIMPLS algorithm.

plsregress uses the SIMPLS algorithm, and first centers X and Y by subtracting off column means to get centered variables. However, it does not rescale the columns. To perform partial least squares regression with standardized variables, use zscore to normalize X and Y.

[xload, yload] = plsregress (X, Y) computes a partial least squares regression of Y on X, using NCOMP PLS components, which by default are calculated as min (size (X, 1) - 1, size(X, 2)), and returns the the predictor and response loadings in xload and yload, respectively.

  • X is an N×P matrix of predictor variables, with rows corresponding to observations, and columns corresponding to variables.
  • Y is an N×M response matrix.
  • xload is a P×NCOMP matrix of predictor loadings, where each row of xload contains coefficients that define a linear combination of PLS components that approximate the original predictor variables.
  • yload is an M×NCOMP matrix of response loadings, where each row of yload contains coefficients that define a linear combination of PLS components that approximate the original response variables.

[xload, yload] = plsregress (X, Y, NCOMP) defines the desired number of PLS components to use in the regression. NCOMP, a scalar positive integer, must not exceed the default calculated value.

[xload, yload, xscore, yscore, coef, pctVar, mse, stats] = plsregress (X, Y, NCOMP) also returns the following arguments:

  • xscore is an N×NCOMP orthonormal matrix with the predictor scores, i.e., the PLS components that are linear combinations of the variables in X, with rows corresponding to observations and columns corresponding to components.
  • yscore is an N×NCOMP orthonormal matrix with the response scores, i.e., the linear combinations of the responses with which the PLS components xscore have maximum covariance, with rows corresponding to observations and columns corresponding to components.
  • coef is a (P+1)×M matrix with the PLS regression coefficients, containing the intercepts in the first row.
  • pctVar is a 2×NCOMP matrix containing the percentage of the variance explained by the model with the first row containing the percentage of exlpained varianced in X by each PLS component and the second row containing the percentage of explained variance in Y.
  • mse is a 2×(NCOMP+1) matrix containing the estimated mean squared errors for PLS models with 0:NCOMP components with the first row containing the squared errors for the predictor variables in X and the second row containing the mean squared errors for the response variable(s) in Y.
  • stats is a structure with the following fields:
    • stats.W is a P×NCOMP matrix of PLS weights.
    • stats.T2 is the T^2 statistics for each point in xscore.
    • stats.Xresiduals is an N×P matrix with the predictor residuals.
    • stats.Yresiduals is an N×M matrix with the response residuals.

[…] = plsregress (…, Name, Value, …) specifies one or more of the following Name/Value pairs:

NameValue
"CV"The method used to compute mse. When Value is a positive integer K, plsregress uses K-fold cross-validation. Set Value to a cross-validation partition, created using cvpartition, to use other forms of cross-validation. Set Value to "resubstitution" to use both X and Y to fit the model and to estimate the mean squared errors, without cross-validation. By default, Value = "resubstitution".
"MCReps"A positive integer indicating the number of Monte-Carlo repetitions for cross-validation. By default, Value = 1. A different "MCReps" value is only meaningful when using the "HoldOut" method for cross-validation, previously set by a cvpartition object. If no cross-validation method is used, then "MCReps" must be 1.

Further information about the PLS regression can be found at https://en.wikipedia.org/wiki/Partial_least_squares_regression

References

  1. SIMPLS: An alternative approach to partial least squares regression. Chemometrics and Intelligent Laboratory Systems (1993)

Source Code: plsregress

Example: 1

  

 ## Perform Partial Least-Squares Regression

 ## Load the spectra data set and use the near infrared (NIR) spectral
 ## intensities (NIR) as the predictor and the corresponding octave
 ## ratings (octave) as the response.
 load spectra

 ## Perform PLS regression with 10 components
 [xload, yload, xscore, yscore, coef, ptcVar] = plsregress (NIR, octane, 10);

 ## Plot the percentage of explained variance in the response variable
 ## (PCTVAR) as a function of the number of components.
 plot (1:10, cumsum (100 * ptcVar(2,:)), "-ro");
 xlim ([1, 10]);
 xlabel ("Number of PLS components");
 ylabel ("Percentage of Explained Variance in octane");
 title ("Explained Variance per PLS components");

 ## Compute the fitted response and display the residuals.
 octane_fitted = [ones(size(NIR,1),1), NIR] * coef;
 residuals = octane - octane_fitted;
 figure
 stem (residuals, "color", "r", "markersize", 4, "markeredgecolor", "r")
 xlabel ("Observations");
 ylabel ("Residuals");
 title ("Residuals in octane's fitted responce");
plotted figure

plotted figure

Example: 2

  

 ## Calculate Variable Importance in Projection (VIP) for PLS Regression

 ## Load the spectra data set and use the near infrared (NIR) spectral
 ## intensities (NIR) as the predictor and the corresponding octave
 ## ratings (octave) as the response.  Variables with a VIP score greater than
 ## 1 are considered important for the projection of the PLS regression model.
 load spectra

 ## Perform PLS regression with 10 components
 [xload, yload, xscore, yscore, coef, pctVar, mse, stats] = ...
                                                 plsregress (NIR, octane, 10);

 ## Calculate the normalized PLS weights
 W0 = stats.W ./ sqrt(sum(stats.W.^2,1));

 ## Calculate the VIP scores for 10 components
 nobs = size (xload, 1);
 SS = sum (xscore .^ 2, 1) .* sum (yload .^ 2, 1);
 VIPscore = sqrt (nobs * sum (SS .* (W0 .^ 2), 2) ./ sum (SS, 2));

 ## Find variables with a VIP score greater than or equal to 1
 VIPidx = find (VIPscore >= 1);

 ## Plot the VIP scores
 scatter (1:length (VIPscore), VIPscore, "xb");
 hold on
 scatter (VIPidx, VIPscore (VIPidx), "xr");
 plot ([1, length(VIPscore)], [1, 1], "--k");
 hold off
 axis ("tight");
 xlabel ("Predictor Variables");
 ylabel ("VIP scores");
 title ("VIP scores for each predictror variable with 10 components");
plotted figure

statistics-release-1.6.3/docs/poisscdf.html000066400000000000000000000141771456127120000207710ustar00rootroot00000000000000 Statistics: poisscdf

Function Reference: poisscdf

statistics: p = poisscdf (x, lambda)
statistics: p = poisscdf (x, lambda, "upper")

Poisson cumulative distribution function (CDF).

For each element of x, compute the cumulative distribution function (CDF) of the Poisson distribution with rate parameter lambda. The size of p is the common size of x and lambda. A scalar input functions as a constant matrix of the same size as the other inputs.

p = poisscdf (x, lambda, "upper") computes the upper tail probability of the Poisson distribution with parameter lambda, at the values in x.

Further information about the Poisson distribution can be found at https://en.wikipedia.org/wiki/Poisson_distribution

See also: poissinv, poisspdf, poissrnd, poissfit, poisslike, poisstat

Source Code: poisscdf

Example: 1

  

 ## Plot various CDFs from the Poisson distribution
 x = 0:20;
 p1 = poisscdf (x, 1);
 p2 = poisscdf (x, 4);
 p3 = poisscdf (x, 10);
 plot (x, p1, "*b", x, p2, "*g", x, p3, "*r")
 grid on
 ylim ([0, 1])
 legend ({"λ = 1", "λ = 4", "λ = 10"}, "location", "southeast")
 title ("Poisson CDF")
 xlabel ("values in x (number of occurences)")
 ylabel ("probability")
plotted figure

statistics-release-1.6.3/docs/poissfit.html000066400000000000000000000177311456127120000210160ustar00rootroot00000000000000 Statistics: poissfit

Function Reference: poissfit

statistics: lambdahat = poissfit (x)
statistics: [lambdahat, lambdaci] = poissfit (x)
statistics: [lambdahat, lambdaci] = poissfit (x, alpha)
statistics: [lambdahat, lambdaci] = poissfit (x, alpha, freq)

Estimate parameter and confidence intervals for the Poisson distribution.

lambdahat = poissfit (x) returns the maximum likelihood estimate of the rate parameter, lambda, of the Poisson distribution given the data in x. x must be a vector of non-negative values.

[lambdahat, lambdaci] = poissfit (x) returns the 95% confidence intervals for the parameter estimate.

[lambdahat, lambdaci] = poissfit (x, alpha) also returns the 100 * (1 - alpha) percent confidence intervals of the estimated parameter. By default, the optional argument alpha is 0.05 corresponding to 95% confidence intervals. Pass in [] for alpha to use the default values.

[…] = poissfit (x, alpha, freq) accepts a frequency vector or matrix, freq, of the same size as x. freq typically contains integer frequencies for the corresponding elements in x. freq cannot contain negative values.

Further information about the Poisson distribution can be found at https://en.wikipedia.org/wiki/Poisson_distribution

See also: poisscdf, poissinv, poisspdf, poissrnd, poisslike, poisstat

Source Code: poissfit

Example: 1

  

 ## Sample 3 populations from 3 different Poisson distibutions
 randp ("seed", 2);    # for reproducibility
 r1 = poissrnd (1, 1000, 1);
 randp ("seed", 2);    # for reproducibility
 r2 = poissrnd (4, 1000, 1);
 randp ("seed", 3);    # for reproducibility
 r3 = poissrnd (10, 1000, 1);
 r = [r1, r2, r3];

 ## Plot them normalized and fix their colors
 hist (r, [0:20], 1);
 h = findobj (gca, "Type", "patch");
 set (h(1), "facecolor", "c");
 set (h(2), "facecolor", "g");
 set (h(3), "facecolor", "r");
 hold on

 ## Estimate their lambda parameter
 lambdahat = poissfit (r);

 ## Plot their estimated PDFs
 x = [0:20];
 y = poisspdf (x, lambdahat(1));
 plot (x, y, "-pr");
 y = poisspdf (x, lambdahat(2));
 plot (x, y, "-sg");
 y = poisspdf (x, lambdahat(3));
 plot (x, y, "-^c");
 xlim ([0, 20])
 ylim ([0, 0.4])
 legend ({"Normalized HIST of sample 1 with λ=1", ...
          "Normalized HIST of sample 2 with λ=4", ...
          "Normalized HIST of sample 3 with λ=10", ...
          sprintf("PDF for sample 1 with estimated λ=%0.2f", ...
                  lambdahat(1)), ...
          sprintf("PDF for sample 2 with estimated λ=%0.2f", ...
                  lambdahat(2)), ...
          sprintf("PDF for sample 3 with estimated λ=%0.2f", ...
                  lambdahat(3))})
 title ("Three population samples from different Poisson distibutions")
 hold off
plotted figure

statistics-release-1.6.3/docs/poissinv.html000066400000000000000000000134621456127120000210250ustar00rootroot00000000000000 Statistics: poissinv

Function Reference: poissinv

statistics: x = poissinv (p, lambda)

Inverse of the Poisson cumulative distribution function (iCDF).

For each element of p, compute the quantile (the inverse of the CDF) of the Poisson distribution with rate parameter lambda. The size of x is the common size of p and lambda. A scalar input functions as a constant matrix of the same size as the other inputs.

Further information about the Poisson distribution can be found at https://en.wikipedia.org/wiki/Poisson_distribution

See also: poisscdf, poisspdf, poissrnd, poissfit, poisslike, poisstat

Source Code: poissinv

Example: 1

  

 ## Plot various iCDFs from the Poisson distribution
 p = 0.001:0.001:0.999;
 x1 = poissinv (p, 13);
 x2 = poissinv (p, 4);
 x3 = poissinv (p, 10);
 plot (p, x1, "-b", p, x2, "-g", p, x3, "-r")
 grid on
 ylim ([0, 20])
 legend ({"λ = 1", "λ = 4", "λ = 10"}, "location", "northwest")
 title ("Poisson iCDF")
 xlabel ("probability")
 ylabel ("values in x (number of occurences)")
plotted figure

statistics-release-1.6.3/docs/poisslike.html000066400000000000000000000125461456127120000211570ustar00rootroot00000000000000 Statistics: poisslike

Function Reference: poisslike

statistics: nlogL = poisslike (lambda, x)
statistics: [nlogL, avar] = poisslike (lambda, x)
statistics: […] = poisslike (lambda, x, freq)

Negative log-likelihood for the Poisson distribution.

nlogL = poisslike (lambda, x) returns the negative log likelihood of the data in x corresponding to the Poisson distribution with rate parameter lambda. x must be a vector of non-negative values.

[nlogL, avar] = poisslike (lambda, x) also returns the inverse of Fisher’s information matrix, avar. If the input rate parameter, lambda, is the maximum likelihood estimate, avar is its asymptotic variance.

[…] = poisslike (lambda, x, freq) accepts a frequency vector, freq, of the same size as x. freq typically contains integer frequencies for the corresponding elements in x, but it can contain any non-integer non-negative values. By default, or if left empty, freq = ones (size (x)).

Further information about the Poisson distribution can be found at https://en.wikipedia.org/wiki/Poisson_distribution

See also: poisscdf, poissinv, poisspdf, poissrnd, poissfit, poisstat

Source Code: poisslike

statistics-release-1.6.3/docs/poisspdf.html000066400000000000000000000134141456127120000207770ustar00rootroot00000000000000 Statistics: poisspdf

Function Reference: poisspdf

statistics: y = poisspdf (x, lambda)

Poisson probability density function (PDF).

For each element of x, compute the probability density function (PDF) of the Poisson distribution with rate parameter lambda. The size of y is the common size of x and lambda. A scalar input functions as a constant matrix of the same size as the other inputs.

Further information about the Poisson distribution can be found at https://en.wikipedia.org/wiki/Poisson_distribution

See also: poisscdf, poissinv, poissrnd, poissfit, poisslike, poisstat

Source Code: poisspdf

Example: 1

  

 ## Plot various PDFs from the Poisson distribution
 x = 0:20;
 y1 = poisspdf (x, 1);
 y2 = poisspdf (x, 4);
 y3 = poisspdf (x, 10);
 plot (x, y1, "*b", x, y2, "*g", x, y3, "*r")
 grid on
 ylim ([0, 0.4])
 legend ({"λ = 1", "λ = 4", "λ = 10"}, "location", "northeast")
 title ("Poisson PDF")
 xlabel ("values in x (number of occurences)")
 ylabel ("density")
plotted figure

statistics-release-1.6.3/docs/poissrnd.html000066400000000000000000000123521456127120000210110ustar00rootroot00000000000000 Statistics: poissrnd

Function Reference: poissrnd

statistics: r = poissrnd (lambda)
statistics: r = poissrnd (lambda, rows)
statistics: r = poissrnd (lambda, rows, cols, …)
statistics: r = poissrnd (lambda, [sz])

Random arrays from the Poisson distribution.

r = normrnd (lambda) returns an array of random numbers chosen from the Poisson distribution with rate parameter lambda. The size of r is the common size of lambda. A scalar input functions as a constant matrix of the same size as the other inputs. lambda must be a finite real number and greater or equal to 0, otherwise NaN is returned.

When called with a single size argument, poissrnd returns a square matrix with the dimension specified. When called with more than one scalar argument, the first two arguments are taken as the number of rows and columns and any further arguments specify additional matrix dimensions. The size may also be specified with a row vector of dimensions, sz.

Further information about the Poisson distribution can be found at https://en.wikipedia.org/wiki/Poisson_distribution

See also: poisscdf, poissinv, poisspdf, poissfit, poisslike, poisstat

Source Code: poissrnd

statistics-release-1.6.3/docs/poisstat.html000066400000000000000000000106131456127120000210140ustar00rootroot00000000000000 Statistics: poisstat

Function Reference: poisstat

statistics: [m, v] = poisstat (lambda)

Compute statistics of the Poisson distribution.

[m, v] = poisstat (lambda) returns the mean and variance of the Poisson distribution with rate parameter lambda.

The size of m (mean) and v (variance) is the same size of the input argument.

Further information about the Poisson distribution can be found at https://en.wikipedia.org/wiki/Poisson_distribution

See also: poisscdf, poissinv, poisspdf, poissrnd, poissfit, poisslike

Source Code: poisstat

statistics-release-1.6.3/docs/ppplot.html000066400000000000000000000116011456127120000204620ustar00rootroot00000000000000 Statistics: ppplot

Function Reference: ppplot

statistics: ppplot (x, dist)
statistics: ppplot (x, dist, params)
statistics: [p, y] = ppplot (x, dist, params)

Perform a PP-plot (probability plot).

If F is the CDF of the distribution dist with parameters params and x a sample vector of length n, the PP-plot graphs ordinate y(i) = F (i-th largest element of x) versus abscissa p(i) = (i - 0.5)/n. If the sample comes from F, the pairs will approximately follow a straight line.

The default for dist is the standard normal distribution.

The optional argument params contains a list of parameters of dist.

For example, for a probability plot of the uniform distribution on [2,4] and x, use

 
 ppplot (x, "uniform", 2, 4)

dist can be any string for which a function distcdf that calculates the CDF of distribution dist exists.

If no output is requested then the data are plotted immediately. See also: qqplot

Source Code: ppplot

statistics-release-1.6.3/docs/princomp.html000066400000000000000000000120131456127120000207710ustar00rootroot00000000000000 Statistics: princomp

Function Reference: princomp

statistics: COEFF = princomp (X)
statistics: [COEFF, SCORE] = princomp (X)
statistics: [COEFF, SCORE, latent] = princomp (X)
statistics: [COEFF, SCORE, latent, tsquare] = princomp (X)
statistics: […] = princomp (X, "econ")

Performs a principal component analysis on a NxP data matrix X.

  • COEFF : returns the principal component coefficients
  • SCORE : returns the principal component scores, the representation of X in the principal component space
  • LATENT : returns the principal component variances, i.e., the eigenvalues of the covariance matrix X.
  • TSQUARE : returns Hotelling’s T-squared Statistic for each observation in X
  • [...] = princomp(X,’econ’) returns only the elements of latent that are not necessarily zero, and the corresponding columns of COEFF and SCORE, that is, when n <= p, only the first n-1. This can be significantly faster when p is much larger than n. In this case the svd will be applied on the transpose of the data matrix X

References

  1. Jolliffe, I. T., Principal Component Analysis, 2nd Edition, Springer, 2002

Source Code: princomp

statistics-release-1.6.3/docs/probit.html000066400000000000000000000073471456127120000204570ustar00rootroot00000000000000 Statistics: probit

Function Reference: probit

statistics: x = probit (p)

Probit transformation

Return the probit (the quantile of the standard normal distribution) for each element of p.

See also: logit

Source Code: probit

statistics-release-1.6.3/docs/procrustes.html000066400000000000000000000364051456127120000213660ustar00rootroot00000000000000 Statistics: procrustes

Function Reference: procrustes

statistics: d = procrustes (X, Y)
statistics: d = procrustes (X, Y, param1, value1, …)
statistics: [d, Z] = procrustes (…)
statistics: [d, Z, transform] = procrustes (…)

Procrustes Analysis.

d = procrustes (X, Y) computes a linear transformation of the points in the matrix Y to best conform them to the points in the matrix X by minimizing the sum of squared errors, as the goodness of fit criterion, which is returned in d as a dissimilarity measure. d is standardized by a measure of the scale of X, given by

  • sum (sum ((X - repmat (mean (X, 1), size (X, 1), 1)) .^ 2, 1))

i.e., the sum of squared elements of a centered version of X. However, if X comprises repetitions of the same point, the sum of squared errors is not standardized.

X and Y must have the same number of points (rows) and procrustes matches the i-th point in Y to the i-th point in X. Points in Y can have smaller dimensions (columns) than those in X, but not the opposite. Missing dimensions in Y are added with padding columns of zeros as necessary to match the the dimensions in X.

[d, Z] = procrustes (X, Y) also returns the transformed values in Y.

[d, Z, transform] = procrustes (X, Y) also returns the transformation that maps Y to Z.

transform is a structure with fields:

cthe translation component
Tthe orthogonal rotation and reflection component
bthe scale component

So that Z = transform.b * Y * transform.T + transform.c

procrustes can take two optional parameters as Name-Value pairs.

[…] = procrustes (…, "Scaling", false) computes a transformation that does not include scaling, that is transform.b = 1. Setting "Scaling" to true includes a scaling component, which is the default.

[…] = procrustes (…, "Reflection", false) computes a transformation that does not include a reflection component, that is transform.T = 1. Setting "Reflection" to true forces the solution to include a reflection component in the computed transformation, that is transform.T = -1.

[…] = procrustes (…, "Reflection", "best") computes the best fit procrustes solution, which may or may not include a reflection component, which is the default.

See also: cmdscale

Source Code: procrustes

Example: 1

  

 ## Create some random points in two dimensions
 n = 10;
 randn ("seed", 1);
 X = normrnd (0, 1, [n, 2]);

 ## Those same points, rotated, scaled, translated, plus some noise
 S = [0.5, -sqrt(3)/2; sqrt(3)/2, 0.5]; # rotate 60 degrees
 Y = normrnd (0.5*X*S + 2, 0.05, n, 2);

 ## Conform Y to X, plot original X and Y, and transformed Y
 [d, Z] = procrustes (X, Y);
 plot (X(:,1), X(:,2), "rx", Y(:,1), Y(:,2), "b.", Z(:,1), Z(:,2), "bx");
plotted figure

Example: 2

  

 ## Find Procrustes distance and plot superimposed shape

 X = [40 88; 51 88; 35 78; 36 75; 39 72; 44 71; 48 71; 52 74; 55 77];
 Y = [36 43; 48 42; 31 26; 33 28; 37 30; 40 31; 45 30; 48 28; 51 24];
 plot (X(:,1),X(:,2),"x");
 hold on
 plot (Y(:,1),Y(:,2),"o");
 xlim ([0 100]);
 ylim ([0 100]);
 legend ("Target shape (X)", "Source shape (Y)");
 [d, Z] = procrustes (X, Y)
 plot (Z(:,1), Z(:,2), "s");
 legend ("Target shape (X)", "Source shape (Y)", "Transformed shape (Z)");
 hold off

d = 0.2026
Z =

   39.769   87.509
   50.562   86.801
   35.549   72.163
   37.313   73.991
   40.873   75.850
   43.552   76.796
   48.058   75.977
   50.783   74.229
   53.541   70.684
plotted figure

Example: 3

  

 ## Apply Procrustes transformation to larger set of points

 ## Create matrices with landmark points for two triangles
 X = [5, 0; 5, 5; 8, 5];   # target
 Y = [0, 0; 1, 0; 1, 1];   # source

 ## Create a matrix with more points on the source triangle
 Y_mp = [linspace(Y(1,1),Y(2,1),10)', linspace(Y(1,2),Y(2,2),10)'; ...
         linspace(Y(2,1),Y(3,1),10)', linspace(Y(2,2),Y(3,2),10)'; ...
         linspace(Y(3,1),Y(1,1),10)', linspace(Y(3,2),Y(1,2),10)'];

 ## Plot both shapes, including the larger set of points for the source shape
 plot ([X(:,1); X(1,1)], [X(:,2); X(1,2)], "bx-");
 hold on
 plot ([Y(:,1); Y(1,1)], [Y(:,2); Y(1,2)], "ro-", "MarkerFaceColor", "r");
 plot (Y_mp(:,1), Y_mp(:,2), "ro");
 xlim ([-1 10]);
 ylim ([-1 6]);
 legend ("Target shape (X)", "Source shape (Y)", ...
         "More points on Y", "Location", "northwest");
 hold off

 ## Obtain the Procrustes transformation
 [d, Z, transform] = procrustes (X, Y)

 ## Use the Procrustes transformation to superimpose the more points (Y_mp)
 ## on the source shape onto the target shape, and then visualize the results.
 Z_mp = transform.b * Y_mp * transform.T + transform.c(1,:);
 figure
 plot ([X(:,1); X(1,1)], [X(:,2); X(1,2)], "bx-");
 hold on
 plot ([Y(:,1); Y(1,1)], [Y(:,2); Y(1,2)], "ro-", "MarkerFaceColor", "r");
 plot (Y_mp(:,1), Y_mp(:,2), "ro");
 xlim ([-1 10]);
 ylim ([-1 6]);
 plot ([Z(:,1); Z(1,1)],[Z(:,2); Z(1,2)],"ks-","MarkerFaceColor","k");
 plot (Z_mp(:,1),Z_mp(:,2),"ks");
 legend ("Target shape (X)", "Source shape (Y)", ...
         "More points on Y", "Transformed source shape (Z)", ...
         "Transformed additional points", "Location", "northwest");
 hold off

d = 0.044118
Z =

   5.0000   0.5000
   4.5000   4.5000
   8.5000   5.0000

transform =

  scalar structure containing the fields:

    T =

      -0.1240   0.9923
       0.9923   0.1240

    b = 4.0311
    c =

       5.0000   0.5000
       5.0000   0.5000
       5.0000   0.5000
plotted figure

plotted figure

Example: 4

  

 ## Compare shapes without reflection

 T = [33, 93; 33, 87; 33, 80; 31, 72; 32, 65; 32, 58; 30, 72; ...
      28, 72; 25, 69; 22, 64; 23, 59; 26, 57; 30, 57];
 S = [48, 83; 48, 77; 48, 70; 48, 65; 49, 59; 49, 56; 50, 66; ...
      52, 66; 56, 65; 58, 61; 57, 57; 54, 56; 51, 55];
 plot (T(:,1), T(:,2), "x-");
 hold on
 plot (S(:,1), S(:,2), "o-");
 legend ("Target shape (d)", "Source shape (b)");
 hold off
 d_false = procrustes (T, S, "reflection", false);
 printf ("Procrustes distance without reflection: %f\n", d_false);
 d_true = procrustes (T, S, "reflection", true);
 printf ("Procrustes distance with reflection: %f\n", d_true);
 d_best = procrustes (T, S, "reflection", "best");
 printf ("Procrustes distance with best fit: %f\n", d_true);

Procrustes distance without reflection: 0.342463
Procrustes distance with reflection: 0.020428
Procrustes distance with best fit: 0.020428
plotted figure

statistics-release-1.6.3/docs/qqplot.html000066400000000000000000000124041456127120000204660ustar00rootroot00000000000000 Statistics: qqplot

Function Reference: qqplot

statistics: [q, s] = qqplot (x)
statistics: [q, s] = qqplot (x, y)
statistics: [q, s] = qqplot (x, dist)
statistics: [q, s] = qqplot (x, y, params)
statistics: qqplot (…)

Perform a QQ-plot (quantile plot).

If F is the CDF of the distribution dist with parameters params and G its inverse, and x a sample vector of length n, the QQ-plot graphs ordinate s(i) = i-th largest element of x versus abscissa q(if) = G((i - 0.5)/n).

If the sample comes from F, except for a transformation of location and scale, the pairs will approximately follow a straight line.

If the second argument is a vector y the empirical CDF of y is used as dist.

The default for dist is the standard normal distribution. The optional argument params contains a list of parameters of dist. For example, for a quantile plot of the uniform distribution on [2,4] and x, use

 
 qqplot (x, "unif", 2, 4)

dist can be any string for which a function distinv or dist_inv exists that calculates the inverse CDF of distribution dist.

If no output arguments are given, the data are plotted directly. See also: ppplot

Source Code: qqplot

statistics-release-1.6.3/docs/random.html000066400000000000000000000326411456127120000204330ustar00rootroot00000000000000 Statistics: random

Function Reference: random

statistics: r = random (name, A)
statistics: r = random (name, A, B)
statistics: r = random (name, A, B, C)
statistics: r = random (name, …, rows, cols)
statistics: r = random (name, …, rows, cols, …)
statistics: r = random (name, …, [sz])

Random arrays from a given one-, two-, or three-parameter distribution.

The variable name must be a string with the name of the distribution to sample from. If this distribution is a one-parameter distribution, A must be supplied, if it is a two-parameter distribution, B must also be supplied, and if it is a three-parameter distribution, C must also be supplied. Any arguments following the distribution parameters will determine the size of the result.

When called with a single size argument, return a square matrix with the dimension specified. When called with more than one scalar argument the first two arguments are taken as the number of rows and columns and any further arguments specify additional matrix dimensions. The size may also be specified with a vector of dimensions sz.

name must be a char string of the name or the abbreviation of the desired probability distribution function as listed in the followng table. The last column shows the required number of parameters that must be passed passed to the desired *rnd distribution function.

Distribution NameAbbreviationInput Parameters
"Beta""beta"2
"Binomial""bino"2
"Birnbaum-Saunders""bisa"2
"Burr""burr"3
"Cauchy""cauchy"2
"Chi-squared""chi2"1
"Extreme Value""ev"2
"Exponential""exp"1
"F-Distribution""f"2
"Gamma""gam"2
"Geometric""geo"1
"Generalized Extreme Value""gev"3
"Generalized Pareto""gp"3
"Gumbel""gumbel"2
"Half-normal""hn"2
"Hypergeometric""hyge"3
"Inverse Gaussian""invg"2
"Laplace""laplace"2
"Logistic""logi"2
"Log-Logistic""logl"2
"Lognormal""logn"2
"Nakagami""naka"2
"Negative Binomial""nbin"2
"Noncentral F-Distribution""ncf"3
"Noncentral Student T""nct"2
"Noncentral Chi-Squared""ncx2"2
"Normal""norm"2
"Poisson""poiss"1
"Rayleigh""rayl"1
"Rician""rice"2
"Student T""t"1
"location-scale T""tls"3
"Triangular""tri"3
"Discrete Uniform""unid"1
"Uniform""unif"2
"Von Mises""vm"2
"Weibull""wbl"2

See also: cdf, icdf, pdf, betarnd, binornd, bisarnd, burrrnd, cauchyrnd, chi2rnd, evrnd, exprnd, frnd, gamrnd, geornd, gevrnd, gprnd, gumbelrnd, hnrnd, hygernd, invgrnd, laplacernd, logirnd, loglrnd, lognrnd, nakarnd, nbinrnd, ncfrnd, nctrnd, ncx2rnd, normrnd, poissrnd, raylrnd, ricernd, trnd, trirnd, unidrnd, unifrnd, vmrnd, wblrnd

Source Code: random

statistics-release-1.6.3/docs/rangesearch.html000066400000000000000000000352141456127120000214340ustar00rootroot00000000000000 Statistics: rangesearch

Function Reference: rangesearch

statistics: idx = rangesearch (X, Y, r)
statistics: [idx, D] = rangesearch (X, Y, r)
statistics: […] = rangesearch (…, name, value)

Find all neighbors within specified distance from input data.

idx = rangesearch (X, Y, r) returns all the points in X that are within distance r from the points in Y. X must be an N×P numeric matrix of input data, where rows correspond to observations and columns correspond to features or variables. Y is an M×P numeric matrix with query points, which must have the same numbers of column as X. r must be a nonnegative scalar value. idx is an M×1 cell array, where M is the number of observations in Y. The vector Idx{j} contains the indices of observations (rows) in X whose distances to Y(j,:) are not greater than r.

[idx, D] = rangesearch (X, Y, r) also returns the distances, D, which correspond to the points in X that are within distance r from the points in Y. D is an M×1 cell array, where M is the number of observations in Y. The vector D{j} contains the indices of observations (rows) in X whose distances to Y(j,:) are not greater than r.

Additional parameters can be specified by Name-Value pair arguments.

NameValue
"P"is the Minkowski distance exponent and it must be a positive scalar. This argument is only valid when the selected distance metric is "minkowski". By default it is 2.
"Scale"is the scale parameter for the standardized Euclidean distance and it must be a nonnegative numeric vector of equal length to the number of columns in X. This argument is only valid when the selected distance metric is "seuclidean", in which case each coordinate of X is scaled by the corresponding element of "scale", as is each query point in Y. By default, the scale parameter is the standard deviation of each coordinate in X.
"Cov"is the covariance matrix for computing the mahalanobis distance and it must be a positive definite matrix matching the the number of columns in X. This argument is only valid when the selected distance metric is "mahalanobis".
"BucketSize"is the maximum number of data points in the leaf node of the Kd-tree and it must be a positive integer. This argument is only valid when the selected search method is "kdtree".
"SortIndices"is a boolean flag to sort the returned indices in ascending order by distance and it is true by default. When the selected search method is "exhaustive" or the "IncludeTies" flag is true, rangesearch always sorts the returned indices.
"Distance"is the distance metric used by rangesearch as specified below:
"euclidean"Euclidean distance.
"seuclidean"standardized Euclidean distance. Each coordinate difference between the rows in X and the query matrix Y is scaled by dividing by the corresponding element of the standard deviation computed from X. To specify a different scaling, use the "Scale" name-value argument.
"cityblock"City block distance.
"chebychev"Chebychev distance (maximum coordinate difference).
"minkowski"Minkowski distance. The default exponent is 2. To specify a different exponent, use the "P" name-value argument.
"mahalanobis"Mahalanobis distance, computed using a positive definite covariance matrix. To change the value of the covariance matrix, use the "Cov" name-value argument.
"cosine"Cosine distance.
"correlation"One minus the sample linear correlation between observations (treated as sequences of values).
"spearman"One minus the sample Spearman’s rank correlation between observations (treated as sequences of values).
"hamming"Hamming distance, which is the percentage of coordinates that differ.
"jaccard"One minus the Jaccard coefficient, which is the percentage of nonzero coordinates that differ.
@distfunCustom distance function handle. A distance function of the form function D2 = distfun (XI, YI), where XI is a 1×P vector containing a single observation in P-dimensional space, YI is an N×P matrix containing an arbitrary number of observations in the same P-dimensional space, and D2 is an N×P vector of distances, where (D2k) is the distance between observations XI and (YIk,:).
"NSMethod"is the nearest neighbor search method used by rangesearch as specified below.
"kdtree"Creates and uses a Kd-tree to find nearest neighbors. "kdtree" is the default value when the number of columns in X is less than or equal to 10, X is not sparse, and the distance metric is "euclidean", "cityblock", "manhattan", "chebychev", or "minkowski". Otherwise, the default value is "exhaustive". This argument is only valid when the distance metric is one of the four aforementioned metrics.
"exhaustive"Uses the exhaustive search algorithm by computing the distance values from all the points in X to each point in Y.

See also: knnsearch, pdist2

Source Code: rangesearch

Example: 1

  

 ## Generate 1000 random 2D points from each of five distinct multivariate
 ## normal distributions that form five separate classes
 N = 1000;
 d = 10;
 randn ("seed", 5);
 X1 = mvnrnd (d * [0, 0], eye (2), 1000);
 randn ("seed", 6);
 X2 = mvnrnd (d * [1, 1], eye (2), 1000);
 randn ("seed", 7);
 X3 = mvnrnd (d * [-1, -1], eye (2), 1000);
 randn ("seed", 8);
 X4 = mvnrnd (d * [1, -1], eye (2), 1000);
 randn ("seed", 8);
 X5 = mvnrnd (d * [-1, 1], eye (2), 1000);
 X = [X1; X2; X3; X4; X5];

 ## For each point in X, find the points in X that are within a radius d
 ## away from the points in X.
 Idx = rangesearch (X, X, d, "NSMethod", "exhaustive");

 ## Select the first point in X (corresponding to the first class) and find
 ## its nearest neighbors within the radius d.  Display these points in
 ## one color and the remaining points in a different color.
 x = X(1,:);
 nearestPoints = X (Idx{1},:);
 nonNearestIdx = true (size (X, 1), 1);
 nonNearestIdx(Idx{1}) = false;

 scatter (X(nonNearestIdx,1), X(nonNearestIdx,2))
 hold on
 scatter (nearestPoints(:,1),nearestPoints(:,2))
 scatter (x(1), x(2), "black", "filled")
 hold off

 ## Select the last point in X (corresponding to the fifth class) and find
 ## its nearest neighbors within the radius d.  Display these points in
 ## one color and the remaining points in a different color.
 x = X(end,:);
 nearestPoints = X (Idx{1},:);
 nonNearestIdx = true (size (X, 1), 1);
 nonNearestIdx(Idx{1}) = false;

 figure
 scatter (X(nonNearestIdx,1), X(nonNearestIdx,2))
 hold on
 scatter (nearestPoints(:,1),nearestPoints(:,2))
 scatter (x(1), x(2), "black", "filled")
 hold off
plotted figure

plotted figure

statistics-release-1.6.3/docs/ranksum.html000066400000000000000000000172031456127120000206300ustar00rootroot00000000000000 Statistics: ranksum

Function Reference: ranksum

statistics: p = ranksum (x, y)
statistics: p = ranksum (x, y, alpha)
statistics: p = ranksum (x, y, alpha, Name, Value)
statistics: p = ranksum (x, y, Name, Value)
statistics: [p, h] = ranksum (x, y, …)
statistics: [p, h, stats] = ranksum (x, y, …)

Wilcoxon rank sum test for equal medians. This test is equivalent to a Mann-Whitney U-test.

p = ranksum (x, y) returns the p-value of a two-sided Wilcoxon rank sum test. It tests the null hypothesis that two independent samples, in the vectors X and Y, come from continuous distributions with equal medians, against the alternative hypothesis that they are not. x and y can have different lengths and the test assumes that they are independent.

ranksum treats NaN in x, y as missing values. The two-sided p-value is computed by doubling the most significant one-sided value.

[p, h] = ranksum (x, y) also returns the result of the hypothesis test with h = 1 indicating a rejection of the null hypothesis at the default alpha = 0.05 significance level, and h = 0 indicating a failure to reject the null hypothesis at the same significance level.

[p, h, stats] = ranksum (x, y) also returns the structure stats with information about the test statistic. It contains the field ranksum with the value of the rank sum test statistic and if computed with the "approximate" method it also contains the value of the z-statistic in the field zval.

[…] = ranksum (x, y, alpha) or alternatively […] = ranksum (x, y, "alpha", alpha) returns the result of the hypothesis test performed at the significance level ALPHA.

[…] = ranksum (x, y, "method", M) defines the computation method of the p-value specified in M, which can be "exact", "approximate", or "oldexact". M must be a single string. When "method" is unspecified, the default is: "exact" when min (length (x), length (y)) < 10 and length (x) + length (y) < 10, otherwise the "approximate" method is used.

  • "exact" method uses full enumeration for small total sample size (< 10), otherwise the network algorithm is used for larger samples.
  • "approximate" uses normal approximation method for computing the p-value.
  • "oldexact" uses full enumeration for any sample size. Note, that this option can lead to out of memory error for large samples. Use with caution!

[…] = ranksum (x, y, "tail", tail) defines the type of test, which can be "both", "right", or "left". tail must be a single string.

  • "both" – "medians are not equal" (two-tailed test, default)
  • "right" – "median of X is greater than median of Y" (right-tailed test)
  • "left" – "median of X is less than median of Y" (left-tailed test)

Note: the rank sum statistic is based on the smaller sample of vectors x and y.

Source Code: ranksum

statistics-release-1.6.3/docs/raylcdf.html000066400000000000000000000143041456127120000205730ustar00rootroot00000000000000 Statistics: raylcdf

Function Reference: raylcdf

statistics: p = raylcdf (x, sigma)
statistics: p = raylcdf (x, sigma, "upper")

Rayleigh cumulative distribution function (CDF).

For each element of x, compute the cumulative distribution function (CDF) of the Rayleigh distribution with scale parameter sigma. The size of p is the common size of x and sigma. A scalar input functions as a constant matrix of the same size as the other inputs.

p = raylcdf (x, sigma, "upper") computes the upper tail probability of the Rayleigh distribution with parameter sigma, at the values in x.

Further information about the Rayleigh distribution can be found at https://en.wikipedia.org/wiki/Rayleigh_distribution

See also: raylinv, raylpdf, raylrnd, raylfit, rayllike, raylstat

Source Code: raylcdf

Example: 1

  

 ## Plot various CDFs from the Rayleigh distribution
 x = 0:0.01:10;
 p1 = raylcdf (x, 0.5);
 p2 = raylcdf (x, 1);
 p3 = raylcdf (x, 2);
 p4 = raylcdf (x, 3);
 p5 = raylcdf (x, 4);
 plot (x, p1, "-b", x, p2, "g", x, p3, "-r", x, p4, "-m", x, p5, "-k")
 grid on
 ylim ([0, 1])
 legend ({"σ = 0.5", "σ = 1", "σ = 2", ...
          "σ = 3", "σ = 4"}, "location", "southeast")
 title ("Rayleigh CDF")
 xlabel ("values in x")
 ylabel ("probability")
plotted figure

statistics-release-1.6.3/docs/raylfit.html000066400000000000000000000210071456127120000206170ustar00rootroot00000000000000 Statistics: raylfit

Function Reference: raylfit

statistics: sigmaA = raylfit (x)
statistics: [sigmaA, sigmaci] = raylfit (x)
statistics: [sigmaA, sigmaci] = raylfit (x, alpha)
statistics: [sigmaA, sigmaci] = raylfit (x, alpha, censor)
statistics: [sigmaA, sigmaci] = raylfit (x, alpha, censor, freq)

Estimate parameter and confidence intervals for the Rayleigh distribution.

sigmaA = raylfit (x) returns the maximum likelihood estimate of the rate parameter, lambda, of the Rayleigh distribution given the data in x. x must be a vector of non-negative values.

[sigmaA, sigmaci] = raylfit (x) returns the 95% confidence intervals for the parameter estimate.

[sigmaA, sigmaci] = raylfit (x, alpha) also returns the 100 * (1 - alpha) percent confidence intervals of the estimated parameter. By default, the optional argument alpha is 0.05 corresponding to 95% confidence intervals. Pass in [] for alpha to use the default values.

[…] = raylfit (x, alpha, censor) accepts a boolean vector, censor, of the same size as x with 1s for observations that are right-censored and 0s for observations that are observed exactly. By default, or if left empty, censor = zeros (size (x)).

[…] = raylfit (x, alpha, censor, freq) accepts a frequency vector or matrix, freq, of the same size as x. freq typically contains integer frequencies for the corresponding elements in x. freq cannot contain negative values.

Further information about the Rayleigh distribution can be found at https://en.wikipedia.org/wiki/Rayleigh_distribution

See also: raylcdf, raylinv, raylpdf, raylrnd, rayllike, raylstat

Source Code: raylfit

Example: 1

  

 ## Sample 3 populations from 3 different Rayleigh distibutions
 rand ("seed", 2);    # for reproducibility
 r1 = raylrnd (1, 1000, 1);
 rand ("seed", 2);    # for reproducibility
 r2 = raylrnd (2, 1000, 1);
 rand ("seed", 3);    # for reproducibility
 r3 = raylrnd (4, 1000, 1);
 r = [r1, r2, r3];

 ## Plot them normalized and fix their colors
 hist (r, [0.5:0.5:10.5], 2);
 h = findobj (gca, "Type", "patch");
 set (h(1), "facecolor", "c");
 set (h(2), "facecolor", "g");
 set (h(3), "facecolor", "r");
 hold on

 ## Estimate their lambda parameter
 sigmaA = raylfit (r(:,1));
 sigmaB = raylfit (r(:,2));
 sigmaC = raylfit (r(:,3));

 ## Plot their estimated PDFs
 x = [0:0.1:10];
 y = raylpdf (x, sigmaA);
 plot (x, y, "-pr");
 y = raylpdf (x, sigmaB);
 plot (x, y, "-sg");
 y = raylpdf (x, sigmaC);
 plot (x, y, "-^c");
 xlim ([0, 10])
 ylim ([0, 0.7])
 legend ({"Normalized HIST of sample 1 with σ=1", ...
          "Normalized HIST of sample 2 with σ=2", ...
          "Normalized HIST of sample 3 with σ=4", ...
          sprintf("PDF for sample 1 with estimated σ=%0.2f", ...
                  sigmaA), ...
          sprintf("PDF for sample 2 with estimated σ=%0.2f", ...
                  sigmaB), ...
          sprintf("PDF for sample 3 with estimated σ=%0.2f", ...
                  sigmaC)})
 title ("Three population samples from different Rayleigh distibutions")
 hold off
plotted figure

statistics-release-1.6.3/docs/raylinv.html000066400000000000000000000135651456127120000206430ustar00rootroot00000000000000 Statistics: raylinv

Function Reference: raylinv

statistics: x = raylinv (p, sigma)

Inverse of the Rayleigh cumulative distribution function (iCDF).

For each element of p, compute the quantile (the inverse of the CDF) of the Rayleigh distribution with scale parameter sigma. The size of x is the common size of p and sigma. A scalar input functions as a constant matrix of the same size as the other inputs.

Further information about the Rayleigh distribution can be found at https://en.wikipedia.org/wiki/Rayleigh_distribution

See also: raylcdf, raylpdf, raylrnd, raylfit, rayllike, raylstat

Source Code: raylinv

Example: 1

  

 ## Plot various iCDFs from the Rayleigh distribution
 p = 0.001:0.001:0.999;
 x1 = raylinv (p, 0.5);
 x2 = raylinv (p, 1);
 x3 = raylinv (p, 2);
 x4 = raylinv (p, 3);
 x5 = raylinv (p, 4);
 plot (p, x1, "-b", p, x2, "g", p, x3, "-r", p, x4, "-m", p, x5, "-k")
 grid on
 ylim ([0, 10])
 legend ({"σ = 0,5", "σ = 1", "σ = 2", ...
          "σ = 3", "σ = 4"}, "location", "northwest")
 title ("Rayleigh iCDF")
 xlabel ("probability")
 ylabel ("values in x")
plotted figure

statistics-release-1.6.3/docs/rayllike.html000066400000000000000000000125171456127120000207670ustar00rootroot00000000000000 Statistics: rayllike

Function Reference: rayllike

statistics: nlogL = rayllike (sigma, x)
statistics: [nlogL, acov] = rayllike (sigma, x)
statistics: […] = rayllike (sigma, x, freq)

Negative log-likelihood for the Rayleigh distribution.

nlogL = rayllike (sigma, x) returns the negative log likelihood of the data in x corresponding to the Rayleigh distribution with rate parameter sigma. x must be a vector of non-negative values.

[nlogL, acov] = rayllike (sigma, x) also returns the inverse of Fisher’s information matrix, acov. If the input rate parameter, sigma, is the maximum likelihood estimate, acov is its asymptotic variance.

[…] = rayllike (sigma, x, freq) accepts a frequency vector, freq, of the same size as x. freq typically contains integer frequencies for the corresponding elements in x, but it can contain any non-integer non-negative values. By default, or if left empty, freq = ones (size (x)).

Further information about the Rayleigh distribution can be found at https://en.wikipedia.org/wiki/Rayleigh_distribution

See also: raylcdf, raylinv, raylpdf, raylrnd, raylfit, raylstat

Source Code: rayllike

statistics-release-1.6.3/docs/raylpdf.html000066400000000000000000000135261456127120000206150ustar00rootroot00000000000000 Statistics: raylpdf

Function Reference: raylpdf

statistics: y = raylpdf (x, sigma)

Rayleigh probability density function (PDF).

For each element of x, compute the probability density function (PDF) of the Rayleigh distribution with scale parameter sigma. The size of p is the common size of x and sigma. A scalar input functions as a constant matrix of the same size as the other inputs.

Further information about the Rayleigh distribution can be found at https://en.wikipedia.org/wiki/Rayleigh_distribution

See also: raylcdf, raylinv, raylrnd, raylfit, rayllike, raylstat

Source Code: raylpdf

Example: 1

  

 ## Plot various PDFs from the Rayleigh distribution
 x = 0:0.01:10;
 y1 = raylpdf (x, 0.5);
 y2 = raylpdf (x, 1);
 y3 = raylpdf (x, 2);
 y4 = raylpdf (x, 3);
 y5 = raylpdf (x, 4);
 plot (x, y1, "-b", x, y2, "g", x, y3, "-r", x, y4, "-m", x, y5, "-k")
 grid on
 ylim ([0, 1.25])
 legend ({"σ = 0,5", "σ = 1", "σ = 2", ...
          "σ = 3", "σ = 4"}, "location", "northeast")
 title ("Rayleigh PDF")
 xlabel ("values in x")
 ylabel ("density")
plotted figure

statistics-release-1.6.3/docs/raylrnd.html000066400000000000000000000123021456127120000206160ustar00rootroot00000000000000 Statistics: raylrnd

Function Reference: raylrnd

statistics: r = raylrnd (sigma)
statistics: r = raylrnd (sigma, rows)
statistics: r = raylrnd (sigma, rows, cols, …)
statistics: r = raylrnd (sigma, [sz])

Random arrays from the Rayleigh distribution.

r = raylrnd (sigma) returns an array of random numbers chosen from the Rayleigh distribution with scale parameter sigma. The size of r is the size of sigma. A scalar input functions as a constant matrix of the same size as the other inputs. sigma must be a finite real number greater than 0, otherwise NaN is returned.

When called with a single size argument, raylrnd returns a square matrix with the dimension specified. When called with more than one scalar argument, the first two arguments are taken as the number of rows and columns and any further arguments specify additional matrix dimensions. The size may also be specified with a row vector of dimensions, sz.

Further information about the Rayleigh distribution can be found at https://en.wikipedia.org/wiki/Rayleigh_distribution

See also: raylcdf, raylinv, raylpdf, raylfit, rayllike, raylstat

Source Code: raylrnd

statistics-release-1.6.3/docs/raylstat.html000066400000000000000000000106021456127120000210070ustar00rootroot00000000000000 Statistics: raylstat

Function Reference: raylstat

statistics: [m, v] = raylstat (sigma)

Compute statistics of the Rayleigh distribution.

[m, v] = raylstat (sigma) returns the mean and variance of the Rayleigh distribution with scale parameter sigma.

The size of m (mean) and v (variance) is the same size of the input argument.

Further information about the Rayleigh distribution can be found at https://en.wikipedia.org/wiki/Rayleigh_distribution

See also: raylcdf, raylinv, raylpdf, raylrnd, raylfit, rayllike

Source Code: raylstat

statistics-release-1.6.3/docs/regress.html000066400000000000000000000126041456127120000206220ustar00rootroot00000000000000 Statistics: regress

Function Reference: regress

statistics: [b, bint, r, rint, stats] = regress (y, X, [alpha])

Multiple Linear Regression using Least Squares Fit of y on X with the model y = X * beta + e.

Here,

  • y is a column vector of observed values
  • X is a matrix of regressors, with the first column filled with the constant value 1
  • beta is a column vector of regression parameters
  • e is a column vector of random errors

Arguments are

  • y is the y in the model
  • X is the X in the model
  • alpha is the significance level used to calculate the confidence intervals bint and rint (see ‘Return values’ below). If not specified, ALPHA defaults to 0.05

Return values are

  • b is the beta in the model
  • bint is the confidence interval for b
  • r is a column vector of residuals
  • rint is the confidence interval for r
  • stats is a row vector containing:
    • The R^2 statistic
    • The F statistic
    • The p value for the full model
    • The estimated error variance

r and rint can be passed to rcoplot to visualize the residual intervals and identify outliers.

NaN values in y and X are removed before calculation begins.

See also: regress_gp, regression_ftest, regression_ttest

Source Code: regress

statistics-release-1.6.3/docs/regress_gp.html000066400000000000000000000572561456127120000213240ustar00rootroot00000000000000 Statistics: regress_gp

Function Reference: regress_gp

statistics: [Yfit, Yint, m, K] = regress_gp (X, Y, Xfit)
statistics: [Yfit, Yint, m, K] = regress_gp (X, Y, Xfit, "linear")
statistics: [Yfit, Yint, Ysd] = regress_gp (X, Y, Xfit, "rbf")
statistics: […] = regress_gp (X, Y, Xfit, "linear", Sp)
statistics: […] = regress_gp (X, Y, Xfit, Sp)
statistics: […] = regress_gp (X, Y, Xfit, "rbf", theta)
statistics: […] = regress_gp (X, Y, Xfit, "rbf", theta, g)
statistics: […] = regress_gp (X, Y, Xfit, "rbf", theta, g, alpha)
statistics: […] = regress_gp (X, Y, Xfit, theta)
statistics: […] = regress_gp (X, Y, Xfit, theta, g)
statistics: […] = regress_gp (X, Y, Xfit, theta, g, alpha)

Regression using Gaussian Processes.

[Yfit, Yint, m, K] = regress_gp (X, Y, Xfit) will estimate a linear Gaussian Process model m in the form Y = X' * m, where X is an N×P matrix with N observations in P dimensional space and Y is an N×1 column vector as the dependent variable. The information about errors of the predictions (interpolation/extrapolation) is given by the covarianve matrix K. By default, the linear model defines the prior covariance of m as Sp = 100 * eye (size (X, 2) + 1). A custom prior covariance matrix can be passed as Sp, which must be a P+1×P+1 positive definite matrix. The model is evaluated for input Xfit, which must have the same columns as X, and the estimates are returned in Yfit along with the estimated variation in Yint. Yint(:,1) contains the upper boundary estimate and Yint(:,1) contains the upper boundary estimate with respect to Yfit.

[Yfit, Yint, Ysd, K] = regress_gp (X, Y, Xfit, "rbf") will estimate a Gaussian Process model with a Radial Basis Function (RBF) kernel with default parameters theta = 5, which corresponds to the characteristic lengthscale, and g = 0.01, which corresponds to the nugget effect, and alpha = 0.05 which defines the confidence level for the estimated intervals returned in Yint. The function also returns the predictive covariance matrix in Ysd. For multidimensional predictors X the function will automatically normalize each column to a zero mean and a standard deviation to one.

Run demo regress_gp to see examples.

See also: regress, regression_ftest, regression_ttest

Source Code: regress_gp

Example: 1

  

 ## Linear fitting of 1D Data
 rand ("seed", 125);
 X = 2 * rand (5, 1) - 1;
 randn ("seed", 25);
 Y = 2 * X - 1 + 0.3 * randn (5, 1);

 ## Points for interpolation/extrapolation
 Xfit = linspace (-2, 2, 10)';

 ## Fit regression model
 [Yfit, Yint, m] = regress_gp (X, Y, Xfit);

 ## Plot fitted data
 plot (X, Y, "xk", Xfit, Yfit, "r-", Xfit, Yint, "b-");
 title ("Gaussian process regression with linear kernel");
plotted figure

Example: 2

  

 ## Linear fitting of 2D Data
 rand ("seed", 135);
 X = 2 * rand (4, 2) - 1;
 randn ("seed", 35);
 Y = 2 * X(:,1) - 3 * X(:,2) - 1 + 1 * randn (4, 1);

 ## Mesh for interpolation/extrapolation
 [x1, x2] = meshgrid (linspace (-1, 1, 10));
 Xfit = [x1(:), x2(:)];

 ## Fit regression model
 [Ypred, Yint, Ysd] = regress_gp (X, Y, Xfit);
 Ypred = reshape (Ypred, 10, 10);
 YintU = reshape (Yint(:,1), 10, 10);
 YintL = reshape (Yint(:,2), 10, 10);

 ## Plot fitted data
 plot3 (X(:,1), X(:,2), Y, ".k", "markersize", 16);
 hold on;
 h = mesh (x1, x2, Ypred, zeros (10, 10));
 set (h, "facecolor", "none", "edgecolor", "yellow");
 h = mesh (x1, x2, YintU, ones (10, 10));
 set (h, "facecolor", "none", "edgecolor", "cyan");
 h = mesh (x1, x2, YintL, ones (10, 10));
 set (h, "facecolor", "none", "edgecolor", "cyan");
 hold off
 axis tight
 view (75, 25)
 title ("Gaussian process regression with linear kernel");
plotted figure

Example: 3

  

 ## Projection over basis function with linear kernel
 pp = [2, 2, 0.3, 1];
 n = 10;
 rand ("seed", 145);
 X = 2 * rand (n, 1) - 1;
 randn ("seed", 45);
 Y = polyval (pp, X) + 0.3 * randn (n, 1);

 ## Powers
 px = [sqrt(abs(X)), X, X.^2, X.^3];

 ## Points for interpolation/extrapolation
 Xfit = linspace (-1, 1, 100)';
 pxi = [sqrt(abs(Xfit)), Xfit, Xfit.^2, Xfit.^3];

 ## Define a prior covariance assuming that the sqrt component is not present
 Sp = 100 * eye (size (px, 2) + 1);
 Sp(2,2) = 1; # We don't believe the sqrt(abs(X)) is present

 ## Fit regression model
 [Yfit, Yint, Ysd] = regress_gp (px, Y, pxi, Sp);

 ## Plot fitted data
 plot (X, Y, "xk;Data;", Xfit, Yfit, "r-;Estimation;", ...
                         Xfit, polyval (pp, Xfit), "g-;True;");
 axis tight
 axis manual
 hold on
 plot (Xfit, Yint(:,1), "m-;Upper bound;", Xfit, Yint(:,2), "b-;Lower bound;");
 hold off
 title ("Linear kernel over basis function with prior covariance");
plotted figure

Example: 4

  

 ## Projection over basis function with linear kernel
 pp = [2, 2, 0.3, 1];
 n = 10;
 rand ("seed", 145);
 X = 2 * rand (n, 1) - 1;
 randn ("seed", 45);
 Y = polyval (pp, X) + 0.3 * randn (n, 1);

 ## Powers
 px = [sqrt(abs(X)), X, X.^2, X.^3];

 ## Points for interpolation/extrapolation
 Xfit = linspace (-1, 1, 100)';
 pxi = [sqrt(abs(Xfit)), Xfit, Xfit.^2, Xfit.^3];

 ## Fit regression model without any assumption on prior covariance
 [Yfit, Yint, Ysd] = regress_gp (px, Y, pxi);

 ## Plot fitted data
 plot (X, Y, "xk;Data;", Xfit, Yfit, "r-;Estimation;", ...
                         Xfit, polyval (pp, Xfit), "g-;True;");
 axis tight
 axis manual
 hold on
 plot (Xfit, Yint(:,1), "m-;Upper bound;", Xfit, Yint(:,2), "b-;Lower bound;");
 hold off
 title ("Linear kernel over basis function without prior covariance");
plotted figure

Example: 5

  

 ## Projection over basis function with rbf kernel
 pp = [2, 2, 0.3, 1];
 n = 10;
 rand ("seed", 145);
 X = 2 * rand (n, 1) - 1;
 randn ("seed", 45);
 Y = polyval (pp, X) + 0.3 * randn (n, 1);

 ## Powers
 px = [sqrt(abs(X)), X, X.^2, X.^3];

 ## Points for interpolation/extrapolation
 Xfit = linspace (-1, 1, 100)';
 pxi = [sqrt(abs(Xfit)), Xfit, Xfit.^2, Xfit.^3];

 ## Fit regression model with RBF kernel (standard parameters)
 [Yfit, Yint, Ysd] = regress_gp (px, Y, pxi, "rbf");

 ## Plot fitted data
 plot (X, Y, "xk;Data;", Xfit, Yfit, "r-;Estimation;", ...
                         Xfit, polyval (pp, Xfit), "g-;True;");
 axis tight
 axis manual
 hold on
 plot (Xfit, Yint(:,1), "m-;Upper bound;", Xfit, Yint(:,2), "b-;Lower bound;");
 hold off
 title ("RBF kernel over basis function with standard parameters");
 text (-0.5, 4, "theta = 5\n g = 0.01");
plotted figure

Example: 6

  

 ## Projection over basis function with rbf kernel
 pp = [2, 2, 0.3, 1];
 n = 10;
 rand ("seed", 145);
 X = 2 * rand (n, 1) - 1;
 randn ("seed", 45);
 Y = polyval (pp, X) + 0.3 * randn (n, 1);

 ## Powers
 px = [sqrt(abs(X)), X, X.^2, X.^3];

 ## Points for interpolation/extrapolation
 Xfit = linspace (-1, 1, 100)';
 pxi = [sqrt(abs(Xfit)), Xfit, Xfit.^2, Xfit.^3];

 ## Fit regression model with RBF kernel with different parameters
 [Yfit, Yint, Ysd] = regress_gp (px, Y, pxi, "rbf", 10, 0.01);

 ## Plot fitted data
 plot (X, Y, "xk;Data;", Xfit, Yfit, "r-;Estimation;", ...
                         Xfit, polyval (pp, Xfit), "g-;True;");
 axis tight
 axis manual
 hold on
 plot (Xfit, Yint(:,1), "m-;Upper bound;", Xfit, Yint(:,2), "b-;Lower bound;");
 hold off
 title ("GP regression with RBF kernel and non default parameters");
 text (-0.5, 4, "theta = 10\n g = 0.01");

 ## Fit regression model with RBF kernel with different parameters
 [Yfit, Yint, Ysd] = regress_gp (px, Y, pxi, "rbf", 50, 0.01);

 ## Plot fitted data
 figure
 plot (X, Y, "xk;Data;", Xfit, Yfit, "r-;Estimation;", ...
                         Xfit, polyval (pp, Xfit), "g-;True;");
 axis tight
 axis manual
 hold on
 plot (Xfit, Yint(:,1), "m-;Upper bound;", Xfit, Yint(:,2), "b-;Lower bound;");
 hold off
 title ("GP regression with RBF kernel and non default parameters");
 text (-0.5, 4, "theta = 50\n g = 0.01");

 ## Fit regression model with RBF kernel with different parameters
 [Yfit, Yint, Ysd] = regress_gp (px, Y, pxi, "rbf", 50, 0.001);

 ## Plot fitted data
 figure
 plot (X, Y, "xk;Data;", Xfit, Yfit, "r-;Estimation;", ...
                         Xfit, polyval (pp, Xfit), "g-;True;");
 axis tight
 axis manual
 hold on
 plot (Xfit, Yint(:,1), "m-;Upper bound;", Xfit, Yint(:,2), "b-;Lower bound;");
 hold off
 title ("GP regression with RBF kernel and non default parameters");
 text (-0.5, 4, "theta = 50\n g = 0.001");

 ## Fit regression model with RBF kernel with different parameters
 [Yfit, Yint, Ysd] = regress_gp (px, Y, pxi, "rbf", 50, 0.05);

 ## Plot fitted data
 figure
 plot (X, Y, "xk;Data;", Xfit, Yfit, "r-;Estimation;", ...
                         Xfit, polyval (pp, Xfit), "g-;True;");
 axis tight
 axis manual
 hold on
 plot (Xfit, Yint(:,1), "m-;Upper bound;", Xfit, Yint(:,2), "b-;Lower bound;");
 hold off
 title ("GP regression with RBF kernel and non default parameters");
 text (-0.5, 4, "theta = 50\n g = 0.05");
plotted figure

plotted figure

plotted figure

plotted figure

Example: 7

  

 ## RBF fitting on noiseless 1D Data
 x = [0:2*pi/7:2*pi]';
 y = 5 * sin (x);

 ## Predictive grid of 500 equally spaced locations
 xi = [-0.5:(2*pi+1)/499:2*pi+0.5]';

 ## Fit regression model with RBF kernel
 [Yfit, Yint, Ysd] = regress_gp (x, y, xi, "rbf");

 ## Plot fitted data
 r = mvnrnd (Yfit, diag (Ysd)', 50);
 plot (xi, r', "c-");
 hold on
 plot (xi, Yfit, "r-;Estimation;", xi, Yint, "b-;Confidence interval;");
 plot (x, y, ".k;Predictor points;", "markersize", 20)
 plot (xi, 5 * sin (xi), "-y;True Function;");
 xlim ([-0.5,2*pi+0.5]);
 ylim ([-10,10]);
 hold off
 title ("GP regression with RBF kernel on noiseless 1D data");
 text (0, -7, "theta = 5\n g = 0.01");
plotted figure

Example: 8

  

 ## RBF fitting on noisy 1D Data
 x = [0:2*pi/7:2*pi]';
 x = [x; x];
 y = 5 * sin (x) + randn (size (x));

 ## Predictive grid of 500 equally spaced locations
 xi = [-0.5:(2*pi+1)/499:2*pi+0.5]';

 ## Fit regression model with RBF kernel
 [Yfit, Yint, Ysd] = regress_gp (x, y, xi, "rbf");

 ## Plot fitted data
 r = mvnrnd (Yfit, diag (Ysd)', 50);
 plot (xi, r', "c-");
 hold on
 plot (xi, Yfit, "r-;Estimation;", xi, Yint, "b-;Confidence interval;");
 plot (x, y, ".k;Predictor points;", "markersize", 20)
 plot (xi, 5 * sin (xi), "-y;True Function;");
 xlim ([-0.5,2*pi+0.5]);
 ylim ([-10,10]);
 hold off
 title ("GP regression with RBF kernel on noisy 1D data");
 text (0, -7, "theta = 5\n g = 0.01");
plotted figure

statistics-release-1.6.3/docs/regression_ftest.html000066400000000000000000000150331456127120000225340ustar00rootroot00000000000000 Statistics: regression_ftest

Function Reference: regression_ftest

statistics: [h, pval, stats] = regression_ftest (y, x, fm)
statistics: […] = regression_ftest (y, x, fm, rm)
statistics: […] = regression_ftest (y, x, fm, rm, Name, Value)
statistics: […] = regression_ftest (y, x, fm, [], Name, Value)

F-test for General Linear Regression Analysis

Perform a general linear regression F test for the null hypothesis that the full model of the form y = b_0 + b_1 * x_1 + b_2 * x_2 + … + b_n * x_n + e, where n is the number of variables in x, does not perform better than a reduced model, such as y = b'_0 + b'_1 * x_1 + b'_2 * x_2 + … + b'_k * x_k + e, where k < n and it corresponds to the first k variables in x. Explanatory (dependent) variable y and response (independent) variables x must not contain any missing values (NaNs).

The full model, fm, must be a vector of length equal to the columns of x, in which case the constant term b_0 is assumed 0, or equal to the columns of x plus one, in which case the first element is the constant b_0.

The reduced model, rm, must include the constant term and a subset of the variables (columns) in x. If rm is not given, then a constant term b’_0 is assumed equal to the constant term, b_0, of the full model or 0, if the full model, fm, does not have a constant term. rm must be a vector or a scalar if only a constant term is passed into the function.

Name-Value pair arguments can be used to set statistical significance. "alpha" can be used to specify the significance level of the test (the default value is 0.05). If you want pass optional Name-Value pair without a reduced model, make sure that the latter is passed as an empty variable.

If h is 1 the null hypothesis is rejected, meaning that the full model explains the variance better than the restricted model. If h is 0, it can be assumed that the full model does NOT explain the variance any better than the restricted model.

The p-value (1 minus the CDF of this distribution at f) is returned in pval.

Under the null, the test statistic f follows an F distribution with ’df1’ and ’df2’ degrees of freedom, which are returned as fields in the stats structure along with the test’s F-statistic, ’fstat’

See also: regression_ttest, regress, regress_gp

Source Code: regression_ftest

statistics-release-1.6.3/docs/regression_ttest.html000066400000000000000000000145531456127120000225600ustar00rootroot00000000000000 Statistics: regression_ttest

Function Reference: regression_ttest

statistics: h = regression_ttest (y, x)
statistics: [h, pval] = regression_ttest (y, x)
statistics: [h, pval, ci] = regression_ttest (y, x)
statistics: [h, pval, ci, stats] = regression_ttest (y, x)
statistics: […] = regression_ttest (y, x, Name, Value)

Perform a linear regression t-test.

h = regression_ttest (y, x) tests the null hypothesis that the slope beta1 of a simple linear regression equals 0. The result is h = 0 if the null hypothesis cannot be rejected at the 5% significance level, or h = 1 if the null hypothesis can be rejected at the 5% level. y and x must be vectors of equal length with finite real numbers.

The p-value of the test is returned in pval. A 100(1-alpha)% confidence interval for beta1 is returned in ci. stats is a structure containing the value of the test statistic (tstat), the degrees of freedom (df), the slope coefficient (beta1), and the intercept (beta0). Under the null, the test statistic stats.tstat follows a T-distribution with stats.df degrees of freedom.

[…] = regression_ttest (…, name, value) specifies one or more of the following name/value pairs:

NameValue
"alpha"the significance level. Default is 0.05.
"tail"a string specifying the alternative hypothesis
"both"beta1 is not 0 (two-tailed, default)
"left"beta1 is less than 0 (left-tailed)
"right"beta1 is greater than 0 (right-tailed)

See also: regression_ftest, regress, regress_gp

Source Code: regression_ttest

statistics-release-1.6.3/docs/ricecdf.html000066400000000000000000000175721456127120000205600ustar00rootroot00000000000000 Statistics: ricecdf

Function Reference: ricecdf

statistics: p = ricecdf (x, nu, sigma)
statistics: p = ricecdf (x, nu, sigma, "upper")

Rician cumulative distribution function (CDF).

For each element of x, compute the cumulative distribution function (CDF) of the Rician distribution with non-centrality (distance) parameter nu and scale parameter sigma. The size of p is the common size of x, nu, and sigma. A scalar input functions as a constant matrix of the same size as the other inputs.

p = ricecdf (x, nu, sigma, "upper") computes the upper tail probability of the Rician distribution with parameters nu and sigma, at the values in x.

Further information about the Rician distribution can be found at https://en.wikipedia.org/wiki/Rice_distribution

See also: riceinv, ricepdf, ricernd, ricefit, ricelike, ricestat

Source Code: ricecdf

Example: 1

  

 ## Plot various CDFs from the Rician distribution
 x = 0:0.01:10;
 p1 = ricecdf (x, 0, 1);
 p2 = ricecdf (x, 0.5, 1);
 p3 = ricecdf (x, 1, 1);
 p4 = ricecdf (x, 2, 1);
 p5 = ricecdf (x, 4, 1);
 plot (x, p1, "-b", x, p2, "g", x, p3, "-r", x, p4, "-m", x, p5, "-k")
 grid on
 ylim ([0, 1])
 xlim ([0, 8])
 legend ({"ν = 0, σ = 1", "ν = 0.5, σ = 1", "ν = 1, σ = 1", ...
          "ν = 2, σ = 1", "ν = 4, σ = 1"}, "location", "southeast")
 title ("Rician CDF")
 xlabel ("values in x")
 ylabel ("probability")
plotted figure

Example: 2

  

 ## Plot various CDFs from the Rician distribution
 x = 0:0.01:10;
 p1 = ricecdf (x, 0, 0.5);
 p2 = ricecdf (x, 0, 2);
 p3 = ricecdf (x, 0, 3);
 p4 = ricecdf (x, 2, 2);
 p5 = ricecdf (x, 4, 2);
 plot (x, p1, "-b", x, p2, "g", x, p3, "-r", x, p4, "-m", x, p5, "-k")
 grid on
 ylim ([0, 1])
 xlim ([0, 8])
 legend ({"ν = 0, σ = 0.5", "ν = 0, σ = 2", "ν = 0, σ = 3", ...
          "ν = 2, σ = 2", "ν = 4, σ = 2"}, "location", "southeast")
 title ("Rician CDF")
 xlabel ("values in x")
 ylabel ("probability")
plotted figure

statistics-release-1.6.3/docs/ricefit.html000066400000000000000000000230321456127120000205720ustar00rootroot00000000000000 Statistics: ricefit

Function Reference: ricefit

statistics: paramhat = ricefit (x)
statistics: [paramhat, paramci] = ricefit (x)
statistics: [paramhat, paramci] = ricefit (x, alpha)
statistics: […] = ricefit (x, alpha, censor)
statistics: […] = ricefit (x, alpha, censor, freq)
statistics: […] = ricefit (x, alpha, censor, freq, options)

Estimate parameters and confidence intervals for the Gamma distribution.

paramhat = ricefit (x) returns the maximum likelihood estimates of the parameters of the Rician distribution given the data in x. paramhat(1) is the non-centrality (distance) parameter, nu, and paramhat(2) is the scale parameter, sigma.

[paramhat, paramci] = ricefit (x) returns the 95% confidence intervals for the parameter estimates.

[…] = ricefit (x, alpha) also returns the 100 * (1 - alpha) percent confidence intervals for the parameter estimates. By default, the optional argument alpha is 0.05 corresponding to 95% confidence intervals. Pass in [] for alpha to use the default values.

[…] = ricefit (x, alpha, censor) accepts a boolean vector, censor, of the same size as x with 1s for observations that are right-censored and 0s for observations that are observed exactly. By default, or if left empty, censor = zeros (size (x)).

[…] = ricefit (x, alpha, censor, freq) accepts a frequency vector, freq, of the same size as x. freq typically contains integer frequencies for the corresponding elements in x, but it can contain any non-integer non-negative values. By default, or if left empty, freq = ones (size (x)).

[…] = ricefit (…, options) specifies control parameters for the iterative algorithm used to compute the maximum likelihood estimates. options is a structure with the following field and its default value:

  • options.Display = "off"
  • options.MaxFunEvals = 1000
  • options.MaxIter = 500
  • options.TolX = 1e-6

Further information about the Rician distribution can be found at https://en.wikipedia.org/wiki/Rice_distribution

See also: ricecdf, ricepdf, riceinv, ricernd, ricelike, ricestat

Source Code: ricefit

Example: 1

  

 ## Sample 3 populations from different Gamma distibutions
 randg ("seed", 5);    # for reproducibility
 randp ("seed", 6);
 r1 = ricernd (1, 2, 3000, 1);
 randg ("seed", 2);    # for reproducibility
 randp ("seed", 8);
 r2 = ricernd (2, 4, 3000, 1);
 randg ("seed", 7);    # for reproducibility
 randp ("seed", 9);
 r3 = ricernd (7.5, 1, 3000, 1);
 r = [r1, r2, r3];

 ## Plot them normalized and fix their colors
 hist (r, 75, 4);
 h = findobj (gca, "Type", "patch");
 set (h(1), "facecolor", "c");
 set (h(2), "facecolor", "g");
 set (h(3), "facecolor", "r");
 ylim ([0, 0.7]);
 xlim ([0, 12]);
 hold on

 ## Estimate their α and β parameters
 nu_sigmaA = ricefit (r(:,1));
 nu_sigmaB = ricefit (r(:,2));
 nu_sigmaC = ricefit (r(:,3));

 ## Plot their estimated PDFs
 x = [0.01,0.1:0.2:18];
 y = ricepdf (x, nu_sigmaA(1), nu_sigmaA(2));
 plot (x, y, "-pr");
 y = ricepdf (x, nu_sigmaB(1), nu_sigmaB(2));
 plot (x, y, "-sg");
 y = ricepdf (x, nu_sigmaC(1), nu_sigmaC(2));
 plot (x, y, "-^c");
 hold off
 legend ({"Normalized HIST of sample 1 with k=1 and θ=2", ...
          "Normalized HIST of sample 2 with k=2 and θ=4", ...
          "Normalized HIST of sample 3 with k=7.5 and θ=1", ...
          sprintf("PDF for sample 1 with estimated k=%0.2f and θ=%0.2f", ...
                  nu_sigmaA(1), nu_sigmaA(2)), ...
          sprintf("PDF for sample 2 with estimated k=%0.2f and θ=%0.2f", ...
                  nu_sigmaB(1), nu_sigmaB(2)), ...
          sprintf("PDF for sample 3 with estimated k=%0.2f and θ=%0.2f", ...
                  nu_sigmaC(1), nu_sigmaC(2))})
 title ("Three population samples from different Rician distibutions")
 hold off
plotted figure

statistics-release-1.6.3/docs/riceinv.html000066400000000000000000000137231456127120000206120ustar00rootroot00000000000000 Statistics: riceinv

Function Reference: riceinv

statistics: x = riceinv (p, nu, sigma)

Inverse of the Rician distribution (iCDF).

For each element of p, compute the quantile (the inverse of the CDF) of the Rician distribution with with with non-centrality (distance) parameter nu and scale parameter sigma. The size of x is the common size of x, nu, and sigma. A scalar input functions as a constant matrix of the same size as the other inputs.

Further information about the Rician distribution can be found at https://en.wikipedia.org/wiki/Rice_distribution

See also: ricecdf, ricepdf, ricernd, ricefit, ricelike, ricestat

Source Code: riceinv

Example: 1

  

 ## Plot various iCDFs from the Beta distribution
 p = 0.001:0.001:0.999;
 x1 = riceinv (p, 0, 1);
 x2 = riceinv (p, 0.5, 1);
 x3 = riceinv (p, 1, 1);
 x4 = riceinv (p, 2, 1);
 x5 = riceinv (p, 4, 1);
 plot (p, x1, "-b", p, x2, "-g", p, x3, "-r", p, x4, "-m", p, x5, "-k")
 grid on
 legend ({"ν = 0, σ = 1", "ν = 0.5, σ = 1", "ν = 1, σ = 1", ...
          "ν = 2, σ = 1", "ν = 4, σ = 1"}, "location", "northwest")
 title ("Beta iCDF")
 xlabel ("probability")
 ylabel ("values in x")
plotted figure

statistics-release-1.6.3/docs/ricelike.html000066400000000000000000000137151456127120000207430ustar00rootroot00000000000000 Statistics: ricelike

Function Reference: ricelike

statistics: nlogL = ricelike (params, x)
statistics: [nlogL, acov] = ricelike (params, x)
statistics: […] = ricelike (params, x, censor)
statistics: […] = ricelike (params, x, censor, freq)

Negative log-likelihood for the Rician distribution.

nlogL = ricelike (params, x) returns the negative log likelihood of the data in x corresponding to the Rician distribution with (1) non-centrality (distance) parameter nu and (2) scale parameter sigma given in the two-element vector params.

[nlogL, acov] = ricelike (params, x) also returns the inverse of Fisher’s information matrix, acov. If the input parameter values in params are the maximum likelihood estimates, the diagonal elements of params are their asymptotic variances.

[…] = ricelike (params, x, censor) accepts a boolean vector, censor, of the same size as x with 1s for observations that are right-censored and 0s for observations that are observed exactly. By default, or if left empty, censor = zeros (size (x)).

[…] = ricelike (params, x, censor, freq) accepts a frequency vector, freq, of the same size as x. freq typically contains integer frequencies for the corresponding elements in x, but it can contain any non-integer non-negative values. By default, or if left empty, freq = ones (size (x)).

Further information about the Rician distribution can be found at https://en.wikipedia.org/wiki/Rice_distribution

See also: ricecdf, riceinv, ricepdf, ricernd, ricefit, ricestat

Source Code: ricelike

statistics-release-1.6.3/docs/ricepdf.html000066400000000000000000000137451456127120000205730ustar00rootroot00000000000000 Statistics: ricepdf

Function Reference: ricepdf

statistics: y = ricepdf (x, nu, sigma)

Rician probability density function (PDF).

For each element of x, compute the probability density function (PDF) of the Rician distribution with with non-centrality (distance) parameter nu and scale parameter sigma. The size of y is the common size of x, nu, and sigma. A scalar input functions as a constant matrix of the same size as the other inputs.

Further information about the Rician distribution can be found at https://en.wikipedia.org/wiki/Rice_distribution

See also: ricecdf, riceinv, ricernd, ricefit, ricelike, ricestat

Source Code: ricepdf

Example: 1

  

 ## Plot various PDFs from the Rician distribution
 x = 0:0.01:8;
 y1 = ricepdf (x, 0, 1);
 y2 = ricepdf (x, 0.5, 1);
 y3 = ricepdf (x, 1, 1);
 y4 = ricepdf (x, 2, 1);
 y5 = ricepdf (x, 4, 1);
 plot (x, y1, "-b", x, y2, "-g", x, y3, "-r", x, y4, "-m", x, y5, "-k")
 grid on
 ylim ([0, 0.65])
 xlim ([0, 8])
 legend ({"ν = 0, σ = 1", "ν = 0.5, σ = 1", "ν = 1, σ = 1", ...
          "ν = 2, σ = 1", "ν = 4, σ = 1"}, "location", "northeast")
 title ("Rician PDF")
 xlabel ("values in x")
 ylabel ("density")
plotted figure

statistics-release-1.6.3/docs/ricernd.html000066400000000000000000000123041456127120000205730ustar00rootroot00000000000000 Statistics: ricernd

Function Reference: ricernd

statistics: r = ricernd (nu, sigma)
statistics: r = ricernd (nu, sigma, rows)
statistics: r = ricernd (nu, sigma, rows, cols, …)
statistics: r = ricernd (nu, sigma, [sz])

Random arrays from the Rician distribution.

r = ricernd (nu, sigma) returns an array of random numbers chosen from the Rician distribution with shape parameters nu and sigma. The size of r is the common size of nu and sigma. A scalar input functions as a constant matrix of the same size as the other inputs.

When called with a single size argument, ricernd returns a square matrix with the dimension specified. When called with more than one scalar argument, the first two arguments are taken as the number of rows and columns and any further arguments specify additional matrix dimensions. The size may also be specified with a row vector of dimensions, sz.

Further information about the Rician distribution can be found at https://en.wikipedia.org/wiki/Rice_distribution

See also: ricecdf, riceinv, ricepdf, ricefit, ricelike, ricestat

Source Code: ricernd

statistics-release-1.6.3/docs/ricestat.html000066400000000000000000000110421456127120000207610ustar00rootroot00000000000000 Statistics: ricestat

Function Reference: ricestat

statistics: [m, v] = ricestat (nu, sigma)

Compute statistics of the Rician distribution.

[m, v] = ricestat (nu, sigma) returns the mean and variance of the Rician distribution with non-centrality (distance) parameter nu and scale parameter sigma.

The size of m (mean) and v (variance) is the common size of the input arguments. A scalar input functions as a constant matrix of the same size as the other inputs.

Further information about the Rician distribution can be found at https://en.wikipedia.org/wiki/Rice_distribution

See also: ricecdf, riceinv, ricepdf, ricernd, ricefit, ricelike

Source Code: ricestat

statistics-release-1.6.3/docs/ridge.html000066400000000000000000000206651456127120000202500ustar00rootroot00000000000000 Statistics: ridge

Function Reference: ridge

statistics: b = ridge (y, X, k)
statistics: b = ridge (y, X, k, scaled)

Ridge regression.

b = ridge (y, X, k) returns the vector of coefficient estimates by applying ridge regression from the predictor matrix X to the response vector y. Each value of b is the coefficient for the respective ridge parameter given k. By default, b is calculated after centering and scaling the predictors to have a zero mean and standard deviation 1.

b = ridge (y, X, k, scaled) performs the regression with the specified scaling of the coefficient estimates b. When scaled = 0, the function restores the coefficients to the scale of the original data thus is more useful for making predictions. When scaled = 1, the coefficient estimates correspond to the scaled centered data.

  • y must be an N×1 numeric vector with the response data.
  • X must be an N×p numeric matrix with the predictor data.
  • k must be a numeric vectir with the ridge parameters.
  • scaled must be a numeric scalar indicating whether the coefficient estimates in b are restored to the scale of the original data. By default, scaled = 1.

Further information about Ridge regression can be found at https://en.wikipedia.org/wiki/Ridge_regression

See also: lasso, stepwisefit, regress

Source Code: ridge

Example: 1

  

 ## Perform ridge regression for a range of ridge parameters and observe
 ## how the coefficient estimates change based on the acetylene dataset.

 load acetylene

 X = [x1, x2, x3];

 x1x2 = x1 .* x2;
 x1x3 = x1 .* x3;
 x2x3 = x2 .* x3;

 D = [x1, x2, x3, x1x2, x1x3, x2x3];

 k = 0:1e-5:5e-3;

 b = ridge (y, D, k);

 figure
 plot (k, b, "LineWidth", 2)
 ylim ([-100, 100])
 grid on
 xlabel ("Ridge Parameter")
 ylabel ("Standardized Coefficient")
 title ("Ridge Trace")
 legend ("x1", "x2", "x3", "x1x2", "x1x3", "x2x3")
plotted figure

Example: 2

  


 load carbig
 X = [Acceleration Weight Displacement Horsepower];
 y = MPG;

 n = length(y);

 rand("seed",1); % For reproducibility

 c = cvpartition(n,'HoldOut',0.3);
 idxTrain = training(c,1);
 idxTest = ~idxTrain;

 idxTrain = training(c,1);
 idxTest = ~idxTrain;

 k = 5;
 b = ridge(y(idxTrain),X(idxTrain,:),k,0);

 % Predict MPG values for the test data using the model.
 yhat = b(1) + X(idxTest,:)*b(2:end);
 scatter(y(idxTest),yhat)

 hold on
 plot(y(idxTest),y(idxTest),"r")
 xlabel('Actual MPG')
 ylabel('Predicted MPG')
 hold off
plotted figure

statistics-release-1.6.3/docs/rmmissing.html000066400000000000000000000130711456127120000211570ustar00rootroot00000000000000 Statistics: rmmissing

Function Reference: rmmissing

statistics: R = rmmissing (A)
statistics: R = rmmissing (A, dim)
statistics: R = rmmissing (…, Name, Value)
statistics: [R TF] = rmmissing (…)

Remove missing or incomplete data from an array.

Given an input vector or matrix (2-D array) A, remove missing data from a vector or missing rows or columns from a matrix. A can be a numeric array, char array, or an array of cell strings. R returns the array after removal of missing data.

The values which represent missing data depend on the data type of A:

  • NaN: single, double.
  • ' ' (white space): char.
  • {''}: string cells.

Choose to remove rows (default) or columns by setting optional input dim:

  • 1: rows.
  • 2: columns.

Note: data types with no default ’missing’ value will always result in R == A and a TF output of false(size(A)).

Additional optional parameters are set by Name-Value pairs. These are:

  • MinNumMissing: minimum number of missing values to remove an entry, row or column, defined as a positive integer number. E.g.: if MinNumMissing is set to 2, remove the row of a numeric matrix only if it includes 2 or more NaN.

Optional return value TF is a logical array where true values represent removed entries, rows or columns from the original data A.

See also: fillmissing, ismissing, standardizeMissing

Source Code: rmmissing

statistics-release-1.6.3/docs/runstest.html000066400000000000000000000153671456127120000210500ustar00rootroot00000000000000 Statistics: runstest

Function Reference: runstest

statistics: h = runstest (x)
statistics: h = runstest (x, v)
statistics: h = runstest (x, "ud")
statistics: h = runstest (…, Name, Value)
statistics: [h, pval, stats] = runstest (…)

Run test for randomness in the vector x.

h = runstest (x) calculates the number of runs of consecutive values above or below the mean of x and tests the null hypothesis that the values in the data vector x come in random order. h is 1 if the test rejects the null hypothesis at the 5% significance level, or 0 otherwise.

h = runstest (x, v) tests the null hypothesis based on the number of runs of consecutive values above or below the specified reference value v. Values exactly equal to v are omitted.

h = runstest (x, "ud") calculates the number of runs up or down and tests the null hypothesis that the values in the data vector x follow a trend. Too few runs indicate a trend, while too many runs indicate an oscillation. Values exactly equal to the preceding value are omitted.

h = runstest (…, Name, Value) specifies additional options to the above tests by one or more Name-Value pair arguments.

NameValue
"alpha"the significance level. Default is 0.05.
"method"a string specifying the method used to compute the p-value of the test. It can be either "exact" to use an exact algorithm, or "approximate" to use a normal approximation. The default is "exact" for runs above/below, and for runs up/down when the length of x is less than or equal to 50. When testing for runs up/down and the length of x is greater than 50, then the default is "approximate", and the "exact" method is not available.
"tail"a string specifying the alternative hypothesis
"both"two-tailed (default)
"left"left-tailed
"right"right-tailed

See also: signrank, signtest

Source Code: runstest

statistics-release-1.6.3/docs/sampsizepwr.html000066400000000000000000000407221456127120000215360ustar00rootroot00000000000000 Statistics: sampsizepwr

Function Reference: sampsizepwr

statistics: n = sampsizepwr (testtype, params, p1)
statistics: n = sampsizepwr (testtype, params, p1, power)
statistics: power = sampsizepwr (testtype, params, p1, [], n)
statistics: p1 = sampsizepwr (testtype, params, [], power, n)
statistics: [n1, n2] = sampsizepwr ("t2", params, p1, power)
statistics: […] = sampsizepwr (testtype, params, p1, power, n, name, value)

Sample size and power calculation for hypothesis test.

sampsizepwr computes the sample size, power, or alternative parameter value for a hypothesis test, given the other two values. For example, you can compute the sample size required to obtain a particular power for a hypothesis test, given the parameter value of the alternative hypothesis.

n = sampsizepwr (testtype, params, p1) returns the sample size N required for a two-sided test of the specified type to have a power (probability of rejecting the null hypothesis when the alternative is true) of 0.90 when the significance level (probability of rejecting the null hypothesis when the null hypothesis is true) is 0.05. params specifies the parameter values under the null hypothesis. P1 specifies the value of the single parameter being tested under the alternative hypothesis. For the two-sample t-test, N is the value of the equal sample size for both samples, params specifies the parameter values of the first sample under the null and alternative hypotheses, and P1 specifies the value of the single parameter from the other sample under the alternative hypothesis.

The following TESTTYPE values are available:

"z"one-sample z-test for normally distributed data with known standard deviation. params is a two-element vector [MU0 SIGMA0] of the mean and standard deviation, respectively, under the null hypothesis. P1 is the value of the mean under the alternative hypothesis.
"t"one-sample t-test or paired t-test for normally distributed data with unknown standard deviation. params is a two-element vector [MU0 SIGMA0] of the mean and standard deviation, respectively, under the null hypothesis. P1 is the value of the mean under the alternative hypothesis.
"t2"two-sample pooled t-test (test for equal means) for normally distributed data with equal unknown standard deviations. params is a two-element vector [MU0 SIGMA0] of the mean and standard deviation of the first sample under the null and alternative hypotheses. P1 is the the mean of the second sample under the alternative hypothesis.
"var"chi-square test of variance for normally distributed data. params is the variance under the null hypothesis. P1 is the variance under the alternative hypothesis.
"p"test of the P parameter (success probability) for a binomial distribution. params is the value of P under the null hypothesis. P1 is the value of P under the alternative hypothesis.
"r"test of the correlation coefficient parameter for significance. params is the value of r under the null hypothesis. P1 is the value of r under the alternative hypothesis.

The "p" test for the binomial distribution is a discrete test for which increasing the sample size does not always increase the power. For N values larger than 200, there may be values smaller than the returned N value that also produce the desired power.

n = sampsizepwr (testtype, params, p1, power) returns the sample size N such that the power is power for the parameter value P1. For the two-sample t-test, N is the equal sample size of both samples.

[n1, n2] = sampsizepwr ("t2", params, p1, power) returns the sample sizes n1 and n2 for the two samples. These values are the same unless the "ratio" parameter, ratio = n2 / n2, is set to a value other than the default (See the name/value pair definition of ratio below).

power = sampsizepwr (testtype, params, p1, [], n) returns the power achieved for a sample size of n when the true parameter value is p1. For the two-sample t-test, n is the smaller one of the two sample sizes.

p1 = sampsizepwr (testtype, params, [], power, n) returns the parameter value detectable with the specified sample size n and power power. For the two-sample t-test, n is the smaller one of the two sample sizes. When computing p1 for the "p" test, if no alternative can be rejected for a given params, n and power value, the function displays a warning message and returns NaN.

[…] = sampsizepwr (…, n, name, value) specifies one or more of the following name / value pairs:

"alpha"significance level of the test (default is 0.05)
"tail"the type of test which can be:
"both"two-sided test for an alternative p1 not equal to params
"right"one-sided test for an alternative p1 larger than params
"left"one-sided test for an alternative p1 smaller than params
"ratio"desired ratio n2 / n2 of the larger sample size n2 to the smaller sample size n1. Used only for the two-sample t-test. The value of ratio is greater than or equal to 1 (default is 1).

sampsizepwr computes the sample size, power, or alternative hypothesis value given values for the other two. Specify one of these as [] to compute it. The remaining parameters (and ALPHA, RATIO) can be scalars or arrays of the same size.

See also: vartest, ttest, ttest2, ztest, binocdf

Source Code: sampsizepwr

Example: 1

  

 ## Compute the mean closest to 100 that can be determined to be
 ## significantly different from 100 using a t-test with a sample size
 ## of 60 and a power of 0.8.
 mu1 = sampsizepwr ("t", [100, 10], [], 0.8, 60);
 disp (mu1);

103.68

Example: 2

  

 ## Compute the sample sizes required to distinguish mu0 = 100 from
 ## mu1 = 110 by a two-sample t-test with a ratio of the larger and the
 ## smaller sample sizes of 1.5 and a power of 0.6.
 [N1,N2] = sampsizepwr ("t2", [100, 10], 110, 0.6, [], "ratio", 1.5)

N1 = 9
N2 = 14

Example: 3

  

 ## Compute the sample size N required to distinguish p=.26 from p=.2
 ## with a binomial test.  The result is approximate, so make a plot to
 ## see if any smaller N values also have the required power of 0.6.
 Napprox = sampsizepwr ("p", 0.2, 0.26, 0.6);
 nn = 1:250;
 pwr = sampsizepwr ("p", 0.2, 0.26, [], nn);
 Nexact = min (nn(pwr >= 0.6));
 plot(nn,pwr,'b-', [Napprox Nexact],pwr([Napprox Nexact]),'ro');
 grid on

warning: sampsizepwr: approximate N.
warning: called from
    sampsizepwr at line 358 column 11
    build_DEMOS at line 94 column 11
    function_texi2html at line 112 column 14
    package_texi2html at line 290 column 9
plotted figure

Example: 4

  

 ## The company must test 52 bottles to detect the difference between a mean
 ## volume of 100 mL and 102 mL with a power of 0.80.  Generate a power curve
 ## to visualize how the sample size affects the power of the test.

 nout = sampsizepwr('t',[100 5],102,0.80);
 nn = 1:100;
 pwrout = sampsizepwr('t',[100 5],102,[],nn);

 figure;
 plot (nn, pwrout, "b-", nout, 0.8, "ro")
 title ("Power versus Sample Size")
 xlabel ("Sample Size")
 ylabel ("Power")
plotted figure

statistics-release-1.6.3/docs/sigma_pts.html000066400000000000000000000151771456127120000211460ustar00rootroot00000000000000 Statistics: sigma_pts

Function Reference: sigma_pts

statistics: pts = sigma_pts (n)
statistics: pts = sigma_pts (n, m)
statistics: pts = sigma_pts (n, m, K)
statistics: pts = sigma_pts (n, m, K, l)

Calculates 2*n+1 sigma points in n dimensions.

Sigma points are used in the unscented transfrom to estimate the result of applying a given nonlinear transformation to a probability distribution that is characterized only in terms of a finite set of statistics.

If only the dimension n is given the resulting points have zero mean and identity covariance matrix. If the mean m or the covaraince matrix K are given, then the resulting points will have those statistics. The factor l scaled the points away from the mean. It is useful to tune the accuracy of the unscented transfrom.

There is no unique way of computing sigma points, this function implements the algorithm described in section 2.6 "The New Filter" pages 40-41 of

Uhlmann, Jeffrey (1995). "Dynamic Map Building and Localization: New Theoretical Foundations". Ph.D. thesis. University of Oxford.

Source Code: sigma_pts

Example: 1

  

 K      = [1 0.5; 0.5 1]; # covaraince matrix
 # calculate and build associated ellipse
 [R,S,~] = svd (K);
 theta   = atan2 (R(2,1), R(1,1));
 v       = sqrt (diag (S));
 v       = v .* [cos(theta) sin(theta); -sin(theta) cos(theta)];
 t       = linspace (0, 2*pi, 100).';
 xe      = v(1,1) * cos (t) + v(2,1) * sin (t);
 ye      = v(1,2) * cos (t) + v(2,2) * sin (t);

 figure(1); clf; hold on
 # Plot ellipse and axes
 line ([0 0; v(:,1).'],[0 0; v(:,2).'])
 plot (xe,ye,'-r');

 col = 'rgb';
 l     = [-1.8 -1 1.5];
 for li = 1:3
  p     = sigma_pts (2, [], K, l(li));
  tmp   = plot (p(2:end,1), p(2:end,2), ['x' col(li)], ...
               p(1,1), p(1,2), ['o' col(li)]);
  h(li) = tmp(1);
 endfor
 hold off
 axis image
 legend (h, arrayfun (@(x) sprintf ("l:%.2g", x), l, "unif", 0));
plotted figure

statistics-release-1.6.3/docs/signtest.html000066400000000000000000000140551456127120000210120ustar00rootroot00000000000000 Statistics: signtest

Function Reference: signtest

statistics: [pval, h, stats] = signtest (x)
statistics: [pval, h, stats] = signtest (x, m)
statistics: [pval, h, stats] = signtest (x, y)
statistics: [pval, h, stats] = signtest (x, y, Name, Value)

Test for median.

Perform a signtest of the null hypothesis that x is from a distribution that has a zero median. x must be a vector.

If the second argument m is a scalar, the null hypothesis is that X has median m.

If the second argument y is a vector, the null hypothesis is that the distribution of x - y has zero median.

The argument "alpha" can be used to specify the significance level of the test (the default value is 0.05). The string argument "tail", can be used to select the desired alternative hypotheses. If "tail" is "both" (default) the null is tested against the two-sided alternative median (x) != m. If "tail" is "right" the one-sided alternative median (x) > m is considered. Similarly for "left", the one-sided alternative median (x) < m is considered.

When "method" is "exact" the p-value is computed using an exact method. When "method" is "approximate" a normal approximation is used for the test statistic. When "method" is not defined as an optional input argument, then for length (x) < 100 the "exact" method is computed, otherwise the "approximate" method is used.

The p-value of the test is returned in pval. If h is 0 the null hypothesis is accepted, if it is 1 the null hypothesis is rejected. stats is a structure containing the value of the test statistic (sign) and the value of the z statistic (zval) (only computed when the ’method’ is ’approximate’.

Source Code: signtest

statistics-release-1.6.3/docs/silhouette.html000066400000000000000000000161421456127120000213360ustar00rootroot00000000000000 Statistics: silhouette

Function Reference: silhouette

statistics: silhouette (X, clust)
statistics: [si, h] = silhouette (X, clust)
statistics: [si, h] = silhouette (…, Metric, MetricArg)

Compute the silhouette values of clustered data and show them on a plot.

X is a n-by-p matrix of n data points in a p-dimensional space. Each datapoint is assigned to a cluster using clust, a vector of n elements, one cluster assignment for each data point.

Each silhouette value of si, a vector of size n, is a measure of the likelihood that a data point is accurately classified to the right cluster. Defining "a" as the mean distance between a point and the other points from its cluster, and "b" as the mean distance between that point and the points from other clusters, the silhouette value of the i-th point is:

$$ S_i = \frac{b_i - a_i}{max(a_1,b_i)} $$

Each element of si ranges from -1, minimum likelihood of a correct classification, to 1, maximum likelihood.

Optional input value Metric is the metric used to compute the distances between data points. Since silhouette uses pdist to compute these distances, Metric is quite similar to the option Metric of pdist and it can be:

  • A known distance metric defined as a string: Euclidean, sqEuclidean (default), cityblock, cosine, correlation, Hamming, Jaccard.
  • A vector as those created by pdist. In this case X does nothing.
  • A function handle that is passed to pdist with MetricArg as optional inputs.

Optional return value h is a handle to the silhouette plot.

Reference Peter J. Rousseeuw, Silhouettes: a Graphical Aid to the Interpretation and Validation of Cluster Analysis. 1987. doi:10.1016/0377-0427(87)90125-7

See also: dendrogram, evalclusters, kmeans, linkage, pdist

Source Code: silhouette

Example: 1

  

 load fisheriris;
 X = meas(:,3:4);
 cidcs = kmeans (X, 3, "Replicates", 5);
 silhouette (X, cidcs);
 y_labels(cidcs([1 51 101])) = unique (species);
 set (gca, "yticklabel", y_labels);
 title ("Fisher's iris data");
plotted figure

statistics-release-1.6.3/docs/slicesample.html000066400000000000000000000241231456127120000214500ustar00rootroot00000000000000 Statistics: slicesample

Function Reference: slicesample

statistics: [smpl, neval] = slicesample (start, nsamples, property, value, …)

Draws nsamples samples from a target stationary distribution pdf using slice sampling of Radford M. Neal.

Input:

  • start is a 1 by dim vector of the starting point of the Markov chain. Each column corresponds to a different dimension.
  • nsamples is the number of samples, the length of the Markov chain.

Next, several property-value pairs can or must be specified, they are:

(Required properties) One of:

  • "pdf": the value is a function handle of the target stationary distribution to be sampled. The function should accept different locations in each row and each column corresponds to a different dimension.

    or

  • logpdf: the value is a function handle of the log of the target stationary distribution to be sampled. The function should accept different locations in each row and each column corresponds to a different dimension.

The following input property/pair values may be needed depending on the desired outut:

  • "burnin" burnin the number of points to discard at the beginning, the default is 0.
  • "thin" thin omitts m-1 of every m points in the generated Markov chain. The default is 1.
  • "width" width the maximum Manhattan distance between two samples. The default is 10.

Outputs:

  • smpl is a nsamples by dim matrix of random values drawn from pdf where the rows are different random values, the columns correspond to the dimensions of pdf.
  • neval is the number of function evaluations per sample.

Example : Sampling from a normal distribution

 
 
 start = 1;
 nsamples = 1e3;
 pdf = @(x) exp (-.5 * x .^ 2) / (pi ^ .5 * 2 ^ .5);
 [smpl, accept] = slicesample (start, nsamples, "pdf", pdf, "thin", 4);
 histfit (smpl);

See also: rand, mhsample, randsample

Source Code: slicesample

Example: 1

  

 ## Define function to sample
 d = 2;
 mu = [-1; 2];
 rand ("seed", 5)  # for reproducibility
 Sigma = rand (d);
 Sigma = (Sigma + Sigma');
 Sigma += eye (d)*abs (eigs (Sigma, 1, "sa")) * 1.1;
 pdf = @(x)(2*pi)^(-d/2)*det(Sigma)^-.5*exp(-.5*sum((x.'-mu).*(Sigma\(x.'-mu)),1));

 ## Inputs
 start = ones (1,2);
 nsamples = 500;
 K = 500;
 m = 10;
 rande ("seed", 4);  rand ("seed", 5)  # for reproducibility
 [smpl, accept] = slicesample (start, nsamples, "pdf", pdf, "burnin", K, "thin", m, "width", [20, 30]);
 figure;
 hold on;
 plot (smpl(:,1), smpl(:,2), 'x');
 [x, y] = meshgrid (linspace (-6,4), linspace(-3,7));
 z = reshape (pdf ([x(:), y(:)]), size(x));
 mesh (x, y, z, "facecolor", "None");

 ## Using sample points to find the volume of half a sphere with radius of .5
 f = @(x) ((.25-(x(:,1)+1).^2-(x(:,2)-2).^2).^.5.*(((x(:,1)+1).^2+(x(:,2)-2).^2)<.25)).';
 int = mean (f (smpl) ./ pdf (smpl));
 errest = std (f (smpl) ./ pdf (smpl)) / nsamples^.5;
 trueerr = abs (2/3*pi*.25^(3/2)-int);
 fprintf ("Monte Carlo integral estimate int f(x) dx = %f\n", int);
 fprintf ("Monte Carlo integral error estimate %f\n", errest);
 fprintf ("The actual error %f\n", trueerr);
 mesh (x,y,reshape (f([x(:), y(:)]), size(x)), "facecolor", "None");

Monte Carlo integral estimate int f(x) dx = 0.228408
Monte Carlo integral error estimate 0.029831
The actual error 0.033392
plotted figure

Example: 2

  

 ## Integrate truncated normal distribution to find normilization constant
 pdf = @(x) exp (-.5*x.^2)/(pi^.5*2^.5);
 nsamples = 1e3;
 rande ("seed", 4);  rand ("seed", 5)  # for reproducibility
 [smpl, accept] = slicesample (1, nsamples, "pdf", pdf, "thin", 4);
 f = @(x) exp (-.5 * x .^ 2) .* (x >= -2 & x <= 2);
 x = linspace (-3, 3, 1000);
 area (x, f(x));
 xlabel ("x");
 ylabel ("f(x)");
 int = mean (f (smpl) ./ pdf (smpl));
 errest = std (f (smpl) ./ pdf (smpl)) / nsamples ^ 0.5;
 trueerr = abs (erf (2 ^ 0.5) * 2 ^ 0.5 * pi ^ 0.5 - int);
 fprintf("Monte Carlo integral estimate int f(x) dx = %f\n", int);
 fprintf("Monte Carlo integral error estimate %f\n", errest);
 fprintf("The actual error %f\n", trueerr);

Monte Carlo integral estimate int f(x) dx = 2.376284
Monte Carlo integral error estimate 0.017608
The actual error 0.016292
plotted figure

statistics-release-1.6.3/docs/squareform.html000066400000000000000000000117551456127120000213420ustar00rootroot00000000000000 Statistics: squareform

Function Reference: squareform

statistics: z = squareform (y)
statistics: y = squareform (z)
statistics: z = squareform (y, "tovector")
statistics: y = squareform (z, "tomatrix")

Interchange between distance matrix and distance vector formats.

Converts between an hollow (diagonal filled with zeros), square, and symmetric matrix and a vector with of the lower triangular part.

Its target application is the conversion of the vector returned by pdist into a distance matrix. It performs the opposite operation if input is a matrix.

If x is a vector, its number of elements must fit into the triangular part of a matrix (main diagonal excluded). In other words, numel (x) = n * (n - 1) / 2 for some integer n. The resulting matrix will be n by n.

If x is a distance matrix, it must be square and the diagonal entries of x must all be zeros. squareform will generate a warning if x is not symmetric.

The second argument is used to specify the output type in case there is a single element. It will defaults to "tomatrix" otherwise.

See also: pdist

Source Code: squareform

statistics-release-1.6.3/docs/standardizeMissing.html000066400000000000000000000122721456127120000230130ustar00rootroot00000000000000 Statistics: standardizeMissing

Function Reference: standardizeMissing

statistics: B = standardizeMissing (A, indicator)

Replace data values specified by indicator in A by the standard ’missing’ data value for that data type.

A can be a numeric scalar or array, a character vector or array, or a cell array of character vectors (a.k.a. string cells).

indicator can be a scalar or an array containing values to be replaced by the ’missing’ value for the class of A, and should have a data type matching A.

’missing’ values are defined as :

  • NaN: single, double
  • " " (white space): char
  • {""} (empty string in cell): string cells.

Compatibility Notes:

  • Octave’s implementation of standardizeMissing does not restrict indicator of type char to row vectors.
  • All numerical and logical inputs for A and indicator may be specified in any combination. The output will be the same class as A, with the indicator converted to that data type for comparison. Only single and double have defined ’missing’ values, so A of other data types will always output B = A.

See also: fillmissing, ismissing, rmmissing

Source Code: standardizeMissing

statistics-release-1.6.3/docs/std.html000066400000000000000000000172741456127120000177520ustar00rootroot00000000000000 Statistics: std

Function Reference: std

statistics: s = std (x)
statistics: s = std (x, w)
statistics: s = std (x, w, "all")
statistics: s = std (x, w, dim)
statistics: s = std (x, w, vecdim)
statistics: s = std (…, nanflag)
statistics: [s, m] = std (…)

Compute the standard deviation of the elements of x.

  • If x is a vector, then std (x) returns the standard deviation of the elements in x defined as $$ {\rm std}(x) = \sqrt{{1\over N-1} \sum_{i=1}^N |x_i - \bar x |^2} $$ where N is the length of the x vector.
  • If x is a matrix, then std (x) returns a row vector with the standard deviation of each column in x.
  • If x is a multi-dimensional array, then std (x) operates along the first non-singleton dimension of x.

std (x, w) specifies a weighting scheme. When w = 0 (default), the standard deviation is normalized by N-1 (population standard deviation), where N is the number of observations. When w = 1, the standard deviation is normalized by the number of observations (sample standard deviation). To use the default value you may pass an empty input argument [] before entering other options.

w can also be an array of non-negative numbers. When w is a vector, it must have the same length as the number of elements in the operating dimension of x. If w is a matrix or n-D array, or the operating dimension is supplied as a vecdim or "all", w must be the same size as x. NaN values are permitted in w, will be multiplied with the associated values in x, and can be excluded by the nanflag option.

std (x, [], dim) returns the standard deviation along the operating dimension dim of x. For dim greater than ndims (x), then s is returned as zeros of the same size as x and m = x.

std (x, [], vecdim) returns the standard deviation over the dimensions specified in the vector vecdim. For example, if x is a 2-by-3-by-4 array, then var (x, [1 2]) returns a 1-by-1-by-4 array. Each element of the output array is the standard deviation of the elements on the corresponding page of x. If vecdim indexes all dimensions of x, then it is equivalent to std (x, "all"). Any dimension in vecdim greater than ndims (x) is ignored.

std (x, "all") returns the standard deviation of all the elements in x. The optional flag "all" cannot be used together with dim or vecdim input arguments.

std (…, nanflag) specifies whether to exclude NaN values from the calculation using any of the input argument combinations in previous syntaxes. The default value for nanflag is "includenan", and keeps NaN values in the calculation. To exclude NaN values, set the value of nanflag to "omitnan".

[s, m] = std (…) also returns the mean of the elements of x used to calculate the standard deviation. If s is the weighted standard deviation, then m is the weighted mean.

See also: var, mean

Source Code: std

statistics-release-1.6.3/docs/stepwisefit.html000066400000000000000000000126031456127120000215150ustar00rootroot00000000000000 Statistics: stepwisefit

Function Reference: stepwisefit

statistics: [X_use, b, bint, r, rint, stats] = stepwisefit (y, X, penter = 0.05, premove = 0.1, method = "corr")

Linear regression with stepwise variable selection.

Arguments

  • y is an n by 1 vector of data to fit.
  • X is an n by k matrix containing the values of k potential predictors. No constant term should be included (one will always be added to the regression automatically).
  • penter is the maximum p-value to enter a new variable into the regression (default: 0.05).
  • premove is the minimum p-value to remove a variable from the regression (default: 0.1).
  • method sets how predictors are selected at each step, either based on their correlation with the residuals ("corr", default) or on the p values of their regression coefficients when they are successively added ("p").

Return values

  • X_use contains the indices of the predictors included in the final regression model. The predictors are listed in the order they were added, so typically the first ones listed are the most significant.
  • b, bint, r, rint, stats are the results of [b, bint, r, rint, stats] = regress(y, [ones(size(y)) X(:, X_use)], penter);

References

  1. N. R. Draper and H. Smith (1966). Applied Regression Analysis. Wiley. Chapter 6.

See also: regress

Source Code: stepwisefit

statistics-release-1.6.3/docs/svmpredict.html000066400000000000000000000143131456127120000213270ustar00rootroot00000000000000 Statistics: svmpredict

Function Reference: svmpredict

statistics: predicted_label = svmpredict (labels, data, model)
statistics: predicted_label = svmpredict (labels, data, model, libsvm_options)
statistics: [predicted_label, accuracy, decision_values] = svmpredict (labels, data, model, libsvm_options)
statistics: [predicted_label, accuracy, prob_estimates] = svmpredict (labels, data, model, libsvm_options)

This function predicts new labels from a testing instance matrtix based on an SVM model created with svmtrain.

  • labels : An m by 1 vector of prediction labels. If labels of test data are unknown, simply use any random values. (type must be double)
  • data : An m by n matrix of m testing instances with n features. It can be dense or sparse. (type must be double)
  • model : The output of svmtrain function.
  • libsvm_options : A string of testing options in the same format as that of LIBSVM.

libsvm_options :

  • -b : probability_estimates; whether to predict probability estimates. For one-class SVM only 0 is supported.
0return decision values. (default)
1return probability estimates.
  • -q : quiet mode. (no outputs)

The svmpredict function has three outputs. The first one, predicted_label, is a vector of predicted labels. The second output, accuracy, is a vector including accuracy (for classification), mean squared error, and squared correlation coefficient (for regression). The third is a matrix containing decision values or probability estimates (if -b 1’ is specified). If k is the number of classes in training data, for decision values, each row includes results of predicting k(k-1)/2 binary-class SVMs. For classification, k = 1 is a special case. Decision value +1 is returned for each testing instance, instead of an empty vector. For probabilities, each row contains k values indicating the probability that the testing instance is in each class. Note that the order of classes here is the same as Label field in the model structure.

Source Code: svmpredict

statistics-release-1.6.3/docs/svmtrain.html000066400000000000000000000177431456127120000210240ustar00rootroot00000000000000 Statistics: svmtrain

Function Reference: svmtrain

statistics: model = svmtrain (labels, data, libsvm_options)

This function trains an SVM model based on known labels and their corresponding data which comprise an instance matrtix.

  • labels : An m by 1 vector of prediction labels. (type must be double)
  • data : An m by n matrix of m testing instances with n features. It can be dense or sparse. (type must be double)
  • libsvm_options : A string of testing options in the same format as that of LIBSVM.

libsvm_options :

  • -s : svm_type; set type of SVM (default 0)
0C-SVC (multi-class classification)
1nu-SVC (multi-class classification)
2one-class SVM
3epsilon-SVR (regression)
4nu-SVR (regression)
  • -t : kernel_type; set type of kernel function (default 2)
0linear: u’*v
1polynomial: (gamma× u’× v + coef0) ^ degree
2radial basis function: e×p(-gamma× |u-v| ^ 2)
3sigmoid: tanh(gamma× u’× v + coef0)
4precomputed kernel (kernel values in training_instance_matrix)
  • -d : degree; set degree in kernel function (default 3)
  • -g : gamma; set gamma in kernel function (default 1/num_features)
  • -r : coef0; set coef0 in kernel function (default 0)
  • -c : cost; set the parameter C of C-SVC, epsilon-SVR, and nu-SVR (default 1)
  • -n : nu; set the parameter nu of nu-SVC, one-class SVM, and nu-SVR (default 0.5)
  • -p : epsilon; set the epsilon in loss function of epsilon-SVR (default 0.1)
  • -m : cachesize; set cache memory size in MB (default 100)
  • -e : epsilon; set tolerance of termination criterion (default 0.001)
  • -h : shrinking; whether to use the shrinking heuristics, 0 or 1 (default 1)
  • -b : probability_estimates; whether to train a SVC or SVR model for probability estimates, 0 or 1 (default 0)
  • -w : weight; set the parameter C of class i to weight*C, for C-SVC (default 1)
  • -v : n; n-fold cross validation mode
  • -q : quiet mode (no outputs)

The function svmtrain function returns a model structure which can be used for future prediction and it contains the following fields:

  • Parameters : parameters
  • nr_class : number of classes; = 2 for regression/one-class svm
  • totalSV : total #SV
  • rho : -b of the decision function(s) w×+b
  • Label : label of each class; empty for regression/one-class SVM
  • sv_indices : values in [1,...,num_traning_data] to indicate SVs in the training set
  • ProbA : pairwise probability information; empty if -b 0 or in one-class SVM
  • ProbB : pairwise probability information; empty if -b 0 or in one-class SVM
  • nSV : number of SVs for each class; empty for regression/one-class SVM
  • sv_coef : coefficients for SVs in decision functions
  • SVs : support vectors

If you do not use the option -b 1, ProbA and ProbB are empty matrices. If the ’-v’ option is specified, cross validation is conducted and the returned model is just a scalar: cross-validation accuracy for classification and mean-squared error for regression.

Source Code: svmtrain

statistics-release-1.6.3/docs/tabulate.html000066400000000000000000000256271456127120000207620ustar00rootroot00000000000000 Statistics: tabulate

Function Reference: tabulate

statistics: tabulate (x)
statistics: table = tabulate (x)

Calculate a frequency table.

tabulate (x) displays a frequency table of the data in the vector x. For each unique value in x, the tabulate function shows the number of instances and percentage of that value in x.

table = tabulate (x) returns the frequency table, table, as a numeric matrix when x is numeric and as a cell array otherwise. When an output argument is requested, tabulate does not print the frequency table in the command window.

If x is numeric, any missing values (NaNs) are ignored.

If all the elements of x are positive integers, then the frequency table includes 0 counts for the integers between 1 and max (x) that do not appear in x.

See also: bar, pareto

Source Code: tabulate

Example: 1

  

 ## Generate a frequency table for a vector of data in a cell array
 load patients

 ## Display the first seven entries of the Gender variable
 gender = Gender(1:7)

 ## Compute the equency table that shows the number and
 ## percentage of Male and Female patients
 tabulate (Gender)

gender =
{
  [1,1] = Male
  [2,1] = Male
  [3,1] = Female
  [4,1] = Female
  [5,1] = Female
  [6,1] = Female
  [7,1] = Female
}

   Value    Count    Percent
    Male       47      47.00%
  Female       53      53.00%

Example: 2

  

 ## Create a frequency table for a vector of positive integers
 load patients

 ## Display the first seven entries of the Gender variable
 height = Height(1:7)

 ## Create a frequency table that shows, in its second and third columns,
 ## the number and percentage of patients with a particular height.
 table = tabulate (Height);

 ## Display the first and last seven entries of the frequency table
 first = table(1:7,:)

 last = table(end-6:end,:)

height =

   71
   69
   64
   67
   64
   68
   64

first =

   1   0   0
   2   0   0
   3   0   0
   4   0   0
   5   0   0
   6   0   0
   7   0   0

last =

   66   15   15
   67    6    6
   68   15   15
   69    8    8
   70   11   11
   71   10   10
   72    4    4

Example: 3

  

 ## Create a frequency table from a character array
 load carsmall

 ## Tabulate the data in the Origin variable, which shows the
 ## country of origin of each car in the data set
 tabulate (Origin)

    Value    Count    Percent
      USA       69      69.00%
   France        4       4.00%
    Japan       15      15.00%
  Germany        9       9.00%
   Sweden        2       2.00%
    Italy        1       1.00%

Example: 4

  

 ## Create a frequency table from a numeric vector with NaN values
 load carsmall

 ## The carsmall dataset contains measurements of 100 cars
 total_cars = length (MPG)
 ## For six cars, the MPG value is missing
 missingMPG = length (MPG(isnan (MPG)))

 ## Create a frequency table using MPG
 tabulate (MPG)
 table = tabulate (MPG);

 ## Only 94 cars were used
 valid_cars = sum (table(:,2))

total_cars = 100
missingMPG = 6
  Value    Count    Percent
     18        4       4.26%
     15        5       5.32%
     16        2       2.13%
     17        1       1.06%
     14        5       5.32%
     24        4       4.26%
     22        4       4.26%
     21        2       2.13%
     27        6       6.38%
     26        4       4.26%
     25        5       5.32%
     10        2       2.13%
     11        1       1.06%
      9        1       1.06%
     28        4       4.26%
   17.5        2       2.13%
   15.5        1       1.06%
   14.5        1       1.06%
   22.5        1       1.06%
     29        3       3.19%
   24.5        1       1.06%
     33        1       1.06%
     20        2       2.13%
   18.5        1       1.06%
   29.5        1       1.06%
     32        4       4.26%
   26.5        1       1.06%
     13        4       4.26%
     19        2       2.13%
   16.5        2       2.13%
     34        2       2.13%
     31        3       3.19%
     23        1       1.06%
     36        5       5.32%
     37        1       1.06%
     38        4       4.26%
     44        1       1.06%
valid_cars = 94
statistics-release-1.6.3/docs/tcdf.html000066400000000000000000000141141456127120000200660ustar00rootroot00000000000000 Statistics: tcdf

Function Reference: tcdf

statistics: p = tcdf (x, df)
statistics: p = tcdf (x, df, "upper")

Student’s T cumulative distribution function (CDF).

For each element of x, compute the cumulative distribution function (CDF) of the Student’s T distribution with df degrees of freedom. The size of p is the common size of x and df. A scalar input functions as a constant matrix of the same size as the other input.

p = tcdf (x, df, "upper") computes the upper tail probability of the Student’s T distribution with df degrees of freedom, at the values in x.

Further information about the Student’s T distribution can be found at https://en.wikipedia.org/wiki/Student%27s_t-distribution

See also: tinv, tpdf, trnd, tstat

Source Code: tcdf

Example: 1

  

 ## Plot various CDFs from the Student's T distribution
 x = -5:0.01:5;
 p1 = tcdf (x, 1);
 p2 = tcdf (x, 2);
 p3 = tcdf (x, 5);
 p4 = tcdf (x, Inf);
 plot (x, p1, "-b", x, p2, "-g", x, p3, "-r", x, p4, "-m")
 grid on
 xlim ([-5, 5])
 ylim ([0, 1])
 legend ({"df = 1", "df = 2", ...
          "df = 5", 'df = \infty'}, "location", "southeast")
 title ("Student's T CDF")
 xlabel ("values in x")
 ylabel ("probability")
plotted figure

statistics-release-1.6.3/docs/tinv.html000066400000000000000000000137111456127120000201300ustar00rootroot00000000000000 Statistics: tinv

Function Reference: tinv

statistics: x = tinv (p, df)

Inverse of the Student’s T cumulative distribution function (iCDF).

For each element of p, compute the quantile (the inverse of the CDF) of the Student’s T distribution with df degrees of freedom. The size of x is the common size of x and df. A scalar input functions as a constant matrix of the same size as the other input.

This function is analogous to looking in a table for the t-value of a single-tailed distribution. For very large df (>10000), the inverse of the standard normal distribution is used.

Further information about the Student’s T distribution can be found at https://en.wikipedia.org/wiki/Student%27s_t-distribution

See also: tcdf, tpdf, trnd, tstat

Source Code: tinv

Example: 1

  

 ## Plot various iCDFs from the Student's T distribution
 p = 0.001:0.001:0.999;
 x1 = tinv (p, 1);
 x2 = tinv (p, 2);
 x3 = tinv (p, 5);
 x4 = tinv (p, Inf);
 plot (p, x1, "-b", p, x2, "-g", p, x3, "-r", p, x4, "-m")
 grid on
 xlim ([0, 1])
 ylim ([-5, 5])
 legend ({"df = 1", "df = 2", ...
          "df = 5", 'df = \infty'}, "location", "northwest")
 title ("Student's T iCDF")
 xlabel ("probability")
 ylabel ("values in x")
plotted figure

statistics-release-1.6.3/docs/tlscdf.html000066400000000000000000000151761456127120000204360ustar00rootroot00000000000000 Statistics: tlscdf

Function Reference: tlscdf

statistics: p = tlscdf (x, mu, sigma, df)
statistics: p = tlscdf (x, mu, sigma, df, "upper")

Location-scale Student’s T cumulative distribution function (CDF).

For each element of x, compute the cumulative distribution function (CDF) of the location-scale Student’s T distribution with location parameter mu, scale parameter sigma, and df degrees of freedom. The size of p is the common size of x, mu, sigma, and df. A scalar input functions as a constant matrix of the same size as the other inputs.

p = tlscdf (x, mu, sigma, df, "upper") computes the upper tail probability of the location-scale Student’s T distribution with parameters mu, sigma, and df, at the values in x.

Further information about the location-scale Student’s T distribution can be found at https://en.wikipedia.org/wiki/Student%27s_t-distribution#Location-scale_t_distribution

See also: tlsinv, tlspdf, tlsrnd, tlsfit, tlslike, tlsstat

Source Code: tlscdf

Example: 1

  

 ## Plot various CDFs from the location-scale Student's T distribution
 x = -8:0.01:8;
 p1 = tlscdf (x, 0, 1, 1);
 p2 = tlscdf (x, 0, 2, 2);
 p3 = tlscdf (x, 3, 2, 5);
 p4 = tlscdf (x, -1, 3, Inf);
 plot (x, p1, "-b", x, p2, "-g", x, p3, "-r", x, p4, "-m")
 grid on
 xlim ([-8, 8])
 ylim ([0, 1])
 legend ({"mu = 0, sigma = 1, df = 1", "mu = 0, sigma = 2, df = 2", ...
          "mu = 3, sigma = 2, df = 5", 'mu = -1, sigma = 3, df = \infty'}, ...
         "location", "northwest")
 title ("Location-scale Student's T CDF")
 xlabel ("values in x")
 ylabel ("probability")
plotted figure

statistics-release-1.6.3/docs/tlsinv.html000066400000000000000000000142671456127120000204760ustar00rootroot00000000000000 Statistics: tlsinv

Function Reference: tlsinv

statistics: x = tlsinv (p, mu, sigma, df)

Inverse of the location-scale Student’s T cumulative distribution function (iCDF).

For each element of p, compute the quantile (the inverse of the CDF) of the location-scale Student’s T distribution with location parameter mu, scale parameter sigma, and df degrees of freedom. The size of x is the common size of p, mu, sigma, and df. A scalar input functions as a constant matrix of the same size as the other inputs.

Further information about the location-scale Student’s T distribution can be found at https://en.wikipedia.org/wiki/Student%27s_t-distribution#Location-scale_t_distribution

See also: tlscdf, tlspdf, tlsrnd, tlsfit, tlslike, tlsstat

Source Code: tlsinv

Example: 1

  

 ## Plot various iCDFs from the location-scale Student's T distribution
 p = 0.001:0.001:0.999;
 x1 = tlsinv (p, 0, 1, 1);
 x2 = tlsinv (p, 0, 2, 2);
 x3 = tlsinv (p, 3, 2, 5);
 x4 = tlsinv (p, -1, 3, Inf);
 plot (p, x1, "-b", p, x2, "-g", p, x3, "-r", p, x4, "-m")
 grid on
 xlim ([0, 1])
 ylim ([-8, 8])
 legend ({"mu = 0, sigma = 1, df = 1", "mu = 0, sigma = 2, df = 2", ...
          "mu = 3, sigma = 2, df = 5", 'mu = -1, sigma = 3, df = \infty'}, ...
         "location", "southeast")
 title ("Location-scale Student's T iCDF")
 xlabel ("probability")
 ylabel ("values in x")
plotted figure

statistics-release-1.6.3/docs/tlspdf.html000066400000000000000000000142121456127120000204410ustar00rootroot00000000000000 Statistics: tlspdf

Function Reference: tlspdf

statistics: p = tlspdf (x, mu, sigma, df)

Location-scale Student’s T probability density function (PDF).

For each element of x, compute the probability density function (PDF) of the location-scale Student’s T distribution with location parameter mu, scale parameter sigma, and df degrees of freedom. The size of y is the common size of x, mu, sigma, and df. A scalar input functions as a constant matrix of the same size as the other inputs.

Further information about the location-scale Student’s T distribution can be found at https://en.wikipedia.org/wiki/Student%27s_t-distribution#Location-scale_t_distribution

See also: tlscdf, tlspdf, tlsrnd, tlsfit, tlslike, tlsstat

Source Code: tlspdf

Example: 1

  

 ## Plot various PDFs from the Student's T distribution
 x = -8:0.01:8;
 y1 = tlspdf (x, 0, 1, 1);
 y2 = tlspdf (x, 0, 2, 2);
 y3 = tlspdf (x, 3, 2, 5);
 y4 = tlspdf (x, -1, 3, Inf);
 plot (x, y1, "-b", x, y2, "-g", x, y3, "-r", x, y4, "-m")
 grid on
 xlim ([-8, 8])
 ylim ([0, 0.41])
 legend ({"mu = 0, sigma = 1, df = 1", "mu = 0, sigma = 2, df = 2", ...
          "mu = 3, sigma = 2, df = 5", 'mu = -1, sigma = 3, df = \infty'}, ...
         "location", "northwest")
 title ("Location-scale Student's T PDF")
 xlabel ("values in x")
 ylabel ("density")
plotted figure

statistics-release-1.6.3/docs/tlsrnd.html000066400000000000000000000131241456127120000204540ustar00rootroot00000000000000 Statistics: tlsrnd

Function Reference: tlsrnd

statistics: r = tlsrnd (mu, sigma, df)
statistics: r = tlsrnd (mu, sigma, df, rows)
statistics: r = tlsrnd (mu, sigma, df, rows, cols, …)
statistics: r = tlsrnd (mu, sigma, df, [sz])

Random arrays from the location-scale Student’s T distribution.

Return a matrix of random samples from the location-scale Student’s T distribution with location parameter mu, scale parameter sigma, and df degrees of freedom.

r = tlsrnd (df) returns an array of random numbers chosen from the location-scale Student’s T distribution with location parameter mu, scale parameter sigma, and df degrees of freedom. The size of r is the common size of mu, sigma, and df. A scalar input functions as a constant matrix of the same size as the other inputs.

When called with a single size argument, tlsrnd returns a square matrix with the dimension specified. When called with more than one scalar argument, the first two arguments are taken as the number of rows and columns and any further arguments specify additional matrix dimensions. The size may also be specified with a row vector of dimensions, sz.

Further information about the location-scale Student’s T distribution can be found at https://en.wikipedia.org/wiki/Student%27s_t-distribution#Location-scale_t_distribution

See also: tlscdf, tlsinv, tlspdf, tlsfit, tlslike, tlsstat

Source Code: tlsrnd

statistics-release-1.6.3/docs/tlsstat.html000066400000000000000000000112441456127120000206450ustar00rootroot00000000000000 Statistics: tlsstat

Function Reference: tlsstat

statistics: [m, v] = tlsstat (mu, sigma, df)

Compute statistics of the location-scale Student’s T distribution.

[m, v] = tlsstat (mu, sigma, df) returns the mean and variance of the location-scale Student’s T distribution with location parameter mu, scale parameter sigma, and df degrees of freedom.

The size of m (mean) and v (variance) is the common size of the input arguments. A scalar input functions as a constant matrix of the same size as the other inputs.

Further information about the location-scale Student’s T distribution can be found at https://en.wikipedia.org/wiki/Student%27s_t-distribution#Location-scale_t_distribution

See also: tlscdf, tlsinv, tlspdf, tlsrnd, tlsfit, tlslike

Source Code: tlsstat

statistics-release-1.6.3/docs/tpdf.html000066400000000000000000000133331456127120000201050ustar00rootroot00000000000000 Statistics: tpdf

Function Reference: tpdf

statistics: p = tpdf (x, df)

Student’s T probability density function (PDF).

For each element of x, compute the probability density function (PDF) of the Student’s T distribution with df degrees of freedom. The size of y is the common size of x and df. A scalar input functions as a constant matrix of the same size as the other input.

Further information about the Student’s T distribution can be found at https://en.wikipedia.org/wiki/Student%27s_t-distribution

See also: tcdf, tpdf, trnd, tstat

Source Code: tpdf

Example: 1

  

 ## Plot various PDFs from the Student's T distribution
 x = -5:0.01:5;
 y1 = tpdf (x, 1);
 y2 = tpdf (x, 2);
 y3 = tpdf (x, 5);
 y4 = tpdf (x, Inf);
 plot (x, y1, "-b", x, y2, "-g", x, y3, "-r", x, y4, "-m")
 grid on
 xlim ([-5, 5])
 ylim ([0, 0.41])
 legend ({"df = 1", "df = 2", ...
          "df = 5", 'df = \infty'}, "location", "northeast")
 title ("Student's T PDF")
 xlabel ("values in x")
 ylabel ("density")
plotted figure

statistics-release-1.6.3/docs/tricdf.html000066400000000000000000000144421456127120000204250ustar00rootroot00000000000000 Statistics: tricdf

Function Reference: tricdf

statistics: p = tricdf (x, a, b, c)
statistics: p = tricdf (x, a, b, c, "upper")

Triangular cumulative distribution function (CDF).

For each element of x, compute the cumulative distribution function (CDF) of the triangular distribution with parameters a, b, and c on the interval [a, b]. The size of p is the common size of the input arguments. A scalar input functions as a constant matrix of the same size as the other inputs.

p = tricdf (x, a, b, c, "upper") computes the upper tail probability of the triangular distribution with parameters a, b, and c, at the values in x.

Further information about the triangular distribution can be found at https://en.wikipedia.org/wiki/Triangular_distribution

See also: triinv, tripdf, trirnd

Source Code: tricdf

Example: 1

  

 ## Plot various CDFs from the triangular distribution
 x = 0.001:0.001:10;
 p1 = tricdf (x, 3, 6, 4);
 p2 = tricdf (x, 1, 5, 2);
 p3 = tricdf (x, 2, 9, 3);
 p4 = tricdf (x, 2, 9, 5);
 plot (x, p1, "-b", x, p2, "-g", x, p3, "-r", x, p4, "-c")
 grid on
 xlim ([0, 10])
 legend ({"a = 3, b = 6, c = 4", "a = 1, b = 5, c = 2", ...
          "a = 2, b = 9, c = 3", "a = 2, b = 9, c = 5"}, ...
         "location", "southeast")
 title ("Triangular CDF")
 xlabel ("values in x")
 ylabel ("probability")
plotted figure

statistics-release-1.6.3/docs/triinv.html000066400000000000000000000135761456127120000204740ustar00rootroot00000000000000 Statistics: triinv

Function Reference: triinv

statistics: x = triinv (p, a, b, c)

Inverse of the triangular cumulative distribution function (iCDF).

For each element of p, compute the quantile (the inverse of the CDF) of the triangular distribution with parameters a, b, and c on the interval [a, b]. The size of x is the common size of the input arguments. A scalar input functions as a constant matrix of the same size as the other inputs.

Further information about the triangular distribution can be found at https://en.wikipedia.org/wiki/Triangular_distribution

See also: tricdf, tripdf, trirnd

Source Code: triinv

Example: 1

  

 ## Plot various iCDFs from the triangular distribution
 p = 0.001:0.001:0.999;
 x1 = triinv (p, 3, 6, 4);
 x2 = triinv (p, 1, 5, 2);
 x3 = triinv (p, 2, 9, 3);
 x4 = triinv (p, 2, 9, 5);
 plot (p, x1, "-b", p, x2, "-g", p, x3, "-r", p, x4, "-c")
 grid on
 ylim ([0, 10])
 legend ({"a = 3, b = 6, c = 4", "a = 1, b = 5, c = 2", ...
          "a = 2, b = 9, c = 3", "a = 2, b = 9, c = 5"}, ...
         "location", "northwest")
 title ("Triangular CDF")
 xlabel ("probability")
 ylabel ("values in x")
plotted figure

statistics-release-1.6.3/docs/trimmean.html000066400000000000000000000172351456127120000207710ustar00rootroot00000000000000 Statistics: trimmean

Function Reference: trimmean

statistics: m = trimmean (x, p)
statistics: m = trimmean (x, p, flag)
statistics: m = trimmean (…, "all")
statistics: m = trimmean (…, dim)
statistics: m = trimmean (…, dim)

Compute the trimmed mean.

The trimmed mean of x is defined as the mean of x excluding the highest and lowest k data values of x, calculated as k = n * (p / 100) / 2), where n is the sample size.

m = trimmean (x, p) returns the mean of x after removing the outliers in x defined by p percent.

  • If x is a vector, then trimmean (x, p) is the mean of all the values of x, computed after removing the outliers.
  • If x is a matrix, then trimmean (x, p) is a row vector of column means, computed after removing the outliers.
  • If x is a multidimensional array, then trimmean operates along the first nonsingleton dimension of x.

To specify the operating dimension(s) when x is a matrix or a multidimensional array, use the dim or vecdim input argument.

trimmean treats NaN values in x as missing values and removes them.

m = trimmean (x, p, flag) specifies how to trim when k, i.e. half the number of outliers, is not an integer. flag can be specified as one of the following values:

ValueDescription
"round"Round k to the nearest integer. This is the default.
"floor"Round k down to the next smaller integer.
"weighted"If k = i + f, where i is an integer and f is a fraction, compute a weighted mean with weight (1 – f) for the (i + 1)-th and (n – i)-th values, and full weight for the values between them.

m = trimmean (…, "all") returns the trimmed mean of all the values in x using any of the input argument combinations in the previous syntaxes.

m = trimmean (…, dim) returns the trimmed mean along the operating dimension dim specified as a positive integer scalar. If not specified, then the default value is the first nonsingleton dimension of x, i.e. whose size does not equal 1. If dim is greater than ndims (X) or if size (x, dim) is 1, then trimmean returns x.

m = trimmean (…, vecdim) returns the trimmed mean over the dimensions specified in the vector vecdim. For example, if x is a 2-by-3-by-4 array, then mean (x, [1 2]) returns a 1-by-1-by-4 array. Each element of the output array is the mean of the elements on the corresponding page of x. If vecdim indexes all dimensions of x, then it is equivalent to mean (x, "all"). Any dimension in vecdim greater than ndims (x) is ignored.

See also: mean

Source Code: trimmean

statistics-release-1.6.3/docs/tripdf.html000066400000000000000000000135471456127120000204470ustar00rootroot00000000000000 Statistics: tripdf

Function Reference: tripdf

statistics: y = tripdf (x, a, b, c)

Triangular probability density function (PDF).

For each element of x, compute the probability density function (PDF) of the triangular distribution with parameters a, b, and c on the interval [a, b]. The size of y is the common size of the input arguments. A scalar input functions as a constant matrix of the same size as the other inputs.

Further information about the triangular distribution can be found at https://en.wikipedia.org/wiki/Triangular_distribution

See also: tricdf, triinv, trirnd

Source Code: tripdf

Example: 1

  

 ## Plot various CDFs from the triangular distribution
 x = 0.001:0.001:10;
 y1 = tripdf (x, 3, 6, 4);
 y2 = tripdf (x, 1, 5, 2);
 y3 = tripdf (x, 2, 9, 3);
 y4 = tripdf (x, 2, 9, 5);
 plot (x, y1, "-b", x, y2, "-g", x, y3, "-r", x, y4, "-c")
 grid on
 xlim ([0, 10])
 legend ({"a = 3, b = 6, c = 4", "a = 1, b = 5, c = 2", ...
          "a = 2, b = 9, c = 3", "a = 2, b = 9, c = 5"}, ...
         "location", "northeast")
 title ("Triangular CDF")
 xlabel ("values in x")
 ylabel ("probability")
plotted figure

statistics-release-1.6.3/docs/trirnd.html000066400000000000000000000122431456127120000204510ustar00rootroot00000000000000 Statistics: trirnd

Function Reference: trirnd

statistics: r = trirnd (a, b, c)
statistics: r = trirnd (a, b, c, rows)
statistics: r = trirnd (a, b, c, rows, cols, …)
statistics: r = trirnd (a, b, c, [sz])

Random arrays from the triangular distribution.

r = trirnd (sigma) returns an array of random numbers chosen from the triangular distribution with parameters a, b, and c on the interval [a, b]. The size of r is the common size of a, b, and c. A scalar input functions as a constant matrix of the same size as the other inputs.

When called with a single size argument, trirnd returns a square matrix with the dimension specified. When called with more than one scalar argument, the first two arguments are taken as the number of rows and columns and any further arguments specify additional matrix dimensions. The size may also be specified with a row vector of dimensions, sz.

Further information about the triangular distribution can be found at https://en.wikipedia.org/wiki/Triangular_distribution

See also: tricdf, triinv, tripdf

Source Code: trirnd

statistics-release-1.6.3/docs/trnd.html000066400000000000000000000122731456127120000201210ustar00rootroot00000000000000 Statistics: trnd

Function Reference: trnd

statistics: r = trnd (df)
statistics: r = trnd (df, rows)
statistics: r = trnd (df, rows, cols, …)
statistics: r = trnd (df, [sz])

Random arrays from the Student’s T distribution.

Return a matrix of random samples from the Students’s T distribution with df degrees of freedom.

r = trnd (df) returns an array of random numbers chosen from the Student’s T distribution with df degrees of freedom. The size of r is the size of df. A scalar input functions as a constant matrix of the same size as the other inputs. df must be a finite real number greater than 0, otherwise NaN is returned.

When called with a single size argument, trnd returns a square matrix with the dimension specified. When called with more than one scalar argument, the first two arguments are taken as the number of rows and columns and any further arguments specify additional matrix dimensions. The size may also be specified with a row vector of dimensions, sz.

Further information about the Student’s T distribution can be found at https://en.wikipedia.org/wiki/Student%27s_t-distribution

See also: tcdf, tpdf, tpdf, tstat

Source Code: trnd

statistics-release-1.6.3/docs/tstat.html000066400000000000000000000104471456127120000203120ustar00rootroot00000000000000 Statistics: tstat

Function Reference: tstat

statistics: [m, v] = tstat (df)

Compute statistics of the Student’s T distribution.

[m, v] = tstat (df) returns the mean and variance of the Student’s T distribution with df degrees of freedom.

The size of m (mean) and v (variance) is the same size of the input argument.

Further information about the Student’s T distribution can be found at https://en.wikipedia.org/wiki/Student%27s_t-distribution

See also: tcdf, tinv, tpdf, trnd

Source Code: tstat

statistics-release-1.6.3/docs/ttest.html000066400000000000000000000156661456127120000203260ustar00rootroot00000000000000 Statistics: ttest

Function Reference: ttest

statistics: [h, pval, ci, stats] = ttest (x)
statistics: [h, pval, ci, stats] = ttest (x, m)
statistics: [h, pval, ci, stats] = ttest (x, y)
statistics: [h, pval, ci, stats] = ttest (x, m, Name, Value)
statistics: [h, pval, ci, stats] = ttest (x, y, Name, Value)

Test for mean of a normal sample with unknown variance.

Perform a t-test of the null hypothesis mean (x) == m for a sample x from a normal distribution with unknown mean and unknown standard deviation. Under the null, the test statistic t has a Student’s t distribution. The default value of m is 0.

If the second argument y is a vector, a paired-t test of the hypothesis mean (x) = mean (y) is performed. If x and y are vectors, they must have the same size and dimensions.

x (and y) can also be matrices. For matrices, ttest performs separate t-tests along each column, and returns a vector of results. x and y must have the same number of columns. The Type I error rate of the resulting vector of pval can be controlled by entering pval as input to the function multcompare.

ttest treats NaNs as missing values, and ignores them.

Name-Value pair arguments can be used to set various options. "alpha" can be used to specify the significance level of the test (the default value is 0.05). "tail", can be used to select the desired alternative hypotheses. If the value is "both" (default) the null is tested against the two-sided alternative mean (x) != m. If it is "right" the one-sided alternative mean (x) > m is considered. Similarly for "left", the one-sided alternative mean (x) < m is considered. When argument x is a matrix, "dim" can be used to select the dimension over which to perform the test. (The default is the first non-singleton dimension).

If h is 1 the null hypothesis is rejected, meaning that the tested sample does not come from a Student’s t distribution. If h is 0, then the null hypothesis cannot be rejected and it can be assumed that x follows a Student’s t distribution. The p-value of the test is returned in pval. A 100(1-alpha)% confidence interval is returned in ci.

stats is a structure containing the value of the test statistic (tstat), the degrees of freedom (df) and the sample’s standard deviation (sd).

See also: hotelling_ttest, ttest2, hotelling_ttest2

Source Code: ttest

statistics-release-1.6.3/docs/ttest2.html000066400000000000000000000143221456127120000203740ustar00rootroot00000000000000 Statistics: ttest2

Function Reference: ttest2

statistics: [h, pval, ci, stats] = ttest2 (x, y)
statistics: [h, pval, ci, stats] = ttest2 (x, y, Name, Value)

Perform a t-test to compare the means of two groups of data under the null hypothesis that the groups are drawn from distributions with the same mean.

x and y can be vectors or matrices. For matrices, ttest2 performs separate t-tests along each column, and returns a vector of results. x and y must have the same number of columns. The Type I error rate of the resulting vector of pval can be controlled by entering pval as input to the function multcompare.

ttest2 treats NaNs as missing values, and ignores them.

For a nested t-test, use anova2.

The argument "alpha" can be used to specify the significance level of the test (the default value is 0.05). The string argument "tail", can be used to select the desired alternative hypotheses. If "tail" is "both" (default) the null is tested against the two-sided alternative mean (x) != m. If "tail" is "right" the one-sided alternative mean (x) > m is considered. Similarly for "left", the one-sided alternative mean (x) < m is considered.

When "vartype" is "equal" the variances are assumed to be equal (this is the default). When "vartype" is "unequal" the variances are not assumed equal.

When argument x and y are matrices the "dim" argument can be used to select the dimension over which to perform the test. (The default is the first non-singleton dimension.)

If h is 0 the null hypothesis is accepted, if it is 1 the null hypothesis is rejected. The p-value of the test is returned in pval. A 100(1-alpha)% confidence interval is returned in ci. stats is a structure containing the value of the test statistic (tstat), the degrees of freedom (df) and the sample standard deviation (sd).

See also: hotelling_ttest2, anova1, hotelling_ttest, ttest

Source Code: ttest2

statistics-release-1.6.3/docs/unidcdf.html000066400000000000000000000150251456127120000205640ustar00rootroot00000000000000 Statistics: unidcdf

Function Reference: unidcdf

statistics: p = unidcdf (x, N)
statistics: p = unidcdf (x, N, "upper")

Discrete uniform cumulative distribution function (CDF).

For each element of x, compute the cumulative distribution function (CDF) of a discrete uniform distribution with parameter N, which corresponds to the maximum observable value. unidcdf assumes the integer values in the range [1,N] with equal probability. The size of p is the common size of x and N. A scalar input functions as a constant matrix of the same size as the other inputs.

The maximum observable values in N must be positive integers, otherwise NaN is returned.

[…] = unidcdf (x, N, "upper") computes the upper tail probability of the discrete uniform distribution with maximum observable value N, at the values in x.

Warning: The underlying implementation uses the double class and will only be accurate for N < flintmax (2^53 on IEEE 754 compatible systems).

Further information about the discrete uniform distribution can be found at https://en.wikipedia.org/wiki/Discrete_uniform_distribution

See also: unidinv, unidpdf, unidrnd, unidfit, unidstat

Source Code: unidcdf

Example: 1

  

 ## Plot various CDFs from the discrete uniform distribution
 x = 0:10;
 p1 = unidcdf (x, 5);
 p2 = unidcdf (x, 9);
 plot (x, p1, "*b", x, p2, "*g")
 grid on
 xlim ([0, 10])
 ylim ([0, 1])
 legend ({"N = 5", "N = 9"}, "location", "southeast")
 title ("Discrete uniform CDF")
 xlabel ("values in x")
 ylabel ("probability")
plotted figure

statistics-release-1.6.3/docs/unidfit.html000066400000000000000000000166641456127120000206240ustar00rootroot00000000000000 Statistics: unidfit

Function Reference: unidfit

statistics: Nhat = unidfit (x)
statistics: [Nhat, Nci] = unidfit (x)
statistics: [Nhat, Nci] = unidfit (x, alpha)
statistics: [Nhat, Nci] = unidfit (x, alpha, freq)

Estimate parameter and confidence intervals for the discrete uniform distribution.

Nhat = unidfit (x) returns the maximum likelihood estimate (MLE) of the maximum observable value for the discrete uniform distribution. x must be a vector.

[Nhat, Nci] = unidfit (x, alpha) also returns the 100 * (1 - alpha) percent confidence intervals of the estimated parameter. By default, the optional argument alpha is 0.05 corresponding to 95% confidence intervals. Pass in [] for alpha to use the default values.

[…] = unidfit (x, alpha, freq) accepts a frequency vector, freq, of the same size as x. freq typically contains integer frequencies for the corresponding elements in x, but it can contain any non-integer non-negative values. By default, or if left empty, freq = ones (size (x)).

Further information about the discrete uniform distribution can be found at https://en.wikipedia.org/wiki/Discrete_uniform_distribution

See also: unidcdf, unidinv, unidpdf, unidrnd, unidstat

Source Code: unidfit

Example: 1

  

 ## Sample 2 populations from different discrete uniform distibutions
 rand ("seed", 1);    # for reproducibility
 r1 = unidrnd (5, 1000, 1);
 rand ("seed", 2);    # for reproducibility
 r2 = unidrnd (9, 1000, 1);
 r = [r1, r2];

 ## Plot them normalized and fix their colors
 hist (r, 0:0.5:20.5, 1);
 h = findobj (gca, "Type", "patch");
 set (h(1), "facecolor", "c");
 set (h(2), "facecolor", "g");
 hold on

 ## Estimate their probability of success
 NhatA = unidfit (r(:,1));
 NhatB = unidfit (r(:,2));

 ## Plot their estimated PDFs
 x = [0:10];
 y = unidpdf (x, NhatA);
 plot (x, y, "-pg");
 y = unidpdf (x, NhatB);
 plot (x, y, "-sc");
 xlim ([0, 10])
 ylim ([0, 0.4])
 legend ({"Normalized HIST of sample 1 with N=5", ...
          "Normalized HIST of sample 2 with N=9", ...
          sprintf("PDF for sample 1 with estimated N=%0.2f", NhatA), ...
          sprintf("PDF for sample 2 with estimated N=%0.2f", NhatB)})
 title ("Two population samples from different discrete uniform distibutions")
 hold off
plotted figure

statistics-release-1.6.3/docs/unidinv.html000066400000000000000000000143021456127120000206210ustar00rootroot00000000000000 Statistics: unidinv

Function Reference: unidinv

statistics: x = unidinv (p, N)

Inverse of the discrete uniform cumulative distribution function (iCDF).

For each element of p, compute the quantile (the inverse of the CDF) of the discrete uniform distribution with parameter N, which corresponds to the maximum observable value. unidinv assumes the integer values in the range [1,N] with equal probability. The size of x is the common size of p and N. A scalar input functions as a constant matrix of the same size as the other inputs.

The maximum observable values in N must be positive integers, otherwise NaN is returned.

Warning: The underlying implementation uses the double class and will only be accurate for N < flintmax (2^53 on IEEE 754 compatible systems).

Further information about the discrete uniform distribution can be found at https://en.wikipedia.org/wiki/Discrete_uniform_distribution

See also: unidcdf, unidpdf, unidrnd, unidfit, unidstat

Source Code: unidinv

Example: 1

  

 ## Plot various iCDFs from the discrete uniform distribution
 p = 0.001:0.001:0.999;
 x1 = unidinv (p, 5);
 x2 = unidinv (p, 9);
 plot (p, x1, "-b", p, x2, "-g")
 grid on
 xlim ([0, 1])
 ylim ([0, 10])
 legend ({"N = 5", "N = 9"}, "location", "northwest")
 title ("Discrete uniform iCDF")
 xlabel ("probability")
 ylabel ("values in x")
plotted figure

statistics-release-1.6.3/docs/unidpdf.html000066400000000000000000000142371456127120000206050ustar00rootroot00000000000000 Statistics: unidpdf

Function Reference: unidpdf

statistics: y = unidpdf (x, N)

Discrete uniform probability density function (PDF).

For each element of x, compute the probability density function (PDF) of the discrete uniform distribution with parameter N, which corresponds to the maximum observable value. unidpdf assumes the integer values in the range [1,N] with equal probability. The size of x is the common size of p and N. A scalar input functions as a constant matrix of the same size as the other inputs.

The maximum observable values in N must be positive integers, otherwise NaN is returned.

Warning: The underlying implementation uses the double class and will only be accurate for N < flintmax (2^53 on IEEE 754 compatible systems).

Further information about the discrete uniform distribution can be found at https://en.wikipedia.org/wiki/Discrete_uniform_distribution

See also: unidcdf, unidinv, unidrnd, unidfit, unidstat

Source Code: unidpdf

Example: 1

  

 ## Plot various PDFs from the discrete uniform distribution
 x = 0:10;
 y1 = unidpdf (x, 5);
 y2 = unidpdf (x, 9);
 plot (x, y1, "*b", x, y2, "*g")
 grid on
 xlim ([0, 10])
 ylim ([0, 0.25])
 legend ({"N = 5", "N = 9"}, "location", "northeast")
 title ("Descrete uniform PDF")
 xlabel ("values in x")
 ylabel ("density")
plotted figure

statistics-release-1.6.3/docs/unidrnd.html000066400000000000000000000130221456127120000206060ustar00rootroot00000000000000 Statistics: unidrnd

Function Reference: unidrnd

statistics: r = unidrnd (N)
statistics: r = unidrnd (N, rows)
statistics: r = unidrnd (N, rows, cols, …)
statistics: r = unidrnd (N, [sz])

Random arrays from the discrete uniform distribution.

r = unidrnd (N) returns an array of random numbers chosen from the discrete uniform distribution with parameter N, which corresponds to the maximum observable value. unidrnd assumes the integer values in the range [1,N] with equal probability. The size of r is the size of N. A scalar input functions as a constant matrix of the same size as the other inputs.

The maximum observable values in N must be positive integers, otherwise NaN is returned.

When called with a single size argument, unidrnd returns a square matrix with the dimension specified. When called with more than one scalar argument, the first two arguments are taken as the number of rows and columns and any further arguments specify additional matrix dimensions. The size may also be specified with a row vector of dimensions, sz.

Warning: The underlying implementation uses the double class and will only be accurate for N < flintmax (2^53 on IEEE 754 compatible systems).

Further information about the discrete uniform distribution can be found at https://en.wikipedia.org/wiki/Discrete_uniform_distribution

See also: unidcdf, unidinv, unidrnd, unidfit, unidstat

Source Code: unidrnd

statistics-release-1.6.3/docs/unidstat.html000066400000000000000000000107421456127120000210040ustar00rootroot00000000000000 Statistics: unidstat

Function Reference: unidstat

statistics: [m, v] = unidstat (df)

Compute statistics of the discrete uniform cumulative distribution.

[m, v] = unidstat (df) returns the mean and variance of the discrete uniform cumulative distribution with parameter N, which corresponds to the maximum observable value and must be a positive natural number.

The size of m (mean) and v (variance) is the same size of the input argument.

Further information about the discrete uniform distribution can be found at https://en.wikipedia.org/wiki/Discrete_uniform_distribution

See also: unidcdf, unidinv, unidpdf, unidrnd, unidfit

Source Code: unidstat

statistics-release-1.6.3/docs/unifcdf.html000066400000000000000000000144261456127120000205720ustar00rootroot00000000000000 Statistics: unifcdf

Function Reference: unifcdf

statistics: p = unifcdf (x, a, b)
statistics: p = unifcdf (x, a, b, "upper")

Continuous uniform cumulative distribution function (CDF).

For each element of x, compute the cumulative distribution function (CDF) of the continuous uniform distribution with parameters a and b, which define the lower and upper bounds of the interval [a, b]. The size of p is the common size of x, a, and b. A scalar input functions as a constant matrix of the same size as the other inputs.

[…] = unifcdf (x, a, b, "upper") computes the upper tail probability of the continuous uniform distribution with parameters a, and b, at the values in x.

Further information about the continuous uniform distribution can be found at https://en.wikipedia.org/wiki/Continuous_uniform_distribution

See also: unifinv, unifpdf, unifrnd, unifit, unifstat

Source Code: unifcdf

Example: 1

  

 ## Plot various CDFs from the continuous uniform distribution
 x = 0:0.1:10;
 p1 = unifcdf (x, 2, 5);
 p2 = unifcdf (x, 3, 9);
 plot (x, p1, "-b", x, p2, "-g")
 grid on
 xlim ([0, 10])
 ylim ([0, 1])
 legend ({"a = 2, b = 5", "a = 3, b = 9"}, "location", "southeast")
 title ("Continuous uniform CDF")
 xlabel ("values in x")
 ylabel ("probability")
plotted figure

statistics-release-1.6.3/docs/unifinv.html000066400000000000000000000136331456127120000206310ustar00rootroot00000000000000 Statistics: unifinv

Function Reference: unifinv

statistics: x = unifinv (p, a, b)

Inverse of the continuous uniform cumulative distribution function (iCDF).

For each element of p, compute the quantile (the inverse of the CDF) of the continuous uniform distribution with parameters a and b, which define the lower and upper bounds of the interval [a, b]. The size of x is the common size of p, a, and b. A scalar input functions as a constant matrix of the same size as the other inputs.

Further information about the continuous uniform distribution can be found at https://en.wikipedia.org/wiki/Continuous_uniform_distribution

See also: unifcdf, unifpdf, unifrnd, unifit, unifstat

Source Code: unifinv

Example: 1

  

 ## Plot various iCDFs from the continuous uniform distribution
 p = 0.001:0.001:0.999;
 x1 = unifinv (p, 2, 5);
 x2 = unifinv (p, 3, 9);
 plot (p, x1, "-b", p, x2, "-g")
 grid on
 xlim ([0, 1])
 ylim ([0, 10])
 legend ({"a = 2, b = 5", "a = 3, b = 9"}, "location", "northwest")
 title ("Continuous uniform iCDF")
 xlabel ("probability")
 ylabel ("values in x")
plotted figure

statistics-release-1.6.3/docs/unifit.html000066400000000000000000000172101456127120000204440ustar00rootroot00000000000000 Statistics: unifit

Function Reference: unifit

statistics: paramhat = unifit (x)
statistics: [paramhat, paramci] = unifit (x)
statistics: [paramhat, paramci] = unifit (x, alpha)
statistics: [paramhat, paramci] = unifit (x, alpha, freq)

Estimate parameter and confidence intervals for the continuous uniform distribution.

paramhat = unifit (x) returns the maximum likelihood estimate (MLE) of the parameters a and b of the continuous uniform distribution given the data in x. x must be a vector.

[paramhat, paramci] = unifit (x, alpha) also returns the 100 * (1 - alpha) percent confidence intervals of the estimated parameter. By default, the optional argument alpha is 0.05 corresponding to 95% confidence intervals. Pass in [] for alpha to use the default values.

[…] = unifit (x, alpha, freq) accepts a frequency vector, freq, of the same size as x. freq typically contains integer frequencies for the corresponding elements in x, but it can contain any non-integer non-negative values. By default, or if left empty, freq = ones (size (x)).

Further information about the continuous uniform distribution can be found at https://en.wikipedia.org/wiki/Discrete_uniform_distribution

See also: unifcdf, unifinv, unifpdf, unifrnd, unifstat

Source Code: unifit

Example: 1

  

 ## Sample 2 populations from different continuous uniform distibutions
 rand ("seed", 5);    # for reproducibility
 r1 = unifrnd (2, 5, 2000, 1);
 rand ("seed", 6);    # for reproducibility
 r2 = unifrnd (3, 9, 2000, 1);
 r = [r1, r2];

 ## Plot them normalized and fix their colors
 hist (r, 0:0.5:10, 2);
 h = findobj (gca, "Type", "patch");
 set (h(1), "facecolor", "c");
 set (h(2), "facecolor", "g");
 hold on

 ## Estimate their probability of success
 a_bA = unifit (r(:,1));
 a_bB = unifit (r(:,2));

 ## Plot their estimated PDFs
 x = [0:10];
 y = unifpdf (x, a_bA(1), a_bA(2));
 plot (x, y, "-pg");
 y = unifpdf (x, a_bB(1), a_bB(2));
 plot (x, y, "-sc");
 xlim ([1, 10])
 ylim ([0, 0.5])
 legend ({"Normalized HIST of sample 1 with a=2 and b=5", ...
          "Normalized HIST of sample 2 with a=3 and b=9", ...
          sprintf("PDF for sample 1 with estimated a=%0.2f and b=%0.2f", ...
                  a_bA(1), a_bA(2)), ...
          sprintf("PDF for sample 2 with estimated a=%0.2f and b=%0.2f", ...
                  a_bB(1), a_bB(2))})
 title ("Two population samples from different continuous uniform distibutions")
 hold off
plotted figure

statistics-release-1.6.3/docs/unifpdf.html000066400000000000000000000135751456127120000206130ustar00rootroot00000000000000 Statistics: unifpdf

Function Reference: unifpdf

statistics: y = unifpdf (x, a, b)

Continuous uniform probability density function (PDF).

For each element of x, compute the probability density function (PDF) of the continuous uniform distribution with parameters a and b, which define the lower and upper bounds of the interval [a, b]. The size of y is the common size of x, a, and b. A scalar input functions as a constant matrix of the same size as the other inputs.

Further information about the continuous uniform distribution can be found at https://en.wikipedia.org/wiki/Continuous_uniform_distribution

See also: unifcdf, unifinv, unifrnd, unifit, unifstat

Source Code: unifpdf

Example: 1

  

 ## Plot various PDFs from the continuous uniform distribution
 x = 0:0.001:10;
 y1 = unifpdf (x, 2, 5);
 y2 = unifpdf (x, 3, 9);
 plot (x, y1, "-b", x, y2, "-g")
 grid on
 xlim ([0, 10])
 ylim ([0, 0.4])
 legend ({"a = 2, b = 5", "a = 3, b = 9"}, "location", "northeast")
 title ("Continuous uniform PDF")
 xlabel ("values in x")
 ylabel ("density")
plotted figure

statistics-release-1.6.3/docs/unifrnd.html000066400000000000000000000124241456127120000206150ustar00rootroot00000000000000 Statistics: unifrnd

Function Reference: unifrnd

statistics: r = unifrnd (a, b)
statistics: r = unifrnd (a, b, rows)
statistics: r = unifrnd (a, b, rows, cols, …)
statistics: r = unifrnd (a, b, [sz])

Random arrays from the continuous uniform distribution.

r = unifrnd (a, b) returns an array of random numbers chosen from the continuous uniform distribution with parameters a and b, which define the lower and upper bounds of the interval [a, b]. The size of r is the common size of a and b. A scalar input functions as a constant matrix of the same size as the other inputs.

When called with a single size argument, unifrnd returns a square matrix with the dimension specified. When called with more than one scalar argument, the first two arguments are taken as the number of rows and columns and any further arguments specify additional matrix dimensions. The size may also be specified with a row vector of dimensions, sz.

Further information about the continuous uniform distribution can be found at https://en.wikipedia.org/wiki/Continuous_uniform_distribution

See also: unifcdf, unifinv, unifpdf, unifit, unifstat

Source Code: unifrnd

statistics-release-1.6.3/docs/unifstat.html000066400000000000000000000111371456127120000210050ustar00rootroot00000000000000 Statistics: unifstat

Function Reference: unifstat

statistics: [m, v] = unifstat (df)

Compute statistics of the continuous uniform cumulative distribution.

[m, v] = unifstat (df) returns the mean and variance of the continuous uniform cumulative distribution with parameters a and b, which define the lower and upper bounds of the interval [a, b].

The size of m (mean) and v (variance) is the common size of the input arguments. A scalar input functions as a constant matrix of the same size as the other inputs.

Further information about the continuous uniform distribution can be found at https://en.wikipedia.org/wiki/Continuous_uniform_distribution

See also: unifcdf, unifinv, unifpdf, unifrnd, unifit

Source Code: unifstat

statistics-release-1.6.3/docs/var.html000066400000000000000000000170511456127120000177410ustar00rootroot00000000000000 Statistics: var

Function Reference: var

statistics: v = var (x)
statistics: v = var (x, w)
statistics: v = var (x, w, "all")
statistics: v = var (x, w, dim)
statistics: v = var (x, w, vecdim)
statistics: v = var (…, nanflag)
statistics: [v, m] = var (…)

Compute the variance of the elements of x.

  • If x is a vector, then var(x) returns the variance of the elements in x defined as $$ {\rm var}(x) = {1\over N-1} \sum_{i=1}^N |x_i - \bar x |^2 $$ where N is the length of the x vector.
  • If x is a matrix, then var (x) returns a row vector with the variance of each column in x.
  • If x is a multi-dimensional array, then var (x) operates along the first non-singleton dimension of x.

var (x, w) specifies a weighting scheme. When w = 0 (default), the variance is normalized by N-1 (population variance) where N is the number of observations. When w = 1, the variance is normalized by the number of observations (sample variance). To use the default value you may pass an empty input argument [] before entering other options.

w can also be an array of non-negative numbers. When w is a vector, it must have the same length as the number of elements in the operating dimension of x. If w is a matrix or n-D array, or the operating dimension is supplied as a vecdim or "all", w must be the same size as x. NaN values are permitted in w, will be multiplied with the associated values in x, and can be excluded by the nanflag option.

var (x, [], dim) returns the variance along the operating dimension dim of x. For dim greater than ndims (x) v is returned as zeros of the same size as x and m = x.

var (x, [], vecdim) returns the variance over the dimensions specified in the vector vecdim. For example, if x is a 2-by-3-by-4 array, then var (x, [1 2]) returns a 1-by-1-by-4 array. Each element of the output array is the variance of the elements on the corresponding page of x. If vecdim indexes all dimensions of x, then it is equivalent to var (x, "all"). Any dimension in vecdim greater than ndims (x) is ignored.

var (x, "all") returns the variance of all the elements in x. The optional flag "all" cannot be used together with dim or vecdim input arguments.

var (…, nanflag) specifies whether to exclude NaN values from the calculation using any of the input argument combinations in previous syntaxes. The default value for nanflag is "includenan", and keeps NaN values in the calculation. To exclude NaN values, set the value of nanflag to "omitnan".

[v, m] = var (…) also returns the mean of the elements of x used to calculate the variance. If v is the weighted variance, then m is the weighted mean.

See also: std, mean

Source Code: var

statistics-release-1.6.3/docs/vartest.html000066400000000000000000000160571456127120000206460ustar00rootroot00000000000000 Statistics: vartest

Function Reference: vartest

statistics: h = vartest (x, v)
statistics: h = vartest (x, v, name, value)
statistics: [h, pval] = vartest (…)
statistics: [h, pval, ci] = vartest (…)
statistics: [h, pval, ci, stats] = vartest (…)

One-sample test of variance.

h = vartest (x, v) performs a chi-square test of the hypothesis that the data in the vector x come from a normal distribution with variance v, against the alternative that x comes from a normal distribution with a different variance. The result is h = 0 if the null hypothesis ("variance is V") cannot be rejected at the 5% significance level, or h = 1 if the null hypothesis can be rejected at the 5% level.

x may also be a matrix or an N-D array. For matrices, vartest performs separate tests along each column of x, and returns a vector of results. For N-D arrays, vartest works along the first non-singleton dimension of x. v must be a scalar.

vartest treats NaNs as missing values, and ignores them.

[h, pval] = vartest (…) returns the p-value. That is the probability of observing the given result, or one more extreme, by chance if the null hypothesisis true.

[h, pval, ci] = vartest (…) returns a 100 * (1 - alpha)% confidence interval for the true variance.

[h, pval, ci, stats] = vartest (…) returns a structure with the following fields:

chisqstatthe value of the test statistic
dfthe degrees of freedom of the test

[…] = vartest (…, name, value), … specifies one or more of the following name/value pairs:

NameValue
"alpha"the significance level. Default is 0.05.
"dim"dimension to work along a matrix or an N-D array.
"tail"a string specifying the alternative hypothesis
"both"variance is not v (two-tailed, default)
"left"variance is less than v (left-tailed)
"right"variance is greater than v (right-tailed)

See also: ttest, ztest, kstest

Source Code: vartest

statistics-release-1.6.3/docs/vartest2.html000066400000000000000000000165431456127120000207300ustar00rootroot00000000000000 Statistics: vartest2

Function Reference: vartest2

statistics: h = vartest2 (x, y)
statistics: h = vartest2 (x, y, name, value)
statistics: [h, pval] = vartest2 (…)
statistics: [h, pval, ci] = vartest2 (…)
statistics: [h, pval, ci, stats] = vartest2 (…)

Two-sample F test for equal variances.

h = vartest2 (x, y) performs an F test of the hypothesis that the independent data in vectors x and y come from normal distributions with equal variance, against the alternative that they come from normal distributions with different variances. The result is h = 0 if the null hypothesis ("variance are equal") cannot be rejected at the 5% significance level, or h = 1 if the null hypothesis can be rejected at the 5% level.

x and y may also be matrices or N-D arrays. For matrices, vartest2 performs separate tests along each column and returns a vector of results. For N-D arrays, vartest2 works along the first non-singleton dimension and x and y must have the same size along all the remaining dimensions.

vartest treats NaNs as missing values, and ignores them.

[h, pval] = vartest (…) returns the p-value. That is the probability of observing the given result, or one more extreme, by chance if the null hypothesisis true.

[h, pval, ci] = vartest (…) returns a 100× (1 - alpha)% confidence interval for the true ratio var(X)/var(Y).

[h, pval, ci, stats] = vartest (…) returns a structure with the following fields:

fstatthe value of the test statistic
df1the numerator degrees of freedom of the test
df2the denominator degrees of freedom of the test

[…] = vartest (…, name, value), … specifies one or more of the following name/value pairs:

NameValue
"alpha"the significance level. Default is 0.05.
"dim"dimension to work along a matrix or an N-D array.
"tail"a string specifying the alternative hypothesis
"both"variance is not v (two-tailed, default)
"left"variance is less than v (left-tailed)
"right"variance is greater than v (right-tailed)

See also: ttest2, kstest2, bartlett_test, levene_test

Source Code: vartest2

statistics-release-1.6.3/docs/vartestn.html000066400000000000000000000351441456127120000210220ustar00rootroot00000000000000 Statistics: vartestn

Function Reference: vartestn

statistics: vartestn (x)
statistics: vartestn (x, group)
statistics: vartestn (…, name, value)
statistics: p = vartestn (…)
statistics: [p, stats] = vartestn (…)
statistics: [p, stats] = vartestn (…, name, value)

Test for equal variances across multiple groups.

h = vartestn (x) performs Bartlett’s test for equal variances for the columns of the matrix x. This is a test of the null hypothesis that the columns of x come from normal distributions with the same variance, against the alternative that they come from normal distributions with different variances. The result is displayed in a summary table of statistics as well as a box plot of the groups.

vartestn (x, group) requires a vector x, and a group argument that is a categorical variable, vector, string array, or cell array of strings with one row for each element of x. Values of x corresponding to the same value of group are placed in the same group.

vartestn treats NaNs as missing values, and ignores them.

p = vartestn (…) returns the probability of observing the given result, or one more extreme, by chance under the null hypothesis that all groups have equal variances. Small values of p cast doubt on the validity of the null hypothesis.

[p, stats] = vartestn (…) returns a structure with the following fields:

chistat– the value of the test statistic
df– the degrees of freedom of the test

[p, stats] = vartestn (…, name, value) specifies one or more of the following name/value pairs:

"display""on" to display a boxplot and table, or "off" to omit these displays. Default "on".
"testtype"One of the following strings to control the type of test to perform
"Bartlett"Bartlett’s test (default).
"LeveneQuadratic"Levene’s test computed by performing anova on the squared deviations of the data values from their group means.
"LeveneAbsolute"Levene’s test computed by performing anova on the absolute deviations of the data values from their group means.
"BrownForsythe"Brown-Forsythe test computed by performing anova on the absolute deviations of the data values from the group medians.
"OBrien"O’Brien’s modification of Levene’s test with W=0.5.

The classical Bartlett’s test is sensitive to the assumption that the distribution in each group is normal. The other test types are more robust to non-normal distributions, especially ones prone to outliers. For these tests, the STATS output structure has a field named fstat containing the test statistic, and df1 and df2 containing its numerator and denominator degrees of freedom.

See also: vartest, vartest2, anova1, bartlett_test, levene_test

Source Code: vartestn

Example: 1

  

 ## Test the null hypothesis that the variances are equal across the five
 ## columns of data in the students’ exam grades matrix, grades.

 load examgrades
 vartestn (grades)


                    Group Summary Table

Group                        Count        Mean       Std Dev
------------------------------------------------------------
1                            120        75.0083     8.720203
2                            120        74.9917     6.542037
3                            120        74.9917     7.430910
4                            120        75.0333     8.601283
5                            120        74.9917     5.258839
Pooled Groups                600        75.0033     7.310655
Pooled valid Groups          600        75.0083     8.720203

Bartlett's statistic             38.73324
Degrees of Freedom               4
p-value                          0.000000

ans = 7.9086e-08
plotted figure

Example: 2

  

 ## Test the null hypothesis that the variances in miles per gallon (MPG) are
 ## equal across different model years.

 load carsmall
 vartestn (MPG, Model_Year)


                    Group Summary Table

Group                        Count        Mean       Std Dev
------------------------------------------------------------
70                            29        17.6897     5.339231
76                            34        21.5735     5.889297
82                            31        31.7097     5.392548
Pooled Groups                 94        23.6576     5.540359
Pooled valid Groups           87        17.6897     5.339231

Bartlett's statistic             0.36619
Degrees of Freedom               2
p-value                          0.832687

ans = 0.8327
plotted figure

Example: 3

  

 ## Use Levene’s test to test the null hypothesis that the variances in miles
 ## per gallon (MPG) are equal across different model years.

 load carsmall
 p = vartestn (MPG, Model_Year, "TestType", "LeveneAbsolute")


                    Group Summary Table

Group                        Count        Mean       Std Dev
------------------------------------------------------------
70                            29        17.6897     5.339231
76                            34        21.5735     5.889297
82                            31        31.7097     5.392548
Pooled Groups                 94        23.6576     5.540359
Pooled valid Groups         2958        23.7181     5.555774

Levene's statistic (absolute)    0.46126
Degrees of Freedom               2,  91
p-value                          0.631954

p = 0.6320
plotted figure

Example: 4

  

 ## Test the null hypothesis that the variances are equal across the five
 ## columns of data in the students’ exam grades matrix, grades, using the
 ## Brown-Forsythe test.  Suppress the display of the summary table of
 ## statistics and the box plot.

 load examgrades
 [p, stats] = vartestn (grades, "TestType", "BrownForsythe", "Display", "off")

p = 1.3121e-06
stats =

  scalar structure containing the fields:

    fstat = 8.4160
    df =

         4   595
statistics-release-1.6.3/docs/violin.html000066400000000000000000000331151456127120000204500ustar00rootroot00000000000000 Statistics: violin

Function Reference: violin

statistics: violin (x)
statistics: h = violin (x)
statistics: h = violin (…, property, value, …)
statistics: h = violin (hax, …)
statistics: h = violin (…, "horizontal")

Produce a Violin plot of the data x.

The input data x can be a N-by-m array containg N observations of m variables. It can also be a cell with m elements, for the case in which the variables are not uniformly sampled.

The following property can be set using property/value pairs (default values in parenthesis). The value of the property can be a scalar indicating that it applies to all the variables in the data. It can also be a cell/array, indicating the property for each variable. In this case it should have m columns (as many as variables).

Color

("y") Indicates the filling color of the violins.

Nbins

(50) Internally, the function calls hist to compute the histogram of the data. This property indicates how many bins to use. See help hist for more details.

SmoothFactor

(4) The fuction performs simple kernel density estimation and automatically finds the bandwith of the kernel function that best approximates the histogram using optimization (sqp). The result is in general very noisy. To smooth the result the bandwidth is multiplied by the value of this property. The higher the value the smoother the violings, but values too high might remove features from the data distribution.

Bandwidth

(NA) If this property is given a value other than NA, it sets the bandwith of the kernel function. No optimization is peformed and the property SmoothFactor is ignored.

Width

(0.5) Sets the maximum width of the violins. Violins are centered at integer axis values. The distance between two violin middle axis is 1. Setting a value higher thna 1 in this property will cause the violins to overlap.

If the string "Horizontal" is among the input arguments, the violin plot is rendered along the x axis with the variables in the y axis.

The returned structure h has handles to the plot elements, allowing customization of the visualization using set/get functions.

Example:

 
 title ("Grade 3 heights");
 axis ([0,3]);
 set (gca, "xtick", 1:2, "xticklabel", {"girls"; "boys"});
 h = violin ({randn(100,1)*5+140, randn(130,1)*8+135}, "Nbins", 10);
 set (h.violin, "linewidth", 2)

See also: boxplot, hist

Source Code: violin

Example: 1

  

 clf
 x = zeros (9e2, 10);
 for i=1:10
   x(:,i) = (0.1 * randn (3e2, 3) * (randn (3,1) + 1) + 2 * randn (1,3))(:);
 endfor
 h = violin (x, "color", "c");
 axis tight
 set (h.violin, "linewidth", 2);
 set (gca, "xgrid", "on");
 xlabel ("Variables")
 ylabel ("Values")
plotted figure

Example: 2

  

 clf
 data = {randn(100,1)*5+140, randn(130,1)*8+135};
 subplot (1,2,1)
 title ("Grade 3 heights - vertical");
 set (gca, "xtick", 1:2, "xticklabel", {"girls"; "boys"});
 violin (data, "Nbins", 10);
 axis tight

 subplot(1,2,2)
 title ("Grade 3 heights - horizontal");
 set (gca, "ytick", 1:2, "yticklabel", {"girls"; "boys"});
 violin (data, "horizontal", "Nbins", 10);
 axis tight
plotted figure

Example: 3

  

 clf
 data = exprnd (0.1, 500,4);
 violin (data, "nbins", {5,10,50,100});
 axis ([0 5 0 max(data(:))])
plotted figure

Example: 4

  

 clf
 data = exprnd (0.1, 500,4);
 violin (data, "color", jet(4));
 axis ([0 5 0 max(data(:))])
plotted figure

Example: 5

  

 clf
 data = repmat(exprnd (0.1, 500,1), 1, 4);
 violin (data, "width", linspace (0.1,0.5,4));
 axis ([0 5 0 max(data(:))])
plotted figure

Example: 6

  

 clf
 data = repmat(exprnd (0.1, 500,1), 1, 4);
 violin (data, "nbins", [5,10,50,100], "smoothfactor", [4 4 8 10]);
 axis ([0 5 0 max(data(:))])
plotted figure

statistics-release-1.6.3/docs/vmcdf.html000066400000000000000000000151101456127120000202420ustar00rootroot00000000000000 Statistics: vmcdf

Function Reference: vmcdf

statistics: p = vmcdf (x, mu, k)
statistics: p = vmcdf (x, mu, k, "upper")

Von Mises probability density function (PDF).

For each element of x, compute the cumulative distribution function (CDF) of the von Mises distribution with location parameter mu and concentration parameter k on the interval [-pi,pi]. The size of p is the common size of x, mu, and k. A scalar input functions as a constant matrix of the same same size as the other inputs.

p = vmcdf (x, mu, k, "upper") computes the upper tail probability of the von Mises distribution with parameters mu and k, at the values in x.

Note: the CDF of the von Mises distribution is not analytic. Hence, it is calculated by integrating its probability density which is expressed as a series of Bessel functions. Balancing between performance and accuracy, the integration uses a step of 1e-5 on the interval [-pi,pi], which results to an accuracy of about 10 significant digits.

Further information about the von Mises distribution can be found at https://en.wikipedia.org/wiki/Von_Mises_distribution

See also: vminv, vmpdf, vmrnd

Source Code: vmcdf

Example: 1

  

 ## Plot various CDFs from the von Mises distribution
 x1 = [-pi:0.1:pi];
 p1 = vmcdf (x1, 0, 0.5);
 p2 = vmcdf (x1, 0, 1);
 p3 = vmcdf (x1, 0, 2);
 p4 = vmcdf (x1, 0, 4);
 plot (x1, p1, "-r", x1, p2, "-g", x1, p3, "-b", x1, p4, "-c")
 grid on
 xlim ([-pi, pi])
 legend ({"μ = 0, k = 0.5", "μ = 0, k = 1", ...
          "μ = 0, k = 2", "μ = 0, k = 4"}, "location", "northwest")
 title ("Von Mises CDF")
 xlabel ("values in x")
 ylabel ("probability")
plotted figure

statistics-release-1.6.3/docs/vminv.html000066400000000000000000000144521456127120000203120ustar00rootroot00000000000000 Statistics: vminv

Function Reference: vminv

statistics: x = vminv (p, mu, k)

Inverse of the von Mises cumulative distribution function (iCDF).

For each element of p, compute the quantile (the inverse of the CDF) of the von Mises distribution with location parameter mu and concentration parameter k on the interval [-pi,pi]. The size of x is the common size of p, mu, and k. A scalar input functions as a constant matrix of the same size as the other inputs.

Note: the quantile of the von Mises distribution is not analytic. Hence, it is approximated by a custom searching algorithm using its CDF until it converges up to a tolerance of 1e-5 or 100 iterations. As a result, balancing between performance and accuracy, the accuracy is about 5e-5 for k = 1 and it drops to 5e-5 as k increases.

Further information about the von Mises distribution can be found at https://en.wikipedia.org/wiki/Von_Mises_distribution

See also: vmcdf, vmpdf, vmrnd

Source Code: vminv

Example: 1

  

 ## Plot various iCDFs from the von Mises distribution
 p1 = [0,0.005,0.01:0.01:0.1,0.15,0.2:0.1:0.8,0.85,0.9:0.01:0.99,0.995,1];
 x1 = vminv (p1, 0, 0.5);
 x2 = vminv (p1, 0, 1);
 x3 = vminv (p1, 0, 2);
 x4 = vminv (p1, 0, 4);
 plot (p1, x1, "-r", p1, x2, "-g", p1, x3, "-b", p1, x4, "-c")
 grid on
 ylim ([-pi, pi])
 legend ({"μ = 0, k = 0.5", "μ = 0, k = 1", ...
          "μ = 0, k = 2", "μ = 0, k = 4"}, "location", "northwest")
 title ("Von Mises iCDF")
 xlabel ("probability")
 ylabel ("values in x")
plotted figure

statistics-release-1.6.3/docs/vmpdf.html000066400000000000000000000134671456127120000202740ustar00rootroot00000000000000 Statistics: vmpdf

Function Reference: vmpdf

statistics: y = vmpdf (x, mu, k)

Von Mises probability density function (PDF).

For each element of x, compute the probability density function (PDF) of the von Mises distribution with location parameter mu and concentration parameter k on the interval [-pi, pi]. The size of y is the common size of x, mu, and k. A scalar input functions as a constant matrix of the same size as the other inputs.

Further information about the von Mises distribution can be found at https://en.wikipedia.org/wiki/Von_Mises_distribution

See also: vmcdf, vminv, vmrnd

Source Code: vmpdf

Example: 1

  

 ## Plot various PDFs from the von Mises distribution
 x1 = [-pi:0.1:pi];
 y1 = vmpdf (x1, 0, 0.5);
 y2 = vmpdf (x1, 0, 1);
 y3 = vmpdf (x1, 0, 2);
 y4 = vmpdf (x1, 0, 4);
 plot (x1, y1, "-r", x1, y2, "-g", x1, y3, "-b", x1, y4, "-c")
 grid on
 xlim ([-pi, pi])
 ylim ([0, 0.8])
 legend ({"μ = 0, k = 0.5", "μ = 0, k = 1", ...
          "μ = 0, k = 2", "μ = 0, k = 4"}, "location", "northwest")
 title ("Von Mises PDF")
 xlabel ("values in x")
 ylabel ("density")
plotted figure

statistics-release-1.6.3/docs/vmrnd.html000066400000000000000000000122651456127120000203010ustar00rootroot00000000000000 Statistics: vmrnd

Function Reference: vmrnd

statistics: r = vmrnd (mu, k)
statistics: r = vmrnd (mu, k, rows)
statistics: r = vmrnd (mu, k, rows, cols, …)
statistics: r = vmrnd (mu, k, [sz])

Random arrays from the von Mises distribution.

r = vmrnd (mu, k) returns an array of random angles chosen from a von Mises distribution with location parameter mu and concentration parameter k on the interval [-pi, pi]. The size of r is the common size of mu and k. A scalar input functions as a constant matrix of the same size as the other inputs. Both parameters must be finite real numbers and k > 0, otherwise NaN is returned.

When called with a single size argument, vmrnd returns a square matrix with the dimension specified. When called with more than one scalar argument, the first two arguments are taken as the number of rows and columns and any further arguments specify additional matrix dimensions. The size may also be specified with a row vector of dimensions, sz.

Further information about the von Mises distribution can be found at https://en.wikipedia.org/wiki/Von_Mises_distribution

See also: vmcdf, vminv, vmpdf

Source Code: vmrnd

statistics-release-1.6.3/docs/wblcdf.html000066400000000000000000000166151456127120000204170ustar00rootroot00000000000000 Statistics: wblcdf

Function Reference: wblcdf

statistics: p = wblcdf (x)
statistics: p = wblcdf (x, lambda)
statistics: p = wblcdf (x, lambda, k)
statistics: p = wblcdf (…, "upper")
statistics: [p, plo, pup] = wblcdf (x, lambda, k, pcov)
statistics: [p, plo, pup] = wblcdf (x, lambda, k, pcov, alpha)
statistics: [p, plo, pup] = wblcdf (…, "upper")

Weibull cumulative distribution function (CDF).

For each element of x, compute the cumulative distribution function (CDF) of the Weibull distribution with scale parameter lambda and shape parameter k. The size of p is the common size of x, lambda and k. A scalar input functions as a constant matrix of the same size as the other inputs.

Default values are lambda = 1, k = 1.

When called with three output arguments, [p, plo, pup] it computes the confidence bounds for p when the input parameters lambda and k are estimates. In such case, pcov, a 2-by-2 matrix containing the covariance matrix of the estimated parameters, is necessary. Optionally, alpha has a default value of 0.05, and specifies 100 * (1 - alpha)% confidence bounds. plo and pup are arrays of the same size as p containing the lower and upper confidence bounds.

[…] = wblcdf (…, "upper") computes the upper tail probability of the lognormal distribution.

Further information about the Weibull distribution can be found at https://en.wikipedia.org/wiki/Weibull_distribution

See also: wblinv, wblpdf, wblrnd, wblstat, wblplot

Source Code: wblcdf

Example: 1

  

 ## Plot various CDFs from the Weibull distribution
 x = 0:0.001:2.5;
 p1 = wblcdf (x, 1, 0.5);
 p2 = wblcdf (x, 1, 1);
 p3 = wblcdf (x, 1, 1.5);
 p4 = wblcdf (x, 1, 5);
 plot (x, p1, "-b", x, p2, "-r", x, p3, "-m", x, p4, "-g")
 grid on
 legend ({"λ = 1, k = 0.5", "λ = 1, k = 1", ...
          "λ = 1, k = 1.5", "λ = 1, k = 5"}, "location", "southeast")
 title ("Weibull CDF")
 xlabel ("values in x")
 ylabel ("probability")
plotted figure

statistics-release-1.6.3/docs/wblfit.html000066400000000000000000000225071456127120000204420ustar00rootroot00000000000000 Statistics: wblfit

Function Reference: wblfit

statistics: paramhat = wblfit (x)
statistics: [paramhat, paramci] = wblfit (x)
statistics: [paramhat, paramci] = wblfit (x, alpha)
statistics: […] = wblfit (x, alpha, censor)
statistics: […] = wblfit (x, alpha, censor, freq)
statistics: […] = wblfit (x, alpha, censor, freq, options)

Estimate mean and confidence intervals for the Weibull distribution.

muhat = wblfit (x) returns the maximum likelihood estimates of the parameters of the Weibull distribution given the data in x. paramhat(1) is the scale parameter, lambda, and paramhat(2) is the shape parameter, k.

[paramhat, paramci] = wblfit (x) returns the 95% confidence intervals for the parameter estimates.

[…] = wblfit (x, alpha) also returns the 100 * (1 - alpha) percent confidence intervals for the parameter estimates. By default, the optional argument alpha is 0.05 corresponding to 95% confidence intervals. Pass in [] for alpha to use the default values.

[…] = wblfit (x, alpha, censor) accepts a boolean vector, censor, of the same size as x with 1s for observations that are right-censored and 0s for observations that are observed exactly. By default, or if left empty, censor = zeros (size (x)).

[…] = wblfit (x, alpha, censor, freq) accepts a frequency vector, freq, of the same size as x. freq typically contains integer frequencies for the corresponding elements in x, but it can contain any non-integer non-negative values. By default, or if left empty, freq = ones (size (x)).

[…] = evfit (…, options) specifies control parameters for the iterative algorithm used to compute ML estimates with the fminsearch function. options is a structure with the following fields and their default values:

  • options.Display = "off"
  • options.TolX = 1e-6

Further information about the Weibull distribution can be found at https://en.wikipedia.org/wiki/Weibull_distribution

See also: wblcdf, wblinv, wblpdf, wblrnd, wbllike, wblstat

Source Code: wblfit

Example: 1

  

 ## Sample 3 populations from 3 different Weibull distibutions
 rande ("seed", 1);    # for reproducibility
 r1 = wblrnd(2, 4, 2000, 1);
 rande ("seed", 2);    # for reproducibility
 r2 = wblrnd(5, 2, 2000, 1);
 rande ("seed", 5);    # for reproducibility
 r3 = wblrnd(1, 5, 2000, 1);
 r = [r1, r2, r3];

 ## Plot them normalized and fix their colors
 hist (r, 30, [2.5 2.1 3.2]);
 h = findobj (gca, "Type", "patch");
 set (h(1), "facecolor", "c");
 set (h(2), "facecolor", "g");
 set (h(3), "facecolor", "r");
 ylim ([0, 2]);
 xlim ([0, 10]);
 hold on

 ## Estimate their lambda parameter
 lambda_kA = wblfit (r(:,1));
 lambda_kB = wblfit (r(:,2));
 lambda_kC = wblfit (r(:,3));

 ## Plot their estimated PDFs
 x = [0:0.1:15];
 y = wblpdf (x, lambda_kA(1), lambda_kA(2));
 plot (x, y, "-pr");
 y = wblpdf (x, lambda_kB(1), lambda_kB(2));
 plot (x, y, "-sg");
 y = wblpdf (x, lambda_kC(1), lambda_kC(2));
 plot (x, y, "-^c");
 hold off
 legend ({"Normalized HIST of sample 1 with λ=2 and k=4", ...
          "Normalized HIST of sample 2 with λ=5 and k=2", ...
          "Normalized HIST of sample 3 with λ=1 and k=5", ...
          sprintf("PDF for sample 1 with estimated λ=%0.2f and k=%0.2f", ...
                  lambda_kA(1), lambda_kA(2)), ...
          sprintf("PDF for sample 2 with estimated λ=%0.2f and k=%0.2f", ...
                  lambda_kB(1), lambda_kB(2)), ...
          sprintf("PDF for sample 3 with estimated λ=%0.2f and k=%0.2f", ...
                  lambda_kC(1), lambda_kC(2))})
 title ("Three population samples from different Weibull distibutions")
 hold off
plotted figure

statistics-release-1.6.3/docs/wblinv.html000066400000000000000000000141431456127120000204510ustar00rootroot00000000000000 Statistics: wblinv

Function Reference: wblinv

statistics: x = wblinv (p)
statistics: x = wblinv (p, lambda)
statistics: x = wblinv (p, lambda, k)

Inverse of the Weibull cumulative distribution function (iCDF).

For each element of p, compute the quantile (the inverse of the CDF) of the Weibull distribution with scale parameter lambda and shape parameter k. The size of x is the common size of p, lambda, and k. A scalar input functions as a constant matrix of the same size as the other inputs.

Default values are lambda = 1, k = 1.

Further information about the Weibull distribution can be found at https://en.wikipedia.org/wiki/Weibull_distribution

See also: wblcdf, wblpdf, wblrnd, wblstat, wblplot

Source Code: wblinv

Example: 1

  

 ## Plot various iCDFs from the Weibull distribution
 p = 0.001:0.001:0.999;
 x1 = wblinv (p, 1, 0.5);
 x2 = wblinv (p, 1, 1);
 x3 = wblinv (p, 1, 1.5);
 x4 = wblinv (p, 1, 5);
 plot (p, x1, "-b", p, x2, "-r", p, x3, "-m", p, x4, "-g")
 ylim ([0, 2.5])
 grid on
 legend ({"λ = 1, k = 0.5", "λ = 1, k = 1",  ...
          "λ = 1, k = 1.5", "λ = 1, k = 5"}, "location", "northwest")
 title ("Weibull iCDF")
 xlabel ("probability")
 ylabel ("x")
plotted figure

statistics-release-1.6.3/docs/wbllike.html000066400000000000000000000140751456127120000206050ustar00rootroot00000000000000 Statistics: wbllike

Function Reference: wbllike

statistics: nlogL = wbllike (params, x)
statistics: [nlogL, acov] = wbllike (params, x)
statistics: […] = wbllike (params, x, alpha, censor)
statistics: […] = wbllike (params, x, alpha, censor, freq)

Negative log-likelihood for the Weibull distribution.

nlogL = wbllike (params, data) returns the negative log-likelihood of the data in x corresponding to the Weibull distribution with (1) scale parameter lambda and (2) shape parameter k given in the two-element vector params.

[nlogL, acov] = wbllike (params, data) also returns the inverse of Fisher’s information matrix, acov. If the input parameter values in params are the maximum likelihood estimates, the diagonal elements of acov are their asymptotic variances. acov is based on the observed Fisher’s information, not the expected information.

[…] = wbllike (params, data, censor) accepts a boolean vector, censor, of the same size as x with 1s for observations that are right-censored and 0s for observations that are observed exactly. By default, or if left empty, censor = zeros (size (x)).

[…] = wbllike (params, data, censor, freq) accepts a frequency vector, freq, of the same size as x. freq typically contains integer frequencies for the corresponding elements in x, but may contain any non-integer non-negative values. By default, or if left empty, freq = ones (size (x)).

Further information about the Weibull distribution can be found at https://en.wikipedia.org/wiki/Weibull_distribution

See also: wblcdf, wblinv, wblpdf, wblrnd, wblfit, wblstat

Source Code: wbllike

statistics-release-1.6.3/docs/wblpdf.html000066400000000000000000000142261456127120000204300ustar00rootroot00000000000000 Statistics: wblpdf

Function Reference: wblpdf

statistics: y = wblinv (x)
statistics: y = wblinv (x, lambda)
statistics: y = wblinv (x, lambda, k)

Weibull probability density function (PDF).

For each element of x, compute the probability density function (PDF) of the Weibull distribution with scale parameter lambda and shpe parameter k. The size of y is the common size of x, lambda, and k. A scalar input functions as a constant matrix of the same size as the other inputs.

Default values are lambda = 1, k = 1.

Further information about the Weibull distribution can be found at https://en.wikipedia.org/wiki/Weibull_distribution

See also: wblcdf, wblinv, wblrnd, wblfit, wbllike, wblstat, wblplot

Source Code: wblpdf

Example: 1

  

 ## Plot various PDFs from the Weibul distribution
 x = 0:0.001:2.5;
 y1 = wblpdf (x, 1, 0.5);
 y2 = wblpdf (x, 1, 1);
 y3 = wblpdf (x, 1, 1.5);
 y4 = wblpdf (x, 1, 5);
 plot (x, y1, "-b", x, y2, "-r", x, y3, "-m", x, y4, "-g")
 grid on
 ylim ([0, 2.5])
 legend ({"λ = 5, k = 0.5", "λ = 9, k = 1", ...
          "λ = 6, k = 1.5", "λ = 2, k = 5"}, "location", "northeast")
 title ("Weibul PDF")
 xlabel ("values in x")
 ylabel ("density")
plotted figure

statistics-release-1.6.3/docs/wblplot.html000066400000000000000000000270341456127120000206360ustar00rootroot00000000000000 Statistics: wblplot

Function Reference: wblplot

statistics: wblplot (data, …)
statistics: handle = wblplot (data, …)
statistics: [handle, param] = wblplot (data)
statistics: [handle, param] = wblplot (data, censor)
statistics: [handle, param] = wblplot (data, censor, freq)
statistics: [handle, param] = wblplot (data, censor, freq, confint)
statistics: [handle, param] = wblplot (data, censor, freq, confint, fancygrid)
statistics: [handle, param] = wblplot (data, censor, freq, confint, fancygrid, showlegend)

Plot a column vector data on a Weibull probability plot using rank regression.

censor: optional parameter is a column vector of same size as data with 1 for right censored data and 0 for exact observation. Pass [] when no censor data are available.

freq: optional vector same size as data with the number of occurences for corresponding data. Pass [] when no frequency data are available.

confint: optional confidence limits for ploting upper and lower confidence bands using beta binomial confidence bounds. If a single value is given this will be used such as LOW = a and HIGH = 1 - a. Pass [] if confidence bounds is not requested.

fancygrid: optional parameter which if set to anything but 1 will turn off the fancy gridlines.

showlegend: optional parameter that when set to zero(0) turns off the legend.

If one output argument is given, a handle for the data marker and plotlines is returned, which can be used for further modification of line and marker style.

If a second output argument is specified, a param vector with scale, shape and correlation factor is returned.

See also: normplot, wblpdf

Source Code: wblplot

Example: 1

  

 x = [16 34 53 75 93 120];
 wblplot (x);
plotted figure

Example: 2

  

 x = [2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67]';
 c = [0 1 0 1 0 1 1 1 0 0 1 0 1 0 1 1 0 1 1]';
 [h, p] = wblplot (x, c);
 p

p =

   82.0192    0.8951    0.9896
plotted figure

Example: 3

  

 x = [16, 34, 53, 75, 93, 120, 150, 191, 240 ,339];
 [h, p] = wblplot (x, [], [], 0.05);
 p
 ## Benchmark Reliasoft eta = 146.2545 beta 1.1973 rho = 0.9999

p =

   146.2545     1.1973     0.9999
plotted figure

Example: 4

  

 x = [46 64 83 105 123 150 150];
 c = [0 0 0 0 0 0 1];
 f = [1 1 1 1 1 1 4];
 wblplot (x, c, f, 0.05);
plotted figure

Example: 5

  

 x = [46 64 83 105 123 150 150];
 c = [0 0 0 0 0 0 1];
 f = [1 1 1 1 1 1 4];
 ## Subtract 30.92 from x to simulate a 3 parameter wbl with gamma = 30.92
 wblplot (x - 30.92, c, f, 0.05);
plotted figure

statistics-release-1.6.3/docs/wblrnd.html000066400000000000000000000124271456127120000204430ustar00rootroot00000000000000 Statistics: wblrnd

Function Reference: wblrnd

statistics: r = wblrnd (lambda, k)
statistics: r = wblrnd (lambda, k, rows)
statistics: r = wblrnd (lambda, k, rows, cols, …)
statistics: r = wblrnd (lambda, k, [sz])

Random arrays from the Weibull distribution.

r = wblrnd (lambda, k) returns an array of random numbers chosen from the Weibull distribution with scale parameter lambda and shape parameter k. The size of r is the common size of lambda and k. A scalar input functions as a constant matrix of the same size as the other inputs. Both parameters must be positive reals.

When called with a single size argument, wblrnd returns a square matrix with the dimension specified. When called with more than one scalar argument, the first two arguments are taken as the number of rows and columns and any further arguments specify additional matrix dimensions. The size may also be specified with a row vector of dimensions, sz.

Further information about the Weibull distribution can be found at https://en.wikipedia.org/wiki/Weibull_distribution

See also: wblcdf, wblinv, wblpdf, wblfit, wbllike, wblstat, wblplot

Source Code: wblrnd

statistics-release-1.6.3/docs/wblstat.html000066400000000000000000000110541456127120000206260ustar00rootroot00000000000000 Statistics: wblstat

Function Reference: wblstat

statistics: [m, v] = wblstat (lambda, k)

Compute statistics of the Weibull distribution.

[m, v] = wblstat (lambda, k) returns the mean and variance of the Weibull distribution with scale parameter lambda and shape parameter k.

The size of m (mean) and v (variance) is the common size of the input arguments. A scalar input functions as a constant matrix of the same size as the other inputs.

Further information about the Weibull distribution can be found at https://en.wikipedia.org/wiki/Weibull_distribution

See also: wblcdf, wblinv, wblpdf, wblrnd, wblfit, wbllike, wblplot

Source Code: wblstat

statistics-release-1.6.3/docs/wienrnd.html000066400000000000000000000100771456127120000206200ustar00rootroot00000000000000 Statistics: wienrnd

Function Reference: wienrnd

statistics: r = wienrnd (t, d, n)

Return a simulated realization of the d-dimensional Wiener Process on the interval [0, t].

If d is omitted, d = 1 is used. The first column of the return matrix contains time, the remaining columns contain the Wiener process.

The optional parameter n defines the number of summands used for simulating the process over an interval of length 1. If n is omitted, n = 1000 is used.

Source Code: wienrnd

statistics-release-1.6.3/docs/wishpdf.html000066400000000000000000000107161456127120000206160ustar00rootroot00000000000000 Statistics: wishpdf

Function Reference: wishpdf

statistics: y = wishpdf (W, Sigma, df, log_y=false)

Compute the probability density function of the Wishart distribution

Inputs: A p x p matrix W where to find the PDF. The p x p positive definite matrix Sigma and scalar degrees of freedom parameter df characterizing the Wishart distribution. (For the density to be finite, need df > (p - 1).)

If the flag log_y is set, return the log probability density – this helps avoid underflow when the numerical value of the density is very small

Output: y is the probability density of Wishart(Sigma, df) at W.

See also: wishrnd, iwishpdf, iwishrnd

Source Code: wishpdf

statistics-release-1.6.3/docs/wishrnd.html000066400000000000000000000120701456127120000206230ustar00rootroot00000000000000 Statistics: wishrnd

Function Reference: wishrnd

statistics: [W, D] = wishrnd (Sigma, df, D, n=1)

Return a random matrix sampled from the Wishart distribution with given parameters

Inputs: the p× p positive definite matrix Sigma (or the lower-triangular Cholesky factor D of Sigma) and scalar degrees of freedom parameter df.

df can be non-integer as long as df > p

Output: a random p× p matrix W from the Wishart(Sigma, df) distribution. If n > 1, then W is p x p x n and holds n such random matrices. (Optionally, the lower-triangular Cholesky factor D of Sigma is also returned.)

Averaged across many samples, the mean of W should approach df*Sigma, and the variance of each element W_ij should approach df*(Sigma_ij^2 + Sigma_ii*Sigma_jj)

References

  1. Yu-Cheng Ku and Peter Bloomfield (2010), Generating Random Wishart Matrices with Fractional Degrees of Freedom in OX, http://www.gwu.edu/~forcpgm/YuChengKu-030510final-WishartYu-ChengKu.pdf

See also: wishpdf, iwishpdf, iwishrnd

Source Code: wishrnd

statistics-release-1.6.3/docs/x2fx.html000066400000000000000000000161061456127120000200400ustar00rootroot00000000000000 Statistics: x2fx

Function Reference: x2fx

statistics: [d, model, termstart, termend] = x2fx (x)
statistics: [d, model, termstart, termend] = x2fx (x, model)
statistics: [d, model, termstart, termend] = x2fx (x, model, categ)
statistics: [d, model, termstart, termend] = x2fx (x, model, categ, catlevels)

Convert predictors to design matrix.

d = x2fx (x, model) converts a matrix of predictors x to a design matrix d for regression analysis. Distinct predictor variables should appear in different columns of x.

The optional input model controls the regression model. By default, x2fx returns the design matrix for a linear additive model with a constant term. model can be any one of the following strings:

"linear"Constant and linear terms (the default)
"interaction"Constant, linear, and interaction terms
"quadratic"Constant, linear, interaction, and squared terms
"purequadratic"Constant, linear, and squared terms

If x has n columns, the order of the columns of d for a full quadratic model is:

  • The constant term.
  • The linear terms (the columns of X, in order 1,2,...,n).
  • The interaction terms (pairwise products of columns of x, in order (1,2), (1,3), ..., (1,n), (2,3), ..., (n-1,n).
  • The squared terms (in the order 1,2,...,n).

Other models use a subset of these terms, in the same order.

Alternatively, MODEL can be a matrix specifying polynomial terms of arbitrary order. In this case, MODEL should have one column for each column in X and one r for each term in the model. The entries in any r of MODEL are powers for the corresponding columns of x. For example, if x has columns X1, X2, and X3, then a row [0 1 2] in model would specify the term (X1.^0).*(X2.^1).*(X3.^2). A row of all zeros in model specifies a constant term, which you can omit.

d = x2fx (x, model, categ) treats columns with numbers listed in the vector categ as categorical variables. Terms involving categorical variables produce dummy variable columns in d. Dummy variables are computed under the assumption that possible categorical levels are completely enumerated by the unique values that appear in the corresponding column of x.

d = x2fx (x, model, categ, catlevels) accepts a vector catlevels the same length as categ, specifying the number of levels in each categorical variable. In this case, values in the corresponding column of x must be integers in the range from 1 to the specified number of levels. Not all of the levels need to appear in x.

Source Code: x2fx

statistics-release-1.6.3/docs/ztest.html000066400000000000000000000155051456127120000203240ustar00rootroot00000000000000 Statistics: ztest

Function Reference: ztest

statistics: h = ztest (x, m, sigma)
statistics: h = ztest (x, m, sigma, Name, Value)
statistics: [h, pval] = ztest (…)
statistics: [h, pval, ci] = ztest (…)
statistics: [h, pval, ci, zvalue] = ztest (…)

One-sample Z-test.

h = ztest (x, v) performs a Z-test of the hypothesis that the data in the vector x come from a normal distribution with mean m, against the alternative that x comes from a normal distribution with a different mean m. The result is h = 0 if the null hypothesis ("mean is M") cannot be rejected at the 5% significance level, or h = 1 if the null hypothesis can be rejected at the 5% level.

x may also be a matrix or an N-D array. For matrices, ztest performs separate tests along each column of x, and returns a vector of results. For N-D arrays, ztest works along the first non-singleton dimension of x. m and sigma must be a scalars.

ztest treats NaNs as missing values, and ignores them.

[h, pval] = ztest (…) returns the p-value. That is the probability of observing the given result, or one more extreme, by chance if the null hypothesisis true.

[h, pval, ci] = ztest (…) returns a 100 * (1 - alpha)% confidence interval for the true mean.

[h, pval, ci, zvalue] = ztest (…) returns the value of the test statistic.

[…] = ztest (…, Name, Value, …) specifies one or more of the following Name/Value pairs:

NameValue
"alpha"the significance level. Default is 0.05.
"dim"dimension to work along a matrix or an N-D array.
"tail"a string specifying the alternative hypothesis:
"both""mean is not m" (two-tailed, default)
"left""mean is less than m" (left-tailed)
"right""mean is greater than m" (right-tailed)

See also: ttest, vartest, signtest, kstest

Source Code: ztest

statistics-release-1.6.3/docs/ztest2.html000066400000000000000000000145431456127120000204070ustar00rootroot00000000000000 Statistics: ztest2

Function Reference: ztest2

statistics: h = ztest2 (x1, n1, x2, n2)
statistics: h = ztest2 (x1, n1, x2, n2, Name, Value)
statistics: [h, pval] = ztest2 (…)
statistics: [h, pval, zvalue] = ztest2 (…)

Two proportions Z-test.

If x1 and n1 are the counts of successes and trials in one sample, and x2 and n2 those in a second one, test the null hypothesis that the success probabilities p1 and p2 are the same. The result is h = 0 if the null hypothesis cannot be rejected at the 5% significance level, or h = 1 if the null hypothesis can be rejected at the 5% level.

Under the null, the test statistic zvalue approximately follows a standard normal distribution.

The size of h, pval, and zvalue is the common size of x, n1, x2, and n2, which must be scalars or of common size. A scalar input functions as a constant matrix of the same size as the other inputs.

[h, pval] = ztest2 (…) returns the p-value. That is the probability of observing the given result, or one more extreme, by chance if the null hypothesisis true.

[h, pval, zvalue] = ztest2 (…) returns the value of the test statistic.

[…] = ztest2 (…, Name, Value, …) specifies one or more of the following Name/Value pairs:

NameValue
"alpha"the significance level. Default is 0.05.
"tail"a string specifying the alternative hypothesis
"both"p1 is not p2 (two-tailed, default)
"left"p1 is less than p2 (left-tailed)
"right"p1 is greater than p2 (right-tailed)

See also: chi2test, fishertest

Source Code: ztest2

statistics-release-1.6.3/inst/000077500000000000000000000000001456127120000163045ustar00rootroot00000000000000statistics-release-1.6.3/inst/@cvpartition/000077500000000000000000000000001456127120000207465ustar00rootroot00000000000000statistics-release-1.6.3/inst/@cvpartition/cvpartition.m000066400000000000000000000206011456127120000234650ustar00rootroot00000000000000## Copyright (C) 2014 Nir Krakauer ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation; either version 3 of the License, or ## (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; If not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{C} =} cvpartition (@var{X}, [@var{partition_type}, [@var{k}]]) ## ## Create a partition object for cross validation. ## ## @var{X} may be a positive integer, interpreted as the number of values ## @var{n} to partition, or a vector of length @var{n} containing class ## designations for the elements, in which case the partitioning types ## @var{KFold} and @var{HoldOut} attempt to ensure each partition represents the ## classes proportionately. ## ## @var{partition_type} must be one of the following: ## ## @table @asis ## @item @samp{KFold} ## Divide set into @var{k} equal-size subsets (this is the default, with ## @var{k}=10). ## @item @samp{HoldOut} ## Divide set into two subsets, "training" and "validation". If @var{k} is a ## fraction, that is the fraction of values put in the validation subset; if it ## is a positive integer, that is the number of values in the validation subset ## (by default @var{k}=0.1). ## @item @samp{LeaveOut} ## Leave-one-out partition (each element is placed in its own subset). ## @item @samp{resubstitution} ## Training and validation subsets that both contain all the original elements. ## @item @samp{Given} ## Subset indices are as given in @var{X}. ## @end table ## ## The following fields are defined for the @samp{cvpartition} class: ## ## @table @asis ## @item @samp{classes} ## Class designations for the elements. ## @item @samp{inds} ## Subset indices for the elements. ## @item @samp{n_classes} ## Number of different classes. ## @item @samp{NumObservations} ## @var{n}, number of elements in data set. ## @item @samp{NumTestSets} ## Number of testing subsets. ## @item @samp{TestSize} ## Number of elements in (each) testing subset. ## @item @samp{TrainSize} ## Number of elements in (each) training subset. ## @item @samp{Type} ## Partition type. ## @end table ## ## @seealso{crossval, @@cvpartition/display} ## @end deftypefn function C = cvpartition (X, partition_type = "KFold", k = []) if (nargin < 1 || nargin > 3 || !isvector(X)) print_usage (); endif if (isscalar (X)) n = X; n_classes = 1; else n = numel (X); endif switch (tolower (partition_type)) case {"kfold", "holdout", "leaveout", "resubstitution", "given"} otherwise warning ("cvpartition: unrecognized type, using KFold.") partition_type = "KFold"; endswitch switch (tolower (partition_type)) case {"kfold", "holdout", "given"} if (! isscalar (X)) [y, ~, j] = unique (X(:)); n_per_class = accumarray (j, 1); n_classes = numel (n_per_class); endif endswitch C = struct ("classes", [], "inds", [], "n_classes", [], "NumObservations", ... [], "NumTestSets", [], "TestSize", [], "TrainSize", [], "Type", []); ## The non-Matlab fields classes, inds, n_classes ## are only useful for some methods switch (tolower (partition_type)) case "kfold" if (isempty (k)) k = 10; endif if (n_classes == 1) inds = floor((0:(n-1))' * (k / n)) + 1; else inds = nan(n, 1); for i = 1:n_classes ## Alternate ordering over classes so that ## the subsets are more nearly the same size if (mod (i, 2)) inds(j == i) = floor((0:(n_per_class(i)-1))' * ... (k / n_per_class(i))) + 1; else inds(j == i) = floor(((n_per_class(i)-1):-1:0)' * ... (k / n_per_class(i))) + 1; endif endfor endif C.inds = inds; C.NumTestSets = k; [~, ~, jj] = unique (inds); n_per_subset = accumarray (jj, 1); C.TrainSize = n - n_per_subset; C.TestSize = n_per_subset; case "given" C.inds = j; C.NumTestSets = n_classes; C.TrainSize = n - n_per_class; C.TestSize = n_per_class; case "holdout" if (isempty (k)) k = 0.1; endif if (k < 1) f = k; # target fraction to sample k = round (k * n); # number of samples else f = k / n; endif inds = zeros (n, 1, "logical"); if (n_classes == 1) inds(randsample(n, k)) = true; # indices for test set else # sample from each class k_check = 0; for i = 1:n_classes ki = round(f*n_per_class(i)); inds(find(j == i)(randsample(n_per_class(i), ki))) = true; k_check += ki; endfor if (k_check < k) # add random elements to test set to make it k inds(find(!inds)(randsample(n - k_check, k - k_check))) = true; elseif (k_check > k) # remove random elements from test set inds(find(inds)(randsample(k_check, k_check - k))) = false; endif C.classes = j; endif C.n_classes = n_classes; C.TrainSize = n - k; C.TestSize = k; C.NumTestSets = 1; C.inds = inds; case "leaveout" C.TrainSize = ones (n, 1); C.TestSize = (n-1) * ones (n, 1); C.NumTestSets = n; case "resubstitution" C.TrainSize = C.TestSize = n; C.NumTestSets = 1; endswitch C.NumObservations = n; C.Type = tolower (partition_type); C = class (C, "cvpartition"); endfunction %!demo %! ## Partition with Fisher iris dataset (n = 150) %! ## Stratified by species %! load fisheriris %! y = species; %! ## 10-fold cross-validation partition %! c = cvpartition (species, 'KFold', 10) %! ## leave-10-out partition %! c1 = cvpartition (species, 'HoldOut', 10) %! idx1 = test (c, 2); %! idx2 = training (c, 2); %! ## another leave-10-out partition %! c2 = repartition (c1) %!test %! C = cvpartition (ones (10, 1)); %! assert (isa (C, "cvpartition"), true); %!test %! C = cvpartition (ones (10, 1), "KFold", 5); %! assert (get (C, "NumObservations"), 10); %! assert (get (C, "NumTestSets"), 5); %! assert (get (C, "TrainSize"), ones(5,1) * 8); %! assert (get (C, "TestSize"), ones (5,1) * 2); %! assert (get (C, "inds"), [1 1 2 2 3 3 4 4 5 5]'); %! assert (get (C, "Type"), "kfold"); %!test %! C = cvpartition (ones (10, 1), "KFold", 2); %! assert (get (C, "NumObservations"), 10); %! assert (get (C, "NumTestSets"), 2); %! assert (get (C, "TrainSize"), [5; 5]); %! assert (get (C, "TestSize"), [5; 5]); %! assert (get (C, "inds"), [1 1 1 1 1 2 2 2 2 2]'); %! assert (get (C, "Type"), "kfold"); %!test %! C = cvpartition (ones (10, 1), "HoldOut", 5); %! assert (get (C, "NumObservations"), 10); %! assert (get (C, "NumTestSets"), 1); %! assert (get (C, "TrainSize"), 5); %! assert (get (C, "TestSize"), 5); %! assert (class (get (C, "inds")), "logical"); %! assert (length (get (C, "inds")), 10); %! assert (get (C, "Type"), "holdout"); %!test %! C = cvpartition ([1 2 3 4 5 6 7 8 9 10], "LeaveOut", 5); %! assert (get (C, "NumObservations"), 10); %! assert (get (C, "NumTestSets"), 10); %! assert (get (C, "TrainSize"), ones (10, 1)); %! assert (get (C, "TestSize"), ones (10, 1) * 9); %! assert (get (C, "inds"), []); %! assert (get (C, "Type"), "leaveout"); %!test %! C = cvpartition ([1 2 3 4 5 6 7 8 9 10], "resubstitution", 5); %! assert (get (C, "NumObservations"), 10); %! assert (get (C, "NumTestSets"), 1); %! assert (get (C, "TrainSize"), 10); %! assert (get (C, "TestSize"), 10); %! assert (get (C, "inds"), []); %! assert (get (C, "Type"), "resubstitution"); %!test %! C = cvpartition ([1 2 3 4 5 6 7 8 9 10], "Given", 2); %! assert (get (C, "NumObservations"), 10); %! assert (get (C, "NumTestSets"), 10); %! assert (get (C, "TrainSize"), ones (10, 1) * 9); %! assert (get (C, "TestSize"), ones (10, 1)); %! assert (get (C, "inds"), [1:10]'); %! assert (get (C, "Type"), "given"); %!warning ... %! C = cvpartition ([1 2 3 4 5 6 7 8 9 10], "some", 2); statistics-release-1.6.3/inst/@cvpartition/display.m000066400000000000000000000050111456127120000225660ustar00rootroot00000000000000## Copyright (C) 2014 Nir Krakauer ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation; either version 3 of the License, or ## (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; If not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {} display (@var{C}) ## ## Display a @samp{cvpartition} object, @var{C}. ## ## @seealso{@@cvpartition/cvpartition} ## @end deftypefn function display (C) if (nargin != 1) print_usage (); endif switch C.Type case "kfold" str = "K-fold"; case "given" str = "Given"; case "holdout" str = "HoldOut"; case "leaveout" str = "Leave-One-Out"; case "resubstitution" str = "Resubstitution"; otherwise str = "Unknown-type"; endswitch disp([str " cross validation partition"]) disp([" N: " num2str(C.NumObservations)]) disp(["NumTestSets: " num2str(C.NumTestSets)]) disp([" TrainSize: " num2str(C.TrainSize')]) disp([" TestSize: " num2str(C.TestSize')]) endfunction %!test %! C = cvpartition (ones (10, 1), "KFold", 5); %! s = evalc ("display (C)"); %! sout = "K-fold cross validation partition"; %! assert (strcmpi (s(1:length (sout)), sout), true); %!test %! C = cvpartition (ones (10, 1), "HoldOut", 5); %! s = evalc ("display (C)"); %! sout = "HoldOut cross validation partition"; %! assert (strcmpi (s(1:length (sout)), sout), true); %!test %! C = cvpartition (ones (10, 1), "LeaveOut", 5); %! s = evalc ("display (C)"); %! sout = "Leave-One-Out cross validation partition"; %! assert (strcmpi (s(1:length (sout)), sout), true); %!test %! C = cvpartition (ones (10, 1), "resubstitution", 5); %! s = evalc ("display (C)"); %! sout = "Resubstitution cross validation partition"; %! assert (strcmpi (s(1:length (sout)), sout), true); %!test %! C = cvpartition (ones (10, 1), "Given", 5); %! s = evalc ("display (C)"); %! sout = "Given cross validation partition"; %! assert (strcmpi (s(1:length (sout)), sout), true); %!error display () statistics-release-1.6.3/inst/@cvpartition/get.m000066400000000000000000000036621456127120000217120ustar00rootroot00000000000000## Copyright (C) 2014 Nir Krakauer ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation; either version 3 of the License, or ## (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; If not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{s} = get} (@var{C}, @var{f}) ## ## Get a field, @var{f}, from a @samp{cvpartition} object, @var{C}. ## ## @seealso{@@cvpartition/cvpartition} ## @end deftypefn function s = get (c, f) if (nargin == 1) s = c; elseif (nargin == 2) if (ischar (f)) switch (f) case {"classes", "inds", "n_classes", "NumObservations", ... "NumTestSets", "TestSize", "TrainSize", "Type"} s = eval (["struct(c)." f]); otherwise error ("get: invalid property %s.", f); endswitch else error ("get: expecting the property to be a string."); endif else print_usage (); endif endfunction %!shared C %! C = cvpartition (ones (10, 1), "KFold", 5); %!assert (get (C, "NumObservations"), 10); %!assert (get (C, "NumTestSets"), 5); %!assert (get (C, "TrainSize"), ones(5,1) * 8); %!assert (get (C, "TestSize"), ones (5,1) * 2); %!assert (get (C, "inds"), [1 1 2 2 3 3 4 4 5 5]'); %!assert (get (C, "Type"), "kfold"); %!error get (C, "some") %!error get (C, 25) %!error get (C, {25}) statistics-release-1.6.3/inst/@cvpartition/repartition.m000066400000000000000000000054101456127120000234640ustar00rootroot00000000000000## Copyright (C) 2014 Nir Krakauer ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation; either version 3 of the License, or ## (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; If not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{Cnew} =} repartition (@var{C}) ## ## Return a new cvpartition object. ## ## @var{C} should be a @samp{cvpartition} object. @var{Cnew} will use the same ## partition_type as @var{C} but redo any randomization performed (currently, ## only the HoldOut type uses randomization). ## ## @seealso{@@cvpartition/cvpartition} ## @end deftypefn function Cnew = repartition (C) if (nargin < 1 || nargin > 2) print_usage (); endif Cnew = C; switch (C.Type) case 'kfold' case 'given' case 'holdout' # Currently, only the HoldOut method uses randomization n = C.NumObservations; k = C.TestSize; n_classes = C.n_classes; if (k < 1) f = k; # target fraction to sample k = round (k * n); #number of samples else f = k / n; endif inds = zeros (n, 1, "logical"); if (n_classes == 1) inds(randsample(n, k)) = true; #indices for test set else # sample from each class j = C.classes; #integer class labels n_per_class = accumarray (j, 1); n_classes = numel (n_per_class); k_check = 0; for i = 1:n_classes ki = round(f*n_per_class(i)); inds(find(j == i)(randsample(n_per_class(i), ki))) = true; k_check += ki; endfor if (k_check < k) # add random elements to test set to make it k inds(find(!inds)(randsample(n - k_check, k - k_check))) = true; elseif (k_check > k) # remove random elements from test set inds(find(inds)(randsample(k_check, k_check - k))) = false; endif endif Cnew.inds = inds; case "leaveout" case "resubstitution" endswitch endfunction %!test %! C = cvpartition (ones (10, 1), "KFold", 5); %! Cnew = repartition (C); %! assert (isa (Cnew, "cvpartition"), true); %!test %! C = cvpartition (ones (100, 1), "HoldOut", 5); %! Cnew = repartition (C); %! indC = get (C, "inds"); %! indCnew = get (Cnew, "inds"); %! assert (isequal (indC, indCnew), false); statistics-release-1.6.3/inst/@cvpartition/set.m000066400000000000000000000040541456127120000217220ustar00rootroot00000000000000## Copyright (C) 2014 Nir Krakauer ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation; either version 3 of the License, or ## (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; If not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{Cnew} =} set (@var{C}, @var{field}, @var{value}) ## ## Set @var{field}, in a @samp{cvpartition} object, @var{C}. ## ## @seealso{@@cvpartition/cvpartition} ## @end deftypefn function s = set (c, varargin) s = struct(c); if (length (varargin) < 2 || rem (length (varargin), 2) != 0) error ("set: expecting field/value pairs."); endif while (length (varargin) > 1) prop = varargin{1}; val = varargin{2}; varargin(1:2) = []; if (ischar (prop)) switch (prop) case {"classes", "inds", "n_classes", "NumObservations", ... "NumTestSets", "TestSize", "TrainSize", "Type"} s = setfield (s, prop, val); otherwise error ("set: invalid field %s.", prop); endswitch else error ("set: expecting the field to be a string."); endif endwhile s = class (s, "cvpartition"); endfunction %!shared C %! C = cvpartition (ones (10, 1), "KFold", 5); %!test %! Cnew = set (C, "inds", [1 2 2 2 3 4 3 4 5 5]'); %! assert (get (Cnew, "inds"), [1 2 2 2 3 4 3 4 5 5]'); %!error set (C) %!error set (C, "NumObservations") %!error set (C, "some", 15) %!error set (C, 15, 15) statistics-release-1.6.3/inst/@cvpartition/test.m000066400000000000000000000040541456127120000221060ustar00rootroot00000000000000## Copyright (C) 2014 Nir Krakauer ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation; either version 3 of the License, or ## (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; If not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{inds} =} test (@var{C}, [@var{i}]) ## ## Return logical vector for testing-subset indices from a @samp{cvpartition} ## object, @var{C}. @var{i} is the fold index (default is 1). ## ## @seealso{@@cvpartition/cvpartition, @@cvpartition/training} ## @end deftypefn function inds = test (C, i = []) if (nargin < 1 || nargin > 2) print_usage (); endif if (nargin < 2 || isempty (i)) i = 1; endif switch (C.Type) case {"kfold", "given"} inds = C.inds == i; case "holdout" inds = C.inds; case "leaveout" inds = zeros(C.NumObservations, 1, "logical"); inds(i) = true; case "resubstitution" inds = ones(C.NumObservations, 1, "logical"); endswitch endfunction %!shared C %! C = cvpartition (ones (10, 1), "KFold", 5); %!assert (test (C, 1), logical ([1 1 0 0 0 0 0 0 0 0]')) %!assert (test (C, 2), logical ([0 0 1 1 0 0 0 0 0 0]')) %!assert (test (C, 3), logical ([0 0 0 0 1 1 0 0 0 0]')) %!assert (test (C, 4), logical ([0 0 0 0 0 0 1 1 0 0]')) %!assert (test (C, 5), logical ([0 0 0 0 0 0 0 0 1 1]')) %!test %! C = set (C, "inds", [1 2 2 2 3 4 3 4 5 5]'); %!assert (test (C), logical ([1 0 0 0 0 0 0 0 0 0]')) %!assert (test (C, 2), logical ([0 1 1 1 0 0 0 0 0 0]')) %!assert (test (C, 3), logical ([0 0 0 0 1 0 1 0 0 0]')) statistics-release-1.6.3/inst/@cvpartition/training.m000066400000000000000000000041271456127120000227430ustar00rootroot00000000000000## Copyright (C) 2014 Nir Krakauer ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation; either version 3 of the License, or ## (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; If not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{inds} =} training (@var{C}, [@var{i}]) ## ## Return logical vector for training-subset indices from a @samp{cvpartition} ## object, @var{C}. @var{i} is the fold index (default is 1). ## ## @seealso{@@cvpartition/cvpartition, @@cvpartition/test} ## @end deftypefn function inds = training (C, i = []) if (nargin < 1 || nargin > 2) print_usage (); endif if (nargin < 2 || isempty (i)) i = 1; endif switch (C.Type) case {"kfold", "given"} inds = C.inds != i; case "holdout" inds = ! C.inds; case "leaveout" inds = ones (C.NumObservations, 1, "logical"); inds(i) = false; case "resubstitution" inds = ones (C.NumObservations, 1, "logical"); endswitch endfunction %!shared C %! C = cvpartition (ones (10, 1), "KFold", 5); %!assert (training (C, 1), logical ([0 0 1 1 1 1 1 1 1 1]')) %!assert (training (C, 2), logical ([1 1 0 0 1 1 1 1 1 1]')) %!assert (training (C, 3), logical ([1 1 1 1 0 0 1 1 1 1]')) %!assert (training (C, 4), logical ([1 1 1 1 1 1 0 0 1 1]')) %!assert (training (C, 5), logical ([1 1 1 1 1 1 1 1 0 0]')) %!test %! C = set (C, "inds", [1 2 2 2 3 4 3 4 5 5]'); %!assert (training (C), logical ([0 1 1 1 1 1 1 1 1 1]')) %!assert (training (C, 2), logical ([1 0 0 0 1 1 1 1 1 1]')) %!assert (training (C, 3), logical ([1 1 1 1 0 1 0 1 1 1]')) statistics-release-1.6.3/inst/Classification/000077500000000000000000000000001456127120000212375ustar00rootroot00000000000000statistics-release-1.6.3/inst/Classification/ClassificationKNN.m000066400000000000000000001514171456127120000247300ustar00rootroot00000000000000## Copyright (C) 2023 Mohammed Azmat Khan ## Copyright (C) 2023-2024 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . classdef ClassificationKNN ## -*- texinfo -*- ## @deftypefn {statistics} {@var{obj} =} ClassificationKNN (@var{X}, @var{Y}) ## @deftypefnx {statistics} {@var{obj} =} ClassificationKNN (@dots{}, @var{name}, @var{value}) ## ## Create a @qcode{ClassificationKNN} class object containing a k-Nearest ## Neighbor classification model. ## ## @code{@var{obj} = ClassificationKNN (@var{X}, @var{Y})} returns a ## ClassificationKNN object, with @var{X} as the predictor data and @var{Y} ## containing the class labels of observations in @var{X}. ## ## @itemize ## @item ## @code{X} must be a @math{NxP} numeric matrix of input data where rows ## correspond to observations and columns correspond to features or variables. ## @var{X} will be used to train the kNN model. ## @item ## @code{Y} is @math{Nx1} matrix or cell matrix containing the class labels of ## corresponding predictor data in @var{X}. @var{Y} can contain any type of ## categorical data. @var{Y} must have same numbers of Rows as @var{X}. ## @item ## @end itemize ## ## @code{@var{obj} = ClassificationKNN (@dots{}, @var{name}, @var{value})} ## returns a ClassificationKNN object with parameters specified by ## @qcode{Name-Value} pair arguments. Type @code{help fitcknn} for more info. ## ## A @qcode{ClassificationKNN} object, @var{obj}, stores the labelled training ## data and various parameters for the k-Nearest Neighbor classification model, ## which can be accessed in the following fields: ## ## @multitable @columnfractions 0.28 0.02 0.7 ## @headitem @var{Field} @tab @tab @var{Description} ## ## @item @qcode{obj.X} @tab @tab Unstandardized predictor data, specified as a ## numeric matrix. Each column of @var{X} represents one predictor (variable), ## and each row represents one observation. ## ## @item @qcode{obj.Y} @tab @tab Class labels, specified as a logical or ## numeric vector, or cell array of character vectors. Each value in @var{Y} is ## the observed class label for the corresponding row in @var{X}. ## ## @item @qcode{obj.NumObservations} @tab @tab Number of observations used in ## training the ClassificationKNN model, specified as a positive integer scalar. ## This number can be less than the number of rows in the training data because ## rows containing @qcode{NaN} values are not part of the fit. ## ## @item @qcode{obj.RowsUsed} @tab @tab Rows of the original training data ## used in fitting the ClassificationKNN model, specified as a numerical vector. ## If you want to use this vector for indexing the training data in @var{X}, you ## have to convert it to a logical vector, i.e ## @qcode{X = obj.X(logical (obj.RowsUsed), :);} ## ## @item @qcode{obj.Standardize} @tab @tab A boolean flag indicating whether ## the data in @var{X} have been standardized prior to training. ## ## @item @qcode{obj.Sigma} @tab @tab Predictor standard deviations, specified ## as a numeric vector of the same length as the columns in @var{X}. If the ## predictor variables have not been standardized, then @qcode{"obj.Sigma"} is ## empty. ## ## @item @qcode{obj.Mu} @tab @tab Predictor means, specified as a numeric ## vector of the same length as the columns in @var{X}. If the predictor ## variables have not been standardized, then @qcode{"obj.Mu"} is empty. ## ## @item @qcode{obj.NumPredictors} @tab @tab The number of predictors ## (variables) in @var{X}. ## ## @item @qcode{obj.PredictorNames} @tab @tab Predictor variable names, ## specified as a cell array of character vectors. The variable names are in ## the same order in which they appear in the training data @var{X}. ## ## @item @qcode{obj.ResponseName} @tab @tab Response variable name, specified ## as a character vector. ## ## @item @qcode{obj.ClassNames} @tab @tab Names of the classes in the training ## data @var{Y} with duplicates removed, specified as a cell array of character ## vectors. ## ## @item @qcode{obj.BreakTies} @tab @tab Tie-breaking algorithm used by predict ## when multiple classes have the same smallest cost, specified as one of the ## following character arrays: @qcode{"smallest"} (default), which favors the ## class with the smallest index among the tied groups, i.e. the one that ## appears first in the training labelled data. @qcode{"nearest"}, which favors ## the class with the nearest neighbor among the tied groups, i.e. the class ## with the closest member point according to the distance metric used. ## @qcode{"nearest"}, which randomly picks one class among the tied groups. ## ## @item @qcode{obj.Prior} @tab @tab Prior probabilities for each class, ## specified as a numeric vector. The order of the elements in @qcode{Prior} ## corresponds to the order of the classes in @qcode{ClassNames}. ## ## @item @qcode{obj.Cost} @tab @tab Cost of the misclassification of a point, ## specified as a square matrix. @qcode{Cost(i,j)} is the cost of classifying a ## point into class @qcode{j} if its true class is @qcode{i} (that is, the rows ## correspond to the true class and the columns correspond to the predicted ## class). The order of the rows and columns in @qcode{Cost} corresponds to the ## order of the classes in @qcode{ClassNames}. The number of rows and columns ## in @qcode{Cost} is the number of unique classes in the response. By default, ## @qcode{Cost(i,j) = 1} if @qcode{i != j}, and @qcode{Cost(i,j) = 0} if ## @qcode{i = j}. In other words, the cost is 0 for correct classification and ## 1 for incorrect classification. ## ## @item @qcode{obj.NumNeighbors} @tab @tab Number of nearest neighbors in ## @var{X} used to classify each point during prediction, specified as a ## positive integer value. ## ## @item @qcode{obj.Distance} @tab @tab Distance metric, specified as a ## character vector. The allowable distance metric names depend on the choice ## of the neighbor-searcher method. See the available distance metrics in ## @code{knnseaarch} for more info. ## ## @item @qcode{obj.DistanceWeight} @tab @tab Distance weighting function, ## specified as a function handle, which accepts a matrix of nonnegative ## distances, and returns a matrix the same size containing nonnegative distance ## weights. ## ## @item @qcode{obj.DistParameter} @tab @tab Parameter for the distance ## metric, specified either as a positive definite covariance matrix (when the ## distance metric is @qcode{"mahalanobis"}, or a positive scalar as the ## Minkowski distance exponent (when the distance metric is @qcode{"minkowski"}, ## or a vector of positive scale values with length equal to the number of ## columns of @var{X} (when the distance metric is @qcode{"seuclidean"}. For ## any other distance metric, the value of @qcode{DistParameter} is empty. ## ## @item @qcode{obj.NSMethod} @tab @tab Nearest neighbor search method, ## specified as either @qcode{"kdtree"}, which creates and uses a Kd-tree to ## find nearest neighbors, or @qcode{"exhaustive"}, which uses the exhaustive ## search algorithm by computing the distance values from all points in @var{X} ## to find nearest neighbors. ## ## @item @qcode{obj.IncludeTies} @tab @tab A boolean flag indicating whether ## prediction includes all the neighbors whose distance values are equal to the ## @math{k^th} smallest distance. If @qcode{IncludeTies} is @qcode{true}, ## prediction includes all of these neighbors. Otherwise, prediction uses ## exactly @math{k} neighbors. ## ## @item @qcode{obj.BucketSize} @tab @tab Maximum number of data points in the ## leaf node of the Kd-tree, specified as positive integer value. This argument ## is meaningful only when @qcode{NSMethod} is @qcode{"kdtree"}. ## ## @end multitable ## ## @seealso{fitcknn, knnsearch, rangesearch, pdist2} ## @end deftypefn properties (Access = public) X = []; # Predictor data Y = []; # Class labels NumObservations = []; # Number of observations in training dataset RowsUsed = []; # Rows used in fitting Standardize = []; # Flag to standardize predictors Sigma = []; # Predictor standard deviations Mu = []; # Predictor means NumPredictors = []; # Number of predictors PredictorNames = []; # Predictor variables names ResponseName = []; # Response variable name ClassNames = []; # Names of classes in Y BreakTies = []; # Tie-breaking algorithm Prior = []; # Prior probability for each class Cost = []; # Cost of misclassification NumNeighbors = []; # Number of nearest neighbors Distance = []; # Distance metric DistanceWeight = []; # Distance weighting function DistParameter = []; # Parameter for distance metric NSMethod = []; # Nearest neighbor search method IncludeTies = []; # Flag for handling ties BucketSize = []; # Maximum data points in each node endproperties methods (Access = public) ## Class object contructor function this = ClassificationKNN (X, Y, varargin) ## Check for sufficient number of input arguments if (nargin < 2) error ("ClassificationKNN: too few input arguments."); endif ## Check X and Y have the same number of observations if (rows (X) != rows (Y)) error ("ClassificationKNN: number of rows in X and Y must be equal."); endif ## Assign original X and Y data to the ClassificationKNN object this.X = X; this.Y = Y; ## Get groups in Y [gY, gnY, glY] = grp2idx (Y); ## Set default values before parsing optional parameters Standardize = false; # Flag to standardize predictors PredictorNames = []; # Predictor variables names ResponseName = []; # Response variable name ClassNames = []; # Names of classes in Y BreakTies = []; # Tie-breaking algorithm Prior = []; # Prior probability for each class Cost = []; # Cost of misclassification Scale = []; # Distance scale for 'seuclidean' Cov = []; # Covariance matrix for 'mahalanobis' Exponent = []; # Exponent for 'minkowski' NumNeighbors = []; # Number of nearest neighbors Distance = []; # Distance metric DistanceWeight = []; # Distance weighting function DistParameter = []; # Parameter for distance metric NSMethod = []; # Nearest neighbor search method IncludeTies = false; # Flag for handling ties BucketSize = 50; # Maximum data points in each node ## Number of parameters for Standardize, Scale, Cov (maximum 1 allowed) SSC = 0; ## Parse extra parameters while (numel (varargin) > 0) switch (tolower (varargin {1})) case "standardize" if (SSC < 1) Standardize = varargin{2}; if (! (Standardize == true || Standardize == false)) error (strcat (["ClassificationKNN: Standardize must"], ... [" be either true or false."])); endif SSC += 1; else error (strcat (["ClassificationKNN: Standardize cannot"], ... [" simultaneously be specified with either"], ... [" Scale or Cov."])); endif case "predictornames" PredictorNames = varargin{2}; if (! iscellstr (PredictorNames)) error (strcat (["ClassificationKNN: PredictorNames must"], ... [" be supplied as a cellstring array."])); elseif (columns (PredictorNames) != columns (X)) error (strcat (["ClassificationKNN: PredictorNames must"], ... [" have the same number of columns as X."])); endif case "responsename" ResponseName = varargin{2}; if (! ischar (ResponseName)) error ("ClassificationKNN: ResponseName must be a character array."); endif case "classnames" ClassNames = varargin{2}; if (! iscellstr (ClassNames)) error (strcat (["ClassificationKNN: ClassNames must"], ... [" be a cellstring array."])); endif ## Check that all class names are available in gnY if (! all (cell2mat (cellfun (@(x) any (strcmp (x, gnY)), ClassNames, "UniformOutput", false)))) error ("ClassificationKNN: not all ClassNames are present in Y."); endif case "breakties" BreakTies = varargin{2}; if (! ischar (BreakTies)) error ("ClassificationKNN: BreakTies must be a character array."); endif ## Check that all class names are available in gnY BTs = {"smallest", "nearest", "random"}; if (! any (strcmpi (BTs, BreakTies))) error ("ClassificationKNN: invalid value for BreakTies."); endif case "prior" Prior = varargin{2}; if (! ((isnumeric (Prior) && isvector (Prior)) || (strcmpi (Prior, "empirical") || strcmpi (Prior, "uniform")))) error (strcat (["ClassificationKNN: Prior must be either a"], ... [" numeric vector or a string."])); endif case "cost" Cost = varargin{2}; if (! (isnumeric (Cost) && issquare (Cost))) error ("ClassificationKNN: Cost must be a numeric square matrix."); endif case "numneighbors" NumNeighbors = varargin{2}; if (! (isnumeric (NumNeighbors) && isscalar (NumNeighbors) && NumNeighbors > 0 && fix (NumNeighbors) == NumNeighbors)) error (strcat (["ClassificationKNN: NumNeighbors must be a"], ... [" positive integer."])); endif case "distance" Distance = varargin{2}; DMs = {"euclidean", "seuclidean", "mahalanobis", "minkowski", ... "cityblock", "manhattan", "chebychev", "cosine", ... "correlation", "spearman", "hamming", "jaccard"}; if (ischar (Distance)) if (! any (strcmpi (DMs, Distance))) error ("ClassificationKNN: unsupported distance metric."); endif elseif (is_function_handle (Distance)) ## Check the input output sizes of the user function D2 = []; try D2 = Distance (X(1,:), Y); catch ME error (strcat (["ClassificationKNN: invalid function"], ... [" handle for distance metric."])); end_try_catch Yrows = rows (Y); if (! isequal (size (D2), [Yrows, 1])) error (strcat (["ClassificationKNN: custom distance"], ... [" function produces wrong output size."])); endif else error ("ClassificationKNN: invalid distance metric."); endif case "distanceweight" DistanceWeight = varargin{2}; DMs = {"equal", "inverse", "squareinverse"}; if (is_function_handle (DistanceWeight)) m = eye (5); if (! isequal (size (m), size (DistanceWeight(m)))) error (strcat (["ClassificationKNN: function handle for"], ... [" distance weight must return the same"], ... [" size as its input."])); endif this.DistanceWeight = DistanceWeight; else if (! any (strcmpi (DMs, DistanceWeight))) error ("ClassificationKNN: invalid distance weight."); endif if (strcmpi ("equal", DistanceWeight)) this.DistanceWeight = @(x) x; endif if (strcmpi ("inverse", DistanceWeight)) this.DistanceWeight = @(x) x.^(-1); endif if (strcmpi ("squareinverse", DistanceWeight)) this.DistanceWeight = @(x) x.^(-2); endif endif case "scale" if (SSC < 1) Scale = varargin{2}; if (! (isnumeric (Scale) && isvector (Scale))) error ("ClassificationKNN: Scale must be a numeric vector."); endif SSC += 1; else error (strcat (["ClassificationKNN: Scale cannot"], ... [" simultaneously be specified with either"], ... [" Standardize or Cov."])); endif case "cov" if (SSC < 1) Cov = varargin{2}; [~, p] = chol (Cov); if (p != 0) error (strcat (["ClassificationKNN: Cov must be a"], ... [" symmetric positive definite matrix."])); endif SSC += 1; else error (strcat (["ClassificationKNN: Cov cannot"], ... [" simultaneously be specified with either"], ... [" Standardize or Scale."])); endif case "exponent" Exponent = varargin{2}; if (! (isnumeric (Exponent) && isscalar (Exponent) && Exponent > 0 && fix (Exponent) == Exponent)) error ("ClassificationKNN: Exponent must be a positive integer."); endif case "nsmethod" NSMethod = varargin{2}; NSM = {"kdtree", "exhaustive"}; if (! ischar (NSMethod)) error ("ClassificationKNN: NSMethod must be a character array."); endif if (! any (strcmpi (NSM, NSMethod))) error (strcat (["ClassificationKNN: NSMethod must"], ... [" be either kdtree or exhaustive."])); endif case "includeties" IncludeTies = varargin{2}; if (! (IncludeTies == true || IncludeTies == false)) error (strcat (["ClassificationKNN: IncludeTies must"], ... [" be either true or false."])); endif case "bucketsize" BucketSize = varargin{2}; if (! (isnumeric (BucketSize) && isscalar (BucketSize) && BucketSize > 0 && fix (BucketSize) == BucketSize)) error (strcat (["ClassificationKNN: BucketSize must be a"], ... [" positive integer."])); endif otherwise error (strcat (["ClassificationKNN: invalid parameter name"],... [" in optional pair arguments."])); endswitch varargin (1:2) = []; endwhile ## Get number of variables in training data ndims_X = columns (X); ## Assign the number of predictors to the ClassificationKNN object this.NumPredictors = ndims_X; ## Generate default predictors and response variabe names (if necessary) if (isempty (PredictorNames)) for i = 1:ndims_X PredictorNames {i} = strcat ("x", num2str (i)); endfor endif if (isempty (ResponseName)) ResponseName = "Y"; endif ## Assign predictors and response variable names this.PredictorNames = PredictorNames; this.ResponseName = ResponseName; ## Handle class names if (isempty (ClassNames)) ClassNames = gnY; else ru = logical (zeros (size (Y))); for i = 1:numel (ClassNames) ac = find (gnY, ClassNames{i}); ru = ru | ac; endfor X = X(ru); Y = Y(ru); gY = gY(ru); endif ## Remove missing values from X and Y RowsUsed = ! logical (sum (isnan ([X, gY]), 2)); Y = Y (RowsUsed); X = X (RowsUsed, :); ## Renew groups in Y [gY, gnY, glY] = grp2idx (Y); this.ClassNames = gnY; ## Check X contains valid data if (! (isnumeric (X) && isfinite (X))) error ("ClassificationKNN: invalid values in X."); endif ## Assign the number of observations and their correspoding indices ## on the original data, which will be used for training the model, ## to the ClassificationKNN object this.NumObservations = rows (X); this.RowsUsed = cast (RowsUsed, "double"); ## Handle Standardize flag if (Standardize) this.Standardize = true; this.Sigma = std (X, [], 1); this.Sigma(this.Sigma == 0) = 1; # predictor is constant this.Mu = mean (X, 1); else this.Standardize = false; this.Sigma = []; this.Mu = []; endif ## Handle BreakTies if (isempty (BreakTies)) this.BreakTies = "smallest"; else this.BreakTies = BreakTies; endif ## Handle Prior and Cost if (isempty (Prior) || strcmpi ("uniform", Prior)) this.Prior = ones (size (gnY)) ./ numel (gnY); elseif (strcmpi ("empirical", Prior)) pr = []; for i = 1:numel (gnY) pr = [pr; sum(gY==i)]; endfor this.Prior = pr ./ sum (pr); elseif (isnumeric (Prior)) if (numel (gnY) != numel (Prior)) error (strcat (["ClassificationKNN: the elements in Prior do not"], ... [" correspond to selected classes in Y."])); endif this.Prior = Prior ./ sum (Prior); endif if (isempty (Cost)) this.Cost = cast (! eye (numel (gnY)), "double"); else if (numel (gnY) != sqrt (numel (Cost))) error (strcat (["ClassificationKNN: the number of rows and"], ... [" columns in Cost must correspond to selected"], ... [" classes in Y."])); endif this.Cost = Cost; endif ## Get number of neighbors if (isempty (NumNeighbors)) this.NumNeighbors = 1; else this.NumNeighbors = NumNeighbors; endif ## Get distance metric if (isempty (Distance)) Distance = "euclidean"; endif this.Distance = Distance; ## Get distance weight if (isempty (DistanceWeight)) this.DistanceWeight = @(x) x; endif ## Handle distance metric parameters (Scale, Cov, Exponent) if (! isempty (Scale)) if (! strcmpi (Distance, "seuclidean")) error (strcat (["ClassificationKNN: Scale is only valid when"], ... [" distance metric is seuclidean."])); endif if (numel (Scale) != ndims_X) error (strcat (["ClassificationKNN: Scale vector must have"], ... [" equal length to the number of columns in X."])); endif if (any (Scale < 0)) error (strcat (["ClassificationKNN: Scale vector must contain"], ... [" nonnegative scalar values."])); endif this.DistParameter = Scale; else if (strcmpi (Distance, "seuclidean")) if (Standardize) this.DistParameter = ones (1, ndims_X); else this.DistParameter = std (X, [], 1); endif endif endif if (! isempty (Cov)) if (! strcmpi (Distance, "mahalanobis")) error (strcat (["ClassificationKNN: Cov is only valid when"], ... [" distance metric is mahalanobis."])); endif if (columns (Cov) != ndims_X) error (strcat (["ClassificationKNN: Cov matrix must have"], ... [" equal columns as X."])); endif this.DistParameter = Cov; else if (strcmpi (Distance, "mahalanobis")) this.DistParameter = cov (X); endif endif if (! isempty (Exponent)) if (! strcmpi (Distance, "minkowski")) error (strcat (["ClassificationKNN: Exponent is only valid when"], ... [" distance metric is minkowski."])); endif this.DistParameter = Exponent; else if (strcmpi (Distance, "minkowski")) this.DistParameter = 2; endif endif ## Get Nearest neighbor search method kdm = {"euclidean", "cityblock", "manhattan", "minkowski", "chebychev"}; if (! isempty (NSMethod)) if (strcmpi ("kdtree", NSMethod) && (! any (strcmpi (kdm, Distance)))) error (strcat (["ClassificationKNN: kdtree method is only valid"], ... [" for euclidean, cityblock, manhattan,"], ... [" minkowski, and chebychev distance metrics."])); endif this.NSMethod = NSMethod; else if (any (strcmpi (kdm, Distance)) && ndims_X <= 10) this.NSMethod = "kdtree"; else this.NSMethod = "exhaustive"; endif endif ## Assign IncludeTies and BucketSize properties this.IncludeTies = IncludeTies; this.BucketSize = BucketSize; endfunction ## -*- texinfo -*- ## @deftypefn {ClassificationKNN} {@var{label} =} predict (@var{obj}, @var{XC}) ## @deftypefnx {ClassificationKNN} {[@var{label}, @var{score}, @var{cost}] =} predict (@var{obj}, @var{XC}) ## ## Classify new data points into categories using the kNN algorithm from a ## k-Nearest Neighbor classification model. ## ## @code{@var{label} = predict (@var{obj}, @var{XC})} returns the matrix of ## labels predicted for the corresponding instances in @var{XC}, using the ## predictor data in @code{obj.X} and corresponding labels, @code{obj.Y}, ## stored in the k-Nearest Neighbor classification model, @var{obj}. ## ## @var{XC} must be an @math{MxP} numeric matrix with the same number of ## features @math{P} as the corresponding predictors of the kNN model in ## @var{obj}. ## ## @code{[@var{label}, @var{score}, @var{cost}] = predict (@var{obj}, @var{XC})} ## also returns @var{score}, which contains the predicted class scores or ## posterior probabilities for each instance of the corresponding unique ## classes, and @var{cost}, which is a matrix containing the expected cost ## of the classifications. ## ## @seealso{fitcknn, ClassificationKNN} ## @end deftypefn function [label, score, cost] = predict (this, XC) ## Check for sufficient input arguments if (nargin < 2) error ("ClassificationKNN.predict: too few input arguments."); endif ## Check for valid XC if (isempty (XC)) error ("ClassificationKNN.predict: XC is empty."); elseif (columns (this.X) != columns (XC)) error (strcat (["ClassificationKNN.predict: XC must have the same"], ... [" number of features (columns) as in the kNN model."])); endif ## Get training data and labels X = this.X(logical (this.RowsUsed),:); Y = this.Y(logical (this.RowsUsed),:); ## Standardize (if necessary) if (this.Standardize) X = (X - this.Mu) ./ this.Sigma; XC = (XC - this.Mu) ./ this.Sigma; endif ## Train kNN if (strcmpi (this.Distance, "seuclidean")) [idx, dist] = knnsearch (X, XC, "k", this.NumNeighbors, ... "NSMethod", this.NSMethod, "Distance", "seuclidean", ... "Scale", this.DistParameter, "sortindices", true, ... "includeties", this.IncludeTies, ... "bucketsize", this.BucketSize); elseif (strcmpi (this.Distance, "mahalanobis")) [idx, dist] = knnsearch (X, XC, "k", this.NumNeighbors, ... "NSMethod", this.NSMethod, "Distance", "mahalanobis", ... "cov", this.DistParameter, "sortindices", true, ... "includeties", this.IncludeTies, ... "bucketsize", this.BucketSize); elseif (strcmpi (this.Distance, "minkowski")) [idx, dist] = knnsearch (X, XC, "k", this.NumNeighbors, ... "NSMethod", this.NSMethod, "Distance", "minkowski", ... "P", this.DistParameter, "sortindices", true, ... "includeties",this.IncludeTies, ... "bucketsize", this.BucketSize); else [idx, dist] = knnsearch (X, XC, "k", this.NumNeighbors, ... "NSMethod", this.NSMethod, "Distance", this.Distance, ... "sortindices", true, "includeties", this.IncludeTies, ... "bucketsize", this.BucketSize); endif ## Make prediction label = {}; score = []; cost = []; ## Get IDs and labels for each point in training data [gY, gnY, glY] = grp2idx (Y); ## Evaluate the K nearest neighbours for each new point for i = 1:rows (idx) ## Get K nearest neighbours if (this.IncludeTies) NN_idx = idx{i}; NNdist = dist{i}; else NN_idx = idx(i,:); NNdist = dist(i,:); endif k = numel (NN_idx); kNNgY = gY(NN_idx); ## Count frequency for each class for c = 1:numel (this.ClassNames) freq(c) = sum (kNNgY == c) / k; endfor ## Get label according to BreakTies if (strcmpi (this.BreakTies, "smallest")) [~, idl] = max (freq); else idl = find (freq == max (freq)); tgn = numel (idl); if (tgn > 1) if (strcmpi (this.BreakTies, "nearest")) for t = 1:tgn tgs(t) = find (gY(NN_idx) == idl(t)); endfor [~, idm] = min (tgs); idl = idl(idm); else # "random" idl = idl(randperm (numel (idl))(1)); endif endif endif label = [label; gnY{idl}]; ## Calculate score and cost score = [score; freq]; cost = [cost; 1-freq]; endfor endfunction endmethods endclassdef %!demo %! ## Create a k-nearest neighbor classifier for Fisher's iris data with k = 5. %! ## Evaluate some model predictions on new data. %! %! load fisheriris %! x = meas; %! y = species; %! xc = [min(x); mean(x); max(x)]; %! obj = fitcknn (x, y, "NumNeighbors", 5, "Standardize", 1); %! [label, score, cost] = predict (obj, xc) %!demo %! ## Train a k-nearest neighbor classifier for k = 10 %! ## and plot the decision boundaries. %! %! load fisheriris %! idx = ! strcmp (species, "setosa"); %! X = meas(idx,3:4); %! Y = cast (strcmpi (species(idx), "virginica"), "double"); %! obj = fitcknn (X, Y, "Standardize", 1, "NumNeighbors", 10, "NSMethod", "exhaustive") %! x1 = [min(X(:,1)):0.03:max(X(:,1))]; %! x2 = [min(X(:,2)):0.02:max(X(:,2))]; %! [x1G, x2G] = meshgrid (x1, x2); %! XGrid = [x1G(:), x2G(:)]; %! pred = predict (obj, XGrid); %! gidx = logical (str2num (cell2mat (pred))); %! %! figure %! scatter (XGrid(gidx,1), XGrid(gidx,2), "markerfacecolor", "magenta"); %! hold on %! scatter (XGrid(!gidx,1), XGrid(!gidx,2), "markerfacecolor", "red"); %! plot (X(Y == 0, 1), X(Y == 0, 2), "ko", X(Y == 1, 1), X(Y == 1, 2), "kx"); %! xlabel ("Petal length (cm)"); %! ylabel ("Petal width (cm)"); %! title ("5-Nearest Neighbor Classifier Decision Boundary"); %! legend ({"Versicolor Region", "Virginica Region", ... %! "Sampled Versicolor", "Sampled Virginica"}, ... %! "location", "northwest") %! axis tight %! hold off ## Test constructor with NSMethod and NumNeighbors parameters %!test %! x = [1, 2, 3; 4, 5, 6; 7, 8, 9; 3, 2, 1]; %! y = ["a"; "a"; "b"; "b"]; %! a = ClassificationKNN (x, y); %! assert (class (a), "ClassificationKNN"); %! assert ({a.X, a.Y, a.NumNeighbors}, {x, y, 1}) %! assert ({a.NSMethod, a.Distance}, {"kdtree", "euclidean"}) %! assert ({a.BucketSize}, {50}) %!test %! x = [1, 2, 3; 4, 5, 6; 7, 8, 9; 3, 2, 1]; %! y = ["a"; "a"; "b"; "b"]; %! a = ClassificationKNN (x, y, "NSMethod", "exhaustive"); %! assert (class (a), "ClassificationKNN"); %! assert ({a.X, a.Y, a.NumNeighbors}, {x, y, 1}) %! assert ({a.NSMethod, a.Distance}, {"exhaustive", "euclidean"}) %! assert ({a.BucketSize}, {50}) %!test %! x = [1, 2, 3; 4, 5, 6; 7, 8, 9; 3, 2, 1]; %! y = ["a"; "a"; "b"; "b"]; %! k = 10; %! a = ClassificationKNN (x, y, "NumNeighbors" ,k); %! assert (class (a), "ClassificationKNN"); %! assert ({a.X, a.Y, a.NumNeighbors}, {x, y, 10}) %! assert ({a.NSMethod, a.Distance}, {"kdtree", "euclidean"}) %! assert ({a.BucketSize}, {50}) %!test %! x = ones (4, 11); %! y = ["a"; "a"; "b"; "b"]; %! k = 10; %! a = ClassificationKNN (x, y, "NumNeighbors" ,k); %! assert (class (a), "ClassificationKNN"); %! assert ({a.X, a.Y, a.NumNeighbors}, {x, y, 10}) %! assert ({a.NSMethod, a.Distance}, {"exhaustive", "euclidean"}) %! assert ({a.BucketSize}, {50}) %!test %! x = [1, 2, 3; 4, 5, 6; 7, 8, 9; 3, 2, 1]; %! y = ["a"; "a"; "b"; "b"]; %! k = 10; %! a = ClassificationKNN (x, y, "NumNeighbors" ,k, "NSMethod", "exhaustive"); %! assert (class (a), "ClassificationKNN"); %! assert ({a.X, a.Y, a.NumNeighbors}, {x, y, 10}) %! assert ({a.NSMethod, a.Distance}, {"exhaustive", "euclidean"}) %! assert ({a.BucketSize}, {50}) %!test %! x = [1, 2, 3; 4, 5, 6; 7, 8, 9; 3, 2, 1]; %! y = ["a"; "a"; "b"; "b"]; %! k = 10; %! a = ClassificationKNN (x, y, "NumNeighbors" ,k, "Distance", "hamming"); %! assert (class (a), "ClassificationKNN"); %! assert ({a.X, a.Y, a.NumNeighbors}, {x, y, 10}) %! assert ({a.NSMethod, a.Distance}, {"exhaustive", "hamming"}) %! assert ({a.BucketSize}, {50}) ## Test constructor with Standardize and DistParameter parameters %!test %! x = [1, 2, 3; 4, 5, 6; 7, 8, 9; 3, 2, 1]; %! y = ["a"; "a"; "b"; "b"]; %! weights = ones (4,1); %! a = ClassificationKNN (x, y, "Standardize", 1); %! assert (class (a), "ClassificationKNN"); %! assert ({a.X, a.Y, a.NumNeighbors}, {x, y, 1}) %! assert ({a.NSMethod, a.Distance}, {"kdtree", "euclidean"}) %! assert ({a.Standardize}, {true}) %! assert ({a.Sigma}, {std(x, [], 1)}) %! assert ({a.Mu}, {[3.75, 4.25, 4.75]}) %!test %! x = [1, 2, 3; 4, 5, 6; 7, 8, 9; 3, 2, 1]; %! y = ["a"; "a"; "b"; "b"]; %! weights = ones (4,1); %! a = ClassificationKNN (x, y, "Standardize", false); %! assert (class (a), "ClassificationKNN"); %! assert ({a.X, a.Y, a.NumNeighbors}, {x, y, 1}) %! assert ({a.NSMethod, a.Distance}, {"kdtree", "euclidean"}) %! assert ({a.Standardize}, {false}) %! assert ({a.Sigma}, {[]}) %! assert ({a.Mu}, {[]}) %!test %! x = [1, 2, 3; 4, 5, 6; 7, 8, 9; 3, 2, 1]; %! y = ["a"; "a"; "b"; "b"]; %! s = ones (1, 3); %! a = ClassificationKNN (x, y, "Scale" , s, "Distance", "seuclidean"); %! assert (class (a), "ClassificationKNN"); %! assert ({a.DistParameter}, {s}) %! assert ({a.NSMethod, a.Distance}, {"exhaustive", "seuclidean"}) %! assert ({a.BucketSize}, {50}) %!test %! x = [1, 2, 3; 4, 5, 6; 7, 8, 9; 3, 2, 1]; %! y = ["a"; "a"; "b"; "b"]; %! a = ClassificationKNN (x, y, "Exponent" , 5, "Distance", "minkowski"); %! assert (class (a), "ClassificationKNN"); %! assert (a.DistParameter, 5) %! assert ({a.NSMethod, a.Distance}, {"kdtree", "minkowski"}) %!test %! x = [1, 2, 3; 4, 5, 6; 7, 8, 9; 3, 2, 1]; %! y = ["a"; "a"; "b"; "b"]; %! a = ClassificationKNN (x, y, "Exponent" , 5, "Distance", "minkowski", ... %! "NSMethod", "exhaustive"); %! assert (class (a), "ClassificationKNN"); %! assert (a.DistParameter, 5) %! assert ({a.NSMethod, a.Distance}, {"exhaustive", "minkowski"}) ## Test constructor with BucketSize and IncludeTies parameters %!test %! x = [1, 2, 3; 4, 5, 6; 7, 8, 9; 3, 2, 1]; %! y = ["a"; "a"; "b"; "b"]; %! a = ClassificationKNN (x, y, "BucketSize" , 20, "distance", "mahalanobis"); %! assert (class (a), "ClassificationKNN"); %! assert ({a.NSMethod, a.Distance}, {"exhaustive", "mahalanobis"}) %! assert ({a.BucketSize}, {20}) %!test %! x = [1, 2, 3; 4, 5, 6; 7, 8, 9; 3, 2, 1]; %! y = ["a"; "a"; "b"; "b"]; %! a = ClassificationKNN (x, y, "IncludeTies", true); %! assert (class (a), "ClassificationKNN"); %! assert (a.IncludeTies, true); %! assert ({a.NSMethod, a.Distance}, {"kdtree", "euclidean"}) %!test %! x = [1, 2, 3; 4, 5, 6; 7, 8, 9; 3, 2, 1]; %! y = ["a"; "a"; "b"; "b"]; %! a = ClassificationKNN (x, y); %! assert (class (a), "ClassificationKNN"); %! assert (a.IncludeTies, false); %! assert ({a.NSMethod, a.Distance}, {"kdtree", "euclidean"}) ## Test constructor with Prior and Cost parameters %!test %! x = [1, 2, 3; 4, 5, 6; 7, 8, 9; 3, 2, 1]; %! y = ["a"; "a"; "b"; "b"]; %! a = ClassificationKNN (x, y); %! assert (class (a), "ClassificationKNN") %! assert (a.Prior, [0.5; 0.5]) %! assert ({a.NSMethod, a.Distance}, {"kdtree", "euclidean"}) %! assert ({a.BucketSize}, {50}) %!test %! x = [1, 2, 3; 4, 5, 6; 7, 8, 9; 3, 2, 1]; %! y = ["a"; "a"; "b"; "b"]; %! prior = [0.5; 0.5]; %! a = ClassificationKNN (x, y, "Prior", "empirical"); %! assert (class (a), "ClassificationKNN") %! assert (a.Prior, prior) %! assert ({a.NSMethod, a.Distance}, {"kdtree", "euclidean"}) %! assert ({a.BucketSize}, {50}) %!test %! x = [1, 2, 3; 4, 5, 6; 7, 8, 9; 3, 2, 1]; %! y = ["a"; "a"; "a"; "b"]; %! prior = [0.75; 0.25]; %! a = ClassificationKNN (x, y, "Prior", "empirical"); %! assert (class (a), "ClassificationKNN") %! assert (a.Prior, prior) %! assert ({a.NSMethod, a.Distance}, {"kdtree", "euclidean"}) %! assert ({a.BucketSize}, {50}) %!test %! x = [1, 2, 3; 4, 5, 6; 7, 8, 9; 3, 2, 1]; %! y = ["a"; "a"; "a"; "b"]; %! prior = [0.5; 0.5]; %! a = ClassificationKNN (x, y, "Prior", "uniform"); %! assert (class (a), "ClassificationKNN") %! assert (a.Prior, prior) %! assert ({a.NSMethod, a.Distance}, {"kdtree", "euclidean"}) %! assert ({a.BucketSize}, {50}) %!test %! x = [1, 2, 3; 4, 5, 6; 7, 8, 9; 3, 2, 1]; %! y = ["a"; "a"; "b"; "b"]; %! cost = eye (2); %! a = ClassificationKNN (x, y, "Cost", cost); %! assert (class (a), "ClassificationKNN") %! assert (a.Cost, [1, 0; 0, 1]) %! assert ({a.NSMethod, a.Distance}, {"kdtree", "euclidean"}) %! assert ({a.BucketSize}, {50}) %!test %! x = [1, 2, 3; 4, 5, 6; 7, 8, 9; 3, 2, 1]; %! y = ["a"; "a"; "b"; "b"]; %! cost = eye (2); %! a = ClassificationKNN (x, y, "Cost", cost, "Distance", "hamming" ); %! assert (class (a), "ClassificationKNN") %! assert (a.Cost, [1, 0; 0, 1]) %! assert ({a.NSMethod, a.Distance}, {"exhaustive", "hamming"}) %! assert ({a.BucketSize}, {50}) ## Test input validation for constructor %!error ClassificationKNN () %!error ... %! ClassificationKNN (ones(4, 1)) %!error ... %! ClassificationKNN (ones (4,2), ones (1,4)) %!error ... %! ClassificationKNN (ones (5,3), ones (5,1), "standardize", "a") %!error ... %! ClassificationKNN (ones (5,2), ones (5,1), "scale", [1 1], "standardize", true) %!error ... %! ClassificationKNN (ones (5,2), ones (5,1), "PredictorNames", ["A"]) %!error ... %! ClassificationKNN (ones (5,2), ones (5,1), "PredictorNames", "A") %!error ... %! ClassificationKNN (ones (5,2), ones (5,1), "PredictorNames", {"A", "B", "C"}) %!error ... %! ClassificationKNN (ones (5,2), ones (5,1), "ResponseName", {"Y"}) %!error ... %! ClassificationKNN (ones (5,2), ones (5,1), "ResponseName", 1) %!error ... %! ClassificationKNN (ones (5,2), ones (5,1), "ClassNames", 1) %!error ... %! ClassificationKNN (ones (5,2), ones (5,1), "ClassNames", ["1"]) %!error ... %! ClassificationKNN (ones (5,2), ones (5,1), "ClassNames", {"1", "2"}) %!error ... %! ClassificationKNN (ones (5,2), ones (5,1), "BreakTies", 1) %!error ... %! ClassificationKNN (ones (5,2), ones (5,1), "BreakTies", {"1"}) %!error ... %! ClassificationKNN (ones (5,2), ones (5,1), "BreakTies", "some") %!error ... %! ClassificationKNN (ones (5,2), ones (5,1), "Prior", {"1", "2"}) %!error ... %! ClassificationKNN (ones (5,2), ones (5,1), "Cost", [1, 2]) %!error ... %! ClassificationKNN (ones (5,2), ones (5,1), "Cost", "string") %!error ... %! ClassificationKNN (ones (5,2), ones (5,1), "Cost", {eye(2)}) %!error ... %! ClassificationKNN (ones (5,2), ones (5,1), "NumNeighbors", 0) %!error ... %! ClassificationKNN (ones (5,2), ones (5,1), "NumNeighbors", 15.2) %!error ... %! ClassificationKNN (ones (5,2), ones (5,1), "NumNeighbors", "asd") %!error ... %! ClassificationKNN (ones (5,2), ones (5,1), "Distance", "somemetric") %!error ... %! ClassificationKNN (ones (5,2), ones (5,1), "Distance", ... %! @(v,m)sqrt(repmat(v,rows(m),1)-m,2)) %!error ... %! ClassificationKNN (ones (5,2), ones (5,1), "Distance", ... %! @(v,m)sqrt(sum(sumsq(repmat(v,rows(m),1)-m,2)))) %!error ... %! ClassificationKNN (ones (5,2), ones (5,1), "Distance", [1 2 3]) %!error ... %! ClassificationKNN (ones (5,2), ones (5,1), "Distance", {"mahalanobis"}) %!error ... %! ClassificationKNN (ones (5,2), ones (5,1), "Distance", logical (5)) %!error ... %! ClassificationKNN (ones (5,2), ones (5,1), "DistanceWeight", @(x)sum(x)) %!error ... %! ClassificationKNN (ones (5,2), ones (5,1), "DistanceWeight", "text") %!error ... %! ClassificationKNN (ones (5,2), ones (5,1), "DistanceWeight", [1 2 3]) %!error ... %! ClassificationKNN (ones (5,2), ones (5,1), "Scale", "scale") %!error ... %! ClassificationKNN (ones (5,2), ones (5,1), "Scale", {[1 2 3]}) %!error ... %! ClassificationKNN (ones (5,2), ones (5,1), "standardize", true, "scale", [1 1]) %!error ... %! ClassificationKNN (ones (5,2), ones (5,1), "Cov", ones (2), "Distance", "mahalanobis") %!error ... %! ClassificationKNN (ones (5,2), ones (5,1), "scale", [1 1], "Cov", ones (2)) %!error ... %! ClassificationKNN (ones (5,2), ones (5,1), "Exponent", 12.5) %!error ... %! ClassificationKNN (ones (5,2), ones (5,1), "Exponent", -3) %!error ... %! ClassificationKNN (ones (5,2), ones (5,1), "Exponent", "three") %!error ... %! ClassificationKNN (ones (5,2), ones (5,1), "Exponent", {3}) %!error ... %! ClassificationKNN (ones (5,2), ones (5,1), "NSMethod", {"kdtree"}) %!error ... %! ClassificationKNN (ones (5,2), ones (5,1), "NSMethod", 3) %!error ... %! ClassificationKNN (ones (5,2), ones (5,1), "NSMethod", "some") %!error ... %! ClassificationKNN (ones (5,2), ones (5,1), "IncludeTies", "some") %!error ... %! ClassificationKNN (ones (5,2), ones (5,1), "BucketSize", 42.5) %!error ... %! ClassificationKNN (ones (5,2), ones (5,1), "BucketSize", -50) %!error ... %! ClassificationKNN (ones (5,2), ones (5,1), "BucketSize", "some") %!error ... %! ClassificationKNN (ones (5,2), ones (5,1), "BucketSize", {50}) %!error ... %! ClassificationKNN (ones (5,2), ones (5,1), "some", "some") %!error ... %! ClassificationKNN ([1;2;3;'a';4], ones (5,1)) %!error ... %! ClassificationKNN ([1;2;3;Inf;4], ones (5,1)) %!error ... %! ClassificationKNN (ones (5,2), ones (5,1), "Prior", [1 2]) %!error ... %! ClassificationKNN (ones (5,2), ones (5,1), "Cost", [1 2; 1 3]) %!error ... %! ClassificationKNN (ones (5,2), ones (5,1), "Scale", [1 1]) %!error ... %! ClassificationKNN (ones (5,2), ones (5,1), "Scale", [1 1 1], "Distance", "seuclidean") %!error ... %! ClassificationKNN (ones (5,2), ones (5,1), "Scale", [1 -1], "Distance", "seuclidean") %!error ... %! ClassificationKNN (ones (5,2), ones (5,1), "Cov", eye (2)) %!error ... %! ClassificationKNN (ones (5,2), ones (5,1), "Cov", eye (3), "Distance", "mahalanobis") %!error ... %! ClassificationKNN (ones (5,2), ones (5,1), "Exponent", 3) %!error ... %! ClassificationKNN (ones (5,2), ones (5,1), "Distance", "hamming", "NSMethod", "kdtree") ## Test output for predict method %!shared x, y %! load fisheriris %! x = meas; %! y = species; %!test %! xc = [min(x); mean(x); max(x)]; %! obj = fitcknn (x, y, "NumNeighbors", 5); %! [l, s, c] = predict (obj, xc); %! assert (l, {"setosa"; "versicolor"; "virginica"}) %! assert (s, [1, 0, 0; 0, 1, 0; 0, 0, 1]) %! assert (c, [0, 1, 1; 1, 0, 1; 1, 1, 0]) %!test %! xc = [min(x); mean(x); max(x)]; %! obj = fitcknn (x, y, "NumNeighbors", 5, "Standardize", 1); %! [l, s, c] = predict (obj, xc); %! assert (l, {"versicolor"; "versicolor"; "virginica"}) %! assert (s, [0.4, 0.6, 0; 0, 1, 0; 0, 0, 1]) %! assert (c, [0.6, 0.4, 1; 1, 0, 1; 1, 1, 0]) %!test %! xc = [min(x); mean(x); max(x)]; %! obj = fitcknn (x, y, "NumNeighbors", 10, "distance", "mahalanobis"); %! [l, s, c] = predict (obj, xc); %! assert (s, [0.3, 0.7, 0; 0, 0.9, 0.1; 0.2, 0.2, 0.6], 1e-4) %! assert (c, [0.7, 0.3, 1; 1, 0.1, 0.9; 0.8, 0.8, 0.4], 1e-4) %!test %! xc = [min(x); mean(x); max(x)]; %! obj = fitcknn (x, y, "NumNeighbors", 10, "distance", "cosine"); %! [l, s, c] = predict (obj, xc); %! assert (l, {"setosa"; "versicolor"; "virginica"}) %! assert (s, [1, 0, 0; 0, 1, 0; 0, 0.3, 0.7], 1e-4) %! assert (c, [0, 1, 1; 1, 0, 1; 1, 0.7, 0.3], 1e-4) %!test %! xc = [5.2, 4.1, 1.5, 0.1; 5.1, 3.8, 1.9, 0.4; ... %! 5.1, 3.8, 1.5, 0.3; 4.9, 3.6, 1.4, 0.1]; %! obj = fitcknn (x, y, "NumNeighbors", 5); %! [l, s, c] = predict (obj, xc); %! assert (l, {"setosa"; "setosa"; "setosa"; "setosa"}) %! assert (s, [1, 0, 0; 1, 0, 0; 1, 0, 0; 1, 0, 0]) %! assert (c, [0, 1, 1; 0, 1, 1; 0, 1, 1; 0, 1, 1]) %!test %! xc = [5, 3, 5, 1.45]; %! obj = fitcknn (x, y, "NumNeighbors", 5); %! [l, s, c] = predict (obj, xc); %! assert (l, {"versicolor"}) %! assert (s, [0, 0.6, 0.4], 1e-4) %! assert (c, [1, 0.4, 0.6], 1e-4) %!test %! xc = [5, 3, 5, 1.45]; %! obj = fitcknn (x, y, "NumNeighbors", 10, "distance", "minkowski", "Exponent", 5); %! [l, s, c] = predict (obj, xc); %! assert (l, {"versicolor"}) %! assert (s, [0, 0.5, 0.5], 1e-4) %! assert (c, [1, 0.5, 0.5], 1e-4) %!test %! xc = [5, 3, 5, 1.45]; %! obj = fitcknn (x, y, "NumNeighbors", 10, "distance", "jaccard"); %! [l, s, c] = predict (obj, xc); %! assert (l, {"setosa"}) %! assert (s, [0.9, 0.1, 0], 1e-4) %! assert (c, [0.1, 0.9, 1], 1e-4) %!test %! xc = [5, 3, 5, 1.45]; %! obj = fitcknn (x, y, "NumNeighbors", 10, "distance", "mahalanobis"); %! [l, s, c] = predict (obj, xc); %! assert (l, {"versicolor"}) %! assert (s, [0.1000, 0.5000, 0.4000], 1e-4) %! assert (c, [0.9000, 0.5000, 0.6000], 1e-4) %!test %! xc = [5, 3, 5, 1.45]; %! obj = fitcknn (x, y, "NumNeighbors", 5, "distance", "jaccard"); %! [l, s, c] = predict (obj, xc); %! assert (l, {"setosa"}) %! assert (s, [0.8, 0.2, 0], 1e-4) %! assert (c, [0.2, 0.8, 1], 1e-4) %!test %! xc = [5, 3, 5, 1.45]; %! obj = fitcknn (x, y, "NumNeighbors", 5, "distance", "seuclidean"); %! [l, s, c] = predict (obj, xc); %! assert (l, {"versicolor"}) %! assert (s, [0, 1, 0], 1e-4) %! assert (c, [1, 0, 1], 1e-4) %!test %! xc = [5, 3, 5, 1.45]; %! obj = fitcknn (x, y, "NumNeighbors", 10, "distance", "chebychev"); %! [l, s, c] = predict (obj, xc); %! assert (l, {"versicolor"}) %! assert (s, [0, 0.7, 0.3], 1e-4) %! assert (c, [1, 0.3, 0.7], 1e-4) %!test %! xc = [5, 3, 5, 1.45]; %! obj = fitcknn (x, y, "NumNeighbors", 10, "distance", "cityblock"); %! [l, s, c] = predict (obj, xc); %! assert (l, {"versicolor"}) %! assert (s, [0, 0.6, 0.4], 1e-4) %! assert (c, [1, 0.4, 0.6], 1e-4) %!test %! xc = [5, 3, 5, 1.45]; %! obj = fitcknn (x, y, "NumNeighbors", 10, "distance", "cosine"); %! [l, s, c] = predict (obj, xc); %! assert (l, {"virginica"}) %! assert (s, [0, 0.1, 0.9], 1e-4) %! assert (c, [1, 0.9, 0.1], 1e-4) %!test %! xc = [5, 3, 5, 1.45]; %! obj = fitcknn (x, y, "NumNeighbors", 10, "distance", "correlation"); %! [l, s, c] = predict (obj, xc); %! assert (l, {"virginica"}) %! assert (s, [0, 0.1, 0.9], 1e-4) %! assert (c, [1, 0.9, 0.1], 1e-4) %!test %! xc = [5, 3, 5, 1.45]; %! obj = fitcknn (x, y, "NumNeighbors", 30, "distance", "spearman"); %! [l, s, c] = predict (obj, xc); %! assert (l, {"versicolor"}) %! assert (s, [0, 1, 0], 1e-4) %! assert (c, [1, 0, 1], 1e-4) %!test %! xc = [5, 3, 5, 1.45]; %! obj = fitcknn (x, y, "NumNeighbors", 30, "distance", "hamming"); %! [l, s, c] = predict (obj, xc); %! assert (l, {"setosa"}) %! assert (s, [0.4333, 0.3333, 0.2333], 1e-4) %! assert (c, [0.5667, 0.6667, 0.7667], 1e-4) %!test %! xc = [5, 3, 5, 1.45]; %! obj = fitcknn (x, y, "NumNeighbors", 5, "distance", "hamming"); %! [l, s, c] = predict (obj, xc); %! assert (l, {"setosa"}) %! assert (s, [0.8, 0.2, 0], 1e-4) %! assert (c, [0.2, 0.8, 1], 1e-4) %!test %! xc = [min(x); mean(x); max(x)]; %! obj = fitcknn (x, y, "NumNeighbors", 10, "distance", "correlation"); %! [l, s, c] = predict (obj, xc); %! assert (l, {"setosa"; "versicolor"; "virginica"}) %! assert (s, [1, 0, 0; 0, 1, 0; 0, 0.4, 0.6], 1e-4) %! assert (c, [0, 1, 1; 1, 0, 1; 1, 0.6, 0.4], 1e-4) %!test %! xc = [min(x); mean(x); max(x)]; %! obj = fitcknn (x, y, "NumNeighbors", 10, "distance", "hamming"); %! [l, s, c] = predict (obj, xc); %! assert (l, {"setosa";"setosa";"setosa"}) %! assert (s, [0.9, 0.1, 0; 1, 0, 0; 0.5, 0, 0.5], 1e-4) %! assert (c, [0.1, 0.9, 1; 0, 1, 1; 0.5, 1, 0.5], 1e-4) ## Test input validation for predict method %!error ... %! predict (ClassificationKNN (ones (4,2), ones (4,1))) %!error ... %! predict (ClassificationKNN (ones (4,2), ones (4,1)), []) %!error ... %! predict (ClassificationKNN (ones (4,2), ones (4,1)), 1) statistics-release-1.6.3/inst/Classification/ConfusionMatrixChart.m000066400000000000000000000715061456127120000255400ustar00rootroot00000000000000## Copyright (C) 2020-2021 Stefano Guidoni ## Copyright (C) 2023-2024 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . classdef ConfusionMatrixChart < handle ## -*- texinfo -*- ## @deftypefn {statistics} {@var{cmc} =} ConfusionMatrixChart () ## Create object @var{cmc}, a Confusion Matrix Chart object. ## ## @table @asis ## @item @qcode{"DiagonalColor"} ## The color of the patches on the diagonal, default is [0.0, 0.4471, 0.7412]. ## ## @item @qcode{"OffDiagonalColor"} ## The color of the patches off the diagonal, default is [0.851, 0.3255, 0.098]. ## ## @item @qcode{"GridVisible"} ## Available values: @qcode{on} (default), @qcode{off}. ## ## @item @qcode{"Normalization"} ## Available values: @qcode{absolute} (default), @qcode{column-normalized}, ## @qcode{row-normalized}, @qcode{total-normalized}. ## ## @item @qcode{"ColumnSummary"} ## Available values: @qcode{off} (default), @qcode{absolute}, ## @qcode{column-normalized},@qcode{total-normalized}. ## ## @item @qcode{"RowSummary"} ## Available values: @qcode{off} (default), @qcode{absolute}, ## @qcode{row-normalized}, @qcode{total-normalized}. ## @end table ## ## MATLAB compatibility -- the not implemented properties are: FontColor, ## PositionConstraint, InnerPosition, Layout. ## ## @seealso{confusionchart} ## @end deftypefn properties (Access = public) ## text properties XLabel = "Predicted Class"; YLabel = "True Class"; Title = ""; FontName = ""; FontSize = 0; ## chart colours DiagonalColor = [0 0.4471 0.7412]; OffDiagonalColor = [0.8510 0.3255 0.0980]; ## data visualization Normalization = "absolute"; ColumnSummary = "off"; RowSummary = "off"; GridVisible = "on"; HandleVisibility = ""; OuterPosition = []; Position = []; Units = ""; endproperties properties (GetAccess = public, SetAccess = private) ClassLabels = {}; # a string cell array of classes NormalizedValues = []; # the normalized confusion matrix Parent = 0; # a handle to the parent object endproperties properties (Access = protected) hax = 0.0; # a handle to the axes ClassN = 0; # the number of classes AbsoluteValues = []; # the original confusion matrix ColumnSummaryAbsoluteValues = []; # default values of the column summary RowSummaryAbsoluteValues = []; # default values of the row summary endproperties methods (Access = public) ## class constructor ## inputs: axis handle, a confusion matrix, a list of class labels, ## an array of optional property-value pairs. function this = ConfusionMatrixChart (hax, cm, cl, args) ## class initialization this.hax = hax; this.Parent = get (this.hax, "parent"); this.ClassLabels = cl; this.NormalizedValues = cm; this.AbsoluteValues = cm; this.ClassN = rows (cm); this.FontName = get (this.hax, "fontname"); this.FontSize = get (this.hax, "fontsize"); set (this.hax, "xlabel", this.XLabel); set (this.hax, "ylabel", this.YLabel); ## draw the chart draw (this); ## apply paired properties if (! isempty (args)) pair_idx = 1; while (pair_idx < length (args)) switch (args{pair_idx}) case "XLabel" this.XLabel = args{pair_idx + 1}; case "YLabel" this.YLabel = args{pair_idx + 1}; case "Title" this.Title = args{pair_idx + 1}; case "FontName" this.FontName = args{pair_idx + 1}; case "FontSize" this.FontSize = args{pair_idx + 1}; case "DiagonalColor" this.DiagonalColor = args{pair_idx + 1}; case "OffDiagonalColor" this.OffDiagonalColor = args{pair_idx + 1}; case "Normalization" this.Normalization = args{pair_idx + 1}; case "ColumnSummary" this.ColumnSummary = args{pair_idx + 1}; case "RowSummary" this.RowSummary = args{pair_idx + 1}; case "GridVisible" this.GridVisible = args{pair_idx + 1}; case "HandleVisibility" this.HandleVisibility = args{pair_idx + 1}; case "OuterPosition" this.OuterPosition = args{pair_idx + 1}; case "Position" this.Position = args{pair_idx + 1}; case "Units" this.Units = args{pair_idx + 1}; otherwise close (this.Parent); error ("confusionchart: invalid property %s", args{pair_idx}); endswitch pair_idx += 2; endwhile endif ## init the color map updateColorMap (this); endfunction ## set functions function set.XLabel (this, string) if (! ischar (string)) close (this.Parent); error ("confusionchart: XLabel must be a string."); endif this.XLabel = updateAxesProperties (this, "xlabel", string); endfunction function set.YLabel (this, string) if (! ischar (string)) close (this.Parent); error ("confusionchart: YLabel must be a string."); endif this.YLabel = updateAxesProperties (this, "ylabel", string); endfunction function set.Title (this, string) if (! ischar (string)) close (this.Parent); error ("confusionchart: Title must be a string."); endif this.Title = updateAxesProperties (this, "title", string); endfunction function set.FontName (this, string) if (! ischar (string)) close (this.Parent); error ("confusionchart: FontName must be a string."); endif this.FontName = updateTextProperties (this, "fontname", string); endfunction function set.FontSize (this, value) if (! isnumeric (value)) close (this.Parent); error ("confusionchart: FontSize must be numeric."); endif this.FontSize = updateTextProperties (this, "fontsize", value); endfunction function set.DiagonalColor (this, color) if (ischar (color)) color = this.convertNamedColor (color); endif if (! (isvector (color) && length (color) == 3 )) close (this.Parent); error ("confusionchart: DiagonalColor must be a color."); endif this.DiagonalColor = color; updateColorMap (this); endfunction function set.OffDiagonalColor (this, color) if (ischar (color)) color = this.convertNamedColor (color); endif if (! (isvector (color) && length (color) == 3)) close (this.Parent); error ("confusionchart: OffDiagonalColor must be a color."); endif this.OffDiagonalColor = color; updateColorMap (this); endfunction function set.Normalization (this, string) if (! any (strcmp (string, {"absolute", "column-normalized",... "row-normalized", "total-normalized"}))) close (this.Parent); error ("confusionchart: invalid value for Normalization."); endif this.Normalization = string; updateChart (this); endfunction function set.ColumnSummary (this, string) if (! any (strcmp (string, {"off", "absolute", "column-normalized",... "total-normalized"}))) close (this.Parent); error ("confusionchart: invalid value for ColumnSummary."); endif this.ColumnSummary = string; updateChart (this); endfunction function set.RowSummary (this, string) if (! any (strcmp (string, {"off", "absolute", "row-normalized",... "total-normalized"}))) close (this.Parent); error ("confusionchart: invalid value for RowSummary."); endif this.RowSummary = string; updateChart (this); endfunction function set.GridVisible (this, string) if (! any (strcmp (string, {"off", "on"}))) close (this.Parent); error ("confusionchart: invalid value for GridVisible."); endif this.GridVisible = string; setGridVisibility (this); endfunction function set.HandleVisibility (this, string) if (! any (strcmp (string, {"off", "on", "callback"}))) close (this.Parent); error ("confusionchart: invalid value for HandleVisibility"); endif set (this.hax, "handlevisibility", string); endfunction function set.OuterPosition (this, vector) if (! isvector (vector) || ! isnumeric (vector) || length (vector) != 4) close (this.Parent); error ("confusionchart: invalid value for OuterPosition"); endif set (this.hax, "outerposition", vector); endfunction function set.Position (this, vector) if (! isvector (vector) || ! isnumeric (vector) || length (vector) != 4) close (this.Parent); error ("confusionchart: invalid value for Position"); endif set (this.hax, "position", vector); endfunction function set.Units (this, string) if (! any (strcmp (string, {"centimeters", "characters", "inches", ... "normalized", "pixels", "points"}))) close (this.Parent); error ("confusionchart: invalid value for Units"); endif set (this.hax, "units", string); endfunction ## -*- texinfo -*- ## @deftypefn {ConfusionMatrixChart} {} disp (@var{cmc}, @var{order}) ## Display the properties of the @code{ConfusionMatrixChart} object ## @var{cmc}. ## ## @end deftypefn function disp (this) nv_sizes = size (this.NormalizedValues); cl_sizes = size (this.ClassLabels); printf ("%s with properties:\n\n", class (this)); printf ("\tNormalizedValues: [ %dx%d %s ]\n", nv_sizes(1), nv_sizes(2),... class (this.NormalizedValues)); printf ("\tClassLabels: { %dx%d %s }\n\n", cl_sizes(1), cl_sizes(2),... class (this.ClassLabels)); endfunction ## -*- texinfo -*- ## @deftypefn {ConfusionMatrixChart} {} sortClasses (@var{cmc}, @var{order}) ## Sort the classes of the @code{ConfusionMatrixChart} object @var{cmc} ## according to @var{order}. ## ## Valid values for @var{order} can be an array or cell array including ## the same class labels as @var{cm}, or a value like @code{"auto"}, ## @code{"ascending-diagonal"}, @code{"descending-diagonal"} and ## @code{"cluster"}. ## ## @end deftypefn ## ## @seealso{confusionchart, linkage, pdist} function sortClasses (this, order) ## check the input parameters if (nargin != 2) print_usage (); endif cl = this.ClassLabels; cm_size = this.ClassN; nv = this.NormalizedValues; av = this.AbsoluteValues; cv = this.ColumnSummaryAbsoluteValues; rv = this.RowSummaryAbsoluteValues; scl = {}; Idx = []; if (strcmp (order, "auto")) [scl, Idx] = sort (cl); elseif (strcmp (order, "ascending-diagonal")) [s, Idx] = sort (diag (nv)); scl = cl(Idx); elseif (strcmp (order, "descending-diagonal")) [s, Idx] = sort (diag (nv)); Idx = flip (Idx); scl = cl(Idx); elseif (strcmp (order, "cluster")) ## the classes are all grouped together ## this way one can visually evaluate which are the most similar classes ## according to the learning algorithm D = zeros (1, ((cm_size - 1) * cm_size / 2)); # a pdist like vector maxD = 2 * max (max (av)); k = 1; # better than computing the index at every cycle for i = 1 : (cm_size - 1) for j = (i + 1) : cm_size D(k++) = maxD - (av(i, j) + av(j, i)); # distance endfor endfor tree = linkage (D, "average"); # clustering ## we could have optimal leaf ordering with Idx = optimalleaforder (tree, D); # optimal clustering ## [sorted_v Idx] = sort (cluster (tree, )); nodes_to_visit = 2 * cm_size - 1; nodecount = 0; while (! isempty (nodes_to_visit)) current_node = nodes_to_visit(1); nodes_to_visit(1) = []; if (current_node > cm_size) node = current_node - cm_size; nodes_to_visit = [tree(node,[2 1]) nodes_to_visit]; end if (current_node <= cm_size) nodecount++; Idx(nodecount) = current_node; end end ## scl = cl(Idx); else ## must be an array or cell array of labels if (! iscellstr (order)) if (! ischar (order)) if (isrow (order)) order = vec (order); endif order = num2str (order); endif scl = cellstr (order); endif if (length (scl) != length (cl)) error ("sortClasses: wrong size for order.") endif Idx = zeros (length (scl), 1); for i = 1 : length (scl) Idx(i) = find (strcmp (cl, scl{i})); endfor endif ## rearrange the normalized values... nv = nv(Idx, :); nv = nv(:, Idx); this.NormalizedValues = nv; ## ...and the absolute values... av = av(Idx, :); av = av(:, Idx); this.AbsoluteValues = av; cv = cv([Idx ( Idx + cm_size )]); this.ColumnSummaryAbsoluteValues = cv; rv = rv([Idx ( Idx + cm_size )]); this.RowSummaryAbsoluteValues = rv; ## ...and the class labels this.ClassLabels = scl; ## update the axes set (this.hax, "xtick", (0.5 : 1 : (cm_size - 0.5)), "xticklabel", scl,... "ytick", (0.5 : 1 : (cm_size - 0.5)), "yticklabel", scl); ## get text and patch handles kids = get (this.hax, "children"); t_kids = kids(find (isprop (kids, "fontname"))); # hack to find texts m_kid = kids(find (strcmp (get (kids, "userdata"), "MainChart"))); c_kid = kids(find (strcmp (get (kids, "userdata"), "ColumnSummary"))); r_kid = kids(find (strcmp (get (kids, "userdata"), "RowSummary"))); ## re-assign colors to the main chart cdata_m = reshape (get (m_kid, "cdata"), cm_size, cm_size); cdata_m = cdata_m(Idx, :); cdata_m = cdata_m(:, Idx); cdata_v = vec (cdata_m); set (m_kid, "cdata", cdata_v); ## re-assign colors to the column summary cdata_m = reshape (transpose (get (c_kid, "cdata")), cm_size, 2); cdata_m = cdata_m(Idx, :); cdata_v = vec (cdata_m); set (c_kid, "cdata", cdata_v); ## re-assign colors to the row summary cdata_m = reshape (get (r_kid, "cdata"), cm_size, 2); cdata_m = cdata_m(Idx, :); cdata_v = vec (cdata_m); set (r_kid, "cdata", cdata_v); ## move the text labels for i = 1:length (t_kids) t_pos = get (t_kids(i), "userdata"); if (t_pos(2) > cm_size) ## row summary t_pos(1) = find (Idx == (t_pos(1) + 1)) - 1; set (t_kids(i), "userdata", t_pos); t_pos = t_pos([2 1]) + 0.5; set (t_kids(i), "position", t_pos); elseif (t_pos(1) > cm_size) ## column summary t_pos(2) = find (Idx == (t_pos(2) + 1)) - 1; set (t_kids(i), "userdata", t_pos); t_pos = t_pos([2 1]) + 0.5; set (t_kids(i), "position", t_pos); else ## main chart t_pos(1) = find (Idx == (t_pos(1) + 1)) - 1; t_pos(2) = find (Idx == (t_pos(2) + 1)) - 1; set (t_kids(i), "userdata", t_pos); t_pos = t_pos([2 1]) + 0.5; set (t_kids(i), "position", t_pos); endif endfor updateChart (this); endfunction endmethods methods (Access = private) ## convertNamedColor ## convert a named colour to a colour triplet function ret = convertNamedColor (this, color) vColorNames = ["ymcrgbwk"]'; vColorTriplets = [1 1 0; 1 0 1; 0 1 1; 1 0 0; 0 1 0; 0 0 1; 1 1 1; 0 0 0]; if (strcmp (color, "black")) color = 'k'; endif index = find (vColorNames == color(1)); if (! isempty (index)) ret = vColorTriplets(index, :); else ret = []; # trigger an error message endif endfunction ## updateAxesProperties ## update the properties of the axes function ret = updateAxesProperties (this, prop, value) set (this.hax, prop, value); ret = value; endfunction ## updateTextProperties ## set the properties of the texts function ret = updateTextProperties (this, prop, value) hax_kids = get (this.hax, "children"); text_kids = hax_kids(isprop (hax_kids , "fontname")); # hack to find texts text_kids(end + 1) = get (this.hax, "xlabel"); text_kids(end + 1) = get (this.hax, "ylabel"); text_kids(end + 1) = get (this.hax, "title"); updateAxesProperties (this, prop, value); set (text_kids, prop, value); ret = value; endfunction ## setGridVisibility ## toggle the visibility of the grid function setGridVisibility (this) kids = get (this.hax, "children"); kids = kids(find (isprop (kids, "linestyle"))); if (strcmp (this.GridVisible, "on")) set (kids, "linestyle", "-"); else set (kids, "linestyle", "none"); endif endfunction ## updateColorMap ## change the colormap and, accordingly, the text colors function updateColorMap (this) cm_size = this.ClassN; d_color = this.DiagonalColor; o_color = this.OffDiagonalColor; ## quick hack d_color(find (d_color == 1.0)) = 0.999; o_color(find (o_color == 1.0)) = 0.999; ## 64 shades for each color cm_colormap(1:64,:) = [1.0 : (-(1.0 - o_color(1)) / 63) : o_color(1);... 1.0 : (-(1.0 - o_color(2)) / 63) : o_color(2);... 1.0 : (-(1.0 - o_color(3)) / 63) : o_color(3)]'; cm_colormap(65:128,:) = [1.0 : (-(1.0 - d_color(1)) / 63) : d_color(1);... 1.0 : (-(1.0 - d_color(2)) / 63) : d_color(2);... 1.0 : (-(1.0 - d_color(3)) / 63) : d_color(3)]'; colormap (this.hax, cm_colormap); ## update text colors kids = get (this.hax, "children"); t_kids = kids(find (isprop (kids, "fontname"))); # hack to find texts m_patch = kids(find (strcmp (get (kids, "userdata"), "MainChart"))); c_patch = kids(find (strcmp (get (kids, "userdata"), "ColumnSummary"))); r_patch = kids(find (strcmp (get (kids, "userdata"), "RowSummary"))); m_colors = get (m_patch, "cdata"); c_colors = get (c_patch, "cdata"); r_colors = get (r_patch, "cdata"); ## when a patch is dark, let's use a pale color for the text for i = 1 : length (t_kids) t_pos = get (t_kids(i), "userdata"); color_idx = 1; if (t_pos(2) > cm_size) ## row summary idx = (t_pos(2) - cm_size - 1) * cm_size + t_pos(1) + 1; color_idx = r_colors(idx) + 1; elseif (t_pos(1) > cm_size) ## column summary idx = (t_pos(1) - cm_size - 1) * cm_size + t_pos(2) + 1; color_idx = c_colors(idx) + 1; else ## main chart idx = t_pos(2) * cm_size + t_pos(1) + 1; color_idx = m_colors(idx) + 1; endif if (sum (cm_colormap(color_idx, :)) < 1.8) set (t_kids(i), "color", [.97 .97 1.0]); else set (t_kids(i), "color", [.15 .15 .15]); endif endfor endfunction ## updateChart ## update the text labels and the NormalizedValues property function updateChart (this) cm_size = this.ClassN; cm = this.AbsoluteValues; l_cs = this.ColumnSummaryAbsoluteValues; l_rs = this.RowSummaryAbsoluteValues; kids = get (this.hax, "children"); t_kids = kids(find (isprop (kids, "fontname"))); # hack to find texts normalization = this.Normalization; column_summary = this.ColumnSummary; row_summary = this.RowSummary; ## normalization for labelling row_totals = sum (cm, 2); col_totals = sum (cm, 1); mat_total = sum (col_totals); cm_labels = cm; add_percent = true; if (strcmp (normalization, "column-normalized")) for i = 1 : cm_size cm_labels(:,i) = cm_labels(:,i) ./ col_totals(i); endfor elseif (strcmp (normalization, "row-normalized")) for i = 1 : cm_size cm_labels(i,:) = cm_labels(i,:) ./ row_totals(i); endfor elseif (strcmp (normalization, "total-normalized")) cm_labels = cm_labels ./ mat_total; else add_percent = false; endif ## update NormalizedValues this.NormalizedValues = cm_labels; ## update axes last_row = cm_size; last_col = cm_size; userdata = cell2mat (get (t_kids, "userdata")); cs_kids = t_kids(find (userdata(:,1) > cm_size)); cs_kids(end + 1) = kids(find (strcmp (get (kids, "userdata"),... "ColumnSummary"))); if (! strcmp ("off", column_summary)) set (cs_kids, "visible", "on"); last_row += 3; else set (cs_kids, "visible", "off"); endif rs_kids = t_kids(find (userdata(:,2) > cm_size)); rs_kids(end + 1) = kids(find (strcmp (get (kids, "userdata"),... "RowSummary"))); if (! strcmp ("off", row_summary)) set (rs_kids, "visible", "on"); last_col += 3; else set (rs_kids, "visible", "off"); endif axis (this.hax, [0 last_col 0 last_row]); ## update column summary data cs_add_percent = true; if (! strcmp (column_summary, "off")) if (strcmp (column_summary, "column-normalized")) for i = 1 : cm_size if (col_totals(i) == 0) ## avoid division by zero l_cs([i (cm_size + i)]) = 0; else l_cs([i, cm_size + i]) = l_cs([i, cm_size + i]) ./ col_totals(i); endif endfor elseif strcmp (column_summary, "total-normalized") l_cs = l_cs ./ mat_total; else cs_add_percent = false; endif endif ## update row summary data rs_add_percent = true; if (! strcmp (row_summary, "off")) if (strcmp (row_summary, "row-normalized")) for i = 1 : cm_size if (row_totals(i) == 0) ## avoid division by zero l_rs([i (cm_size + i)]) = 0; else l_rs([i, cm_size + i]) = l_rs([i, cm_size + i]) ./ row_totals(i); endif endfor elseif (strcmp (row_summary, "total-normalized")) l_rs = l_rs ./ mat_total; else rs_add_percent = false; endif endif ## update text label_list = vec (cm_labels); for i = 1 : length (t_kids) t_pos = get (t_kids(i), "userdata"); new_string = ""; if (t_pos(2) > cm_size) ## this is the row summary idx = (t_pos(2) - cm_size - 1) * cm_size + t_pos(1) + 1; if (rs_add_percent) new_string = num2str (100.0 * l_rs(idx), "%3.1f"); new_string = [new_string "%"]; else new_string = num2str (l_rs(idx)); endif elseif (t_pos(1) > cm_size) ## this is the column summary idx = (t_pos(1) - cm_size - 1) * cm_size + t_pos(2) + 1; if (cs_add_percent) new_string = num2str (100.0 * l_cs(idx), "%3.1f"); new_string = [new_string "%"]; else new_string = num2str (l_cs(idx)); endif else ## this is the main chart idx = t_pos(2) * cm_size + t_pos(1) + 1; if (add_percent) new_string = num2str (100.0 * label_list(idx), "%3.1f"); new_string = [new_string "%"]; else new_string = num2str (label_list(idx)); endif endif set (t_kids(i), "string", new_string); endfor endfunction ## draw ## draw the chart function draw (this) cm = this.AbsoluteValues; cl = this.ClassLabels; cm_size = this.ClassN; ## set up the axes set (this.hax, "xtick", (0.5 : 1 : (cm_size - 0.5)), "xticklabel", cl,... "ytick", (0.5 : 1 : (cm_size - 0.5)), "yticklabel", cl ); axis ("ij"); axis (this.hax, [0 cm_size 0 cm_size]); ## prepare the patches indices_b = 0 : (cm_size -1); indices_v = repmat (indices_b, cm_size, 1); indices_vx = transpose (vec (indices_v)); indices_vy = vec (indices_v', 2); indices_ex = vec ((cm_size + 1) * [1; 2] .* ones (2, cm_size), 2); ## normalization for colorization ## it is used a colormap of 128 shades of two colors, 64 shades for each ## color normal = max (max (cm)); cm_norm = round (63 * cm ./ normal); cm_norm = cm_norm + 64 * eye (cm_size); ## default normalization: absolute cm_labels = vec (cm); ## the patches of the main chart x_patch = [indices_vx; ( indices_vx + 1 ); ( indices_vx + 1 ); indices_vx]; y_patch = [indices_vy; indices_vy; ( indices_vy + 1 ); ( indices_vy + 1 )]; c_patch = vec (cm_norm(1 : cm_size, 1 : cm_size)); ## display the patches ph = patch (this.hax, x_patch, y_patch, c_patch); set (ph, "userdata", "MainChart"); ## display the labels userdata = [indices_vy; indices_vx]'; nonzero_idx = find (cm_labels != 0); th = text ((x_patch(1, nonzero_idx) + 0.5), (y_patch(1, nonzero_idx) +... 0.5), num2str (cm_labels(nonzero_idx)), "parent", this.hax ); set (th, "horizontalalignment", "center"); for i = 1 : length (nonzero_idx) set (th(i), "userdata", userdata(nonzero_idx(i), :)); endfor ## patches for the summaries main_values = diag (cm); ct_values = sum (cm)'; rt_values = sum (cm, 2); cd_values = ct_values - main_values; rd_values = rt_values - main_values; ## column summary x_cs = [[indices_b indices_b]; ( [indices_b indices_b] + 1 ); ( [indices_b indices_b] + 1 ); [indices_b indices_b]]; y_cs = [(repmat ([1 1 2 2]', 1, cm_size)) (repmat ([2 2 3 3]', 1, cm_size))] +... cm_size; c_cs = [(round (63 * (main_values ./ ct_values)) + 64); (round (63 * (cd_values ./ ct_values)))]; c_cs(isnan (c_cs)) = 0; l_cs = [main_values; cd_values]; ph = patch (this.hax, x_cs, y_cs, c_cs); set (ph, "userdata", "ColumnSummary"); set (ph, "visible", "off" ); userdata = [y_cs(1,:); x_cs(1,:)]'; nonzero_idx = find (l_cs != 0); th = text ((x_cs(1,nonzero_idx) + 0.5), (y_cs(1,nonzero_idx) + 0.5),... num2str (l_cs(nonzero_idx)), "parent", this.hax); set (th, "horizontalalignment", "center"); for i = 1 : length (nonzero_idx) set (th(i), "userdata", userdata(nonzero_idx(i), :)); endfor set (th, "visible", "off"); ## row summary x_rs = y_cs; y_rs = x_cs; c_rs = [(round (63 * (main_values ./ rt_values)) + 64); (round (63 * (rd_values ./ rt_values)))]; c_rs(isnan (c_rs)) = 0; l_rs = [main_values; rd_values]; ph = patch (this.hax, x_rs, y_rs, c_rs); set (ph, "userdata", "RowSummary"); set (ph, "visible", "off"); userdata = [y_rs(1,:); x_rs(1,:)]'; nonzero_idx = find (l_rs != 0); th = text ((x_rs(1,nonzero_idx) + 0.5), (y_rs(1,nonzero_idx) + 0.5),... num2str (l_rs(nonzero_idx)), "parent", this.hax); set (th, "horizontalalignment", "center"); for i = 1 : length (nonzero_idx) set (th(i), "userdata", userdata(nonzero_idx(i), :)); endfor set (th, "visible", "off"); this.ColumnSummaryAbsoluteValues = l_cs; this.RowSummaryAbsoluteValues = l_rs; endfunction endmethods endclassdef %!demo %! ## Create a simple ConfusionMatrixChart Object %! %! cm = ConfusionMatrixChart (gca, [1 2; 1 2], {"A","B"}, {"XLabel","LABEL A"}) %! NormalizedValues = cm.NormalizedValues %! ClassLabels = cm.ClassLabels ## Test plotting %!test %! hf = figure ("visible", "off"); %! unwind_protect %! cm = ConfusionMatrixChart (gca, [1 2; 1 2], {"A","B"}, {"XLabel","LABEL A"}); %! assert (isa (cm, "ConfusionMatrixChart"), true); %! unwind_protect_cleanup %! close (hf); %! end_unwind_protect statistics-release-1.6.3/inst/Clustering/000077500000000000000000000000001456127120000204235ustar00rootroot00000000000000statistics-release-1.6.3/inst/Clustering/CalinskiHarabaszEvaluation.m000066400000000000000000000227721456127120000260540ustar00rootroot00000000000000## Copyright (C) 2021 Stefano Guidoni ## Copyright (C) 2024 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . classdef CalinskiHarabaszEvaluation < ClusterCriterion ## -*- texinfo -*- ## @deftypefn {statistics} {@var{obj} =} evalclusters (@var{x}, @var{clust}, @qcode{CalinskiHarabasz}) ## @deftypefnx {statistics} {@var{obj} =} evalclusters (@dots{}, @qcode{Name}, @qcode{Value}) ## ## A Calinski-Harabasz object to evaluate clustering solutions. ## ## A @code{CalinskiHarabaszEvaluation} object is a @code{ClusterCriterion} ## object used to evaluate clustering solutions using the Calinski-Harabasz ## criterion. ## ## The Calinski-Harabasz index is based on the ratio between SSb and SSw. ## SSb is the overall variance between clusters, that is the variance of the ## distances between the centroids. ## SSw is the overall variance within clusters, that is the sum of the ## variances of the distances between each datapoint and its centroid. ## ## The best solution according to the Calinski-Harabasz criterion is the one ## that scores the highest value. ## ## @seealso{evalclusters, ClusterCriterion, DaviesBouldinEvaluation, ## GapEvaluation, SilhouetteEvaluation} ## @end deftypefn properties (GetAccess = public, SetAccess = private) endproperties properties (Access = protected) Centroids = {}; # a list of the centroids for every solution endproperties methods (Access = public) ## constructor function this = CalinskiHarabaszEvaluation (x, clust, KList) this@ClusterCriterion(x, clust, KList); this.CriterionName = "CalinskiHarabasz"; this.evaluate(this.InspectedK); # evaluate the list of cluster numbers endfunction ## -*- texinfo -*- ## @deftypefn {CalinskiHarabaszEvaluation} {@var{obj} =} addK (@var{obj}, @var{K}) ## ## Add a new cluster array to inspect the CalinskiHarabaszEvaluation object. ## ## @end deftypefn function this = addK (this, K) addK@ClusterCriterion(this, K); ## if we have new data, we need a new evaluation if (this.OptimalK == 0) Centroids_tmp = {}; pS = 0; # position shift of the elements of Centroids for iter = 1 : length (this.InspectedK) ## reorganize Centroids according to the new list of cluster numbers if (any (this.InspectedK(iter) == K)) pS += 1; else Centroids_tmp{iter} = this.Centroids{iter - pS}; endif endfor this.Centroids = Centroids_tmp; this.evaluate(K); # evaluate just the new cluster numbers endif endfunction ## -*- texinfo -*- ## @deftypefn {CalinskiHarabaszEvaluation} {} plot (@var{obj}) ## @deftypefnx {CalinskiHarabaszEvaluation} {@var{h} =} plot (@var{obj}) ## ## Plot the evaluation results. ## ## Plot the CriterionValues against InspectedK from the ## CalinskiHarabaszEvaluation, @var{obj}, to the current plot. It can also ## return a handle to the current plot. ## ## @end deftypefn function h = plot (this) yLabel = sprintf ("%s value", this.CriterionName); h = gca (); hold on; plot (this.InspectedK, this.CriterionValues, "bo-"); plot (this.OptimalK, this.CriterionValues(this.OptimalIndex), "b*"); xlabel ("number of clusters"); ylabel (yLabel); hold off; endfunction ## -*- texinfo -*- ## @deftypefn {CalinskiHarabaszEvaluation} {@var{obj} =} compact (@var{obj}) ## ## Return a compact CalinskiHarabaszEvaluation object (not implemented yet). ## ## @end deftypefn function this = compact (this) warning (["CalinskiHarabaszEvaluation.compact: this"... " method is not yet implemented."]); endfunction endmethods methods (Access = protected) ## evaluate ## do the evaluation function this = evaluate (this, K) ## use complete observations only UsableX = this.X(find (this.Missing == false), :); if (! isempty (this.ClusteringFunction)) ## build the clusters for iter = 1 : length (this.InspectedK) ## do it only for the specified K values if (any (this.InspectedK(iter) == K)) if (isa (this.ClusteringFunction, "function_handle")) ## custom function ClusteringSolution = ... this.ClusteringFunction(UsableX, this.InspectedK(iter)); if (ismatrix (ClusteringSolution) && ... rows (ClusteringSolution) == this.NumObservations && ... columns (ClusteringSolution) == this.P) ## the custom function returned a matrix: ## we take the index of the maximum value for every row [~, this.ClusteringSolutions(:, iter)] = ... max (ClusteringSolution, [], 2); elseif (iscolumn (ClusteringSolution) && length (ClusteringSolution) == this.NumObservations) this.ClusteringSolutions(:, iter) = ClusteringSolution; elseif (isrow (ClusteringSolution) && length (ClusteringSolution) == this.NumObservations) this.ClusteringSolutions(:, iter) = ClusteringSolution'; else error (["CalinskiHarabaszEvaluation: invalid return value "... "from custom clustering function"]); endif this.ClusteringSolutions(:, iter) = ... this.ClusteringFunction(UsableX, this.InspectedK(iter)); else switch (this.ClusteringFunction) case "kmeans" [this.ClusteringSolutions(:, iter), this.Centroids{iter}] =... kmeans (UsableX, this.InspectedK(iter), ... "Distance", "sqeuclidean", "EmptyAction", "singleton", ... "Replicates", 5); case "linkage" ## use clusterdata this.ClusteringSolutions(:, iter) = clusterdata (UsableX, ... "MaxClust", this.InspectedK(iter), ... "Distance", "euclidean", "Linkage", "ward"); this.Centroids{iter} = this.computeCentroids (UsableX, iter); case "gmdistribution" gmm = fitgmdist (UsableX, this.InspectedK(iter), ... "SharedCov", true, "Replicates", 5); this.ClusteringSolutions(:, iter) = cluster (gmm, UsableX); this.Centroids{iter} = gmm.mu; otherwise error (["CalinskiHarabaszEvaluation: unexpected error, " ... "report this bug"]); endswitch endif endif endfor endif ## get the criterion values for every clustering solution for iter = 1 : length (this.InspectedK) ## do it only for the specified K values if (any (this.InspectedK(iter) == K)) ## not defined for one cluster if (this.InspectedK(iter) == 1) this.CriterionValues(iter) = NaN; continue; endif ## CaliÅ„ski-Harabasz index ## reference: calinhara function from the fpc package of R, ## by Christian Hennig ## https://CRAN.R-project.org/package=fpc W = zeros (columns (UsableX)); # between clusters covariance for i = 1 : this.InspectedK(iter) vIndicesI = find (this.ClusteringSolutions(:, iter) == i); ni = length (vIndicesI); # size of cluster i if (ni == 1) ## if the cluster has just one member the covariance is zero continue; endif ## weighted update of the covariance matrix W += cov (UsableX(vIndicesI, :)) * (ni - 1); endfor S = (this.NumObservations - 1) * cov (UsableX); # within clusters cov. B = S - W; # between clusters means ## tr(B) / tr(W) * (N-k) / (k-1) this.CriterionValues(iter) = (this.NumObservations - ... this.InspectedK(iter)) * trace (B) / ... ((this.InspectedK(iter) - 1) * trace (W)); endif endfor [~, this.OptimalIndex] = max (this.CriterionValues); this.OptimalK = this.InspectedK(this.OptimalIndex(1)); this.OptimalY = this.ClusteringSolutions(:, this.OptimalIndex(1)); endfunction endmethods methods (Access = private) ## computeCentroids ## compute the centroids if they are not available by other means function C = computeCentroids (this, X, index) C = zeros (this.InspectedK(index), columns (X)); for iter = 1 : this.InspectedK(index) vIndicesI = find (this.ClusteringSolutions(:, index) == iter); C(iter, :) = mean (X(vIndicesI, :)); endfor endfunction endmethods endclassdef statistics-release-1.6.3/inst/Clustering/ClusterCriterion.m000066400000000000000000000202151456127120000241010ustar00rootroot00000000000000## Copyright (C) 2021 Stefano Guidoni ## Copyright (C) 2024 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . classdef ClusterCriterion < handle ## -*- texinfo -*- ## @deftypefn {statistics} {@var{obj} =} ClusterCriterion (@var{x}, @var{clust}, @var{criterion}) ## ## A clustering evaluation object as created by @code{evalclusters}. ## ## @code{ClusterCriterion} is a superclass for clustering evaluation objects ## as created by @code{evalclusters}. ## ## List of public properties: ## @table @code ## @item @qcode{ClusteringFunction} ## a valid clustering funtion name or function handle. It can be empty if ## the clustering solutions are passed as an input matric. ## ## @item @qcode{CriterionName} ## a valid criterion name to evaluate the clustering solutions. ## ## @item @qcode{CriterionValues} ## a vector of values as generated by the evaluation criterion for each ## clustering solution. ## ## @item @qcode{InspectedK} ## the list of proposed cluster numbers. ## ## @item @qcode{Missing} ## a logical vector of missing observations. When there are @code{NaN} ## values in the data matrix, the corresponding observation is excluded. ## ## @item @qcode{NumObservations} ## the number of non-missing observations in the data matrix. ## ## @item @qcode{OptimalK} ## the optimal number of clusters. ## ## @item @qcode{OptimalY} ## the clustering solution corresponding to @code{OptimalK}. ## ## @item @qcode{X} ## the data matrix. ## ## @end table ## ## List of public methods: ## @table @code ## @item @qcode{addK} ## add a list of numbers of clusters to evaluate. ## ## @item @qcode{compact} ## return a compact clustering evaluation object. Not implemented ## ## @item @qcode{plot} ## plot the clustering evaluation values against the corresponding number of ## clusters. ## ## @end table ## ## @seealso{evalclusters, CalinskiHarabaszEvaluation, DaviesBouldinEvaluation, ## GapEvaluation, SilhouetteEvaluation} ## @end deftypefn properties (Access = public) ## public properties endproperties properties (GetAccess = public, SetAccess = protected) ClusteringFunction = ""; CriterionName = ""; CriterionValues = []; InspectedK = []; Missing = []; NumObservations = 0; OptimalK = 0; OptimalY = []; X = []; endproperties properties (Access = protected) N = 0; # number of observations P = 0; # number of variables ClusteringSolutions = []; # OptimalIndex = 0; # index of the optimal K endproperties methods (Access = public) ## constructor function this = ClusterCriterion (x, clust, KList) ## parsing input data if ((! ismatrix (x)) || (! isnumeric (x))) error ("ClusterCriterion: 'x' must be a numeric matrix"); endif this.X = x; this.N = rows (this.X); this.P = columns (this.X); ## look for missing values for iter = 1 : this.N if (any (find (x(iter, :) == NaN))) this.Missing(iter) = true; else this.Missing(iter) = false; endif endfor ## number of usable observations this.NumObservations = sum (this.Missing == false); ## parsing the clustering algorithm if (ischar (clust)) if (any (strcmpi (clust, {"kmeans", "linkage", "gmdistribution"}))) this.ClusteringFunction = lower (clust); else error ("ClusterCriterion: unknown clustering algorithm '%s'", clust); endif elseif (isa (clust, "function_handle")) this.ClusteringFunction = clust; elseif (ismatrix (clust)) if (isnumeric (clust) && (length (size (clust)) == 2) && ... (rows (clust) == this.N)) this.ClusteringFunction = ""; this.ClusteringSolutions = clust(find (this.Missing == false), :); else error ("ClusterCriterion: invalid matrix of clustering solutions"); endif else error ("ClusterCriterion: invalid argument"); endif ## parsing the list of cluster sizes to inspect this.InspectedK = parseKList (this, KList); endfunction ## -*- texinfo -*- ## @deftypefn {ClusterCriterion} {@var{obj} =} addK (@var{obj}, @var{K}) ## ## Add a new cluster array to inspect the ClusterCriterion object. ## ## @end deftypefn function this = addK (this, k) ## if there is not a clustering function, then we are using a predefined ## set of clustering solutions, hence we cannot redefine the number of ## solutions if (isempty (this.ClusteringFunction)) warning (["ClusterCriterion.addK: cannot redefine the list of cluster"... "numbers to evaluate when there is not a clustering function"]); return; endif ## otherwise go on newList = this.parseKList ([this.InspectedK k]); ## check if the list has changed if (length (newList) == length (this.InspectedK)) warning ("ClusterCriterion.addK: the list has not changed"); else ## update ClusteringSolutions and CriterionValues ClusteringSolutions_tmp = zeros (this.NumObservations, ... length (newList)); CriterionValues_tmp = zeros (length (newList), 1); for iter = 1 : length (this.InspectedK) idx = find (newList == this.InspectedK(iter)); if (! isempty (idx)) ClusteringSolutions_tmp(:, idx) = this.ClusteringSolutions(:, iter); CriterionValues_tmp(idx) = this.CriterionValues(iter); endif endfor this.ClusteringSolutions = ClusteringSolutions_tmp; this.CriterionValues = CriterionValues_tmp; ## reset the old results this.OptimalK = 0; this.OptimalY = []; this.OptimalIndex = 0; ## update the list of cluster numbers to evaluate this.InspectedK = newList; endif endfunction ## -*- texinfo -*- ## @deftypefn {ClusterCriterion} {} plot (@var{obj}) ## @deftypefnx {ClusterCriterion} {@var{h} =} plot (@var{obj}) ## ## Plot the evaluation results. ## ## Plot the CriterionValues against InspectedK from the ClusterCriterion, ## @var{obj}, to the current plot. It can also return a handle to the ## current plot. ## ## @end deftypefn function h = plot (this) yLabel = sprintf ("%s value", this.CriterionName); h = gca (); hold on; plot (this.InspectedK, this.CriterionValues, "bo-"); plot (this.OptimalK, this.CriterionValues(this.OptimalIndex), "b*"); xlabel ("number of clusters"); ylabel (yLabel); hold off; endfunction ## -*- texinfo -*- ## @deftypefn {ClusterCriterion} {@var{obj} =} compact (@var{obj}) ## ## Return a compact ClusterCriterion object (not implemented yet). ## ## @end deftypefn function this = compact (this) warning ("ClusterCriterion.compact: this method is not yet implemented."); endfunction endmethods methods (Access = private) ## check if a list of cluster sizes is correct function retList = parseKList (this, KList) if (isnumeric (KList) && isvector (KList) && all (find (KList > 0)) && ... all (floor (KList) == KList)) retList = unique (KList); else error (["ClusterCriterion: the list of cluster sizes must be an " ... "array of positive integer numbers"]); endif endfunction endmethods endclassdef statistics-release-1.6.3/inst/Clustering/DaviesBouldinEvaluation.m000066400000000000000000000230521456127120000253630ustar00rootroot00000000000000## Copyright (C) 2021 Stefano Guidoni ## Copyright (C) 2024 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . classdef DaviesBouldinEvaluation < ClusterCriterion ## -*- texinfo -*- ## @deftypefn {Function File} {@var{obj} =} evalclusters (@var{x}, @var{clust}, @qcode{DaviesBouldin}) ## @deftypefnx {Function File} {@var{obj} =} evalclusters (@dots{}, @qcode{Name}, @qcode{Value}) ## ## A Davies-Bouldin object to evaluate clustering solutions. ## ## A @code{DaviesBouldinEvaluation} object is a @code{ClusterCriterion} ## object used to evaluate clustering solutions using the Davies-Bouldin ## criterion. ## ## The Davies-Bouldin criterion is based on the ratio between the distances ## between clusters and within clusters, that is the distances between the ## centroids and the distances between each datapoint and its centroid. ## ## The best solution according to the Davies-Bouldin criterion is the one ## that scores the lowest value. ## ## @seealso{evalclusters, ClusterCriterion, CalinskiHarabaszEvaluation, ## GapEvaluation, SilhouetteEvaluation} ## @end deftypefn properties (GetAccess = public, SetAccess = private) endproperties properties (Access = protected) Centroids = {}; # a list of the centroids for every solution endproperties methods (Access = public) ## constructor function this = DaviesBouldinEvaluation (x, clust, KList) this@ClusterCriterion(x, clust, KList); this.CriterionName = "DaviesBouldin"; this.evaluate(this.InspectedK); # evaluate the list of cluster numbers endfunction ## -*- texinfo -*- ## @deftypefn {DaviesBouldinEvaluation} {@var{obj} =} addK (@var{obj}, @var{K}) ## ## Add a new cluster array to inspect the DaviesBouldinEvaluation object. ## ## @end deftypefn function this = addK (this, K) addK@ClusterCriterion(this, K); ## if we have new data, we need a new evaluation if (this.OptimalK == 0) Centroids_tmp = {}; pS = 0; # position shift of the elements of Centroids for iter = 1 : length (this.InspectedK) ## reorganize Centroids according to the new list of cluster numbers if (any (this.InspectedK(iter) == K)) pS += 1; else Centroids_tmp{iter} = this.Centroids{iter - pS}; endif endfor this.Centroids = Centroids_tmp; this.evaluate(K); # evaluate just the new cluster numbers endif endfunction ## -*- texinfo -*- ## @deftypefn {DaviesBouldinEvaluation} {} plot (@var{obj}) ## @deftypefnx {DaviesBouldinEvaluation} {@var{h} =} plot (@var{obj}) ## ## Plot the evaluation results. ## ## Plot the CriterionValues against InspectedK from the ## DaviesBouldinEvaluation ClusterCriterion, @var{obj}, to the current plot. ## It can also return a handle to the current plot. ## ## @end deftypefn function h = plot (this) yLabel = sprintf ("%s value", this.CriterionName); h = gca (); hold on; plot (this.InspectedK, this.CriterionValues, "bo-"); plot (this.OptimalK, this.CriterionValues(this.OptimalIndex), "b*"); xlabel ("number of clusters"); ylabel (yLabel); hold off; endfunction ## -*- texinfo -*- ## @deftypefn {DaviesBouldinEvaluation} {@var{obj} =} compact (@var{obj}) ## ## Return a compact DaviesBouldinEvaluation object (not implemented yet). ## ## @end deftypefn function this = compact (this) warning (["DaviesBouldinEvaluation.compact: this"... " method is not yet implemented."]); endfunction endmethods methods (Access = protected) ## evaluate ## do the evaluation function this = evaluate (this, K) ## use complete observations only UsableX = this.X(find (this.Missing == false), :); if (! isempty (this.ClusteringFunction)) ## build the clusters for iter = 1 : length (this.InspectedK) ## do it only for the specified K values if (any (this.InspectedK(iter) == K)) if (isa (this.ClusteringFunction, "function_handle")) ## custom function ClusteringSolution = ... this.ClusteringFunction(UsableX, this.InspectedK(iter)); if (ismatrix (ClusteringSolution) && ... rows (ClusteringSolution) == this.NumObservations && ... columns (ClusteringSolution) == this.P) ## the custom function returned a matrix: ## we take the index of the maximum value for every row [~, this.ClusteringSolutions(:, iter)] = ... max (ClusteringSolution, [], 2); elseif (iscolumn (ClusteringSolution) && length (ClusteringSolution) == this.NumObservations) this.ClusteringSolutions(:, iter) = ClusteringSolution; elseif (isrow (ClusteringSolution) && length (ClusteringSolution) == this.NumObservations) this.ClusteringSolutions(:, iter) = ClusteringSolution'; else error (["DaviesBouldinEvaluation: invalid return value "... "from custom clustering function"]); endif this.ClusteringSolutions(:, iter) = ... this.ClusteringFunction(UsableX, this.InspectedK(iter)); else switch (this.ClusteringFunction) case "kmeans" [this.ClusteringSolutions(:, iter), this.Centroids{iter}] =... kmeans (UsableX, this.InspectedK(iter), ... "Distance", "sqeuclidean", "EmptyAction", "singleton", ... "Replicates", 5); case "linkage" ## use clusterdata this.ClusteringSolutions(:, iter) = clusterdata (UsableX, ... "MaxClust", this.InspectedK(iter), ... "Distance", "euclidean", "Linkage", "ward"); this.Centroids{iter} = this.computeCentroids (UsableX, iter); case "gmdistribution" gmm = fitgmdist (UsableX, this.InspectedK(iter), ... "SharedCov", true, "Replicates", 5); this.ClusteringSolutions(:, iter) = cluster (gmm, UsableX); this.Centroids{iter} = gmm.mu; otherwise error (["DaviesBouldinEvaluation: unexpected error, " ... "report this bug"]); endswitch endif endif endfor endif ## get the criterion values for every clustering solution for iter = 1 : length (this.InspectedK) ## do it only for the specified K values if (any (this.InspectedK(iter) == K)) ## not defined for one cluster if (this.InspectedK(iter) == 1) this.CriterionValues(iter) = NaN; continue; endif ## Davies-Bouldin value ## an evaluation of the ratio between within-cluster and ## between-cluster distances ## mean distances between cluster members and their centroid vD = zeros (this.InspectedK(iter), 1); for i = 1 : this.InspectedK(iter) vIndicesI = find (this.ClusteringSolutions(:, iter) == i); vD(i) = mean (vecnorm (UsableX(vIndicesI, :) - ... this.Centroids{iter}(i, :), 2, 2)); endfor ## within-to-between cluster distance ratio Dij = zeros (this.InspectedK(iter)); for i = 1 : (this.InspectedK(iter) - 1) for j = (i + 1) : this.InspectedK(iter) ## centroid to centroid distance dij = vecnorm (this.Centroids{iter}(i, :) - ... this.Centroids{iter}(j, :)); ## within-to-between cluster distance ratio for clusters i and j Dij(i, j) = (vD(i) + vD(j)) / dij; endfor endfor ## ( max_j D1j + max_j D2j + ... + max_j Dkj) / k this.CriterionValues(iter) = sum (max (Dij(i, :), [], 2)) / ... this.InspectedK(iter); endif endfor [~, this.OptimalIndex] = min (this.CriterionValues); this.OptimalK = this.InspectedK(this.OptimalIndex(1)); this.OptimalY = this.ClusteringSolutions(:, this.OptimalIndex(1)); endfunction endmethods methods (Access = private) ## computeCentroids ## compute the centroids if they are not available by other means function C = computeCentroids (this, X, index) C = zeros (this.InspectedK(index), columns (X)); for iter = 1 : this.InspectedK(index) vIndicesI = find (this.ClusteringSolutions(:, index) == iter); C(iter, :) = mean (X(vIndicesI, :)); endfor endfunction endmethods endclassdef statistics-release-1.6.3/inst/Clustering/GapEvaluation.m000066400000000000000000000365141456127120000233510ustar00rootroot00000000000000## Copyright (C) 2021 Stefano Guidoni ## Copyright (C) 2024 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . classdef GapEvaluation < ClusterCriterion ## -*- texinfo -*- ## @deftypefn {statistics} {@var{obj} =} evalclusters (@var{x}, @var{clust}, @qcode{gap}) ## @deftypefnx {statistics} {@var{obj} =} evalclusters (@dots{}, @qcode{Name}, @qcode{Value}) ## ## A gap object to evaluate clustering solutions. ## ## A @code{GapEvaluation} object is a @code{ClusterCriterion} ## object used to evaluate clustering solutions using the gap criterion, ## which is a mathematical formalization of the elbow method. ## ## List of public properties specific to @code{SilhouetteEvaluation}: ## @table @code ## @item @qcode{B} ## the number of reference datasets to generate. ## ## @item @qcode{Distance} ## a valid distance metric name, or a function handle as accepted by the ## @code{pdist} function. ## ## @item @qcode{ExpectedLogW} ## a vector of the expected values for the logarithm of the within clusters ## dispersion. ## ## @item @qcode{LogW} ## a vector of the values of the logarithm of the within clusters dispersion. ## ## @item @qcode{ReferenceDistribution} ## a valid name for the reference distribution, namely: @code{PCA} (default) ## or @code{uniform}. ## ## @item @qcode{SE} ## a vector of the standard error of the expected values for the logarithm ## of the within clusters dispersion. ## ## @item @qcode{SearchMethod} ## a valid name for the search method to use: @code{globalMaxSE} (default) or ## @code{firstMaxSE}. ## ## @item @qcode{StdLogW} ## a vector of the standard deviation of the expected values for the logarithm ## of the within clusters dispersion. ## @end table ## ## The best solution according to the gap criterion depends on the chosen ## search method. When the search method is @code{globalMaxSE}, the chosen ## gap value is the smaller one which is inside a standard error from the ## max gap value; when the search method is @code{firstMaxSE}, the chosen ## gap value is the first one which is inside a standard error from the next ## gap value. ## ## @seealso{evalclusters, ClusterCriterion, CalinskiHarabaszEvaluation, ## DaviesBouldinEvaluation, SilhouetteEvaluation} ## @end deftypefn properties (GetAccess = public, SetAccess = private) B = 0; # number of reference datasets Distance = ""; # pdist parameter ReferenceDistribution = ""; # distribution to use as reference SearchMethod = ""; # the method do identify the optimal number of clusters ExpectedLogW = []; # expected value for the natural logarithm of W LogW = []; # natural logarithm of W SE = []; # standard error for the natural logarithm of W StdLogW = []; # standard deviation of the natural logarithm of W endproperties properties (Access = protected) DistanceVector = []; # vector of pdist distances mExpectedLogW = []; # the result of the Monte-Carlo simulations endproperties methods (Access = public) ## constructor function this = GapEvaluation (x, clust, KList, b = 100, ... distanceMetric = "sqeuclidean", ... referenceDistribution = "pca", searchMethod = "globalmaxse") this@ClusterCriterion(x, clust, KList); ## parsing the distance criterion if (ischar (distanceMetric)) if (any (strcmpi (distanceMetric, {"sqeuclidean", "euclidean", ... "cityblock", "cosine", "correlation", "hamming", "jaccard"}))) this.Distance = lower (distanceMetric); ## kmeans can use only a subset if (strcmpi (clust, "kmeans") && any (strcmpi (this.Distance, ... {"euclidean", "jaccard"}))) error (["GapEvaluation: invalid distance criterion '%s' "... "for 'kmeans'"], distanceMetric); endif else error ("GapEvaluation: unknown distance criterion '%s'", ... distanceMetric); endif elseif (isa (distanceMetric, "function_handle")) this.Distance = distanceMetric; ## kmeans cannot use a function handle if (strcmpi (clust, "kmeans")) error ("GapEvaluation: invalid distance criterion for 'kmeans'"); endif elseif (isvector (distanceMetric) && isnumeric (distanceMetric)) this.Distance = ""; this.DistanceVector = distanceMetric; # the validity check is delegated ## kmeans cannot use a distance vector if (strcmpi (clust, "kmeans")) error (["GapEvaluation: invalid distance criterion for "... "'kmeans'"]); endif else error ("GapEvaluation: invalid distance metric"); endif ## B: number of Monte-Carlo iterations if (! isnumeric (b) || ! isscalar (b) || b != floor (b) || b < 1) error ("GapEvaluation: b must a be positive integer number"); endif this.B = b; ## reference distribution if (! ischar (referenceDistribution) || ! any (strcmpi ... (referenceDistribution, {"pca", "uniform"}))) error (["GapEvaluation: the reference distribution must be either" ... "'PCA' or 'uniform'"]); elseif (strcmpi (referenceDistribution, "pca")) warning (["GapEvaluation: 'PCA' distribution not implemented, " ... "using 'uniform'"]); endif this.ReferenceDistribution = lower (referenceDistribution); if (! ischar (searchMethod) || ! any (strcmpi (searchMethod, ... {"globalmaxse", "firstmaxse"}))) error (["evalclusters: the search method must be either" ... "'globalMaxSE' or 'firstMaxSE'"]); endif this.SearchMethod = lower (searchMethod); ## a matrix to store the results from the Monte-Carlo runs this.mExpectedLogW = zeros (this.B, length (this.InspectedK)); this.CriterionName = "gap"; this.evaluate(this.InspectedK); # evaluate the list of cluster numbers endfunction ## -*- texinfo -*- ## @deftypefn {GapEvaluation} {@var{obj} =} addK (@var{obj}, @var{K}) ## ## Add a new cluster array to inspect the GapEvaluation object. ## ## @end deftypefn function this = addK (this, K) addK@ClusterCriterion(this, K); ## if we have new data, we need a new evaluation if (this.OptimalK == 0) mExpectedLogW_tmp = zeros (this.B, length (this.InspectedK)); pS = 0; # position shift for iter = 1 : length (this.InspectedK) ## reorganize all the arrays according to the new list ## of cluster numbers if (any (this.InspectedK(iter) == K)) pS += 1; else mExpectedLogW_tmp(:, iter) = this.mExpectedLogW(:, iter - pS); endif endfor this.mExpectedLogW = mExpectedLogW_tmp; this.evaluate(K); # evaluate just the new cluster numbers endif endfunction ## -*- texinfo -*- ## @deftypefn {ClusterCriterion} {} plot (@var{obj}) ## @deftypefnx {ClusterCriterion} {@var{h} =} plot (@var{obj}) ## ## Plot the evaluation results. ## ## Plot the CriterionValues against InspectedK from the GapEvaluation ## ClusterCriterion, @var{obj}, and show the standard deviation to the ## current plot. It can also return a handle to the current plot. ## ## @end deftypefn function h = plot (this) yLabel = sprintf ("%s value", this.CriterionName); h = gca (); hold on; errorbar (this.InspectedK, this.CriterionValues, this.StdLogW); plot (this.InspectedK, this.CriterionValues, "bo"); plot (this.OptimalK, this.CriterionValues(this.OptimalIndex), "b*"); xlabel ("number of clusters"); ylabel (yLabel); hold off; endfunction ## -*- texinfo -*- ## @deftypefn {GapEvaluation} {@var{obj} =} compact (@var{obj}) ## ## Return a compact GapEvaluation object (not implemented yet). ## ## @end deftypefn function this = compact (this) warning ("GapEvaluation.compact: this method is not yet implemented."); endfunction endmethods methods (Access = protected) ## evaluate ## do the evaluation function this = evaluate (this, K) ## Monte-Carlo runs for mcrun = 1 : (this.B + 1) ## use complete observations only UsableX = this.X(find (this.Missing == false), :); ## the last run use tha actual data, ## the others are Monte-Carlo runs with reconstructed data if (mcrun <= this.B) ## uniform distribution colMins = min (UsableX); colMaxs = max (UsableX); for col = 1 : columns (UsableX) UsableX(:, col) = colMins(col) + rand (this.NumObservations, 1) *... (colMaxs(col) - colMins(col)); endfor endif if (! isempty (this.ClusteringFunction)) ## build the clusters for iter = 1 : length (this.InspectedK) ## do it only for the specified K values if (any (this.InspectedK(iter) == K)) if (isa (this.ClusteringFunction, "function_handle")) ## custom function ClusteringSolution = ... this.ClusteringFunction(UsableX, this.InspectedK(iter)); if (ismatrix (ClusteringSolution) && ... rows (ClusteringSolution) == this.NumObservations && ... columns (ClusteringSolution) == this.P) ## the custom function returned a matrix: ## we take the index of the maximum value for every row [~, this.ClusteringSolutions(:, iter)] = ... max (ClusteringSolution, [], 2); elseif (iscolumn (ClusteringSolution) && length (ClusteringSolution) == this.NumObservations) this.ClusteringSolutions(:, iter) = ClusteringSolution; elseif (isrow (ClusteringSolution) && length (ClusteringSolution) == this.NumObservations) this.ClusteringSolutions(:, iter) = ClusteringSolution'; else error (["GapEvaluation: invalid return value from " ... "custom clustering function"]); endif this.ClusteringSolutions(:, iter) = ... this.ClusteringFunction(UsableX, this.InspectedK(iter)); else switch (this.ClusteringFunction) case "kmeans" this.ClusteringSolutions(:, iter) = kmeans (UsableX, ... this.InspectedK(iter), "Distance", this.Distance, ... "EmptyAction", "singleton", "Replicates", 5); case "linkage" if (! isempty (this.Distance)) ## use clusterdata Distance_tmp = this.Distance; LinkageMethod = "average"; # for non euclidean methods if (strcmpi (this.Distance, "sqeuclidean")) ## pdist uses different names for its algorithms Distance_tmp = "squaredeuclidean"; LinkageMethod = "ward"; elseif (strcmpi (this.Distance, "euclidean")) LinkageMethod = "ward"; endif this.ClusteringSolutions(:, iter) = clusterdata ... (UsableX, "MaxClust", this.InspectedK(iter), ... "Distance", Distance_tmp, "Linkage", LinkageMethod); else ## use linkage Z = linkage (this.DistanceVector, "average"); this.ClusteringSolutions(:, iter) = ... cluster (Z, "MaxClust", this.InspectedK(iter)); endif case "gmdistribution" gmm = fitgmdist (UsableX, this.InspectedK(iter), ... "SharedCov", true, "Replicates", 5); this.ClusteringSolutions(:, iter) = cluster (gmm, UsableX); otherwise ## this should not happen error (["GapEvaluation: unexpected error, " ... "report this bug"]); endswitch endif endif endfor endif ## get the gap values for every clustering distance_pdist = this.Distance; if (strcmpi (distance_pdist, "sqeuclidean")) distance_pdist = "squaredeuclidean"; endif ## compute LogW for iter = 1 : length (this.InspectedK) ## do it only for the specified K values if (any (this.InspectedK(iter) == K)) wk = 0; for r = 1 : this.InspectedK(iter) vIndicesR = find (this.ClusteringSolutions(:, iter) == r); nr = length (vIndicesR); Dr = pdist (UsableX(vIndicesR, :), distance_pdist); wk += sum (Dr) / (2 * nr); endfor if (mcrun <= this.B) this.mExpectedLogW(mcrun, iter) = log (wk); else this.LogW(iter) = log (wk); endif endif endfor endfor this.ExpectedLogW = mean (this.mExpectedLogW); this.SE = sqrt ((1 + 1 / this.B) * sumsq (this.mExpectedLogW - ... this.ExpectedLogW) / this.B); this.StdLogW = std (this.mExpectedLogW); this.CriterionValues = this.ExpectedLogW - this.LogW; this.OptimalIndex = this.gapSearch (); this.OptimalK = this.InspectedK(this.OptimalIndex(1)); this.OptimalY = this.ClusteringSolutions(:, this.OptimalIndex(1)); endfunction ## gapSearch ## find the best solution according to the gap method function ind = gapSearch (this) if (strcmpi (this.SearchMethod, "globalmaxse")) [gapmax, indgp] = max (this.CriterionValues); for iter = 1 : length (this.InspectedK) ind = iter; if (this.CriterionValues(iter) > (gapmax - this.SE(indgp))) return endif endfor elseif (strcmpi (this.SearchMethod, "firstmaxse")) for iter = 1 : (length (this.InspectedK) - 1) ind = iter; if (this.CriterionValues(iter) > (this.CriterionValues(iter + 1) - ... this.SE(iter + 1))) return endif endfor else ## this should not happen error (["GapEvaluation: unexpected error, please report this bug"]); endif endfunction endmethods endclassdef statistics-release-1.6.3/inst/Clustering/SilhouetteEvaluation.m000066400000000000000000000274131456127120000247650ustar00rootroot00000000000000## Copyright (C) 2021 Stefano Guidoni ## Copyright (C) 2024 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . classdef SilhouetteEvaluation < ClusterCriterion ## -*- texinfo -*- ## @deftypefn {Function File} {@var{obj} =} evalclusters (@var{x}, @var{clust}, @qcode{silhouette}) ## @deftypefnx {Function File} {@var{obj} =} evalclusters (@dots{}, @qcode{Name}, @qcode{Value}) ## ## A silhouette object to evaluate clustering solutions. ## ## A @code{SilhouetteEvaluation} object is a @code{ClusterCriterion} ## object used to evaluate clustering solutions using the silhouette ## criterion. ## ## List of public properties specific to @code{SilhouetteEvaluation}: ## @table @code ## @item @qcode{Distance} ## a valid distance metric name, or a function handle or a numeric array as ## generated by the @code{pdist} function. ## ## @item @qcode{ClusterPriors} ## a valid name for the evaluation of silhouette values: @code{empirical} ## (default) or @code{equal}. ## ## @item @qcode{ClusterSilhouettes} ## a cell array with the silhouette values of each data point for each cluster ## number. ## ## @end table ## ## The best solution according to the silhouette criterion is the one ## that scores the highest average silhouette value. ## ## @seealso{evalclusters, ClusterCriterion, CalinskiHarabaszEvaluation, ## DaviesBouldinEvaluation, GapEvaluation} ## @end deftypefn properties (GetAccess = public, SetAccess = private) Distance = ""; # pdist parameter ClusterPriors = ""; # evaluation of silhouette values: equal or empirical ClusterSilhouettes = {}; # results of the silhoutte function for each K endproperties properties (Access = protected) DistanceVector = []; # vector of pdist distances endproperties methods (Access = public) ## constructor function this = SilhouetteEvaluation (x, clust, KList, ... distanceMetric = "sqeuclidean", clusterPriors = "empirical") this@ClusterCriterion(x, clust, KList); ## parsing the distance criterion if (ischar (distanceMetric)) if (any (strcmpi (distanceMetric, {"sqeuclidean", ... "euclidean", "cityblock", "cosine", "correlation", ... "hamming", "jaccard"}))) this.Distance = lower (distanceMetric); ## kmeans can use only a subset if (strcmpi (clust, "kmeans") && any (strcmpi (this.Distance, ... {"euclidean", "jaccard"}))) error (["SilhouetteEvaluation: invalid distance criterion '%s' "... "for 'kmeans'"], distanceMetric); endif else error ("SilhouetteEvaluation: unknown distance criterion '%s'", ... distanceMetric); endif elseif (isa (distanceMetric, "function_handle")) this.Distance = distanceMetric; ## kmeans cannot use a function handle if (strcmpi (clust, "kmeans")) error (["SilhouetteEvaluation: invalid distance criterion for "... "'kmeans'"]); endif elseif (isvector (distanceMetric) && isnumeric (distanceMetric)) this.Distance = ""; this.DistanceVector = distanceMetric; # the validity check is delegated ## kmeans cannot use a distance vector if (strcmpi (clust, "kmeans")) error (["SilhouetteEvaluation: invalid distance criterion for "... "'kmeans'"]); endif else error ("SilhouetteEvaluation: invalid distance metric"); endif ## parsing the prior probabilities of each cluster if (ischar (distanceMetric)) if (any (strcmpi (clusterPriors, {"empirical", "equal"}))) this.ClusterPriors = lower (clusterPriors); else error (["SilhouetteEvaluation: unknown prior probability criterion"... " '%s'"], clusterPriors); endif else error ("SilhouetteEvaluation: invalid prior probabilities"); endif this.CriterionName = "silhouette"; this.evaluate(this.InspectedK); # evaluate the list of cluster numbers endfunction ## -*- texinfo -*- ## @deftypefn {SilhouetteEvaluation} {@var{obj} =} addK (@var{obj}, @var{K}) ## ## Add a new cluster array to inspect the SilhouetteEvaluation object. ## ## @end deftypefn function this = addK (this, K) addK@ClusterCriterion(this, K); ## if we have new data, we need a new evaluation if (this.OptimalK == 0) ClusterSilhouettes_tmp = {}; pS = 0; # position shift of the elements of ClusterSilhouettes for iter = 1 : length (this.InspectedK) ## reorganize ClusterSilhouettes according to the new list ## of cluster numbers if (any (this.InspectedK(iter) == K)) pS += 1; else ClusterSilhouettes_tmp{iter} = this.ClusterSilhouettes{iter - pS}; endif endfor this.ClusterSilhouettes = ClusterSilhouettes_tmp; this.evaluate(K); # evaluate just the new cluster numbers endif endfunction ## -*- texinfo -*- ## @deftypefn {SilhouetteEvaluation} {} plot (@var{obj}) ## @deftypefnx {SilhouetteEvaluation} {@var{h} =} plot (@var{obj}) ## ## Plot the evaluation results. ## ## Plot the CriterionValues against InspectedK from the ## SilhouetteEvaluation ClusterCriterion, @var{obj}, to the current plot. ## It can also return a handle to the current plot. ## ## @end deftypefn function h = plot (this) yLabel = sprintf ("%s value", this.CriterionName); h = gca (); hold on; plot (this.InspectedK, this.CriterionValues, "bo-"); plot (this.OptimalK, this.CriterionValues(this.OptimalIndex), "b*"); xlabel ("number of clusters"); ylabel (yLabel); hold off; endfunction ## -*- texinfo -*- ## @deftypefn {SilhouetteEvaluation} {@var{obj} =} compact (@var{obj}) ## ## Return a compact SilhouetteEvaluation object (not implemented yet). ## ## @end deftypefn function this = compact (this) warning (["SilhouetteEvaluation.compact: this"... " method is not yet implemented."]); endfunction endmethods methods (Access = protected) ## evaluate ## do the evaluation function this = evaluate (this, K) ## use complete observations only UsableX = this.X(find (this.Missing == false), :); if (! isempty (this.ClusteringFunction)) ## build the clusters for iter = 1 : length (this.InspectedK) ## do it only for the specified K values if (any (this.InspectedK(iter) == K)) if (isa (this.ClusteringFunction, "function_handle")) ## custom function ClusteringSolution = ... this.ClusteringFunction(UsableX, this.InspectedK(iter)); if (ismatrix (ClusteringSolution) && ... rows (ClusteringSolution) == this.NumObservations && ... columns (ClusteringSolution) == this.P) ## the custom function returned a matrix: ## we take the index of the maximum value for every row [~, this.ClusteringSolutions(:, iter)] = ... max (ClusteringSolution, [], 2); elseif (iscolumn (ClusteringSolution) && length (ClusteringSolution) == this.NumObservations) this.ClusteringSolutions(:, iter) = ClusteringSolution; elseif (isrow (ClusteringSolution) && length (ClusteringSolution) == this.NumObservations) this.ClusteringSolutions(:, iter) = ClusteringSolution'; else error (["SilhouetteEvaluation: invalid return value from " ... "custom clustering function"]); endif this.ClusteringSolutions(:, iter) = ... this.ClusteringFunction(UsableX, this.InspectedK(iter)); else switch (this.ClusteringFunction) case "kmeans" this.ClusteringSolutions(:, iter) = kmeans (UsableX, ... this.InspectedK(iter), "Distance", this.Distance, ... "EmptyAction", "singleton", "Replicates", 5); case "linkage" if (! isempty (this.Distance)) ## use clusterdata Distance_tmp = this.Distance; LinkageMethod = "average"; # for non euclidean methods if (strcmpi (this.Distance, "sqeuclidean")) ## pdist uses different names for its algorithms Distance_tmp = "squaredeuclidean"; LinkageMethod = "ward"; elseif (strcmpi (this.Distance, "euclidean")) LinkageMethod = "ward"; endif this.ClusteringSolutions(:, iter) = clusterdata (UsableX,... "MaxClust", this.InspectedK(iter), ... "Distance", Distance_tmp, "Linkage", LinkageMethod); else ## use linkage Z = linkage (this.DistanceVector, "average"); this.ClusteringSolutions(:, iter) = ... cluster (Z, "MaxClust", this.InspectedK(iter)); endif case "gmdistribution" gmm = fitgmdist (UsableX, this.InspectedK(iter), ... "SharedCov", true, "Replicates", 5); this.ClusteringSolutions(:, iter) = cluster (gmm, UsableX); otherwise error (["SilhouetteEvaluation: unexpected error, " ... "report this bug"]); endswitch endif endif endfor endif ## get the silhouette values for every clustering set (0, 'DefaultFigureVisible', 'off'); # temporarily disable figures for iter = 1 : length (this.InspectedK) ## do it only for the specified K values if (any (this.InspectedK(iter) == K)) this.ClusterSilhouettes{iter} = silhouette (UsableX, ... this.ClusteringSolutions(:, iter)); if (strcmpi (this.ClusterPriors, "empirical")) this.CriterionValues(iter) = mean (this.ClusterSilhouettes{iter}); else ## equal this.CriterionValues(iter) = 0; si = this.ClusterSilhouettes{iter}; for k = 1 : this.InspectedK(iter) this.CriterionValues(iter) += mean (si(find ... (this.ClusteringSolutions(:, iter) == k))); endfor this.CriterionValues(iter) /= this.InspectedK(iter); endif endif endfor set (0, 'DefaultFigureVisible', 'on'); # enable figures again [~, this.OptimalIndex] = max (this.CriterionValues); this.OptimalK = this.InspectedK(this.OptimalIndex(1)); this.OptimalY = this.ClusteringSolutions(:, this.OptimalIndex(1)); endfunction endmethods endclassdef statistics-release-1.6.3/inst/PKG_ADD000066400000000000000000000015431456127120000173230ustar00rootroot00000000000000if (compare_versions (version (), "9", "<")) a1_e324kporit985_itogj3_dirlist = ... {"Classification", "Clustering", "datasets", "dist_fit", "dist_fun", ... "dist_stat", "Regression", "shadow9"}; else a1_e324kporit985_itogj3_dirlist = ... {"Classification", "Clustering", "datasets", "dist_fit", "dist_fun", ... "dist_stat", "Regression"}; endif d_2seRTE546_oyi_795jg09_dirname = fileparts (canonicalize_file_name ... (mfilename ("fullpath"))); for iiII123DRT_idx = 1:length (a1_e324kporit985_itogj3_dirlist) addpath (fullfile (d_2seRTE546_oyi_795jg09_dirname, ... a1_e324kporit985_itogj3_dirlist{iiII123DRT_idx})); endfor clear a1_e324kporit985_itogj3_dirlist clear d_2seRTE546_oyi_795jg09_dirname iiII123DRT_idx warning ("off", "Octave:data-file-in-path") statistics-release-1.6.3/inst/PKG_DEL000066400000000000000000000015231456127120000173350ustar00rootroot00000000000000clear -f libsvmread libsvmwrite svmpredict svmtrain if (compare_versions (version (), "9", "<")) a1_e324kporit985_itogj3_dirlist = ... {"Classification", "Clustering", "datasets", "dist_fit", "dist_fun", ... "dist_stat", "Regression", "shadow9"}; else a1_e324kporit985_itogj3_dirlist = ... {"Classification", "Clustering", "datasets", "dist_fit", "dist_fun", ... "dist_stat", "Regression"}; endif d_2seRTE546_oyi_795jg09_dirname = fileparts (canonicalize_file_name ... (mfilename ("fullpath"))); for iiII123DRT_idx = 1:length (a1_e324kporit985_itogj3_dirlist) rmpath (fullfile (d_2seRTE546_oyi_795jg09_dirname, ... a1_e324kporit985_itogj3_dirlist{iiII123DRT_idx})); endfor clear a1_e324kporit985_itogj3_dirlist clear d_2seRTE546_oyi_795jg09_dirname iiII123DRT_idx statistics-release-1.6.3/inst/Regression/000077500000000000000000000000001456127120000204245ustar00rootroot00000000000000statistics-release-1.6.3/inst/Regression/RegressionGAM.m000066400000000000000000001112461456127120000232540ustar00rootroot00000000000000## Copyright (C) 2023 Mohammed Azmat Khan ## Copyright (C) 2023-2024 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . classdef RegressionGAM ## -*- texinfo -*- ## @deftypefn {statistics} {@var{obj} =} RegressionGAM (@var{X}, @var{Y}) ## @deftypefnx {statistics} {@var{obj} =} RegressionGAM (@dots{}, @var{name}, @var{value}) ## ## Create a @qcode{RegressionGAM} class object containing a Generalised Additive ## Model (GAM) for regression. ## ## A @qcode{RegressionGAM} class object can store the predictors and response ## data along with various parameters for the GAM model. It is recommended to ## use the @code{fitrgam} function to create a @qcode{RegressionGAM} object. ## ## @code{@var{obj} = RegressionGAM (@var{X}, @var{Y})} returns an object of ## class RegressionGAM, with matrix @var{X} containing the predictor data and ## vector @var{Y} containing the continuous response data. ## ## @itemize ## @item ## @var{X} must be a @math{NxP} numeric matrix of input data where rows ## correspond to observations and columns correspond to features or variables. ## @var{X} will be used to train the GAM model. ## @item ## @var{Y} must be @math{Nx1} numeric vector containing the response data ## corresponding to the predictor data in @var{X}. @var{Y} must have same ## number of rows as @var{X}. ## @end itemize ## ## @code{@var{obj} = RegressionGAM (@dots{}, @var{name}, @var{value})} returns ## an object of class RegressionGAM with additional properties specified by ## @qcode{Name-Value} pair arguments listed below. ## ## @multitable @columnfractions 0.05 0.2 0.75 ## @headitem @tab @var{Name} @tab @var{Value} ## ## @item @tab @qcode{"predictors"} @tab Predictor Variable names, specified as ## a row vector cell of strings with the same length as the columns in @var{X}. ## If omitted, the program will generate default variable names ## @qcode{(x1, x2, ..., xn)} for each column in @var{X}. ## ## @item @tab @qcode{"responsename"} @tab Response Variable Name, specified as ## a string. If omitted, the default value is @qcode{"Y"}. ## ## @item @tab @qcode{"formula"} @tab a model specification given as a string in ## the form @qcode{"Y ~ terms"} where @qcode{Y} represents the reponse variable ## and @qcode{terms} the predictor variables. The formula can be used to ## specify a subset of variables for training model. For example: ## @qcode{"Y ~ x1 + x2 + x3 + x4 + x1:x2 + x2:x3"} specifies four linear terms ## for the first four columns of for predictor data, and @qcode{x1:x2} and ## @qcode{x2:x3} specify the two interaction terms for 1st-2nd and 3rd-4th ## columns respectively. Only these terms will be used for training the model, ## but @var{X} must have at least as many columns as referenced in the formula. ## If Predictor Variable names have been defined, then the terms in the formula ## must reference to those. When @qcode{"formula"} is specified, all terms used ## for training the model are referenced in the @qcode{IntMatrix} field of the ## @var{obj} class object as a matrix containing the column indexes for each ## term including both the predictors and the interactions used. ## ## @item @tab @qcode{"interactions"} @tab a logical matrix, a positive integer ## scalar, or the string @qcode{"all"} for defining the interactions between ## predictor variables. When given a logical matrix, it must have the same ## number of columns as @var{X} and each row corresponds to a different ## interaction term combining the predictors indexed as @qcode{true}. Each ## interaction term is appended as a column vector after the available predictor ## column in @var{X}. When @qcode{"all"} is defined, then all possible ## combinations of interactions are appended in @var{X} before training. At the ## moment, parsing a positive integer has the same effect as the @qcode{"all"} ## option. When @qcode{"interactions"} is specified, only the interaction terms ## appended to @var{X} are referenced in the @qcode{IntMatrix} field of the ## @var{obj} class object. ## ## @item @tab @qcode{"knots"} @tab a scalar or a row vector with the same ## columns as @var{X}. It defines the knots for fitting a polynomial when ## training the GAM. As a scalar, it is expanded to a row vector. The default ## value is 5, hence expanded to @qcode{ones (1, columns (X)) * 5}. You can ## parse a row vector with different number of knots for each predictor ## variable to be fitted with, although not recommended. ## ## @item @tab @qcode{"order"} @tab a scalar or a row vector with the same ## columns as @var{X}. It defines the order of the polynomial when training the ## GAM. As a scalar, it is expanded to a row vector. The default values is 3, ## hence expanded to @qcode{ones (1, columns (X)) * 3}. You can parse a row ## vector with different number of polynomial order for each predictor variable ## to be fitted with, although not recommended. ## ## @item @tab @qcode{"dof"} @tab a scalar or a row vector with the same columns ## as @var{X}. It defines the degrees of freedom for fitting a polynomial when ## training the GAM. As a scalar, it is expanded to a row vector. The default ## value is 8, hence expanded to @qcode{ones (1, columns (X)) * 8}. You can ## parse a row vector with different degrees of freedom for each predictor ## variable to be fitted with, although not recommended. ## ## @item @tab @qcode{"tol"} @tab a positive scalar to set the tolerance for ## covergence during training. By defaul, it is set to @qcode{1e-3}. ## @end multitable ## ## You can parse either a @qcode{"formula"} or an @qcode{"interactions"} ## optional parameter. Parsing both parameters will result an error. ## Accordingly, you can only pass up to two parameters among @qcode{"knots"}, ## @qcode{"order"}, and @qcode{"dof"} to define the required polynomial for ## training the GAM model. ## ## @seealso{fitrgam, regress, regress_gp} ## @end deftypefn properties (Access = public) X = []; # Predictor data Y = []; # Response data BaseModel = []; # Base model parameters (no interactions) ModelwInt = []; # Model parameters with interactions IntMatrix = []; # Interactions matrix applied to predictor data NumObservations = []; # Number of observations in training dataset RowsUsed = []; # Rows used in fitting NumPredictors = []; # Number of predictors PredictorNames = []; # Predictor variable names ResponseName = []; # Response variable name Formula = []; # Formula for GAM model Interactions = []; # Number or matrix of interaction terms Knots = []; # Knots of spline fitting Order = []; # Order of spline fitting DoF = []; # Degrees of freedom for fitting spline Tol = []; # Tolerence for convergence endproperties methods (Access = public) ## Class object contructor function this = RegressionGAM (X, Y, varargin) ## Check for sufficient number of input arguments if (nargin < 2) error ("RegressionGAM: too few input arguments."); endif ## Get training sample size and number of variables in training data nsample = rows (X); ndims_X = columns (X); ## Check correspodence between predictors and response if (nsample != rows (Y)) error ("RegressionGAM: number of rows in X and Y must be equal."); endif ## Set default values before parsing optional parameters PredictorNames = {}; # Predictor variable names ResponseName = []; # Response variable name Formula = []; # Formula for GAM model Interactions = []; # Interaction terms DoF = ones (1, ndims_X) * 8; # Degrees of freedom Order = ones (1, ndims_X) * 3; # Order of spline Knots = ones (1, ndims_X) * 5; # Knots Tol = 1e-3; # Tolerence for convergence ## Number of parameters for Knots, DoF, Order (maximum 2 allowed) KOD = 0; ## Number of parameters for Formula, Ineractions (maximum 1 allowed) F_I = 0; ## Parse extra parameters while (numel (varargin) > 0) switch (tolower (varargin {1})) case "predictors" PredictorNames = varargin{2}; if (! isempty (PredictorNames)) if (! iscellstr (PredictorNames)) error (strcat (["RegressionGAM: PredictorNames must"], ... [" be a cellstring array."])); elseif (columns (PredictorNames) != columns (X)) error (strcat (["RegressionGAM: PredictorNames must"], ... [" have same number of columns as X."])); endif endif case "responsename" ResponseName = varargin{2}; if (! ischar (ResponseName)) error ("RegressionGAM: ResponseName must be a char string."); endif case "formula" if (F_I < 1) Formula = varargin{2}; if (! ischar (Formula) && ! islogical (Formula)) error ("RegressionGAM: Formula must be a string."); endif F_I += 1; else error ("RegressionGAM: Interactions have been already defined."); endif case "interactions" if (F_I < 1) tmp = varargin{2}; if (isnumeric (tmp) && isscalar (tmp) && tmp == fix (tmp) && tmp >= 0) Interactions = tmp; elseif (islogical (tmp)) Interactions = tmp; elseif (ischar (tmp) && strcmpi (tmp, "all")) Interactions = tmp; else error ("RegressionGAM: invalid Interactions parameter."); endif F_I += 1; else error ("RegressionGAM: Formula has been already defined."); endif case "knots" if (KOD < 2) Knots = varargin{2}; if (! isnumeric (Knots) || ! (isscalar (Knots) || isequal (size (Knots), [1, ndims_X]))) error ("RegressionGAM: invalid value for Knots."); endif DoF = Knots + Order; Order = DoF - Knots; KOD += 1; else error ("RegressionGAM: DoF and Order have been set already."); endif case "order" if (KOD < 2) Order = varargin{2}; if (! isnumeric (Order) || ! (isscalar (Order) || isequal (size (Order), [1, ndims_X]))) error ("RegressionGAM: invalid value for Order."); endif DoF = Knots + Order; Knots = DoF - Order; KOD += 1; else error ("RegressionGAM: DoF and Knots have been set already."); endif case "dof" if (KOD < 2) DoF = varargin{2}; if (! isnumeric (DoF) || ! (isscalar (DoF) || isequal (size (DoF), [1, ndims_X]))) error ("RegressionGAM: invalid value for DoF."); endif Knots = DoF - Order; Order = DoF - Knots; KOD += 1; else error ("RegressionGAM: Knots and Order have been set already."); endif case "tol" Tol = varargin{2}; if (! (isnumeric (Tol) && isscalar (Tol) && (Tol > 0))) error ("RegressionGAM: Tolerance must be a Positive scalar."); endif otherwise error (strcat (["RegressionGAM: invalid parameter name"],... [" in optional pair arguments."])); endswitch varargin (1:2) = []; endwhile ## Assign original X and Y data to the RegressionGAM object this.X = X; this.Y = Y; ## Remove nans from X and Y RowsUsed = ! logical (sum (isnan ([Y, X]), 2)); Y = Y (RowsUsed); X = X (RowsUsed, :); ## Check X and Y contain valid data if (! isnumeric (X) || ! isfinite (X)) error ("RegressionGAM: invalid values in X."); endif if (! isnumeric (Y) || ! isfinite (Y)) error ("RegressionGAM: invalid values in Y."); endif ## Assign the number of observations and their correspoding indices ## on the original data, which will be used for training the model, ## to the RegressionGAM object this.NumObservations = rows (X); this.RowsUsed = cast (RowsUsed, "double"); ## Assign the number of original predictors to the RegressionGAM object this.NumPredictors = ndims_X; ## Generate default predictors and response variabe names (if necessary) if (isempty (PredictorNames)) for i = 1:ndims_X PredictorNames {i} = strcat ("x", num2str (i)); endfor endif if (isempty (ResponseName)) ResponseName = "Y"; endif ## Assign predictors and response variable names this.PredictorNames = PredictorNames; this.ResponseName = ResponseName; ## Assign remaining optional parameters this.Formula = Formula; this.Interactions = Interactions; this.Knots = Knots; this.Order = Order; this.DoF = DoF; this.Tol = Tol; ## Fit the basic model Inter = mean (Y); [iter, param, res, RSS] = this.fitGAM (X, Y, Inter, Knots, Order); this.BaseModel.Intercept = Inter; this.BaseModel.Parameters = param; this.BaseModel.Iterations = iter; this.BaseModel.Residuals = res; this.BaseModel.RSS = RSS; ## Handle interaction terms (if given) if (F_I > 0) if (isempty (this.Formula)) ## Analyze Interactions optional parameter this.IntMatrix = this.parseInteractions (); ## Append interaction terms to the predictor matrix for i = 1:rows (this.IntMatrix) tindex = logical (this.IntMatrix(i,:)); Xterms = X(:,tindex); Xinter = ones (this.NumObservations, 1); for c = 1:sum (tindex) Xinter = Xinter .* Xterms(:,c); endfor ## Append interaction terms X = [X, Xinter]; endfor else ## Analyze Formula optional parameter this.IntMatrix = this.parseFormula (); ## Add selected predictors and interaction terms XN = []; for i = 1:rows (this.IntMatrix) tindex = logical (this.IntMatrix(i,:)); Xterms = X(:,tindex); Xinter = ones (this.NumObservations, 1); for c = 1:sum (tindex) Xinter = Xinter .* Xterms(:,c); endfor ## Append selected predictors and interaction terms XN = [XN, Xinter]; endfor X = XN; endif ## Update length of Knots, Order, and DoF vectors to match ## the columns of X with the interaction terms Knots = ones (1, columns (X)) * Knots(1); # Knots Order = ones (1, columns (X)) * Order(1); # Order of spline DoF = ones (1, columns (X)) * DoF(1); # Degrees of freedom ## Fit the model with interactions [iter, param, res, RSS] = this.fitGAM (X, Y, Inter, Knots, Order); this.ModelwInt.Intercept = Inter; this.ModelwInt.Parameters = param; this.ModelwInt.Iterations = iter; this.ModelwInt.Residuals = res; this.ModelwInt.RSS = RSS; endif endfunction ## -*- texinfo -*- ## @deftypefn {RegressionGAM} {@var{yFit} =} predict (@var{obj}, @var{Xfit}) ## @deftypefnx {RegressionGAM} {@var{yFit} =} predict (@dots{}, @var{Name}, @var{Value}) ## @deftypefnx {RegressionGAM} {[@var{yFit}, @var{ySD}, @var{yInt}] =} predict (@dots{}) ## ## Predict new data points using generalized additive model regression object. ## ## @code{@var{yFit} = predict (@var{obj}, @var{Xfit}} returns a vector of ## predicted responses, @var{yFit}, for the predictor data in matrix @var{Xfit} ## based on the Generalized Additive Model in @var{obj}. @var{Xfit} must have ## the same number of features/variables as the training data in @var{obj}. ## ## @itemize ## @item ## @var{obj} must be a @qcode{RegressionGAM} class object. ## @end itemize ## ## @code{[@var{yFit}, @var{ySD}, @var{yInt}] = predict (@var{obj}, @var{Xfit}} ## also returns the standard deviations, @var{ySD}, and prediction intervals, ## @var{yInt}, of the response variable @var{yFit}, evaluated at each ## observation in the predictor data @var{Xfit}. ## ## @code{@var{yFit} = predict (@dots{}, @var{Name}, @var{Value})} returns the ## aforementioned results with additional properties specified by ## @qcode{Name-Value} pair arguments listed below. ## ## @multitable @columnfractions 0.28 0.02 0.7 ## @headitem @var{Name} @tab @tab @var{Value} ## ## @item @qcode{"alpha"} @tab @tab significance level of the prediction ## intervals @var{yInt}, specified as scalar in range @qcode{[0,1]}. The default ## value is 0.05, which corresponds to 95% prediction intervals. ## ## @item @qcode{"includeinteractions"} @tab @tab a boolean flag to include ## interactions to predict new values based on @var{Xfit}. By default, ## @qcode{"includeinteractions"} is @qcode{true} when the GAM model in @var{obj} ## contains a @qcode{obj.Formula} or @qcode{obj.Interactions} fields. Otherwise, ## is set to @qcode{false}. If set to @qcode{true} when no interactions are ## present in the trained model, it will result to an error. If set to ## @qcode{false} when using a model that includes interactions, the predictions ## will be made on the basic model without any interaction terms. This way you ## can make predictions from the same GAM model without having to retrain it. ## @end multitable ## ## @seealso{fitrgam, RegressionGAM} ## @end deftypefn function [yFit, ySD, yInt] = predict (this, Xfit, varargin) ## Check for sufficient input arguments if (nargin < 2) error ("RegressionGAM.predict: too few arguments."); endif ## Check for valid XC if (isempty (Xfit)) error ("RegressionGAM.predict: Xfit is empty."); elseif (columns (this.X) != columns (Xfit)) error (strcat (["@RegressionGAM/predict: Xfit must have the same"], ... [" number of features (columns) as in the GAM model."])); endif ## Clean Xfit data notnansf = ! logical (sum (isnan (Xfit), 2)); Xfit = Xfit (notnansf, :); ## Default values for Name-Value Pairs alpha = 0.05; if (isempty (this.IntMatrix)) incInt = false; else incInt = true; endif ## Parse optional arguments while (numel (varargin) > 0) switch (tolower (varargin {1})) case "includeinteractions" tmpInt = varargin{2}; if (! islogical (tmpInt) || (tmpInt != 0 && tmpInt != 1)) error (strcat (["RegressionGAM.predict: includeinteractions"], ... [" must be a logical value."])); endif ## Check model for interactions if (tmpInt && isempty (this.IntMat)) error (strcat (["RegressionGAM.predict: trained model"], ... [" does not include any interactions."])); endif incInt = tmpInt; case "alpha" alpha = varargin{2}; if (! (isnumeric (alpha) && isscalar (alpha) && alpha > 0 && alpha < 1)) error (strcat (["RegressionGAM.predict: alpha must be a"], ... [" scalar value between 0 and 1."])); endif otherwise error (strcat(["RegressionGAM.predict: invalid NAME in"], ... [" optional pairs of arguments."])); endswitch varargin (1:2) = []; endwhile ## Choose whether interactions must be included if (incInt) if (! isempty (this.Interactions)) ## Append interaction terms to the predictor matrix for i = 1:rows (this.IntMat) tindex = logical (this.IntMat(i,:)); Xterms = Xfit(:,tindex); Xinter = ones (rows (Xfit), 1); for c = 1:sum (tindex) Xinter = Xinter .* Xterms(:,c); endfor ## Append interaction terms Xfit = [Xfit, Xinter]; endfor else ## Add selected predictors and interaction terms XN = []; for i = 1:rows (this.IntMat) tindex = logical (this.IntMat(i,:)); Xterms = Xfit(:,tindex); Xinter = ones (rows (Xfit), 1); for c = 1:sum (tindex) Xinter = Xinter .* Xterms(:,c); endfor ## Append selected predictors and interaction terms XN = [XN, Xinter]; endfor Xfit = XN; endif ## Get parameters and intercept vectors from model with interactions params = this.ModelwInt.Parameters; Interc = this.ModelwInt.Intercept; ## Update length of DoF vector DoF = ones (1, columns (Xfit)) * this.DoF(1); else ## Get parameters and intercept vectors from base model params = this.BaseModel.Parameters; Interc = this.BaseModel.Intercept; ## Get DoF from model DoF = this.DoF; endif ## Predict values from testing data yFit = predict_val (params, Xfit, Interc); ## Predict Standard Deviation and Intervals of estimated data if requested if (nargout > 0) ## Ensure that RowsUsed in the model are selected Y = this.Y(logical (this.RowsUsed)); X = this.X(logical (this.RowsUsed), :); ## Predict response from training predictor data with the trained model yrs = predict_val (params, X , Interc); ## Get the residuals between predicted and actual response data rs = Y - yrs; var_rs = var (rs); var_pr = var (yFit); # var is calculated here instead take sqrt(SD) t_mul = tinv (1 - alpha / 2, this.DoF); ydev = (yFit - mean (yFit)) .^ 2; ySD = sqrt (ydev / (rows (yFit) - 1)); varargout{1} = ySD; if (nargout > 1) moe = t_mul (1) * ySD; lower = (yFit - moe); upper = (yFit + moe); yInt = [lower, upper]; varargout{2} = yInt; endif endif endfunction endmethods ## Helper functions methods (Access = private) ## Determine interactions from Interactions optional parameter function intMat = parseInteractions (this) if (islogical (this.Interactions)) ## Check that interaction matrix corresponds to predictors if (numel (this.PredictorNames) != columns (this.Interactions)) error (strcat (["RegressionGAM: columns in Interactions logical"], ... [" matrix must equal to the number of predictors."])); endif intMat = this.Interactions elseif (isnumeric (this.Interactions)) ## Need to measure the effect of all interactions to keep the best ## performing. Just check that the given number is not higher than ## p*(p-1)/2, where p is the number of predictors. p = this.NumPredictors; if (this.Interactions > p * (p - 1) / 2) error (strcat (["RegressionGAM: number of interaction terms"], ... [" requested is larger than all possible"], ... [" combinations of predictors in X."])); endif ## Get all combinations except all zeros allMat = flip (fullfact(p)([2:end],:), 2); ## Only keep interaction terms iterms = find (sum (allMat, 2) != 1); intMat = allMat(iterms); elseif (strcmpi (this.Interactions, "all")) ## Calculate all p*(p-1)/2 interaction terms allMat = flip (fullfact(p)([2:end],:), 2); ## Only keep interaction terms iterms = find (sum (allMat, 2) != 1); intMat = allMat(iterms); endif endfunction ## Determine interactions from formula function intMat = parseFormula (this) intMat = []; ## Check formula for syntax if (isempty (strfind (this.Formula, '~'))) error ("RegressionGAM: invalid syntax in Formula."); endif ## Split formula and keep predictor terms formulaParts = strsplit (this.Formula, '~'); ## Check there is some string after '~' if (numel (formulaParts) < 2) error ("RegressionGAM: no predictor terms in Formula."); endif predictorString = strtrim (formulaParts{2}); if (isempty (predictorString)) error ("RegressionGAM: no predictor terms in Formula."); endif ## Spit additive terms (between + sign) aterms = strtrim (strsplit (predictorString, '+')); ## Process all terms for i = 1:numel (aterms) ## Find individual terms (string missing ':') if (isempty (strfind (aterms(i), ':'){:})) ## Search PredictorNames to associate with column in X sterms = strcmp (this.PredictorNames, aterms(i)); ## Append to interactions matrix intMat = [intMat; sterms]; else ## Split interaction terms (string contains ':') mterms = strsplit (aterms{i}, ':'); ## Add each individual predictor to interaction term vector iterms = logical (zeros (1, this.NumPredictors)); for t = 1:numel (mterms) iterms = iterms | strcmp (this.PredictorNames, mterms(t)); endfor ## Check that all predictors have been identified if (sum (iterms) != t) error ("RegressionGAM: some predictors have not been identified."); endif ## Append to interactions matrix intMat = [intMat; iterms]; endif endfor ## Check that all terms have been identified if (! all (sum (intMat, 2) > 0)) error ("RegressionGAM: some terms have not been identified."); endif endfunction ## Fit the model function [iter, param, res, RSS] = fitGAM (this, X, Y, Inter, Knots, Order) ## Initialize variables converged = false; iter = 0; RSS = zeros (1, columns (X)); res = Y - Inter; ns = rows (X); Tol = this.Tol; ## Start training while (! converged) iter += 1; ## Single cycle of backfit for j = 1:columns (X) ## Calculate residuals to fit spline if (iter > 1) res = res + ppval (param(j), X(:, j)); endif ## Fit an spline to the data gk = splinefit (X(:,j), res, Knots(j), "order", Order(j)); ## This might be wrong! We need to check this out RSSk(j) = abs (sum (abs (Y - ppval (gk, X(:,j)) - Inter)) .^ 2) / ns; param(j) = gk; res = res - ppval (param(j), X(:,j)); endfor ## Check if RSS is less than the tolerence if (all (abs (RSS - RSSk) <= Tol)) converged = true; endif ## Update RSS RSS = RSSk; endwhile endfunction endmethods endclassdef ## Helper function for making prediction of new data based on GAM model function ypred = predict_val (params, X, intercept) [nsample, ndims_X] = size (X); ypred = ones (nsample, 1) * intercept; ## Add the remaining terms for j = 1:ndims_X ypred = ypred + ppval (params(j), X (:,j)); endfor endfunction %!demo %! ## Train a RegressionGAM Model for synthetic values %! f1 = @(x) cos (3 * x); %! f2 = @(x) x .^ 3; %! x1 = 2 * rand (50, 1) - 1; %! x2 = 2 * rand (50, 1) - 1; %! y = f1(x1) + f2(x2); %! y = y + y .* 0.2 .* rand (50,1); %! X = [x1, x2]; %! a = fitrgam (X, y, "tol", 1e-3) %!demo %! ## Declare two different functions %! f1 = @(x) cos (3 * x); %! f2 = @(x) x .^ 3; %! %! ## Generate 80 samples for f1 and f2 %! x = [-4*pi:0.1*pi:4*pi-0.1*pi]'; %! X1 = f1 (x); %! X2 = f2 (x); %! %! ## Create a synthetic response by adding noise %! rand ("seed", 3); %! Ytrue = X1 + X2; %! Y = Ytrue + Ytrue .* 0.2 .* rand (80,1); %! %! ## Assemble predictor data %! X = [X1, X2]; %! %! ## Train the GAM and test on the same data %! a = fitrgam (X, Y, "order", [5, 5]); %! [ypred, ySDsd, yInt] = predict (a, X); %! %! ## Plot the results %! figure %! [sortedY, indY] = sort (Ytrue); %! plot (sortedY, "r-"); %! xlim ([0, 80]); %! hold on %! plot (ypred(indY), "g+") %! plot (yInt(indY,1), "k:") %! plot (yInt(indY,2), "k:") %! xlabel ("Predictor samples"); %! ylabel ("Response"); %! title ("actual vs predicted values for function f1(x) = cos (3x) "); %! legend ({"Theoretical Response", "Predicted Response", "Prediction Intervals"}); %! %! ## Use 30% Holdout partitioning for training and testing data %! C = cvpartition (80, "HoldOut", 0.3); %! [ypred, ySDsd, yInt] = predict (a, X(test(C),:)); %! %! ## Plot the results %! figure %! [sortedY, indY] = sort (Ytrue(test(C))); %! plot (sortedY, 'r-'); %! xlim ([0, sum(test(C))]); %! hold on %! plot (ypred(indY), "g+") %! plot (yInt(indY,1),'k:') %! plot (yInt(indY,2),'k:') %! xlabel ("Predictor samples"); %! ylabel ("Response"); %! title ("actual vs predicted values for function f1(x) = cos (3x) "); %! legend ({"Theoretical Response", "Predicted Response", "Prediction Intervals"}); ## Test constructor %!test %! x = [1, 2, 3; 4, 5, 6; 7, 8, 9; 3, 2, 1]; %! y = [1; 2; 3; 4]; %! a = RegressionGAM (x, y); %! assert ({a.X, a.Y}, {x, y}) %! assert ({a.BaseModel.Intercept}, {2.5000}) %! assert ({a.Knots, a.Order, a.DoF}, {[5, 5, 5], [3, 3, 3], [8, 8, 8]}) %! assert ({a.NumObservations, a.NumPredictors}, {4, 3}) %! assert ({a.ResponseName, a.PredictorNames}, {"Y", {"x1", "x2", "x3"}}) %! assert ({a.Formula}, {[]}) %!test %! x = [1, 2, 3, 4; 4, 5, 6, 7; 7, 8, 9, 1; 3, 2, 1, 2]; %! y = [1; 2; 3; 4]; %! pnames = {"A", "B", "C", "D"}; %! formula = "Y ~ A + B + C + D + A:C"; %! intMat = logical ([1,0,0,0;0,1,0,0;0,0,1,0;0,0,0,1;1,0,1,0]); %! a = RegressionGAM (x, y, "predictors", pnames, "formula", formula); %! assert ({a.IntMatrix}, {intMat}) %! assert ({a.ResponseName, a.PredictorNames}, {"Y", pnames}) %! assert ({a.Formula}, {formula}) ## Test input validation for constructor %!error RegressionGAM () %!error RegressionGAM (ones(10,2)) %!error ... %! RegressionGAM (ones(10,2), ones (5,1)) %!error ... %! RegressionGAM ([1;2;3;"a";4], ones (5,1)) %!error ... %! RegressionGAM (ones(10,2), ones (10,1), "some", "some") %!error %! RegressionGAM (ones(10,2), ones (10,1), "formula", {"y~x1+x2"}) %!error %! RegressionGAM (ones(10,2), ones (10,1), "formula", [0, 1, 0]) %!error ... %! RegressionGAM (ones(10,2), ones (10,1), "formula", "something") %!error ... %! RegressionGAM (ones(10,2), ones (10,1), "formula", "something~") %!error ... %! RegressionGAM (ones(10,2), ones (10,1), "formula", "something~") %!error ... %! RegressionGAM (ones(10,2), ones (10,1), "formula", "something~x1:") %!error ... %! RegressionGAM (ones(10,2), ones (10,1), "interactions", "some") %!error ... %! RegressionGAM (ones(10,2), ones (10,1), "interactions", -1) %!error ... %! RegressionGAM (ones(10,2), ones (10,1), "interactions", [1 2 3 4]) %!error ... %! RegressionGAM (ones(10,2), ones (10,1), "interactions", 3) %!error ... %! RegressionGAM (ones(10,2), ones (10,1), "formula", "y ~ x1 + x2", "interactions", 1) %!error ... %! RegressionGAM (ones(10,2), ones (10,1), "interactions", 1, "formula", "y ~ x1 + x2") %!error ... %! RegressionGAM (ones(10,2), ones (10,1), "knots", "a") %!error ... %! RegressionGAM (ones(10,2), ones (10,1), "order", 3, "dof", 2, "knots", 5) %!error ... %! RegressionGAM (ones(10,2), ones (10,1), "dof", 'a') %!error ... %! RegressionGAM (ones(10,2), ones (10,1), "knots", 5, "order", 3, "dof", 2) %!error ... %! RegressionGAM (ones(10,2), ones (10,1), "order", 'a') %!error ... %! RegressionGAM (ones(10,2), ones (10,1), "knots", 5, "dof", 2, "order", 2) %!error ... %! RegressionGAM (ones(10,2), ones (10,1), "tol", -1) %!error ... %! RegressionGAM (ones(10,2), ones (10,1), "responsename", -1) %!error ... %! RegressionGAM (ones(10,2), ones (10,1), "predictors", -1) %!error ... %! RegressionGAM (ones(10,2), ones (10,1), "predictors", ['a','b','c']) %!error ... %! RegressionGAM (ones(10,2), ones (10,1), "predictors", {'a','b','c'}) ## Test input validation for predict method %!error ... %! predict (RegressionGAM (ones(10,1), ones(10,1))) %!error ... %! predict (RegressionGAM (ones(10,1), ones(10,1)), []) %!error ... %! predict (RegressionGAM(ones(10,2), ones(10,1)), 2) %!error ... %! predict (RegressionGAM(ones(10,2), ones(10,1)), ones (10,2), "some", "some") %!error ... %! predict (RegressionGAM(ones(10,2), ones(10,1)), ones (10,2), "includeinteractions", "some") %!error ... %! predict (RegressionGAM(ones(10,2), ones(10,1)), ones (10,2), "includeinteractions", 5) %!error ... %! predict (RegressionGAM(ones(10,2), ones(10,1)), ones (10,2), "alpha", 5) %!error ... %! predict (RegressionGAM(ones(10,2), ones(10,1)), ones (10,2), "alpha", -1) %!error ... %! predict (RegressionGAM(ones(10,2), ones(10,1)), ones (10,2), "alpha", 'a') statistics-release-1.6.3/inst/adtest.m000066400000000000000000001007271456127120000177550ustar00rootroot00000000000000## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{h} =} adtest (@var{x}) ## @deftypefnx {statistics} {@var{h} =} adtest (@var{x}, @var{Name}, @var{Value}) ## @deftypefnx {statistics} {[@var{h}, @var{pval}] =} adtest (@dots{}) ## @deftypefnx {statistics} {[@var{h}, @var{pval}, @var{adstat}, @var{cv}] =} adtest (@dots{}) ## ## Anderson-Darling goodness-of-fit hypothesis test. ## ## @code{@var{h} = adtest (@var{x})} returns a test decision for the null ## hypothesis that the data in vector @var{x} is from a population with a normal ## distribution, using the Anderson-Darling test. The alternative hypothesis is ## that x is not from a population with a normal distribution. The result ## @var{h} is 1 if the test rejects the null hypothesis at the 5% significance ## level, or 0 otherwise. ## ## @code{@var{h} = adtest (@var{x}, @var{Name}, @var{Value})} returns a test ## decision for the Anderson-Darling test with additional options specified by ## one or more Name-Value pair arguments. For example, you can specify a null ## distribution other than normal, or select an alternative method for ## calculating the p-value, such as a Monte Carlo simulation. ## ## The following parameters can be parsed as Name-Value pair arguments. ## ## @multitable @columnfractions 0.25 0.75 ## @headitem Name @tab Description ## @item "Distribution" @tab The distribution being tested for. It tests ## whether @var{x} could have come from the specified distribution. There are ## two choise available for parsing distribution parameters: ## @end multitable ## ## @itemize ## @item ## One of the following char strings: "norm", "exp", "ev", "logn", "weibull", ## for defining either the 'normal', 'exponential', 'extreme value', lognormal, ## or 'Weibull' distribution family, accordingly. In this case, @var{x} is ## tested against a composite hypothesis for the specified distribution family ## and the required distribution parameters are estimated from the data in ## @var{x}. The default is "norm". ## ## @item ## A cell array defining a distribution in which the first cell contains a char ## string with the distribution name, as mentioned above, and the consecutive ## cells containing all specified parameters of the null distribution. In this ## case, @var{x} is tested against a simple hypothesis. ## @end itemize ## ## @multitable @columnfractions 0.25 0.75 ## @item "Alpha" @tab Significance level alpha for the test. Any scalar ## numeric value between 0 and 1. The default is 0.05 corresponding to the 5% ## significance level. ## ## @item "MCTol" @tab Monte-Carlo standard error for the p-value, ## @var{pval}, value. which must be a positive scalar value. In this case, an ## approximation for the p-value is computed directly, using Monte-Carlo ## simulations. ## ## @item "Asymptotic" @tab Method for calculating the p-value of the ## Anderson-Darling test, which can be either true or false logical value. If ## you specify 'true', adtest estimates the p-value using the limiting ## distribution of the Anderson-Darling test statistic. If you specify 'false', ## adtest calculates the p-value based on an analytical formula. For sample ## sizes greater than 120, the limiting distribution estimate is likely to be ## more accurate than the small sample size approximation method. ## @end multitable ## ## @itemize ## @item ## If you specify a distribution family with unknown parameters for the ## distribution Name-Value pair (i.e. composite distribution hypothesis test), ## the "Asymptotic" option must be false. ## @item ## ## If you use MCTol to calculate the p-value using a Monte Carlo simulation, ## the "Asymptotic" option must be false. ## @end itemize ## ## @code{[@var{h}, @var{pval}] = adtest (@dots{})} also returns the p-value, ## @var{pval}, of the Anderson-Darling test, using any of the input arguments ## from the previous syntaxes. ## ## @code{[@var{h}, @var{pval}, @var{adstat}, @var{cv}] = adtest (@dots{})} also ## returns the test statistic, @var{adstat}, and the critical value, @var{cv}, ## for the Anderson-Darling test. ## ## The Anderson-Darling test statistic belongs to the family of Quadratic ## Empirical Distribution Function statistics, which are based on the weighted ## sum of the difference @math{[Fn(x)-F(x)]^2} over the ordered sample values ## @math{X1 < X2 < ... < Xn}, where @math{F} is the hypothesized continuous ## distribution and @math{Fn} is the empirical CDF based on the data sample with ## @math{n} sample points. ## ## @seealso{kstest} ## @end deftypefn function [H, pVal, ADStat, CV] = adtest (x, varargin) ## Check for valid input data if (nargin < 1) print_usage; endif ## Ensure the sample data is a real vector. if (! isvector (x) || ! isreal(x)) error ("adtest: X must be a vector of real numbers."); endif ## Add defaults distribution = "norm"; isCompositeTest = true; alpha = 0.05; MCTol = []; asymptotic = false; ## Parse arguments and check if parameter/value pairs are valid i = 1; while (i <= length (varargin)) switch lower (varargin{i}) case "distribution" i = i + 1; distribution = varargin{i}; ## Check for char string or cell array ## If distribution is a char string, then X is tested against a ## composite hypothesis for the specified distribution family and ## distribution parameters are estimated from the sample data. valid_dists = {"norm", "exp", "ev", "logn", "weibull"}; if (ischar (distribution)) if (! any (strcmpi (distribution, valid_dists))) error ("adtest: invalid distribution family in char string."); endif ## If distribution is a cell array, then X is tested against a ## simple hypothesis for the specified distribution and ## distribution parameters given in the cell array. elseif (iscell (distribution)) if (! any (strcmpi (distribution(1), valid_dists))) error ("adtest: invalid distribution family in cell array."); endif ## Check for valid distribution parameter in cell array err_msg = "adtest: invalid distribution parameters in cell array."; if (strcmpi (distribution(1), "norm")) if (numel (distribution) != 3) error (err_msg); endif z = normcdf (x, distribution{2}, distribution{3}); elseif (strcmpi (distribution(1), "exp")) if (numel (distribution) != 2) error (err_msg); endif z = expcdf (x, distribution{2}); elseif (strcmpi (distribution(1), "ev")) if (numel (distribution) != 3) error (err_msg); endif z = evcdf (x, distribution{2}, distribution{3}); elseif (strcmpi (distribution(1), "logn")) if (numel (distribution) != 3) error (err_msg); endif z = logncdf (x, distribution{2}, distribution{3}); elseif (strcmpi (distribution(1), "weibull")) if (numel (distribution) != 3) error (err_msg); endif z = wblcdf (x, distribution{2}, distribution{3}); endif isCompositeTest = false; else error ("adtest: invalid distribution option."); endif case "alpha" i = i + 1; alpha = varargin{i}; ## Check for valid alpha if (! isscalar (alpha) || ! isnumeric (alpha) || ... alpha <= 0 || alpha >= 1) error ("adtest: invalid value for alpha."); endif case "mctol" i = i + 1; MCTol = varargin{i}; ## Check Monte Carlo tolerance is a numeric scalar greater than 0 if (! isempty (MCTol) && (! isscalar (MCTol) || MCTol <= 0)) error ("adtest: invalid Monte Carlo Tolerance."); endif case "asymptotic" i = i + 1; asymptotic = varargin{i}; ## Check that it is either true or false if (! isbool (asymptotic)) error ("adtest: asymptotic option must be boolean."); endif otherwise error ("adtest: invalid Name argument."); endswitch i = i + 1; endwhile ## Check conflicts with asymptotic option if (asymptotic && isCompositeTest) error (strcat (["adtest: asymptotic option is not valid"], ... [" for the composite distribution test."])); elseif (asymptotic && ! isempty (MCTol)) error (strcat (["adtest: asymptotic option is not valid"], ... [" for the Monte Carlo simulation test."])); endif ## Remove missing values. x = x(! isnan (x)); ## Compute sample size n. n = length (x); ## For composite tests if (isCompositeTest) ## Abort for sample size less than 4 if (n < 4) error ("adtest: not enough data for composite testing."); endif ## If data follow a lognormal distribution, log(x) is normally distributed if (strcmpi (distribution, "logn")) x = log (x); distribution = "norm"; ## If data follow a Weibull distribution, log(x) has a type I extreme-value ## distribution elseif (strcmpi (distribution, "weibull")) x = log (x); distribution = "ev"; endif ## Compute ADStat switch distribution case "norm" ## Check for complex numbers due to log (x) if (any (! isreal (x))) ## Data is not compatible with logn distribution test warning ("adtest: bad data for lognormal distribution."); ADStat = NaN; else z = normcdf (x, mean (x), std (x)); ADStat = ComputeADStat (z, n); endif case "exp" z = expcdf (x, mean (x)); ADStat = ComputeADStat (z, n); case "ev" ## Check for complex numbers due to log (x) if (any (! isreal (x))) ## Data is not compatible with Weibull distribution test warning ("adtest: bad data for Weibull distribution."); ADStat = NaN; else params = evfit (x); z = evcdf (x, params (1), params (2)); ADStat = ComputeADStat (z, n); endif endswitch ## Compute p-value and critical values without Monte Carlo simulation if (isempty (MCTol)) alphas = [0.0005, 0.0010, 0.0015, 0.0020, 0.0050, 0.0100, 0.0250, ... 0.0500, 0.1000, 0.1500, 0.2000, 0.2500, 0.3000, 0.3500, ... 0.4000, 0.4500, 0.5000, 0.5500, 0.6000, 0.6500, 0.7000, ... 0.7500, 0.8000, 0.8500, 0.9000, 0.9500, 0.9900]; switch distribution case "norm" CVs = computeCriticalValues_norm (n); case "exp" CVs = computeCriticalValues_exp (n); case "ev" CVs = computeCriticalValues_ev (n); endswitch ## 1-D interpolation into the tabulated results pp = pchip (log (alphas), CVs); CV = ppval (pp, log (alpha)); ## If alpha is not within the lookup table, throw a warning ## Hypothesis result is computed by comparing the p-value with ## alpha, rather than CV with ADStat if alpha < alphas(1) CV = CVs(1); warning ("adtest: alpha not within the lookup table."); elseif alpha > alphas(end) CV = CVs(end); warning ("adtest: alpha not within the lookup table."); endif if (ADStat > CVs(1)) ## P value is smaller than smallest tabulated value warning (strcat (["adtest: out of range min p-value:"], ... sprintf (" %g", alphas(1)))); pVal = alphas(1); elseif (ADStat < CVs(end)) ## P value is larger than largest tabulated value warning (strcat (["adtest: out of range max p-value:"], ... sprintf (" %g", alphas(end)))); pVal = alphas(end); elseif (isnan (ADStat)) ## Handle certain cases of negative data (ADStat == NaN) pVal = 0; else ## Find p-value by inverse interpolation i = find (ADStat > CVs, 1, "first"); logPVal = fzero (@(x)ppval(pp,x) - ADStat, log(alphas([i-1,i]))); pVal = exp (logPVal); endif ## Compute p-value and critical values without Monte Carlo simulation else [CV, pVal] = adtestMC (ADStat, n, alpha, distribution, mctol); endif ## Calculate H if (isnan (ADStat)) H = true; else if (isempty (MCTol)) if (alpha < alphas(1) || alpha > alphas(end)) H = (pVal < alpha); else H = (ADStat > CV); endif else H = (ADStat > CV); endif endif ## For simple tests else ## Compute the Anderson-Darling statistic ADStat = ComputeADStat (z, n); ## Compute p-value and critical values without Monte Carlo simulation if (isempty (MCTol)) alphas = [0.0005, 0.0010, 0.0015, 0.0020, 0.0050, 0.0100, 0.0250, ... 0.0500, 0.1000, 0.1500, 0.2000, 0.2500, 0.3000, 0.3500, ... 0.4000, 0.4500, 0.5000, 0.5500, 0.6000, 0.6500, 0.7000, ... 0.7500, 0.8000, 0.8500, 0.9000, 0.9500, 0.9900]; if (asymptotic) if (n <= 120) warning ("adtest: asymptotic distribution with small sample size."); endif pVal = 1 - ADInf (ADStat); ## For extra output arguments if (nargout > 3) ## Make sure alpha is within the lookup table validateAlpha (alpha, alphas); ## Find critical values critVals = findAsymptoticDistributionCriticalValues; i = find (alphas > alpha, 1, "first"); startVal = critVals(i-1); CV = fzero(@(ad)1-ADInf(ad)-alpha, startVal); endif else if (n == 1) pVal = 1 - sqrt (1 - 4 * exp (-1 - ADStat)); else if (n < 4) warning ("adtest: small sample size."); endif pVal = 1 - ADn (n, ADStat); endif ## For extra output arguments if (nargout > 3) ## Make sure alpha is within the lookup table validateAlpha (alpha, alphas); ## Find critical values [CVs, sampleSizes] = findCriticalValues; [OneOverSampleSizes, LogAlphas] = meshgrid (1 ./ sampleSizes, ... log (alphas)); CV = interp2 (OneOverSampleSizes, LogAlphas, CVs', 1./n, log (alpha)); endif endif ## Compute p-value and critical values with Monte Carlo simulation else [CV, pVal] = adtestMC (ADStat, n, alpha, "unif", mctol); endif ## Calculate H H = (pVal < alpha); endif endfunction ## Compute Anderson-Darling Statistic function ADStat = ComputeADStat (z, n) ## Sort the data and compute the statistic z = reshape (z, n, 1); z = sort (z); w = 2 * (1:n) - 1; ADStat = - w * (log (z)+ log (1-z(end:-1:1))) / n - n; endfunction ## Anderson-Darling distribution Pr(An= 2); x(ad >= 2) = exp (-exp (1.0776 - (2.30695 - (0.43424 - (0.082433 - ... (0.008056 - 0.0003146 .* adh) .* adh) .* adh) .* adh) .* adh)); ## Compute error function defined between 0 and 1 if (any (x < 0 | x > 1)) error ("adtest: invalid values for error function."); endif e = zeros (size (x)); c = 0.01265 + 0.1757/n; ## Define function by intervals using 3 fixed functions g1, g2, g3. xc1 = x(x < c) / c; g1 = sqrt (xc1) .* (1 - xc1) .* (49 * xc1 - 102); e(x < c) = (0.0037 / n ^ 3 + 0.00078 / n ^ 2 + 0.00006 / n) * g1; xc2 = (x(x >= c & x < 0.8) - c) ./ (0.8 - c); g2 = -0.00022633 + (6.54034 - (14.6538 - (14.458 - (8.259 -... 1.91864 .* xc2) .* xc2) .* xc2) .* xc2) .* xc2; e(x >= c & x < 0.8) = (0.04213 / n + 0.01365 / n ^ 2) * g2; xc3 = x(x >= 0.8); e(x >= 0.8) = 1 / n * (-130.2137 + (745.2337 - (1705.091 - (1950.646 -... (1116.360 - 255.7844 .* ... xc3) .* xc3) .* xc3) .* xc3) .* xc3) .* xc3; p = x + e; endfunction ## Evaluate the Anderson-Darling limit distribution function ad = ADInf (z) ## Distribution is invalid for negative values if (z < 0) error ("adtest: invalid X for asymptotic distribution."); endif ## Due to floating point precision: ## Return 0 below a certain threshold if (z < 0.02) ad = 0; return; ## Return 1 above the following threshold elseif (z >= 32.4) ad = 1; return; endif n = 1:500; K = 1/z*[1, ((4*n + 1).*cumprod((1/2 - n)./n))]; ADTerms = arrayfun(@(j)ADf(z,j),0:500); ad = ADTerms*K'; endfunction ## Series expansion for f(z,j) called by ADInf function f = ADf(z,j) ## Compute t=tj=(4j+1)^2*pi^2/(8z) t = (4 * j + 1) ^ 2 * 1.233700550136170 / z; ## First 2 terms in recursive series ## c0=pi*exp(-t)/(sqrt(2t)) ## c1=pi*sqrt(pi/2)*erfc(sqrt(t)) c0 = 2.221441469079183 * exp (-t) / sqrt (t); c1 = 3.937402486430604 * erfc (sqrt (t)); r = z / 8; f = c0 + c1 * r; ## Evaluate the recursion for n = 2:500 c = 1 / (n - 1) * ((n - 3 / 2 - t) * c1 + t * c0); r = r * (z / 8) * (1 / n); fn = f + c * r; c0 = c1; c1 = c; if (f == fn) return; endif f = fn; endfor endfunction ## An improved version of the Petitt method for the composite normal case. function CVs = computeCriticalValues_norm (n) CVs = [1.5649, 1.4407, 1.3699, 1.3187, 1.1556, 1.0339, 0.8733, ... 0.7519, 0.6308, 0.5598, 0.5092, 0.4694, 0.4366, 0.4084, ... 0.3835, 0.3611, 0.3405, 0.3212, 0.3029, 0.2852, 0.2679, ... 0.2506, 0.2330, 0.2144, 0.1935, 0.1673, 0.1296] + ... [-0.9362, -0.9029, -0.8906, -0.8865, -0.8375, -0.7835, -0.6746, ... -0.5835, -0.4775, -0.4094, -0.3679, -0.3327, -0.3099, -0.2969, ... -0.2795, -0.2623, -0.2464, -0.2325, -0.2164, -0.1994, -0.1784, ... -0.1569, -0.1377, -0.1201, -0.0989, -0.0800, -0.0598] ./ n + ... [-8.3249, -6.6022, -5.6461, -4.9685, -3.2208, -2.1647, -1.2460, ... -0.7803, -0.4627, -0.3672, -0.2833, -0.2349, -0.1442, -0.0229, ... 0.0377, 0.0817, 0.1150, 0.1583, 0.1801, 0.1887, 0.1695, ... 0.1513, 0.1533, 0.1724, 0.2027, 0.3158, 0.6431] ./ n ^ 2; endfunction ## An improved version of the Petitt method for the composite exponential case. function CVs = computeCriticalValues_exp (n) CVs = [3.2371, 2.9303, 2.7541, 2.6307, 2.2454, 1.9621, 1.5928, ... 1.3223, 1.0621, 0.9153, 0.8134, 0.7355, 0.6725, 0.6194, ... 0.5734, 0.5326, 0.4957, 0.4617, 0.4301, 0.4001, 0.3712, ... 0.3428, 0.3144, 0.2849, 0.2527, 0.2131, 0.1581] + ... [1.6146, 0.8716, 0.4715, 0.2066, -0.4682, -0.7691, -0.7388, ... -0.5758, -0.4036, -0.3142, -0.2564, -0.2152, -0.1845, -0.1607, ... -0.1409, -0.1239, -0.1084, -0.0942, -0.0807, -0.0674, -0.0537, ... -0.0401, -0.0261, -0.0116, 0.0047, 0.0275, 0.0780] ./ n; endfunction ## An improved version of the Petitt method for the composite extreme value case function CVs = computeCriticalValues_ev(n) CVs = [1.6473, 1.5095, 1.4301, 1.3742, 1.1974, 1.0667, 0.8961, ... 0.7683, 0.6416, 0.5680, 0.5156, 0.4744, 0.4405, 0.4115, ... 0.3858, 0.3626, 0.3415, 0.3217, 0.3029, 0.2848, 0.2672, ... 0.2496, 0.2315, 0.2124, 0.1909, 0.1633, 0.1223] + ... [-0.7097, -0.5934, -0.5328, -0.4930, -0.3708, -0.2973, -0.2075, ... -0.1449, -0.0892, -0.0619, -0.0442, -0.0302, -0.0196, -0.0112, ... -0.0039, 0.0024, 0.0074, 0.0122, 0.0167, 0.0207, 0.0245, ... 0.0282, 0.0323, 0.0371, 0.0436, 0.0549, 0.0813] ./ n .^ (1 / 2); endfunction ## Find rows of critical values at relevant significance levels function [CVs, sampleSizes] = findCriticalValues CVs = [7.2943, 6.6014, 6.1962, 5.9088, 4.9940, 4.3033, 3.3946, 2.7142, ... 2.0470, 1.6682, 1.4079, 1.2130, 1.0596, 0.9353, 0.8326, 0.7465, ... 0.6740, 0.6126, 0.5606, 0.5170, 0.4806, 0.4508, 0.4271, 0.4091, ... NaN, NaN, NaN; ... %n=1 7.6624, 6.4955, 5.9916, 5.6682, 4.7338, 4.0740, 3.2247, 2.5920, ... 1.9774, 1.6368, 1.4078, 1.2329, 1.0974, 0.9873, 0.8947, 0.8150, ... 0.7448, 0.6820, 0.6251, 0.5727, 0.5240, 0.4779, 0.4337, 0.3903, ... 0.3462, 0.3030, 0.2558; ... %n=2 7.2278, 6.3094, 5.8569, 5.5557, 4.6578, 4.0111, 3.1763, 2.5581, ... 1.9620, 1.6314, 1.4079, 1.2390, 1.1065, 0.9979, 0.9060, 0.8264, ... 0.7560, 0.6928, 0.6350, 0.5816, 0.5314, 0.4835, 0.4371, 0.3907, ... 0.3424, 0.2885, 0.2255; ... %n=3 7.0518, 6.2208, 5.7904, 5.4993, 4.6187, 3.9788, 3.1518, 2.5414, ... 1.9545, 1.6288, 1.4080, 1.2416, 1.1104, 1.0025, 0.9110, 0.8315, ... 0.7611, 0.6977, 0.6397, 0.5859, 0.5352, 0.4868, 0.4395, 0.3920, ... 0.3421, 0.2845, 0.2146; ... %n=4 6.9550, 6.1688, 5.7507, 5.4653, 4.5949, 3.9591, 3.1370, 2.5314, ... 1.9501, 1.6272, 1.4080, 1.2430, 1.1126, 1.0051, 0.9138, 0.8344, ... 0.7640, 0.7005, 0.6424, 0.5884, 0.5375, 0.4888, 0.4411, 0.3930, ... 0.3424, 0.2833, 0.2097; ... %n=5 6.8935, 6.1345, 5.7242, 5.4426, 4.5789, 3.9459, 3.1271, 2.5248, ... 1.9472, 1.6262, 1.4081, 1.2439, 1.1140, 1.0067, 0.9156, 0.8362, ... 0.7658, 0.7023, 0.6441, 0.5901, 0.5391, 0.4901, 0.4422, 0.3938, ... 0.3427, 0.2828, 0.2071; ... %n=6 6.8509, 6.1102, 5.7053, 5.4264, 4.5674, 3.9364, 3.1201, 2.5201, ... 1.9451, 1.6255, 1.4081, 1.2445, 1.1149, 1.0079, 0.9168, 0.8375, ... 0.7671, 0.7036, 0.6454, 0.5912, 0.5401, 0.4911, 0.4430, 0.3944, ... 0.3430, 0.2826, 0.2056; ... %n=7 6.8196, 6.0920, 5.6912, 5.4142, 4.5588, 3.9293, 3.1148, 2.5166, ... 1.9436, 1.6249, 1.4081, 1.2450, 1.1156, 1.0087, 0.9177, 0.8384, ... 0.7681, 0.7045, 0.6463, 0.5921, 0.5409, 0.4918, 0.4436, 0.3949, ... 0.3433, 0.2825, 0.2046; ... %n=8 6.7486, 6.0500, 5.6582, 5.3856, 4.5384, 3.9124, 3.1024, 2.5084, ... 1.9400, 1.6237, 1.4081, 1.2460, 1.1171, 1.0106, 0.9197, 0.8406, ... 0.7702, 0.7066, 0.6483, 0.5941, 0.5428, 0.4935, 0.4451, 0.3961, ... 0.3441, 0.2826, 0.2029; ... %n=12 6.7140, 6.0292, 5.6417, 5.3713, 4.5281, 3.9040, 3.0962, 2.5044, ... 1.9382, 1.6230, 1.4081, 1.2465, 1.1179, 1.0115, 0.9207, 0.8416, ... 0.7712, 0.7077, 0.6493, 0.5950, 0.5437, 0.4944, 0.4459, 0.3968, ... 0.3445, 0.2827, 0.2023; ... %n=16 6.6801, 6.0084, 5.6252, 5.3569, 4.5178, 3.8955, 3.0900, 2.5003, ... 1.9365, 1.6224, 1.4081, 1.2470, 1.1186, 1.0123, 0.9217, 0.8426, ... 0.7723, 0.7087, 0.6503, 0.5960, 0.5446, 0.4952, 0.4466, 0.3974, ... 0.3450, 0.2829, 0.2019; ... %n=24 6.6468, 5.9877, 5.6087, 5.3425, 4.5075, 3.8869, 3.0837, 2.4963, ... 1.9347, 1.6218, 1.4082, 1.2474, 1.1193, 1.0132, 0.9226, 0.8436, ... 0.7732, 0.7097, 0.6513, 0.5969, 0.5455, 0.4960, 0.4474, 0.3980, ... 0.3455, 0.2832, 0.2016; ... %n=48 6.6634, 5.9980, 5.6169, 5.3497, 4.5127, 3.8912, 3.0868, 2.4983, ... 1.9356, 1.6221, 1.4081, 1.2472, 1.1190, 1.0128, 0.9222, 0.8431, ... 0.7728, 0.7092, 0.6508, 0.5965, 0.5451, 0.4956, 0.4470, 0.3977, ... 0.3453, 0.2830, 0.2017; ... %n=32 6.6385, 5.9825, 5.6046, 5.3389, 4.5049, 3.8848, 3.0822, 2.4953, ... 1.9343, 1.6217, 1.4082, 1.2475, 1.1195, 1.0134, 0.9228, 0.8438, ... 0.7735, 0.7099, 0.6516, 0.5972, 0.5458, 0.4962, 0.4476, 0.3982, ... 0.3456, 0.2833, 0.2016; ... %n=64 6.6318, 5.9783, 5.6012, 5.3360, 4.5028, 3.8830, 3.0809, 2.4944, ... 1.9339, 1.6215, 1.4082, 1.2476, 1.1197, 1.0136, 0.9230, 0.8440, ... 0.7737, 0.7101, 0.6517, 0.5974, 0.5459, 0.4964, 0.4477, 0.3984, ... 0.3457, 0.2833, 0.2015; ... %n=88 6.6297, 5.9770, 5.6001, 5.3350, 4.5021, 3.8825, 3.0805, 2.4942, ... 1.9338, 1.6215, 1.4082, 1.2476, 1.1197, 1.0136, 0.9231, 0.8441, ... 0.7738, 0.7102, 0.6518, 0.5974, 0.5460, 0.4965, 0.4478, 0.3984, ... 0.3458, 0.2834, 0.2015; ... %n=100 6.6262, 5.9748, 5.5984, 5.3335, 4.5010, 3.8816, 3.0798, 2.4937, ... 1.9336, 1.6214, 1.4082, 1.2477, 1.1198, 1.0137, 0.9232, 0.8442, ... 0.7739, 0.7103, 0.6519, 0.5975, 0.5461, 0.4966, 0.4479, 0.3985, ... 0.3458, 0.2834, 0.2015; ... %n=128 6.6201, 5.9709, 5.5953, 5.3308, 4.4990, 3.8800, 3.0787, 2.4930, ... 1.9333, 1.6213, 1.4082, 1.2478, 1.1199, 1.0139, 0.9234, 0.8443, ... 0.7740, 0.7105, 0.6521, 0.5977, 0.5463, 0.4967, 0.4480, 0.3986, ... 0.3459, 0.2834, 0.2015; ... %n=256 6.6127, 5.9694, 5.5955, 5.3314, 4.4982, 3.8781, 3.0775, 2.4924, ... 1.9330, 1.6212, 1.4082, 1.2479, 1.1201, 1.0140, 0.9235, 0.8445, ... 0.7742, 0.7106, 0.6523, 0.5979, 0.5464, 0.4969, 0.4481, 0.3987, ... 0.3460, 0.2835, 0.2015]; %n=Inf sampleSizes = [1 2 3 4 5 6 7 8 12 16 24 32 48 64 88 100 128 256 Inf]; endfunction % ------------------------------------------ function critVals = findAsymptoticDistributionCriticalValues critVals = [6.6127034546551, 5.9694013422151, 5.5954643397078, ... 5.3313658857909, 4.4981996466091, 3.8781250216054, ... 3.0774641787107, 2.4923671600494, 1.9329578327416, ... 1.6212385363175, 1.4081977005506, 1.2478596347253, ... 1.1200136586965, 1.0140004020016, 0.9235137094902, ... 0.8445069178452, 0.7742142410993, 0.7106405935247, ... 0.6522701010084, 0.5978828157471, 0.5464229310982, ... 0.4968804113119, 0.4481425895777, 0.3987228486242, ... 0.3460480234939, 0.2835161264344, 0.2014922164166]; %n=Inf endfunction ## Make sure alpha is within the lookup table function validateAlpha (alpha, alphas) if (alpha < alphas(1) || alpha > alphas(end)) error (strcat (["adtest: out of range invalid alpha -"], ... sprintf (" lower limit: %g", alphas(1)),... sprintf (" upper limit: %g", alphas(end)))); endif endfunction ## Simulated critical values and p-values for Anderson-Darling test function [crit, p] = adtestMC (ADStat, n, alpha, distribution, MCTol) ## Initial values vartol = mctol^2; crit = 0; p = 0; mcRepsTot = 0; mcRepsMin = 1000; ## Monte Carlo loop while true mcRepsOld = mcRepsTot; mcReps = ceil(mcRepsMin - mcRepsOld); ADstatMC = zeros(mcReps,1); ## Switch to selected distribution switch distribution case "norm" mu0 = 0; sigma0 = 1; for rep = 1:length (ADstatMC) x = normrnd (mu0, sigma0, n, 1); xCDF = sort (x); nullCDF = normcdf (xCDF, mean (x), std (x)); w = 2 * (1:n) - 1 ; ADstatMC(rep) = - w * (log (nullCDF) + ... log (1 - nullCDF(end:-1:1))) / n - n; endfor case "exp" beta0 = 1; for rep = 1:length (ADstatMC) x = exprnd (beta0, n, 1); xCDF = sort (x); nullCDF = expcdf (xCDF, mean (x)); w = 2 * (1:n) - 1 ; ADstatMC(rep) = - w * (log (nullCDF) + ... log (1 - nullCDF(end:-1:1))) / n - n; endfor case "ev" mu0 = 0; sigma0 = 1; for rep = 1:length (ADstatMC) x = evrnd (mu0, sigma0, n, 1); pHat = evfit (x); xCDF = sort (x); nullCDF = evcdf (xCDF, pHat(1), pHat(2)); w = 2 * (1:n) - 1 ; ADstatMC(rep) = - w * (log (nullCDF) + ... log (1 - nullCDF(end:-1:1))) / n - n; endfor case "unif" for rep = 1:length(ADstatMC) z = sort (rand (n, 1)); w = 2 * (1:n) - 1 ; ADstatMC(rep) = - w * (log (z) + ... log (1 - z(end:-1:1))) / n - n; endfor endswitch critMC = prctile (ADstatMC, 100 * (1 - alpha)); pMC = sum (ADstatMC > ADStat) ./ mcReps; mcRepsTot = mcRepsOld + mcReps; crit = (mcRepsOld * crit + mcReps * critMC) / mcRepsTot; p = (mcRepsOld * p + mcReps * pMC) / mcRepsTot; ## Compute a std err for p, with lower bound (1/N)*(1-1/N)/N when p==0. sepsq = max (p * (1 - p) / mcRepsTot, 1 / mcRepsTot ^ 2); if (sepsq < MCTol ^ 2) break endif ## Based on the current estimate, find the number of trials needed to ## make the MC std err less than the specified tolerance. mcRepsMin = 1.2 * (mcRepsTot * sepsq) / (MCTol ^ 2); endwhile endfunction ## Test input validation %!error adtest (); %!error adtest (ones (20,2)); %!error adtest ([1+i,0-3i]); %!error ... %! adtest (ones (20,1), "Distribution", "normal"); %!error ... %! adtest (rand (20,1), "Distribution", {"normal", 5, 3}); %!error ... %! adtest (rand (20,1), "Distribution", {"norm", 5}); %!error ... %! adtest (rand (20,1), "Distribution", {"exp", 5, 4}); %!error ... %! adtest (rand (20,1), "Distribution", {"ev", 5}); %!error ... %! adtest (rand (20,1), "Distribution", {"logn", 5, 3, 2}); %!error ... %! adtest (rand (20,1), "Distribution", {"Weibull", 5}); %!error ... %! adtest (rand (20,1), "Distribution", 35); %!error ... %! adtest (rand (20,1), "Name", "norm"); %!error ... %! adtest (rand (20,1), "Name", {"norm", 75, 10}); %!error ... %! adtest (rand (20,1), "Distribution", "norm", "Asymptotic", true); %!error ... %! adtest (rand (20,1), "MCTol", 0.001, "Asymptotic", true); %!error ... %! adtest (rand (20,1), "Distribution", {"norm", 5, 3}, "MCTol", 0.001, ... %! "Asymptotic", true); %!error ... %! [h, pval, ADstat, CV] = adtest (ones (20,1), "Distribution", {"norm",5,3},... %! "Alpha", 0.000000001); %!error ... %! [h, pval, ADstat, CV] = adtest (ones (20,1), "Distribution", {"norm",5,3},... %! "Alpha", 0.999999999); %!error ... %! adtest (10); ## Test warnings %!warning ... %! randn ("seed", 34); %! adtest (ones (20,1), "Alpha", 0.000001); %!warning ... %! randn ("seed", 34); %! adtest (normrnd(0,1,100,1), "Alpha", 0.99999); %!warning ... %! randn ("seed", 34); %! adtest (normrnd(0,1,100,1), "Alpha", 0.00001); ## Test results %!test %! load examgrades %! x = grades(:,1); %! [h, pval, adstat, cv] = adtest (x); %! assert (h, false); %! assert (pval, 0.1854, 1e-4); %! assert (adstat, 0.5194, 1e-4); %! assert (cv, 0.7470, 1e-4); %!test %! load examgrades %! x = grades(:,1); %! [h, pval, adstat, cv] = adtest (x, "Distribution", "ev"); %! assert (h, false); %! assert (pval, 0.071363, 1e-6); %!test %! load examgrades %! x = grades(:,1); %! [h, pval, adstat, cv] = adtest (x, "Distribution", {"norm", 75, 10}); %! assert (h, false); %! assert (pval, 0.4687, 1e-4); statistics-release-1.6.3/inst/anova1.m000066400000000000000000000316151456127120000176550ustar00rootroot00000000000000## Copyright (C) 2021-2022 Andreas Bertsatos ## Copyright (C) 2022 Andrew Penn ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{p} =} anova1 (@var{x}) ## @deftypefnx {statistics} {@var{p} =} anova1 (@var{x}, @var{group}) ## @deftypefnx {statistics} {@var{p} =} anova1 (@var{x}, @var{group}, @var{displayopt}) ## @deftypefnx {statistics} {@var{p} =} anova1 (@var{x}, @var{group}, @var{displayopt}, @var{vartype}) ## @deftypefnx {statistics} {[@var{p}, @var{atab}] =} anova1 (@var{x}, @dots{}) ## @deftypefnx {statistics} {[@var{p}, @var{atab}, @var{stats}] =} anova1 (@var{x}, @dots{}) ## ## Perform a one-way analysis of variance (ANOVA) for comparing the means of two ## or more groups of data under the null hypothesis that the groups are drawn ## from distributions with the same mean. For planned contrasts and/or ## diagnostic plots, use @qcode{anovan} instead. ## ## anova1 can take up to three input arguments: ## ## @itemize ## @item ## @var{x} contains the data and it can either be a vector or matrix. ## If @var{x} is a matrix, then each column is treated as a separate group. ## If @var{x} is a vector, then the @var{group} argument is mandatory. ## ## @item ## @var{group} contains the names for each group. If @var{x} is a matrix, then ## @var{group} can either be a cell array of strings of a character array, with ## one row per column of @var{x}. If you want to omit this argument, enter an ## empty array ([]). If @var{x} is a vector, then @var{group} must be a vector ## of the same length, or a string array or cell array of strings with one row ## for each element of @var{x}. @var{x} values corresponding to the same value ## of @var{group} are placed in the same group. ## ## @item ## @var{displayopt} is an optional parameter for displaying the groups contained ## in the data in a boxplot. If omitted, it is 'on' by default. If group names ## are defined in @var{group}, these are used to identify the groups in the ## boxplot. Use 'off' to omit displaying this figure. ## ## @item ## @var{vartype} is an optional parameter to used to indicate whether the ## groups can be assumed to come from populations with equal variance. When ## @qcode{vartype} is @qcode{"equal"} the variances are assumed to be equal ## (this is the default). When @qcode{vartype} is @qcode{"unequal"} the ## population variances are not assumed to be equal and Welch's ANOVA test is ## used instead. ## @end itemize ## ## anova1 can return up to three output arguments: ## ## @itemize ## @item ## @var{p} is the p-value of the null hypothesis that all group means are equal. ## ## @item ## @var{atab} is a cell array containing the results in a standard ANOVA table. ## ## @item ## @var{stats} is a structure containing statistics useful for performing ## a multiple comparison of means with the MULTCOMPARE function. ## @end itemize ## ## If anova1 is called without any output arguments, then it prints the results ## in a one-way ANOVA table to the standard output. It is also printed when ## @var{displayopt} is 'on'. ## ## ## Examples: ## ## @example ## x = meshgrid (1:6); ## x = x + normrnd (0, 1, 6, 6); ## anova1 (x, [], 'off'); ## [p, atab] = anova1(x); ## @end example ## ## ## @example ## x = ones (50, 4) .* [-2, 0, 1, 5]; ## x = x + normrnd (0, 2, 50, 4); ## groups = @{"A", "B", "C", "D"@}; ## anova1 (x, groups); ## @end example ## ## @seealso{anova2, anovan, multcompare} ## @end deftypefn function [p, anovatab, stats] = anova1 (x, group, displayopt, vartype) ## Check for valid number of input arguments if (nargin < 1 || nargin > 4) error ("anova1: invalid number of input arguments."); endif ## Add defaults if (nargin < 2) group = []; endif if (nargin < 3) displayopt = "on"; endif if (nargin < 4) vartype = "equal"; endif plotdata = ! (strcmp (displayopt, 'off')); ## Convert group to cell array from character array, make it a column if (! isempty (group) && ischar (group)) group = cellstr (group); endif if (size (group, 1) == 1) group = group'; endif ## If x is a matrix, convert it to column vector and create a ## corresponging column vector for groups if (length (x) < prod (size (x))) [n, m] = size (x); x = x(:); gi = reshape (repmat ((1:m), n, 1), n*m, 1); if (length (group) == 0) ## no group names are provided group = gi; elseif (size (group, 1) == m) ## group names exist and match columns group = group(gi,:); else error ("anova1: columns in X and GROUP length do not match."); endif endif ## Check that x and group are the same size if (! all (numel (x) == numel (group))) error ("anova1: GROUP must be a vector with the same number of rows as x."); endif ## Identify NaN values (if any) and remove them from X along with ## their corresponding values from group vector nonan = ! isnan (x); x = x(nonan); group = group(nonan, :); ## Convert group to indices and separate names [group_id, group_names] = grp2idx (group); group_id = group_id(:); named = 1; ## Center data to improve accuracy and keep uncentered data for ploting xorig = x; mu = mean(x); x = x - mu; xr = x; ## Get group size and mean for each group groups = size (group_names, 1); xs = zeros (1, groups); xm = xs; xv = xs; for j = 1:groups group_size = find (group_id == j); xs(j) = length (group_size); xm(j) = mean (xr(group_size)); xv(j) = var (xr(group_size), 0); endfor ## Calculate statistics lx = length (xr); ## Number of samples in groups gm = mean (xr); ## Grand mean of groups dfm = length (xm) - 1; ## degrees of freedom for model dfe = lx - dfm - 1; ## degrees of freedom for error SSM = xs .* (xm - gm) * (xm - gm)'; ## Sum of Squares for Model SST = (xr(:) - gm)' * (xr(:) - gm); ## Sum of Squares Total SSE = SST - SSM; ## Sum of Squares Error if (dfm > 0) MSM = SSM / dfm; ## Mean Square for Model else MSM = NaN; endif if (dfe > 0) MSE = SSE / dfe; ## Mean Square for Error else MSE = NaN; endif ## Calculate F statistic if (SSE != 0) ## Regular Matrix case. switch (lower (vartype)) case "equal" ## Assume equal variances (Fisher's One-way ANOVA) F = (SSM / dfm) / MSE; case "unequal" ## Accomodate for unequal variances (Welch's One-way ANOVA) ## Calculate the sampling variance for each group (i.e. the square of the SEM) sv = xv ./ xs; ## Calculate weights as the reciprocal of the sampling variance w = 1 ./ sv; ## Calculate the origin ori = sum (w .* xm) ./ sum (w); ## Calculate Welch's F statistic F = (groups - 1)^-1 * sum (w .* (xm - ori).^2) /... (1 + ((2 * (groups - 2)/(groups^2 - 1)) * ... sum ((1 - w / sum (w)).^2 .* (xs - 1).^-1))); ## Welch's test does not use a pooled error term MSE = NaN; ## Correct the error degrees of freedom dfe = (3 /(groups^2 - 1) * sum ((1 - w / sum (w)).^2 .* (xs-1).^-1))^-1; otherwise error ("anova1: invalid fourth (vartype) argument to anova1."); endswitch p = 1 - fcdf (F, dfm, dfe); ## Probability of F given equal means. elseif (SSM == 0) ## Constant Matrix case. F = 0; p = 1; else ## Perfect fit case. F = Inf; p = 0; end ## Create results table (if requested) if (nargout > 1) switch (lower (vartype)) case "equal" anovatab = {"Source", "SS", "df", "MS", "F", "Prob>F"; ... "Groups", SSM, dfm, MSM, F, p; ... "Error", SSE, dfe, MSE, "", ""; ... "Total", SST, dfm + dfe, "", "", ""}; case "unequal" anovatab = {"Source", "F", "df", "dfe", "F", "Prob>F"; ... "Groups", SSM, dfm, dfe, F, p}; endswitch endif ## Create stats structure (if requested) for MULTCOMPARE if (nargout > 2) if (length (group_names) > 0) stats.gnames = group_names; else stats.gnames = strjust (num2str ((1:length (xm))'), 'left'); end stats.n = xs; stats.source = 'anova1'; stats.vartype = vartype; stats.means = xm + mu; stats.vars = xv; stats.df = dfe; stats.s = sqrt (MSE); endif ## Print results table on screen if no output argument was requested if (nargout == 0 || plotdata) switch (lower (vartype)) case "equal" printf("\n ANOVA Table\n\n"); printf("Source SS df MS F Prob>F\n"); printf("------------------------------------------------------\n"); printf("Groups %10.4f %5.0f %10.4f %8.2f %9.4f\n", SSM, dfm, MSM, F, p); printf("Error %10.4f %5.0f %10.4f\n", SSE, dfe, MSE); printf("Total %10.4f %5.0f\n\n", SST, dfm + dfe); case "unequal" printf("\n Welch's ANOVA Table\n\n"); printf("Source F df dfe Prob>F\n"); printf("-----------------------------------------\n"); printf("Groups %8.2f %5.0f %7.2f %10.4f\n\n", F, dfm, dfe, p); endswitch endif ## Plot data using BOXPLOT (unless opted out) if (plotdata) boxplot (x, group_id, "Notch", "on", "Labels", group_names); endif endfunction %!demo %! x = meshgrid (1:6); %! randn ("seed", 15); # for reproducibility %! x = x + normrnd (0, 1, 6, 6); %! anova1 (x, [], 'off'); %!demo %! x = meshgrid (1:6); %! randn ("seed", 15); # for reproducibility %! x = x + normrnd (0, 1, 6, 6); %! [p, atab] = anova1(x); %!demo %! x = ones (50, 4) .* [-2, 0, 1, 5]; %! randn ("seed", 13); # for reproducibility %! x = x + normrnd (0, 2, 50, 4); %! groups = {"A", "B", "C", "D"}; %! anova1 (x, groups); %!demo %! y = [54 87 45; 23 98 39; 45 64 51; 54 77 49; 45 89 50; 47 NaN 55]; %! g = [1 2 3 ; 1 2 3 ; 1 2 3 ; 1 2 3 ; 1 2 3 ; 1 2 3 ]; %! anova1 (y(:), g(:), "on", "unequal"); ## testing against GEAR.DAT data file and results for one-factor ANOVA from ## https://www.itl.nist.gov/div898/handbook/eda/section3/eda354.htm %!test %! data = [1.006, 0.996, 0.998, 1.000, 0.992, 0.993, 1.002, 0.999, 0.994, 1.000, ... %! 0.998, 1.006, 1.000, 1.002, 0.997, 0.998, 0.996, 1.000, 1.006, 0.988, ... %! 0.991, 0.987, 0.997, 0.999, 0.995, 0.994, 1.000, 0.999, 0.996, 0.996, ... %! 1.005, 1.002, 0.994, 1.000, 0.995, 0.994, 0.998, 0.996, 1.002, 0.996, ... %! 0.998, 0.998, 0.982, 0.990, 1.002, 0.984, 0.996, 0.993, 0.980, 0.996, ... %! 1.009, 1.013, 1.009, 0.997, 0.988, 1.002, 0.995, 0.998, 0.981, 0.996, ... %! 0.990, 1.004, 0.996, 1.001, 0.998, 1.000, 1.018, 1.010, 0.996, 1.002, ... %! 0.998, 1.000, 1.006, 1.000, 1.002, 0.996, 0.998, 0.996, 1.002, 1.006, ... %! 1.002, 0.998, 0.996, 0.995, 0.996, 1.004, 1.004, 0.998, 0.999, 0.991, ... %! 0.991, 0.995, 0.984, 0.994, 0.997, 0.997, 0.991, 0.998, 1.004, 0.997]; %! group = [1:10] .* ones (10,10); %! group = group(:); %! [p, tbl] = anova1 (data, group, "off"); %! assert (p, 0.022661, 1e-6); %! assert (tbl{2,5}, 2.2969, 1e-4); %! assert (tbl{2,3}, 9, 0); %! assert (tbl{4,2}, 0.003903, 1e-6); %! data = reshape (data, 10, 10); %! [p, tbl, stats] = anova1 (data, [], "off"); %! assert (p, 0.022661, 1e-6); %! assert (tbl{2,5}, 2.2969, 1e-4); %! assert (tbl{2,3}, 9, 0); %! assert (tbl{4,2}, 0.003903, 1e-6); %! means = [0.998, 0.9991, 0.9954, 0.9982, 0.9919, 0.9988, 1.0015, 1.0004, 0.9983, 0.9948]; %! N = 10 * ones (1, 10); %! assert (stats.means, means, 1e-6); %! assert (length (stats.gnames), 10, 0); %! assert (stats.n, N, 0); ## testing against one-way ANOVA example dataset from GraphPad Prism 8 %!test %! y = [54 87 45; 23 98 39; 45 64 51; 54 77 49; 45 89 50; 47 NaN 55]; %! g = [1 2 3 ; 1 2 3 ; 1 2 3 ; 1 2 3 ; 1 2 3 ; 1 2 3 ]; %! [p, tbl] = anova1 (y(:), g(:), "off", "equal"); %! assert (p, 0.00004163, 1e-6); %! assert (tbl{2,5}, 22.573418, 1e-6); %! assert (tbl{2,3}, 2, 0); %! assert (tbl{3,3}, 14, 0); %! [p, tbl] = anova1 (y(:), g(:), "off", "unequal"); %! assert (p, 0.00208877, 1e-8); %! assert (tbl{2,5}, 15.523192, 1e-6); %! assert (tbl{2,3}, 2, 0); %! assert (tbl{2,4}, 7.5786897, 1e-6); statistics-release-1.6.3/inst/anova2.m000066400000000000000000000370561456127120000176630ustar00rootroot00000000000000## Copyright (C) 2022 Andreas Bertsatos ## Copyright (C) 2022 Andrew Penn ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{p} =} anova2 (@var{x}, @var{reps}) ## @deftypefnx {statistics} {@var{p} =} anova2 (@var{x}, @var{reps}, @var{displayopt}) ## @deftypefnx {statistics} {@var{p} =} anova2 (@var{x}, @var{reps}, @var{displayopt}, @var{model}) ## @deftypefnx {statistics} {[@var{p}, @var{atab}] =} anova2 (@dots{}) ## @deftypefnx {statistics} {[@var{p}, @var{atab}, @var{stats}] =} anova2 (@dots{}) ## ## Performs two-way factorial (crossed) or a nested analysis of variance ## (ANOVA) for balanced designs. For unbalanced factorial designs, diagnostic ## plots and/or planned contrasts, use @qcode{anovan} instead. ## ## @qcode{anova2} requires two input arguments with an optional third and fourth: ## ## @itemize ## @item ## @var{x} contains the data and it must be a matrix of at least two columns and ## two rows. ## ## @item ## @var{reps} is the number of replicates for each combination of factor groups. ## ## @item ## @var{displayopt} is an optional parameter for displaying the ANOVA table, ## when it is 'on' (default) and suppressing the display when it is 'off'. ## ## @item ## @var{model} is an optional parameter to specify the model type as either: ## ## @itemize ## @item ## "interaction" or "full" (default): compute both main effects and their ## interaction ## ## @item ## "linear": compute both main effects without an interaction. When @var{reps} ## > 1 the test is suitable for a balanced randomized block design. When ## @var{reps} == 1, the test becomes a One-way Repeated Measures (RM)-ANOVA ## with Greenhouse-Geisser correction to the column factor degrees of freedom ## to make the test robust to violations of sphericity ## ## @item ## "nested": treat the row factor as nested within columns. Note that the row ## factor is considered a random factor in the calculation of the statistics. ## ## @end itemize ## @end itemize ## ## @qcode{anova2} returns up to three output arguments: ## ## @itemize ## @item ## @var{p} is the p-value of the null hypothesis that all group means are equal. ## ## @item ## @var{atab} is a cell array containing the results in a standard ANOVA table. ## ## @item ## @var{stats} is a structure containing statistics useful for performing ## a multiple comparison of means with the MULTCOMPARE function. ## @end itemize ## ## If anova2 is called without any output arguments, then it prints the results ## in a one-way ANOVA table to the standard output as if @var{displayopt} is ## 'on'. ## ## Examples: ## ## @example ## load popcorn; ## anova2 (popcorn, 3); ## @end example ## ## ## @example ## [p, anovatab, stats] = anova2 (popcorn, 3, "off"); ## disp (p); ## @end example ## ## @seealso{anova1, anovan, multcompare} ## @end deftypefn function [p, anovatab, stats] = anova2 (x, reps, displayopt, model) ## Check for valid number of input arguments if (nargin < 1 || nargin >4) error ("anova2: invalid number of input arguments."); endif ## Check for NaN values in X if (any (isnan( x(:)))) error ("anova2: NaN values in input are not allowed. Use anovan instead."); endif ## Add defaults if (nargin == 1) reps = 1; endif if (nargin < 3) displayopt = "on"; endif if (nargin < 4) model = "interaction"; endif epsilonhat = []; plotdata = ! (strcmp (displayopt, "off")); ## Calculate group numbers FFGn = size (x, 1) / reps; ## Number of groups in Row Factor SFGn = size (x, 2); ## Number of groups in Column Factor ## Check for valid repetitions if (! (int16 (FFGn) == FFGn)) error ("anova2: the number of rows in X must be a multiple of REPS."); else idx_s = 1; idx_e = reps; for i = 1:FFGn RIdx(i,:) = [idx_s:idx_e]; idx_s += reps; idx_e += reps; endfor endif ## Calculate group sample sizes GTsz = length (x(:)); ## Number of total samples FFGs = prod (size (x(RIdx(1,:),:))); ## Number of group samples of Row Factor SFGs = size (x, 1); ## Number of group samples of Column Factor ## Calculate group means GTmu = sum (x(:)) / GTsz; ## Grand mean of groups for i = 1:FFGn ## Group means of Row Factor FFGm(i) = mean (x(RIdx(i,:),:), "all"); endfor for i = 1:SFGn ## Group means of Column Factor SFGm(i) = mean (x(:,i)); endfor ## Calculate Sum of Squares for Row and Column Factors SSR = sum (FFGs * ((FFGm - GTmu) .^ 2)); ## Rows Sum of Squares SSC = sum (SFGs * ((SFGm - GTmu) .^ 2)); ## Columns Sum of Squares ## Calculate Total Sum of Squares SST = (x(:) - GTmu)' * (x(:) - GTmu); ## Calculate Sum of Squares Error (Within) if (reps > 1) SSE = 0; for i = 1:FFGn for j = 1:SFGn SSE += sum ((x(RIdx(i,:),j) - mean (x(RIdx(i,:),j))) .^ 2); endfor endfor else SSE = SST - SSC - SSR; endif ## Calculate degrees of freedom and Sum of Squares Interaction (if applicable) df_SSR = FFGn - 1; ## Row Factor df_SSC = SFGn - 1; ## Column Factor if (reps > 1) df_SSE = GTsz - (FFGn * SFGn); ## Error with replication df_SSI = df_SSR * df_SSC; ## Interaction: Degrees of Freedom SSI = SST - SSR - SSC - SSE; ## Interaction: Sum of Squares else df_SSE = df_SSR * df_SSC; ## No replication, assuming additive model df_SSI = 0; SSI = 0; endif df_tot = GTsz - 1; ## Total ## Model-specific calculations of sums-of-squares, mean squares and degrees of ## freedom. The calculations are based on equalities for the partitioning of ## variance in fully balanced designs. switch (lower (model)) case {"interaction", "full"} ## TWO-WAY ANOVA WITH INTERACTION (full factorial model) ## Sums--of-squares are already partitioned into main effects and ## interaction. Just calculate mean-squares and degrees of fredom model = "interaction"; MSE = SSE / df_SSE; ## Mean Square for Error (Within) MSR = SSR / df_SSR; ## Mean Square for Row Factor MS_DENOM = MSE; df_DENOM = df_SSE; case "linear" ## TWO-WAY ANOVA WITHOUT INTERACTION (additive, linear model) ## Pool Error and Interaction term model = "linear"; SSE += SSI; df_SSE += df_SSI; SSI = 0; df_SSI = 0; if (reps == 1) ## Assume one-way repeated measures ANOVA. Perform calculations for a ## correction factor (epsilonhat) to make tests of the Column factor ## robust to violations of sphericity vcov = cov (x); N = SFGn^2 * (mean (diag (vcov)) - mean (mean (vcov)))^2; D = (SFGn - 1) * ... (sum (sumsq (vcov)) - 2 * SFGn * sum ((mean (vcov, 2).^2)) + ... SFGn^2 * mean (mean (vcov))^2); epsilonhat = N / D; dfN_GG = epsilonhat * (SFGn - 1); dfD_GG = epsilonhat * (FFGn - 1) * (SFGn - 1); endif reps = 1; ## Set reps to 1 to avoid printing interaction MSE = SSE / df_SSE; ## Mean Square for Error (Within) MSR = SSR / df_SSR; ## Mean Square for Row Factor MS_DENOM = MSE; df_DENOM = df_SSE; case "nested" ## NESTED ANOVA ## Row Factor is nested within Column Factor. Treat Row factor as random. ## Pool Row Factor and Interaction term model = "nested"; SSR += SSI; df_SSR += df_SSI; SSI = 0; df_SSI = 0; reps = 1; ## Set reps to 1 to avoid printing interaction MSE = SSE / df_SSE; ## Mean Square for Error (Within) MSR = SSR / df_SSR; ## Mean Square for Row Factor MS_DENOM = MSR; ## Row factor is random so MSR is denominator df_DENOM = df_SSR; ## Row factor is random so df_SSR is denominator otherwise error ("anova2: model type not recognised"); endswitch ## Calculate F statistics and p values F_MSR = MSR / MSE; ## F statistic for Row Factor p_MSR = 1 - fcdf (F_MSR, df_SSR, df_SSE); MSC = SSC / df_SSC; ## Mean Square for Column Factor F_MSC = MSC / MS_DENOM; ## F statistic for Column Factor if (isempty(epsilonhat)) p_MSC = 1 - fcdf (F_MSC, df_SSC, df_DENOM); else ## Apply correction for sphericity to the p-value of the column factor p_MSC = 1 - fcdf (F_MSC, dfN_GG, dfD_GG); endif ## With replication if (reps > 1) MSI = SSI / df_SSI; ## Mean Square for Interaction F_MSI = MSI / MSE; ## F statistic for Interaction p_MSI = 1 - fcdf (F_MSI, df_SSI, df_SSE); else MSI = 0; F_MSI = 0; p_MSI = NaN; endif ## Create p output (if requested) if (nargout > 0) if (reps > 1) p = [p_MSC, p_MSR, p_MSI]; else p = [p_MSC, p_MSR]; endif endif ## Create results table (if requested) if (nargout > 1 && reps > 1) anovatab = {"Source", "SS", "df", "MS", "F", "Prob>F"; ... "Columns", SSC, df_SSC, MSC, F_MSC, p_MSC; ... "Rows", SSR, df_SSR, MSR, F_MSR, p_MSR; ... "Interaction", SSI, df_SSI, MSI, F_MSI, p_MSI; ... "Error", SSE, df_SSE, MSE, "", ""; ... "Total", SST, df_tot, "", "", ""}; elseif (nargout > 1 && reps == 1) anovatab = {"Source", "SS", "df", "MS", "F", "Prob>F"; ... "Columns", SSC, df_SSC, MSC, F_MSC, p_MSC; ... "Rows", SSR, df_SSR, MSR, F_MSR, p_MSR; ... "Error", SSE, df_SSE, MSE, "", ""; ... "Total", SST, df_tot, "", "", ""}; endif ## Create stats structure (if requested) for MULTCOMPARE if (nargout > 2) stats.source = "anova2"; stats.sigmasq = MS_DENOM; ## MS used to calculate F relating to stats.pval stats.colmeans = SFGm(:)'; stats.coln = SFGs; stats.rowmeans = FFGm(:)'; stats.rown = FFGs; stats.inter = (reps > 1); if stats.inter stats.pval = p_MSI; ## Interaction p-value if stats.inter is true else stats.pval = p_MSC; ## Column Factor p-value if stats.inter is false end stats.df = df_DENOM; ## Degrees of freedom used to calculate stats.pval stats.model = model; endif ## Print results table on screen if no output argument was requested if (nargout == 0 || plotdata) printf("\n ANOVA Table\n\n"); printf("Source SS df MS F Prob>F\n"); printf("-----------------------------------------------------------\n"); printf("Columns %10.4f %5.0f %10.4f %8.2f %9.4f\n", ... SSC, df_SSC, MSC, F_MSC, p_MSC); printf("Rows %10.4f %5.0f %10.4f %8.2f %9.4f\n", ... SSR, df_SSR, MSR, F_MSR, p_MSR); if (reps > 1) printf("Interaction %10.4f %5.0f %10.4f %8.2f %9.4f\n", ... SSI, df_SSI, MSI, F_MSI, p_MSI); endif printf("Error %10.4f %5.0f %10.4f\n", SSE, df_SSE, MSE); printf("Total %10.4f %5.0f\n\n", SST, df_tot); if (! isempty (epsilonhat)) printf (strcat (["Note: Greenhouse-Geisser's correction was applied to the\n"], ... ["degrees of freedom for the Column factor: F(%.2f,%.2f)\n\n"]),... dfN_GG, dfD_GG); endif if (strcmpi (model, "nested")) printf (strcat (["Note: Rows are a random factor nested within the columns.\n"], ... ["The Column F statistic uses the Row MS instead of the MSE.\n\n"])); endif endif endfunction %!demo %! %! # Factorial (Crossed) Two-way ANOVA with Interaction %! %! popcorn = [5.5, 4.5, 3.5; 5.5, 4.5, 4.0; 6.0, 4.0, 3.0; ... %! 6.5, 5.0, 4.0; 7.0, 5.5, 5.0; 7.0, 5.0, 4.5]; %! %! [p, atab, stats] = anova2(popcorn, 3, "on"); %!demo %! %! # One-way Repeated Measures ANOVA (Rows are a crossed random factor) %! %! data = [54, 43, 78, 111; %! 23, 34, 37, 41; %! 45, 65, 99, 78; %! 31, 33, 36, 35; %! 15, 25, 30, 26]; %! %! [p, atab, stats] = anova2 (data, 1, "on", "linear"); %!demo %! %! # Balanced Nested One-way ANOVA (Rows are a nested random factor) %! %! data = [4.5924 7.3809 21.322; -0.5488 9.2085 25.0426; ... %! 6.1605 13.1147 22.66; 2.3374 15.2654 24.1283; ... %! 5.1873 12.4188 16.5927; 3.3579 14.3951 10.2129; ... %! 6.3092 8.5986 9.8934; 3.2831 3.4945 10.0203]; %! %! [p, atab, stats] = anova2 (data, 4, "on", "nested"); ## testing against popcorn data and results from Matlab %!test %! ## Test for anova2 ("interaction") %! ## comparison with results from Matlab for column effect %! popcorn = [5.5, 4.5, 3.5; 5.5, 4.5, 4.0; 6.0, 4.0, 3.0; ... %! 6.5, 5.0, 4.0; 7.0, 5.5, 5.0; 7.0, 5.0, 4.5]; %! [p, atab, stats] = anova2 (popcorn, 3, "off"); %! assert (p(1), 7.678957383294716e-07, 1e-14); %! assert (p(2), 0.0001003738963050171, 1e-14); %! assert (p(3), 0.7462153966366274, 1e-14); %! assert (atab{2,5}, 56.700, 1e-14); %! assert (atab{2,3}, 2, 0); %! assert (atab{4,2}, 0.08333333333333348, 1e-14); %! assert (atab{5,4}, 0.1388888888888889, 1e-14); %! assert (atab{5,2}, 1.666666666666667, 1e-14); %! assert (atab{6,2}, 22); %! assert (stats.source, "anova2"); %! assert (stats.colmeans, [6.25, 4.75, 4]); %! assert (stats.inter, 1, 0); %! assert (stats.pval, 0.7462153966366274, 1e-14); %! assert (stats.df, 12); %!test %! ## Test for anova2 ("linear") - comparison with results from GraphPad Prism 8 %! data = [54, 43, 78, 111; %! 23, 34, 37, 41; %! 45, 65, 99, 78; %! 31, 33, 36, 35; %! 15, 25, 30, 26]; %! [p, atab, stats] = anova2 (data, 1, "off", "linear"); %! assert (atab{2,2}, 2174.95, 1e-10); %! assert (atab{3,2}, 8371.7, 1e-10); %! assert (atab{4,2}, 2404.3, 1e-10); %! assert (atab{5,2}, 12950.95, 1e-10); %! assert (atab{2,4}, 724.983333333333, 1e-10); %! assert (atab{3,4}, 2092.925, 1e-10); %! assert (atab{4,4}, 200.358333333333, 1e-10); %! assert (atab{2,5}, 3.61843363972882, 1e-10); %! assert (atab{3,5}, 10.445909412303, 1e-10); %! assert (atab{2,6}, 0.087266112738617, 1e-10); %! assert (atab{3,6}, 0.000698397753556, 1e-10); %!test %! ## Test for anova2 ("nested") - comparison with results from GraphPad Prism 8 %! data = [4.5924 7.3809 21.322; -0.5488 9.2085 25.0426; ... %! 6.1605 13.1147 22.66; 2.3374 15.2654 24.1283; ... %! 5.1873 12.4188 16.5927; 3.3579 14.3951 10.2129; ... %! 6.3092 8.5986 9.8934; 3.2831 3.4945 10.0203]; %! [p, atab, stats] = anova2 (data, 4, "off", "nested"); %! assert (atab{2,2}, 745.360306290833, 1e-10); %! assert (atab{3,2}, 278.01854140125, 1e-10); %! assert (atab{4,2}, 180.180377467501, 1e-10); %! assert (atab{5,2}, 1203.55922515958, 1e-10); %! assert (atab{2,4}, 372.680153145417, 1e-10); %! assert (atab{3,4}, 92.67284713375, 1e-10); %! assert (atab{4,4}, 10.0100209704167, 1e-10); %! assert (atab{2,5}, 4.02146005730833, 1e-10); %! assert (atab{3,5}, 9.25800729165627, 1e-10); %! assert (atab{2,6}, 0.141597630656771, 1e-10); %! assert (atab{3,6}, 0.000636643812875719, 1e-10); statistics-release-1.6.3/inst/anovan.m000066400000000000000000002223731456127120000177550ustar00rootroot00000000000000## Copyright (C) 2003-2005 Andy Adler ## Copyright (C) 2021 Christian Scholz ## Copyright (C) 2022 Andreas Bertsatos ## Copyright (C) 2022 Andrew Penn ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{p} =} anovan (@var{Y}, @var{GROUP}) ## @deftypefnx {statistics} {@var{p} =} anovan (@var{Y}, @var{GROUP}, @var{name}, @var{value}) ## @deftypefnx {statistics} {[@var{p}, @var{atab}] =} anovan (@dots{}) ## @deftypefnx {statistics} {[@var{p}, @var{atab}, @var{stats}] =} anovan (@dots{}) ## @deftypefnx {statistics} {[@var{p}, @var{atab}, @var{stats}, @var{terms}] =} anovan (@dots{}) ## ## Perform a multi (N)-way analysis of (co)variance (ANOVA or ANCOVA) to ## evaluate the effect of one or more categorical or continuous predictors (i.e. ## independent variables) on a continuous outcome (i.e. dependent variable). The ## algorithms used make @code{anovan} suitable for balanced or unbalanced ## factorial (crossed) designs. By default, @code{anovan} treats all factors ## as fixed. Examples of function usage can be found by entering the command ## @code{demo anovan}. A bootstrap resampling variant of this function, ## @code{bootlm}, is available in the statistics-resampling package and has ## similar usage. ## ## Data is a single vector @var{Y} with groups specified by a corresponding ## matrix or cell array of group labels @var{GROUP}, where each column of ## @var{GROUP} has the same number of rows as @var{Y}. For example, if ## @code{@var{Y} = [1.1;1.2]; @var{GROUP} = [1,2,1; 1,5,2];} then observation ## 1.1 was measured under conditions 1,2,1 and observation 1.2 was measured ## under conditions 1,5,2. If the @var{GROUP} provided is empty, then the linear ## model is fit with just the intercept (no predictors). ## ## @code{anovan} can take a number of optional parameters as name-value pairs. ## ## @code{[@dots{}] = anovan (@var{Y}, @var{GROUP}, "continuous", @var{continuous})} ## ## @itemize ## @item ## @var{continuous} is a vector of indices indicating which of the columns (i.e. ## factors) in @var{GROUP} should be treated as continuous predictors rather ## than as categorical predictors. The relationship between continuous ## predictors and the outcome should be linear. ## @end itemize ## ## @code{[@dots{}] = anovan (@var{Y}, @var{GROUP}, "random", @var{random})} ## ## @itemize ## @item ## @var{random} is a vector of indices indicating which of the columns (i.e. ## factors) in @var{GROUP} should be treated as random effects rather than ## fixed effects. Octave @code{anovan} provides only basic support for random ## effects. Specifically, since all F-statistics in @code{anovan} are ## calculated using the mean-squared error (MSE), any interaction terms ## containing a random effect are dropped from the model term definitions and ## their associated variance is pooled with the residual, unexplained variance ## making up the MSE. In effect, the model then fitted equates to a linear mixed ## model with random intercept(s). Variable names for random factors are ## appended with a ' symbol. ## @end itemize ## ## @code{[@dots{}] = anovan (@var{Y}, @var{GROUP}, "model", @var{modeltype})} ## ## @itemize ## @item ## @var{modeltype} can specified as one of the following: ## ## @itemize ## @item ## "linear" (default) : compute @math{N} main effects with no interactions. ## ## @item ## "interaction" : compute @math{N} effects and @math{N*(N-1)} two-factor ## interactions ## ## @item ## "full" : compute the @math{N} main effects and interactions at all levels ## ## @item ## a scalar integer : representing the maximum interaction order ## ## @item ## a matrix of term definitions : each row is a term and each column is a factor ## @end itemize ## ## @example ## -- Example: ## A two-way ANOVA with interaction would be: [1 0; 0 1; 1 1] ## @end example ## ## @end itemize ## ## @code{[@dots{}] = anovan (@var{Y}, @var{GROUP}, "sstype", @var{sstype})} ## ## @itemize ## @item ## @var{sstype} can specified as one of the following: ## ## @itemize ## @item ## 1 : Type I sequential sums-of-squares. ## ## @item ## 2 or "h" : Type II partially sequential (or hierarchical) sums-of-squares ## ## @item ## 3 (default) : Type III partial, constrained or marginal sums-of-squares ## ## @end itemize ## @end itemize ## ## @code{[@dots{}] = anovan (@var{Y}, @var{GROUP}, "varnames", @var{varnames})} ## ## @itemize ## @item ## @var{varnames} must be a cell array of strings with each element containing a ## factor name for each column of @var{GROUP}. By default (if not parsed as ## optional argument), @var{varnames} are "X1","X2","X3", etc. ## @end itemize ## ## @code{[@dots{}] = anovan (@var{Y}, @var{GROUP}, "alpha", @var{alpha})} ## ## @itemize ## @item ## @var{alpha} must be a scalar value between 0 and 1 requesting ## @math{100*(1-@var{alpha})%} confidence bounds for the regression coefficients ## returned in @var{stats}.coeffs (default 0.05 for 95% confidence). ## @end itemize ## ## @code{[@dots{}] = anovan (@var{Y}, @var{GROUP}, "display", @var{dispopt})} ## ## @itemize ## @item ## @var{dispopt} can be either "on" (default) or "off" and controls the display ## of the model formula, table of model parameters, the ANOVA table and the ## diagnostic plots. The F-statistic and p-values are formatted in APA-style. ## To avoid p-hacking, the table of model parameters is only displayed if we set ## planned contrasts (see below). ## @end itemize ## ## @code{[@dots{}] = anovan (@var{Y}, @var{GROUP}, "contrasts", @var{contrasts})} ## ## @itemize ## @item ## @var{contrasts} can be specified as one of the following: ## ## @itemize ## @item ## A string corresponding to one of the built-in contrasts listed below: ## ## @itemize ## @item ## "simple" or "anova" (default): Simple (ANOVA) contrast coding. (The first ## level appearing in the @var{GROUP} column is the reference level) ## ## @item ## "poly": Polynomial contrast coding for trend analysis. ## ## @item ## "helmert": Helmert contrast coding: the difference between each level with ## the mean of the subsequent levels. ## ## @item ## "effect": Deviation effect coding. (The first level appearing in the ## @var{GROUP} column is omitted). ## ## @item ## "sdif" or "sdiff": Successive differences contrast coding: the difference ## between each level with the previous level. ## ## @item ## "treatment": Treatment contrast (or dummy) coding. (The first level appearing ## in the @var{GROUP} column is the reference level). These contrasts are not ## compatible with @var{sstype} = 3. ## ## @end itemize ## ## @item ## A matrix containing a custom contrast coding scheme (i.e. the generalized ## inverse of contrast weights). Rows in the contrast matrices correspond to ## factor levels in the order that they first appear in the @var{GROUP} column. ## The matrix must contain the same number of columns as there are the number of ## factor levels minus one. ## @end itemize ## ## If the anovan model contains more than one factor and a built-in contrast ## coding scheme was specified, then those contrasts are applied to all factors. ## To specify different contrasts for different factors in the model, ## @var{contrasts} should be a cell array with the same number of cells as there ## are columns in @var{GROUP}. Each cell should define contrasts for the ## respective column in @var{GROUP} by one of the methods described above. If ## cells are left empty, then the default contrasts are applied. Contrasts for ## cells corresponding to continuous factors are ignored. ## @end itemize ## ## @code{[@dots{}] = anovan (@var{Y}, @var{GROUP}, "weights", @var{weights})} ## ## @itemize ## @item ## @var{weights} is an optional vector of weights to be used when fitting the ## linear model. Weighted least squares (WLS) is used with weights (that is, ## minimizing @code{sum (@var{weights} * @var{residuals} .^ 2))}; otherwise ## ordinary least squares (OLS) is used (default is empty for OLS). ## @end itemize ## ## @code{anovan} can return up to four output arguments: ## ## @code{@var{p} = anovan (@dots{})} returns a vector of p-values, one for each ## term. ## ## @code{[@var{p}, @var{atab}] = anovan (@dots{})} returns a cell array ## containing the ANOVA table. ## ## @code{[@var{p}, @var{atab}, @var{stats}] = anovan (@dots{})} returns a ## structure containing additional statistics, including degrees of freedom and ## effect sizes for each term in the linear model, the design matrix, the ## variance-covariance matrix, (weighted) model residuals, and the mean squared ## error. The columns of @var{stats}.coeffs (from left-to-right) report the ## model coefficients, standard errors, lower and upper @math{100*(1-alpha)%} ## confidence interval bounds, t-statistics, and p-values relating to the ## contrasts. The number appended to each term name in @var{stats}.coeffnames ## corresponds to the column number in the relevant contrast matrix for that ## factor. The @var{stats} structure can be used as input for @code{multcompare}. ## ## @code{[@var{p}, @var{atab}, @var{stats}, @var{terms}] = anovan (@dots{})} ## returns the model term definitions. ## ## @seealso{anova1, anova2, multcompare, fitlm} ## @end deftypefn function [P, T, STATS, TERMS] = anovan (Y, GROUP, varargin) if (nargin < 2) error (strcat (["anovan usage: ""anovan (Y, GROUP)""; "], ... [" atleast 2 input arguments required"])); endif ## Check supplied parameters if ((numel (varargin) / 2) != fix (numel (varargin) / 2)) error ("anovan: wrong number of arguments.") endif MODELTYPE = "linear"; DISPLAY = "on"; SSTYPE = 3; VARNAMES = []; CONTINUOUS = []; RANDOM = []; CONTRASTS = {}; ALPHA = 0.05; WEIGHTS = []; for idx = 3:2:nargin name = varargin{idx-2}; value = varargin{idx-1}; switch (lower (name)) case "model" MODELTYPE = value; case "continuous" CONTINUOUS = value; case "random" RANDOM = value; case "nested" error (strcat (["anovan: nested ANOVA is not supported. Please use"], ... [" anova2 for fully balanced nested ANOVA designs."])); case "sstype" SSTYPE = value; case "varnames" VARNAMES = value; case {"display","displayopt"} DISPLAY = value; case "contrasts" CONTRASTS = value; case "alpha" ALPHA = value; case "weights" WEIGHTS = value; otherwise error (sprintf ("anovan: parameter %s is not supported", name)); endswitch endfor ## Evaluate continuous input argument if (isnumeric (CONTINUOUS)) if (any (CONTINUOUS != abs (fix (CONTINUOUS)))) error (strcat (["anovan: the value provided for the CONTINUOUS"], ... [" parameter must be a positive integer"])); endif else error (strcat (["anovan: the value provided for the CONTINUOUS"], ... [" parameter must be numeric"])); endif ## Accomodate for different formats for GROUP ## GROUP can be a matrix of numeric identifiers of a cell arrays ## of strings or numeric idenitiers N = size (GROUP, 2); # number of anova "ways" n = numel (Y); # total number of observations if (prod (size (Y)) != n) error ("anovan: for ""anovan (Y, GROUP)"", Y must be a vector"); endif if (numel (unique (CONTINUOUS)) > N) error (strcat (["anovan: the number of factors assigned as continuous"], ... [" cannot exceed the number of factors in GROUP"])); endif if (any ((CONTINUOUS > N) || any (CONTINUOUS <= 0))) error (strcat (["anovan: one or more indices provided in the value"], ... [" for the continuous parameter are out of range"])); endif cont_vec = false (1, N); cont_vec(CONTINUOUS) = true; if (iscell (GROUP)) if (size (GROUP, 1) == 1) tmp = cell (n, N); for j = 1:N if (isnumeric (GROUP{j})) if (ismember (j, CONTINUOUS)) tmp(:,j) = num2cell (GROUP{j}); else tmp(:,j) = cellstr (num2str (GROUP{j})); endif else if (ismember (j, CONTINUOUS)) error ("anovan: continuous factors must be a numeric datatype"); endif tmp(:,j) = GROUP{j}; endif endfor GROUP = tmp; endif endif if (! isempty (GROUP)) if (size (GROUP,1) != n) error ("anovan: GROUP must be a matrix with the same number of rows as Y"); endif endif if (! isempty (VARNAMES)) if (iscell (VARNAMES)) if (all (cellfun (@ischar, VARNAMES))) nvarnames = numel(VARNAMES); else error (strcat (["anovan: all variable names must be character"], ... [" or character arrays"])); endif elseif (ischar (VARNAMES)) nvarnames = 1; VARNAMES = {VARNAMES}; elseif (isstring (VARNAMES)) nvarnames = 1; VARNAMES = {char(VARNAMES)}; else error (strcat (["anovan: varnames is not of a valid type. Must be a cell"], ... [" array of character arrays, character array or string"])); endif else nvarnames = N; VARNAMES = arrayfun(@(x) ["X",num2str(x)], 1:N, "UniformOutput", 0); endif if (nvarnames != N) error (strcat (["anovan: number of variable names is not equal"], ... [" to the number of grouping variables"])); endif ## Evaluate random argument (if applicable) if (! isempty(RANDOM)) if (isnumeric (RANDOM)) if (any (RANDOM != abs (fix (RANDOM)))) error (strcat (["anovan: the value provided for the RANDOM"], ... [" parameter must be a positive integer"])); endif else error (strcat (["anovan: the value provided for the RANDOM"], ... [" parameter must be numeric"])); endif if (numel (RANDOM) > N) error (strcat (["anovan: the number of elements in RANDOM cannot"], ... [" exceed the number of columns in GROUP."])); endif if (max (RANDOM) > N) error (strcat (["anovan: the indices listed in RANDOM cannot"], ... [" exceed the number of columns in GROUP."])); endif for v = 1:N if (ismember (v, RANDOM)) VARNAMES{v} = strcat (VARNAMES{v},"'"); endif endfor endif ## Evaluate contrasts (if applicable) if isempty (CONTRASTS) CONTRASTS = cell (1, N); planned = false; else if (ischar(CONTRASTS)) contr_str = CONTRASTS; CONTRASTS = cell (1, N); CONTRASTS(:) = {contr_str}; endif if (! iscell (CONTRASTS)) CONTRASTS = {CONTRASTS}; endif for i = 1:N if (! isempty (CONTRASTS{i})) msg = strcat(["columns in CONTRASTS must sum to"], ... [" 0 for SSTYPE 3. Switching to SSTYPE 2 instead."]); if (isnumeric(CONTRASTS{i})) ## Check whether all the columns sum to 0 if (any (abs (sum (CONTRASTS{i})) > eps ('single'))) warning (sprintf ( ... 'Note that the CONTRASTS for predictor %u do not sum to zero', i)); endif ## Check whether contrasts are orthogonal if (any (abs (reshape (corr (CONTRASTS{i}) - ... eye (size (CONTRASTS{i}, 2)), [], 1))... > eps ('single'))) warning (sprintf ( ... 'Note that the CONTRASTS for predictor %u are not orthogonal', i)); endif else if (! ismember (lower (CONTRASTS{i}), ... {"simple","anova","poly","helmert","effect",... "sdif","sdiff","treatment"})) error (strcat(["anovan: valid built-in contrasts are:"], ... [" ""simple"", ""poly"", ""helmert"","],... ["""effect"", ""sdif"" or ""treatment"""])); endif if (strcmpi (CONTRASTS{i}, "treatment") && (SSTYPE==3)) warning (msg); SSTYPE = 2; endif endif endif endfor planned = true; endif ## Evaluate alpha input argument if (! isa (ALPHA,'numeric') || numel (ALPHA) != 1) error("anovan: alpha must be a numeric scalar value"); endif if ((ALPHA <= 0) || (ALPHA >= 1)) error("anovan: alpha must be a value between 0 and 1"); endif ## Remove NaN or non-finite observations if (isempty (GROUP)) excl = any ([isnan(Y), isinf(Y)], 2); else XC = GROUP(:,CONTINUOUS); if iscell(XC) XC = cell2mat (XC); endif excl = any ([isnan(Y), isinf(Y), any(isnan(XC),2), any(isinf(XC),2)], 2); GROUP(excl,:) = []; endif Y(excl) = []; if (size (Y, 1) == 1) Y = Y.'; # if Y is a row vector, make it a column vector endif n = numel (Y); # recalculate total number of observations ## Evaluate weights input argument if (! isempty(WEIGHTS)) if (! isnumeric(WEIGHTS)) error ("anovan: WEIGHTS must be a numeric datatype"); endif if (any (size (WEIGHTS) != [n,1])) error ("anovan: WEIGHTS must be a vector with the same dimensions as Y"); endif if (any(!(WEIGHTS > 0)) || any (isinf (WEIGHTS))) error ("anovan: WEIGHTS must be a vector of positive finite values"); endif # Create diaganal matrix of normalized weights W = diag (WEIGHTS / mean (WEIGHTS)); else # Create identity matrix W = eye (n);; endif ## Evaluate model type input argument and create terms matrix if not provided msg = strcat (["anovan: the number of columns in the term definitions"], ... [" cannot exceed the number of columns of GROUP"]); if (ischar (MODELTYPE)) switch (lower (MODELTYPE)) case "linear" MODELTYPE = 1; case {"interaction","interactions"} MODELTYPE = 2; case "full" MODELTYPE = N; otherwise error ("anovan: model type not recognised"); endswitch endif if (isscalar (MODELTYPE)) TERMS = cell (MODELTYPE,1); v = false (1, N); switch (lower (MODELTYPE)) case 1 ## Create term definitions for an additive linear model TERMS = eye (N); case 2 ## Create term definitions for a model with two factor interactions if (N > 1) Nx = nchoosek (N, 2); else Nx = 0; endif TERMS = zeros (N + Nx, N); TERMS(1:N,:) = eye (N); for j = 1:N for i = j:N-1 TERMS(N+j+i-1,j) = 1; TERMS(N+j+i-1,i+1) = 1; endfor endfor otherwise if (MODELTYPE > N) error (msg); endif ## Create term definitions for a full model Nx = zeros (1, N-1); Nx = 0; for k = 1:N Nx = Nx + nchoosek(N,k); endfor for j = 1:MODELTYPE v(1:j) = 1; TERMS(j) = flipud (unique (perms (v), "rows")); endfor TERMS = cell2mat (TERMS); endswitch TERMS = logical (TERMS); else ## Assume that the user provided a suitable matrix of term definitions if (size (MODELTYPE, 2) > N) error (msg); endif if (! all (ismember (MODELTYPE(:), [0,1]))) error (strcat (["anovan: elements of the model terms matrix"], ... [" must be either 0 or 1"])); endif TERMS = logical (MODELTYPE); endif ## Evaluate terms matrix Ng = sum (TERMS, 2); if (any (diff (Ng) < 0)) error (strcat (["anovan: the model terms matrix must list main"], ... [" effects above/before interactions"])); endif ## Drop terms that include interactions with factors specified as random effects. drop = any (bsxfun (@and, TERMS(:,RANDOM), (Ng > 1)), 2); TERMS (drop, :) = []; Ng(drop) = []; ## Evaluate terms Nm = sum (Ng == 1); Nx = sum (Ng > 1); Nt = Nm + Nx; if (any (any (TERMS(1:Nm,:), 1) != any (TERMS, 1))) error (strcat (["anovan: all factors involved in interactions"], ... [" must have a main effect"])); endif ## Calculate total sum-of-squares ct = sum (Y)^2 / n; % correction term sst = sum (Y.^2) - ct; dft = n - 1; ## Create design matrix mDesignMatrix (); ## Fit linear models, and calculate sums-of-squares for ANOVA switch (lower (SSTYPE)) case 1 ## Type I sequential sums-of-squares (SSTYPE = 1) R = sst; ss = zeros (Nt,1); for j = 1:Nt XS = cell2mat (X(1:j+1)); [b, sse] = lmfit (XS, Y, W); ss(j) = R - sse; R = sse; endfor [b, sse, resid, ucov, hat] = lmfit (XS, Y, W); sstype_char = "I"; case {2,'h'} ## Type II (partially sequential, or hierarchical) sums-of-squares ss = zeros (Nt,1); for j = 1:Nt i = find (TERMS(j,:)); k = cat (1, 1, 1 + find (any (!TERMS(:,i),2))); XS = cell2mat (X(k)); [jnk, R1] = lmfit (XS, Y, W); k = cat (1, j+1, k); XS = cell2mat (X(k)); [jnk, R2] = lmfit (XS, Y, W); ss(j) = R1 - R2; endfor [b, sse, resid, ucov, hat] = lmfit (cell2mat (X), Y, W); sstype_char = "II"; case 3 ## Type III (partial, constrained or marginal) sums-of-squares ss = zeros (Nt, 1); [b, sse, resid, ucov, hat] = lmfit (cell2mat (X), Y, W); for j = 1:Nt XS = cell2mat (X(1:Nt+1 != j+1)); [jnk, R] = lmfit (XS, Y, W); ss(j) = R - sse; endfor sstype_char = "III"; otherwise error ("anovan: sstype value not supported"); endswitch ss = max (0, ss); # Truncate negative SS at 0 dfe = dft - sum (df); ms = ss ./ df; mse = sse / dfe; eta_sq = ss ./ sst; partial_eta_sq = ss ./ (ss + sse); F = ms / mse; P = 1 - fcdf (F, df, dfe); ## Prepare model formula and cell array containing the ANOVA table T = cell (Nt + 3, 7); T(1,:) = {"Source", "Sum Sq.", "d.f.", "Mean Sq.", "Eta Sq.", "F", "Prob>F"}; T(2:Nt+1,2:7) = num2cell ([ss df ms partial_eta_sq F P]); T(end-1,1:4) = {"Error", sse, dfe, mse}; T(end,1:3) = {"Total", sst, dft}; formula = sprintf ("Y ~ 1"); # Initialise model formula for i = 1:Nt str = sprintf ("%s*", VARNAMES{find (TERMS(i,:))}); T(i+1,1) = str(1:end-1); ## Append model term to formula str = regexprep (str, "\\*", ":"); if (strcmp (str(end-1), "'")) ## Random intercept term formula = sprintf ("%s + (1|%s)", formula, str(1:end-2)); ## Remove statistics for random factors from the ANOVA table #T(RANDOM+1,4:7) = cell(1,4); #P(RANDOM) = NaN; else ## Fixed effect term formula = sprintf ("%s + %s", formula, str(1:end-1)); endif endfor ## Calculate a standard error, t-statistic and p-value for each ## of the regression coefficients (fixed effects only) t_crit = tinv (1 - ALPHA / 2, dfe); se = sqrt (diag (ucov) * mse); t = b ./ se; p = 2 * (1 - (tcdf (abs (t), dfe))); coeff_stats = zeros (1 + sum (df), 4); coeff_stats(:,1) = b; # coefficients coeff_stats(:,2) = se; # standard errors coeff_stats(:,3) = b - se * t_crit; # Lower CI bound coeff_stats(:,4) = b + se * t_crit; # Upper CI bound coeff_stats(:,5) = t; # t-statistics coeff_stats(:,6) = p; # p-values ## Assign NaN to p-value to avoid printing statistics relating to ## coefficients for 'random' effects hi = 1 + cumsum(df); for ignore = RANDOM p(hi(ignore)-df(ignore)+1:hi(ignore)) = NaN; endfor ## Compute leverage values and Cook's distance h = diag (hat); % Leverage values D = resid.^2 / ((1 + sum (df)) * mse) ... .* h ./ (1 - h).^2; % Cook's distance ## Create STATS structure for MULTCOMPARE STATS = struct ("source","anovan", ... "resid", resid, ... # These are weighted (not raw) residuals "coeffs", coeff_stats, ... "Rtr", [], ... # Not used by Octave "rowbasis", [], ... # Not used by Octave "dfe", dfe, ... "mse", mse, ... "nullproject", [], ... # Not used by Octave "terms", TERMS, ... "nlevels", nlevels, ... "continuous", cont_vec, ... "vmeans", vmeans, ... "termcols", termcols, ... "coeffnames", {cellstr(char(coeffnames{:}))}, ... "vars", [], ... # Not used by Octave "varnames", {VARNAMES}, ... "grpnames", {levels}, ... "vnested", [], ... # Not used since "nested" argument name is not supported "ems", [], ... # Not used since "nested" argument name is not supported "denom", [], ... # Not used since interactions with random factors is not supported "dfdenom", [], ... # Not used since interactions with random factors is not supported "msdenom", [], ... # Not used since interactions with random factors is not supported "varest", [], ... # Not used since interactions with random factors is not supported "varci", [], ... # Not used since interactions with random factors is not supported "txtdenom", [], ... # Not used since interactions with random factors is not supported "txtems", [], ... # Not used since interactions with random factors is not supported "rtnames", [], ... # Not used since interactions with random factors is not supported ## Additional STATS fields used exclusively by Octave "center_continuous", center_continuous, ... "random", RANDOM, ... "formula", formula, ... "alpha", ALPHA, ... "df", df, ... "contrasts", {CONTRASTS}, ... "X", sparse (cell2mat (X)), ... "Y", Y, ... "W", sparse (W), ... "lmfit", @lmfit, ... "vcov", sparse (ucov * mse), ... "CooksD", D, ... "grps", gid, ... "eta_squared", eta_sq, ... "partial_eta_squared", partial_eta_sq); ## Print ANOVA table switch (lower (DISPLAY)) case {"on", true} ## Print model formula fprintf("\nMODEL FORMULA (based on Wilkinson's notation):\n\n%s\n", formula); ## If applicable, print parameter estimates (a.k.a contrasts) for fixed effects if (planned && ! isempty(GROUP)) ## Parameter estimates correspond to the contrasts we set. To avoid ## p-hacking, don't print contrasts if we don't specify them to start with fprintf("\nMODEL PARAMETERS (contrasts for the fixed effects)\n\n"); fprintf("Parameter Estimate SE Lower.CI Upper.CI t Prob>|t|\n"); fprintf("--------------------------------------------------------------------------------\n"); for j = 1:size (coeff_stats, 1) if (p(j) < 0.001) fprintf ("%-20s %10.3g %9.3g %9.3g %9.3g %8.2f <.001 \n", ... STATS.coeffnames{j}, STATS.coeffs(j,1:end-1)); elseif (p(j) < 0.9995) fprintf ("%-20s %10.3g %9.3g %9.3g %9.3g %8.2f .%03u \n", ... STATS.coeffnames{j}, STATS.coeffs(j,1:end-1), round (p(j) * 1e+03)); elseif (isnan(p(j))) ## Don't display coefficients for 'random' effects since they were ## treated as fixed effects else fprintf ("%-20s %10.3g %9.3g %9.3g %9.3g %8.2f 1.000 \n", ... STATS.coeffnames{j}, STATS.coeffs(j,1:end-1)); endif endfor endif ## Print ANOVA table [nrows, ncols] = size (T); fprintf("\nANOVA TABLE (Type %s sums-of-squares):\n\n", sstype_char); fprintf("Source Sum Sq. d.f. Mean Sq. R Sq. F Prob>F\n"); fprintf("--------------------------------------------------------------------------------\n"); for i = 1:Nt str = T{i+1,1}; l = numel(str); # Needed to truncate source term name at 18 characters ## Format and print the statistics for each model term ## Format F statistics and p-values in APA style if (P(i) < 0.001) fprintf ("%-20s %10.5g %6d %10.5g %4.3f %11.2f <.001 \n", ... str(1:min(18,l)), T{i+1,2:end-1}); elseif (P(i) < 0.9995) fprintf ("%-20s %10.5g %6d %10.5g %4.3f %11.2f .%03u \n", ... str(1:min(18,l)), T{i+1,2:end-1}, round (P(i) * 1e+03)); elseif (isnan(P(i))) fprintf ("%-20s %10.5g %6d \n", str(1:min(18,l)), T{i+1,2:3}); else fprintf ("%-20s %10.5g %6d %10.5g %4.3f %11.2f 1.000 \n", ... str(1:min(18,l)), T{i+1,2:end-1}); endif endfor fprintf("Error %10.5g %6d %10.5g\n", T{end-1,2:4}); fprintf("Total %10.5g %6d \n", T{end,2:3}); fprintf("\n"); ## Make figure of diagnostic plots figure ("Name", "Diagnostic Plots: Model Residuals"); t = STATS.resid ./ (sqrt (mse * (1 - h))); % Studentized residuals fit = STATS.X * STATS.coeffs(:,1); % Fitted values [jnk, DI] = sort (D, "descend"); % Indices of sorted D nk = 4; % Top nk residuals with largest D ## Normal quantile-quantile plot subplot (2, 2, 1); x = ((1 : n)' - .5) / n; [ts, I] = sort (t); q = norminv (x); plot (q, ts, "ok", "markersize", 3); box off; grid on; xlabel ("Theoretical quantiles"); ylabel ("Studentized Residuals"); title ("Normal Q-Q Plot"); arrayfun (@(i) text (q(I == DI(i)), t(DI(i)), ... sprintf (" %u", DI(i))), [1:min(nk,n)]) iqr = [0.25; 0.75]; yl = quantile (t, iqr, 1, 6); xl = norminv (iqr); slope = diff (yl) / diff (xl); int = yl(1) - slope * xl(1); ax1_xlim = get (gca, "XLim"); hold on; plot (ax1_xlim, slope * ax1_xlim + int, "k-"); hold off; set (gca, "Xlim", ax1_xlim); ## Spread-Location Plot subplot (2, 2, 2); plot (fit, sqrt (abs (t)), "ko", "markersize", 3); box off; xlabel ("Fitted values"); ylabel ("sqrt ( | Studentized Residuals | )"); title ("Spread-Location Plot") ax2_xlim = get (gca, "XLim"); hold on; plot (ax2_xlim, ones (1, 2) * sqrt (2), "k:"); plot (ax2_xlim, ones (1, 2) * sqrt (3), "k-."); plot (ax2_xlim, ones (1, 2) * sqrt (4), "k--"); hold off; arrayfun (@(i) text (fit(DI(i)), sqrt (abs (t(DI(i)))), ... sprintf (" %u", DI(i))), [1:min(nk,n)]); xlim (ax2_xlim); ## Residual-Leverage plot subplot (2, 2, 3); plot (h, t, "ko", "markersize", 3); box off; xlabel ("Leverage") ylabel ("Studentized Residuals"); title ("Residual-Leverage Plot") ax3_xlim = get (gca, "XLim"); ax3_ylim = get (gca, "YLim"); hold on; plot (ax3_xlim, zeros (1, 2), "k-"); hold off; arrayfun (@(i) text (h(DI(i)), t(DI(i)), ... sprintf (" %u", DI(i))), [1:min(nk,n)]); set (gca, "ygrid", "on"); xlim (ax3_xlim); ylim (ax3_ylim); ## Cook's distance stem plot subplot (2, 2, 4); stem (D, "ko", "markersize", 3); box off; xlabel ("Obs. number") ylabel ("Cook's distance") title ("Cook's Distance Stem Plot") xlim ([0, n]); ax4_xlim = get (gca, "XLim"); ax4_ylim = get (gca, "YLim"); hold on; plot (ax4_xlim, ones (1, 2) * 4 / dfe, "k:"); plot (ax4_xlim, ones (1, 2) * 0.5, "k-."); plot (ax4_xlim, ones (1, 2), "k--"); hold off; arrayfun (@(i) text (DI(i), D(DI(i)), ... sprintf (" %u", DI(i))), [1:min(nk,n)]); xlim (ax4_xlim); ylim (ax4_ylim); set (findall ( gcf, "-property", "FontSize"), "FontSize", 7) case {"off", false} ## do nothing otherwise error ("anovan: wrong value for 'display' parameter."); endswitch function mDesignMatrix () ## Nested function that returns a cell array of the design matrix for ## each term in the model ## Input variables it uses: ## GROUP, TERMS, CONTINUOUS, CONTRASTS, VARNAMES, n, Nm, Nx, Ng ## Variables it creates or modifies: ## X, grpnames, nlevels, df, termcols, coeffnames, vmeans, gid, CONTRASTS ## EVALUATE FACTOR LEVELS levels = cell (Nm, 1); gid = zeros (n, Nm); nlevels = zeros (Nm, 1); df = zeros (Nm + Nx, 1); termcols = ones (1 + Nm + Nx, 1); for j = 1:Nm if (any (j == CONTINUOUS)) ## CONTINUOUS PREDICTOR nlevels(j) = 1; termcols(j+1) = 1; df(j) = 1; if iscell (GROUP(:,j)) gid(:,j) = cell2mat ([GROUP(:,j)]); else gid(:,j) = GROUP(:,j); end else ## CATEGORICAL PREDICTOR levels{j} = unique (GROUP(:,j), "stable"); if isnumeric (levels{j}) levels{j} = num2cell (levels{j}); endif nlevels(j) = numel (levels{j}); for k = 1:nlevels(j) gid(ismember (GROUP(:,j),levels{j}{k}),j) = k; endfor termcols(j+1) = nlevels(j); df(j) = nlevels(j) - 1; endif endfor ## MAKE DESIGN MATRIX ## MAIN EFFECTS X = cell (1, 1 + Nm + Nx); X(1) = ones (n, 1); coeffnames = cell (1, 1 + Nm + Nx); coeffnames(1) = "(Intercept)"; vmeans = zeros (Nm, 1); center_continuous = cont_vec; for j = 1:Nm if (any (j == CONTINUOUS)) ## CONTINUOUS PREDICTOR if iscell (GROUP(:,j)) X(1+j) = cell2mat (GROUP(:,j)); else X(1+j) = GROUP(:,j); end if (strcmpi (CONTRASTS{j}, 'treatment')) ## Don't center continuous variables if contrasts are 'treatment' center_continuous(j) = false; CONTRASTS{j} = []; else center_continuous(j) = true; vmeans(j) = mean ([X{1+j}]); X(1+j) = [X{1+j}] - vmeans(j); endif ## Create names of the coefficients relating to continuous main effects coeffnames{1+j} = VARNAMES{j}; else ## CATEGORICAL PREDICTOR if (isempty (CONTRASTS{j})) CONTRASTS{j} = contr_simple (nlevels(j)); else switch (lower (CONTRASTS{j})) case {"simple","anova"} ## SIMPLE EFFECT CODING (DEFAULT) ## The first level is the reference level CONTRASTS{j} = contr_simple (nlevels(j)); case "poly" ## POLYNOMIAL CONTRAST CODING CONTRASTS{j} = contr_poly (nlevels(j)); case "helmert" ## HELMERT CONTRAST CODING CONTRASTS{j} = contr_helmert (nlevels(j)); case "effect" ## DEVIATION EFFECT CONTRAST CODING CONTRASTS{j} = contr_sum (nlevels(j)); case {"sdif","sdiff"} ## SUCCESSIVE DEVIATIONS CONTRAST CODING CONTRASTS{j} = contr_sdif (nlevels(j)); case "treatment" ## The first level is the reference level CONTRASTS{j} = contr_treatment (nlevels(j)); otherwise ## EVALUATE CUSTOM CONTRAST MATRIX ## Check that the contrast matrix provided is the correct size if (! all (size (CONTRASTS{j},1) == nlevels(j))) error (strcat (["anovan: the number of rows in the contrast"], ... [" matrices should equal the number of factor levels"])); endif if (! all (size (CONTRASTS{j},2) == df(j))) error (strcat (["anovan: the number of columns in each contrast"], ... [" matrix should equal the degrees of freedom (i.e."], ... [" number of levels minus 1) for that factor"])); endif if (! all (any (CONTRASTS{j}))) error (strcat (["anovan: a contrast must be coded in each"], ... [" column of the contrast matrices"])); endif endswitch endif C = CONTRASTS{j}; func = @(x) x(gid(:,j)); X(1+j) = cell2mat (cellfun (func, num2cell (C, 1), "UniformOutput", false)); ## Create names of the coefficients relating to continuous main effects coeffnames{1+j} = cell (df(j),1); for v = 1:df(j) coeffnames{1+j}{v} = sprintf ("%s_%u", VARNAMES{j}, v); endfor endif endfor ## INTERACTION TERMS if (Nx > 0) row = TERMS((Ng > 1),:); for i = 1:Nx I = 1 + find (row(i,:)); df(Nm+i) = prod (df(I-1)); termcols(1+Nm+i) = prod (df(I-1) + 1); tmp = ones (n,1); for j = 1:numel(I); tmp = num2cell (tmp, 1); for k = 1:numel(tmp) tmp(k) = bsxfun (@times, tmp{k}, X{I(j)}); endfor tmp = cell2mat (tmp); endfor X{1+Nm+i} = tmp; coeffnames{1+Nm+i} = cell (df(Nm+i),1); for v = 1:df(Nm+i) str = sprintf ("%s:", VARNAMES{I-1}); coeffnames{1+Nm+i}{v} = strcat (str(1:end-1), "_", num2str (v)); endfor endfor endif endfunction endfunction ## BUILT IN CONTRAST CODING FUNCTIONS function C = contr_simple (N) ## Create contrast matrix (of doubles) using simple (ANOVA) contrast coding ## These contrasts are centered (i.e. sum to 0) ## Ideal for unordered factors, with comparison to a reference level ## The first factor level is the reference level C = cat (1, zeros (1,N-1), eye(N-1)) - 1/N; endfunction function C = contr_poly (N) ## Create contrast matrix (of doubles) using polynomial contrast coding ## for trend analysis of ordered categorical factor levels ## These contrasts are orthogonal and centered (i.e. sum to 0) ## Ideal for ordered factors [C, jnk] = qr (bsxfun (@power, [1:N]' - mean ([1:N]'), [0:N-1])); C(:,1) = []; s = ones (1, N-1); s(1:2:N-1) *= -1; f = (sign(C(1,:)) != s); C(:,f) *= -1; endfunction function C = contr_helmert (N) ## Create contrast matrix (of doubles) using Helmert coding contrasts ## These contrasts are orthogonal and centered (i.e. sum to 0) C = cat (1, tril (-ones (N-1), -1) + diag (N-1:-1:1), ... -ones (1, N-1)) ./ (N:-1:2); endfunction function C = contr_sum (N) ## Create contrast matrix (of doubles) using deviation effect coding ## These contrasts are centered (i.e. sum to 0) C = cat (1, - (ones (1,N-1)), eye (N-1)); endfunction function C = contr_sdif (N) ## Create contrast matrix (of doubles) using successive differences coding ## These contrasts are centered (i.e. sum to 0) C = tril (ones (N, N - 1), -1) - ones (N, 1) / N * [N - 1 : -1 : 1]; endfunction function C = contr_treatment (N) ## Create contrast matrix (of doubles) using treatment contrast coding ## Not compatible with SSTYPE 3 since contrasts are not centered ## Ideal for unordered factors, with comparison to a reference level ## The first factor level is the reference level C = cat (1, zeros (1,N-1), eye(N-1)); endfunction ## FUNCTION TO FIT THE LINEAR MODEL function [b, sse, resid, ucov, hat] = lmfit (X, Y, W) ## Get model coefficients by solving the linear equation by QR decomposition ## The number of free parameters (i.e. intercept + coefficients) is equal ## to n - dfe. If optional arument W is provided, it should be a diagonal ## matrix of weights or a positive definite covariance matrix if (nargin < 3) ## If no weights are provided, create an identity matrix n = numel (Y); W = eye (n); endif C = chol (W); XW = C*X; YW = C*Y; [Q, R] = qr (XW, 0); b = R \ Q' * YW; ## Get fitted values fit = Q'\R * b; # This is equivalent to fit = XW * b; ## Get residuals from the fit resid = YW - fit; ## Calculate the residual sums-of-squares sse = sum (resid.^2); ## Calculate the unscaled covariance matrix (i.e. inv (X'*X )) if (nargout > 3) ucov = R \ Q' / XW'; endif ## Calculate the Hat matrix if (nargout > 4) w = diag (W); rw = sqrt (w); Q1 = diag (1 ./ rw) * Q; Q2 = diag (rw) * Q; hat = Q1 * Q2'; endif endfunction %!demo %! %! # Two-sample unpaired test on independent samples (equivalent to Student's %! # t-test). Note that the absolute value of t-statistic can be obtained by %! # taking the square root of the reported F statistic. In this example, %! # t = sqrt (1.44) = 1.20. %! %! score = [54 23 45 54 45 43 34 65 77 46 65]'; %! gender = {"male" "male" "male" "male" "male" "female" "female" "female" ... %! "female" "female" "female"}'; %! %! [P, ATAB, STATS] = anovan (score, gender, "display", "on", "varnames", "gender"); %!demo %! %! # Two-sample paired test on dependent or matched samples equivalent to a %! # paired t-test. As for the first example, the t-statistic can be obtained by %! # taking the square root of the reported F statistic. Note that the interaction %! # between treatment x subject was dropped from the full model by assigning %! # subject as a random factor ('). %! %! score = [4.5 5.6; 3.7 6.4; 5.3 6.4; 5.4 6.0; 3.9 5.7]'; %! treatment = {"before" "after"; "before" "after"; "before" "after"; %! "before" "after"; "before" "after"}'; %! subject = {"GS" "GS"; "JM" "JM"; "HM" "HM"; "JW" "JW"; "PS" "PS"}'; %! %! [P, ATAB, STATS] = anovan (score(:), {treatment(:), subject(:)}, ... %! "model", "full", "random", 2, "sstype", 2, ... %! "varnames", {"treatment", "subject"}, ... %! "display", "on"); %!demo %! %! # One-way ANOVA on the data from a study on the strength of structural beams, %! # in Hogg and Ledolter (1987) Engineering Statistics. New York: MacMillan %! %! strength = [82 86 79 83 84 85 86 87 74 82 ... %! 78 75 76 77 79 79 77 78 82 79]'; %! alloy = {"st","st","st","st","st","st","st","st", ... %! "al1","al1","al1","al1","al1","al1", ... %! "al2","al2","al2","al2","al2","al2"}'; %! %! [P, ATAB, STATS] = anovan (strength, alloy, "display", "on", ... %! "varnames", "alloy"); %!demo %! %! # One-way repeated measures ANOVA on the data from a study on the number of %! # words recalled by 10 subjects for three time condtions, in Loftus & Masson %! # (1994) Psychon Bull Rev. 1(4):476-490, Table 2. Note that the interaction %! # between seconds x subject was dropped from the full model by assigning %! # subject as a random factor ('). %! %! words = [10 13 13; 6 8 8; 11 14 14; 22 23 25; 16 18 20; ... %! 15 17 17; 1 1 4; 12 15 17; 9 12 12; 8 9 12]; %! seconds = [1 2 5; 1 2 5; 1 2 5; 1 2 5; 1 2 5; ... %! 1 2 5; 1 2 5; 1 2 5; 1 2 5; 1 2 5;]; %! subject = [ 1 1 1; 2 2 2; 3 3 3; 4 4 4; 5 5 5; ... %! 6 6 6; 7 7 7; 8 8 8; 9 9 9; 10 10 10]; %! %! [P, ATAB, STATS] = anovan (words(:), {seconds(:), subject(:)}, ... %! "model", "full", "random", 2, "sstype", 2, ... %! "display", "on", "varnames", {"seconds", "subject"}); %!demo %! %! # Balanced two-way ANOVA with interaction on the data from a study of popcorn %! # brands and popper types, in Hogg and Ledolter (1987) Engineering Statistics. %! # New York: MacMillan %! %! popcorn = [5.5, 4.5, 3.5; 5.5, 4.5, 4.0; 6.0, 4.0, 3.0; ... %! 6.5, 5.0, 4.0; 7.0, 5.5, 5.0; 7.0, 5.0, 4.5]; %! brands = {"Gourmet", "National", "Generic"; ... %! "Gourmet", "National", "Generic"; ... %! "Gourmet", "National", "Generic"; ... %! "Gourmet", "National", "Generic"; ... %! "Gourmet", "National", "Generic"; ... %! "Gourmet", "National", "Generic"}; %! popper = {"oil", "oil", "oil"; "oil", "oil", "oil"; "oil", "oil", "oil"; ... %! "air", "air", "air"; "air", "air", "air"; "air", "air", "air"}; %! %! [P, ATAB, STATS] = anovan (popcorn(:), {brands(:), popper(:)}, ... %! "display", "on", "model", "full", ... %! "varnames", {"brands", "popper"}); %!demo %! %! # Unbalanced two-way ANOVA (2x2) on the data from a study on the effects of %! # gender and having a college degree on salaries of company employees, %! # in Maxwell, Delaney and Kelly (2018): Chapter 7, Table 15 %! %! salary = [24 26 25 24 27 24 27 23 15 17 20 16, ... %! 25 29 27 19 18 21 20 21 22 19]'; %! gender = {"f" "f" "f" "f" "f" "f" "f" "f" "f" "f" "f" "f"... %! "m" "m" "m" "m" "m" "m" "m" "m" "m" "m"}'; %! degree = [1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 0 0 0 0 0 0 0]'; %! %! [P, ATAB, STATS] = anovan (salary, {gender, degree}, "model", "full", ... %! "sstype", 3, "display", "on", "varnames", ... %! {"gender", "degree"}); %!demo %! %! # Unbalanced two-way ANOVA (3x2) on the data from a study of the effect of %! # adding sugar and/or milk on the tendency of coffee to make people babble, %! # in from Navarro (2019): 16.10 %! %! sugar = {"real" "fake" "fake" "real" "real" "real" "none" "none" "none" ... %! "fake" "fake" "fake" "real" "real" "real" "none" "none" "fake"}'; %! milk = {"yes" "no" "no" "yes" "yes" "no" "yes" "yes" "yes" ... %! "no" "no" "yes" "no" "no" "no" "no" "no" "yes"}'; %! babble = [4.6 4.4 3.9 5.6 5.1 5.5 3.9 3.5 3.7... %! 5.6 4.7 5.9 6.0 5.4 6.6 5.8 5.3 5.7]'; %! %! [P, ATAB, STATS] = anovan (babble, {sugar, milk}, "model", "full", ... %! "sstype", 3, "display", "on", ... %! "varnames", {"sugar", "milk"}); %!demo %! %! # Unbalanced three-way ANOVA (3x2x2) on the data from a study of the effects %! # of three different drugs, biofeedback and diet on patient blood pressure, %! # adapted* from Maxwell, Delaney and Kelly (2018): Chapter 8, Table 12 %! # * Missing values introduced to make the sample sizes unequal to test the %! # calculation of different types of sums-of-squares %! %! drug = {"X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" ... %! "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X"; %! "Y" "Y" "Y" "Y" "Y" "Y" "Y" "Y" "Y" "Y" "Y" "Y" ... %! "Y" "Y" "Y" "Y" "Y" "Y" "Y" "Y" "Y" "Y" "Y" "Y"; %! "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" ... %! "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z"}; %! feedback = [1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0; %! 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0; %! 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0]; %! diet = [0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1 1; %! 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1 1; %! 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1 1]; %! BP = [170 175 165 180 160 158 161 173 157 152 181 190 ... %! 173 194 197 190 176 198 164 190 169 164 176 175; %! 186 194 201 215 219 209 164 166 159 182 187 174 ... %! 189 194 217 206 199 195 171 173 196 199 180 NaN; %! 180 187 199 170 204 194 162 184 183 156 180 173 ... %! 202 228 190 206 224 204 205 199 170 160 NaN NaN]; %! %! [P, ATAB, STATS] = anovan (BP(:), {drug(:), feedback(:), diet(:)}, ... %! "model", "full", "sstype", 3, ... %! "display", "on", ... %! "varnames", {"drug", "feedback", "diet"}); %!demo %! %! # Balanced three-way ANOVA (2x2x2) with one of the factors being a blocking %! # factor. The data is from a randomized block design study on the effects %! # of antioxidant treatment on glutathione-S-transferase (GST) levels in %! # different mouse strains, from Festing (2014), ILAR Journal, 55(3):427-476. %! # Note that all interactions involving block were dropped from the full model %! # by assigning block as a random factor ('). %! %! measurement = [444 614 423 625 408 856 447 719 ... %! 764 831 586 782 609 1002 606 766]'; %! strain= {"NIH","NIH","BALB/C","BALB/C","A/J","A/J","129/Ola","129/Ola", ... %! "NIH","NIH","BALB/C","BALB/C","A/J","A/J","129/Ola","129/Ola"}'; %! treatment={"C" "T" "C" "T" "C" "T" "C" "T" "C" "T" "C" "T" "C" "T" "C" "T"}'; %! block = [1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2]'; %! %! [P, ATAB, STATS] = anovan (measurement/10, {strain, treatment, block}, ... %! "sstype", 2, "model", "full", "random", 3, ... %! "display", "on", ... %! "varnames", {"strain", "treatment", "block"}); %!demo %! %! # One-way ANCOVA on data from a study of the additive effects of species %! # and temperature on chirpy pulses of crickets, from Stitch, The Worst Stats %! # Text eveR %! %! pulse = [67.9 65.1 77.3 78.7 79.4 80.4 85.8 86.6 87.5 89.1 ... %! 98.6 100.8 99.3 101.7 44.3 47.2 47.6 49.6 50.3 51.8 ... %! 60 58.5 58.9 60.7 69.8 70.9 76.2 76.1 77 77.7 84.7]'; %! temp = [20.8 20.8 24 24 24 24 26.2 26.2 26.2 26.2 28.4 ... %! 29 30.4 30.4 17.2 18.3 18.3 18.3 18.9 18.9 20.4 ... %! 21 21 22.1 23.5 24.2 25.9 26.5 26.5 26.5 28.6]'; %! species = {"ex" "ex" "ex" "ex" "ex" "ex" "ex" "ex" "ex" "ex" "ex" ... %! "ex" "ex" "ex" "niv" "niv" "niv" "niv" "niv" "niv" "niv" ... %! "niv" "niv" "niv" "niv" "niv" "niv" "niv" "niv" "niv" "niv"}; %! %! [P, ATAB, STATS] = anovan (pulse, {species, temp}, "model", "linear", ... %! "continuous", 2, "sstype", "h", "display", "on", ... %! "varnames", {"species", "temp"}); %!demo %! %! # Factorial ANCOVA on data from a study of the effects of treatment and %! # exercise on stress reduction score after adjusting for age. Data from R %! # datarium package). %! %! score = [95.6 82.2 97.2 96.4 81.4 83.6 89.4 83.8 83.3 85.7 ... %! 97.2 78.2 78.9 91.8 86.9 84.1 88.6 89.8 87.3 85.4 ... %! 81.8 65.8 68.1 70.0 69.9 75.1 72.3 70.9 71.5 72.5 ... %! 84.9 96.1 94.6 82.5 90.7 87.0 86.8 93.3 87.6 92.4 ... %! 100. 80.5 92.9 84.0 88.4 91.1 85.7 91.3 92.3 87.9 ... %! 91.7 88.6 75.8 75.7 75.3 82.4 80.1 86.0 81.8 82.5]'; %! treatment = {"yes" "yes" "yes" "yes" "yes" "yes" "yes" "yes" "yes" "yes" ... %! "yes" "yes" "yes" "yes" "yes" "yes" "yes" "yes" "yes" "yes" ... %! "yes" "yes" "yes" "yes" "yes" "yes" "yes" "yes" "yes" "yes" ... %! "no" "no" "no" "no" "no" "no" "no" "no" "no" "no" ... %! "no" "no" "no" "no" "no" "no" "no" "no" "no" "no" ... %! "no" "no" "no" "no" "no" "no" "no" "no" "no" "no"}'; %! exercise = {"lo" "lo" "lo" "lo" "lo" "lo" "lo" "lo" "lo" "lo" ... %! "mid" "mid" "mid" "mid" "mid" "mid" "mid" "mid" "mid" "mid" ... %! "hi" "hi" "hi" "hi" "hi" "hi" "hi" "hi" "hi" "hi" ... %! "lo" "lo" "lo" "lo" "lo" "lo" "lo" "lo" "lo" "lo" ... %! "mid" "mid" "mid" "mid" "mid" "mid" "mid" "mid" "mid" "mid" ... %! "hi" "hi" "hi" "hi" "hi" "hi" "hi" "hi" "hi" "hi"}'; %! age = [59 65 70 66 61 65 57 61 58 55 62 61 60 59 55 57 60 63 62 57 ... %! 58 56 57 59 59 60 55 53 55 58 68 62 61 54 59 63 60 67 60 67 ... %! 75 54 57 62 65 60 58 61 65 57 56 58 58 58 52 53 60 62 61 61]'; %! %! [P, ATAB, STATS] = anovan (score, {treatment, exercise, age}, ... %! "model", [1 0 0; 0 1 0; 0 0 1; 1 1 0], ... %! "continuous", 3, "sstype", "h", "display", "on", ... %! "varnames", {"treatment", "exercise", "age"}); %!demo %! %! # Unbalanced one-way ANOVA with custom, orthogonal contrasts. The statistics %! # relating to the contrasts are shown in the table of model parameters, and %! # can be retrieved from the STATS.coeffs output. %! %! dv = [ 8.706 10.362 11.552 6.941 10.983 10.092 6.421 14.943 15.931 ... %! 22.968 18.590 16.567 15.944 21.637 14.492 17.965 18.851 22.891 ... %! 22.028 16.884 17.252 18.325 25.435 19.141 21.238 22.196 18.038 ... %! 22.628 31.163 26.053 24.419 32.145 28.966 30.207 29.142 33.212 ... %! 25.694 ]'; %! g = [1 1 1 1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 3 ... %! 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5]'; %! C = [ 0.4001601 0.3333333 0.5 0.0 %! 0.4001601 0.3333333 -0.5 0.0 %! 0.4001601 -0.6666667 0.0 0.0 %! -0.6002401 0.0000000 0.0 0.5 %! -0.6002401 0.0000000 0.0 -0.5]; %! %! [P,ATAB, STATS] = anovan (dv, g, "contrasts", C, "varnames", "score", ... %! "alpha", 0.05, "display", "on"); %!demo %! %! # One-way ANOVA with the linear model fit by weighted least squares to %! # account for heteroskedasticity. In this example, the variance appears %! # proportional to the outcome, so weights have been estimated by initially %! # fitting the model without weights and regressing the absolute residuals on %! # the fitted values. Although this data could have been analysed by Welch's %! # ANOVA test, the approach here can generalize to ANOVA models with more than %! # one factor. %! %! g = [1, 1, 1, 1, 1, 1, 1, 1, ... %! 2, 2, 2, 2, 2, 2, 2, 2, ... %! 3, 3, 3, 3, 3, 3, 3, 3]'; %! y = [13, 16, 16, 7, 11, 5, 1, 9, ... %! 10, 25, 66, 43, 47, 56, 6, 39, ... %! 11, 39, 26, 35, 25, 14, 24, 17]'; %! %! [P,ATAB,STATS] = anovan(y, g, "display", "off"); %! fitted = STATS.X * STATS.coeffs(:,1); # fitted values %! b = polyfit (fitted, abs (STATS.resid), 1); %! v = polyval (b, fitted); # Variance as a function of the fitted values %! figure("Name", "Regression of the absolute residuals on the fitted values"); %! plot (fitted, abs (STATS.resid),'ob');hold on; plot(fitted,v,'-r'); hold off; %! xlabel("Fitted values"); ylabel("Absolute residuals"); %! %! [P,ATAB,STATS] = anovan (y, g, "weights", v.^-1); ## Test 1 for anovan example 1 ## Test compares anovan to results from MATLAB's anovan and ttest2 functions %!test %! score = [54 23 45 54 45 43 34 65 77 46 65]'; %! gender = {'male' 'male' 'male' 'male' 'male' 'female' 'female' 'female' ... %! 'female' 'female' 'female'}'; %! %! [P, T, STATS] = anovan (score,gender,'display','off'); %! assert (P(1), 0.2612876773271042, 1e-09); # compared to p calculated by MATLAB anovan %! assert (sqrt(T{2,6}), abs(1.198608733288208), 1e-09); # compared to abs(t) calculated from sqrt(F) by MATLAB anovan %! assert (P(1), 0.2612876773271047, 1e-09); # compared to p calculated by MATLAB ttest2 %! assert (sqrt(T{2,6}), abs(-1.198608733288208), 1e-09); # compared to abs(t) calculated by MATLAB ttest2 ## Test 2 for anovan example 2 ## Test compares anovan to results from MATLAB's anovan and ttest functions %!test %! score = [4.5 5.6; 3.7 6.4; 5.3 6.4; 5.4 6.0; 3.9 5.7]'; %! treatment = {'before' 'after'; 'before' 'after'; 'before' 'after'; %! 'before' 'after'; 'before' 'after'}'; %! subject = {'GS' 'GS'; 'JM' 'JM'; 'HM' 'HM'; 'JW' 'JW'; 'PS' 'PS'}'; %! %! [P, ATAB, STATS] = anovan (score(:),{treatment(:),subject(:)},'display','off','sstype',2); %! assert (P(1), 0.016004356735364, 1e-09); # compared to p calculated by MATLAB anovan %! assert (sqrt(ATAB{2,6}), abs(4.00941576558195), 1e-09); # compared to abs(t) calculated from sqrt(F) by MATLAB anovan %! assert (P(1), 0.016004356735364, 1e-09); # compared to p calculated by MATLAB ttest2 %! assert (sqrt(ATAB{2,6}), abs(-4.00941576558195), 1e-09); # compared to abs(t) calculated by MATLAB ttest2 ## Test 3 for anovan example 3 ## Test compares anovan to results from MATLAB's anovan and anova1 functions %!test %! strength = [82 86 79 83 84 85 86 87 74 82 ... %! 78 75 76 77 79 79 77 78 82 79]'; %! alloy = {'st','st','st','st','st','st','st','st', ... %! 'al1','al1','al1','al1','al1','al1', ... %! 'al2','al2','al2','al2','al2','al2'}'; %! %! [P, ATAB, STATS] = anovan (strength,{alloy},'display','off'); %! assert (P(1), 0.000152643638830491, 1e-09); %! assert (ATAB{2,6}, 15.4, 1e-09); ## Test 4 for anovan example 4 ## Test compares anovan to results from MATLAB's anovan function %!test %! words = [10 13 13; 6 8 8; 11 14 14; 22 23 25; 16 18 20; ... %! 15 17 17; 1 1 4; 12 15 17; 9 12 12; 8 9 12]; %! subject = [ 1 1 1; 2 2 2; 3 3 3; 4 4 4; 5 5 5; ... %! 6 6 6; 7 7 7; 8 8 8; 9 9 9; 10 10 10]; %! seconds = [1 2 5; 1 2 5; 1 2 5; 1 2 5; 1 2 5; ... %! 1 2 5; 1 2 5; 1 2 5; 1 2 5; 1 2 5;]; %! %! [P, ATAB, STATS] = anovan (words(:),{seconds(:),subject(:)},'model','full','random',2,'sstype',2,'display','off'); %! assert (P(1), 1.51865926758752e-07, 1e-09); %! assert (ATAB{2,2}, 52.2666666666667, 1e-09); %! assert (ATAB{3,2}, 942.533333333333, 1e-09); %! assert (ATAB{4,2}, 11.0666666666667, 1e-09); ## Test 5 for anovan example 5 ## Test compares anovan to results from MATLAB's anovan function %!test %! popcorn = [5.5, 4.5, 3.5; 5.5, 4.5, 4.0; 6.0, 4.0, 3.0; ... %! 6.5, 5.0, 4.0; 7.0, 5.5, 5.0; 7.0, 5.0, 4.5]; %! brands = {'Gourmet', 'National', 'Generic'; ... %! 'Gourmet', 'National', 'Generic'; ... %! 'Gourmet', 'National', 'Generic'; ... %! 'Gourmet', 'National', 'Generic'; ... %! 'Gourmet', 'National', 'Generic'; ... %! 'Gourmet', 'National', 'Generic'}; %! popper = {'oil', 'oil', 'oil'; 'oil', 'oil', 'oil'; 'oil', 'oil', 'oil'; ... %! 'air', 'air', 'air'; 'air', 'air', 'air'; 'air', 'air', 'air'}; %! %! [P, ATAB, STATS] = anovan (popcorn(:),{brands(:),popper(:)},'display','off','model','full'); %! assert (P(1), 7.67895738278171e-07, 1e-09); %! assert (P(2), 0.000100373896304998, 1e-09); %! assert (P(3), 0.746215396636649, 1e-09); %! assert (ATAB{2,6}, 56.7, 1e-09); %! assert (ATAB{3,6}, 32.4, 1e-09); %! assert (ATAB{4,6}, 0.29999999999997, 1e-09); ## Test 6 for anovan example 6 ## Test compares anovan to results from MATLAB's anovan function %!test %! salary = [24 26 25 24 27 24 27 23 15 17 20 16, ... %! 25 29 27 19 18 21 20 21 22 19]'; %! gender = {'f' 'f' 'f' 'f' 'f' 'f' 'f' 'f' 'f' 'f' 'f' 'f'... %! 'm' 'm' 'm' 'm' 'm' 'm' 'm' 'm' 'm' 'm'}'; %! degree = [1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 0 0 0 0 0 0 0]'; %! %! [P, ATAB, STATS] = anovan (salary,{gender,degree},'model','full','sstype',1,'display','off'); %! assert (P(1), 0.747462549227232, 1e-09); %! assert (P(2), 1.03809316857694e-08, 1e-09); %! assert (P(3), 0.523689833702691, 1e-09); %! assert (ATAB{2,2}, 0.296969696969699, 1e-09); %! assert (ATAB{3,2}, 272.391841491841, 1e-09); %! assert (ATAB{4,2}, 1.17482517482512, 1e-09); %! assert (ATAB{5,2}, 50.0000000000001, 1e-09); %! [P, ATAB, STATS] = anovan (salary,{degree,gender},'model','full','sstype',1,'display','off'); %! assert (P(1), 2.53445097305047e-08, 1e-09); %! assert (P(2), 0.00388133678528749, 1e-09); %! assert (P(3), 0.523689833702671, 1e-09); %! assert (ATAB{2,2}, 242.227272727273, 1e-09); %! assert (ATAB{3,2}, 30.4615384615384, 1e-09); %! assert (ATAB{4,2}, 1.17482517482523, 1e-09); %! assert (ATAB{5,2}, 50.0000000000001, 1e-09); %! [P, ATAB, STATS] = anovan (salary,{gender,degree},'model','full','sstype',2,'display','off'); %! assert (P(1), 0.00388133678528743, 1e-09); %! assert (P(2), 1.03809316857694e-08, 1e-09); %! assert (P(3), 0.523689833702691, 1e-09); %! assert (ATAB{2,2}, 30.4615384615385, 1e-09); %! assert (ATAB{3,2}, 272.391841491841, 1e-09); %! assert (ATAB{4,2}, 1.17482517482512, 1e-09); %! assert (ATAB{5,2}, 50.0000000000001, 1e-09); %! [P, ATAB, STATS] = anovan (salary,{gender,degree},'model','full','sstype',3,'display','off'); %! assert (P(1), 0.00442898146583742, 1e-09); %! assert (P(2), 1.30634252053587e-08, 1e-09); %! assert (P(3), 0.523689833702691, 1e-09); %! assert (ATAB{2,2}, 29.3706293706294, 1e-09); %! assert (ATAB{3,2}, 264.335664335664, 1e-09); %! assert (ATAB{4,2}, 1.17482517482512, 1e-09); %! assert (ATAB{5,2}, 50.0000000000001, 1e-09); ## Test 7 for anovan example 7 ## Test compares anovan to results from MATLAB's anovan function %!test %! sugar = {'real' 'fake' 'fake' 'real' 'real' 'real' 'none' 'none' 'none' ... %! 'fake' 'fake' 'fake' 'real' 'real' 'real' 'none' 'none' 'fake'}'; %! milk = {'yes' 'no' 'no' 'yes' 'yes' 'no' 'yes' 'yes' 'yes' ... %! 'no' 'no' 'yes' 'no' 'no' 'no' 'no' 'no' 'yes'}'; %! babble = [4.6 4.4 3.9 5.6 5.1 5.5 3.9 3.5 3.7... %! 5.6 4.7 5.9 6.0 5.4 6.6 5.8 5.3 5.7]'; %! %! [P, ATAB, STATS] = anovan (babble,{sugar,milk},'model','full','sstype',1,'display','off'); %! assert (P(1), 0.0108632139833963, 1e-09); %! assert (P(2), 0.0810606976703546, 1e-09); %! assert (P(3), 0.00175433329935627, 1e-09); %! assert (ATAB{2,2}, 3.55752380952381, 1e-09); %! assert (ATAB{3,2}, 0.956108477471702, 1e-09); %! assert (ATAB{4,2}, 5.94386771300448, 1e-09); %! assert (ATAB{5,2}, 3.1625, 1e-09); %! [P, ATAB, STATS] = anovan (babble,{milk,sugar},'model','full','sstype',1,'display','off'); %! assert (P(1), 0.0373333189297505, 1e-09); %! assert (P(2), 0.017075098787169, 1e-09); %! assert (P(3), 0.00175433329935627, 1e-09); %! assert (ATAB{2,2}, 1.444, 1e-09); %! assert (ATAB{3,2}, 3.06963228699552, 1e-09); %! assert (ATAB{4,2}, 5.94386771300448, 1e-09); %! assert (ATAB{5,2}, 3.1625, 1e-09); %! [P, ATAB, STATS] = anovan (babble,{sugar,milk},'model','full','sstype',2,'display','off'); %! assert (P(1), 0.017075098787169, 1e-09); %! assert (P(2), 0.0810606976703546, 1e-09); %! assert (P(3), 0.00175433329935627, 1e-09); %! assert (ATAB{2,2}, 3.06963228699552, 1e-09); %! assert (ATAB{3,2}, 0.956108477471702, 1e-09); %! assert (ATAB{4,2}, 5.94386771300448, 1e-09); %! assert (ATAB{5,2}, 3.1625, 1e-09); %! [P, ATAB, STATS] = anovan (babble,{sugar,milk},'model','full','sstype',3,'display','off'); %! assert (P(1), 0.0454263063473954, 1e-09); %! assert (P(2), 0.0746719907091438, 1e-09); %! assert (P(3), 0.00175433329935627, 1e-09); %! assert (ATAB{2,2}, 2.13184977578476, 1e-09); %! assert (ATAB{3,2}, 1.00413461538462, 1e-09); %! assert (ATAB{4,2}, 5.94386771300448, 1e-09); %! assert (ATAB{5,2}, 3.1625, 1e-09); ## Test 8 for anovan example 8 ## Test compares anovan to results from MATLAB's anovan function %!test %! drug = {'X' 'X' 'X' 'X' 'X' 'X' 'X' 'X' 'X' 'X' 'X' 'X' ... %! 'X' 'X' 'X' 'X' 'X' 'X' 'X' 'X' 'X' 'X' 'X' 'X'; %! 'Y' 'Y' 'Y' 'Y' 'Y' 'Y' 'Y' 'Y' 'Y' 'Y' 'Y' 'Y' ... %! 'Y' 'Y' 'Y' 'Y' 'Y' 'Y' 'Y' 'Y' 'Y' 'Y' 'Y' 'Y'; %! 'Z' 'Z' 'Z' 'Z' 'Z' 'Z' 'Z' 'Z' 'Z' 'Z' 'Z' 'Z' ... %! 'Z' 'Z' 'Z' 'Z' 'Z' 'Z' 'Z' 'Z' 'Z' 'Z' 'Z' 'Z'}; %! feedback = [1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0; %! 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0; %! 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0]; %! diet = [0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1 1; %! 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1 1; %! 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1 1]; %! BP = [170 175 165 180 160 158 161 173 157 152 181 190 ... %! 173 194 197 190 176 198 164 190 169 164 176 175; %! 186 194 201 215 219 209 164 166 159 182 187 174 ... %! 189 194 217 206 199 195 171 173 196 199 180 NaN; %! 180 187 199 170 204 194 162 184 183 156 180 173 ... %! 202 228 190 206 224 204 205 199 170 160 NaN NaN]; %! %! [P, ATAB, STATS] = anovan (BP(:),{drug(:),feedback(:),diet(:)},'model','full','sstype', 1,'display','off'); %! assert (P(1), 7.02561843825325e-05, 1e-09); %! assert (P(2), 0.000425806013389362, 1e-09); %! assert (P(3), 6.16780773446401e-07, 1e-09); %! assert (P(4), 0.261347622678438, 1e-09); %! assert (P(5), 0.0542278432357043, 1e-09); %! assert (P(6), 0.590353225626655, 1e-09); %! assert (P(7), 0.0861628249564267, 1e-09); %! assert (ATAB{2,2}, 3614.70355731226, 1e-09); %! assert (ATAB{3,2}, 2227.46639771024, 1e-09); %! assert (ATAB{4,2}, 5008.25614451819, 1e-09); %! assert (ATAB{5,2}, 437.066007908781, 1e-09); %! assert (ATAB{6,2}, 976.180770397332, 1e-09); %! assert (ATAB{7,2}, 46.616653365254, 1e-09); %! assert (ATAB{8,2}, 814.345251396648, 1e-09); %! assert (ATAB{9,2}, 9065.8, 1e-09); %! [P, ATAB, STATS] = anovan (BP(:),{drug(:),feedback(:),diet(:)},'model','full','sstype',2,'display','off'); %! assert (P(1), 9.4879638470754e-05, 1e-09); %! assert (P(2), 0.00124177666315809, 1e-09); %! assert (P(3), 6.86162012732911e-07, 1e-09); %! assert (P(4), 0.260856132341256, 1e-09); %! assert (P(5), 0.0523758623892078, 1e-09); %! assert (P(6), 0.590353225626655, 1e-09); %! assert (P(7), 0.0861628249564267, 1e-09); %! assert (ATAB{2,2}, 3481.72176560122, 1e-09); %! assert (ATAB{3,2}, 1837.08812970469, 1e-09); %! assert (ATAB{4,2}, 4957.20277938622, 1e-09); %! assert (ATAB{5,2}, 437.693674777847, 1e-09); %! assert (ATAB{6,2}, 988.431929811402, 1e-09); %! assert (ATAB{7,2}, 46.616653365254, 1e-09); %! assert (ATAB{8,2}, 814.345251396648, 1e-09); %! assert (ATAB{9,2}, 9065.8, 1e-09); %! [P, ATAB, STATS] = anovan (BP(:),{drug(:),feedback(:),diet(:)},'model','full','sstype', 3,'display','off'); %! assert (P(1), 0.000106518678028207, 1e-09); %! assert (P(2), 0.00125371366571508, 1e-09); %! assert (P(3), 5.30813260778464e-07, 1e-09); %! assert (P(4), 0.308353667232981, 1e-09); %! assert (P(5), 0.0562901327343161, 1e-09); %! assert (P(6), 0.599091042141092, 1e-09); %! assert (P(7), 0.0861628249564267, 1e-09); %! assert (ATAB{2,2}, 3430.88156424581, 1e-09); %! assert (ATAB{3,2}, 1833.68031496063, 1e-09); %! assert (ATAB{4,2}, 5080.48346456693, 1e-09); %! assert (ATAB{5,2}, 382.07709497207, 1e-09); %! assert (ATAB{6,2}, 963.037988826813, 1e-09); %! assert (ATAB{7,2}, 44.4519685039322, 1e-09); %! assert (ATAB{8,2}, 814.345251396648, 1e-09); %! assert (ATAB{9,2}, 9065.8, 1e-09); ## Test 9 for anovan example 9 ## Test compares anovan to results from MATLAB's anovan function %!test %! measurement = [444 614 423 625 408 856 447 719 ... %! 764 831 586 782 609 1002 606 766]'; %! strain= {'NIH','NIH','BALB/C','BALB/C','A/J','A/J','129/Ola','129/Ola', ... %! 'NIH','NIH','BALB/C','BALB/C','A/J','A/J','129/Ola','129/Ola'}'; %! treatment={'C' 'T' 'C' 'T' 'C' 'T' 'C' 'T' 'C' 'T' 'C' 'T' 'C' 'T' 'C' 'T'}'; %! block = [1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2]'; %! %! [P, ATAB, STATS] = anovan (measurement/10,{strain,treatment,block},'model','full','random',3,'display','off'); %! assert (P(1), 0.0914352969909372, 1e-09); %! assert (P(2), 5.04077373924908e-05, 1e-09); %! assert (P(4), 0.0283196918836667, 1e-09); %! assert (ATAB{2,2}, 286.132500000002, 1e-09); %! assert (ATAB{3,2}, 2275.29, 1e-09); %! assert (ATAB{4,2}, 1242.5625, 1e-09); %! assert (ATAB{5,2}, 495.905000000001, 1e-09); %! assert (ATAB{6,2}, 207.007499999999, 1e-09); ## Test 10 for anovan example 10 ## Test compares anovan to results from MATLAB's anovan function %!test %! pulse = [67.9 65.1 77.3 78.7 79.4 80.4 85.8 86.6 87.5 89.1 ... %! 98.6 100.8 99.3 101.7 44.3 47.2 47.6 49.6 50.3 51.8 ... %! 60 58.5 58.9 60.7 69.8 70.9 76.2 76.1 77 77.7 84.7]'; %! temp = [20.8 20.8 24 24 24 24 26.2 26.2 26.2 26.2 28.4 ... %! 29 30.4 30.4 17.2 18.3 18.3 18.3 18.9 18.9 20.4 ... %! 21 21 22.1 23.5 24.2 25.9 26.5 26.5 26.5 28.6]'; %! species = {'ex' 'ex' 'ex' 'ex' 'ex' 'ex' 'ex' 'ex' 'ex' 'ex' 'ex' ... %! 'ex' 'ex' 'ex' 'niv' 'niv' 'niv' 'niv' 'niv' 'niv' 'niv' ... %! 'niv' 'niv' 'niv' 'niv' 'niv' 'niv' 'niv' 'niv' 'niv' 'niv'}; %! %! [P, ATAB, STATS] = anovan (pulse,{species,temp},'model','linear','continuous',2,'sstype','h','display','off'); %! assert (P(1), 6.27153318786007e-14, 1e-09); %! assert (P(2), 2.48773241196644e-25, 1e-09); %! assert (ATAB{2,2}, 598.003953318404, 1e-09); %! assert (ATAB{3,2}, 4376.08256843712, 1e-09); %! assert (ATAB{4,2}, 89.3498685376726, 1e-09); %! assert (ATAB{2,6}, 187.399388123951, 1e-09); %! assert (ATAB{3,6}, 1371.35413763454, 1e-09); ## Test 11 for anovan example 11 ## Test compares anovan to results from MATLAB's anovan function %!test %! score = [95.6 82.2 97.2 96.4 81.4 83.6 89.4 83.8 83.3 85.7 ... %! 97.2 78.2 78.9 91.8 86.9 84.1 88.6 89.8 87.3 85.4 ... %! 81.8 65.8 68.1 70.0 69.9 75.1 72.3 70.9 71.5 72.5 ... %! 84.9 96.1 94.6 82.5 90.7 87.0 86.8 93.3 87.6 92.4 ... %! 100. 80.5 92.9 84.0 88.4 91.1 85.7 91.3 92.3 87.9 ... %! 91.7 88.6 75.8 75.7 75.3 82.4 80.1 86.0 81.8 82.5]'; %! treatment = {'yes' 'yes' 'yes' 'yes' 'yes' 'yes' 'yes' 'yes' 'yes' 'yes' ... %! 'yes' 'yes' 'yes' 'yes' 'yes' 'yes' 'yes' 'yes' 'yes' 'yes' ... %! 'yes' 'yes' 'yes' 'yes' 'yes' 'yes' 'yes' 'yes' 'yes' 'yes' ... %! 'no' 'no' 'no' 'no' 'no' 'no' 'no' 'no' 'no' 'no' ... %! 'no' 'no' 'no' 'no' 'no' 'no' 'no' 'no' 'no' 'no' ... %! 'no' 'no' 'no' 'no' 'no' 'no' 'no' 'no' 'no' 'no'}'; %! exercise = {'lo' 'lo' 'lo' 'lo' 'lo' 'lo' 'lo' 'lo' 'lo' 'lo' ... %! 'mid' 'mid' 'mid' 'mid' 'mid' 'mid' 'mid' 'mid' 'mid' 'mid' ... %! 'hi' 'hi' 'hi' 'hi' 'hi' 'hi' 'hi' 'hi' 'hi' 'hi' ... %! 'lo' 'lo' 'lo' 'lo' 'lo' 'lo' 'lo' 'lo' 'lo' 'lo' ... %! 'mid' 'mid' 'mid' 'mid' 'mid' 'mid' 'mid' 'mid' 'mid' 'mid' ... %! 'hi' 'hi' 'hi' 'hi' 'hi' 'hi' 'hi' 'hi' 'hi' 'hi'}'; %! age = [59 65 70 66 61 65 57 61 58 55 62 61 60 59 55 57 60 63 62 57 ... %! 58 56 57 59 59 60 55 53 55 58 68 62 61 54 59 63 60 67 60 67 ... %! 75 54 57 62 65 60 58 61 65 57 56 58 58 58 52 53 60 62 61 61]'; %! %! [P, ATAB, STATS] = anovan (score,{treatment,exercise,age},'model','full','continuous',3,'sstype','h','display','off'); %! assert (P(5), 0.9245630968248468, 1e-09); %! assert (P(6), 0.791115159521822, 1e-09); %! assert (P(7), 0.9296668751457956, 1e-09); %! [P, ATAB, STATS] = anovan (score,{treatment,exercise,age},'model',[1 0 0; 0 1 0; 0 0 1; 1 1 0],'continuous',3,'sstype','h','display','off'); %! assert (P(1), 0.00158132928938933, 1e-09); %! assert (P(2), 2.12537505039986e-07, 1e-09); %! assert (P(3), 0.00390292555160047, 1e-09); %! assert (P(4), 0.0164086580775543, 1e-09); %! assert (ATAB{2,6}, 11.0956027650549, 1e-09); %! assert (ATAB{3,6}, 20.8195665467178, 1e-09); %! assert (ATAB{4,6}, 9.10966630720186, 1e-09); %! assert (ATAB{5,6}, 4.4457923698584, 1e-09); ## Test 12 for anovan example 12 ## Test compares anovan regression coefficients to R: ## https://www.uvm.edu/~statdhtx/StatPages/Unequal-ns/Unequal_n%27s_contrasts.html %!test %! dv = [ 8.706 10.362 11.552 6.941 10.983 10.092 6.421 14.943 15.931 ... %! 22.968 18.590 16.567 15.944 21.637 14.492 17.965 18.851 22.891 ... %! 22.028 16.884 17.252 18.325 25.435 19.141 21.238 22.196 18.038 ... %! 22.628 31.163 26.053 24.419 32.145 28.966 30.207 29.142 33.212 ... %! 25.694 ]'; %! g = [1 1 1 1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5]'; %! C = [ 0.4001601 0.3333333 0.5 0.0 %! 0.4001601 0.3333333 -0.5 0.0 %! 0.4001601 -0.6666667 0.0 0.0 %! -0.6002401 0.0000000 0.0 0.5 %! -0.6002401 0.0000000 0.0 -0.5]; %! %! [P,ATAB,STATS] = anovan (dv,g,'contrasts',{C},'display','off'); %! assert (STATS.coeffs(1,1), 19.4001, 1e-04); %! assert (STATS.coeffs(2,1), -9.3297, 1e-04); %! assert (STATS.coeffs(3,1), -5.0000, 1e-04); %! assert (STATS.coeffs(4,1), -8.0000, 1e-04); %! assert (STATS.coeffs(5,1), -8.0000, 1e-04); %! assert (STATS.coeffs(1,2), 0.4831, 1e-04); %! assert (STATS.coeffs(2,2), 0.9694, 1e-04); %! assert (STATS.coeffs(3,2), 1.3073, 1e-04); %! assert (STATS.coeffs(4,2), 1.6411, 1e-04); %! assert (STATS.coeffs(5,2), 1.4507, 1e-04); %! assert (STATS.coeffs(1,5), 40.161, 1e-03); %! assert (STATS.coeffs(2,5), -9.624, 1e-03); %! assert (STATS.coeffs(3,5), -3.825, 1e-03); %! assert (STATS.coeffs(4,5), -4.875, 1e-03); %! assert (STATS.coeffs(5,5), -5.515, 1e-03); %! assert (STATS.coeffs(2,6), 5.74e-11, 1e-12); %! assert (STATS.coeffs(3,6), 0.000572, 1e-06); %! assert (STATS.coeffs(4,6), 2.86e-05, 1e-07); %! assert (STATS.coeffs(5,6), 4.44e-06, 1e-08); statistics-release-1.6.3/inst/bartlett_test.m000066400000000000000000000217621456127120000213520ustar00rootroot00000000000000## Copyright (C) 1995-2017 Kurt Hornik ## Copyright (C) 2022 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{h} =} bartlett_test (@var{x}) ## @deftypefnx {statistics} {@var{h} =} bartlett_test (@var{x}, @var{group}) ## @deftypefnx {statistics} {@var{h} =} bartlett_test (@var{x}, @var{alpha}) ## @deftypefnx {statistics} {@var{h} =} bartlett_test (@var{x}, @var{group}, @var{alpha}) ## @deftypefnx {statistics} {[@var{h}, @var{pval}] =} bartlett_test (@dots{}) ## @deftypefnx {statistics} {[@var{h}, @var{pval}, @var{chisq}] =} bartlett_test (@dots{}) ## @deftypefnx {statistics} {[@var{h}, @var{pval}, @var{chisq}, @var{df}] =} bartlett_test (@dots{}) ## ## Perform a Bartlett test for the homogeneity of variances. ## ## Under the null hypothesis of equal variances, the test statistic @var{chisq} ## approximately follows a chi-square distribution with @var{df} degrees of ## freedom. ## ## The p-value (1 minus the CDF of this distribution at @var{chisq}) is ## returned in @var{pval}. @var{h} = 1 if the null hypothesis is rejected at ## the significance level of @var{alpha}. Otherwise @var{h} = 0. ## ## Input Arguments: ## ## @itemize ## @item ## @var{x} contains the data and it can either be a vector or matrix. ## If @var{x} is a matrix, then each column is treated as a separate group. ## If @var{x} is a vector, then the @var{group} argument is mandatory. ## NaN values are omitted. ## ## @item ## @var{group} contains the names for each group. If @var{x} is a vector, then ## @var{group} must be a vector of the same length, or a string array or cell ## array of strings with one row for each element of @var{x}. @var{x} values ## corresponding to the same value of @var{group} are placed in the same group. ## If @var{x} is a matrix, then @var{group} can either be a cell array of ## strings of a character array, with one row per column of @var{x} in the same ## way it is used in @code{anova1} function. If @var{x} is a matrix, then ## @var{group} can be omitted either by entering an empty array ([]) or by ## parsing only @var{alpha} as a second argument (if required to change its ## default value). ## ## @item ## @var{alpha} is the statistical significance value at which the null ## hypothesis is rejected. Its default value is 0.05 and it can be parsed ## either as a second argument (when @var{group} is omitted) or as a third ## argument. ## @end itemize ## ## @seealso{levene_test, vartest2, vartestn} ## @end deftypefn function [h, pval, chisq, df] = bartlett_test (x, varargin) ## Check for valid number of input arguments if (nargin < 1 || nargin > 3) error ("bartlett_test: invalid number of input arguments."); endif ## Add defaults group = []; alpha = 0.05; ## Check for 2nd argument being ALPHA or GROUP if (nargin > 1) if (isscalar (varargin{1}) && isnumeric (varargin{1}) ... && numel (varargin{1}) == 1) alpha = varargin{1}; ## Check for valid alpha value if (alpha <= 0 || alpha >= 1) error ("bartlett_test: wrong value for alpha."); endif elseif (isvector (varargin{1}) && numel (varargin{1} > 1)) if ((size (x, 2) == 1 && size (x, 1) == numel (varargin{1})) || ... (size (x, 2) > 1 && size (x, 2) == numel (varargin{1}))) group = varargin{1}; else error ("bartlett_test: GROUP and X mismatch."); endif elseif (isempty (varargin{1})) ## Do nothing else error ("bartlett_test: invalid second input argument."); endif endif ## Check for 3rd argument if (nargin > 2) alpha = varargin{2}; ## Check for valid alpha value if (! isscalar (alpha) || ! isnumeric (alpha) || alpha <= 0 || alpha >= 1) error ("bartlett_test: wrong value for alpha."); endif endif ## Convert group to cell array from character array, make it a column if (! isempty (group) && ischar (group)) group = cellstr (group); endif if (size (group, 1) == 1) group = group'; endif ## If x is a matrix, convert it to column vector and create a ## corresponging column vector for groups if (length (x) < prod (size (x))) [n, m] = size (x); x = x(:); gi = reshape (repmat ((1:m), n, 1), n*m, 1); if (length (group) == 0) ## no group names are provided group = gi; elseif (size (group, 1) == m) ## group names exist and match columns group = group(gi,:); else error ("bartlett_test: columns in X and GROUP length do not match."); endif endif ## Check that x and group are the same size if (! all (numel (x) == numel (group))) error (srtcat (["bartlett_test: GROUP must be a vector with the same"], ... [" number of rows as x."])); endif ## Identify NaN values (if any) and remove them from X along with ## their corresponding values from group vector nonan = ! isnan (x); x = x(nonan); group = group(nonan, :); ## Convert group to indices and separate names [group_id, group_names] = grp2idx (group); group_id = group_id(:); ## Get sample size (n_i) and var (s^2_i) for each group with n_i > 1 groups = size (group_names, 1); rgroup = []; n_i = zeros (1, groups); s_i = n_i; for k = 1:groups group_size = find (group_id == k); if (length (group_size) > 1) n_i(k) = length (group_size); s_i(k) = var (x(group_size)); else warning (strcat (sprintf ("bartlett_test: GROUP %s has a single", ... group_names{k}), [" sample and is not included in the test.\n"])); rgroup = [rgroup, k]; n_i(k) = 1; s_i(k) = NaN; endif endfor ## Remove groups with a single sample if (! isempty (rgroup)) n_i(rgroup) = []; s_i(rgroup) = []; k = k - numel (rgroup); endif ## Compute total sample size (N) and pooled variance (S) N = sum (n_i); S = (1 / (N - k)) * sum ((n_i - 1) .* s_i); ## Calculate B statistic. That is, B ~ X^2(k-1) B_nom = (N - k) * log (S) - sum ((n_i - 1) .* log (s_i)); B_den = 1 + (1 / (3 * (k - 1))) * (sum (1 ./ (n_i - 1)) - (1 / (N - k))); chisq = B_nom / B_den; ## Calculate p-value from the chi-square distribution df = k - 1; pval = 1 - chi2cdf (chisq, df); ## Determine the test outcome h = double (pval < alpha); endfunction ## Test input validation %!error bartlett_test () %!error ... %! bartlett_test (1, 2, 3, 4); %!error bartlett_test (randn (50, 2), 0); %!error ... %! bartlett_test (randn (50, 2), [1, 2, 3]); %!error ... %! bartlett_test (randn (50, 1), ones (55, 1)); %!error ... %! bartlett_test (randn (50, 1), ones (50, 2)); %!error ... %! bartlett_test (randn (50, 2), [], 1.2); %!error ... %! bartlett_test (randn (50, 2), [], "alpha"); %!error ... %! bartlett_test (randn (50, 1), [ones(25, 1); 2*ones(25, 1)], 1.2); %!error ... %! bartlett_test (randn (50, 1), [ones(25, 1); 2*ones(25, 1)], "err"); %!warning ... %! bartlett_test (randn (50, 1), [ones(24, 1); 2*ones(25, 1); 3]); ## Test results %!test %! load examgrades %! [h, pval, chisq, df] = bartlett_test (grades); %! assert (h, 1); %! assert (pval, 7.908647337018238e-08, 1e-14); %! assert (chisq, 38.73324, 1e-5); %! assert (df, 4); %!test %! load examgrades %! [h, pval, chisq, df] = bartlett_test (grades(:,[2:4])); %! assert (h, 1); %! assert (pval, 0.01172, 1e-5); %! assert (chisq, 8.89274, 1e-5); %! assert (df, 2); %!test %! load examgrades %! [h, pval, chisq, df] = bartlett_test (grades(:,[1,4])); %! assert (h, 0); %! assert (pval, 0.88118, 1e-5); %! assert (chisq, 0.02234, 1e-5); %! assert (df, 1); %!test %! load examgrades %! grades = [grades; nan(10, 5)]; %! [h, pval, chisq, df] = bartlett_test (grades(:,[1,4])); %! assert (h, 0); %! assert (pval, 0.88118, 1e-5); %! assert (chisq, 0.02234, 1e-5); %! assert (df, 1); %!test %! load examgrades %! [h, pval, chisq, df] = bartlett_test (grades(:,[2,5]), 0.01); %! assert (h, 0); %! assert (pval, 0.01791, 1e-5); %! assert (chisq, 5.60486, 1e-5); %! assert (df, 1); statistics-release-1.6.3/inst/barttest.m000066400000000000000000000142061456127120000203150ustar00rootroot00000000000000## Copyright (C) 2022 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{ndim} =} barttest (@var{x}) ## @deftypefnx {statistics} {@var{ndim} =} barttest (@var{x}, @var{alpha}) ## @deftypefnx {statistics} {[@var{ndim}, @var{pval}] =} barttest (@var{x}, @var{alpha}) ## @deftypefnx {statistics} {[@var{ndim}, @var{pval}, @var{chisq}] =} barttest (@var{x}, @var{alpha}) ## ## Bartlett's test of sphericity for correlation. ## ## It compares an observed correlation matrix to the identity matrix in order to ## check if there is a certain redundancy between the variables that we can ## summarize with a few number of factors. A statistically significant test ## shows that the variables (columns) in @var{x} are correlated, thus it makes ## sense to perform some dimensionality reduction of the data in @var{x}. ## ## @code{@var{ndim} = barttest (@var{x}, @var{alpha})} returns the number of ## dimensions necessary to explain the nonrandom variation in the data matrix ## @var{x} at the @var{alpha} significance level. @var{alpha} is an optional ## input argument and, when not provided, it is 0.05 by default. ## ## @code{[@var{ndim}, @var{pval}, @var{chisq}] = barttest (@dots{})} also ## returns the significance values @var{pval} for the hypothesis test for each ## dimension as well as the associated chi^2 values in @var{chisq} ## ## @end deftypefn function [ndim, pval, chisq] = barttest (x, alpha); ## Check for valid number of input arguments if (nargin < 1 || nargin >2) error ("barttest: invalid number of input arguments."); endif ## Check for NaN values in X if (any (isnan( x(:)))) error ("barttest: NaN values in input are not allowed."); endif ## Add default value for alpha if not supplied if (nargin == 1) alpha = 0.05; endif ## Check for valid value of alpha if (! isscalar (alpha) || ! isnumeric (alpha) || alpha <= 0 || alpha >= 1) error ("barttest: wrong value for alpha."); endif ## Check size of data [row, col] = size (x); if (col <= 1 || row <= 1) error ("barttest: not enough data in X."); endif ## Compute the eigenvalues of X in a more efficient way latent = sort ((svd (x - repmat (mean (x, 1), row, 1)) .^ 2) / (row - 1)); ## The degrees of ffredom should be N-1, where N is the sample size row -= 1; k = (0:col - 2)'; pk = col - k; loglatent = flipud (cumsum (log (latent))); ## Compute the chi-square statistic logsum = log (flipud ((latent(1) + cumsum (latent(2:col))) ./ flipud(pk))); chisq = (pk .* logsum - loglatent(1:col - 1)) * row; ## Calculate the degrees of freedom df = (pk - 1) .* (pk + 2) / 2; ## Find the corresponding p-values pval = 1 - chi2cdf (chisq, df); ## Get ndim dim = min (find (pval > alpha)); if (isempty (dim)) ndim = col; return; endif if (dim == 1) ndim = NaN; warning ("barttest: heuristics are violated."); else ndim = dim - 1; endif endfunction ## Test input validation %!error barttest () %!error barttest ([2,NaN;3,4]) %!error barttest (ones (30, 4), "alpha") %!error barttest (ones (30, 4), 0) %!error barttest (ones (30, 4), 1.2) %!error barttest (ones (30, 4), [0.2, 0.05]) %!error barttest (ones (30, 1)) %!error barttest (ones (30, 1), 0.05) ## Test results %!test %! x = [2, 3, 4, 5, 6, 7, 8, 9; 1, 2, 3, 4, 5, 6, 7, 8]'; %! [ndim, pval, chisq] = barttest (x); %! assert (ndim, 2); %! assert (pval, 0); %! ## assert (chisq, 512.0558, 1e-4); Result differs between octave 6 and 7 ? %!test %! x = [0.53767, 0.62702, -0.10224, -0.25485, 1.4193, 1.5237 ; ... %! 1.8339, 1.6452, -0.24145, -0.23444, 0.29158, 0.1634 ; ... %! -2.2588, -2.1351, 0.31286, 0.39396, 0.19781, 0.20995 ; ... %! 0.86217, 1.0835, 0.31286, 0.46499, 1.5877, 1.495 ; ... %! 0.31877, 0.38454, -0.86488, -0.63839, -0.80447, -0.7536 ; ... %! -1.3077, -1.1487, -0.030051, -0.017629, 0.69662, 0.60497 ; ... %! -0.43359, -0.32672, -0.16488, -0.37364, 0.83509, 0.89586 ; ... %! 0.34262, 0.29639, 0.62771, 0.51672, -0.24372, -0.13698 ; ... %! 3.5784, 3.5841, 1.0933, 0.93258, 0.21567, 0.455 ; ... %! 2.7694, 2.6307, 1.1093, 1.4298, -1.1658, -1.1816 ; ... %! -1.3499, -1.2111, -0.86365, -0.94186, -1.148, -1.4381 ; ... %! 3.0349, 2.8428, 0.077359, 0.18211, 0.10487, -0.014613; ... %! 0.7254, 0.56737, -1.2141, -1.2291, 0.72225, 0.90612 ; ... %! -0.063055,-0.17662, -1.1135, -0.97701, 2.5855, 2.4084 ; ... %! 0.71474, 0.29225, -0.0068493, -0.11468, -0.66689, -0.52466 ; ... %! -0.20497, -7.8874e-06, 1.5326, 1.3195, 0.18733, 0.20296 ; ... %! -0.12414, -0.077029, -0.76967, -0.96262, -0.082494, 0.121 ; ... %! 1.4897, 1.3683, 0.37138, 0.43653, -1.933, -2.1903 ; ... %! 1.409, 1.5882, -0.22558, -0.24835, -0.43897, -0.46247 ; ... %! 1.4172, 1.1616, 1.1174, 1.0785, -1.7947, -1.9471 ]; %! [ndim, pval, chisq] = barttest (x); %! assert (ndim, 3); %! assert (pval, [0; 0; 0; 0.52063; 0.34314], 1e-5); %! chisq_out = [251.6802; 210.2670; 153.1773; 4.2026; 2.1392]; %! assert (chisq, chisq_out, 1e-4); statistics-release-1.6.3/inst/binotest.m000066400000000000000000000122021456127120000203060ustar00rootroot00000000000000## Copyright (C) 2016 Andreas Stahel ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {[@var{h}, @var{pval}, @var{ci}] =} binotest (@var{pos}, @var{N}, @var{p0}) ## @deftypefnx {statistics} {[@var{h}, @var{pval}, @var{ci}] =} binotest (@var{pos}, @var{N}, @var{p0}, @var{Name}, @var{Value}) ## ## Test for probability @var{p} of a binomial sample ## ## Perform a test of the null hypothesis @var{p} == @var{p0} for a sample ## of size @var{N} with @var{pos} positive results. ## ## ## Name-Value pair arguments can be used to set various options. ## @qcode{"alpha"} can be used to specify the significance level ## of the test (the default value is 0.05). The option @qcode{"tail"}, ## can be used to select the desired alternative hypotheses. If the ## value is @qcode{"both"} (default) the null is tested against the two-sided ## alternative @code{@var{p} != @var{p0}}. The value of @var{pval} is ## determined by adding the probabilities of all event less or equally ## likely than the observed number @var{pos} of positive events. ## If the value of @qcode{"tail"} is @qcode{"right"} ## the one-sided alternative @code{@var{p} > @var{p0}} is considered. ## Similarly for @qcode{"left"}, the one-sided alternative ## @code{@var{p} < @var{p0}} is considered. ## ## If @var{h} is 0 the null hypothesis is accepted, if it is 1 the null ## hypothesis is rejected. The p-value of the test is returned in @var{pval}. ## A 100(1-alpha)% confidence interval is returned in @var{ci}. ## ## @end deftypefn function [h, p, ci] = binotest(pos,n,p0,varargin) % Set default arguments alpha = 0.05; tail = 'both'; i = 1; while ( i <= length(varargin) ) switch lower(varargin{i}) case 'alpha' i = i + 1; alpha = varargin{i}; case 'tail' i = i + 1; tail = varargin{i}; otherwise error('Invalid Name argument.',[]); end i = i + 1; end if ~isa(tail,'char') error('tail argument to vartest must be a string\n',[]); end if (n<=0) error('binotest: required n>0\n',[]); end if (p0<0)|(p0>1) error('binotest: required 0<= p0 <= 1\n',[]); end if (pos<0)|(pos>n) error('binotest: required 0<= pos <= n\n',[]); end % Based on the "tail" argument determine the P-value, the critical values, % and the confidence interval. switch lower(tail) case 'both' A_low = binoinv(alpha/2,n,p0)/n; A_high = binoinv(1-alpha/2,n,p0)/n; p_pos = binopdf(pos,n,p0); p_all = binopdf([0:n],n,p0); ind = find(p_all <=p_pos); % p = min(1,sum(p_all(ind))); p = sum(p_all(ind)); if pos==0 p_low = 0; else p_low = fzero(@(pl)1-binocdf(pos-1,n,pl)-alpha/2,[0 1]); endif if pos==n p_high = 1; else p_high = fzero(@(ph) binocdf(pos,n,ph) -alpha/2,[0,1]); endif ci = [p_low,p_high]; case 'left' p = 1-binocdf(pos-1,n,p0); if pos==n p_high = 1; else p_high = fzero(@(ph) binocdf(pos,n,ph) -alpha,[0,1]); endif ci = [0, p_high]; case 'right' p = binocdf(pos,n,p0); if pos==0 p_low = 0; else p_low = fzero(@(pl)1-binocdf(pos-1,n,pl)-alpha,[0 1]); endif ci = [p_low 1]; otherwise error('Invalid fifth (tail) argument to binotest\n',[]); end % Determine the test outcome % MATLAB returns this a double instead of a logical array h = double(p < alpha); end %!demo %! % flip a coin 1000 times, showing 475 heads %! % Hypothesis: coin is fair, i.e. p=1/2 %! [h,p_val,ci] = binotest(475,1000,0.5) %! % Result: h = 0 : null hypothesis not rejected, coin could be fair %! % P value 0.12, i.e. hypothesis not rejected for alpha up to 12% %! % 0.444 <= p <= 0.506 with 95% confidence %!demo %! % flip a coin 100 times, showing 65 heads %! % Hypothesis: coin shows less than 50% heads, i.e. p<=1/2 %! [h,p_val,ci] = binotest(65,100,0.5,'tail','left','alpha',0.01) %! % Result: h = 1 : null hypothesis is rejected, i.e. coin shows more heads than tails %! % P value 0.0018, i.e. hypothesis not rejected for alpha up to 0.18% %! % 0 <= p <= 0.76 with 99% confidence %!test #example from https://en.wikipedia.org/wiki/Binomial_test %! [h,p_val,ci] = binotest (51,235,1/6); %! assert (p_val, 0.0437, 0.00005) %! [h,p_val,ci] = binotest (51,235,1/6,'tail','left'); %! assert (p_val, 0.027, 0.0005) statistics-release-1.6.3/inst/boxplot.m000066400000000000000000001041131456127120000201510ustar00rootroot00000000000000## Copyright (C) 2002 Alberto Terruzzi ## Copyright (C) 2006 Alberto Pose ## Copyright (C) 2011 Pascal Dupuis ## Copyright (C) 2012 Juan Pablo Carbajal ## Copyright (C) 2016 Pascal Dupuis ## Copyright (C) 2020 Andreas Bertsatos ## Copyright (C) 2020 Philip Nienhuis (prnienhuis@users.sf.net) ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{s} =} boxplot (@var{data}) ## @deftypefnx {statistics} {@var{s} =} boxplot (@var{data}, @var{group}) ## @deftypefnx {statistics} {@var{s} =} boxplot (@var{data}, @var{notched}, @var{symbol}, @var{orientation}, @var{whisker}, @dots{}) ## @deftypefnx {statistics} {@var{s} =} boxplot (@var{data}, @var{group}, @var{notched}, @var{symbol}, @var{orientation}, @var{whisker}, @dots{}) ## @deftypefnx {statistics} {@var{s} =} boxplot (@var{data}, @var{options}) ## @deftypefnx {statistics} {@var{s} =} boxplot (@var{data}, @var{group}, @var{options}, @dots{}) ## @deftypefnx {statistics} {[@dots{}, @var{h}] =} boxplot (@var{data}, @dots{}) ## ## Produce a box plot. ## ## A box plot is a graphical display that simultaneously describes several ## important features of a data set, such as center, spread, departure from ## symmetry, and identification of observations that lie unusually far from ## the bulk of the data. ## ## Input arguments (case-insensitive) recognized by boxplot are: ## ## @itemize ## @item ## @var{data} is a matrix with one column for each data set, or a cell vector ## with one cell for each data set. Each cell must contain a numerical row or ## column vector (NaN and NA are ignored) and not a nested vector of cells. ## ## @item ## @var{notched} = 1 produces a notched-box plot. Notches represent a robust ## estimate of the uncertainty about the median. ## ## @var{notched} = 0 (default) produces a rectangular box plot. ## ## @var{notched} within the interval (0,1) produces a notch of the specified ## depth. Notched values outside (0,1) are amusing if not exactly impractical. ## ## @item ## @var{symbol} sets the symbol for the outlier values. The default symbol ## for points that lie outside 3 times the interquartile range is 'o'; ## the default symbol for points between 1.5 and 3 times the interquartile ## range is '+'. @* ## Alternative @var{symbol} settings: ## ## @var{symbol} = '.': points between 1.5 and 3 times the IQR are marked with ## '.' and points outside 3 times IQR with 'o'. ## ## @var{symbol} = ['x','*']: points between 1.5 and 3 times the IQR are marked ## with 'x' and points outside 3 times IQR with '*'. ## ## @item ## @var{orientation} = 0 makes the boxes horizontally. @* ## @var{orientation} = 1 plots the boxes vertically (default). Alternatively, ## orientation can be passed as a string, e.g., 'vertical' or 'horizontal'. ## ## @item ## @var{whisker} defines the length of the whiskers as a function of the IQR ## (default = 1.5). If @var{whisker} = 0 then @code{boxplot} displays all data ## values outside the box using the plotting symbol for points that lie ## outside 3 times the IQR. ## ## @item ## @var{group} may be passed as an optional argument only in the second ## position after @var{data}. @var{group} contains a numerical vector defining ## separate categories, each plotted in a different box, for each set of ## @var{DATA} values that share the same @var{group} value or values. With ## the formalism (@var{data}, @var{group}), both must be vectors of the same ## length. ## ## @item ## @var{options} are additional paired arguments passed with the formalism ## (Name, Value) that provide extra functionality as listed below. ## @var{options} can be passed at any order after the initial arguments and ## are case-insensitive. ## ## @multitable {Name} {Value} {description} @columnfractions .2 .2 .6 ## @item 'Notch' @tab 'on' @tab Notched by 0.25 of the boxes width. ## @item @tab 'off' @tab Produces a straight box. ## @item @tab scalar @tab Proportional width of the notch. ## ## @item 'Symbol' @tab '.' @tab Defines only outliers between 1.5 and 3 IQR. ## @item @tab ['x','*'] @tab 2nd character defines outliers > 3 IQR ## ## @item 'Orientation' @tab 'vertical' @tab Default value, can also be defined ## with numerical 1. ## @item @tab 'horizontal' @tab Can also be defined with numerical 0. ## ## @item 'Whisker' @tab scalar @tab Multiplier of IQR (default is 1.5). ## ## @item 'OutlierTags' @tab 'on' or 1 @tab Plot the vector index of the outlier ## value next to its point. ## @item @tab 'off' or 0 @tab No tags are plotted (default value). ## ## @item 'Sample_IDs' @tab 'cell' @tab A cell vector with one cell for each ## data set containing a nested cell vector with each sample's ID (should be ## a string). If this option is passed, then all outliers are tagged with ## their respective sample's ID string instead of their vector's index. ## ## @item 'BoxWidth' @tab 'proportional' @tab Create boxes with their width ## proportional to the number of samples in their respective dataset (default ## value). ## @item @tab 'fixed' @tab Make all boxes with equal width. ## ## @item 'Widths' @tab scalar @tab Scaling factor for box widths (default ## value is 0.4). ## ## @item 'CapWidths' @tab scalar @tab Scaling factor for whisker cap widths ## (default value is 1, which results to 'Widths'/8 halflength) ## ## @item 'BoxStyle' @tab 'outline' @tab Draw boxes as outlines (default value). ## @item @tab 'filled' @tab Fill boxes with a color (outlines are still ## plotted). ## ## @item 'Positions' @tab vector @tab Numerical vector that defines the ## position of each data set. It must have the same length as the number of ## groups in a desired manner. This vector merely defines the points along ## the group axis, which by default is [1:number of groups]. ## ## @item 'Labels' @tab cell @tab A cell vector of strings containing the names ## of each group. By default each group is labeled numerically according to ## its order in the data set ## ## @item 'Colors' @tab character string or Nx3 numerical matrix @tab If just ## one character or 1x3 vector of RGB values, specify the fill color of all ## boxes when BoxStyle = 'filled'. If a character string or Nx3 matrix is ## entered, box #1's fill color corrresponds to the first character or first ## matrix row, and the next boxes' fill colors corresponds to the next ## characters or rows. If the char string or Nx3 array is exhausted the color ## selection wraps around. ## @end multitable ## @end itemize ## ## Supplemental arguments not described above (@dots{}) are concatenated and ## passed to the plot() function. ## ## The returned matrix @var{s} has one column for each data set as follows: ## ## @multitable @columnfractions .1 .8 ## @item 1 @tab Minimum ## @item 2 @tab 1st quartile ## @item 3 @tab 2nd quartile (median) ## @item 4 @tab 3rd quartile ## @item 5 @tab Maximum ## @item 6 @tab Lower confidence limit for median ## @item 7 @tab Upper confidence limit for median ## @end multitable ## ## The returned structure @var{h} contains handles to the plot elements, ## allowing customization of the visualization using set/get functions. ## ## Example ## ## @example ## title ("Grade 3 heights"); ## axis ([0,3]); ## set(gca (), "xtick", [1 2], "xticklabel", @{"girls", "boys"@}); ## boxplot (@{randn(10,1)*5+140, randn(13,1)*8+135@}); ## @end example ## ## @end deftypefn function [s_o, hs_o] = boxplot (data, varargin) ## Assign parameter defaults if (nargin < 1) print_usage; endif ## Check data if (! (isnumeric (data) || iscell (data))) error ("boxplot: numerical array or cell array containing data expected."); elseif (iscell (data)) ## Check if cell contain numerical data if (! all (cellfun ("isnumeric", data))) error ("boxplot: data cells must contain numerical data."); endif endif ## Default values maxwhisker = 1.5; orientation = 1; symbol = ["+", "o"]; notched = 0; plot_opts = {}; groups = []; sample_IDs = {}; outlier_tags = 0; box_width = "proportional"; widths = 0.4; capwid = 1; box_style = 0; positions = []; labels = {}; nug = 0; bcolor = "y"; ## Optional arguments analysis numarg = nargin - 1; indopt = 1; group_exists = 0; while (numarg) dummy = varargin{indopt++}; if ((! ischar (dummy) || iscellstr (dummy)) && indopt < 6) ## MATLAB allows passing the second argument as a grouping vector if (length (dummy) > 1) if (2 != indopt) error ("boxplot: grouping vector may only be passed as second arg."); endif if (isnumeric (dummy)) groups = dummy; group_exists = 1; else error ("boxplot: grouping vector must be numerical"); endif elseif (length (dummy) == 1) ## Old way: positional argument switch indopt - group_exists case 2 notched = dummy; case 4 orientation = dummy; case 5 maxwhisker = dummy; otherwise error("boxplot: no positional argument allowed at position %d", ... --indopt); endswitch endif numarg--; continue; else if (3 == (indopt - group_exists) && length (dummy) <= 2) symbol = dummy; numarg--; continue; else ## Check for additional paired arguments switch lower (dummy) case "notch" notched = varargin{indopt}; ## Check for string input: "on" or "off" if (ischar (notched)) if (strcmpi (notched, "on")) notched = 1; elseif (strcmpi (notched, "off")) notched = 0; else msg = ["boxplot: 'Notch' input argument accepts only 'on',", ... " 'off' or a numeric scalar as value"]; error (msg); endif elseif (! (isnumeric (notched) && isreal (notched))) error ("boxplot: illegal Notch value"); endif case "symbol" symbol = varargin{indopt}; if (! ischar (symbol)) error ("boxplot; Symbol(s) must be character(s)"); endif case "orientation" orientation = varargin{indopt}; if (ischar (orientation)) ## Check for string input: "vertical" or "horizontal" if (strcmpi (orientation, "vertical")) orientation = 1; elseif (strcmpi (orientation, "horizontal")) orientation = 0; else msg = ["boxplot: 'Orientation' input argument accepts only", ... " 'vertical' (or 1) or 'horizontal' (or 0) as value"]; error (msg); endif elseif (! (isnumeric (orientation) && isreal (orientation))) error ("boxplot: illegal Orientation value"); endif case "whisker" maxwhisker = varargin{indopt}; if (! isscalar (maxwhisker) || ... ! (isnumeric (maxwhisker) && isreal (maxwhisker))) msg = ["boxplot: 'Whisker' input argument accepts only", ... " a real scalar value as input parameter"]; error(msg); endif case "outliertags" outlier_tags = varargin{indopt}; ## Check for string input: "on" or "off" if (ischar (outlier_tags)) if (strcmpi (outlier_tags, "on")) outlier_tags = 1; elseif (strcmpi (outlier_tags, "off")) outlier_tags = 0; else msg = ["boxplot: 'OutlierTags' input argument accepts only", ... " 'on' (or 1) or 'off' (or 0) as value"]; error (msg); endif elseif (! (isnumeric (outlier_tags) && isreal (outlier_tags))) error ("boxplot: illegal OutlierTags value"); endif case "sample_ids" sample_IDs = varargin{indopt}; if (! iscell (sample_IDs)) msg = ["boxplot: 'Sample_IDs' input argument accepts only", ... " a cell array as value"]; error (msg); endif outlier_tags = 1; case "boxwidth" box_width = varargin{indopt}; ## Check for string input: "fixed" or "proportional" if (! ischar (box_width) || ... ! ismember (lower (box_width), {"fixed", "proportional"})) msg = ["boxplot: 'BoxWidth' input argument accepts only", ... " 'fixed' or 'proportional' as value"]; error (msg); endif box_width = lower (box_width); case "widths" widths = varargin{indopt}; if (! isscalar (widths) || ! (isnumeric (widths) && isreal (widths))) msg = ["boxplot: 'Widths' input argument accepts only", ... " a real scalar value as value"]; error (msg); endif case "capwidths" capwid = varargin{indopt}; if (! isscalar (capwid) || ! (isnumeric (capwid) && isreal (capwid))) msg = ["boxplot: 'CapWidths' input argument accepts only", ... " a real scalar value as value"]; error (msg); endif case "boxstyle" box_style = varargin{indopt}; ## Check for string input: "outline" or "filled" if (! ischar (box_style) || ... ! ismember (lower (box_style), {"outline", "filled"})) msg = ["boxplot: 'BoxStyle' input argument accepts only", ... " 'outline' or 'filled' as value"]; error (msg); endif box_style = lower (box_style); case "positions" positions = varargin{indopt}; if (! isvector (positions) || ! isnumeric (positions)) msg = ["boxplot: 'Positions' input argument accepts only", ... " a numeric vector as value"]; error (msg); endif case "labels" labels = varargin{indopt}; if (! iscellstr (labels)) msg = ["boxplot: 'Labels' input argument accepts only", ... " a cellstr array as value"]; error (msg); endif case "colors" bcolor = varargin{indopt}; if (! (ischar (bcolor) || ... (isnumeric (bcolor) && size (bcolor, 2) == 3))) msg = ["boxplot: 'Colors' input argument accepts only", ... " a character (string) or Nx3 numeric array as value"]; error (msg); endif otherwise ## Take two args and append them to plot_opts plot_opts(1, end+1:end+2) = {dummy, varargin{indopt}}; endswitch endif numarg -= 2; indopt++; endif endwhile if (1 == length (symbol)) symbol(2) = symbol(1); endif if (1 == notched) notched = 0.25; endif a = 1-notched; ## Figure out how many data sets we have if (isempty (groups)) if (iscell (data)) nc = nug = length (data); for ind_c = (1:nc) lc(ind_c) = length (data{ind_c}); endfor else if (isvector (data)) data = data(:); endif nc = nug = columns (data); lc = ones (1, nc) * rows (data); endif groups = (1:nc); ## In case sample_IDs exists. check that it has same size as data if (! isempty (sample_IDs) && length (sample_IDs) == 1) for ind_c = (1:nc) if (lc(ind_c) != length (sample_IDs)) error ("boxplot: Sample_IDs must match the data"); endif endfor elseif (! isempty (sample_IDs) && length (sample_IDs) == nc) for ind_c = (1:nc) if (lc(ind_c) != length (sample_IDs{ind_c})) error ("boxplot: Sample_IDs must match the data"); endif endfor elseif (! isempty (sample_IDs) && length (sample_IDs) != nc) error ("boxplot: Sample_IDs must match the data"); endif ## Create labels according to number of datasets as ordered in data ## in case they are not provided by the user as optional argument if (isempty (labels)) for i = 1:nc column_label = num2str (groups(i)); labels(i) = {column_label}; endfor endif else if (! isvector (data)) error ("boxplot: with the formalism (data, group), both must be vectors"); endif ## If sample IDs given, check that their size matches the data if (! isempty (sample_IDs)) if (length (sample_IDs) != 1 || length (sample_IDs{1}) != length (data)) error ("boxplot: Sample_IDs must match the data"); endif nug = unique (groups); dummy_data = cell (1, length (nug)); dummy_sIDs = cell (1, length (nug)); ## Check if groups are parsed as a numeric vector if (isnumeric (groups)) for ind_c = (1:length (nug)) dummy_data(ind_c) = data(groups == nug(ind_c)); dummy_sIDs(ind_c) = {sample_IDs{1}(groups == nug(ind_c))}; endfor ## Create labels according to unique numeric groups in case ## they are not provided by the user as optional argument if (isempty (labels)) for i = 1:nug column_label = num2str (groups(i)); labels(i) = {column_label}; endfor endif ## Check if groups are parsed as a cell string vector elseif iscellstr (groups) for ind_c = (1:length (nug)) dummy_data(ind_c) = data(ismember (group, nug(ind_c))); dummy_sIDs(ind_c) = {sample_IDs{1}(ismember (group, nug(ind_c)))}; endfor ## Create labels according to unique cell string groups in case ## they are not provided by the user as optional argument if (isempty (labels)) labels = nug; endif else error ("boxplot: group argument must be numeric or cell string vector"); endif data = dummy_data; groups = nug(:).'; nc = length (nug); sample_IDs = dummy_sIDs; else nug = unique (groups); dummy_data = cell (1, length (nug)); ## Check if groups are parsed as a numeric vector if (isnumeric (groups)) for ind_c = (1:length (nug)) dummy_data(ind_c) = data(groups == nug(ind_c)); endfor ## Create labels according to unique numeric groups in case ## they are not provided by the user as optional argument if (isempty (labels)) for i = 1:nug column_label = num2str (groups(i)); labels(i) = {column_label}; endfor endif ## Check if groups are parsed as a cell string vector elseif (iscellstr (groups)) for ind_c = (1:length (nug)) dummy_data(ind_c) = data(ismember (group, nug(ind_c))); endfor ## Create labels according to unique cell string groups in case ## they are not provided by the user as optional argument if (isempty (labels)) labels = nug; endif else error ("boxplot: group argument must be numeric vector or cell string"); endif data = dummy_data; nc = length (nug); if (iscell (groups)) groups = [1:nc]; else groups = nug(:).'; endif endif endif ## Compute statistics. ## s will contain ## 1,5 min and max ## 2,3,4 1st, 2nd and 3rd quartile ## 6,7 lower and upper confidence intervals for median s = zeros (7, nc); box = zeros (1, nc); ## Arrange the boxes into desired positions (if requested, otherwise leave ## default 1:nc) if (! isempty (positions)) groups = positions; endif ## Initialize whisker matrices to correct size and all necessary outlier ## variables whisker_x = ones (2, 1) * [groups, groups]; whisker_y = zeros (2, 2 * nc); outliers_x = []; outliers_y = []; outliers_idx = []; outliers_IDs = {}; outliers2_x = []; outliers2_y = []; outliers2_idx = []; outliers2_IDs = {}; for indi = (1:nc) ## Get the next data set from the array or cell array if (iscell (data)) col = data{indi}(:); if (!isempty (sample_IDs)) sIDs = sample_IDs{indi}; else sIDs = num2cell([1:length(col)]); endif else col = data(:, indi); sIDs = num2cell([1:length(col)]); endif ## Skip missing data (NaN, NA) and remove respective sample IDs. ## Do this only on nonempty data if (length (col) > 0) remove_samples = find (isnan (col) | isna (col)); if (length (remove_samples) > 0) col(remove_samples) = []; sIDs(remove_samples) = []; endif endif ## Remember data length nd = length (col); box(indi) = nd; if (nd > 1) ## Min, max and quartiles s(1:5, indi) = statistics (col)(1:5); ## Confidence interval for the median est = 1.57 * (s(4, indi) - s(2, indi)) / sqrt (nd); s(6, indi) = max ([s(3, indi) - est, s(2, indi)]); s(7, indi) = min ([s(3, indi) + est, s(4, indi)]); ## Whiskers out to the last point within the desired inter-quartile range IQR = maxwhisker * (s(4, indi) - s(2, indi)); whisker_y(:, indi) = [min(col(col >= s(2, indi) - IQR)); s(2, indi)]; whisker_y(:, nc+indi) = [max(col(col <= s(4, indi) + IQR)); s(4, indi)]; ## Outliers beyond 1 and 2 inter-quartile ranges outliers = col((col < s(2, indi) - IQR & col >= s(2, indi) - 2 * IQR) | ... (col > s(4, indi) + IQR & col <= s(4, indi) + 2 * IQR)); outliers2 = col(col < s(2, indi) - 2 * IQR | col > s(4, indi) + 2 * IQR); ## Get outliers indices from this dataset if (length (outliers) > 0) for out_i = 1:length (outliers) outliers_idx = [outliers_idx; (find (col == outliers(out_i)))]; outliers_IDs = {outliers_IDs{:}, sIDs{(find (col == outliers(out_i)))}}; endfor endif if (length (outliers2) > 0) for out_i = 1:length (outliers2) outliers2_idx = [outliers2_idx; find(col == outliers2(out_i))]; outliers2_IDs = {outliers2_IDs{:}, sIDs{find(col == outliers2(out_i))}}; endfor endif outliers_x = [outliers_x; (groups(indi) * ones (size (outliers)))]; outliers_y = [outliers_y; outliers]; outliers2_x = [outliers2_x; (groups(indi) * ones (size (outliers2)))]; outliers2_y = [outliers2_y; outliers2]; elseif (1 == nd) ## All statistics collapse to the value of the point s(:, indi) = col; ## Single point data sets are plotted as outliers. outliers_x = [outliers_x; groups(indi)]; outliers_y = [outliers_y; col]; ## Append the single point's index to keep the outliers' vector aligned outliers_idx = [outliers_idx; 1]; outliers_IDs = {outliers_IDs{:}, sIDs{:}}; else ## No statistics if no points s(:, indi) = NaN; endif endfor ## Note which boxes don't have enough stats chop = find (box <= 1); ## Replicate widths (if scalar or shorter vector) to match the number of boxes widths = widths(repmat (1:length (widths), 1, nc)); ## Truncate just in case :) widths([nc+1:end]) = []; ## Draw a box around the quartiles, with box width being fixed or proportional ## to the number of items in the box. if (strcmpi (box_width, "proportional")) box = box .* (widths ./ max (box)); else box = box .* (widths ./ box); endif ## Draw notches if desired. quartile_x = ones (11, 1) * groups + ... [-a; -1; -1; 1 ; 1; a; 1; 1; -1; -1; -a] * box; quartile_y = s([3, 7, 4, 4, 7, 3, 6, 2, 2, 6, 3], :); ## Draw a line through the median median_x = ones (2, 1) * groups + [-a; +a] * box; median_y = s([3, 3], :); ## Chop all boxes which don't have enough stats quartile_x(:, chop) = []; quartile_y(:, chop) = []; whisker_x(:, [chop, chop + nc]) = []; whisker_y(:, [chop, chop + nc]) = []; median_x(:, chop) = []; median_y(:, chop) = []; box(chop) = []; ## Add caps to the remaining whiskers cap_x = whisker_x; if (strcmpi (box_width, "proportional")) cap_x(1, :) -= repmat (((capwid * box .* (widths ./ max (box))) / 8), 1, 2); cap_x(2, :) += repmat (((capwid * box .* (widths ./ max (box))) / 8), 1, 2); else cap_x(1, :) -= repmat ((capwid * widths / 8), 1, 2); cap_x(2, :) += repmat ((capwid * widths / 8), 1, 2); endif cap_y = whisker_y([1, 1], :); ## Calculate coordinates for outlier tags outliers_tags_x = outliers_x + 0.08; outliers_tags_y = outliers_y; outliers2_tags_x = outliers2_x + 0.08; outliers2_tags_y = outliers2_y; ## Do the plot if (orientation) ## Define outlier_tags' vertical alignment outlier_tags_alignment = {"horizontalalignment", "left"}; if (box_style) f = fillbox (quartile_x, quartile_y, bcolor); endif h = plot (quartile_x, quartile_y, "b;;", whisker_x, whisker_y, "b;;", cap_x, cap_y, "b;;", median_x, median_y, "r;;", outliers_x, outliers_y, [symbol(1), "r;;"], outliers2_x, outliers2_y, [symbol(2), "r;;"], plot_opts{:}); ## Print outlier tags if (outlier_tags == 1 && outliers_x > 0) t1 = plot_tags (outliers_tags_x, outliers_tags_y, outliers_idx, outliers_IDs, sample_IDs, outlier_tags_alignment); endif if (outlier_tags == 1 && outliers2_x > 0) t2 = plot_tags (outliers2_tags_x, outliers2_tags_y, outliers2_idx, outliers2_IDs, sample_IDs, outlier_tags_alignment); endif else ## Define outlier_tags' horizontal alignment outlier_tags_alignment = {"horizontalalignment", "left", "rotation", 90}; if (box_style) f = fillbox (quartile_y, quartile_x, bcolor); endif h = plot (quartile_y, quartile_x, "b;;", whisker_y, whisker_x, "b;;", cap_y, cap_x, "b;;", median_y, median_x, "r;;", outliers_y, outliers_x, [symbol(1), "r;;"], outliers2_y, outliers2_x, [symbol(2), "r;;"], plot_opts{:}); ## Print outlier tags if (outlier_tags == 1 && outliers_x > 0) t1 = plot_tags (outliers_tags_y, outliers_tags_x, outliers_idx, outliers_IDs, sample_IDs, outlier_tags_alignment); endif if (outlier_tags == 1 && outliers2_x > 0) t2 = plot_tags (outliers2_tags_y, outliers2_tags_x, outliers2_idx, outliers2_IDs, sample_IDs, outlier_tags_alignment); endif endif ## Distribute handles for box outlines and box fill (if any) nq = 1 : size (quartile_x, 2); hs.box = h(nq); if (box_style) nf = 1 : length (groups); hs.box_fill = f(nf); else hs.box_fill = []; endif ## Distribute handles for whiskers (including caps) and median lines nw = nq(end) + [1 : 2 * (size (whisker_x, 2))]; hs.whisker = h(nw); nm = nw(end)+ [1 : (size (median_x, 2))]; hs.median = h(nm); ## Distribute handles for outliers (if any) and their respective tags ## (if applicable) no = nm; if (! isempty (outliers_y)) no = nm(end) + [1 : size(outliers_y, 2)]; hs.outliers = h(no); if (outlier_tags == 1) nt = 1 : length (outliers_tags_y); hs.out_tags = t1(nt); else hs.out_tags = []; endif else hs.outliers = []; hs.out_tags = []; endif ## Distribute handles for extreme outliers (if any) and their respective tags ## (if applicable) if (! isempty (outliers2_y)) no2 = no(end) + [1 : size(outliers2_y, 2)]; hs.outliers2 = h(no2); if (outlier_tags == 1) nt2 = 1 : length (outliers2_tags_y); hs.out_tags2 = t2(nt2); else hs.out_tags2 = []; endif else hs.outliers2 = []; hs.out_tags2 = []; end ## Redraw the median lines to avoid colour overlapping in case of 'filled' ## BoxStyle if (box_style) set (hs.median, "color", "r"); endif ## Print labels according to orientation and return handle if (orientation) set (gca(), "xtick", groups, "xticklabel", labels); hs.labels = get (gcf, "currentaxes"); else set (gca(), "ytick", groups, "yticklabel", labels); hs.labels = get (gcf, "currentaxes"); endif hold off; ## Return output arguments if desired if (nargout >= 1) s_o = s; endif if (nargout == 2) hs_o = hs; endif endfunction function htags = plot_tags (out_tags_x, out_tags_y, out_idx, out_IDs, ... sample_IDs, opt) for i=1 : length (out_tags_x) if (! isempty (sample_IDs)) htags(i) = text (out_tags_x(i), out_tags_y(i), out_IDs{i}, opt{:}); else htags(i) = text (out_tags_x(i), out_tags_y(i), num2str (out_idx(i)), ... opt{:}); endif endfor endfunction function f = fillbox (quartile_y, quartile_x, bcolor) f = []; for icol = 1 : columns (quartile_x) if (ischar (bcolor)) f = [ f; fill(quartile_y(:, icol), quartile_x(:, icol), ... bcolor(mod (icol-1, numel (bcolor))+1)) ]; else f = [ f; fill(quartile_y(:, icol), quartile_x(:, icol), ... bcolor(mod (icol-1, size (bcolor, 1))+1, :)) ]; endif hold on; endfor endfunction %!demo %! axis ([0, 3]); %! randn ("seed", 1); # for reproducibility %! girls = randn (10, 1) * 5 + 140; %! randn ("seed", 2); # for reproducibility %! boys = randn (13, 1) * 8 + 135; %! boxplot ({girls, boys}); %! set (gca (), "xtick", [1 2], "xticklabel", {"girls", "boys"}) %! title ("Grade 3 heights"); %!demo %! randn ("seed", 7); # for reproducibility %! A = randn (10, 1) * 5 + 140; %! randn ("seed", 8); # for reproducibility %! B = randn (25, 1) * 8 + 135; %! randn ("seed", 9); # for reproducibility %! C = randn (20, 1) * 6 + 165; %! data = [A; B; C]; %! groups = [(ones (10, 1)); (ones (25, 1) * 2); (ones (20, 1) * 3)]; %! labels = {"Team A", "Team B", "Team C"}; %! pos = [2, 1, 3]; %! boxplot (data, groups, "Notch", "on", "Labels", labels, "Positions", pos, ... %! "OutlierTags", "on", "BoxStyle", "filled"); %! title ("Example of Group splitting with paired vectors"); %!demo %! randn ("seed", 1); # for reproducibility %! data = randn (100, 9); %! boxplot (data, "notch", "on", "boxstyle", "filled", ... %! "colors", "ygcwkmb", "whisker", 1.2); %! title ("Example of different colors specified with characters"); %!demo %! randn ("seed", 5); # for reproducibility %! data = randn (100, 13); %! colors = [0.7 0.7 0.7; ... %! 0.0 0.4 0.9; ... %! 0.7 0.4 0.3; ... %! 0.7 0.1 0.7; ... %! 0.8 0.7 0.4; ... %! 0.1 0.8 0.5; ... %! 0.9 0.9 0.2]; %! boxplot (data, "notch", "on", "boxstyle", "filled", ... %! "colors", colors, "whisker", 1.3, "boxwidth", "proportional"); %! title ("Example of different colors specified as RGB values"); ## Input data validation %!error boxplot ("a") %!error boxplot ({[1 2 3], "a"}) %!error boxplot ([1 2 3], 1, {2, 3}) %!error boxplot ([1 2 3], {"a", "b"}) %!error <'Notch' input argument accepts> boxplot ([1:10], "notch", "any") %!error boxplot ([1:10], "notch", i) %!error boxplot ([1:10], "notch", {}) %!error boxplot (1, "symbol", 1) %!error <'Orientation' input argument accepts only> boxplot (1, "orientation", "diagonal") %!error boxplot (1, "orientation", {}) %!error <'Whisker' input argument accepts only> boxplot (1, "whisker", "a") %!error <'Whisker' input argument accepts only> boxplot (1, "whisker", [1 3]) %!error <'OutlierTags' input argument accepts only> boxplot (3, "OutlierTags", "maybe") %!error boxplot (3, "OutlierTags", {}) %!error <'Sample_IDs' input argument accepts only> boxplot (1, "sample_IDs", 1) %!error <'BoxWidth' input argument accepts only> boxplot (1, "boxwidth", 2) %!error <'BoxWidth' input argument accepts only> boxplot (1, "boxwidth", "anything") %!error <'Widths' input argument accepts only> boxplot (5, "widths", "a") %!error <'Widths' input argument accepts only> boxplot (5, "widths", [1:4]) %!error <'Widths' input argument accepts only> boxplot (5, "widths", []) %!error <'CapWidths' input argument accepts only> boxplot (5, "capwidths", "a") %!error <'CapWidths' input argument accepts only> boxplot (5, "capwidths", [1:4]) %!error <'CapWidths' input argument accepts only> boxplot (5, "capwidths", []) %!error <'BoxStyle' input argument accepts only> boxplot (1, "Boxstyle", 1) %!error <'BoxStyle' input argument accepts only> boxplot (1, "Boxstyle", "garbage") %!error <'Positions' input argument accepts only> boxplot (1, "positions", "aa") %!error <'Labels' input argument accepts only> boxplot (3, "labels", [1 5]) %!error <'Colors' input argument accepts only> boxplot (1, "colors", {}) %!error <'Colors' input argument accepts only> boxplot (2, "colors", [1 2 3 4]) %!error boxplot (randn (10, 3), 'Sample_IDs', {"a", "b"}) %!error boxplot (rand (3, 3), [1 2]) ## Test plotting %!test %! hf = figure ("visible", "off"); %! unwind_protect %! [a, b] = boxplot (rand (10, 3)); %! assert (size (a), [7, 3]); %! assert (numel (b.box), 3); %! assert (numel (b.whisker), 12); %! assert (numel (b.median), 3); %! unwind_protect_cleanup %! close (hf); %! end_unwind_protect %!test %! hf = figure ("visible", "off"); %! unwind_protect %! [~, b] = boxplot (rand (10, 3), "BoxStyle", "filled", "colors", "ybc"); %! assert (numel (b.box_fill), 3); %! unwind_protect_cleanup %! close (hf); %! end_unwind_protect statistics-release-1.6.3/inst/canoncorr.m000066400000000000000000000064411456127120000204530ustar00rootroot00000000000000## Copyright (C) 2016-2019 by Nir Krakauer ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {[@var{A}, @var{B}, @var{r}, @var{U}, @var{V}] =} canoncorr (@var{X}, @var{Y}) ## ## Canonical correlation analysis. ## ## Given @var{X} (size @var{k}*@var{m}) and @var{Y} (@var{k}*@var{n}), returns ## projection matrices of canonical coefficients @var{A} (size @var{m}*@var{d}, ## where @var{d} is the smallest of @var{m}, @var{n}, @var{d}) and @var{B} ## (size @var{m}*@var{d}); the canonical correlations @var{r} (1*@var{d}, ## arranged in decreasing order); the canonical variables @var{U}, @var{V} ## (both @var{k}*@var{d}, with orthonormal columns); and @var{stats}, ## a structure containing results from Bartlett's chi-square and Rao's F tests ## of significance. ## ## @seealso{princomp} ## @end deftypefn function [A,B,r,U,V,stats] = canoncorr (X,Y) k = size (X, 1); # should also be size (Y, 1) m = size (X, 2); n = size (Y, 2); d = min ([k m n]); X = center (X); Y = center (Y); [Qx Rx] = qr (X, 0); [Qy Ry] = qr (Y, 0); [U S V] = svd (Qx' * Qy, "econ"); A = Rx \ U(:, 1:d); B = Ry \ V(:, 1:d); ## A, B are scaled to make the covariance matrices of the outputs U, V ## identity matrices f = sqrt (k-1); A .*= f; B .*= f; if (nargout > 2) r = max(0, min(diag(S), 1))'; endif if (nargout > 3) U = X * A; endif if (nargout > 4) V = Y * B; endif if (nargout > 5) Wilks = fliplr(cumprod(fliplr((1 - r .^ 2)))); chisq = - (k - 1 - (m + n + 1)/2) * log(Wilks); df1 = (m - (1:d) + 1) .* (n - (1:d) + 1); pChisq = 1 - chi2cdf (chisq, df1); s = sqrt((df1.^2 - 4) ./ ((m - (1:d) + 1).^2 + (n - (1:d) + 1).^2 - 5)); df2 = (k - 1 - (m + n + 1)/2) * s - df1/2 + 1; ls = Wilks .^ (1 ./ s); F = (1 ./ ls - 1) .* (df2 ./ df1); pF = 1 - fcdf (F, df1, df2); stats.Wilks = Wilks; stats.df1 = df1; stats.df2 = df2; stats.F = F; stats.pF = pF; stats.chisq = chisq; stats.pChisq = pChisq; endif endfunction %!shared X,Y,A,B,r,U,V,k %! k = 10; %! X = [1:k; sin(1:k); cos(1:k)]'; Y = [tan(1:k); tanh((1:k)/k)]'; %! [A,B,r,U,V,stats] = canoncorr (X,Y); %!assert (A, [-0.329229 0.072908; 0.074870 1.389318; -0.069302 -0.024109], 1E-6); %!assert (B, [-0.017086 -0.398402; -4.475049 -0.824538], 1E-6); %!assert (r, [0.99590 0.26754], 1E-5); %!assert (U, center(X) * A, 10*eps); %!assert (V, center(Y) * B, 10*eps); %!assert (cov(U), eye(size(U, 2)), 10*eps); %!assert (cov(V), eye(size(V, 2)), 10*eps); %! rand ("state", 1); [A,B,r] = canoncorr (rand(5, 10),rand(5, 20)); %!assert (r, ones(1, 5), 10*eps); statistics-release-1.6.3/inst/cdf.m000066400000000000000000000372221456127120000172240ustar00rootroot00000000000000## Copyright (C) 2013 Pantxo Diribarne ## Copyright (C) 2022-2024 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it ## under the terms of the GNU General Public License as published by ## the Free Software Foundation; either version 3 of the License, or ## (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program. If not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{p} =} cdf (@var{name}, @var{x}, @var{A}) ## @deftypefnx {statistics} {@var{p} =} cdf (@var{name}, @var{x}, @var{A}, @var{B}) ## @deftypefnx {statistics} {@var{p} =} cdf (@var{name}, @var{x}, @var{A}, @var{B}, @var{C}) ## @deftypefnx {statistics} {@var{p} =} cdf (@dots{}, @qcode{"upper"}) ## ## Return the CDF of a univariate distribution evaluated at @var{x}. ## ## @code{cdf} is a wrapper for the univariate cumulative distribution functions ## available in the statistics package. See the corresponding functions' help ## to learn the signification of the parameters after @var{x}. ## ## @code{@var{p} = cdf (@var{name}, @var{x}, @var{A})} returns the CDF for the ## one-parameter distribution family specified by @var{name} and the ## distribution parameter @var{A}, evaluated at the values in @var{x}. ## ## @code{@var{p} = cdf (@var{name}, @var{x}, @var{A}, @var{B})} returns the CDF ## for the two-parameter distribution family specified by @var{name} and the ## distribution parameters @var{A} and @var{B}, evaluated at the values in ## @var{x}. ## ## @code{@var{p} = cdf (@var{name}, @var{x}, @var{A}, @var{B}, @var{C})} returns ## the CDF for the three-parameter distribution family specified by @var{name} ## and the distribution parameters @var{A}, @var{B}, and @var{C}, evaluated at ## the values in @var{x}. ## ## @code{@var{p} = cdf (@dots{}, @qcode{"upper"})} returns the complement of the ## CDF using an algorithm that more accurately computes the extreme upper-tail ## probabilities. @qcode{"upper"} can follow any of the input arguments in the ## previous syntaxes. ## ## @var{name} must be a char string of the name or the abbreviation of the ## desired cumulative distribution function as listed in the followng table. ## The last column shows the number of required parameters that should be parsed ## after @var{x} to the desired CDF. The optional input argument ## @qcode{"upper"} does not count in the required number of parameters. ## ## @multitable @columnfractions 0.4 0.05 0.2 0.05 0.3 ## @headitem Distribution Name @tab @tab Abbreviation @tab @tab Input Parameters ## @item @qcode{"Beta"} @tab @tab @qcode{"beta"} @tab @tab 2 ## @item @qcode{"Binomial"} @tab @tab @qcode{"bino"} @tab @tab 2 ## @item @qcode{"Birnbaum-Saunders"} @tab @tab @qcode{"bisa"} @tab @tab 2 ## @item @qcode{"Burr"} @tab @tab @qcode{"burr"} @tab @tab 3 ## @item @qcode{"Cauchy"} @tab @tab @qcode{"cauchy"} @tab @tab 2 ## @item @qcode{"Chi-squared"} @tab @tab @qcode{"chi2"} @tab @tab 1 ## @item @qcode{"Extreme Value"} @tab @tab @qcode{"ev"} @tab @tab 2 ## @item @qcode{"Exponential"} @tab @tab @qcode{"exp"} @tab @tab 1 ## @item @qcode{"F-Distribution"} @tab @tab @qcode{"f"} @tab @tab 2 ## @item @qcode{"Gamma"} @tab @tab @qcode{"gam"} @tab @tab 2 ## @item @qcode{"Geometric"} @tab @tab @qcode{"geo"} @tab @tab 1 ## @item @qcode{"Generalized Extreme Value"} @tab @tab @qcode{"gev"} @tab @tab 3 ## @item @qcode{"Generalized Pareto"} @tab @tab @qcode{"gp"} @tab @tab 3 ## @item @qcode{"Gumbel"} @tab @tab @qcode{"gumbel"} @tab @tab 2 ## @item @qcode{"Half-normal"} @tab @tab @qcode{"hn"} @tab @tab 2 ## @item @qcode{"Hypergeometric"} @tab @tab @qcode{"hyge"} @tab @tab 3 ## @item @qcode{"Inverse Gaussian"} @tab @tab @qcode{"invg"} @tab @tab 2 ## @item @qcode{"Laplace"} @tab @tab @qcode{"laplace"} @tab @tab 2 ## @item @qcode{"Logistic"} @tab @tab @qcode{"logi"} @tab @tab 2 ## @item @qcode{"Log-Logistic"} @tab @tab @qcode{"logl"} @tab @tab 2 ## @item @qcode{"Lognormal"} @tab @tab @qcode{"logn"} @tab @tab 2 ## @item @qcode{"Nakagami"} @tab @tab @qcode{"naka"} @tab @tab 2 ## @item @qcode{"Negative Binomial"} @tab @tab @qcode{"nbin"} @tab @tab 2 ## @item @qcode{"Noncentral F-Distribution"} @tab @tab @qcode{"ncf"} @tab @tab 3 ## @item @qcode{"Noncentral Student T"} @tab @tab @qcode{"nct"} @tab @tab 2 ## @item @qcode{"Noncentral Chi-Squared"} @tab @tab @qcode{"ncx2"} @tab @tab 2 ## @item @qcode{"Normal"} @tab @tab @qcode{"norm"} @tab @tab 2 ## @item @qcode{"Poisson"} @tab @tab @qcode{"poiss"} @tab @tab 1 ## @item @qcode{"Rayleigh"} @tab @tab @qcode{"rayl"} @tab @tab 1 ## @item @qcode{"Rician"} @tab @tab @qcode{"rice"} @tab @tab 2 ## @item @qcode{"Student T"} @tab @tab @qcode{"t"} @tab @tab 1 ## @item @qcode{"location-scale T"} @tab @tab @qcode{"tls"} @tab @tab 3 ## @item @qcode{"Triangular"} @tab @tab @qcode{"tri"} @tab @tab 3 ## @item @qcode{"Discrete Uniform"} @tab @tab @qcode{"unid"} @tab @tab 1 ## @item @qcode{"Uniform"} @tab @tab @qcode{"unif"} @tab @tab 2 ## @item @qcode{"Von Mises"} @tab @tab @qcode{"vm"} @tab @tab 2 ## @item @qcode{"Weibull"} @tab @tab @qcode{"wbl"} @tab @tab 2 ## @end multitable ## ## @seealso{icdf, pdf, cdf, betacdf, binocdf, bisacdf, burrcdf, cauchycdf, ## chi2cdf, evcdf, expcdf, fcdf, gamcdf, geocdf, gevcdf, gpcdf, gumbelcdf, ## hncdf, hygecdf, invgcdf, laplacecdf, logicdf, loglcdf, logncdf, nakacdf, ## nbincdf, ncfcdf, nctcdf, ncx2cdf, normcdf, poisscdf, raylcdf, ricecdf, ## tcdf, tricdf, unidcdf, unifcdf, vmcdf, wblcdf} ## @end deftypefn function p = cdf (name, x, varargin) ## implemented functions persistent allDF = { ... {"beta" , "Beta"}, @betacdf, 2, ... {"bino" , "Binomial"}, @binocdf, 2, ... {"bisa" , "Birnbaum-Saunders"}, @bisacdf, 2, ... {"burr" , "Burr"}, @burrcdf, 3, ... {"cauchy" , "Cauchy"}, @cauchycdf, 2, ... {"chi2" , "Chi-squared"}, @chi2cdf, 1, ... {"ev" , "Extreme Value"}, @evcdf, 2, ... {"exp" , "Exponential"}, @expcdf, 1, ... {"f" , "F-Distribution"}, @fcdf, 2, ... {"gam" , "Gamma"}, @gamcdf, 2, ... {"geo" , "Geometric"}, @geocdf, 1, ... {"gev" , "Generalized Extreme Value"}, @gevcdf, 3, ... {"gp" , "Generalized Pareto"}, @gpcdf, 3, ... {"gumbel" , "Gumbel"}, @gumbelcdf, 2, ... {"hn" , "Half-normal"}, @hncdf, 2, ... {"hyge" , "Hypergeometric"}, @hygecdf, 3, ... {"invg" , "Inverse Gaussian"}, @invgcdf, 2, ... {"laplace" , "Laplace"}, @laplacecdf, 2, ... {"logi" , "Logistic"}, @logicdf, 2, ... {"logl" , "Log-Logistic"}, @loglcdf, 2, ... {"logn" , "Lognormal"}, @logncdf, 2, ... {"naka" , "Nakagami"}, @nakacdf, 2, ... {"nbin" , "Negative Binomial"}, @nbincdf, 2, ... {"ncf" , "Noncentral F-Distribution"}, @ncfcdf, 3, ... {"nct" , "Noncentral Student T"}, @nctcdf, 2, ... {"ncx2" , "Noncentral Chi-squared"}, @ncx2cdf, 2, ... {"norm" , "Normal"}, @normcdf, 2, ... {"poiss" , "Poisson"}, @poisscdf, 1, ... {"rayl" , "Rayleigh"}, @raylcdf, 1, ... {"rice" , "Rician"}, @ricecdf, 2, ... {"t" , "Student T"}, @tcdf, 1, ... {"tls" , "location-scale T"}, @tlscdf, 3, ... {"tri" , "Triangular"}, @tricdf, 3, ... {"unid" , "Discrete Uniform"}, @unidcdf, 1, ... {"unif" , "Uniform"}, @unifcdf, 2, ... {"vm" , "Von Mises"}, @vmcdf, 2, ... {"wbl" , "Weibull"}, @wblcdf, 2}; ## Check NAME being a char string if (! ischar (name)) error ("cdf: distribution NAME must a char string."); endif ## Check X being numeric and real if (! isnumeric (x)) error ("cdf: X must be numeric."); elseif (! isreal (x)) error ("cdf: values in X must be real."); endif ## Get number of arguments nargs = numel (varargin); ## Get available functions cdfnames = allDF(1:3:end); cdfhandl = allDF(2:3:end); cdf_args = allDF(3:3:end); ## Search for CDF function idx = cellfun (@(x)any(strcmpi (name, x)), cdfnames); if (any (idx)) if (nargs == cdf_args{idx} + 1) ## Check for "upper" option if (! strcmpi (varargin{nargs}, "upper")) error ("cdf: invalid argument for upper tail."); else ## Check that all remaining distribution parameters are numeric if (! all (cellfun (@(x)isnumeric(x), (varargin([1:nargs-1]))))) error ("cdf: distribution parameters must be numeric."); endif ## Call appropriate CDF with "upper" flag p = feval (cdfhandl{idx}, x, varargin{:}); endif elseif (nargs == cdf_args{idx}) ## Check that all distribution parameters are numeric if (! all (cellfun (@(x)isnumeric(x), (varargin)))) error ("cdf: distribution parameters must be numeric."); endif ## Call appropriate CDF without "upper" flag p = feval (cdfhandl{idx}, x, varargin{:}); else if (cdf_args{idx} == 1) error ("cdf: %s distribution requires 1 parameter.", name); else error ("cdf: %s distribution requires %d parameters.", ... name, cdf_args{idx}); endif endif else error ("cdf: %s distribution is not implemented in Statistics.", name); endif endfunction ## Test results %!shared x %! x = [1:5]; %!assert (cdf ("Beta", x, 5, 2), betacdf (x, 5, 2)) %!assert (cdf ("beta", x, 5, 2, "upper"), betacdf (x, 5, 2, "upper")) %!assert (cdf ("Binomial", x, 5, 2), binocdf (x, 5, 2)) %!assert (cdf ("bino", x, 5, 2, "upper"), binocdf (x, 5, 2, "upper")) %!assert (cdf ("Birnbaum-Saunders", x, 5, 2), bisacdf (x, 5, 2)) %!assert (cdf ("bisa", x, 5, 2, "upper"), bisacdf (x, 5, 2, "upper")) %!assert (cdf ("Burr", x, 5, 2, 2), burrcdf (x, 5, 2, 2)) %!assert (cdf ("burr", x, 5, 2, 2, "upper"), burrcdf (x, 5, 2, 2, "upper")) %!assert (cdf ("Cauchy", x, 5, 2), cauchycdf (x, 5, 2)) %!assert (cdf ("cauchy", x, 5, 2, "upper"), cauchycdf (x, 5, 2, "upper")) %!assert (cdf ("Chi-squared", x, 5), chi2cdf (x, 5)) %!assert (cdf ("chi2", x, 5, "upper"), chi2cdf (x, 5, "upper")) %!assert (cdf ("Extreme Value", x, 5, 2), evcdf (x, 5, 2)) %!assert (cdf ("ev", x, 5, 2, "upper"), evcdf (x, 5, 2, "upper")) %!assert (cdf ("Exponential", x, 5), expcdf (x, 5)) %!assert (cdf ("exp", x, 5, "upper"), expcdf (x, 5, "upper")) %!assert (cdf ("F-Distribution", x, 5, 2), fcdf (x, 5, 2)) %!assert (cdf ("f", x, 5, 2, "upper"), fcdf (x, 5, 2, "upper")) %!assert (cdf ("Gamma", x, 5, 2), gamcdf (x, 5, 2)) %!assert (cdf ("gam", x, 5, 2, "upper"), gamcdf (x, 5, 2, "upper")) %!assert (cdf ("Geometric", x, 5), geocdf (x, 5)) %!assert (cdf ("geo", x, 5, "upper"), geocdf (x, 5, "upper")) %!assert (cdf ("Generalized Extreme Value", x, 5, 2, 2), gevcdf (x, 5, 2, 2)) %!assert (cdf ("gev", x, 5, 2, 2, "upper"), gevcdf (x, 5, 2, 2, "upper")) %!assert (cdf ("Generalized Pareto", x, 5, 2, 2), gpcdf (x, 5, 2, 2)) %!assert (cdf ("gp", x, 5, 2, 2, "upper"), gpcdf (x, 5, 2, 2, "upper")) %!assert (cdf ("Gumbel", x, 5, 2), gumbelcdf (x, 5, 2)) %!assert (cdf ("gumbel", x, 5, 2, "upper"), gumbelcdf (x, 5, 2, "upper")) %!assert (cdf ("Half-normal", x, 5, 2), hncdf (x, 5, 2)) %!assert (cdf ("hn", x, 5, 2, "upper"), hncdf (x, 5, 2, "upper")) %!assert (cdf ("Hypergeometric", x, 5, 2, 2), hygecdf (x, 5, 2, 2)) %!assert (cdf ("hyge", x, 5, 2, 2, "upper"), hygecdf (x, 5, 2, 2, "upper")) %!assert (cdf ("Inverse Gaussian", x, 5, 2), invgcdf (x, 5, 2)) %!assert (cdf ("invg", x, 5, 2, "upper"), invgcdf (x, 5, 2, "upper")) %!assert (cdf ("Laplace", x, 5, 2), laplacecdf (x, 5, 2)) %!assert (cdf ("laplace", x, 5, 2, "upper"), laplacecdf (x, 5, 2, "upper")) %!assert (cdf ("Logistic", x, 5, 2), logicdf (x, 5, 2)) %!assert (cdf ("logi", x, 5, 2, "upper"), logicdf (x, 5, 2, "upper")) %!assert (cdf ("Log-Logistic", x, 5, 2), loglcdf (x, 5, 2)) %!assert (cdf ("logl", x, 5, 2, "upper"), loglcdf (x, 5, 2, "upper")) %!assert (cdf ("Lognormal", x, 5, 2), logncdf (x, 5, 2)) %!assert (cdf ("logn", x, 5, 2, "upper"), logncdf (x, 5, 2, "upper")) %!assert (cdf ("Nakagami", x, 5, 2), nakacdf (x, 5, 2)) %!assert (cdf ("naka", x, 5, 2, "upper"), nakacdf (x, 5, 2, "upper")) %!assert (cdf ("Negative Binomial", x, 5, 2), nbincdf (x, 5, 2)) %!assert (cdf ("nbin", x, 5, 2, "upper"), nbincdf (x, 5, 2, "upper")) %!assert (cdf ("Noncentral F-Distribution", x, 5, 2, 2), ncfcdf (x, 5, 2, 2)) %!assert (cdf ("ncf", x, 5, 2, 2, "upper"), ncfcdf (x, 5, 2, 2, "upper")) %!assert (cdf ("Noncentral Student T", x, 5, 2), nctcdf (x, 5, 2)) %!assert (cdf ("nct", x, 5, 2, "upper"), nctcdf (x, 5, 2, "upper")) %!assert (cdf ("Noncentral Chi-Squared", x, 5, 2), ncx2cdf (x, 5, 2)) %!assert (cdf ("ncx2", x, 5, 2, "upper"), ncx2cdf (x, 5, 2, "upper")) %!assert (cdf ("Normal", x, 5, 2), normcdf (x, 5, 2)) %!assert (cdf ("norm", x, 5, 2, "upper"), normcdf (x, 5, 2, "upper")) %!assert (cdf ("Poisson", x, 5), poisscdf (x, 5)) %!assert (cdf ("poiss", x, 5, "upper"), poisscdf (x, 5, "upper")) %!assert (cdf ("Rayleigh", x, 5), raylcdf (x, 5)) %!assert (cdf ("rayl", x, 5, "upper"), raylcdf (x, 5, "upper")) %!assert (cdf ("Rician", x, 5, 1), ricecdf (x, 5, 1)) %!assert (cdf ("rice", x, 5, 1, "upper"), ricecdf (x, 5, 1, "upper")) %!assert (cdf ("Student T", x, 5), tcdf (x, 5)) %!assert (cdf ("t", x, 5, "upper"), tcdf (x, 5, "upper")) %!assert (cdf ("location-scale T", x, 5, 1, 2), tlscdf (x, 5, 1, 2)) %!assert (cdf ("tls", x, 5, 1, 2, "upper"), tlscdf (x, 5, 1, 2, "upper")) %!assert (cdf ("Triangular", x, 5, 2, 2), tricdf (x, 5, 2, 2)) %!assert (cdf ("tri", x, 5, 2, 2, "upper"), tricdf (x, 5, 2, 2, "upper")) %!assert (cdf ("Discrete Uniform", x, 5), unidcdf (x, 5)) %!assert (cdf ("unid", x, 5, "upper"), unidcdf (x, 5, "upper")) %!assert (cdf ("Uniform", x, 5, 2), unifcdf (x, 5, 2)) %!assert (cdf ("unif", x, 5, 2, "upper"), unifcdf (x, 5, 2, "upper")) %!assert (cdf ("Von Mises", x, 5, 2), vmcdf (x, 5, 2)) %!assert (cdf ("vm", x, 5, 2, "upper"), vmcdf (x, 5, 2, "upper")) %!assert (cdf ("Weibull", x, 5, 2), wblcdf (x, 5, 2)) %!assert (cdf ("wbl", x, 5, 2, "upper"), wblcdf (x, 5, 2, "upper")) ## Test input validation %!error cdf (1) %!error cdf ({"beta"}) %!error cdf ("beta", {[1 2 3 4 5]}) %!error cdf ("beta", "text") %!error cdf ("beta", 1+i) %!error ... %! cdf ("Beta", x, "a", 2) %!error ... %! cdf ("Beta", x, 5, "") %!error ... %! cdf ("Beta", x, 5, {2}) %!error cdf ("chi2", x) %!error cdf ("Beta", x, 5) %!error cdf ("Burr", x, 5) %!error cdf ("Burr", x, 5, 2) statistics-release-1.6.3/inst/cdfcalc.m000066400000000000000000000057571456127120000200570ustar00rootroot00000000000000## Copyright (C) 2022 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {[@var{yCDF}, @var{xCDF}, @var{n}, @var{emsg}, @var{eid}] =} cdfcalc (@var{x}) ## ## Calculate an empirical cumulative distribution function. ## ## @code{[@var{yCDF}, @var{xCDF}] = cdfcalc (@var{x})} calculates an empirical ## cumulative distribution function (CDF) of the observations in the data sample ## vector @var{x}. @var{x} may be a row or column vector, and represents a ## random sample of observations from some underlying distribution. On return ## @var{xCDF} is the set of @var{x} values at which the CDF increases. ## At XCDF(i), the function increases from YCDF(i) to YCDF(i+1). ## ## @code{[@var{yCDF}, @var{xCDF}, @var{n}] = cdfcalc (@var{x})} also returns ## @var{n}, the sample size. ## ## @code{[@var{yCDF}, @var{xCDF}, @var{n}, @var{emsg}, @var{eid}] = cdfcalc ## (@var{x})} also returns an error message and error id if @var{x} is not a ## vector or if it contains no values other than NaN. ## ## @seealso{cdfplot} ## @end deftypefn function [yCDF, xCDF, n, emsg, eid] = cdfcalc (x) ## Check number of input and output argument narginchk (1,1); nargoutchk (2,5); ## Add defaults yCDF = []; xCDF = []; n = 0; ## Check that x is a vector if (! isvector (x)) warning ("cdfcalc: vector required as input."); emsg = "VectorRequired"; eid = "VectorRequired"; return endif ## Remove NaNs and check if there are remaining data to calculate ecdf x = x(! isnan (x)); n = length (x); if (n == 0) warning ("cdfcalc: not enough data."); emsg = "NotEnoughData"; eid = "NotEnoughData"; return endif ## Sort data in ascending order x = sort (x(:)); ## Get cumulative sums yCDF = (1:n)' / n; ## Remove duplicates, keep the last one keep_idx = ([diff(x(:)); 1] > 0); xCDF = x(keep_idx); yCDF = [0; yCDF(keep_idx)]; emsg = ''; eid = ''; endfunction %!test %! x = [2, 4, 3, 2, 4, 3, 2, 5, 6, 4]; %! [yCDF, xCDF, n, emsg, eid] = cdfcalc (x); %! assert (yCDF, [0, 0.3, 0.5, 0.8, 0.9, 1]'); %! assert (xCDF, [2, 3, 4, 5, 6]'); %! assert (n, 10); %!shared x %! x = [2, 4, 3, 2, 4, 3, 2, 5, 6, 4]; %!error yCDF = cdfcalc (x); %!error [yCDF, xCDF] = cdfcalc (); %!error [yCDF, xCDF] = cdfcalc (x, x); %!warning [yCDF, xCDF] = cdfcalc (ones(10,2)); statistics-release-1.6.3/inst/cdfplot.m000066400000000000000000000074071456127120000201250ustar00rootroot00000000000000## Copyright (C) 2022 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{hCDF} =} cdfplot (@var{x}) ## @deftypefnx {statistics} {[@var{hCDF}, @var{stats}] =} cdfplot (@var{x}) ## ## Display an empirical cumulative distribution function. ## ## @code{@var{hCDF} = cdfplot (@var{x})} plots an empirical cumulative ## distribution function (CDF) of the observations in the data sample vector ## @var{x}. @var{x} may be a row or column vector, and represents a random ## sample of observations from some underlying distribution. ## ## @code{cdfplot} plots F(x), the empirical (or sample) CDF versus the ## observations in @var{x}. The empirical CDF, F(x), is defined as follows: ## ## F(x) = (Number of observations <= x) / (Total number of observations) ## ## for all values in the sample vector @var{x}. NaNs are ignored. @var{hCDF} ## is the handle of the empirical CDF curve (a handle hraphics 'line' object). ## ## @code{[@var{hCDF}, @var{stats}] = cdfplot (@var{x})} also returns a structure ## with the following fields as a statistical summary. ## ## @multitable @columnfractions 0.05 0.3 0.65 ## @item @tab STATS.min @tab minimum value of @var{x} ## @item @tab STATS.max @tab maximum value of @var{x} ## @item @tab STATS.mean @tab sample mean of @var{x} ## @item @tab STATS.median @tab sample median (50th percentile) of @var{x} ## @item @tab STATS.std @tab sample standard deviation of @var{x} ## @end multitable ## ## @seealso{qqplot, cdfcalc} ## @end deftypefn function [hCDF, stats] = cdfplot (x) ## Check number of input arguments narginchk (1,1); ## Calculate sample cdf [yy, xx, ~, ~, eid] = cdfcalc (x); ## Check for errors returned from cdfcalc if (strcmpi (eid, "VectorRequired")) error ("cdfplot: vector required as input."); elseif (strcmpi (eid, "NotEnoughData")) error("cdfplot: not enough data."); endif ## Create vectors for plotting k = length (xx); n = reshape (repmat (1:k, 2, 1), 2*k, 1); xCDF = [-Inf; xx(n); Inf]; yCDF = [0; 0; yy(1+n)]; ## Plot cdf h = plot (xCDF, yCDF); grid ('on') xlabel ("x") ylabel ("F(x)") title ("CDF plot of x"); ## Return requested output arguments if (nargout > 0) hCDF = h; endif if (nargout > 1) stats.min = nanmin (x); stats.max = nanmax (x); stats.mean = mean (x, "omitnan"); stats.median = median (x, "omitnan"); stats.std = std (x, "omitnan"); endif endfunction %!demo %! x = randn(100,1); %! cdfplot (x); ## Test results %!test %! hf = figure ("visible", "off"); %! unwind_protect %! x = [2, 4, 3, 2, 4, 3, 2, 5, 6, 4]; %! [hCDF, stats] = cdfplot (x); %! assert (stats.min, 2); %! assert (stats.max, 6); %! assert (stats.median, 3.5); %! assert (stats.std, 1.35400640077266, 1e-14); %! unwind_protect_cleanup %! close (hf); %! end_unwind_protect %!test %! hf = figure ("visible", "off"); %! unwind_protect %! x = randn(100,1); %! cdfplot (x); %! unwind_protect_cleanup %! close (hf); %! end_unwind_protect ## Test input validation %!error cdfplot (); %!error cdfplot ([x',x']); %!error cdfplot ([NaN, NaN, NaN, NaN]); statistics-release-1.6.3/inst/chi2gof.m000066400000000000000000000413061456127120000200070ustar00rootroot00000000000000## Copyright (C) 2022 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{h} =} chi2gof (@var{x}) ## @deftypefnx {statistics} {[@var{h}, @var{p}] =} chi2gof (@var{x}) ## @deftypefnx {statistics} {[@var{p}, @var{h}, @var{stats}] =} chi2gof (@var{x}) ## @deftypefnx {statistics} {[@dots{}] =} chi2gof (@var{x}, @var{Name}, @var{Value}, @dots{}) ## ## Chi-square goodness-of-fit test. ## ## @code{chi2gof} performs a chi-square goodness-of-fit test for discrete or ## continuous distributions. The test is performed by grouping the data into ## bins, calculating the observed and expected counts for those bins, and ## computing the chi-square test statistic ## @tex ## $$ \chi ^ 2 = \sum_{i=1}^N \left (O_i - E_i \right) ^ 2 / E_i $$ ## @end tex ## @ifnottex ## SUM((O-E).^2./E), ## @end ifnottex ## where O is the observed counts and E is the expected counts. This test ## statistic has an approximate chi-square distribution when the counts are ## sufficiently large. ## ## Bins in either tail with an expected count less than 5 are pooled with ## neighboring bins until the count in each extreme bin is at least 5. If ## bins remain in the interior with counts less than 5, @code{chi2gof} displays ## a warning. In that case, you should use fewer bins, or provide bin centers ## or binedges, to increase the expected counts in all bins. ## ## @code{@var{h} = chi2gof (@var{x})} performs a chi-square goodness-of-fit test ## that the data in the vector X are a random sample from a normal distribution ## with mean and variance estimated from @var{x}. The result is @var{h} = 0 if ## the null hypothesis (that @var{x} is a random sample from a normal ## distribution) cannot be rejected at the 5% significance level, or @var{h} = 1 ## if the nullhypothesis can be rejected at the 5% level. @code{chi2gof} uses ## by default 10 bins (@qcode{"nbins"}), and compares the test statistic to a ## chi-square distribution with @qcode{@var{nbins} - 3} degrees of freedom, to ## take into account that two parameters were estimated. ## ## @code{[@var{h}, @var{p}] = chi2gof (@var{x})} also returns the p-value @var{p}, ## which is the probability of observing the given result, or one more extreme, ## by chance if the null hypothesis is true. If there are not enough degrees of ## freedom to carry out the test, @var{p} is NaN. ## ## @code{[@var{h}, @var{p}, @var{stats}] = chi2gof (@var{x})} also returns a ## @var{stats} structure with the following fields: ## ## @multitable @columnfractions 0.05 0.3 0.65 ## @item @tab "chi2stat" @tab Chi-square statistic ## @item @tab "df" @tab Degrees of freedom ## @item @tab "binedges" @tab Vector of bin binedges after pooling ## @item @tab "O" @tab Observed count in each bin ## @item @tab "E" @tab Expected count in each bin ## @end multitable ## ## @code{[@dots{}] = chi2gof (@var{x}, @var{Name}, @var{Value}, @dots{})} ## specifies optional Name/Value pair arguments chosen from the following list. ## ## @multitable @columnfractions 0.05 0.2 0.75 ## @headitem @tab Name @tab Value ## @item @tab @qcode{"nbins"} @tab The number of bins to use. Default is 10. ## @item @tab @qcode{"binctrs"} @tab A vector of bin centers. ## @item @tab @qcode{"binedges"} @tab A vector of bin binedges. ## @item @tab @qcode{"cdf"} @tab A fully specified cumulative distribution ## function or a function handle provided in a cell array whose first element is ## a function handle, and all later elements are its parameter values. The ## function must take @var{x} values as its first argument, and other parameters ## as later arguments. ## @item @tab @qcode{"expected"} @tab A vector with one element per bin ## specifying the expected counts for each bin. ## @item @tab @qcode{"nparams"} @tab The number of estimated parameters; used to ## adjust the degrees of freedom to be @qcode{@var{nbins} - 1 - @var{nparams}}, ## where @var{nbins} is the number of bins. ## @item @tab @qcode{"emin"} @tab The minimum allowed expected value for a bin; ## any bin in either tail having an expected value less than this amount is ## pooled with a neighboring bin. Use the value 0 to prevent pooling. Default ## is 5. ## @item @tab @qcode{"frequency"} @tab A vector of the same length as @var{x} ## containing the frequency of the corresponding @var{x} values. ## @item @tab @qcode{"alpha"} @tab An @var{alpha} value such that the hypothesis ## is rejected if @qcode{@var{p} < @var{alpha}}. Default is ## @qcode{@var{alpha} = 0.05}. ## @end multitable ## ## You should specify either @qcode{"cdf"} or @qcode{"expected"} parameters, but ## not both. If your @qcode{"cdf"} input contains extra parameters, these are ## accounted for automatically and there is no need to specify @qcode{"nparams"}. ## If your @qcode{"expected"} input depends on estimated parameters, you should ## use the @qcode{"nparams"} parameter to ensure that the degrees of freedom for ## the test is correct. ## ## @end deftypefn function [h, p, stats] = chi2gof (x, varargin) ## Check imput arguments if (nargin < 1) error ("chi2gof: At least one imput argument is required."); endif if (! isvector(x) || ! isreal(x)) error ("chi2gof: X must ba a vector of real numbers."); endif ## Add initial parameters nbins = []; binctrs = []; binedges = []; cdf_spec = []; expected = []; nparams = []; emin = 5; frequency = []; alpha = 0.05; ## Parse additional arguments numarg = nargin - 1; argpos = 1; while (numarg) argname = varargin{argpos}; switch (lower (argname)) case "nbins" nbins = varargin{argpos + 1}; case "ctrs" binctrs = varargin{argpos + 1}; case "edges" binedges = varargin{argpos + 1}; case "cdf" cdf_spec = varargin{argpos + 1}; case "expected" expected = varargin{argpos + 1}; case "nparams" nparams = varargin{argpos + 1}; case "emin" emin = varargin{argpos + 1}; case "frequency" frequency = varargin{argpos + 1}; case "alpha" alpha = varargin{argpos + 1}; endswitch numarg -= 2; argpos += 2; endwhile ## Check additional arguments for errors if ((! isempty (nbins) + ! isempty (binctrs) + ! isempty (binedges)) > 1) error ("chi2gof: Inconsistent Arguments."); endif if ((! isempty (cdf_spec) + ! isempty (expected)) > 1) error ("chi2gof: Conflicted Arguments."); endif if (! isempty (frequency)) if (! isvector (frequency) || numel (frequency) != numel (x)) error ("chi2gof: X and Frequency vectors mismatch."); endif if (any (frequency < 0)) error ("chi2gof: Frequency vector contains negative numbers."); endif endif if (! isscalar (emin) || emin < 0 || emin != round (emin) || ! isreal (emin)) error("chi2gof: 'emin' must be a positive integer."); endif if (! isempty (nparams)) if (! isscalar (nparams) || nparams < 0 || nparams != round (nparams) ... || ! isreal (nparams)) error ("chi2gof: Wrong number of parameters."); endif endif if (! isscalar (alpha) || ! isreal (alpha) || alpha <= 0 || alpha >= 1) error ("chi2gof: Wrong value of alpha."); endif ## Make X a column vector x = x(:); ## Parse or create a frequeny vector if (isempty (frequency)) frequency = ones (size (x)); else frequency = frequency(:); endif ## Remove NaNs if any remove_NaNs = isnan (frequency) | isnan (x); if (any (remove_NaNs)) x(remove_NaNs) = []; frequency(remove_NaNs) = []; endif ## Check for bin numbers, centers, or edges and calculate bins accordingly if (! isempty (binctrs)) [Observed, binedges] = calculatebins (x, frequency, "ctrs", binctrs); elseif (! isempty (binedges)) [Observed, binedges] = calculatebins (x, frequency, "edges", binedges); else if (isempty (nbins)) if (isempty (expected)) nbins = 10; ## default number of bins else nbins = length (expected); ## determined by Expected vector endif endif [Observed, binedges] = calculatebins (x, frequency, "nbins", nbins); endif Observed = Observed(:); nbins = length (Observed); ## Calculate expected vector cdfargs = {}; if (! isempty (expected)) ## Provided as input argument if (! isvector (expected) || numel (expected) != nbins) error ("chi2gof: Expected counts vector is the wrong size."); endif if (any (expected < 0)) error ("chi2gof: Expected counts vector has negative values."); endif Expected = expected(:); else ## Calculate from the cdf if (isempty (cdf_spec)) ## Use estimated normal as default cdffunc = @normcdf; sumfreq = sum (frequency); mu = sum (x.*frequency)/sumfreq; sigma = sqrt (sum ((x.*frequency - mu) .^ 2) / (sumfreq-1)); cdfargs = {mu, sigma}; if (isempty (nparams)) nparams = 2; endif elseif (isa (cdf_spec, "function_handle")) ## Split function handle to get function name and optional parameters cstr = ostrsplit (func2str (cdf_spec), ","); ## Simple function handle, no parameters: e.g. @normcdf if (isempty (strfind (cstr, "@")) && numel (cstr) == 1) cdffunc = str2func (char (strcat ("@", cstr))); if (isempty (nparams)) nparams = numel (cdfargs); endif ## Complex function handle, no parameters: e.g. @(x) normcdf(x) elseif (! isempty (strfind (cstr, "@")) && numel (cstr) == 1) ## Remove white spaces cstr = char (cstr); cstr(strfind (cstr, " ")) = []; ## Remove input argument in parentheses while (length (strfind (cstr,"("))) cstr(index (cstr, "("):index (cstr, ")")) = []; endwhile cdffunc = str2func (cstr); if (isempty (nparams)) nparams = numel (cdfargs); endif elseif (! isempty (strfind (cstr, "@")) && numel (cstr) > 1) ## Evaluate function name in first cell cstr_f = char (cstr(1)); cstr_f(strfind (cstr_f, " ")) = []; cstr_f(index (cstr_f, "("):index (cstr_f, ")")) = []; cstr_f(index (cstr_f, "("):end) = []; cdffunc = str2func (cstr_f); ## Evaluate optional parameters in remaining cells cstr_idx = 2; while (cstr_idx <= numel (cstr)) cstr_p = char (cstr(cstr_idx)); cstr_p(strfind (cstr_p, " ")) = []; ## Check for numerical value if (isscalar (str2num (cstr_p))) cdfargs{cstr_idx - 1} = cstr_p; else ## Get function handle: e.g. mean cstr_p(index (cstr_p, "("):end) = []; cdfargs{cstr_idx - 1} = feval (str2func (cstr_p), x .* frequency); cstr_idx += 1; endif endwhile if (isempty (nparams)) nparams = numel (cdfargs); endif endif elseif (iscell (cdf_spec)) % Get function and args from cell array cdffunc = cdf_spec{1}; cdfargs = cdf_spec(2:end); if (isempty (nparams)) nparams = numel(cdfargs); endif endif if (! is_function_handle (cdffunc)) error ("chi2gof: Poorly specified cumulative distribution function."); else cdfname = func2str (cdffunc); endif ## Calculate only inner bins, since tail probabilitiyis included in the ## calculation of expected counts for the first and last bins interioredges = binedges(2:end-1); ## Compute the cumulative probabilities Fcdf = feval (cdffunc, interioredges, cdfargs{:}); if (! isvector(Fcdf) || numel (Fcdf) != (nbins - 1)) msg = sprintf("chi2gof: Wrong number of outputs from: %s\n", cdfname); error (msg); endif % Compute the expected values Expected = sum(Observed) * diff([0;Fcdf(:);1]); endif ## Avoid too small expected values if (any (Expected < emin)) [Expected, Observed, binedges] = poolbins (Expected, Observed, binedges, emin); nbins = length (Expected); end ## Compute test statistic cstat = sum(((Observed - Expected) .^ 2) ./ Expected); ## Calculate degrees of freedom if (isempty (nparams)) nparams = 0; endif df = nbins - 1 - nparams; if (df > 0) p = 1 - chi2cdf (cstat, df); else df = 0; p = NaN; endif h = cast (p <= alpha, "double"); ## Create 3rd output argument if necessary if (nargout > 2) stats.chi2stat = cstat; stats.df = df; stats.edges = binedges; stats.O = Observed'; stats.E = Expected'; endif endfunction function [Expected, Observed, binedges] = poolbins (Expected, ... Observed, binedges, emin) i = 1; j = length(Expected); while (i < j - 1 && (Expected(i) < emin || Expected(i + 1) < emin || ... Expected(j) < emin || Expected(j - 1) < emin)) if (Expected(i) < Expected(j)) Expected(i+1) = Expected(i+1) + Expected(i); Observed(i+1) = Observed(i+1) + Observed(i); i = i + 1; else Expected(j-1) = Expected(j-1) + Expected(j); Observed(j-1) = Observed(j-1) + Observed(j); j = j - 1; endif endwhile ## Keep only pooled bins Expected = Expected(i:j); Observed = Observed(i:j); binedges(j+1:end-1) = []; binedges(2:i) = []; endfunction function [Observed, binedges] = calculatebins (x, frequency, binspec, specval) lo = double (min (x(:))); hi = double (max (x(:))); ## Check binspec for bin count, bin centers, or bin edges. switch (binspec) case "nbins" nbins = specval; if (isempty (x)) lo = 0; hi = 1; endif if (lo == hi) lo = lo - floor (nbins / 2) - 0.5; hi = hi + ceil (nbins / 2) - 0.5; endif binwidth = (hi - lo) ./ nbins; binedges = lo + binwidth * (0:nbins); binedges(length (binedges)) = hi; case "ctrs" binctrs = specval(:)'; binwidth = diff (binctrs); binwidth = [binwidth binwidth(end)]; binedges = [binctrs(1)-binwidth(1)/2 binctrs+binwidth/2]; case "edges" binedges = specval(:)'; endswitch ## Update bins nbins = length (binedges) - 1; ## Calculate bin numbers if (isempty (x)) binnum = x; elseif (! isequal (binspec, "edges")) binedges = binedges + eps(binedges); [ignore, binnum] = histc (x, [-Inf binedges(2:end-1) Inf]); else [ignore, binnum] = histc (x, binedges); binnum(binnum == nbins + 1) = nbins; end ## Remove empty bins if (any (binnum == 0)) frequency(binnum == 0) = []; binnum(binnum == 0) = []; end ## Compute Observed vector binnum = binnum(:); Observed = accumarray ([ones(size(binnum)), binnum], frequency, [1, nbins]); endfunction %!demo %! x = normrnd (50, 5, 100, 1); %! [h, p, stats] = chi2gof (x) %! [h, p, stats] = chi2gof (x, "cdf", @(x)normcdf (x, mean(x), std(x))) %! [h, p, stats] = chi2gof (x, "cdf", {@normcdf, mean(x), std(x)}) %!demo %! x = rand (100,1 ); %! n = length (x); %! binedges = linspace (0, 1, 11); %! expectedCounts = n * diff (binedges); %! [h, p, stats] = chi2gof (x, "binedges", binedges, "expected", expectedCounts) %!demo %! bins = 0:5; %! obsCounts = [6 16 10 12 4 2]; %! n = sum(obsCounts); %! lambdaHat = sum(bins.*obsCounts) / n; %! expCounts = n * poisspdf(bins,lambdaHat); %! [h, p, stats] = chi2gof (bins, "binctrs", bins, "frequency", obsCounts, ... %! "expected", expCounts, "nparams",1) ## Test input validation %!error chi2gof () %!error chi2gof ([2,3;3,4]) %!error chi2gof ([1,2,3,4], "nbins", 3, "ctrs", [2,3,4]) %!error chi2gof ([1,2,3,4], "frequency", [2,3,2]) %!error chi2gof ([1,2,3,4], "frequency", [2,3,2,-2]) %!error chi2gof ([1,2,3,4], "frequency", [2,3,2,2], "nparams", i) %!error chi2gof ([1,2,3,4], "frequency", [2,3,2,2], "alpha", 1.3) %!error chi2gof ([1,2,3,4], "expected", [-3,2,2]) %!error chi2gof ([1,2,3,4], "expected", [3,2,2], "nbins", 5) %!error chi2gof ([1,2,3,4], "cdf", @normcdff) %!test %! x = [1 2 1 3 2 4 3 2 4 3 2 2]; %! [h, p, stats] = chi2gof (x); %! assert (h, 0); %! assert (p, NaN); %! assert (stats.chi2stat, 0.1205375022748029, 1e-14); %! assert (stats.df, 0); %! assert (stats.edges, [1, 2.5, 4], 1e-14); %! assert (stats.O, [7, 5], 1e-14); %! assert (stats.E, [6.399995519909668, 5.600004480090332], 1e-14); statistics-release-1.6.3/inst/chi2test.m000066400000000000000000000417621456127120000202210ustar00rootroot00000000000000## Copyright (C) 2022 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{pval} =} chi2test (@var{x}) ## @deftypefnx {statistics} {[@var{pval}, @var{chisq}] =} chi2test (@var{x}) ## @deftypefnx {statistics} {[@var{pval}, @var{chisq}, @var{dF}] =} chi2test (@var{x}) ## @deftypefnx {statistics} {[@var{pval}, @var{chisq}, @var{dF}, @var{E}] =} chi2test (@var{x}) ## @deftypefnx {statistics} {[@dots{}] =} chi2test (@var{x}, @var{name}, @var{value}) ## ## Perform a chi-squared test (for independence or homogeneity). ## ## For 2-way contingency tables, @code{chi2test} performs and a chi-squared test ## for independence or homogeneity, according to the sampling scheme and related ## question. Independence means that the the two variables forming the 2-way ## table are not associated, hence you cannot predict from one another. ## Homogeneity refers to the concept of similarity, hence they all come from the ## same distribution. ## ## Both tests are computationally identical and will produce the same result. ## Nevertheless, they anwser to different questions. Consider two variables, ## one for gender and another for smoking. To test independence (whether gender ## and smoking is associated), we would randomly sample from the general ## population and break them down into categories in the table. To test ## homogeneity (whether men and women share the same smoking habits), we would ## sample individuals from within each gender, and then measure their smoking ## habits (e.g. smokers vs non-smokers). ## ## When @code{chi2test} is called without any output arguments, it will print ## the result in the terminal including p-value, chi^2 statistic, and degrees of ## freedom. Otherwise it can return the following output arguments: ## ## @multitable @columnfractions 0.05 0.1 0.85 ## @item @tab @var{pval} @tab the p-value of the relevant test. ## @item @tab @var{chisq} @tab the chi^2 statistic of the relevant test. ## @item @tab @var{dF} @tab the degrees of freedom of the relevant test. ## @item @tab @var{E} @tab the EXPECTED values of the original contigency table. ## @end multitable ## ## Unlike MATLAB, in GNU Octave @code{chi2test} also supports 3-way tables, ## which involve three categorical variables (each in a different dimension of ## @var{x}. In its simplest form, @code{[@dots{}] = chi2test (@var{x})} will ## will test for mutual independence among the three variables. Alternatively, ## when called in the form @code{[@dots{}] = chi2test (@var{x}, @var{name}, ## @var{value})}, it can perform the following tests: ## ## @multitable @columnfractions 0.2 0.1 0.7 ## @headitem @var{name} @tab @var{value} @tab Description ## @item "mutual" @tab [] @tab Mutual independence. All variables are ## independent from each other, (A, B, C). Value must be an empty matrix. ## @item "joint" @tab scalar @tab Joint independence. Two variables are jointly ## independent of the third, (AB, C). The scalar value corresponds to the ## dimension of the independent variable (i.e. 3 for C). ## @item "marginal" @tab scalar @tab Marginal independence. Two variables are ## independent if you ignore the third, (A, C). The scalar value corresponds ## to the dimension of the variable to be ignored (i.e. 2 for B). ## @item "conditional" @tab scalar @tab Conditional independence. Two variables ## are independent given the third, (AC, BC). The scalar value corresponds to ## the dimension of the variable that forms the conditional dependence ## (i.e. 3 for C). ## @item "homogeneous" @tab [] @tab Homogeneous associations. Conditional ## (partial) odds-ratios are not related on the value of the third, ## (AB, AC, BC). Value must be an empty matrix. ## @end multitable ## ## When testing for homogeneous associations in 3-way tables, the iterative ## proportional fitting procedure is used. For small samples it is better to ## use the Cochran-Mantel-Haenszel Test. K-way tables for k > 3 are supported ## only for testing mutual independence. Similar to 2-way tables, no optional ## parameters are required for k > 3 multi-way tables. ## ## @code{chi2test} produces a warning if any cell of a 2x2 table has an expected ## frequency less than 5 or if more than 20% of the cells in larger 2-way tables ## have expected frequencies less than 5 or any cell with expected frequency ## less than 1. In such cases, use @code{fishertest}. ## ## @seealso{crosstab, fishertest, mcnemar_test} ## @end deftypefn function [pval, chisq, df, E] = chi2test (x, varargin) ## Check input arguments if (nargin < 1) print_usage (); endif if (isvector (x)) error ("chi2test: X must be a matrix."); endif if (! isreal (x)) error ("chi2test: values in X must be real numbers."); endif if (any (isnan (x(:)))) error ("chi2test: X must not have missing values (NaN)."); endif ## Get size and dimensions of contigency table sz = size (x); dim = length (sz); ## Check optional arguments if (dim == 2 && nargin > 1) error ("chi2test: optional arguments are not supported for 2-way tables."); endif if (dim == 3 && mod (numel (varargin(:)), 2) != 0) error ("chi2test: optional arguments must be in pairs."); endif if (dim == 3 && nargin > 1 && ! isnumeric (varargin{2})) error (strcat (["chi2test: value must be numeric in optional argument"], ... [" name/value pair, for 3-way tables."])); endif if (dim == 3 && nargin > 1 && numel (varargin{2}) > 1) error (strcat (["chi2test: value must be empty or scalar in optional"], ... [" argument name/value pair, for 3-way tables."])); endif if (dim >= 4 && nargin > 1) error ("chi2test: optional arguments are not supported for k>3."); endif ## Calculate total sample size n = sum (x(:)); ## For 2-way contigency table if (length (sz) == 2) ## Calculate degrees of freedom df = prod (sz - 1); ## Calculate expected values E = sum (x')' * sum (x) / n; ## For 3-way contigency table elseif (length (sz) == 3) ## Check optional arguments if (nargin == 1 || strcmpi (varargin{1}, "mutual")) ## Calculate degrees of freedom df = prod (sz) - sum (sz) + 2; ## Calculate marginal table sums q1 = sum (sum (x, 2), 3); q2 = sum (sum (x, 1), 3); q3 = sum (sum (x, 1), 2); ns = sum (x(:)) ^ 2; for d1 = 1:size (x, 1) for d2 = 1:size (x, 2) for d3 = 1:size (x, 3) E(d1,d2,d3) = q1(d1,:,:) * q2(:,d2,:) * q3(:,:,d3) / ns; endfor endfor endfor elseif (strcmpi (varargin{1}, "joint")) ## Get dimension of independent variable (dim) c_dim = varargin{2}; ## Calculate degrees of freedom c_sz = sz; c_sz(c_dim) = []; df = (sz(c_dim) - 1) * (prod (c_sz) - 1); ## Rearrange dimensions so that independent variable goes in dim 1 dm = [1, 2, 3]; dm(c_dim) = []; x = permute (x, [c_dim, dm]); ## Calculate partial table sums q1 = sum (sum (x, 1), 1); q2 = sum (sum (x, 2), 3); n = sum (x(:)); for d1 = 1:size (x, 1) for d2 = 1:size (x, 2) for d3 = 1:size (x, 3) E(d1,d2,d3) = q1(:,d2,d3) * q2(d1) / n; endfor endfor endfor ## Rearrange OBSERVED and EXPECTED matrices in original dimensions x = permute (x, [c_dim, dm]); x = permute (x, [c_dim, dm]); E = permute (E, [c_dim, dm]); E = permute (E, [c_dim, dm]); elseif (strcmpi (varargin{1}, "marginal")) ## Get dimension of marginal variable (dim) c_dim = varargin{2}; ## Calculate degrees of freedom c_sz = sz; c_sz(c_dim) = []; df = prod (sz) - sum (c_sz) + 1; ## Rearrange dimensions so that marginal variable goes in dim 1 dm = [1, 2, 3]; dm(c_dim) = []; x = permute (x, [c_dim, dm]); ## Calculate partial table sums q1 = sum (sum (x, 1), 3); q2 = sum (sum (x, 1), 2); n2 = sz(c_dim) * sum (x(:)); ## Calculate expected values for d1 = 1:size (x, 1) for d2 = 1:size (x, 2) for d3 = 1:size (x, 3) E(d1,d2,d3) = q1(:,d2) * q2(:,:,d3) / n2; endfor endfor endfor ## Rearrange OBSERVED and EXPECTED matrices in original dimensions x = permute (x, [c_dim, dm]); E = permute (E, [c_dim, dm]); elseif (strcmpi (varargin{1}, "conditional")) ## Get dimension of conditional variable (dim) c_dim = varargin{2}; ## Calculate degrees of freedom c_sz = sz; c_sz(c_dim) = []; df = prod (c_sz - 1) * sz(c_dim); ## Rearrange dimensions so that conditional variable goes in dim 1 dm = [1, 2, 3]; dm(c_dim) = []; x = permute (x, [c_dim, dm]); ## Calculate partial table sums q1 = sum (sum (x, 3), 3); q2 = sum (sum (x, 2), 2); q3 = sum (sum (x, 2), 3); ## Calculate expected values for d1 = 1:size (x, 1) for d2 = 1:size (x, 2) for d3 = 1:size (x, 3) E(d1,d2,d3) = q1(d1,d2) * q2(d1,:,d3) / q3(d1); endfor endfor endfor ## Rearrange OBSERVED and EXPECTED matrices in original dimensions x = permute (x, [c_dim, dm]); x = permute (x, [c_dim, dm]); E = permute (E, [c_dim, dm]); E = permute (E, [c_dim, dm]); elseif (strcmpi (varargin{1}, "homogeneous")) ## Calculate degrees of freedom df = prod(sz - 1); ## Compute observed marginal totals for any two dimensions omt12 = sum (sum (x, 3), 3); omt13 = sum (sum (x, 2), 2); omt23 = sum (sum (x, 1), 1); ## Produce initial seed 3-way table S = ones (sz); ## Calculate initial expected marginal totals emt12 = sum (sum (S, 3), 3); emt13 = sum (sum (S, 2), 2); emt23 = sum (sum (S, 1), 1); ## Compute difference to converge within certain tolerance or iterations OEdiff = sum (omt12(:) - emt12(:)) + sum (omt13(:) - emt13(:)) + ... sum (omt23(:) - emt23(:)); iter = 1; tol = 1e-6; ## Start Iterative Proportional Fitting Procedure while (OEdiff > tol || iter > 50) ## Rows x Columns for d1 = 1:size (x, 1) for d2 = 1:size (x, 2) for d3 = 1:size (x, 3) E(d1,d2,d3) = S(d1,d2,d3) * omt12(d1,d2) / emt12(d1,d2); endfor endfor endfor ## Update seed and recalculate Rows x Layers expected marginal totals S = E; emt13 = sum (sum (S, 2), 2); ## Rows x Layers for d1 = 1:size (x, 1) for d2 = 1:size (x, 2) for d3 = 1:size (x, 3) E(d1,d2,d3) = S(d1,d2,d3) * omt13(d1,:,d3) / emt13(d1,:,d3); endfor endfor endfor ## Update seed and recalculate Columns x Layers expected marginal totals S = E; emt23 = sum (sum (S, 1), 1); ## Columns x Layers for d1 = 1:size (x, 1) for d2 = 1:size (x, 2) for d3 = 1:size (x, 3) E(d1,d2,d3) = S(d1,d2,d3) * omt23(:,d2,d3) / emt23(:,d2,d3); endfor endfor endfor ## Update seed and recalculate Rows x Layers expected marginal totals S = E; emt12 = sum (sum (S, 3), 3); ## Update difference between OBSERVED and EXPECTED tables OEdiff = sum (omt12(:) - emt12(:)) + sum (omt13(:) - emt13(:)) + ... sum (omt23(:) - emt23(:)); iter += 1; endwhile else error ("chi2test: invalid model name for testing a 3-way table."); endif ## For k-way contigency table, where k > 3 else ## Calculate degrees of freedom df = prod (sz) - sum (sz) + 2; ## Calculate squared sample size ns = sum (x(:)) ^ (dim - 1); ## Calculate marginal table sums for each available dimension for i = 1:dim qi(i) = {x}; remdim = [1:dim]; remdim(remdim == i) = []; for j = 1:length (remdim) qi(i) = sum (qi{i}, remdim(j)); endfor qi(i) = squeeze (qi{i}); endfor ## Iterate through all cells cn = numel (x); for i = 1:cn E(i) = 1; cid = i; ## Keep track of indexing for d = dim - 1:-1:1 idx(d+1) = ceil (cid / prod (sz(1:d))); if (idx(d+1) > 1) cid -= (idx(d+1) - 1) * prod (sz(1:d)); endif endfor idx(1) = cid; ## Calculate the expected value for j = 1:dim E(i) = E(i) * qi{j}(idx(j)); endfor E(i) = E(i) / ns; endfor ## Reshape to original dimensions E = reshape (E, sz); endif ## Check expected values and display warnings if ((dim == 2 && isequal (sz, [2, 2]) && any (E(:) < 5)) || ... (dim == 2 && any (sz > 2) && sum (E(:) < 5) > 0.2 * numel (E)) || ... (dim > 2 && sum (E(:) < 5) > 0.2 * numel (E))) warning ("chi2test: Expected values less than 5."); endif if (any (E(:) < 1)) warning ("chi2test: Expected values less than 1."); endif ## Calculate chi-squared and p-value cells = ((x - E) .^2) ./ E; chisq = sum (cells(:)); pval = 1 - chi2cdf (chisq, df); ## Print results if no output requested if (nargout == 0) printf ("p-val = %f with chi^2 statistic = %f and d.f. = %d.\n", ... pval, chisq, df); endif endfunction ## Input validation tests %!error chi2test (); %!error chi2test ([1, 2, 3, 4, 5]); %!error chi2test ([1, 2; 2, 1+3i]); %!error chi2test ([NaN, 6; 34, 12]); %!error ... %! p = chi2test (ones (3, 3), "mutual", []); %!error ... %! p = chi2test (ones (3, 3, 3), "testtype", 2); %!error ... %! p = chi2test (ones (3, 3, 3), "mutual"); %!error ... %! p = chi2test (ones (3, 3, 3), "joint", ["a"]); %!error ... %! p = chi2test (ones (3, 3, 3), "joint", [2, 3]); %!error ... %! p = chi2test (ones (3, 3, 3, 4), "mutual", []) ## Check warning %!warning p = chi2test (ones (2)); %!warning p = chi2test (ones (3, 2)); %!warning p = chi2test (0.4 * ones (3)); ## Output validation tests %!test %! x = [11, 3, 8; 2, 9, 14; 12, 13, 28]; %! p = chi2test (x); %! assert (p, 0.017787, 1e-6); %!test %! x = [11, 3, 8; 2, 9, 14; 12, 13, 28]; %! [p, chisq] = chi2test (x); %! assert (chisq, 11.9421, 1e-4); %!test %! x = [11, 3, 8; 2, 9, 14; 12, 13, 28]; %! [p, chisq, df] = chi2test (x); %! assert (df, 4); %!test %!shared x %! x(:,:,1) = [59, 32; 9,16]; %! x(:,:,2) = [55, 24;12,33]; %! x(:,:,3) = [107,80;17,56];%! %!assert (chi2test (x), 2.282063427117009e-11, 1e-14); %!assert (chi2test (x, "mutual", []), 2.282063427117009e-11, 1e-14); %!assert (chi2test (x, "joint", 1), 1.164834895206468e-11, 1e-14); %!assert (chi2test (x, "joint", 2), 7.771350230001417e-11, 1e-14); %!assert (chi2test (x, "joint", 3), 0.07151361728026107, 1e-14); %!assert (chi2test (x, "marginal", 1), 0, 1e-14); %!assert (chi2test (x, "marginal", 2), 6.347555814301131e-11, 1e-14); %!assert (chi2test (x, "marginal", 3), 0, 1e-14); %!assert (chi2test (x, "conditional", 1), 0.2303114201312508, 1e-14); %!assert (chi2test (x, "conditional", 2), 0.0958810684407079, 1e-14); %!assert (chi2test (x, "conditional", 3), 2.648037344954446e-11, 1e-14); %!assert (chi2test (x, "homogeneous", []), 0.4485579470993741, 1e-14); %!test %! [pval, chisq, df, E] = chi2test (x); %! assert (chisq, 64.0982, 1e-4); %! assert (df, 7); %! assert (E(:,:,1), [42.903, 39.921; 17.185, 15.991], ones (2, 2) * 1e-3); %!test %! [pval, chisq, df, E] = chi2test (x, "joint", 2); %! assert (chisq, 56.0943, 1e-4); %! assert (df, 5); %! assert (E(:,:,2), [40.922, 23.310; 38.078, 21.690], ones (2, 2) * 1e-3); %!test %! [pval, chisq, df, E] = chi2test (x, "marginal", 3); %! assert (chisq, 146.6058, 1e-4); %! assert (df, 9); %! assert (E(:,1,1), [61.642; 57.358], ones (2, 1) * 1e-3); %!test %! [pval, chisq, df, E] = chi2test (x, "conditional", 3); %! assert (chisq, 52.2509, 1e-4); %! assert (df, 3); %! assert (E(:,:,1), [53.345, 37.655; 14.655, 10.345], ones (2, 2) * 1e-3); %!test %! [pval, chisq, df, E] = chi2test (x, "homogeneous", []); %! assert (chisq, 1.6034, 1e-4); %! assert (df, 2); %! assert (E(:,:,1), [60.827, 31.382; 7.173, 16.618], ones (2, 2) * 1e-3); statistics-release-1.6.3/inst/cholcov.m000066400000000000000000000101661456127120000201230ustar00rootroot00000000000000## Copyright (C) 2022 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{T} =} cholcov (@var{sigma}) ## @deftypefnx {statistics} {[@var{T}, @var{p} =} cholcov (@var{sigma}) ## @deftypefnx {statistics} {[@dots{}] =} cholcov (@var{sigma}, @var{flag}) ## ## Cholesky-like decomposition for covariance matrix. ## ## @code{@var{T} = cholcov (@var{sigma})} computes matrix @var{T} such that ## @var{sigma} = @var{T}' @var{T}. @var{sigma} must be square, symmetric, and ## positive semi-definite. ## ## If @var{sigma} is positive definite, then @var{T} is the square, upper ## triangular Cholesky factor. If @var{sigma} is not positive definite, @var{T} ## is computed with an eigenvalue decomposition of @var{sigma}, but in this case ## @var{T} is not necessarily triangular or square. Any eigenvectors whose ## corresponding eigenvalue is close to zero (within a tolerance) are omitted. ## If any remaining eigenvalues are negative, @var{T} is empty. ## ## The tolerance is calculated as @code{10 * eps (max (abs (diag (sigma))))}. ## ## @code{[@var{T}, @var{p} = cholcov (@var{sigma})} returns in @var{p} the ## number of negative eigenvalues of @var{sigma}. If @var{p} > 0, then @var{T} ## is empty, whereas if @var{p} = 0, @var{sigma}) is positive semi-definite. ## ## If @var{sigma} is not square and symmetric, P is NaN and T is empty. ## ## @code{[@var{T}, @var{p} = cholcov (@var{sigma}, 0)} returns @var{p} = 0 if ## @var{sigma} is positive definite, in which case @var{T} is the Cholesky ## factor. If @var{sigma} is not positive definite, @var{p} is a positive ## integer and @var{T} is empty. ## ## @code{[@dots{}] = cholcov (@var{sigma}, 1)} is equivalent to ## @code{ [@dots{}] = cholcov (@var{sigma})}. ## ## @seealso{chov} ## @end deftypefn function [T, p] = cholcov (sigma, flag) ## Check number of input arguments narginchk (1,2) ## Add default flag if not givens if (nargin < 2) flag = 1; endif ## Check if sigma is a sparse matrix is_sparse = issparse (sigma); ## Check if sigma is single or double class is_type = "double"; if (isa (sigma, "single")) is_type = "single"; endif ## Test for sigma being square and symmetric [col, row] = size (sigma); ## Add tolerance Tol = 10 * eps (max (abs (diag (sigma)))); if ((row == col) && all (all (abs (sigma - sigma') < col * Tol))) ## Check if positive definite [T, p] = chol (sigma); if (p > 0) ## Check flag for factoring using eigenvalue decomposition if (flag) [V, LAMBDA] = eig (full ((sigma + sigma') / 2)); [~, EIGMAX] = max (abs (V), [], 1); neg_idx = (V(EIGMAX + (0:row:(col-1)*row)) < 0); V(:,neg_idx) = -V(:,neg_idx); LAMBDA = diag(LAMBDA); Tol = eps (max (LAMBDA)) * length (LAMBDA); t = (abs (LAMBDA) > Tol); LAMBDA = LAMBDA(t); p = sum (LAMBDA < 0); ## Check for negative eigenvalues if (p == 0) T = diag (sqrt (LAMBDA)) * V(:,t)'; else T = zeros (0, is_type); endif else T = zeros (0, is_type); endif endif else T = zeros (0, is_type); p = NaN (is_type); endif if (is_sparse) T = sparse(T); endif endfunction %!demo %! C1 = [2, 1, 1, 2; 1, 2, 1, 2; 1, 1, 2, 2; 2, 2, 2, 3] %! T = cholcov (C1) %! C2 = T'*T %!test %! C1 = [2, 1, 1, 2; 1, 2, 1, 2; 1, 1, 2, 2; 2, 2, 2, 3]; %! T = cholcov (C1); %! assert (C1, T'*T, 1e-15 * ones (size (C1))); statistics-release-1.6.3/inst/cl_multinom.m000066400000000000000000000131771456127120000210150ustar00rootroot00000000000000## Copyright (C) 2009 Levente Torok ## Copyright (C) 2023 Andreas Bertsatos ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{CL} =} cl_multinom (@var{X}, @var{N}, @var{b}) ## @deftypefnx {statistics} {@var{CL} =} cl_multinom (@var{X}, @var{N}, @var{b}, @var{method}) ## ## Confidence level of multinomial portions. ## ## @code{cl_multinom} returns confidence level of multinomial parameters ## estimated as @math{p = X / sum(X)} with predefined confidence interval ## @var{b}. Finite population is also considered. ## ## This function calculates the level of confidence at which the samples ## represent the true distribution given that there is a predefined tolerance ## (confidence interval). This is the upside down case of the typical excercises ## at which we want to get the confidence interval given the confidence level ## (and the estimated parameters of the underlying distribution). ## But once we accept (lets say at elections) that we have a standard predefined ## maximal acceptable error rate (e.g. @var{b}=0.02 ) in the estimation and we ## just want to know that how sure we can be that the measured proportions are ## the same as in the entire population (ie. the expected value and mean of the ## samples are roughly the same) we need to use this function. ## ## @subheading Arguments ## @multitable @columnfractions 0.1 0.01 0.10 0.01 0.78 ## @headitem Variable @tab @tab Type @tab @tab Description ## @item @var{X} @tab @tab int vector @tab @tab sample frequencies bins. ## @item @var{N} @tab @tab int scalar @tab @tab Population size that was sampled ## by @var{X}. If @qcode{N < sum (@var{X})}, infinite number assumed. ## @item @var{b} @tab @tab real vector @tab @tab confidence interval. If vector, ## it should be the size of @var{X} containing confence interval for each cells. ## If scalar, each cell will have the same value of b unless it is zero or -1. ## If value is 0, @var{b} = 0.02 is assumed which is standard choice at ## elections otherwise it is calculated in a way that one sample in a cell ## alteration defines the confidence interval. ## @item @var{method} @tab @tab string @tab @tab An optional argument ## for defining the calculation method. Available choices are ## @qcode{"bromaghin"} (default), @qcode{"cochran"}, and @qcode{agresti_cull}. ## @end multitable ## ## Note! The @qcode{agresti_cull} method is not exactly the solution at ## reference given below but an adjustment of the solutions above. ## ## @subheading Returns ## Confidence level. ## ## @subheading Example ## CL = cl_multinom ([27; 43; 19; 11], 10000, 0.05) ## returns 0.69 confidence level. ## ## @subheading References ## @enumerate ## @item ## "bromaghin" calculation type (default) is based on the article: ## ## Jeffrey F. Bromaghin, "Sample Size Determination for Interval Estimation ## of Multinomial Probabilities", The American Statistician vol 47, 1993, ## pp 203-206. ## ## @item ## "cochran" calculation type is based on article: ## ## Robert T. Tortora, "A Note on Sample Size Estimation for Multinomial ## Populations", The American Statistician, , Vol 32. 1978, pp 100-102. ## ## @item ## "agresti_cull" calculation type is based on article: ## ## A. Agresti and B.A. Coull, "Approximate is better than 'exact' for ## interval estimation of binomial portions", The American Statistician, ## Vol. 52, 1998, pp 119-126 ## @end enumerate ## ## @end deftypefn function CL = cl_multinom (X, N, b = 0.05, method = "bromaghin") if (nargin < 2 || nargin > 4) print_usage; elseif (! ischar (method)) error ("cl_multinom: argument method must be a string."); endif k = rows (X); nn = sum (X); p = X / nn; if (isscalar (b)) if (b==0) b=0.02; endif b = ones (rows (X), 1 ) * b; if (b<0) b = 1 ./ max (X, 1); endif endif bb = b .* b; if (N == nn) CL = 1; return; endif if (N < nn) fpc = 1; else fpc = (N - 1) / (N - nn); # finite population correction tag endif beta = p .* (1 - p); switch lower (method) case "cochran" t = sqrt (fpc * nn * bb ./ beta); alpha = (1 - normcdf (t)) * 2; case "bromaghin" t = sqrt (fpc * (nn * 2 * bb ) ./ ... (beta - 2 * bb + sqrt (beta .* beta - bb .* (4 * beta - 1)))); alpha = (1 - normcdf (t)) * 2; case "agresti_cull" ts = fpc * nn * bb ./ beta ; if (k <= 2) alpha = 1 - chi2cdf (ts, k - 1); # adjusted Wilson interval else alpha = 1 - chi2cdf (ts / k, 1); # Goodman interval with Bonferroni arg. endif otherwise error ("cl_multinom: unknown calculation type '%s'.", method); endswitch CL = 1 - max( alpha ); endfunction %!demo %! CL = cl_multinom ([27; 43; 19; 11], 10000, 0.05) ## Test input validation %!error cl_multinom (); %!error cl_multinom (1, 2, 3, 4, 5); %!error ... %! cl_multinom (1, 2, 3, 4); %!error ... %! cl_multinom (1, 2, 3, "some string"); statistics-release-1.6.3/inst/cluster.m000066400000000000000000000147631456127120000201560ustar00rootroot00000000000000## Copyright (C) 2021 Stefano Guidoni ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{T} =} cluster (@var{Z}, "Cutoff", @var{C}) ## @deftypefnx {statistics} {@var{T} =} cluster (@var{Z}, "Cutoff", @var{C}, "Depth", @var{D}) ## @deftypefnx {statistics} {@var{T} =} cluster (@var{Z}, "Cutoff", @var{C}, "Criterion", @var{criterion}) ## @deftypefnx {statistics} {@var{T} =} cluster (@var{Z}, "MaxClust", @var{N}) ## ## Define clusters from an agglomerative hierarchical cluster tree. ## ## Given a hierarchical cluster tree @var{Z} generated by the @code{linkage} ## function, @code{cluster} defines clusters, using a threshold value @var{C} to ## identify new clusters ('Cutoff') or according to a maximum number of desired ## clusters @var{N} ('MaxClust'). ## ## @var{criterion} is used to choose the criterion for defining clusters, which ## can be either "inconsistent" (default) or "distance". When using ## "inconsistent", @code{cluster} compares the threshold value @var{C} to the ## inconsistency coefficient of each link; when using "distance", @code{cluster} ## compares the threshold value @var{C} to the height of each link. ## @var{D} is the depth used to evaluate the inconsistency coefficient, its ## default value is 2. ## ## @code{cluster} uses "distance" as a criterion for defining new clusters when ## it is used the 'MaxClust' method. ## ## @seealso{clusterdata, dendrogram, inconsistent, kmeans, linkage, pdist} ## @end deftypefn function T = cluster (Z, opt, varargin) switch (lower (opt)) ## check the input case "cutoff" if (nargin < 3) print_usage (); else C = varargin{1}; D = 2; criterion = "inconsistent"; if (nargin > 3) pair_index = 2; while (pair_index < (nargin - 2)) switch (lower (varargin{pair_index})) case "depth" D = varargin{pair_index + 1}; case "criterion" criterion = varargin{pair_index + 1}; otherwise error ("cluster: unknown property %s", varargin{pair_index}); endswitch pair_index += 2; endwhile endif endif if ((! (isscalar (C) || isvector (C))) || (C < 0)) error ... (["cluster: C must be a positive scalar or a vector of positive"... "numbers"]); endif case "maxclust" if (nargin != 3) print_usage (); else N = varargin{1}; C = []; endif if ((! (isscalar (N) || isvector (N))) || (N < 0)) error ... (["cluster: N must be a positive number or a vector of positive"... "numbers"]); endif otherwise error ("cluster: unknown option %s", opt); endswitch if ((columns (Z) != 3) || (! isnumeric (Z)) ... (! (max (Z(end, 1:2)) == rows (Z) * 2))) error ("cluster: Z must be a matrix generated by the linkage function"); endif ## number of observations n = rows (Z) + 1; ## vector of values used by the threshold check vThresholds = []; ## starting number of clusters nClusters = 1; ## the return value is the matrix T, constituted by one or more vector vT T = []; vT = zeros (1, n); ## main logic ## a few checks and computations before launching the recursive function switch (lower (opt)) case "cutoff" switch (lower (criterion)) case "inconsistent" vThresholds = inconsistent (Z, D)(:, 4); case "distance" vThresholds = Z(:, 3); otherwise error ("cluster: unkown criterion %s", criterion); endswitch case "maxclust" ## the MaxClust case can be regarded as a Cutoff case with distance ## criterion, where the threshold is set to the height of the highest node ## that allows us to have N different clusters vThresholds = Z(:, 3); ## let's build a vector with the apt threshold values for k = 1:length (N); if (N(k) > n) C(end+1) = 0; elseif (N(k) < 2) C(end+1) = Z(end, 3) + 1; else C(end+1) = Z((end + 2 - N(k)), 3); endif endfor endswitch for c_index = 1:length (C) cluster_cutoff_recursive (rows (Z), nClusters, c_index); T = [T; vT]; endfor T = T'; # return value ## recursive function ## for each link check if the cutoff criteria (a threshold value) are met, ## then call recursively this function for every node below that; ## when we find a leaf, we add the index of its cluster to the return value function cluster_cutoff_recursive (index, cluster_number, c_index) vClusterNumber = [cluster_number, cluster_number]; ## check the threshold value if (vThresholds(index) >= C(c_index)) ## create a new cluster nClusters++; vClusterNumber(2) = nClusters; endif; ## go on, down the tree for j = 1:2 if (Z(index,j) > n) new_index = Z(index,j) - n; cluster_cutoff_recursive (new_index, vClusterNumber(j), c_index); else ## if the next node is a leaf, add the index of its cluster to the ## result at the correct position, i.e. the leaf number; ## if leaf 14 belongs to cluster 3: ## vT(14) = 3; vT(Z(index,j)) = vClusterNumber(j); endif endfor endfunction endfunction ## Test input validation %!error cluster () %!error cluster ([1 1], "Cutoff", 1) %!error cluster ([1 2 1], "Bogus", 1) %!error cluster ([1 2 1], "Cutoff", -1) %!error cluster ([1 2 1], "Cutoff", 1, "Bogus", 1) ## Test output %!test % X = [(randn (10, 2) * 0.25) + 1; (randn (10, 2) * 0.25) - 1]; % Z = linkage(X, "ward"); % T = [ones (10, 1); 2 * ones (10, 1)]; % assert (cluster (Z, "MaxClust", 2), T); statistics-release-1.6.3/inst/clusterdata.m000066400000000000000000000105201456127120000207730ustar00rootroot00000000000000## Copyright (C) 2021 Stefano Guidoni ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{T} =} clusterdata (@var{X}, @var{cutoff}) ## @deftypefnx {statistics} {@var{T} =} clusterdata (@var{X}, @var{Name}, @var{Value}) ## ## Wrapper function for @code{linkage} and @code{cluster}. ## ## If @var{cutoff} is used, then @code{clusterdata} calls @code{linkage} and ## @code{cluster} with default value, using @var{cutoff} as a threshold value ## for @code{cluster}. If @var{cutoff} is an integer and greater or equal to 2, ## then @var{cutoff} is interpreted as the maximum number of cluster desired ## and the "MaxClust" option is used for @code{cluster}. ## ## If @var{cutoff} is not used, then @code{clusterdata} expects a list of pair ## arguments. Then you must specify either the "Cutoff" or "MaxClust" option ## for @code{cluster}. The method and metric used by @code{linkage}, are ## defined through the "linkage" and "distance" arguments. ## ## @seealso{cluster, dendrogram, inconsistent, kmeans, linkage, pdist} ## @end deftypefn function T = clusterdata (X, varargin) ## Check for valid number of input arguments if (nargin < 2) error ("clusterdata: function called with too few input arguments."); endif linkage_criterion = "single"; distance_method = "euclidean"; savememory = "off"; clustering_method = []; criterion = "inconsistent"; D = 2; if (isnumeric (varargin{1})) # clusterdata (X, cutoff) if (isinteger (varargin{1}) && (varargin{1} >= 2)) clustering_method = "MaxClust"; else clustering_method = "Cutoff"; endif C = varargin{1}; else # clusterdata (Name, Value) pair_index = 1; while (pair_index < (nargin - 1)) switch (lower (varargin{pair_index})) case "criterion" criterion = varargin{pair_index + 1}; case "cutoff" clustering_method = "Cutoff"; C = varargin{pair_index + 1}; case "depth" D = varargin{pair_index + 1}; case "distance" distance_method = varargin{pair_index + 1}; case "linkage" linkage_criterion = varargin{pair_index + 1}; case "maxclust" clustering_method = "MaxClust"; C = varargin{pair_index + 1}; case "savememory" savememory = varargin{pair_index + 1}; otherwise error ("clusterdata: unknown property %s", varargin{pair_index}); endswitch pair_index += 2; endwhile endif if (isempty (clustering_method)) error (strcat (["clusterdata: you must specify either 'MaxClust'"], ... [" or 'Cutoff' when using name-value arguments."])); endif ## main body Z = linkage (X, linkage_criterion, distance_method, "savememory"); if (strcmp (lower (clustering_method), "cutoff")) T = cluster (Z, clustering_method, C, "Criterion", criterion, "Depth", D); else T = cluster (Z, clustering_method, C); endif endfunction %!demo %! randn ("seed", 1) # for reproducibility %! r1 = randn (10, 2) * 0.25 + 1; %! randn ("seed", 5) # for reproducibility %! r2 = randn (20, 2) * 0.5 - 1; %! X = [r1; r2]; %! %! wnl = warning ("off", "Octave:linkage_savemem", "local"); %! T = clusterdata (X, "linkage", "ward", "MaxClust", 2); %! scatter (X(:,1), X(:,2), 36, T, "filled"); ## Test input validation %!error ... %! clusterdata () %!error ... %! clusterdata (1) %!error clusterdata ([1 1], "Bogus", 1) %!error clusterdata ([1 1], "Depth", 1) statistics-release-1.6.3/inst/cmdscale.m000066400000000000000000000132121456127120000202340ustar00rootroot00000000000000## Copyright (C) 2014 JD Walsh ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation; either version 3 of the License, or ## (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{Y} =} cmdscale (@var{D}) ## @deftypefnx {statistics} {[@var{Y}, @var{e}] =} cmdscale (@var{D}) ## ## Classical multidimensional scaling of a matrix. ## ## Takes an @var{n} by @var{n} distance (or difference, similarity, or ## dissimilarity) matrix @var{D}. Returns @var{Y}, a matrix of @var{n} points ## with coordinates in @var{p} dimensional space which approximate those ## distances (or differences, similarities, or dissimilarities). Also returns ## the eigenvalues @var{e} of ## @code{@var{B} = -1/2 * @var{J} * (@var{D}.^2) * @var{J}}, where ## @code{J = eye(@var{n}) - ones(@var{n},@var{n})/@var{n}}. @var{p}, the number ## of columns of @var{Y}, is equal to the number of positive real eigenvalues of ## @var{B}. ## ## @var{D} can be a full or sparse matrix or a vector of length ## @code{@var{n}*(@var{n}-1)/2} containing the upper triangular elements (like ## the output of the @code{pdist} function). It must be symmetric with ## non-negative entries whose values are further restricted by the type of ## matrix being represented: ## ## * If @var{D} is either a distance, dissimilarity, or difference matrix, then ## it must have zero entries along the main diagonal. In this case the points ## @var{Y} equal or approximate the distances given by @var{D}. ## ## * If @var{D} is a similarity matrix, the elements must all be less than or ## equal to one, with ones along the the main diagonal. In this case the points ## @var{Y} equal or approximate the distances given by ## @code{@var{D} = sqrt(ones(@var{n},@var{n})-@var{D})}. ## ## @var{D} is a Euclidean matrix if and only if @var{B} is positive ## semi-definite. When this is the case, then @var{Y} is an exact representation ## of the distances given in @var{D}. If @var{D} is non-Euclidean, @var{Y} only ## approximates the distance given in @var{D}. The approximation used by ## @code{cmdscale} minimizes the statistical loss function known as ## @var{strain}. ## ## The returned @var{Y} is an @var{n} by @var{p} matrix showing possible ## coordinates of the points in @var{p} dimensional space ## (@code{@var{p} < @var{n}}). The columns are correspond to the positive ## eigenvalues of @var{B} in descending order. A translation, rotation, or ## reflection of the coordinates given by @var{Y} will satisfy the same distance ## matrix up to the limits of machine precision. ## ## For any @code{@var{k} <= @var{p}}, if the largest @var{k} positive ## eigenvalues of @var{B} are significantly greater in absolute magnitude than ## its other eigenvalues, the first @var{k} columns of @var{Y} provide a ## @var{k}-dimensional reduction of @var{Y} which approximates the distances ## given by @var{D}. The optional return @var{e} can be used to consider various ## values of @var{k}, or to evaluate the accuracy of specific dimension ## reductions (e.g., @code{@var{k} = 2}). ## ## Reference: Ingwer Borg and Patrick J.F. Groenen (2005), Modern ## Multidimensional Scaling, Second Edition, Springer, ISBN: 978-0-387-25150-9 ## (Print) 978-0-387-28981-6 (Online) ## ## @seealso{pdist} ## @end deftypefn function [Y, e] = cmdscale (D) % Check for matrix input if ((nargin ~= 1) || ... (~any(strcmp ({'matrix' 'scalar' 'range'}, typeinfo(D))))) usage ('cmdscale: input must be vector or matrix; see help'); endif % If vector, convert to matrix; otherwise, check for square symmetric input if (isvector (D)) D = squareform (D); elseif ((~issquare (D)) || (norm (D - D', 1) > 0)) usage ('cmdscale: matrix input must be square symmetric; see help'); endif n = size (D,1); % Check for valid format (see help above); If similarity matrix, convert if (any (any (D < 0))) usage ('cmdscale: entries must be nonnegative; see help'); elseif (trace (D) ~= 0) if ((~all (diag (D) == 1)) || (~all (D <= 1))) usage ('cmdscale: input must be distance vector or matrix; see help'); endif D = sqrt (ones (n,n) - D); endif % Build centering matrix, perform double centering, extract eigenpairs J = eye (n) - ones (n,n) / n; B = -1 / 2 * J * (D .^ 2) * J; [Q, e] = eig (B); e = diag (e); etmp = e; e = sort(e, 'descend'); % Remove complex eigenpairs (only possible due to machine approximation) if (iscomplex (etmp)) for i = 1 : size (etmp,1) cmp(i) = (isreal (etmp(i))); endfor etmp = etmp(cmp); Q = Q(:,cmp); endif % Order eigenpairs [etmp, ord] = sort (etmp, 'descend'); Q = Q(:,ord); % Remove negative eigenpairs cmp = (etmp > 0); etmp = etmp(cmp); Q = Q(:,cmp); % Test for n-dimensional results if (size(etmp,1) == n) etmp = etmp(1:n-1); Q = Q(:, 1:n-1); endif % Build output matrix Y Y = Q * diag (sqrt (etmp)); endfunction %!shared m, n, X, D %! m = randi(100) + 1; n = randi(100) + 1; X = rand(m, n); D = pdist(X); %!assert(norm(pdist(cmdscale(D))), norm(D), sqrt(eps)) %!assert(norm(pdist(cmdscale(squareform(D)))), norm(D), sqrt(eps)) statistics-release-1.6.3/inst/combnk.m000066400000000000000000000050051456127120000177330ustar00rootroot00000000000000## Copyright (C) 2010 Soren Hauberg ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{c} =} combnk (@var{data}, @var{k}) ## ## Return all combinations of @var{k} elements in @var{data}. ## ## @end deftypefn function retval = combnk (data, k) ## Check input if (nargin != 2) print_usage; elseif (! isvector (data)) error ("combnk: first input argument must be a vector"); elseif (!isreal (k) || k != round (k) || k < 0) error ("combnk: second input argument must be a non-negative integer"); endif ## Simple checks n = numel (data); if (k == 0 || k > n) retval = resize (data, 0, k); elseif (k == n) retval = data (:).'; else retval = __combnk__ (data, k); endif ## For some odd reason Matlab seems to treat strings differently compared to ## other data-types... if (ischar (data)) retval = flipud (retval); endif endfunction function retval = __combnk__ (data, k) ## Recursion stopping criteria if (k == 1) retval = data (:); else ## Process data n = numel (data); if (iscell (data)) retval = {}; else retval = []; endif for j = 1:n C = __combnk__ (data ((j+1):end), k-1); C = cat (2, repmat (data (j), rows (C), 1), C); if (! isempty (C)) if (isempty (retval)) retval = C; else retval = [retval; C]; endif endif endfor endif endfunction %!demo %! c = combnk (1:5, 2); %! disp ("All pairs of integers between 1 and 5:"); %! disp (c); %!test %! c = combnk (1:3, 2); %! assert (c, [1, 2; 1, 3; 2, 3]); %!test %! c = combnk (1:3, 6); %! assert (isempty (c)); %!test %! c = combnk ({1, 2, 3}, 2); %! assert (c, {1, 2; 1, 3; 2, 3}); %!test %! c = combnk ("hello", 2); %! assert (c, ["lo"; "lo"; "ll"; "eo"; "el"; "el"; "ho"; "hl"; "hl"; "he"]); statistics-release-1.6.3/inst/confusionchart.m000066400000000000000000000250031456127120000215070ustar00rootroot00000000000000## Copyright (C) 2020-2021 Stefano Guidoni ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {} confusionchart (@var{trueLabels}, @var{predictedLabels}) ## @deftypefnx {statistics} {} confusionchart (@var{m}) ## @deftypefnx {statistics} {} confusionchart (@var{m}, @var{classLabels}) ## @deftypefnx {statistics} {} confusionchart (@var{parent}, @dots{}) ## @deftypefnx {statistics} {} confusionchart (@dots{}, @var{prop}, @var{val}, @dots{}) ## @deftypefnx {statistics} {@var{cm} =} confusionchart (@dots{}) ## ## Display a chart of a confusion matrix. ## ## The two vectors of values @var{trueLabels} and @var{predictedLabels}, which ## are used to compute the confusion matrix, must be defined with the same ## format as the inputs of @code{confusionmat}. ## Otherwise a confusion matrix @var{m} as computed by @code{confusionmat} can ## be given. ## ## @var{classLabels} is an array of labels, i.e. the list of the class names. ## ## If the first argument is a handle to a @code{figure} or to a @code{uipanel}, ## then the confusion matrix chart is displayed inside that object. ## ## Optional property/value pairs are passed directly to the underlying objects, ## e.g. @qcode{"xlabel"}, @qcode{"ylabel"}, @qcode{"title"}, @qcode{"fontname"}, ## @qcode{"fontsize"} etc. ## ## The optional return value @var{cm} is a @code{ConfusionMatrixChart} object. ## Specific properties of a @code{ConfusionMatrixChart} object are: ## @itemize @bullet ## @item @qcode{"DiagonalColor"} ## The color of the patches on the diagonal, default is [0.0, 0.4471, 0.7412]. ## ## @item @qcode{"OffDiagonalColor"} ## The color of the patches off the diagonal, default is [0.851, 0.3255, 0.098]. ## ## @item @qcode{"GridVisible"} ## Available values: @qcode{on} (default), @qcode{off}. ## ## @item @qcode{"Normalization"} ## Available values: @qcode{absolute} (default), @qcode{column-normalized}, ## @qcode{row-normalized}, @qcode{total-normalized}. ## ## @item @qcode{"ColumnSummary"} ## Available values: @qcode{off} (default), @qcode{absolute}, ## @qcode{column-normalized},@qcode{total-normalized}. ## ## @item @qcode{"RowSummary"} ## Available values: @qcode{off} (default), @qcode{absolute}, ## @qcode{row-normalized}, @qcode{total-normalized}. ## @end itemize ## ## Run @code{demo confusionchart} to see some examples. ## ## @seealso{confusionmat, sortClasses} ## @end deftypefn function cm = confusionchart (varargin) ## check the input parameters if (nargin < 1) print_usage (); endif p_i = 1; if (ishghandle (varargin{p_i})) ## parameter is a parent figure handle_type = get (varargin{p_i}, "type"); if (strcmp (handle_type, "figure")) h = figure (varargin{p_i}); hax = axes ("parent", h); elseif (strcmp (handle_type, "uipanel")) h = varargin{p_i}; hax = axes ("parent", varargin{p_i}); else ## MATLAB compatibility: on MATLAB are also available Tab objects, ## TiledChartLayout objects, GridLayout objects error ("confusionchart: invalid handle to parent object"); endif p_i++; else h = figure (); hax = axes ("parent", h); endif if (ismatrix (varargin{p_i}) && rows (varargin{p_i}) == ... columns (varargin{p_i})) ## parameter is a confusion matrix conmat = varargin{p_i}; p_i++; if (p_i <= nargin && ((isvector (varargin{p_i}) && ... length (varargin{p_i}) == rows (conmat)) || ... (ischar ( varargin{p_i}) && rows (varargin{p_i}) == rows (conmat)) ... || iscellstr (varargin{p_i}))) ## parameter is an array of labels labarr = varargin{p_i}; if (isrow (labarr)) labarr = vec (labarr); endif p_i++; else labarr = [1 : (rows (conmat))]'; endif elseif (isvector (varargin{p_i})) ## parameter must be a group for confusionmat [conmat, labarr] = confusionmat (varargin{p_i}, varargin{p_i + 1}); p_i = p_i + 2; else close (h); error ("confusionchart: invalid argument"); endif ## remaining parameters are stored i = p_i; args = {}; while (i <= nargin) args{end + 1} = varargin{i++}; endwhile ## prepare the labels if (! iscellstr (labarr)) if (! ischar (labarr)) labarr = num2str (labarr); endif labarr = cellstr (labarr); endif ## MATLAB compatibility: labels are sorted [labarr, I] = sort (labarr); conmat = conmat(I, :); conmat = conmat(:, I); cm = ConfusionMatrixChart (hax, conmat, labarr, args); endfunction ## Demonstration using the confusion matrix example from ## R.Bonnin, "Machine Learning for Developers", pp. 55-56 %!demo %! ## Setting the chart properties %! Yt = [8 5 6 8 5 3 1 6 4 2 5 3 1 4]'; %! Yp = [8 5 6 8 5 2 3 4 4 5 5 7 2 6]'; %! confusionchart (Yt, Yp, "Title", ... %! "Demonstration with summaries","Normalization",... %! "absolute","ColumnSummary", "column-normalized","RowSummary",... %! "row-normalized") ## example: confusion matrix and class labels %!demo %! ## Cellstr as inputs %! Yt = {"Positive", "Positive", "Positive", "Negative", "Negative"}; %! Yp = {"Positive", "Positive", "Negative", "Negative", "Negative"}; %! m = confusionmat (Yt, Yp); %! confusionchart (m, {"Positive", "Negative"}); ## example: editing the properties of an existing ConfusionMatrixChart object %!demo %! ## Editing the object properties %! Yt = {"Positive", "Positive", "Positive", "Negative", "Negative"}; %! Yp = {"Positive", "Positive", "Negative", "Negative", "Negative"}; %! cm = confusionchart (Yt, Yp); %! cm.Title = "This is an example with a green diagonal"; %! cm.DiagonalColor = [0.4660, 0.6740, 0.1880]; ## example: drawing the chart inside a uipanel %!demo %! ## Confusion chart in a uipanel %! h = uipanel (); %! Yt = {"Positive", "Positive", "Positive", "Negative", "Negative"}; %! Yp = {"Positive", "Positive", "Negative", "Negative", "Negative"}; %! cm = confusionchart (h, Yt, Yp); ## example: sortClasses %!demo %! ## Sorting classes %! Yt = [8 5 6 8 5 3 1 6 4 2 5 3 1 4]'; %! Yp = [8 5 6 8 5 2 3 4 4 5 5 7 2 6]'; %! cm = confusionchart (Yt, Yp, "Title", ... %! "Classes are sorted in ascending order"); %! cm = confusionchart (Yt, Yp, "Title", ... %! "Classes are sorted according to clusters"); %! sortClasses (cm, "cluster"); ## Test input validation ## Get current figure visibility so it can be restored after tests %!shared visibility_setting %! visibility_setting = get (0, "DefaultFigureVisible"); %!test %! set (0, "DefaultFigureVisible", "off"); %! fail ("confusionchart ()", "Invalid call"); %! set (0, "DefaultFigureVisible", visibility_setting); %!test %! set (0, "DefaultFigureVisible", "off"); %! fail ("confusionchart ([1 1; 2 2; 3 3])", "invalid argument"); %! set (0, "DefaultFigureVisible", visibility_setting); %!test %! set (0, "DefaultFigureVisible", "off"); %! fail ("confusionchart ([1 2], [0 1], 'xxx', 1)", "invalid property"); %! set (0, "DefaultFigureVisible", visibility_setting); %!test %! set (0, "DefaultFigureVisible", "off"); %! fail ("confusionchart ([1 2], [0 1], 'XLabel', 1)", "XLabel .* string"); %! set (0, "DefaultFigureVisible", visibility_setting); %!test %! set (0, "DefaultFigureVisible", "off"); %! fail ("confusionchart ([1 2], [0 1], 'YLabel', [1 0])", ... %! ".* YLabel .* string"); %! set (0, "DefaultFigureVisible", visibility_setting); %!test %! set (0, "DefaultFigureVisible", "off"); %! fail ("confusionchart ([1 2], [0 1], 'Title', .5)", ".* Title .* string"); %! set (0, "DefaultFigureVisible", visibility_setting); %!test %! set (0, "DefaultFigureVisible", "off"); %! fail ("confusionchart ([1 2], [0 1], 'FontName', [])", ... %! ".* FontName .* string"); %! set (0, "DefaultFigureVisible", visibility_setting); %!test %! set (0, "DefaultFigureVisible", "off"); %! fail ("confusionchart ([1 2], [0 1], 'FontSize', 'b')", ... %! ".* FontSize .* numeric"); %! set (0, "DefaultFigureVisible", visibility_setting); %!test %! set (0, "DefaultFigureVisible", "off"); %! fail ("confusionchart ([1 2], [0 1], 'DiagonalColor', 'h')", ... %! ".* DiagonalColor .* color"); %! set (0, "DefaultFigureVisible", visibility_setting); %!test %! set (0, "DefaultFigureVisible", "off"); %! fail ("confusionchart ([1 2], [0 1], 'OffDiagonalColor', [])", ... %! ".* OffDiagonalColor .* color"); %! set (0, "DefaultFigureVisible", visibility_setting); %!test %! set (0, "DefaultFigureVisible", "off"); %! fail ("confusionchart ([1 2], [0 1], 'Normalization', '')", ... %! ".* invalid .* Normalization"); %! set (0, "DefaultFigureVisible", visibility_setting); %!test %! set (0, "DefaultFigureVisible", "off"); %! fail ("confusionchart ([1 2], [0 1], 'ColumnSummary', [])", ... %! ".* invalid .* ColumnSummary"); %! set (0, "DefaultFigureVisible", visibility_setting); %!test %! set (0, "DefaultFigureVisible", "off"); %! fail ("confusionchart ([1 2], [0 1], 'RowSummary', 1)", ... %! ".* invalid .* RowSummary"); %! set (0, "DefaultFigureVisible", visibility_setting); %!test %! set (0, "DefaultFigureVisible", "off"); %! fail ("confusionchart ([1 2], [0 1], 'GridVisible', .1)", ... %! ".* invalid .* GridVisible"); %! set (0, "DefaultFigureVisible", visibility_setting); %!test %! set (0, "DefaultFigureVisible", "off"); %! fail ("confusionchart ([1 2], [0 1], 'HandleVisibility', .1)", ... %! ".* invalid .* HandleVisibility"); %! set (0, "DefaultFigureVisible", visibility_setting); %!test %! set (0, "DefaultFigureVisible", "off"); %! fail ("confusionchart ([1 2], [0 1], 'OuterPosition', .1)", ... %! ".* invalid .* OuterPosition"); %! set (0, "DefaultFigureVisible", visibility_setting); %!test %! set (0, "DefaultFigureVisible", "off"); %! fail ("confusionchart ([1 2], [0 1], 'Position', .1)", ... %! ".* invalid .* Position"); %! set (0, "DefaultFigureVisible", visibility_setting); %!test %! set (0, "DefaultFigureVisible", "off"); %! fail ("confusionchart ([1 2], [0 1], 'Units', .1)", ".* invalid .* Units"); %! set (0, "DefaultFigureVisible", visibility_setting); statistics-release-1.6.3/inst/confusionmat.m000066400000000000000000000153521456127120000211750ustar00rootroot00000000000000## Copyright (C) 2020 Stefano Guidoni ## Copyright (C) 2022 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{C} =} confusionmat (@var{group}, @var{grouphat}) ## @deftypefnx {statistics} {@var{C} =} confusionmat (@var{group}, @var{grouphat}, "Order", @var{grouporder}) ## @deftypefnx {statistics} {[@var{C}, @var{order}] =} confusionmat (@var{group}, @var{grouphat}) ## ## Compute a confusion matrix for classification problems ## ## @code{confusionmat} returns the confusion matrix @var{C} for the group of ## actual values @var{group} and the group of predicted values @var{grouphat}. ## The row indices of the confusion matrix represent actual values, while the ## column indices represent predicted values. The indices are the same for both ## actual and predicted values, so the confusion matrix is a square matrix. ## Each element of the matrix represents the number of matches between a given ## actual value (row index) and a given predicted value (column index), hence ## correct matches lie on the main diagonal of the matrix. ## The order of the rows and columns is returned in @var{order}. ## ## @var{group} and @var{grouphat} must have the same number of observations ## and the same data type. ## Valid data types are numeric vectors, logical vectors, character arrays, ## string arrays (not implemented yet), cell arrays of strings. ## ## The order of the rows and columns can be specified by setting the ## @var{grouporder} variable. The data type of @var{grouporder} must be the ## same of @var{group} and @var{grouphat}. ## ## MATLAB compatibility: Octave misses string arrays and categorical vectors. ## ## @seealso{crosstab} ## @end deftypefn function [C, order] = confusionmat (group, grouphat, opt = "Order", grouporder) ## check the input parameters if ((nargin < 2) || (nargin > 4)) print_usage(); endif y_true = group; y_pred = grouphat; if (class (y_true) != class (y_pred)) error ("confusionmat: group and grouphat must be of the same data type"); endif if (length (y_true) != length (y_pred)) error ("confusionmat: group and grouphat must be of the same length"); endif if ((nargin > 3) && strcmp (opt, "Order")) unique_tokens = grouporder; if class( y_true ) != class( unique_tokens ) error ("confusionmat: group and grouporder must be of the same data type"); endif endif if (isvector (y_true)) if (isrow (y_true)) y_true = vec(y_true); endif else error ("confusionmat: group must be a vector or array"); endif if (isvector (y_pred)) if (isrow (y_pred)) y_pred = vec(y_pred); endif else error ("confusionmat: grouphat must be a vector or array"); endif if (exist ( "unique_tokens", "var")) if (isvector (unique_tokens)) if (isrow (unique_tokens)) unique_tokens = vec(unique_tokens); endif else error ("confusionmat: grouporder must be a vector or array"); endif endif ## compute the confusion matrix if (isa (y_true, "numeric") || isa (y_true, "logical")) ## numeric or boolean vector ## MATLAB compatibility: ## remove NaN values from grouphat nan_indices = find (isnan (y_pred)); y_pred(nan_indices) = []; ## MATLAB compatibility: ## numeric and boolean values ## are sorted in ascending order if (! exist ("unique_tokens", "var")) unique_tokens = union (y_true, y_pred); endif y_true(nan_indices) = []; C_size = length (unique_tokens); C = zeros (C_size); for i = 1:length (y_true) row_index = find (unique_tokens == y_true(i)); col_index = find (unique_tokens == y_pred(i)); C(row_index, col_index)++; endfor elseif (iscellstr (y_true)) ## string cells ## MATLAB compatibility: ## remove empty values from grouphat empty_indices = []; for i = 1:length (y_pred) if (isempty (y_pred{i})) empty_indices = [empty_indices; i]; endif endfor y_pred(empty_indices) = []; ## MATLAB compatibility: ## string values are sorted according to their ## first appearance in group and grouphat if (! exist ("unique_tokens", "var")) all_tokens = vertcat (y_true, y_pred); unique_tokens = [all_tokens(1)]; for i = 2:length( all_tokens ) if (! any (strcmp (all_tokens(i), unique_tokens))) unique_tokens = [unique_tokens; all_tokens(i)]; endif endfor endif y_true(empty_indices) = []; C_size = length (unique_tokens); C = zeros (C_size); for i = 1:length (y_true) row_index = find (strcmp (y_true{i}, unique_tokens)); col_index = find (strcmp (y_pred{i}, unique_tokens)); C(row_index, col_index)++; endfor elseif (ischar (y_true)) ## character array ## MATLAB compatibility: ## character values are sorted according to their ## first appearance in group and grouphat if (! exist ("unique_tokens", "var")) all_tokens = vertcat (y_true, y_pred); unique_tokens = [all_tokens(1)]; for i = 2:length (all_tokens) if (! any (find (unique_tokens == all_tokens(i)))) unique_tokens = [unique_tokens; all_tokens(i)]; endif endfor endif C_size = length ( unique_tokens ); C = zeros ( C_size ); for i = 1:length( y_true) row_index = find (unique_tokens == y_true(i)); col_index = find (unique_tokens == y_pred(i)); C(row_index, col_index)++; endfor elseif (isstring (y_true)) ## string array ## FIXME: not implemented yet error ("confusionmat: string array not implemented yet"); else error ("confusionmat: invalid data type"); endif order = unique_tokens; endfunction ## Test the confusion matrix example from ## R.Bonnin, "Machine Learning for Developers", pp. 55-56 %!test %! Yt = [8 5 6 8 5 3 1 6 4 2 5 3 1 4]'; %! Yp = [8 5 6 8 5 2 3 4 4 5 5 7 2 6]'; %! C = [0 1 1 0 0 0 0 0; 0 0 0 0 1 0 0 0; 0 1 0 0 0 0 1 0; 0 0 0 1 0 1 0 0; ... %! 0 0 0 0 3 0 0 0; 0 0 0 1 0 1 0 0; 0 0 0 0 0 0 0 0; 0 0 0 0 0 0 0 2]; %! assert (confusionmat (Yt, Yp), C) statistics-release-1.6.3/inst/cophenet.m000066400000000000000000000111341456127120000202670ustar00rootroot00000000000000## Copyright (C) 2021 Stefano Guidoni ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {[@var{c}, @var{d}] =} cophenet (@var{Z}, @var{y}) ## ## Compute the cophenetic correlation coefficient. ## ## The cophenetic correlation coefficient @var{C} of a hierarchical cluster tree ## @var{Z} is the linear correlation coefficient between the cophenetic ## distances @var{d} and the euclidean distances @var{y}. ## @tex ## \def\frac#1#2{{\begingroup#1\endgroup\over#2}} ## $$ c = \frac {\sum_{i < j}(Y_{ij}-{\bar {y}})(Z_{ij}-{\bar{z}})} ## {\sqrt{\sum_{i < j}(Y_{ij}-{\bar {y}})^2(Z_{ij}-{\bar{z}})^2}} $$ ## @end tex ## ## It is a measure of the similarity between the distance of the leaves, as seen ## in the tree, and the distance of the original data points, which were used to ## build the tree. When this similarity is greater, that is the coefficient is ## closer to 1, the tree renders an accurate representation of the distances ## between the original data points. ## ## @var{Z} is a hierarchical cluster tree, as the output of @code{linkage}. ## @var{y} is a vector of euclidean distances, as the output of @code{pdist}. ## ## The optional output @var{d} is a vector of cophenetic distances, in the same ## lower triangular format as @var{y}. The cophenetic distance between two data ## points is the height of the lowest common node of the tree. ## ## @seealso{cluster, dendrogram, inconsistent, linkage, pdist, squareform} ## @end deftypefn function [c, d] = cophenet (Z, y) ## Check input arguments if (nargin < 2) error ("cophenet: function called with too few input arguments."); endif ## Z must be a tree [m w] = size (Z); if ((w != 3) || (! isnumeric (Z)) || (! (max (Z(end,1:2)) == m * 2))) error ("cophenet: Z must be a matrix as generated by the linkage function."); endif ## Data set size n = m + 1; ## Y must be a vector of distances if ((! isnumeric (y)) || (length (y) != (n - 1) * n / 2)) error ("cophenet: Y must be a vector of euclidean distances."); endif ## Compute the cophenetic distances d d = zeros (1, length (y)); N = sparse ((m - 1), m); # to keep track of the leaves from each branch for i = 1 : m l_n = Z(i, 1); r_n = Z(i, 2); if (l_n > n) l_v = nonzeros (N(l_n - n, :)); # the list of leaves from the left branch else l_v = l_n; endif if (r_n > n) r_v = nonzeros (N(r_n - n, :)); # the list of leaves from the right branch else r_v = r_n; endif j_max = length (l_v); k_max = length (r_v); ## Keep track of the leaves in each sub-branch, i.e. node; ## this does not matter for the last node, which includes all leaves if (i < m) N(i, 1 : (j_max + k_max)) = [l_v' r_v']; endif for j = 1 : j_max for k = 1: k_max ## d is in the same format as y if (l_v(j) < r_v(k)) index = (l_v(j) - 1) * m - sum (1 : (l_v(j) - 2)) + (r_v(k) - l_v(j)); else index = (r_v(k) - 1) * m - sum (1 : (r_v(k) - 2)) + (l_v(j) - r_v(k)); endif d(index) = Z(i, 3); endfor endfor endfor ## Compute the cophenetic correlation c y_mean = mean (y); z_mean = mean (d); Y_sigma = y - y_mean; Z_sigma = d - z_mean; c = sum (Z_sigma .* Y_sigma) / sqrt (sum (Y_sigma .^ 2) * sum (Z_sigma .^ 2)); endfunction %!demo %! randn ("seed", 5) # for reproducibility %! X = randn (10,2); %! y = pdist (X); %! Z = linkage (y, "average"); %! cophenet (Z, y) ## Test input validation %!error cophenet () %!error cophenet (1) %!error ... %! cophenet (ones (2,2), 1) %!error ... %! cophenet ([1 2 1], "a") %!error ... %! cophenet ([1 2 1], [1 2]) statistics-release-1.6.3/inst/correlation_test.m000066400000000000000000000213741456127120000220510ustar00rootroot00000000000000## Copyright (C) 1995-2017 Kurt Hornik ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{h} =} correlation_test (@var{x}, @var{y}) ## @deftypefnx {statistics} {[@var{h}, @var{pval}] =} correlation_test (@var{y}, @var{x}) ## @deftypefnx {statistics} {[@var{h}, @var{pval}, @var{stats}] =} correlation_test (@var{y}, @var{x}) ## @deftypefnx {statistics} {[@dots{}] =} correlation_test (@var{y}, @var{x}, @var{Name}, @var{Value}) ## ## Perform a correlation coefficient test whether two samples @var{x} and ## @var{y} come from uncorrelated populations. ## ## @code{@var{h} = correlation_test (@var{y}, @var{x})} tests the null ## hypothesis that the two samples @var{x} and @var{y} come from uncorrelated ## populations. The result is @var{h} = 0 if the null hypothesis cannot be ### rejected at the 5% significance level, or @var{h} = 1 if the null hypothesis ## can be rejected at the 5% level. @var{y} and @var{x} must be vectors of ## equal length with finite real numbers. ## ## The p-value of the test is returned in @var{pval}. @var{stats} is a ## structure with the following fields: ## @multitable @columnfractions 0.05 0.2 0.05 0.70 ## @headitem @tab Field @tab @tab Value ## @item @tab @qcode{method} @tab @tab the type of correlation coefficient used ## for the test ## @item @tab @qcode{df} @tab @tab the degrees of freedom (where applicable) ## @item @tab @qcode{corrcoef} @tab @tab the correlation coefficient ## @item @tab @qcode{stat} @tab @tab the test's statistic ## @item @tab @qcode{dist} @tab @tab the respective distribution for the test ## @item @tab @qcode{alt} @tab @tab the alternative hypothesis for the test ## @end multitable ## ## ## @code{[@dots{}] = correlation_test (@dots{}, @var{name}, @var{value})} ## specifies one or more of the following name/value pairs: ## ## @multitable @columnfractions 0.05 0.2 0.75 ## @headitem @tab Name @tab Value ## @item @tab @qcode{"alpha"} @tab the significance level. Default is 0.05. ## ## @item @tab @qcode{"tail"} @tab a string specifying the alternative hypothesis ## @end multitable ## @multitable @columnfractions 0.1 0.25 0.65 ## @item @tab @qcode{"both"} @tab @math{corrcoef} is not 0 (two-tailed, default) ## @item @tab @qcode{"left"} @tab @math{corrcoef} is less than 0 (left-tailed) ## @item @tab @qcode{"right"} @tab @math{corrcoef} is greater than 0 ## (right-tailed) ## @end multitable ## ## @multitable @columnfractions 0.05 0.2 0.75 ## @item @tab @qcode{"method"} @tab a string specifying the correlation ## coefficient used for the test ## @end multitable ## @multitable @columnfractions 0.1 0.25 0.65 ## @item @tab @qcode{"pearson"} @tab Pearson's product moment correlation ## (Default) ## @item @tab @qcode{"kendall"} @tab Kendall's rank correlation tau ## @item @tab @qcode{"spearman"} @tab Spearman's rank correlation rho ## @end multitable ## ## @seealso{regression_ftest, regression_ttest} ## @end deftypefn function [h, pval, stats] = correlation_test (x, y, varargin) if (nargin < 2) print_usage (); endif if (! isvector (x) || ! isvector (y) || length (x) != length (y)) error ("correlation_test: X and Y must be vectors of equal length."); endif ## Force to column vectors x = x(:); y = y(:); ## Check for finite real numbers in X and Y if (! all (isfinite (x)) || ! isreal (x)) error ("correlation_test: X must contain finite real numbers."); endif if (! all (isfinite (y(:))) || ! isreal (y)) error ("correlation_test: Y must contain finite real numbers."); endif ## Set default arguments alpha = 0.05; tail = "both"; method = "pearson"; ## Check additional options i = 1; while (i <= length (varargin)) switch lower (varargin{i}) case "alpha" i = i + 1; alpha = varargin{i}; ## Check for valid alpha if (! isscalar (alpha) || ! isnumeric (alpha) || ... alpha <= 0 || alpha >= 1) error ("correlation_test: invalid value for alpha."); endif case "tail" i = i + 1; tail = varargin{i}; if (! any (strcmpi (tail, {"both", "left", "right"}))) error ("correlation_test: invalid value for tail."); endif case "method" i = i + 1; method = varargin{i}; if (! any (strcmpi (method, {"pearson", "kendall", "spearman"}))) error ("correlation_test: invalid value for method."); endif otherwise error ("correlation_test: invalid Name argument."); endswitch i = i + 1; endwhile n = length (x); if (strcmpi (method, "pearson")) r = corr (x, y); stats.method = "Pearson's product moment correlation"; stats.df = n - 2; stats.corrcoef = r; stats.stat = sqrt (stats.df) .* r / sqrt (1 - r.^2); stats.dist = "Student's t"; cdf = tcdf (stats.stat, stats.df); elseif (strcmpi (method, "kendall")) tau = kendall (x, y); stats.method = "Kendall's rank correlation tau"; stats.df = []; stats.corrcoef = tau; stats.stat = tau / sqrt ((2 * (2*n+5)) / (9*n*(n-1))); stats.dist = "standard normal"; cdf = stdnormal_cdf (stats.stat); else # spearman rho = spearman (x, y); stats.method = "Spearman's rank correlation rho"; stats.df = []; stats.corrcoef = rho; stats.stat = sqrt (n-1) * (rho - 6/(n^3-n)); stats.dist = "standard normal"; cdf = stdnormal_cdf (stats.stat); endif ## Based on the "tail" argument determine the P-value switch lower (tail) case "both" pval = 2 * min (cdf, 1 - cdf); case "right" pval = 1 - cdf; case "left" pval = cdf; endswitch stats.alt = tail; ## Determine the test outcome h = double (pval < alpha); endfunction ## Test input validation %!error correlation_test (); %!error correlation_test (1); %!error ... %! correlation_test ([1 2 NaN]', [2 3 4]'); %!error ... %! correlation_test ([1 2 Inf]', [2 3 4]'); %!error ... %! correlation_test ([1 2 3+i]', [2 3 4]'); %!error ... %! correlation_test ([1 2 3]', [2 3 NaN]'); %!error ... %! correlation_test ([1 2 3]', [2 3 Inf]'); %!error ... %! correlation_test ([1 2 3]', [3 4 3+i]'); %!error ... %! correlation_test ([1 2 3]', [3 4 4 5]'); %!error ... %! correlation_test ([1 2 3]', [2 3 4]', "alpha", 0); %!error ... %! correlation_test ([1 2 3]', [2 3 4]', "alpha", 1.2); %!error ... %! correlation_test ([1 2 3]', [2 3 4]', "alpha", [.02 .1]); %!error ... %! correlation_test ([1 2 3]', [2 3 4]', "alpha", "a"); %!error ... %! correlation_test ([1 2 3]', [2 3 4]', "some", 0.05); %!error ... %! correlation_test ([1 2 3]', [2 3 4]', "tail", "val"); %!error ... %! correlation_test ([1 2 3]', [2 3 4]', "alpha", 0.01, "tail", "val"); %!error ... %! correlation_test ([1 2 3]', [2 3 4]', "method", 0.01); %!error ... %! correlation_test ([1 2 3]', [2 3 4]', "method", "some"); %!test %! x = [6 7 7 9 10 12 13 14 15 17]; %! y = [19 22 27 25 30 28 30 29 25 32]; %! [h, pval, stats] = correlation_test (x, y); %! assert (stats.corrcoef, corr (x', y'), 1e-14); %! assert (pval, 0.0223, 1e-4); %!test %! x = [6 7 7 9 10 12 13 14 15 17]'; %! y = [19 22 27 25 30 28 30 29 25 32]'; %! [h, pval, stats] = correlation_test (x, y); %! assert (stats.corrcoef, corr (x, y), 1e-14); %! assert (pval, 0.0223, 1e-4); statistics-release-1.6.3/inst/crosstab.m000066400000000000000000000107641456127120000203120ustar00rootroot00000000000000## Copyright (C) 1995-2017 Kurt Hornik ## Copyright (C) 2018 John Donoghue ## Copyright (C) 2021 Stefano Guidoni ## Copyright (C) 2022 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{t} =} crosstab (@var{x1}, @var{x2}) ## @deftypefnx {statistics} {@var{t} =} crosstab (@var{x1}, @dots{}, @var{xn}) ## @deftypefnx {statistics} {[@var{t}, @var{chisq}, @var{p}, @var{labels}] =} crosstab (@dots{}) ## ## Create a cross-tabulation (contingency table) @var{t} from data vectors. ## ## The inputs @var{x1}, @var{x2}, ... @var{xn} must be vectors of equal length ## with a data type of numeric, logical, char, or string (cell array). ## ## As additional return values @code{crosstab} returns the chi-square statistics ## @var{chisq}, its p-value @var{p} and a cell array @var{labels}, containing ## the labels of each input argument. ## ## @seealso{grp2idx, tabulate} ## @end deftypefn function [t, chisq, p, labels] = crosstab (varargin) ## check input if (nargin < 2) print_usage (); endif v_length = size (varargin{1}, 1); ## main - begin v_reshape = []; # vector of the dimensions of t X = []; # matrix of the indexed input values labels = {}; # cell array of labels for i = 1 : nargin ## If char array, convert to numerical vector try [vector, gnames] = grp2idx (varargin{i}); catch error ("crosstab: x1, x2 ... xn must be vectors."); end_try_catch if ((! isvector (vector)) || (v_length != length (vector))) error ("crosstab: x1, x2 ... xn must be vectors of the same length."); endif X = [X, vector]; for h = 1 : length (gnames) labels{h, i} = gnames{h, 1}; endfor v_reshape(i) = length (unique (vector)); endfor v = unique (X(:, nargin)); t = []; ## core logic, this employs a recursive function "crosstab_recursive" ## given (x1, x2, x3, ... xn) as inputs ## t(i,j,k,...) = sum (x1(:) == v1(i) & x2(:) == v2(j) & ...) for i = 1 : length (v) t = [t, (crosstab_recursive (nargin - 1,... (X(:, nargin) == v(i) | isnan (v(i)) * isnan (X(:, nargin)))))]; endfor t = reshape(t, v_reshape); # build the nargin-dimensional matrix ## additional statistics if (length (v_reshape) > 1) [p, chisq] = chi2test (t); endif ## main - end ## function: crosstab_recursive ## while there are input vectors, let's do iterations over them function t_partial = crosstab_recursive (x_idx, t_parent) y = X(:, x_idx); w = unique (y); t_partial = []; if (x_idx == 1) ## we have reached the last vector, ## let the computation begin for j = 1 : length (w) t_partial = [t_partial, ... sum(t_parent & (y == w(j) | isnan (w(j)) * isnan (y)));]; endfor else ## if there are more vectors, ## just add data and pass it through to the next iteration for j = 1 : length (w) t_partial = [t_partial, ... (crosstab_recursive (x_idx - 1, ... (t_parent & (y == w(j) | isnan (w(j)) * isnan (y)))))]; endfor endif endfunction endfunction ## Test input validation %!error crosstab () %!error crosstab (1) %!error crosstab (ones (2), [1 1]) %!error crosstab ([1 1], ones (2)) %!error crosstab ([1], [1 2]) %!error crosstab ([1 2], [1]) %!test %! load carbig %! [table, chisq, p, labels] = crosstab (cyl4, when, org); %! assert (table(2,3,1), 38); %! assert (labels{3,3}, "Japan"); %!test %! load carbig %! [table, chisq, p, labels] = crosstab (cyl4, when, org); %! assert (table(2,3,2), 17); %! assert (labels{1,3}, "USA"); statistics-release-1.6.3/inst/crossval.m000066400000000000000000000141421456127120000203200ustar00rootroot00000000000000## Copyright (C) 2014 Nir Krakauer ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation; either version 3 of the License, or ## (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; If not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{results} =} crossval (@var{f}, @var{X}, @var{y}) ## @deftypefnx {statistics} {@var{results} =} crossval (@var{f}, @var{X}, @var{y}, @var{name}, @var{value}) ## ## Perform cross validation on given data. ## ## @var{f} should be a function that takes 4 inputs @var{xtrain}, @var{ytrain}, ## @var{xtest}, @var{ytest}, fits a model based on @var{xtrain}, @var{ytrain}, ## applies the fitted model to @var{xtest}, and returns a goodness of fit ## measure based on comparing the predicted and actual @var{ytest}. ## @code{crossval} returns an array containing the values returned by @var{f} ## for every cross-validation fold or resampling applied to the given data. ## ## @var{X} should be an @var{n} by @var{m} matrix of predictor values ## ## @var{y} should be an @var{n} by @var{1} vector of predicand values ## ## Optional arguments may include name-value pairs as follows: ## ## @table @asis ## @item @qcode{"KFold"} ## Divide set into @var{k} equal-size subsets, using each one successively ## for validation. ## ## @item @qcode{"HoldOut"} ## Divide set into two subsets, training and validation. If the value ## @var{k} is a fraction, that is the fraction of values put in the ## validation subset (by default @var{k}=0.1); if it is a positive integer, ## that is the number of values in the validation subset. ## ## @item @qcode{"LeaveOut"} ## Leave-one-out partition (each element is placed in its own subset). ## The value is ignored. ## ## @item @qcode{"Partition"} ## The value should be a @var{cvpartition} object. ## ## @item @qcode{"Given"} ## The value should be an @var{n} by @var{1} vector specifying in which ## partition to put each element. ## ## @item @qcode{"stratify"} ## The value should be an @var{n} by @var{1} vector containing class ## designations for the elements, in which case the @qcode{"KFold"} and ## @qcode{"HoldOut"} partitionings attempt to ensure each partition ## represents the classes proportionately. ## ## @item @qcode{"mcreps"} ## The value should be a positive integer specifying the number of times ## to resample based on different partitionings. Currently only works with ## the partition type @qcode{"HoldOut"}. ## ## @end table ## ## Only one of @qcode{"KFold"}, @qcode{"HoldOut"}, @qcode{"LeaveOut"}, ## @qcode{"Given"}, @qcode{"Partition"} should be specified. If none is ## specified, the default is @qcode{"KFold"} with @var{k} = 10. ## ## @seealso{cvpartition} ## @end deftypefn function results = crossval (f, X, y, varargin) [n, m] = size (X); if (numel (y) != n) error ("X, y sizes incompatible") endif ## extract optional parameter-value argument pairs if (numel (varargin) > 1) vargs = varargin; nargs = numel (vargs); values = vargs(2:2:nargs); names = vargs(1:2:nargs)(1:numel(values)); validnames = {"KFold", "HoldOut", "LeaveOut", "Partition", ... "Given", "stratify", "mcreps"}; for i = 1:numel (names) names(i) = validatestring (names(i){:}, validnames); end for i = 1:numel(validnames) name = validnames(i){:}; name_pos = strmatch (name, names); if (! isempty (name_pos)) eval ([name " = values(name_pos){:};"]) endif endfor endif ## construct CV partition if exist ("Partition", "var") P = Partition; elseif exist ("Given", "var") P = cvpartition (Given, "Given"); elseif exist ("KFold", "var") if (! exist ("stratify", "var")) stratify = n; endif P = cvpartition (stratify, "KFold", KFold); elseif (exist ("HoldOut", "var")) if (! exist ("stratify", "var")) stratify = n; endif P = cvpartition (stratify, "HoldOut", HoldOut); if (! exist ("mcreps", "var") || isempty (mcreps)) mcreps = 1; endif elseif (exist ("LeaveOut", "var")) P = cvpartition (n, "LeaveOut"); else #KFold if (! exist ("stratify", "var")) stratify = n; endif P = cvpartition (stratify, "KFold"); endif nr = get (P, "NumTestSets"); # number of test sets to do cross validation on nreps = 1; if (strcmp (get (P, "Type"), "holdout") && exist ("mcreps", "var") && mcreps > 1) nreps = mcreps; endif results = nan (nreps, nr); for rep = 1:nreps if (rep > 1) P = repartition (P); endif for i = 1:nr inds_train = training (P, i); inds_test = test (P, i); result = f (X(inds_train, :), y(inds_train), X(inds_test, :), y(inds_test)); results(rep, i) = result; endfor endfor endfunction %!test %! load fisheriris %! y = meas(:, 1); %! X = [ones(size(y)) meas(:, 2:4)]; %! f = @(X1, y1, X2, y2) meansq (y2 - X2*regress(y1, X1)); %! results0 = crossval (f, X, y); %! results1 = crossval (f, X, y, 'KFold', 10); %! folds = 5; %! results2 = crossval (f, X, y, 'KFold', folds); %! results3 = crossval (f, X, y, 'Partition', cvpartition (numel (y), 'KFold', folds)); %! results4 = crossval (f, X, y, 'LeaveOut', 1); %! mcreps = 2; n_holdout = 20; %! results5 = crossval (f, X, y, 'HoldOut', n_holdout, 'mcreps', mcreps); %! %! ## ensure equal representation of iris species in the training set -- tends %! ## to slightly reduce cross-validation mean square error %! results6 = crossval (f, X, y, 'KFold', 5, 'stratify', grp2idx(species)); %! %! assert (results0, results1, 2e-15); %! assert (results2, results3, 5e-17); %! assert (size(results4), [1 numel(y)]); %! assert (mean(results4), 0.1018, 1e-4); %! assert (size(results5), [mcreps 1]); statistics-release-1.6.3/inst/datasample.m000066400000000000000000000164271456127120000206070ustar00rootroot00000000000000## Copyright (C) 2021 Stefano Guidoni ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{y} =} datasample (@var{data}, @var{k}) ## @deftypefnx {statistics} {@var{y} =} datasample (@var{data}, @var{k}, @var{dim}) ## @deftypefnx {statistics} {@var{y} =} datasample (@dots{}, @var{Name}, @var{Value}) ## @deftypefnx {statistics} {[@var{y} @var{idcs}] =} datasample (@dots{}) ## ## Randomly sample data. ## ## Return @var{k} observations randomly sampled from @var{data}. @var{data} can ## be a vector or a matrix of any data. When @var{data} is a matrix or a ## n-dimensional array, the samples are the subarrays of size n - 1, taken along ## the dimension @var{dim}. The default value for @var{dim} is 1, that is the ## row vectors when sampling a matrix. ## ## Output @var{y} is the returned sampled data. Optional output @var{idcs} is ## the vector of the indices to build @var{y} from @var{data}. ## ## Additional options are set through pairs of parameter name and value. ## Available parameters are: ## ## @table @code ## @item @qcode{Replace} ## a logical value that can be @code{true} (default) or @code{false}: when set ## to @code{true}, @code{datasample} returns data sampled with replacement. ## ## @item @qcode{Weigths} ## a vector of positive numbers that sets the probability of each element. It ## must have the same size as @var{data} along dimension @var{dim}. ## ## @end table ## ## ## @end deftypefn ## ## @seealso{rand, randi, randperm, randsample} function [y, idcs] = datasample (data, k, varargin) ## check input if ( nargin < 2 ) print_usage (); endif ## data: some data, any type, any format but cell ## MATLAB compatibility: there are no "table" or "dataset array" types in ## Octave if (iscell (data)) error ("datasample: data must be a vector or matrix"); endif ## k, a positive integer if ((! isnumeric (k) || ! isscalar (k)) || (! (floor (k) == k)) || (k <= 0)) error ("datasample: k must be a positive integer scalar"); endif dim = 1; replace = true; weights = []; if ( nargin > 2 ) pair_index = 1; if (! ischar (varargin{1})) ## it must be dim dim = varargin{1}; ## the (Name, Value) pairs start further pair_index += 1; ## dim, another positive integer if ((! isscalar (dim)) || (! (floor (dim) == dim)) || (dim <= 0)) error ("datasample: DIM must be a positive integer scalar"); endif endif ## (Name, Value) pairs while (pair_index < (nargin - 2)) switch (lower (varargin{pair_index})) case "replace" if (! islogical (varargin{pair_index + 1})) error ("datasample: expected a logical value for 'Replace'"); endif replace = varargin{pair_index + 1}; case "weights" if ((! isnumeric (varargin{pair_index + 1})) || (! isvector (varargin{pair_index + 1})) || (any (varargin{pair_index + 1} < 0))) error (["datasample: the sampling weights must be defined as a " ... "vector of positive values"]); endif weights = varargin{pair_index + 1}; otherwise error ("datasample: unknown property %s", varargin{pair_index}); endswitch pair_index += 2; endwhile endif ## get the size of the population to sample if (isvector (data)) imax = length (data); else imax = size (data, dim); endif if (isempty (weights)) ## all elements have the same probability of being chosen ## this is easy ## with or without replacement if (replace) idcs = randi (imax, k, 1); else idcs = randperm (imax, k); endif else ## first check if the weights vector is right if (imax != length (weights)) error (["datasample: the size of the vector of sampling weights must"... " be equal to the size of the sampled data"]); endif if (replace) ## easy case: ## normalize the weights, weights_n = cumsum (weights ./ sum (weights)); weights_n(end) = 1; # just to be sure ## then choose k numbers uniformly between 0 and 1 samples = rand (k, 1); ## we have subdivided the space between 0 and 1 accordingly to the ## weights vector: we have just to map back the random numbers to the ## indices of the orginal dataset for iter = 1 : k idcs(iter) = find (weights_n >= samples(iter), 1); endfor else ## complex case ## choose k numbers uniformly between 0 and 1 samples = rand (k, 1); for iter = 1 : k ## normalize the weights weights_n = cumsum (weights ./ sum (weights)); weights_n(end) = 1; # just to be sure idcs(iter) = find (weights_n >= samples(iter), 1); ## remove the element from the set, i. e. set its probability to zero weights(idcs(iter)) = 0; endfor endif endif ## let's get the actual data from the original set if (isvector (data)) ## data is a vector y = data(idcs); else vS = size (data); if (length (vS) == 2) ## data is a 2-dimensional matrix if (dim == 1) y = data(idcs, :); else y = data(:, idcs); endif else ## data is an n-dimensional matrix s = "y = data("; for iter = 1 : length (vS) if (iter == dim) s = [s "idcs,"]; else s = [s ":,"]; endif endfor s = [s ":);"]; eval (s); endif endif endfunction ## some tests %!error datasample(); %!error datasample(1); %!error datasample({1, 2, 3}, 1); %!error datasample([1 2], -1); %!error datasample([1 2], 1.5); %!error datasample([1 2], [1 1]); %!error datasample([1 2], 'g', [1 1]); %!error datasample([1 2], 1, -1); %!error datasample([1 2], 1, 1.5); %!error datasample([1 2], 1, [1 1]); %!error datasample([1 2], 1, 1, "Replace", -2); %!error datasample([1 2], 1, 1, "Weights", "abc"); %!error datasample([1 2], 1, 1, "Weights", [1 -2 3]); %!error datasample([1 2], 1, 1, "Weights", ones (2)); %!error datasample([1 2], 1, 1, "Weights", [1 2 3]); %!test %! dat = randn (10, 4); %! assert (size (datasample (dat, 3, 1)), [3 4]); %!test %! dat = randn (10, 4); %! assert (size (datasample (dat, 3, 2)), [10 3]); statistics-release-1.6.3/inst/datasets/000077500000000000000000000000001456127120000201145ustar00rootroot00000000000000statistics-release-1.6.3/inst/datasets/acetylene.mat000066400000000000000000000045241456127120000225750ustar00rootroot00000000000000Octave-1-L Descriptionÿ sq_stringþÿÿÿi= M xxxy SKC RMT= u 123: ouh eah l ::: une freA t C rum eq c i RRCo cgi ruAe p eaon eic eamt l atnv :,a nrey e cete Tl cdrl t ar . etie r oocs ,E :,cn e rfti n ae g o Tg Dn r tHtn .i . D e e2i n ,Sa s m mo Te tt s ptef ae Raa i eo mr .t o r (n ui s= n ans- rn St= t-eh ag ni d uhce , ec a reop P ei t epnt "r ,a a tda No n (asn eg ", w dn)e wr R i ee e iv t g t As d. h r( o cs g2 em e, e9 c eo a t o sl c yv Rn r e e l. eo r c t e5 g. e er y n7 r1 l na l e e a tt e ( s( t ii n P1 s1 e go e r9 i9 d r) o6 o7 a ( c1 n5 p d % e) ) r e ) s, i, e ) s n d p p i Up Pp c s. r. t e4 a3 o s3 c- r - t2 s H4 i0 y9 c. d. e r , o " g e n D i l u t i o n , " x1ÿmatrixþÿÿÿP”@P”@P”@P”@P”@P”@À’@À’@À’@À’@À’@À’@0‘@0‘@0‘@0‘@x2ÿmatrixþÿÿÿ@"@&@+@1@7@333333@@&@+@1@7@333333@@&@1@x3ÿmatrixþÿÿÿú~j¼t“ˆ?ú~j¼t“ˆ?Zd;ßO‡?9´Èv¾ŸŠ?ÙÎ÷S㥋?ú~j¼t“ˆ?{®Gáz¤?Ûù~j¼t£?ü©ñÒMb ?9´Èv¾Ÿš?œÄ °rh¡?Ë¡E¶óý¤?/Ý$µ?J +‡¹?Zd;ßO·?j¼t“¶?yÿmatrixþÿÿÿ€H@š™™™™I@@I@@H@ÀG@@F@<@€?@@A@€A@C@@C@.@1@€4@€=@statistics-release-1.6.3/inst/datasets/arrhythmia.mat000066400000000000000000037416671456127120000230170ustar00rootroot00000000000000Octave-1-L Descriptionÿ sq_stringþÿÿÿEChSY Nate ortei mdp s121ii:V -6na/at=1 ac/rh 5=l aNen ara o=uarcmc ntrhelavcthisaralryv srraitefshisbh.o yosumiratuitic thsfeasntm is .arice dumialdaaceb a rtisu scea. t sa eoe escfdf seoru r sdo/2a o emm7n f d l9g t/ i a ahdin r seang r tp h nUauf y uCstr t mIe o h e tvm m rmsa i ia/r1 a ccAi hrat virbo anhl reye1 i ts6 alh.: bem lai era sn i wn ig t hr e vp ao ls ui et so r 0y : a n d 1VarNamesÿcellþÿÿÿÿ sq_stringþÿÿÿ Age, yearsÿ sq_stringþÿÿÿSex (0=male, 1=female)ÿ sq_stringþÿÿÿ Height, cmÿ sq_stringþÿÿÿ Weight, kgÿ sq_stringþÿÿÿ QRS durationÿ sq_stringþÿÿÿ P-R intervalÿ sq_stringþÿÿÿ Q-T intervalÿ sq_stringþÿÿÿ T intervalÿ sq_stringþÿÿÿ P intervalÿ sq_stringþÿÿÿ QRS angleÿ sq_stringþÿÿÿT angleÿ sq_stringþÿÿÿP angleÿ sq_stringþÿÿÿ QRST angleÿ sq_stringþÿÿÿJ angleÿ sq_stringþÿÿÿHeart rate per minuteÿ sq_stringþÿÿÿDI Q wave widthÿ sq_stringþÿÿÿDI R wave widthÿ sq_stringþÿÿÿDI S wave widthÿ sq_stringþÿÿÿDI R' wave widthÿ sq_stringþÿÿÿDI S' wave widthÿ sq_stringþÿÿÿDI N of intrinsic deflectionsÿ sq_stringþÿÿÿDI Existence of ragged R waveÿ sq_stringþÿÿÿ-DI Existence of diphasic derivation of R waveÿ sq_stringþÿÿÿDI Existence of ragged P waveÿ sq_stringþÿÿÿ-DI Existence of diphasic derivation of P waveÿ sq_stringþÿÿÿDI Existence of ragged T waveÿ sq_stringþÿÿÿ-DI Existence of diphasic derivation of T waveÿ sq_stringþÿÿÿDII Q wave widthÿ sq_stringþÿÿÿDII R wave widthÿ sq_stringþÿÿÿDII S wave widthÿ sq_stringþÿÿÿDII R' wave widthÿ sq_stringþÿÿÿDII S' wave widthÿ sq_stringþÿÿÿDII N of intrinsic deflectionsÿ sq_stringþÿÿÿDII Existence of ragged R waveÿ sq_stringþÿÿÿ.DII Existence of diphasic derivation of R waveÿ sq_stringþÿÿÿDII Existence of ragged P waveÿ sq_stringþÿÿÿ.DII Existence of diphasic derivation of P waveÿ sq_stringþÿÿÿDII Existence of ragged T waveÿ sq_stringþÿÿÿ.DII Existence of diphasic derivation of T waveÿ sq_stringþÿÿÿDIII Q wave widthÿ sq_stringþÿÿÿDIII R wave widthÿ sq_stringþÿÿÿDIII S wave widthÿ sq_stringþÿÿÿDIII R' wave widthÿ sq_stringþÿÿÿDIII S' wave widthÿ sq_stringþÿÿÿDIII N of intrinsic deflectionsÿ sq_stringþÿÿÿDIII Existence of ragged R waveÿ sq_stringþÿÿÿ/DIII Existence of diphasic derivation of R waveÿ sq_stringþÿÿÿDIII Existence of ragged P waveÿ sq_stringþÿÿÿ/DIII Existence of diphasic derivation of P waveÿ sq_stringþÿÿÿDIII Existence of ragged T waveÿ sq_stringþÿÿÿ/DIII Existence of diphasic derivation of T waveÿ sq_stringþÿÿÿAVR Q wave widthÿ sq_stringþÿÿÿAVR R wave widthÿ sq_stringþÿÿÿAVR S wave widthÿ sq_stringþÿÿÿAVR R' wave widthÿ sq_stringþÿÿÿAVR S' wave widthÿ sq_stringþÿÿÿAVR N of intrinsic deflectionsÿ sq_stringþÿÿÿAVR Existence of ragged R waveÿ sq_stringþÿÿÿ.AVR Existence of diphasic derivation of R waveÿ sq_stringþÿÿÿAVR Existence of ragged P waveÿ sq_stringþÿÿÿ.AVR Existence of diphasic derivation of P waveÿ sq_stringþÿÿÿAVR Existence of ragged T waveÿ sq_stringþÿÿÿ.AVR Existence of diphasic derivation of T waveÿ sq_stringþÿÿÿAVL Q wave widthÿ sq_stringþÿÿÿAVL R wave widthÿ sq_stringþÿÿÿAVL S wave widthÿ sq_stringþÿÿÿAVL R' wave widthÿ sq_stringþÿÿÿAVL S' wave widthÿ sq_stringþÿÿÿAVL N of intrinsic deflectionsÿ sq_stringþÿÿÿAVL Existence of ragged R waveÿ sq_stringþÿÿÿ.AVL Existence of diphasic derivation of R waveÿ sq_stringþÿÿÿAVL Existence of ragged P waveÿ sq_stringþÿÿÿ.AVL Existence of diphasic derivation of P waveÿ sq_stringþÿÿÿAVL Existence of ragged T waveÿ sq_stringþÿÿÿ.AVL Existence of diphasic derivation of T waveÿ sq_stringþÿÿÿAVF Q wave widthÿ sq_stringþÿÿÿAVF R wave widthÿ sq_stringþÿÿÿAVF S wave widthÿ sq_stringþÿÿÿAVF R' wave widthÿ sq_stringþÿÿÿAVF S' wave widthÿ sq_stringþÿÿÿAVF N of intrinsic deflectionsÿ sq_stringþÿÿÿAVF Existence of ragged R waveÿ sq_stringþÿÿÿ.AVF Existence of diphasic derivation of R waveÿ sq_stringþÿÿÿAVF Existence of ragged P waveÿ sq_stringþÿÿÿ.AVF Existence of diphasic derivation of P waveÿ sq_stringþÿÿÿAVF Existence of ragged T waveÿ sq_stringþÿÿÿ.AVF Existence of diphasic derivation of T waveÿ sq_stringþÿÿÿV1 Q wave widthÿ sq_stringþÿÿÿV1 R wave widthÿ sq_stringþÿÿÿV1 S wave widthÿ sq_stringþÿÿÿV1 R' wave widthÿ sq_stringþÿÿÿV1 S' wave widthÿ sq_stringþÿÿÿV1 N of intrinsic deflectionsÿ sq_stringþÿÿÿV1 Existence of ragged R waveÿ sq_stringþÿÿÿ-V1 Existence of diphasic derivation of R waveÿ sq_stringþÿÿÿV1 Existence of ragged P waveÿ sq_stringþÿÿÿ-V1 Existence of diphasic derivation of P waveÿ sq_stringþÿÿÿV1 Existence of ragged T waveÿ sq_stringþÿÿÿ-V1 Existence of diphasic derivation of T waveÿ sq_stringþÿÿÿV2 Q wave widthÿ sq_stringþÿÿÿV2 R wave widthÿ sq_stringþÿÿÿV2 S wave widthÿ sq_stringþÿÿÿV2 R' wave widthÿ sq_stringþÿÿÿV2 S' wave widthÿ sq_stringþÿÿÿV2 N of intrinsic deflectionsÿ sq_stringþÿÿÿV2 Existence of ragged R waveÿ sq_stringþÿÿÿ-V2 Existence of diphasic derivation of R waveÿ sq_stringþÿÿÿV2 Existence of ragged P waveÿ sq_stringþÿÿÿ-V2 Existence of diphasic derivation of P waveÿ sq_stringþÿÿÿV2 Existence of ragged T waveÿ sq_stringþÿÿÿ-V2 Existence of diphasic derivation of T waveÿ sq_stringþÿÿÿV3 Q wave widthÿ sq_stringþÿÿÿV3 R wave widthÿ sq_stringþÿÿÿV3 S wave widthÿ sq_stringþÿÿÿV3 R' wave widthÿ sq_stringþÿÿÿV3 S' wave widthÿ sq_stringþÿÿÿV3 N of intrinsic deflectionsÿ sq_stringþÿÿÿV3 Existence of ragged R waveÿ sq_stringþÿÿÿ-V3 Existence of diphasic derivation of R waveÿ sq_stringþÿÿÿV3 Existence of ragged P waveÿ sq_stringþÿÿÿ-V3 Existence of diphasic derivation of P waveÿ sq_stringþÿÿÿV3 Existence of ragged T waveÿ sq_stringþÿÿÿ-V3 Existence of diphasic derivation of T waveÿ sq_stringþÿÿÿV4 Q wave widthÿ sq_stringþÿÿÿV4 R wave widthÿ sq_stringþÿÿÿV4 S wave widthÿ sq_stringþÿÿÿV4 R' wave widthÿ sq_stringþÿÿÿV4 S' wave widthÿ sq_stringþÿÿÿV4 N of intrinsic deflectionsÿ sq_stringþÿÿÿV4 Existence of ragged R waveÿ sq_stringþÿÿÿ-V4 Existence of diphasic derivation of R waveÿ sq_stringþÿÿÿV4 Existence of ragged P waveÿ sq_stringþÿÿÿ-V4 Existence of diphasic derivation of P waveÿ sq_stringþÿÿÿV4 Existence of ragged T waveÿ sq_stringþÿÿÿ-V4 Existence of diphasic derivation of T waveÿ sq_stringþÿÿÿV5 Q wave widthÿ sq_stringþÿÿÿV5 R wave widthÿ sq_stringþÿÿÿV5 S wave widthÿ sq_stringþÿÿÿV5 R' wave widthÿ sq_stringþÿÿÿV5 S' wave widthÿ sq_stringþÿÿÿV5 N of intrinsic deflectionsÿ sq_stringþÿÿÿV5 Existence of ragged R waveÿ sq_stringþÿÿÿ-V5 Existence of diphasic derivation of R waveÿ sq_stringþÿÿÿV5 Existence of ragged P waveÿ sq_stringþÿÿÿ-V5 Existence of diphasic derivation of P waveÿ sq_stringþÿÿÿV5 Existence of ragged T waveÿ sq_stringþÿÿÿ-V5 Existence of diphasic derivation of T waveÿ sq_stringþÿÿÿV6 Q wave widthÿ sq_stringþÿÿÿV6 R wave widthÿ sq_stringþÿÿÿV6 S wave widthÿ sq_stringþÿÿÿV6 R' wave widthÿ sq_stringþÿÿÿV6 S' wave widthÿ sq_stringþÿÿÿV6 N of intrinsic deflectionsÿ sq_stringþÿÿÿV6 Existence of ragged R waveÿ sq_stringþÿÿÿ-V6 Existence of diphasic derivation of R waveÿ sq_stringþÿÿÿV6 Existence of ragged P waveÿ sq_stringþÿÿÿ-V6 Existence of diphasic derivation of P waveÿ sq_stringþÿÿÿV6 Existence of ragged T waveÿ sq_stringþÿÿÿ-V6 Existence of diphasic derivation of T waveÿ sq_stringþÿÿÿDI JJ wave amplitudeÿ sq_stringþÿÿÿDI Q wave amplitudeÿ sq_stringþÿÿÿDI R wave amplitudeÿ sq_stringþÿÿÿDI S wave amplitudeÿ sq_stringþÿÿÿDI R' wave amplitudeÿ sq_stringþÿÿÿDI S' wave amplitudeÿ sq_stringþÿÿÿDI P wave amplitudeÿ sq_stringþÿÿÿDI T wave amplitudeÿ sq_stringþÿÿÿDI QRSAÿ sq_stringþÿÿÿDI QRSTAÿ sq_stringþÿÿÿDII JJ wave amplitudeÿ sq_stringþÿÿÿDII Q wave amplitudeÿ sq_stringþÿÿÿDII R wave amplitudeÿ sq_stringþÿÿÿDII S wave amplitudeÿ sq_stringþÿÿÿDII R' wave amplitudeÿ sq_stringþÿÿÿDII S' wave amplitudeÿ sq_stringþÿÿÿDII P wave amplitudeÿ sq_stringþÿÿÿDII T wave amplitudeÿ sq_stringþÿÿÿDII QRSAÿ sq_stringþÿÿÿ DII QRSTAÿ sq_stringþÿÿÿDIII JJ wave amplitudeÿ sq_stringþÿÿÿDIII Q wave amplitudeÿ sq_stringþÿÿÿDIII R wave amplitudeÿ sq_stringþÿÿÿDIII S wave amplitudeÿ sq_stringþÿÿÿDIII R' wave amplitudeÿ sq_stringþÿÿÿDIII S' wave amplitudeÿ sq_stringþÿÿÿDIII P wave amplitudeÿ sq_stringþÿÿÿDIII T wave amplitudeÿ sq_stringþÿÿÿ DIII QRSAÿ sq_stringþÿÿÿ DIII QRSTAÿ sq_stringþÿÿÿAVR JJ wave amplitudeÿ sq_stringþÿÿÿAVR Q wave amplitudeÿ sq_stringþÿÿÿAVR R wave amplitudeÿ sq_stringþÿÿÿAVR S wave amplitudeÿ sq_stringþÿÿÿAVR R' wave amplitudeÿ sq_stringþÿÿÿAVR S' wave amplitudeÿ sq_stringþÿÿÿAVR P wave amplitudeÿ sq_stringþÿÿÿAVR T wave amplitudeÿ sq_stringþÿÿÿAVR QRSAÿ sq_stringþÿÿÿ AVR QRSTAÿ sq_stringþÿÿÿAVL JJ wave amplitudeÿ sq_stringþÿÿÿAVL Q wave amplitudeÿ sq_stringþÿÿÿAVL R wave amplitudeÿ sq_stringþÿÿÿAVL S wave amplitudeÿ sq_stringþÿÿÿAVL R' wave amplitudeÿ sq_stringþÿÿÿAVL S' wave amplitudeÿ sq_stringþÿÿÿAVL P wave amplitudeÿ sq_stringþÿÿÿAVL T wave amplitudeÿ sq_stringþÿÿÿAVL QRSAÿ sq_stringþÿÿÿ AVL QRSTAÿ sq_stringþÿÿÿAVF JJ wave amplitudeÿ sq_stringþÿÿÿAVF Q wave amplitudeÿ sq_stringþÿÿÿAVF R wave amplitudeÿ sq_stringþÿÿÿAVF S wave amplitudeÿ sq_stringþÿÿÿAVF R' wave amplitudeÿ sq_stringþÿÿÿAVF S' wave amplitudeÿ sq_stringþÿÿÿAVF P wave amplitudeÿ sq_stringþÿÿÿAVF T wave amplitudeÿ sq_stringþÿÿÿAVF QRSAÿ sq_stringþÿÿÿ AVF QRSTAÿ sq_stringþÿÿÿV1 JJ wave amplitudeÿ sq_stringþÿÿÿV1 Q wave amplitudeÿ sq_stringþÿÿÿV1 R wave amplitudeÿ sq_stringþÿÿÿV1 S wave amplitudeÿ sq_stringþÿÿÿV1 R' wave amplitudeÿ sq_stringþÿÿÿV1 S' wave amplitudeÿ sq_stringþÿÿÿV1 P wave amplitudeÿ sq_stringþÿÿÿV1 T wave amplitudeÿ sq_stringþÿÿÿV1 QRSAÿ sq_stringþÿÿÿV1 QRSTAÿ sq_stringþÿÿÿV2 JJ wave amplitudeÿ sq_stringþÿÿÿV2 Q wave amplitudeÿ sq_stringþÿÿÿV2 R wave amplitudeÿ sq_stringþÿÿÿV2 S wave amplitudeÿ sq_stringþÿÿÿV2 R' wave amplitudeÿ sq_stringþÿÿÿV2 S' wave amplitudeÿ sq_stringþÿÿÿV2 P wave amplitudeÿ sq_stringþÿÿÿV2 T wave amplitudeÿ sq_stringþÿÿÿV2 QRSAÿ sq_stringþÿÿÿV2 QRSTAÿ sq_stringþÿÿÿV3 JJ wave amplitudeÿ sq_stringþÿÿÿV3 Q wave amplitudeÿ sq_stringþÿÿÿV3 R wave amplitudeÿ sq_stringþÿÿÿV3 S wave amplitudeÿ sq_stringþÿÿÿV3 R' wave amplitudeÿ sq_stringþÿÿÿV3 S' wave amplitudeÿ sq_stringþÿÿÿV3 P wave amplitudeÿ sq_stringþÿÿÿV3 T wave amplitudeÿ sq_stringþÿÿÿV3 QRSAÿ sq_stringþÿÿÿV3 QRSTAÿ sq_stringþÿÿÿV4 JJ wave amplitudeÿ sq_stringþÿÿÿV4 Q wave amplitudeÿ sq_stringþÿÿÿV4 R wave amplitudeÿ sq_stringþÿÿÿV4 S wave amplitudeÿ sq_stringþÿÿÿV4 R' wave amplitudeÿ sq_stringþÿÿÿV4 S' wave amplitudeÿ sq_stringþÿÿÿV4 P wave amplitudeÿ sq_stringþÿÿÿV4 T wave amplitudeÿ sq_stringþÿÿÿV4 QRSAÿ sq_stringþÿÿÿV4 QRSTAÿ sq_stringþÿÿÿV5 JJ wave amplitudeÿ sq_stringþÿÿÿV5 Q wave amplitudeÿ sq_stringþÿÿÿV5 R wave amplitudeÿ sq_stringþÿÿÿV5 S wave amplitudeÿ sq_stringþÿÿÿV5 R' wave amplitudeÿ sq_stringþÿÿÿV5 S' wave amplitudeÿ sq_stringþÿÿÿV5 P wave amplitudeÿ sq_stringþÿÿÿV5 T wave amplitudeÿ sq_stringþÿÿÿV5 QRSAÿ sq_stringþÿÿÿV5 QRSTAÿ sq_stringþÿÿÿV6 JJ wave amplitudeÿ sq_stringþÿÿÿV6 Q wave amplitudeÿ sq_stringþÿÿÿV6 R wave amplitudeÿ sq_stringþÿÿÿV6 S wave amplitudeÿ sq_stringþÿÿÿV6 R' wave amplitudeÿ sq_stringþÿÿÿV6 S' wave amplitudeÿ sq_stringþÿÿÿV6 P wave amplitudeÿ sq_stringþÿÿÿV6 T wave amplitudeÿ sq_stringþÿÿÿV6 QRSAÿ sq_stringþÿÿÿV6 QRSTAXÿmatrixþÿÿÿÄÀR@L@K@€K@ÀR@*@D@€H@F@I@O@€F@K@>@F@€G@€G@G@@R@€L@<@€F@B@€L@D@F@A@?@L@€I@€J@M@I@J@@Q@F@I@€A@O@€F@€E@D@>@A@D@ÀR@@Q@>@€D@A@ÀR@<@€C@8@J@R@C@E@G@@@ð?€A@B@;@H@F@K@:@€F@P@€B@€B@?@G@€A@A@€F@€B@L@ÀQ@€B@€L@ÀS@€I@€O@ÀR@3@D@Q@@R@J@P@€O@O@9@€K@€@@€B@J@B@€I@€C@3@;@B@€O@G@€L@1@I@ÀQ@€E@€@@"@=@€C@€R@<@8@M@@Q@I@R@ÀQ@€B@9@M@€N@€J@G@L@H@€C@G@€D@D@€G@€G@€A@€I@€A@ð?€H@?@A@B@2@A@€C@1@H@@@B@A@;@€F@€F@N@€M@G@E@E@K@8@D@9@Q@3@@R@D@F@I@@@L@€K@K@P@;@€P@L@€O@2@F@€L@@P@€N@€G@O@6@€P@€H@N@@P@B@F@;@5@G@*@M@€Q@P@I@€A@€@@G@€J@O@E@€G@&@2@€O@€J@€H@>@1@€P@I@@S@€G@E@L@ÀT@E@ÀP@€B@€E@B@€H@R@T@6@€L@G@€F@€E@B@L@C@€E@O@G@€H@R@N@H@J@F@G@.@M@G@€M@€P@Q@€J@€F@H@F@€@@€A@€C@C@A@€A@M@I@€E@I@R@H@€N@€O@€N@P@@@M@H@?@€D@€H@€L@€A@@R@@P@€@@€A@€L@I@N@€F@D@€K@F@€@@€M@@(@€I@€M@€O@€R@ÀP@?@,@€B@€J@O@D@€D@T@@Q@€Q@P@€L@J@€K@€B@@:@€M@K@C@H@M@ÀP@P@€M@€M@@P@E@€B@3@€G@?@€@@€C@E@K@F@H@R@B@@@4@@T@A@O@P@€J@H@:@€F@€G@€G@L@D@K@O@D@€G@€L@C@G@€E@€M@P@>@Q@G@H@R@.@€Q@ÀP@K@ÀP@ @€P@€L@B@€G@€A@ÀR@€S@€J@F@€Q@€P@B@€I@0@€M@€N@€F@>@€G@ÀQ@€K@"@€P@@O@5@S@9@M@€P@€L@P@€D@€J@ÀP@>@;@@P@I@;@M@€O@=@€I@@B@€A@M@P@ @&@€G@&@€Q@4@€C@@@€A@€B@€H@€B@€B@@P@€D@=@€F@4@€J@€B@B@@@€S@ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?Àg@ d@€e@àe@Àg@ e@d@@d@e@àd@@e@ d@€e@@e@d@Àb@`e@Àc@ d@Àd@d@ e@ c@ d@ c@ e@@e@d@€d@d@àe@`d@d@`c@f@d@€e@€d@`d@àe@ c@d@d@ d@d@€c@d@Àc@àc@àc@€c@€c@€e@`d@ d@d@d@e@ d@d@€[@d@Àc@@d@`c@`c@d@d@Àb@€c@`c@Àf@ d@`d@Àc@ d@ f@`e@ d@€c@€c@ f@Àb@ d@d@àc@ d@@d@ d@Àc@`c@`c@`d@ d@e@ g@Àb@ c@`c@d@€c@`c@ c@@e@d@àe@ d@ 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aerdnedcryerceelndrncltyrlyuyrderdeeendnrclzzyrsnynntidrryderldrevcm ddvmt rvdm gmdvcom dskgib dvg1tldgavg dmvd vvkdgdt ksodvmgocivdgvvvtgtd kddmcsdoddscsydogvdkgdvrkor roiaor oreo rktoh use 2g re2 eurem or hrrsae ipsut ooetelr errriei csaaouaataaukms ter se ro uetgouameotue umo auomnwo190rfo 011c lo aguomooow gaawnap uoadlof oooa afow urn a n olga omw rolstboaltcbnlotbctule tra at090e2ld039actlctrtlarllacarcca i2tt eal1dlllcacanaggty a cc cber cluaralekherleh a erhehhseschnvpg 0e2m5e2d10po eoaahevneegisa egbcn4h msce01eee r icgllh sccii3eirachesganety liat csdti la ttto eele5 l0t0 0rl1trdn teettevpnvre-ot5 5a- t80tttjiproecc ltcovv1nl neattemgt ls nx fass ns l'a ardtr5n0l i 0 it2 oo v rt nieae n2ro v0rbs 0 2ehmrn hyaorii0telalr anpe caasoiiuta-coaslcnmtou i1 4s n c tcnrtani cw cndnd 1o o4keec ccc0sood cconnrocc u edigcn arshrts emrad2h tseugoens c01 2 i lhe oloc hor at/r0l l nv1 aaa0 en ruurxzdl grcbaceagpg -eket pylo1e(etnd n at k 1 0 i et rivk eoa s ula l a izi0 vvv0sntdassi a l (xyua arm ie 1v l 5a ir vsl gabtw e 3 v iaa vdb egr b a r i l aaa ei lbttzl a a trl agc 0e3l 0lin pewl(e oeamr 1 eb nn eyb ha b e 2l llls xf booo x u llo g2rlk l2i 0aiadal)isr3s ga lr ot t i i i 8e iiie u immn e t ian t.o u l0t i pll tw 4scor d lo t t as t p 0 eee t t o ms 2 p e e lle e)s0 ank e eu e j r s rrrh u l m ) ism a e 3r l g e a r l i t e m sc ( 0l(i u mh m w2t a s ecd a o s 2osi x aa i a-c e dia l n w w e lm e gdh r el i c ) ) i r oob rl b o s b noa ai u u e u v rc o r d 8 k (n s a c d n l i ( a e s s s w s e ) i l c ) Model_Yearÿmatrixþÿÿÿd€Q@€Q@€Q@€Q@€Q@€Q@€Q@€Q@€Q@€Q@€Q@€Q@€Q@€Q@€Q@€Q@€Q@€Q@€Q@€Q@€Q@€Q@€Q@€Q@€Q@€Q@€Q@€Q@€Q@€Q@€Q@€Q@€Q@€Q@€Q@S@S@S@S@S@S@S@S@S@S@S@S@S@S@S@S@S@S@S@S@S@S@S@S@S@S@S@S@S@S@S@S@S@S@€T@€T@€T@€T@€T@€T@€T@€T@€T@€T@€T@€T@€T@€T@€T@€T@€T@€T@€T@€T@€T@€T@€T@€T@€T@€T@€T@€T@€T@€T@€T@Originÿ sq_stringþÿÿÿdUUUUUUUUUUFUUUUUUUUUJUUUJGFGSGUUUUUIGUUFUUUUUUUUUUGJUUUUGJJUSUFJGUUUUUUUUUUUUGJJUUJJJJJJUUUUJUUUGUUUSSSSSSSSSSrSSSSSSSSSaSSSaereweSSSSSteSSrSSSSSSSSSSeaSSSSeaaSwSraeSSSSSSSSSSSSeaaSSaaaaaaSSSSaSSSeSSSAAAAAAAAAAaAAAAAAAAApAAAprarerAAAAAarAAaAAAAAAAAAArpAAAArppAeAaprAAAAAAAAAAAArppAAppppppAAAApAAArAAA n a amnmdm lm n ma maa d nam maa aaaaaa a m c n nacaea ya c an ann e cna ann nnnnnn n a e nennn n e n n n e n n n y y y y y y y y y Weightÿmatrixþÿÿÿd`«@Ú¬@ت@Òª@òª@õ°@±@Ø°@I±@®@$¨@.°@„¯@F°@®@Ö«@2¬@2ª@b­@¨@ˆ¢@"¦@¬¥@6¤@¤ @¬œ@à¤@ü¢@Ž¢@t¡@°¤@²@±@±@|²@@£@X¡@¤@ž¡@4¡@w°@^°@ô®@w°@B©@2ª@ˆ§@¨@ÌŸ@è @Dž@ œ@†¬@ì«@z¬@ò¨@„œ@Ÿ@Ö @ ¤@œ¨@È®@Œ©@ä¦@Ø­@±@®¯@<®@V­@Z¤@ ¤@¶¢@¤@º£@^¥@b¦@¶§@ðž@¤Ÿ@Èž@š @š @à @:¡@Š¡@´ž@´ž@,Ÿ@§@Ž§@2¤@&¦@Ò¤@„¢@ §@Ì¥@¤ @î¡@‚¤@@¥@statistics-release-1.6.3/inst/datasets/cereal.mat000066400000000000000000000363571456127120000220700ustar00rootroot00000000000000Octave-1-LCaloriesÿmatrixþÿÿÿM€Q@^@€Q@I@€[@€[@€[@@`@€V@€V@^@€[@^@€[@€[@€[@Y@€[@€[@€[@Y@€[@Y@Y@€[@€[@Y@^@^@€[@Y@€[@Y@€[@^@^@€[@€[@€[@€a@€[@Y@€[@Y@Àb@Àb@d@Y@^@€a@€V@@`@^@Y@I@I@Y@Y@^@Y@€V@€[@€[@T@€V@€V@€[@€[@€V@€[@€a@Y@€[@€[@Y@Y@€[@CarboÿmatrixþÿÿÿM@ @@ @,@%@&@2@.@*@(@1@*@*@(@6@5@*@(@$@5@5@&@2@&@,@,@(@,@*@&@.@.@1@*@(@'@,@1@4@5@(@(@0@0@0@1@.@.@5@2@+@&@4@*@$@,@ð¿,@%@.@7@6@0@3@4@"@0@.@5@.@0@5@*@1@1@0@CupsÿmatrixþÿÿÿM…ëQ¸Õ?ð¿…ëQ¸Õ?à?è?è?ð?è?q= ×£på?q= ×£på?è?ô?è?à?ð?ð?ð?ð?ð?à?ð?ð?è?è?ð?è?š™™™™™é?q= ×£på?q= ×£på?è?)\Âõ(ì?è?)\Âõ(ì?Ð?…ëQ¸Õ?ð?è?Ház®Gõ?ð¿è?ø?q= ×£på?ð?ð¿ð¿ð¿q= ×£på?ð?q= ×£på?q= ×£på?ð¿à?q= ×£på?ð?ð?ð¿à?q= ×£på?è?à?à?®Gázò?ð?ð¿q= ×£på?q= ×£på?è?ð?ð¿ð?ð?ð?è?ð?q= ×£på?ð?è?FatÿmatrixþÿÿÿMð?@ð?@@@ð?@@@@ð?ð?@ð?ð?@ð?ð?ð?@@ð?ð?ð?ð?@ð?ð?@@@ð?ð?@@ð?ð?@ð?@ð?ð?ð?ð?ð?ð?ð?ð?ð?FiberÿmatrixþÿÿÿM$@@"@,@ð?ø?ð?@@@@@ð?ð?@ð?ð?@ð?ð?ð?@@@@@@ð?ø?ð?@@@@@@@@ø?@ð?ð?@š™™™™™@@@@@@@ð?ð?@@@@@ð?Mfgÿ sq_stringþÿÿÿMNQKKRGKGRPQGGGGRKKGKNKGRKKKPKPPGPPPQGPKKGQGARRKGKKKGPKQQQQKGKRKNNNKKNGGGGGRGGNameÿcellþÿÿÿMÿ sq_stringþÿÿÿ 100% Branÿ sq_stringþÿÿÿ100% Natural Branÿ sq_stringþÿÿÿAll-Branÿ sq_stringþÿÿÿAll-Bran with Extra Fiberÿ sq_stringþÿÿÿAlmond Delightÿ sq_stringþÿÿÿApple Cinnamon Cheeriosÿ sq_stringþÿÿÿ Apple Jacksÿ sq_stringþÿÿÿBasic 4ÿ sq_stringþÿÿÿ Bran Chexÿ sq_stringþÿÿÿ Bran Flakesÿ sq_stringþÿÿÿ Cap n Crunchÿ sq_stringþÿÿÿCheeriosÿ sq_stringþÿÿÿCinnamon Toast Crunchÿ sq_stringþÿÿÿClustersÿ sq_stringþÿÿÿ Cocoa Puffsÿ sq_stringþÿÿÿ Corn Chexÿ sq_stringþÿÿÿ Corn Flakesÿ sq_stringþÿÿÿ Corn Popsÿ sq_stringþÿÿÿ Count Choculaÿ sq_stringþÿÿÿCracklin Oat Branÿ sq_stringþÿÿÿCream of Wheat (Quick)ÿ sq_stringþÿÿÿCrispixÿ sq_stringþÿÿÿCrispy Wheat & Raisinsÿ sq_stringþÿÿÿ Double Chexÿ sq_stringþÿÿÿ Froot Loopsÿ sq_stringþÿÿÿFrosted Flakesÿ sq_stringþÿÿÿFrosted Mini-Wheatsÿ sq_stringþÿÿÿ&Fruit & Fibre Dates, Walnuts, and Oatsÿ sq_stringþÿÿÿ Fruitful Branÿ sq_stringþÿÿÿFruity Pebblesÿ sq_stringþÿÿÿ Golden Crispÿ sq_stringþÿÿÿGolden Grahamsÿ sq_stringþÿÿÿGrape Nuts Flakesÿ sq_stringþÿÿÿ Grape-Nutsÿ sq_stringþÿÿÿGreat Grains Pecanÿ sq_stringþÿÿÿHoney Graham Ohsÿ sq_stringþÿÿÿHoney Nut Cheeriosÿ sq_stringþÿÿÿ Honey-combÿ sq_stringþÿÿÿJust Right Crunchy Nuggetsÿ sq_stringþÿÿÿJust Right Fruit & Nutÿ sq_stringþÿÿÿKixÿ sq_stringþÿÿÿLifeÿ sq_stringþÿÿÿ Lucky Charmsÿ sq_stringþÿÿÿMaypoÿ sq_stringþÿÿÿ Muesli Raisins, Dates, & Almondsÿ sq_stringþÿÿÿ!Muesli Raisins, Peaches, & Pecansÿ sq_stringþÿÿÿMueslix Crispy Blendÿ sq_stringþÿÿÿMulti-Grain Cheeriosÿ sq_stringþÿÿÿNut&Honey Crunchÿ sq_stringþÿÿÿNutri-Grain Almond-Raisinÿ sq_stringþÿÿÿNutri-grain Wheatÿ sq_stringþÿÿÿOatmeal Raisin Crispÿ sq_stringþÿÿÿPost Nat. Raisin Branÿ sq_stringþÿÿÿ Product 19ÿ sq_stringþÿÿÿ Puffed Riceÿ sq_stringþÿÿÿ Puffed Wheatÿ sq_stringþÿÿÿQuaker Oat Squaresÿ sq_stringþÿÿÿQuaker Oatmealÿ sq_stringþÿÿÿ Raisin Branÿ sq_stringþÿÿÿRaisin Nut Branÿ sq_stringþÿÿÿRaisin Squaresÿ sq_stringþÿÿÿ Rice Chexÿ sq_stringþÿÿÿ Rice Krispiesÿ sq_stringþÿÿÿShredded Wheatÿ sq_stringþÿÿÿShredded Wheat n Branÿ sq_stringþÿÿÿShredded Wheat spoon sizeÿ sq_stringþÿÿÿSmacksÿ sq_stringþÿÿÿ Special Kÿ sq_stringþÿÿÿStrawberry Fruit Wheatsÿ sq_stringþÿÿÿTotal Corn Flakesÿ sq_stringþÿÿÿTotal Raisin Branÿ sq_stringþÿÿÿTotal Whole Grainÿ sq_stringþÿÿÿTriplesÿ sq_stringþÿÿÿTrixÿ sq_stringþÿÿÿ Wheat Chexÿ sq_stringþÿÿÿWheatiesÿ sq_stringþÿÿÿWheaties Honey GoldPotassÿmatrixþÿÿÿM€q@à`@t@ t@ð¿€Q@>@Y@@_@Àg@€A@@Z@€F@@Z@€K@9@€A@4@@P@d@ð¿>@^@T@>@9@Y@i@Àg@9@D@€F@@U@€V@Y@€F@€V@€A@N@ÀW@D@ÀW@€K@ÀW@@e@@e@d@€V@D@@`@€V@^@@p@€F@.@I@€[@€[@n@€a@€[@>@€A@ÀW@€a@^@D@€K@€V@€A@Àl@€[@N@9@À\@€[@N@ProteinÿmatrixþÿÿÿM@@@@@@@@@@ð?@ð?@ð?@@ð?ð?@@@@@@ð?@@@ð?@ð?@@@ð?@ð?@@@@@@@@@@@@@@@@ð?@@@@@@ð?@@@@@@@@@@@ð?@@@ShelfÿmatrixþÿÿÿM@@@@@ð?@@ð?@@ð?@@@ð?ð?@@@@@@@@ð?@@@@ð?@@@@@ð?ð?@@@@@@@@@ð?@@@@@@@@@ð?@@@ð?ð?ð?ð?ð?@ð?@@@@@@ð?ð?ð?SodiumÿmatrixþÿÿÿM@`@.@@p@€a@i@€f@@_@@j@i@@j@€k@ r@@j@€a@€f@€q@ r@€V@€f@€a@T@€k@€a@Àg@@_@i@d@n@à`@€F@€q@€a@@e@ÀR@€k@@o@€f@@e@@e@@p@Àb@€f@ÀW@Àb@Àb@€k@Àg@€k@@e@@e@i@t@à`@@j@€a@n@ r@€Q@Àl@.@i@Àg@i@@o@€a@Àl@i@i@SugarsÿmatrixþÿÿÿM@ @@ @$@,@ @@@(@ð?"@@*@@@(@*@@@$@@*@&@@$@(@(@.@"@@@@&@$@&@@"@@@(@@&@&@*@@"@@@$@,@@@ð¿(@ @@@@.@@@@,@@@(@@@ @TypeÿmatrixþÿÿÿMð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð?ð? Variablesÿcellþÿÿÿÿ sq_stringþÿÿÿNameÿ sq_stringþÿÿÿMfgÿ sq_stringþÿÿÿTypeÿ sq_stringþÿÿÿCaloriesÿ sq_stringþÿÿÿProteinÿ sq_stringþÿÿÿFatÿ sq_stringþÿÿÿSodiumÿ sq_stringþÿÿÿFiberÿ sq_stringþÿÿÿCarboÿ sq_stringþÿÿÿSugarsÿ sq_stringþÿÿÿShelfÿ sq_stringþÿÿÿPotassÿ sq_stringþÿÿÿVitaminsÿ sq_stringþÿÿÿWeightÿ sq_stringþÿÿÿcupsÿ sq_stringþÿÿÿ cereal nameÿ sq_stringþÿÿÿmanufacturer (e.g., Kellogg's)ÿ sq_stringþÿÿÿtype of cereal (cold/hot)ÿ sq_stringþÿÿÿcalories (number)ÿ sq_stringþÿÿÿ protein(g)ÿ sq_stringþÿÿÿfat(g)ÿ sq_stringþÿÿÿ sodium(mg)ÿ sq_stringþÿÿÿdietary fiber(g)ÿ sq_stringþÿÿÿcomplex carbohydrates(g)ÿ sq_stringþÿÿÿ sugars(g)ÿ sq_stringþÿÿÿ3display shelf (1, 2, or 3, counting from the floor)ÿ sq_stringþÿÿÿ potassium(mg)ÿ sq_stringþÿÿÿvitamins & minerals (0, 25, or 100,respectively indicating 'none added'; 'enriched, often to 25% FDA recommended'; '100% of FDA recommended')ÿ sq_stringþÿÿÿ0weight (in ounces) of one serving (serving size)ÿ sq_stringþÿÿÿcups per 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Rizzo and Gabor J. Szekely ## Copyright (C) 2014 Juan Pablo Carbajal ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This progrm is free software; you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation; either version 3 of the License, or ## (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program. If not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {[@var{dCor}, @var{dCov}, @var{dVarX}, @var{dVarY}] =} dcov (@var{x}, @var{y}) ## ## Distance correlation, covariance and correlation statistics. ## ## It returns the distance correlation (@var{dCor}) and the distance covariance ## (@var{dCov}) between @var{x} and @var{y}, the distance variance of @var{x} ## in (@var{dVarX}) and the distance variance of @var{y} in (@var{dVarY}). ## ## @var{x} and @var{y} must have the same number of observations (rows) but they ## can have different number of dimensions (columns). Rows with missing values ## (@qcode{NaN}) in either @var{x} or @var{y} are omitted. ## ## The Brownian covariance is the same as the distance covariance: ## ## @tex ## $$ cov_W (X, Y) = dCov(X, Y) $$ ## ## @end tex ## @ifnottex ## @math{cov_W (@var{x}, @var{y}) = dCov (@var{x}, @var{y})} ## @end ifnottex ## ## and thus Brownian correlation is the same as distance correlation. ## ## @seealso{corr, cov} ## @end deftypefn function [dCor, dCov, dVarX, dVarY] = dcov (x, y) ## Validate input size if (size (x, 1) != size (y, 1)) error ("dcov: Sample sizes (rows) in X and Y must agree."); endif ## Exclude missing values is_nan = any ([isnan(x) isnan(y)], 2); x(is_nan,:) = []; y(is_nan,:) = []; ## Calculate double centered distance A = pdist2 (x, x); A_col = mean (A, 1); A_row = mean (A, 2); Acbar = ones (size (A_row)) * A_col; Arbar = A_row * ones (size (A_col)); A_bar = mean (A(:)) * ones (size (A)); A = A - Acbar - Arbar + A_bar; B = pdist2 (y, y); B_col = mean (B, 1); B_row = mean (B, 2); Bcbar = ones (size (B_row)) * B_col; Brbar = B_row * ones (size (B_col)); B_bar = mean (B(:)) * ones (size (B)); B = B - Bcbar - Brbar + B_bar; ## Calculate distance covariance and variances dCov = sqrt (mean (A(:) .* B(:))); dVarX = sqrt (mean (A(:) .^ 2)); dVarY = sqrt (mean (B(:) .^ 2)); ## Calculate distance correlation V = sqrt (dVarX .* dVarY); if V > 0 dCor = dCov / V; else dCor = 0; end endfunction %!demo %! base=@(x) (x- min(x))./(max(x)-min(x)); %! N = 5e2; %! x = randn (N,1); x = base (x); %! z = randn (N,1); z = base (z); %! # Linear relations %! cy = [1 0.55 0.3 0 -0.3 -0.55 -1]; %! ly = x .* cy; %! ly(:,[1:3 5:end]) = base (ly(:,[1:3 5:end])); %! # Correlated Gaussian %! cz = 1 - abs (cy); %! gy = base ( ly + cz.*z); %! # Shapes %! sx = repmat (x,1,7); %! sy = zeros (size (ly)); %! v = 2 * rand (size(x,1),2) - 1; %! sx(:,1) = v(:,1); sy(:,1) = cos(2*pi*sx(:,1)) + 0.5*v(:,2).*exp(-sx(:,1).^2/0.5); %! R =@(d) [cosd(d) sind(d); -sind(d) cosd(d)]; %! tmp = R(35) * v.'; %! sx(:,2) = tmp(1,:); sy(:,2) = tmp(2,:); %! tmp = R(45) * v.'; %! sx(:,3) = tmp(1,:); sy(:,3) = tmp(2,:); %! sx(:,4) = v(:,1); sy(:,4) = sx(:,4).^2 + 0.5*v(:,2); %! sx(:,5) = v(:,1); sy(:,5) = 3*sign(v(:,2)).*(sx(:,5)).^2 + v(:,2); %! sx(:,6) = cos (2*pi*v(:,1)) + 0.5*(x-0.5); %! sy(:,6) = sin (2*pi*v(:,1)) + 0.5*(z-0.5); %! sx(:,7) = x + sign(v(:,1)); sy(:,7) = z + sign(v(:,2)); %! sy = base (sy); %! sx = base (sx); %! # scaled shape %! sc = 1/3; %! ssy = (sy-0.5) * sc + 0.5; %! n = size (ly,2); %! ym = 1.2; %! xm = 0.5; %! fmt={'horizontalalignment','center'}; %! ff = "% .2f"; %! figure (1) %! for i=1:n %! subplot(4,n,i); %! plot (x, gy(:,i), '.b'); %! axis tight %! axis off %! text (xm,ym,sprintf (ff, dcov (x,gy(:,i))),fmt{:}) %! %! subplot(4,n,i+n); %! plot (x, ly(:,i), '.b'); %! axis tight %! axis off %! text (xm,ym,sprintf (ff, dcov (x,ly(:,i))),fmt{:}) %! %! subplot(4,n,i+2*n); %! plot (sx(:,i), sy(:,i), '.b'); %! axis tight %! axis off %! text (xm,ym,sprintf (ff, dcov (sx(:,i),sy(:,i))),fmt{:}) %! v = axis (); %! %! subplot(4,n,i+3*n); %! plot (sx(:,i), ssy(:,i), '.b'); %! axis (v) %! axis off %! text (xm,ym,sprintf (ff, dcov (sx(:,i),ssy(:,i))),fmt{:}) %! endfor %!error dcov (randn (30, 5), randn (25,5)) statistics-release-1.6.3/inst/dendrogram.m000066400000000000000000000335611456127120000206140ustar00rootroot00000000000000## Copyright (c) 2012 Juan Pablo Carbajal ## Copyright (C) 2021 Stefano Guidoni ## Copyright (C) 2022 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {} dendrogram (@var{tree}) ## @deftypefnx {statistics} {} dendrogram (@var{tree}, @var{p}) ## @deftypefnx {statistics} {} dendrogram (@var{tree}, @var{prop}, @var{val}) ## @deftypefnx {statistics} {} dendrogram (@var{tree}, @var{p}, @var{prop}, @var{val} ) ## @deftypefnx {statistics} {@var{h} =} dendrogram (@dots{}) ## @deftypefnx {statistics} {[@var{h}, @var{t}, @var{perm}] =} dendrogram (@dots{}) ## ## Plot a dendrogram of a hierarchical binary cluster tree. ## ## Given @var{tree}, a hierarchical binary cluster tree as the output of ## @code{linkage}, plot a dendrogram of the tree. The number of leaves shown by ## the dendrogram plot is limited to @var{p}. The default value for @var{p} is ## 30. Set @var{p} to 0 to plot all leaves. ## ## The optional outputs are @var{h}, @var{t} and @var{perm}: ## @itemize @bullet ## @item @var{h} is a handle to the lines of the plot. ## ## @item @var{t} is the vector with the numbers assigned to each leaf. ## Each element of @var{t} is a leaf of @var{tree} and its value is the number ## shown in the plot. ## When the dendrogram plot is collapsed, that is when the number of shown ## leaves @var{p} is inferior to the total number of leaves, a single leaf of ## the plot can represent more than one leaf of @var{tree}: in that case ## multiple elements of @var{t} share the same value, that is the same leaf of ## the plot. ## When the dendrogram plot is not collapsed, each leaf of the plot is the leaf ## of @var{tree} with the same number. ## ## @item @var{perm} is the vector list of the leaves as ordered as in the plot. ## @end itemize ## ## Additional input properties can be specified by pairs of properties and ## values. Known properties are: ## @itemize @bullet ## @item @qcode{"Reorder"} ## Reorder the leaves of the dendrogram plot using a numerical vector of size n, ## the number of leaves. When @var{p} is smaller than @var{n}, the reordering ## cannot break the @var{p} groups of leaves. ## ## @item @qcode{"Orientation"} ## Change the orientation of the plot. Available values: @qcode{top} (default), ## @qcode{bottom}, @qcode{left}, @qcode{right}. ## ## @item @qcode{"CheckCrossing"} ## Check if the lines of a reordered dendrogram cross each other. Available ## values: @qcode{true} (default), @qcode{false}. ## ## @item @qcode{"ColorThreshold"} ## Not implemented. ## ## @item @qcode{"Labels"} ## Use a char, string or cellstr array of size @var{n} to set the label for each ## leaf; the label is dispayed only for nodes with just one leaf. ## @end itemize ## ## @seealso{cluster, clusterdata, cophenet, inconsistent, linkage, pdist} ## @end deftypefn function [H, T, perm] = dendrogram (tree, varargin) [m d] = size (tree); if ((d != 3) || (! isnumeric (tree)) || ... (! (max (tree(end, 1:2)) == m * 2))) error (["dendrogram: tree must be a matrix as generated by the " ... "linkage function"]); end pair_index = 1; ## node count n = m + 1; P = 30; # default value vReorder = []; csLabels = {}; checkCrossing = 1; orientation = "top"; if (nargin > 1) if (isnumeric (varargin{1}) && isscalar (varargin{1})) ## dendrogram (tree, P) P = varargin{1}; pair_index++; endif ## dendrogram (..., Name, Value) while (pair_index < (nargin - 1)) switch (lower (varargin{pair_index})) case "reorder" if (isvector (varargin{pair_index + 1}) && ... isnumeric (varargin{pair_index + 1}) && ... length (varargin{pair_index + 1}) == n ) vReorder = varargin{pair_index + 1}; else error (["dendrogram: 'reorder' must be a numeric vector of size" ... "n, the number of leaves"]); endif case "checkcrossing" if (ischar (varargin{pair_index + 1})) switch (lower (varargin{pair_index + 1})) case "true" checkCrossing = 1; case "false" checkCrossing = 0; otherwise error ("dendrogram: unknown value '%s' for CheckCrossing", ... varargin{pair_index + 1}); endswitch elseif error (["dendrogram: the value of property CheckCrossing must ", ... "be either 'true' or 'false'"]); endif case "colorthreshold" warning ("dendrogram: property '%s' not implemented",... varargin{pair_index}); case "orientation" orientation = varargin{pair_index + 1}; # validity check below case "labels" if (ischar (varargin{pair_index + 1}) && ... (isvector (varargin{pair_index + 1}) && ... length (varargin{pair_index + 1}) == n) || ... (ismatrix (varargin{pair_index + 1}) && ... rows (varargin{pair_index + 1}) == n)) csLabels = cellstr (varargin{pair_index + 1}); elseif (iscellstr (varargin{pair_index + 1}) && length (varargin{pair_index + 1}) == n) csLabels = varargin{pair_index + 1}; else error (["dendrogram: labels must be a char or string or" ... "cellstr array of size n"]); endif otherwise error ("dendrogram: unknown property '%s'", varargin{pair_index}); endswitch pair_index += 2; endwhile endif ## MATLAB compatibility: ## P <= 0 to plot all leaves if (P < 1) P = n; endif if (n > P) level_0 = tree((n - P), 3); else P = n; level_0 = 0; endif vLeafPosition = zeros((n + m), 1); T = (1:n)'; nodecnt = 1; ## main dendrogram_recursive (m, 0); ## T reordering ## MATLAB compatibility: when n > P, each node group is renamed with a number ## between 1 and P, according to the smallest node index of each group; ## the group with the node 1 is always group 1, while group 2 is the group ## with the smallest node index outside of group 1, and group 3 is the group ## with the smallest node index outside of groups 1 and 2... newT = 1 : (length (T)); if (n > P) uniqueT = unique (T); minT = zeros (uniqueT, 1); counter = 1; for i = 1 : length (uniqueT) # it should be exactly equal to P idcs = find (T == uniqueT(i)); minT(i) = min (idcs); endfor minT = minT(find (minT > 0)); # to prevent a strange bug [minT, minTidcs] = sort (minT); uniqueT = uniqueT(minTidcs); for i = 1 : length (uniqueT) idcs = find (T == uniqueT(i)); newT(idcs) = counter++; endfor endif ## leaf reordering if (! isempty(vReorder)) if (P < n) checkT = newT(vReorder(:)); for i = 1 : P idcs = find (checkT == i); if (length (idcs) > 1) if (max (idcs) - min (idcs) >= length (idcs)) error (["dendrogram: invalid reordering that redefines the 'P'"... "groups of leaves"]); endif endif endfor checkT = unique (checkT, "stable"); vNewLeafPosition = zeros (n, 1); uT = unique (T, "stable"); for i = 1 : P vNewLeafPosition(uT(checkT(i))) = i; endfor vLeafPosition = vNewLeafPosition; else for i = 1 : length (vReorder) vLeafPosition(vReorder(i)) = i; endfor endif endif ## figure x = []; ## ticks and tricks xticks = 1:P; perm = zeros (P, 1); for i = 1 : length (vLeafPosition) if (vLeafPosition(i) != 0) idcs = find (T == i); perm(vLeafPosition(i)) = newT(idcs(1)); endif endfor T = newT; # this should be unnecessary for n <= P ## lines for i = (n - P + 1) : m vLeafPosition(n + i) = mean (vLeafPosition(tree(i, 1:2), 1)); x(end + 1,1:4) = [vLeafPosition(tree(i, 1:2))' tree(i, [3 3])]; for j = 1 : 2 x0 = 0; if (tree(i,j) > (2 * n - P)) x0 = tree(tree(i, j) - n, 3); endif x(end + 1, 1:4) = [vLeafPosition(tree(i, [j j]))' x0 tree(i, 3)]; endfor endfor ## plot stuff if (strcmp (orientation, "top")) H = line (x(:, 1:2)', x(:, 3:4)', "color", "blue"); set (gca, "xticklabel", perm, "xtick", xticks); elseif (strcmp (orientation, "bottom")) H = line (x(:, 1:2)', x(:, 3:4)', "color", "blue"); set (gca, "xticklabel", perm, "xtick", xticks, "xaxislocation", "top"); axis ("ij"); elseif (strcmp (orientation, "left")) H = line (x(:, 3:4)', x(:, 1:2)', "color", "blue"); set (gca, "yticklabel", perm, "ytick", xticks, "xdir", "reverse",... "yaxislocation", "right"); elseif (strcmp (orientation, "right")) H = line (x(:, 3:4)', x(:, 1:2)', "color", "blue"); set (gca, "yticklabel", perm, "ytick", xticks); else close (H); error ("dendrogram: invalid orientation '%s'", orientation); endif ## labels if (! isempty (csLabels)) csCurrent = cellstr (num2str (perm)); for i = 1 : n ## when there is just one leaf, use the named label for that leaf if (1 == length (find (T == i))) csCurrent(find (perm == i)) = csLabels(find (T == i)); endif endfor switch (orientation) case {"top", "bottom"} xticklabels (csCurrent); case {"left", "right"} yticklabels (csCurrent); endswitch endif ## check crossings if (checkCrossing && ! isempty(vReorder)) for j = 1 : rows (x) if (x(j, 3) == x(j, 4)) # an horizontal line for i = 1 : rows (x) if (x(i, 1) == x(i, 2) && ... # orthogonal lines (x(i, 1) > x(j, 1) && x(i, 1) < x(j, 2)) && ... (x(j, 3) > x(i, 3) && x(j, 3) < x(i, 4))) warning ("dendrogram: line intersection detected"); endif endfor endif endfor endif ## dendrogram_recursive function dendrogram_recursive (k, cn) if (tree(k, 3) > level_0) for j = 1:2 if (tree(k, j) > n) dendrogram_recursive (tree(k, j) - n, 0) else vLeafPosition(tree(k, j)) = nodecnt++; T(tree(k, j)) = tree(k, j); endif endfor else for j = 1:2 if (cn == 0) cn = n + k; vLeafPosition(cn) = nodecnt++; endif if (tree(k, j) > n) dendrogram_recursive (tree(k, j) - n, cn) else T(tree(k, j)) = cn; endif endfor endif endfunction endfunction %!demo %! ## simple dendrogram %! y = [4, 5; 2, 6; 3, 7; 8, 9; 1, 10]; %! y(:,3) = 1:5; %! dendrogram (y); %! title ("simple dendrogram"); %!demo %! ## another simple dendrogram %! v = 2 * rand (30, 1) - 1; %! d = abs (bsxfun (@minus, v(:, 1), v(:, 1)')); %! y = linkage (squareform (d, "tovector")); %! dendrogram (y); %! title ("another simple dendrogram"); %!demo %! ## collapsed tree, find all the leaves of node 5 %! X = randn (60, 2); %! D = pdist (X); %! y = linkage (D, "average"); %! subplot (2, 1, 1); %! title ("original tree"); %! dendrogram (y, 0); %! subplot (2, 1, 2); %! title ("collapsed tree"); %! [~, t] = dendrogram (y, 20); %! find(t == 5) %!demo %! ## optimal leaf order %! X = randn (30, 2); %! D = pdist (X); %! y = linkage (D, "average"); %! order = optimalleaforder (y, D); %! subplot (2, 1, 1); %! title ("original leaf order"); %! dendrogram (y); %! subplot (2, 1, 2); %! title ("optimal leaf order"); %! dendrogram (y, "Reorder", order); %!demo %! ## horizontal orientation and labels %! X = randn (8, 2); %! D = pdist (X); %! L = ["Snow White"; "Doc"; "Grumpy"; "Happy"; "Sleepy"; "Bashful"; ... %! "Sneezy"; "Dopey"]; %! y = linkage (D, "average"); %! dendrogram (y, "Orientation", "left", "Labels", L); %! title ("horizontal orientation and labels"); ## Test plotting %!shared visibility_setting %! visibility_setting = get (0, "DefaultFigureVisible"); %!test %! hf = figure ("visible", "off"); %! unwind_protect %! y = [4, 5; 2, 6; 3, 7; 8, 9; 1, 10]; %! y(:,3) = 1:5; %! dendrogram (y); %! unwind_protect_cleanup %! close (hf); %! end_unwind_protect %!test %! hf = figure ("visible", "off"); %! unwind_protect %! y = [4, 5; 2, 6; 3, 7; 8, 9; 1, 10]; %! y(:,3) = 1:5; %! dendrogram (y); %! unwind_protect_cleanup %! close (hf); %! end_unwind_protect %!test %! hf = figure ("visible", "off"); %! unwind_protect %! v = 2 * rand (30, 1) - 1; %! d = abs (bsxfun (@minus, v(:, 1), v(:, 1)')); %! y = linkage (squareform (d, "tovector")); %! dendrogram (y); %! unwind_protect_cleanup %! close (hf); %! end_unwind_protect %!test %! hf = figure ("visible", "off"); %! unwind_protect %! X = randn (30, 2); %! D = pdist (X); %! y = linkage (D, "average"); %! order = optimalleaforder (y, D); %! subplot (2, 1, 1); %! title ("original leaf order"); %! dendrogram (y); %! subplot (2, 1, 2); %! title ("optimal leaf order"); %! dendrogram (y, "Reorder", order); %! unwind_protect_cleanup %! close (hf); %! end_unwind_protect ## Test input validation %!error dendrogram (); %!error dendrogram (ones (2, 2), 1); %!error dendrogram ([1 2 1], 1, "xxx", "xxx"); %!error dendrogram ([1 2 1], "Reorder", "xxx"); %!error dendrogram ([1 2 1], "Reorder", [1 2 3 4]); %! fail ('dendrogram ([1 2 1], "Orientation", "north")', "invalid orientation .*") statistics-release-1.6.3/inst/dist_fit/000077500000000000000000000000001456127120000201115ustar00rootroot00000000000000statistics-release-1.6.3/inst/dist_fit/betafit.m000066400000000000000000000212651456127120000217130ustar00rootroot00000000000000## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{paramhat} =} betafit (@var{x}) ## @deftypefnx {statistics} {[@var{paramhat}, @var{paramci}] =} betafit (@var{x}) ## @deftypefnx {statistics} {[@var{paramhat}, @var{paramci}] =} betafit (@var{x}, @var{alpha}) ## @deftypefnx {statistics} {[@var{paramhat}, @var{paramci}] =} betafit (@var{x}, @var{alpha}, @var{options}) ## ## Estimate parameters and confidence intervals for the Beta distribution. ## ## @code{@var{paramhat} = betafit (@var{x})} returns the maximum likelihood ## estimates of the parameters of the Beta distribution given the data in vector ## @var{x}. @qcode{@var{paramhat}([1, 2])} corresponds to the @math{α} and ## @math{β} shape parameters, respectively. Missing values, @qcode{NaNs}, are ## ignored. ## ## @code{[@var{paramhat}, @var{paramci}] = betafit (@var{x})} returns the 95% ## confidence intervals for the parameter estimates. ## ## @code{[@dots{}] = betafit (@var{x}, @var{alpha})} also returns the ## @qcode{100 * (1 - @var{alpha})} percent confidence intervals of the estimated ## parameter. By default, the optional argument @var{alpha} is 0.05 ## corresponding to 95% confidence intervals. ## ## @code{[@var{paramhat}, @var{paramci}] = nbinfit (@var{x}, @var{alpha}, ## @var{options})} specifies control parameters for the iterative algorithm used ## to compute ML estimates with the @code{fminsearch} function. @var{options} ## is a structure with the following fields and their default values: ## @itemize ## @item @qcode{@var{options}.Display = "off"} ## @item @qcode{@var{options}.MaxFunEvals = 400} ## @item @qcode{@var{options}.MaxIter = 200} ## @item @qcode{@var{options}.TolX = 1e-6} ## @end itemize ## ## The Beta distribution is defined on the open interval @math{(0,1)}. However, ## @code{betafit} can also compute the unbounded beta likelihood function for ## data that include exact zeros or ones. In such cases, zeros and ones are ## treated as if they were values that have been left-censored at ## @qcode{sqrt (realmin)} or right-censored at @qcode{1 - eps/2}, respectively. ## ## Further information about the Beta distribution can be found at ## @url{https://en.wikipedia.org/wiki/Beta_distribution} ## ## @seealso{betacdf, betainv, betapdf, betarnd, betalike, betastat} ## @end deftypefn function [paramhat, paramci] = betafit (x, alpha, options) ## Check X for being a vector if (isempty (x)) phat = nan (1, 2, class (x)); pci = nan (2, 2, class (x)); return elseif (! isvector (x) || ! isreal (x)) error ("betafit: X must be a vector of real values."); endif ## Remove missing values x(isnan (x)) = []; ## Check that X contains values in the range [0,1] if (any (x < 0) || any (x > 1)) error ("betafit: X must be in the range [0,1]."); endif ## Check X being a constant vector if (min (x) == max(x)) error ("betafit: X must contain distinct values."); endif ## Parse ALPHA argument or add default if (nargin > 1) if (! isscalar (alpha) || ! isreal (alpha) || alpha <= 0 || alpha >= 1) error ("betafit: wrong value for ALPHA."); endif else alpha = 0.05; endif ## Get options structure or add defaults if (nargin < 3) options.Display = "off"; options.MaxFunEvals = 400; options.MaxIter = 200; options.TolX = 1e-6; else if (! isstruct (options) || ! isfield (options, "Display") || ! isfield (options, "MaxFunEvals") || ! isfield (options, "MaxIter") || ! isfield (options, "TolX")) error (strcat (["gamfit: 'options' 5th argument must be a"], ... [" structure with 'Display', 'MaxFunEvals',"], ... [" 'MaxIter', and 'TolX' fields present."])); endif endif ## Estimate initial parameters numx = length (x); tmp1 = prod ((1 - x) .^ (1 / numx)); tmp2 = prod (x .^ (1 / numx)); tmp3 = (1 - tmp1 - tmp2); ahat = 0.5 * (1 - tmp1) / tmp3; bhat = 0.5 * (1 - tmp2) / tmp3; init = log ([ahat, bhat]); ## Add tolerance for boundary conditions x_lo = sqrt (realmin (class (x))); x_hi = 1 - eps (class (x)) / 2; ## All values are strictly within the interval (0,1) if (all (x > x_lo) && all (x < x_hi)) sumlogx = sum (log (x)); sumlog1px = sum (log1p (-x)); paramhat = fminsearch (@cont_negloglike, init, options); paramhat = exp (paramhat); ## Find boundary elements and process them separately else num0 = sum (x < x_lo); num1 = sum (x > x_hi); x_ct = x (x > x_lo & x < x_hi); numx = length (x_ct); sumlogx = sum (log (x_ct)); sumlog1px = sum (log1p (-x_ct)); paramhat = fminsearch (@mixed_negloglike, init, options); paramhat = exp (paramhat); endif ## Compute confidence intervals if (nargout == 2) [~, acov] = betalike (paramhat,x); logphat = log (paramhat); serrlog = sqrt (diag (acov))' ./ paramhat; p_int = [alpha/2; 1-alpha/2]; paramci = exp (norminv ([p_int p_int], ... [logphat; logphat], [serrlog; serrlog])); endif ## Continuous Negative log-likelihood function function nll = cont_negloglike (params) params = exp (params); nll = numx * betaln (params(1), params(2)) - (params(1) - 1) ... * sumlogx - (params(2) - 1) * sumlog1px; endfunction ## Unbounded Negative log-likelihood function function nll = mixed_negloglike (params) params = exp (params); nll = numx * betaln (params(1), params(2)) - (params(1) - 1) ... * sumlogx - (params(2) - 1) * sumlog1px; ## Handle zeros if (num0 > 0) nll = nll - num0 * log (betainc (x_lo, params(1), params(2), "lower")); endif ## Handle ones if (num1 > 0) nll = nll - num1 * log (betainc (x_hi, params(1), params(2), "upper")); endif endfunction endfunction %!demo %! ## Sample 2 populations from different Beta distibutions %! randg ("seed", 1); # for reproducibility %! r1 = betarnd (2, 5, 500, 1); %! randg ("seed", 2); # for reproducibility %! r2 = betarnd (2, 2, 500, 1); %! r = [r1, r2]; %! %! ## Plot them normalized and fix their colors %! hist (r, 12, 15); %! h = findobj (gca, "Type", "patch"); %! set (h(1), "facecolor", "c"); %! set (h(2), "facecolor", "g"); %! hold on %! %! ## Estimate their shape parameters %! a_b_A = betafit (r(:,1)); %! a_b_B = betafit (r(:,2)); %! %! ## Plot their estimated PDFs %! x = [min(r(:)):0.01:max(r(:))]; %! y = betapdf (x, a_b_A(1), a_b_A(2)); %! plot (x, y, "-pr"); %! y = betapdf (x, a_b_B(1), a_b_B(2)); %! plot (x, y, "-sg"); %! ylim ([0, 4]) %! legend ({"Normalized HIST of sample 1 with α=2 and β=5", ... %! "Normalized HIST of sample 2 with α=2 and β=2", ... %! sprintf("PDF for sample 1 with estimated α=%0.2f and β=%0.2f", ... %! a_b_A(1), a_b_A(2)), ... %! sprintf("PDF for sample 2 with estimated α=%0.2f and β=%0.2f", ... %! a_b_B(1), a_b_B(2))}) %! title ("Two population samples from different Beta distibutions") %! hold off ## Test output %!test %! x = 0.01:0.02:0.99; %! [paramhat, paramci] = betafit (x); %! paramhat_out = [1.0199, 1.0199]; %! paramci_out = [0.6947, 0.6947; 1.4974, 1.4974]; %! assert (paramhat, paramhat_out, 1e-4); %! assert (paramci, paramci_out, 1e-4); %!test %! x = 0.01:0.02:0.99; %! [paramhat, paramci] = betafit (x, 0.01); %! paramci_out = [0.6157, 0.6157; 1.6895, 1.6895]; %! assert (paramci, paramci_out, 1e-4); %!test %! x = 0.00:0.02:1; %! [paramhat, paramci] = betafit (x); %! paramhat_out = [0.0875, 0.1913]; %! paramci_out = [0.0822, 0.1490; 0.0931, 0.2455]; %! assert (paramhat, paramhat_out, 1e-4); %! assert (paramci, paramci_out, 1e-4); ## Test input validation %!error betafit ([0.2, 0.5+i]); %!error betafit (ones (2,2) * 0.5); %!error betafit ([0.5, 1.2]); %!error betafit ([0.1, 0.1]); %!error betafit ([0.01:0.1:0.99], 1.2); statistics-release-1.6.3/inst/dist_fit/betalike.m000066400000000000000000000135121456127120000220510ustar00rootroot00000000000000## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{nlogL} =} betalike (@var{params}, @var{x}) ## @deftypefnx {statistics} {[@var{nlogL}, @var{avar}] =} betalike (@var{params}, @var{x}) ## ## Negative log-likelihood for the Beta distribution. ## ## @code{@var{nlogL} = betalike (@var{params}, @var{x})} returns the negative ## log likelihood of the data in @var{x} corresponding to the Beta distribution ## with (1) shape parameter @math{α} and (2) shape parameter @math{β} given in ## the two-element vector @var{params}. Both parameters must be positive real ## numbers and the data in the range @math{[0,1]}. Out of range parameters or ## data return @qcode{NaN}. ## ## @code{[@var{nlogL}, @var{avar}] = betalike (@var{params}, @var{x})} returns ## the inverse of Fisher's information matrix, @var{avar}. If the input ## parameter values in @var{params} are the maximum likelihood estimates, the ## diagonal elements of @var{params} are their asymptotic variances. ## ## The Beta distribution is defined on the open interval @math{(0,1)}. However, ## @code{betafit} can also compute the unbounded beta likelihood function for ## data that include exact zeros or ones. In such cases, zeros and ones are ## treated as if they were values that have been left-censored at ## @qcode{sqrt (realmin)} or right-censored at @qcode{1 - eps/2}, respectively. ## ## Further information about the Beta distribution can be found at ## @url{https://en.wikipedia.org/wiki/Beta_distribution} ## ## @seealso{betacdf, betainv, betapdf, betarnd, betafit, betastat} ## @end deftypefn function [nlogL, avar] = betalike (params, x) ## Check input arguments and add defaults if (nargin < 2) error ("betalike: function called with too few input arguments."); endif if (numel (params) != 2) error ("betalike: wrong parameters length."); endif ## Get α and β parameters a = params(1); b = params(2); ## Force X to column vector x = x(:); ## Return NaN for out of range parameters or data. a(a <= 0) = NaN; b(b <= 0) = NaN; xmin = min (x); xmax = max (x); x(! (0 <= x & x <= 1)) = NaN; ## Add tolerance for boundary conditions x_lo = sqrt (realmin (class (x))); x_hi = 1 - eps (class (x)) / 2; ## All values are strictly within the interval (0,1) if (all (x > x_lo) && all (x < x_hi)) num0 = 0; num1 = 0; x_ct = x; numx = length (x_ct); ## Find boundary elements and process them separately else num0 = sum (x < x_lo); num1 = sum (x > x_hi); x_ct = x (x > x_lo & x < x_hi); numx = length (x_ct); endif ## Compute continuous log likelihood logx = log (x_ct); log1px = log1p (-x_ct); sumlogx = sum (logx); sumlog1px = sum (log1px); nlogL = numx * betaln (a, b) - (a - 1) * sumlogx - (b - 1) * sumlog1px; ## Include log likelihood for zeros if (num0 > 0) nlogL = nlogL - num0 * log (betainc (x_lo, a, b, "lower")); endif ## Include log likelihood for ones if (num1 > 0) nlogL = nlogL - num1 * log (betainc (x_hi, a, b, "upper")); endif ## Compute the asymptotic covariance if (nargout > 1) if (numel (x) < 2) error ("betalike: not enough data in X."); endif ## Compute the Jacobian of the likelihood for values (0,1) psiab = psi (a + b); psi_a = psi (a); psi_b = psi (b); J = [logx+psiab-psi_a, log1px+psiab-psi_b]; ## Add terms into the Jacobian for the zero and one values. if (num0 > 0 || num1 > 0) dd = sqrt (eps (class (x))); aa = a + a * dd * [1, -1]; bb = b + b * dd * [1, -1]; ad = 2 * a *dd; bd = 2 * b *dd; if (num0 > 0) da = diff (log (betainc (x_lo, aa, b, "lower"))) / ad; db = diff (log (betainc (x_lo, a, bb, "lower"))) / bd; J = [J; repmat([da, db], num0, 1)]; endif if num1 > 0 da = diff (log (betainc (x_hi, aa, b, "upper"))) / ad; db = diff (log (betainc (x_hi, a, bb, "upper"))) / bd; J = [J; repmat([da, db], num1, 1)]; endif endif ## Invert the inner product of the Jacobian to get the asymptotic covariance [~, R] = qr (J, 0); if (any (isnan (R(:)))) avar = [NaN, NaN; NaN, NaN]; else Rinv = R \ eye (2); avar = Rinv * Rinv'; endif endif endfunction ## Test output %!test %! x = 0.01:0.02:0.99; %! [nlogL, avar] = betalike ([2.3, 1.2], x); %! avar_out = [0.03691678, 0.02803056; 0.02803056, 0.03965629]; %! assert (nlogL, 17.873477715879040, 3e-14); %! assert (avar, avar_out, 1e-7); %!test %! x = 0.01:0.02:0.99; %! [nlogL, avar] = betalike ([1, 4], x); %! avar_out = [0.02793282, 0.02717274; 0.02717274, 0.03993361]; %! assert (nlogL, 79.648061114839550, 1e-13); %! assert (avar, avar_out, 1e-7); %!test %! x = 0.00:0.02:1; %! [nlogL, avar] = betalike ([1, 4], x); %! avar_out = [0.00000801564765, 0.00000131397245; ... %! 0.00000131397245, 0.00070827639442]; %! assert (nlogL, 573.2008434477486, 1e-10); %! assert (avar, avar_out, 1e-14); ## Test input validation %!error ... %! betalike ([12, 15]); %!error betalike ([12, 15, 3], [1:50]); statistics-release-1.6.3/inst/dist_fit/binofit.m000066400000000000000000000134261456127120000217270ustar00rootroot00000000000000## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{pshat} =} binofit (@var{x}, @var{n}) ## @deftypefnx {statistics} {[@var{pshat}, @var{psci}] =} binofit (@var{x}, @var{n}) ## @deftypefnx {statistics} {[@var{pshat}, @var{psci}] =} binofit (@var{x}, @var{n}, @var{alpha}) ## ## Estimate parameter and confidence intervals for the binomial distribution. ## ## @code{@var{pshat} = binofit (@var{x}, @var{n})} returns the maximum ## likelihood estimate (MLE) of the probability of success for the binomial ## distribution. @var{x} and @var{n} are scalars containing the number of ## successes and the number of trials, respectively. If @var{x} and @var{n} are ## vectors, @code{binofit} returns a vector of estimates whose @math{i}-th ## element is the parameter estimate for @var{x}(i) and @var{n}(i). A scalar ## value for @var{x} or @var{n} is expanded to the same size as the other input. ## ## @code{[@var{pshat}, @var{psci}] = binofit (@var{x}, @var{n}, @var{alpha})} ## also returns the @qcode{100 * (1 - @var{alpha})} percent confidence intervals ## of the estimated parameter. By default, the optional argument @var{alpha} ## is 0.05 corresponding to 95% confidence intervals. ## ## @code{binofit} treats a vector @var{x} as a collection of measurements from ## separate samples, and returns a vector of estimates. If you want to treat ## @var{x} as a single sample and compute a single parameter estimate and ## confidence interval, use @qcode{binofit (sum (@var{x}), sum (@var{n}))} when ## @var{n} is a vector, and ## @qcode{binofit (sum (@var{x}), @var{n} * length (@var{x}))} when @var{n} is a ## scalar. ## ## Further information about the binomial distribution can be found at ## @url{https://en.wikipedia.org/wiki/Binomial_distribution} ## ## @seealso{binocdf, binoinv, binopdf, binornd, binolike, binostat} ## @end deftypefn function [pshat, psci] = binofit (x, n, alpha) ## Check input arguments if (nargin < 2) error ("binofit: function called with too few input arguments."); endif if (any (x < 0)) error ("binofit: X cannot have negative values."); endif if (! isvector (x)) error ("binofit: X must be a vector."); endif if (any (n < 0) || any (n != round (n)) || any (isinf (n))) error ("binofit: N must be a non-negative integer."); endif if (any (x > n)) error ("binofit: N cannot be greater than X."); endif if (nargin < 3 || isempty (alpha)) alpha = 0.05; elseif (! isscalar (alpha) || ! isreal (alpha) || alpha <= 0 || alpha >= 1) error ("binofit: wrong value for ALPHA."); endif ## Compute pshat pshat = x ./ n; ## Compute lower confidence interval nu1 = 2 * x; nu2 = 2 * (n - x + 1); F = finv (alpha / 2, nu1, nu2); lb = (nu1 .* F) ./ (nu2 + nu1 .* F); x0 = find (x == 0); if (! isempty (x0)) lb(x0) = 0; endif ## Compute upper confidence interval nu1 = 2 * (x + 1); nu2 = 2 * (n - x); F = finv (1 - alpha / 2, nu1, nu2); ub = (nu1 .* F) ./ (nu2 + nu1 .* F); xn = find (x == n); if (! isempty (xn)) ub(xn) = 1; endif psci = [lb(:), ub(:)]; endfunction %!demo %! ## Sample 2 populations from different binomial distibutions %! rand ("seed", 1); # for reproducibility %! r1 = binornd (50, 0.15, 1000, 1); %! rand ("seed", 2); # for reproducibility %! r2 = binornd (100, 0.5, 1000, 1); %! r = [r1, r2]; %! %! ## Plot them normalized and fix their colors %! hist (r, 23, 0.35); %! h = findobj (gca, "Type", "patch"); %! set (h(1), "facecolor", "c"); %! set (h(2), "facecolor", "g"); %! hold on %! %! ## Estimate their probability of success %! pshatA = binofit (r(:,1), 50); %! pshatB = binofit (r(:,2), 100); %! %! ## Plot their estimated PDFs %! x = [min(r(:,1)):max(r(:,1))]; %! y = binopdf (x, 50, mean (pshatA)); %! plot (x, y, "-pg"); %! x = [min(r(:,2)):max(r(:,2))]; %! y = binopdf (x, 100, mean (pshatB)); %! plot (x, y, "-sc"); %! ylim ([0, 0.2]) %! legend ({"Normalized HIST of sample 1 with ps=0.15", ... %! "Normalized HIST of sample 2 with ps=0.50", ... %! sprintf("PDF for sample 1 with estimated ps=%0.2f", ... %! mean (pshatA)), ... %! sprintf("PDF for sample 2 with estimated ps=%0.2f", ... %! mean (pshatB))}) %! title ("Two population samples from different binomial distibutions") %! hold off ## Test output %!test %! x = 0:3; %! [pshat, psci] = binofit (x, 3); %! assert (pshat, [0, 0.3333, 0.6667, 1], 1e-4); %! assert (psci(1,:), [0, 0.7076], 1e-4); %! assert (psci(2,:), [0.0084, 0.9057], 1e-4); %! assert (psci(3,:), [0.0943, 0.9916], 1e-4); %! assert (psci(4,:), [0.2924, 1.0000], 1e-4); ## Test input validation %!error ... %! binofit ([1 2 3 4]) %!error binofit (-1, [1 2 3 3]) %!error binofit (1, [1 2 -1 3]) %!error binofit (1, [1 2 3], 0) %!error binofit (1, [1 2 3], 1.2) %!error binofit (1, [1 2 3], [0.02 0.05]) statistics-release-1.6.3/inst/dist_fit/binolike.m000066400000000000000000000105551456127120000220710ustar00rootroot00000000000000## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{nlogL} =} binolike (@var{params}, @var{x}) ## @deftypefnx {statistics} {[@var{nlogL}, @var{acov}] =} binolike (@var{params}, @var{x}) ## ## Negative log-likelihood for the binomial distribution. ## ## @code{@var{nlogL} = binolike (@var{params}, @var{x})} returns the negative ## log likelihood of the binomial distribution with (1) parameter @var{n} and ## (2) parameter @var{ps}, given in the two-element vector @var{params}, where ## @var{n} is the number of trials and @var{ps} is the probability of success, ## given the number of successes in @var{x}. Unlike @code{binofit}, which ## handles each element in @var{x} independently, @code{binolike} returns the ## negative log likelihood of the entire vector @var{x}. ## ## @code{[@var{nlogL}, @var{acov}] = binolike (@var{params}, @var{x})} also ## returns the inverse of Fisher's information matrix, @var{acov}. If the input ## parameter values in @var{params} are the maximum likelihood estimates, the ## diagonal elements of @var{params} are their asymptotic variances. ## ## Further information about the binomial distribution can be found at ## @url{https://en.wikipedia.org/wiki/Binomial_distribution} ## ## @seealso{binocdf, binoinv, binopdf, binornd, binofit, binostat} ## @end deftypefn function [nlogL, acov] = binolike (params, x) ## Check input arguments if (nargin < 2) error ("binolike: function called with too few input arguments."); endif if (! isvector (x)) error ("binolike: X must be a vector."); endif if (any (x < 0)) error ("binolike: X cannot have negative values."); endif if (length (params) != 2) error ("binolike: PARAMS must be a two-element vector."); endif if (params(1) < 0 || params(1) != round (params(1)) || isinf (params(1))) error (strcat (["binolike: number of trials, PARAMS(1), must be a"], ... [" finite non-negative integer."])); endif if (params(2) < 0 || params(2) > 1) error (strcat (["binolike: probability of success, PARAMS(2), must be"], ... [" in the range [0,1]."])); endif if (any (x > params(1))) error (strcat (["binolike: number of successes, X, must be at least"], ... [" as large as the number of trials, N."])); endif ## Compute negative log-likelihood and asymptotic covariance n = params(1); ps = params(2); numx = length (x); nlogL = -sum (log (binopdf (x, n, ps))); tmp = ps * (1 - ps) / (n * numx); acov = [0, 0; 0, tmp]; endfunction ## Test output %!assert (binolike ([3, 0.333], [0:3]), 6.8302, 1e-4) %!assert (binolike ([3, 0.333], 0), 1.2149, 1e-4) %!assert (binolike ([3, 0.333], 1), 0.8109, 1e-4) %!assert (binolike ([3, 0.333], 2), 1.5056, 1e-4) %!assert (binolike ([3, 0.333], 3), 3.2988, 1e-4) %!test %! [nlogL, acov] = binolike ([3, 0.333], 3); %! assert (acov(4), 0.0740, 1e-4) ## Test input validation %!error binolike (3.25) %!error binolike ([5, 0.2], ones (2)) %!error binolike ([5, 0.2], [-1, 3]) %!error ... %! binolike ([1, 0.2, 3], [1, 3, 5, 7]) %!error binolike ([1.5, 0.2], 1) %!error binolike ([-1, 0.2], 1) %!error binolike ([Inf, 0.2], 1) %!error binolike ([5, 1.2], [3, 5]) %!error binolike ([5, -0.2], [3, 5]) %!error binolike ([5, 0.2], [3, 5, 7]) statistics-release-1.6.3/inst/dist_fit/bisafit.m000066400000000000000000000224551456127120000217200ustar00rootroot00000000000000## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{paramhat} =} bisafit (@var{x}) ## @deftypefnx {statistics} {[@var{paramhat}, @var{paramci}] =} bisafit (@var{x}) ## @deftypefnx {statistics} {[@var{paramhat}, @var{paramci}] =} bisafit (@var{x}, @var{alpha}) ## @deftypefnx {statistics} {[@dots{}] =} bisafit (@var{x}, @var{alpha}, @var{censor}) ## @deftypefnx {statistics} {[@dots{}] =} bisafit (@var{x}, @var{alpha}, @var{censor}, @var{freq}) ## @deftypefnx {statistics} {[@dots{}] =} bisafit (@var{x}, @var{alpha}, @var{censor}, @var{freq}, @var{options}) ## ## Estimate mean and confidence intervals for the Birnbaum-Saunders distribution. ## ## @code{@var{muhat} = bisafit (@var{x})} returns the maximum likelihood ## estimates of the parameters of the Birnbaum-Saunders distribution given the ## data in @var{x}. @qcode{@var{paramhat}(1)} is the scale parameter, ## @var{beta}, and @qcode{@var{paramhat}(2)} is the shape parameter, ## @var{gamma}. ## ## @code{[@var{paramhat}, @var{paramci}] = bisafit (@var{x})} returns the 95% ## confidence intervals for the parameter estimates. ## ## @code{[@dots{}] = bisafit (@var{x}, @var{alpha})} also returns the ## @qcode{100 * (1 - @var{alpha})} percent confidence intervals for the ## parameter estimates. By default, the optional argument @var{alpha} is ## 0.05 corresponding to 95% confidence intervals. Pass in @qcode{[]} for ## @var{alpha} to use the default values. ## ## @code{[@dots{}] = bisafit (@var{x}, @var{alpha}, @var{censor})} accepts a ## boolean vector, @var{censor}, of the same size as @var{x} with @qcode{1}s for ## observations that are right-censored and @qcode{0}s for observations that are ## observed exactly. By default, or if left empty, ## @qcode{@var{censor} = zeros (size (@var{x}))}. ## ## @code{[@dots{}] = bisafit (@var{x}, @var{alpha}, @var{censor}, @var{freq})} ## accepts a frequency vector, @var{freq}, of the same size as @var{x}. ## @var{freq} typically contains integer frequencies for the corresponding ## elements in @var{x}, but it can contain any non-integer non-negative values. ## By default, or if left empty, @qcode{@var{freq} = ones (size (@var{x}))}. ## ## @code{[@dots{}] = bisafit (@dots{}, @var{options})} specifies control ## parameters for the iterative algorithm used to compute ML estimates with the ## @code{fminsearch} function. @var{options} is a structure with the following ## fields and their default values: ## @itemize ## @item @qcode{@var{options}.Display = "off"} ## @item @qcode{@var{options}.MaxFunEvals = 400} ## @item @qcode{@var{options}.MaxIter = 200} ## @item @qcode{@var{options}.TolX = 1e-6} ## @end itemize ## ## Further information about the Birnbaum-Saunders distribution can be found at ## @url{https://en.wikipedia.org/wiki/Birnbaum%E2%80%93Saunders_distribution} ## ## @seealso{bisacdf, bisainv, bisapdf, bisarnd, bisalike, bisastat} ## @end deftypefn function [paramhat, paramci] = bisafit (x, alpha, censor, freq, options) ## Check input arguments if (! isvector (x)) error ("bisafit: X must be a vector."); elseif (any (x <= 0)) error ("bisafit: X must contain only positive values."); endif ## Check alpha if (nargin < 2 || isempty (alpha)) alpha = 0.05; else if (! isscalar (alpha) || ! isreal (alpha) || alpha <= 0 || alpha >= 1) error ("bisafit: wrong value for ALPHA."); endif endif ## Check censor vector if (nargin < 3 || isempty (censor)) censor = zeros (size (x)); elseif (! isequal (size (x), size (censor))) error ("bisafit: X and CENSOR vectors mismatch."); endif ## Check frequency vector if (nargin < 4 || isempty (freq)) freq = ones (size (x)); elseif (! isequal (size (x), size (freq))) error ("bisafit: X and FREQ vectors mismatch."); endif ## Get options structure or add defaults if (nargin < 5) options.Display = "off"; options.MaxFunEvals = 400; options.MaxIter = 200; options.TolX = 1e-6; else if (! isstruct (options) || ! isfield (options, "Display") || ! isfield (options, "MaxFunEvals") || ! isfield (options, "MaxIter") || ! isfield (options, "TolX")) error (strcat (["bisafit: 'options' 5th argument must be a"], ... [" structure with 'Display', 'MaxFunEvals',"], ... [" 'MaxIter', and 'TolX' fields present."])); endif endif ## Starting points as suggested by Birnbaum and Saunders x_uncensored = x(censor==0); xubar = mean (x_uncensored); xuinv = mean (1 ./ x_uncensored); beta = sqrt (xubar ./ xuinv); gamma = 2 .* sqrt (sqrt (xubar .* xuinv) - 1); x0 = [beta, gamma]; ## Minimize negative log-likelihood to estimate parameters f = @(params) bisalike (params, x, censor, freq); [paramhat, ~, err, output] = fminsearch (f, x0, options); ## Force positive parameter values paramhat = abs (paramhat); ## Handle errors if (err == 0) if (output.funcCount >= options.MaxFunEvals) warning ("bisafit: maximum number of function evaluations are exceeded."); elseif (output.iterations >= options.MaxIter) warning ("bisafit: maximum number of iterations are exceeded."); endif elseif (err < 0) error ("bisafit: no solution."); endif ## Compute CIs using a log normal approximation for parameters. if (nargout > 1) ## Compute asymptotic covariance [~, acov] = bisalike (paramhat, x, censor, freq); ## Get standard errors stderr = sqrt (diag (acov))'; stderr = stderr ./ paramhat; ## Apply log transform phatlog = log (paramhat); ## Compute normal quantiles z = norminv (alpha / 2); ## Compute CI paramci = [phatlog; phatlog] + [stderr; stderr] .* [z, z; -z, -z]; ## Inverse log transform paramci = exp (paramci); endif endfunction %!demo %! ## Sample 3 populations from different Birnbaum-Saunders distibutions %! rand ("seed", 5); # for reproducibility %! r1 = bisarnd (1, 0.5, 2000, 1); %! rand ("seed", 2); # for reproducibility %! r2 = bisarnd (2, 0.3, 2000, 1); %! rand ("seed", 7); # for reproducibility %! r3 = bisarnd (4, 0.5, 2000, 1); %! r = [r1, r2, r3]; %! %! ## Plot them normalized and fix their colors %! hist (r, 80, 4.2); %! h = findobj (gca, "Type", "patch"); %! set (h(1), "facecolor", "c"); %! set (h(2), "facecolor", "g"); %! set (h(3), "facecolor", "r"); %! ylim ([0, 1.1]); %! xlim ([0, 8]); %! hold on %! %! ## Estimate their α and β parameters %! beta_gammaA = bisafit (r(:,1)); %! beta_gammaB = bisafit (r(:,2)); %! beta_gammaC = bisafit (r(:,3)); %! %! ## Plot their estimated PDFs %! x = [0:0.1:8]; %! y = bisapdf (x, beta_gammaA(1), beta_gammaA(2)); %! plot (x, y, "-pr"); %! y = bisapdf (x, beta_gammaB(1), beta_gammaB(2)); %! plot (x, y, "-sg"); %! y = bisapdf (x, beta_gammaC(1), beta_gammaC(2)); %! plot (x, y, "-^c"); %! hold off %! legend ({"Normalized HIST of sample 1 with β=1 and γ=0.5", ... %! "Normalized HIST of sample 2 with β=2 and γ=0.3", ... %! "Normalized HIST of sample 3 with β=4 and γ=0.5", ... %! sprintf("PDF for sample 1 with estimated β=%0.2f and γ=%0.2f", ... %! beta_gammaA(1), beta_gammaA(2)), ... %! sprintf("PDF for sample 2 with estimated β=%0.2f and γ=%0.2f", ... %! beta_gammaB(1), beta_gammaB(2)), ... %! sprintf("PDF for sample 3 with estimated β=%0.2f and γ=%0.2f", ... %! beta_gammaC(1), beta_gammaC(2))}) %! title ("Three population samples from different Birnbaum-Saunders distibutions") %! hold off ## Test output %!test %! paramhat = bisafit ([1:50]); %! paramhat_out = [16.2649, 1.0156]; %! assert (paramhat, paramhat_out, 1e-4); %!test %! paramhat = bisafit ([1:5]); %! paramhat_out = [2.5585, 0.5839]; %! assert (paramhat, paramhat_out, 1e-4); ## Test input validation %!error bisafit (ones (2,5)); %!error bisafit ([-1 2 3 4]); %!error bisafit ([1, 2, 3, 4, 5], 1.2); %!error bisafit ([1, 2, 3, 4, 5], 0); %!error bisafit ([1, 2, 3, 4, 5], "alpha"); %!error ... %! bisafit ([1, 2, 3, 4, 5], 0.05, [1 1 0]); %!error ... %! bisafit ([1, 2, 3, 4, 5], [], [1 1 0 1 1]'); %!error ... %! bisafit ([1, 2, 3, 4, 5], 0.05, zeros (1,5), [1 1 0]); %!error ... %! bisafit ([1, 2, 3, 4, 5], [], [], [1 1 0 1 1]'); %!error ... %! bisafit ([1, 2, 3, 4, 5], 0.05, [], [], 2); statistics-release-1.6.3/inst/dist_fit/bisalike.m000066400000000000000000000153661456127120000220650ustar00rootroot00000000000000## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{nlogL} =} bisalike (@var{params}, @var{x}) ## @deftypefnx {statistics} {[@var{nlogL}, @var{acov}] =} bisalike (@var{params}, @var{x}) ## @deftypefnx {statistics} {[@dots{}] =} bisalike (@var{params}, @var{x}, @var{censor}) ## @deftypefnx {statistics} {[@dots{}] =} bisalike (@var{params}, @var{x}, @var{censor}, @var{freq}) ## ## Negative log-likelihood for the Birnbaum-Saunders distribution. ## ## @code{@var{nlogL} = bisalike (@var{params}, @var{x})} returns the negative ## log likelihood of the data in @var{x} corresponding to the Birnbaum-Saunders ## distribution with (1) scale parameter @var{beta} and (2) shape parameter ## @var{gamma} given in the two-element vector @var{params}. ## ## @code{[@var{nlogL}, @var{acov}] = bisalike (@var{params}, @var{x})} also ## returns the inverse of Fisher's information matrix, @var{acov}. If the input ## parameter values in @var{params} are the maximum likelihood estimates, the ## diagonal elements of @var{params} are their asymptotic variances. ## ## @code{[@dots{}] = bisalike (@var{params}, @var{x}, @var{censor})} accepts a ## boolean vector, @var{censor}, of the same size as @var{x} with @qcode{1}s for ## observations that are right-censored and @qcode{0}s for observations that are ## observed exactly. By default, or if left empty, ## @qcode{@var{censor} = zeros (size (@var{x}))}. ## ## @code{[@dots{}] = bisalike (@var{params}, @var{x}, @var{censor}, @var{freq})} ## accepts a frequency vector, @var{freq}, of the same size as @var{x}. ## @var{freq} typically contains integer frequencies for the corresponding ## elements in @var{x}, but it can contain any non-integer non-negative values. ## By default, or if left empty, @qcode{@var{freq} = ones (size (@var{x}))}. ## ## Further information about the Birnbaum-Saunders distribution can be found at ## @url{https://en.wikipedia.org/wiki/Birnbaum%E2%80%93Saunders_distribution} ## ## @seealso{bisacdf, bisainv, bisapdf, bisarnd, bisafit, bisastat} ## @end deftypefn function [nlogL, acov] = bisalike (params, x, censor, freq) ## Check input arguments if (nargin < 2) error ("bisalike: function called with too few input arguments."); endif if (! isvector (x)) error ("bisalike: X must be a vector."); endif if (any (x < 0)) error ("bisalike: X cannot have negative values."); endif if (length (params) != 2) error ("bisalike: PARAMS must be a two-element vector."); endif ## Check censor vector if (nargin < 3 || isempty (censor)) censor = zeros (size (x)); elseif (! isequal (size (x), size (censor))) error ("bisalike: X and CENSOR vector mismatch."); endif ## Check frequency vector if (nargin < 4 || isempty (freq)) freq = ones (size (x)); elseif (! isequal (size (x), size (freq))) error ("bisalike: X and FREQ vector mismatch."); endif beta = params(1); gamma = params(2); z = (sqrt (x ./ beta) - sqrt (beta ./ x)) ./ gamma; w = (sqrt (x ./ beta) + sqrt (beta ./ x)) ./ gamma; L = -0.5 .* (z .^ 2 + log (2 .* pi)) + log (w) - log (2 .* x); n_censored = sum (freq .* censor); if (n_censored > 0) censored = (censor == 1); z_censored = z(censored); Scen = 0.5 * erfc (z_censored ./ sqrt(2)); L(censored) = log (Scen); endif nlogL = -sum (freq .* L); ## Compute asymptotic covariance if (nargout > 1) ## Compute first order central differences of the log-likelihood gradient dp = 0.0001 .* max (abs (params), 1); ngrad_p1 = bisa_ngrad (params + [dp(1), 0], x, censor, freq); ngrad_m1 = bisa_ngrad (params - [dp(1), 0], x, censor, freq); ngrad_p2 = bisa_ngrad (params + [0, dp(2)], x, censor, freq); ngrad_m2 = bisa_ngrad (params - [0, dp(2)], x, censor, freq); ## Compute negative Hessian by normalizing the differences by the increment nH = [(ngrad_p1(:) - ngrad_m1(:))./(2 * dp(1)), ... (ngrad_p2(:) - ngrad_m2(:))./(2 * dp(2))]; ## Force neg Hessian being symmetric nH = 0.5 .* (nH + nH'); ## Check neg Hessian is positive definite [R, p] = chol (nH); if (p > 0) warning ("bisalike: non positive definite Hessian matrix."); acov = NaN (2); return endif ## ACOV estimate is the negative inverse of the Hessian. Rinv = inv (R); acov = Rinv * Rinv; endif endfunction ## Helper function for computing negative gradient function ngrad = bisa_ngrad (params, x, censor, freq) beta = params(1); gamma = params(2); z = (sqrt (x ./ beta) - sqrt (beta ./ x)) ./ gamma; w = (sqrt (x ./ beta) + sqrt (beta ./ x)) ./ gamma; logphi = -0.5 .* (z .^ 2 + log (2 .* pi)); n_censored = sum (freq .* censor); if (n_censored > 0) censored = (censor == 1); z_censored = z(censored); Scen = 0.5 * erfc (z_censored ./ sqrt(2)); endif dL1 = (w .^ 2 - 1) .* 0.5 .* z ./ (w .* beta); dL2 = (z .^ 2 - 1) ./ gamma; if (n_censored > 0) phi_censored = exp (logphi(censored)); wcen = w(censored); d1Scen = phi_censored .* 0.5 .* wcen ./ beta; d2Scen = phi_censored .* z_censored ./ gamma; dL1(censored) = d1Scen ./ Scen; dL2(censored) = d2Scen ./ Scen; endif ngrad = -[sum(freq .* dL1), sum(freq .* dL2)]; endfunction ## Test results %!test %! nlogL = bisalike ([16.2649, 1.0156], [1:50]); %! assert (nlogL, 215.5905, 1e-4); %!test %! nlogL = bisalike ([2.5585, 0.5839], [1:5]); %! assert (nlogL, 8.9950, 1e-4); ## Test input validation %!error bisalike (3.25) %!error bisalike ([5, 0.2], ones (2)) %!error bisalike ([5, 0.2], [-1, 3]) %!error ... %! bisalike ([1, 0.2, 3], [1, 3, 5, 7]) %!error ... %! bisalike ([1.5, 0.2], [1:5], [0, 0, 0]) %!error ... %! bisalike ([1.5, 0.2], [1:5], [0, 0, 0, 0, 0], [1, 1, 1]) %!error ... %! bisalike ([1.5, 0.2], [1:5], [], [1, 1, 1]) statistics-release-1.6.3/inst/dist_fit/burrfit.m000066400000000000000000000351731456127120000217550ustar00rootroot00000000000000## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{paramhat} =} burrfit (@var{x}) ## @deftypefnx {statistics} {[@var{paramhat}, @var{paramci}] =} burrfit (@var{x}) ## @deftypefnx {statistics} {[@var{paramhat}, @var{paramci}] =} burrfit (@var{x}, @var{alpha}) ## @deftypefnx {statistics} {[@dots{}] =} burrfit (@var{x}, @var{alpha}, @var{censor}) ## @deftypefnx {statistics} {[@dots{}] =} burrfit (@var{x}, @var{alpha}, @var{censor}, @var{freq}) ## @deftypefnx {statistics} {[@dots{}] =} burrfit (@var{x}, @var{alpha}, @var{censor}, @var{freq}, @var{options}) ## ## Estimate mean and confidence intervals for the Burr type XII distribution. ## ## @code{@var{muhat} = burrfit (@var{x})} returns the maximum likelihood ## estimates of the parameters of the Burr type XII distribution given the data ## in @var{x}. @qcode{@var{paramhat}(1)} is the scale parameter, @var{lambda}, ## @qcode{@var{paramhat}(2)} is the first shape parameter, @var{c}, and ## @qcode{@var{paramhat}(3)} is the second shape parameter, @var{k} ## ## @code{[@var{paramhat}, @var{paramci}] = burrfit (@var{x})} returns the 95% ## confidence intervals for the parameter estimates. ## ## @code{[@dots{}] = burrfit (@var{x}, @var{alpha})} also returns the ## @qcode{100 * (1 - @var{alpha})} percent confidence intervals for the ## parameter estimates. By default, the optional argument @var{alpha} is ## 0.05 corresponding to 95% confidence intervals. Pass in @qcode{[]} for ## @var{alpha} to use the default values. ## ## @code{[@dots{}] = burrfit (@var{x}, @var{alpha}, @var{censor})} accepts a ## boolean vector, @var{censor}, of the same size as @var{x} with @qcode{1}s for ## observations that are right-censored and @qcode{0}s for observations that are ## observed exactly. By default, or if left empty, ## @qcode{@var{censor} = zeros (size (@var{x}))}. ## ## @code{[@dots{}] = burrfit (@var{x}, @var{alpha}, @var{censor}, @var{freq})} ## accepts a frequency vector, @var{freq}, of the same size as @var{x}. ## @var{freq} typically contains integer frequencies for the corresponding ## elements in @var{x}, but it can contain any non-integer non-negative values. ## By default, or if left empty, @qcode{@var{freq} = ones (size (@var{x}))}. ## ## @code{[@dots{}] = burrfit (@dots{}, @var{options})} specifies control ## parameters for the iterative algorithm used to compute ML estimates with the ## @code{fminsearch} function. @var{options} is a structure with the following ## fields and their default values: ## @itemize ## @item @qcode{@var{options}.Display = "off"} ## @item @qcode{@var{options}.MaxFunEvals = 1000} ## @item @qcode{@var{options}.MaxIter = 500} ## @item @qcode{@var{options}.TolX = 1e-6} ## @end itemize ## ## Further information about the Burr type XII distribution can be found at ## @url{https://en.wikipedia.org/wiki/Burr_distribution} ## ## @seealso{burrcdf, burrinv, burrpdf, burrrnd, burrlike, burrstat} ## @end deftypefn function [paramhat, paramci] = burrfit (x, alpha, censor, freq, options) ## Check input arguments if (! isvector (x)) error ("burrfit: X must be a vector."); elseif (any (x <= 0)) error ("burrfit: X must contain only positive values."); endif ## Check alpha if (nargin < 2 || isempty (alpha)) alpha = 0.05; else if (! isscalar (alpha) || ! isreal (alpha) || alpha <= 0 || alpha >= 1) error ("burrfit: wrong value for ALPHA."); endif endif ## Check censor vector if (nargin < 3 || isempty (censor)) censor = zeros (size (x)); elseif (! isequal (size (x), size (censor))) error ("burrfit: X and CENSOR vectors mismatch."); endif ## Check frequency vector if (nargin < 4 || isempty (freq)) freq = ones (size (x)); elseif (! isequal (size (x), size (freq))) error ("burrfit: X and FREQ vectors mismatch."); endif ## Get options structure or add defaults if (nargin < 5) options.Display = "off"; options.MaxFunEvals = 1000; options.MaxIter = 500; options.TolX = 1e-6; else if (! isstruct (options) || ! isfield (options, "Display") || ! isfield (options, "MaxFunEvals") || ! isfield (options, "MaxIter") || ! isfield (options, "TolX")) error (strcat (["burrfit: 'options' 5th argument must be a"], ... [" structure with 'Display', 'MaxFunEvals',"], ... [" 'MaxIter', and 'TolX' fields present."])); endif endif ## Force censoring vector into logical notc = ! censor; cens = ! notc; ## Check for identical data in X if (! isscalar (x) && max (abs (diff (x)) ./ x(2:end)) <= sqrt (eps)) warning ("burrfit: X must not contain identical data."); ## Return some sensical values for estimated parameters lambda = x(1); c = Inf; k = sum (notc .* freq) / sum (freq) / log (2); paramhat = [lambda, c, k]; if (nargout > 1) paramci = [paramhat; paramhat]; endif return endif ## Fit a Pareto distribution [paramhat_prt, nlogL_prt] = prtfit (x, cens, freq); ## Fit a Weibull distribution paramhat_wbl = wblfit (x, alpha, cens, freq); nlogL_wbl = wbllike (paramhat_wbl, x, cens, freq); ## Calculate the discriminator x_lambda = x ./ paramhat_wbl(1); x_lambdk = x_lambda .^ paramhat_wbl(2); discrimi = sum (freq .* (0.5 * x_lambdk .^ 2 - x_lambdk .* notc)); ## Compute Burr distribution if (discrimi > 0) ## Expand data (if necessary) if (any (freq != 1)) ## Preserve class x_expand = zeros (1, sum (freq), class (x)); id0 = 1; for idx = 1:numel (x) x_expand(id0:id0 + freq(idx) - 1) = x_lambda(idx); id0 += freq(idx); endfor else x_expand = x_lambda; endif ## Calculate median and 3rd quartile to estimate LAMBDA and C parameters Q = prctile (x_expand); xl_median = Q(3); xl_upperq = Q(4); ## Avoid median and upper quartile being too close together IRDdist = sqrt (eps (xl_median)) * xl_median; if ((xl_upperq - xl_median) < IRDdist) if (any (x_lambda > xl_upperq)) xl_upperq = min (x_lambda(x_lambda > xl_upperq)); elseif (any (x_lambda < xl_upperq)) xl_median = max (x_lambda(x_lambda < xl_median)); endif endif ## Compute starting LAMBDA and C, either directly or by minimization if (xl_median >= xl_upperq / xl_median) l0 = xl_median; c0 = log(3)/log(xl_upperq/xl_median); else l0 = 1; opts = optimset ("fzero"); opts = optimset (opts, "Display", "off"); cmax = log(realmax)/(2*log(xl_upperq/xl_median)); c0 = fzero (@(c)(xl_upperq/xl_median).^c-xl_median.^c-2, [0, cmax], opts); endif ## Calculate starting K from other starting parameters and scaled data k0 = exp (compute_logk (x_lambda, l0, c0, censor, freq)); ## Estimate parameters by minimizing the negative log-likelihood function f = @(params) burrlike (params, x_lambda, censor, freq); [paramhat, ~, err, output] = fminsearch (f, [l0, c0, k0], options); ## Force positive parameter values paramhat = abs (paramhat); ## Handle errors if (err == 0) if (output.funcCount >= options.MaxFunEvals) warning (strcat (["burrfit: maximum number of function"], ... [" evaluations are exceeded."])); elseif (output.iterations >= options.MaxIter) warning ("burrfit: maximum number of iterations are exceeded."); endif endif ## Scale back LAMBDA parameter paramhat(1) = paramhat(1) * paramhat_wbl(1); ## Compute negative log-likelihood with estimated parameters nlogL_burr = burrlike (paramhat, x, censor, freq); ## Check if fitting a Burr distribution is better than fitting a Pareto ## according to step 5 of the algorithmic implementation in Shao, 2004 if (paramhat(3) > 1e-6 && nlogL_burr < nlogL_prt) ## Compute CIs using a log normal approximation for phat. if (nargout > 1) ## Compute asymptotic covariance [~, acov] = burrlike (paramhat, x, censor, freq); ## Get standard errors stderr = sqrt (diag (acov))'; stderr = stderr ./ paramhat; ## Apply log transform phatlog = log (paramhat); ## Compute normal quantiles z = norminv (alpha / 2); ## Compute CI paramci = [phatlog; phatlog] + ... [stderr; stderr] .* [z, z, z; -z, -z, -z]; ## Inverse log transform paramci = exp (paramci); endif else if (nlogL_prt < nlogL_wbl) error ("burrfit: Pareto distribution fits better in X."); else error ("burrfit: Weibull distribution fits better in X."); endif endif else if (nlogL_prt < nlogL_wbl) error ("burrfit: Pareto distribution fits better in X."); else error ("burrfit: Weibull distribution fits better in X."); endif endif endfunction ## Helper function for fitting a Pareto distribution function [paramhat, nlogL] = prtfit (x, censor, freq) ## Force censoring vector into logical notc = ! censor; cens = ! notc; ## Compute MLE for x_m xm = x(notc); xm = min (xm); ## Handle case with all data censored if (all (cens)) paramhat = [max(x), NaN]; nlogL_prt = 0; return endif ## Compute some values logx = log (x); suml = sum (freq .* (logx - log (xm)) .* (x > xm)); sumf = sum (freq .* notc); ## Compute MLE for alpha a = sumf ./ suml; ## Add MLEs to returning vector paramhat = [xm, a]; ## Compute negative log-likelihood nlogL = a .* suml + sum (freq .* notc .* logx) - log (a) .* sumf; endfunction ## Helper function for computing K from X, LAMBDA, and C function logk = compute_logk (x, lambda, c, censor, freq) ## Force censoring vector into logical notc = ! censor; cens = ! notc; ## Precalculate some values xl = x ./ lambda; l1_xlc = log1p (xl .^ c); ## Avoid realmax overflow by approximation is_inf = isinf (l1_xlc); l1_xlc(is_inf) = c .* log (xl(is_inf)); if (sum (freq .* l1_xlc) < eps) lsxc = log (sum (freq .* (x .^ c))); if (isinf (lsxc)) [maxx, idx] = max (x); lsxc = c * log (freq(idx) * maxx); endif logk = log (sum (freq .* notc)) + c * log (lambda) - lsxc; else logk = log (sum (freq .* notc)) - log (sum (freq .* l1_xlc)); endif endfunction %!demo %! ## Sample 3 populations from different Burr type XII distibutions %! rand ("seed", 4); # for reproducibility %! r1 = burrrnd (3.5, 2, 2.5, 10000, 1); %! rand ("seed", 2); # for reproducibility %! r2 = burrrnd (1, 3, 1, 10000, 1); %! rand ("seed", 9); # for reproducibility %! r3 = burrrnd (0.5, 2, 3, 10000, 1); %! r = [r1, r2, r3]; %! %! ## Plot them normalized and fix their colors %! hist (r, [0.1:0.2:20], [18, 5, 3]); %! h = findobj (gca, "Type", "patch"); %! set (h(1), "facecolor", "c"); %! set (h(2), "facecolor", "g"); %! set (h(3), "facecolor", "r"); %! ylim ([0, 3]); %! xlim ([0, 5]); %! hold on %! %! ## Estimate their α and β parameters %! lambda_c_kA = burrfit (r(:,1)); %! lambda_c_kB = burrfit (r(:,2)); %! lambda_c_kC = burrfit (r(:,3)); %! %! ## Plot their estimated PDFs %! x = [0.01:0.15:15]; %! y = burrpdf (x, lambda_c_kA(1), lambda_c_kA(2), lambda_c_kA(3)); %! plot (x, y, "-pr"); %! y = burrpdf (x, lambda_c_kB(1), lambda_c_kB(2), lambda_c_kB(3)); %! plot (x, y, "-sg"); %! y = burrpdf (x, lambda_c_kC(1), lambda_c_kC(2), lambda_c_kC(3)); %! plot (x, y, "-^c"); %! hold off %! legend ({"Normalized HIST of sample 1 with λ=3.5, c=2, and k=2.5", ... %! "Normalized HIST of sample 2 with λ=1, c=3, and k=1", ... %! "Normalized HIST of sample 3 with λ=0.5, c=2, and k=3", ... %! sprintf("PDF for sample 1 with estimated λ=%0.2f, c=%0.2f, and k=%0.2f", ... %! lambda_c_kA(1), lambda_c_kA(2), lambda_c_kA(3)), ... %! sprintf("PDF for sample 2 with estimated λ=%0.2f, c=%0.2f, and k=%0.2f", ... %! lambda_c_kB(1), lambda_c_kB(2), lambda_c_kB(3)), ... %! sprintf("PDF for sample 3 with estimated λ=%0.2f, c=%0.2f, and k=%0.2f", ... %! lambda_c_kC(1), lambda_c_kC(2), lambda_c_kC(3))}) %! title ("Three population samples from different Burr type XII distibutions") %! hold off ## Test output %!test %! l = 1; c = 2; k = 3; %! r = burrrnd (l, c, k, 100000, 1); %! lambda_c_kA = burrfit (r); %! assert (lambda_c_kA(1), l, 0.2); %! assert (lambda_c_kA(2), c, 0.2); %! assert (lambda_c_kA(3), k, 0.3); %!test %! l = 0.5; c = 1; k = 3; %! r = burrrnd (l, c, k, 100000, 1); %! lambda_c_kA = burrfit (r); %! assert (lambda_c_kA(1), l, 0.2); %! assert (lambda_c_kA(2), c, 0.2); %! assert (lambda_c_kA(3), k, 0.3); %!test %! l = 1; c = 3; k = 1; %! r = burrrnd (l, c, k, 100000, 1); %! lambda_c_kA = burrfit (r); %! assert (lambda_c_kA(1), l, 0.2); %! assert (lambda_c_kA(2), c, 0.2); %! assert (lambda_c_kA(3), k, 0.3); %!test %! l = 3; c = 2; k = 1; %! r = burrrnd (l, c, k, 100000, 1); %! lambda_c_kA = burrfit (r); %! assert (lambda_c_kA(1), l, 0.2); %! assert (lambda_c_kA(2), c, 0.2); %! assert (lambda_c_kA(3), k, 0.3); %!test %! l = 4; c = 2; k = 4; %! r = burrrnd (l, c, k, 100000, 1); %! lambda_c_kA = burrfit (r); %! assert (lambda_c_kA(1), l, 0.2); %! assert (lambda_c_kA(2), c, 0.2); %! assert (lambda_c_kA(3), k, 0.3); ## Test input validation %!error burrfit (ones (2,5)); %!error burrfit ([-1 2 3 4]); %!error burrfit ([1, 2, 3, 4, 5], 1.2); %!error burrfit ([1, 2, 3, 4, 5], 0); %!error burrfit ([1, 2, 3, 4, 5], "alpha"); %!error ... %! burrfit ([1, 2, 3, 4, 5], 0.05, [1 1 0]); %!error ... %! burrfit ([1, 2, 3, 4, 5], [], [1 1 0 1 1]'); %!error ... %! burrfit ([1, 2, 3, 4, 5], 0.05, zeros (1,5), [1 1 0]); %!error ... %! burrfit ([1, 2, 3, 4, 5], [], [], [1 1 0 1 1]'); %!error ... %! burrfit ([1, 2, 3, 4, 5], 0.05, [], [], 2); statistics-release-1.6.3/inst/dist_fit/burrlike.m000066400000000000000000000153731456127120000221170ustar00rootroot00000000000000## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{nlogL} =} burrlike (@var{params}, @var{x}) ## @deftypefnx {statistics} {[@var{nlogL}, @var{acov}] =} burrlike (@var{params}, @var{x}) ## @deftypefnx {statistics} {[@dots{}] =} burrlike (@var{params}, @var{x}, @var{censor}) ## @deftypefnx {statistics} {[@dots{}] =} burrlike (@var{params}, @var{x}, @var{censor}, @var{freq}) ## ## Negative log-likelihood for the Burr type XII distribution. ## ## @code{@var{nlogL} = burrlike (@var{params}, @var{x})} returns the negative ## log likelihood of the data in @var{x} corresponding to the Burr type XII ## distribution with (1) scale parameter @var{lambda}, (2) first shape parameter ## @var{c}, and (3) second shape parameter @var{k} given in the three-element ## vector @var{params}. ## ## @code{[@var{nlogL}, @var{acov}] = burrlike (@var{params}, @var{x})} also ## returns the inverse of Fisher's information matrix, @var{acov}. If the input ## parameter values in @var{params} are the maximum likelihood estimates, the ## diagonal elements of @var{acov} are their asymptotic variances. ## ## @code{[@dots{}] = burrlike (@var{params}, @var{x}, @var{censor})} accepts a ## boolean vector, @var{censor}, of the same size as @var{x} with @qcode{1}s for ## observations that are right-censored and @qcode{0}s for observations that are ## observed exactly. By default, or if left empty, ## @qcode{@var{censor} = zeros (size (@var{x}))}. ## ## @code{[@dots{}] = burrlike (@var{params}, @var{x}, @var{censor}, @var{freq})} ## accepts a frequency vector, @var{freq}, of the same size as @var{x}. ## @var{freq} typically contains integer frequencies for the corresponding ## elements in @var{x}, but it can contain any non-integer non-negative values. ## By default, or if left empty, @qcode{@var{freq} = ones (size (@var{x}))}. ## ## Further information about the Burr type XII distribution can be found at ## @url{https://en.wikipedia.org/wiki/Burr_distribution} ## ## @seealso{burrcdf, burrinv, burrpdf, burrrnd, burrfit, burrstat} ## @end deftypefn function [nlogL, acov] = burrlike (params, x, censor, freq) ## Check input arguments if (nargin < 2) error ("burrlike: function called with too few input arguments."); endif if (! isvector (x)) error ("burrlike: X must be a vector."); endif if (any (x < 0)) error ("burrlike: X cannot have negative values."); endif if (length (params) != 3) error ("burrlike: PARAMS must be a three-element vector."); endif ## Check censor vector if (nargin < 3 || isempty (censor)) censor = zeros (size (x)); elseif (! isequal (size (x), size (censor))) error ("burrlike: X and CENSOR vector mismatch."); endif ## Check frequency vector if (nargin < 4 || isempty (freq)) freq = ones (size (x)); elseif (! isequal (size (x), size (freq))) error ("burrlike: X and FREQ vector mismatch."); endif ## Get parameters lambda = params(1); c = params(2); k = params(3); ## Precalculate some values xl = x ./ lambda; log_xl = log (xl); l1_xlc = log1p (xl .^ c); ## Avoid realmax overflow by approximation is_inf = isinf (l1_xlc); l1_xlc(is_inf) = c .* log (xl(is_inf)); ## Force censoring vector into logical notc = ! censor; cens = ! notc; ## Compute neg-loglikelihood likeL = zeros (size (x)); likeL(notc) = (c - 1) .* log_xl(notc) - (k + 1) .* l1_xlc(notc); likeL(cens) = -k .* l1_xlc(cens); nlogL = sum (freq (notc)) * log (lambda / k / c) - sum (freq .* likeL); ## Compute asymptotic covariance if (nargout > 1) ## Preallocate variables nH = zeros (3); d2V1 = zeros (size (x)); d2V2 = d2V1; ## Precalculate some more values xlc = xl .^ c; log_xl = log (xl); xlc1 = (1 + xlc); xlc1sq = xlc1.^2; invxlc1sq = (1 + 1./xlc).^2; ## Find realmax overflow is_inf = isinf (xlc); is_fin = ! is_inf; ## Compute each element of the negative Hessian d2V1(is_fin) = -((1 + c) ./ xlc(is_fin) + 1) ./ invxlc1sq(is_fin); d2V1(is_inf) = -1; d2V2(notc) = d2V1(notc) .* (k + 1) + 1; d2V2(cens) = d2V1(cens) .* k; nH(1,1) = c ./ lambda .^ 2 .* sum (freq .* d2V2); d2V1(is_fin) = xlc(is_fin) .* (c .* log_xl(is_fin) + xlc1(is_fin)) ... ./ xlc1sq(is_fin); d2V1(is_inf) = 1; d2V2(notc) = (k + 1) .* d2V1(notc) - 1; d2V2(cens) = k .* d2V1(cens); nH(1,2) = sum (freq ./ lambda .* d2V2); nH(2,1) = nH(1,2); d2V1(is_fin) = xlc(is_fin) .* log_xl(is_fin) .^ 2 ./ xlc1sq(is_fin); d2V1(is_inf) = 0; d2V2(notc) = d2V1(notc) .* k + d2V1(notc); d2V2(cens) = d2V1(cens) .* k; nH(2,2) = -(sum (freq (notc))) ./ c .^ 2 - sum (freq .* d2V2); d2V1(is_fin) = xlc(is_fin) ./ xlc1(is_fin); d2V1(is_inf) = 1; nH(1,3) = (c ./ lambda) .* sum (freq .* d2V1); nH(3,1) = nH(1,3); nH(2,3) = -sum (freq .* d2V1 .* log_xl); nH(3,2) = nH(2,3); nH(3,3) = -(sum (freq (notc))) ./ k .^ 2; nH = -nH; ## Check negative Hessian is positive definite [R, p] = chol (nH); if (p > 0) warning ("burrlike: non positive definite Hessian matrix."); acov = NaN (3); return endif ## ACOV estimate is the negative inverse of the Hessian. Rinv = inv (R); acov = Rinv * Rinv; endif endfunction ## Test output ## Test input validation %!error burrlike (3.25) %!error burrlike ([1, 2, 3], ones (2)) %!error burrlike ([1, 2, 3], [-1, 3]) %!error ... %! burrlike ([1, 2], [1, 3, 5, 7]) %!error ... %! burrlike ([1, 2, 3, 4], [1, 3, 5, 7]) %!error ... %! burrlike ([1, 2, 3], [1:5], [0, 0, 0]) %!error ... %! burrlike ([1, 2, 3], [1:5], [0, 0, 0, 0, 0], [1, 1, 1]) %!error ... %! burrlike ([1, 2, 3], [1:5], [], [1, 1, 1]) statistics-release-1.6.3/inst/dist_fit/evfit.m000066400000000000000000000326421456127120000214130ustar00rootroot00000000000000## Copyright (C) 2022-2023 Andreas Bertsatos ## Copyright (C) 2022 Andrew Penn ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{paramhat} =} evfit (@var{x}) ## @deftypefnx {statistics} {[@var{paramhat}, @var{paramci}] =} evfit (@var{x}) ## @deftypefnx {statistics} {[@var{paramhat}, @var{paramci}] =} evfit (@var{x}, @var{alpha}) ## @deftypefnx {statistics} {[@dots{}] =} evfit (@var{x}, @var{alpha}, @var{censor}) ## @deftypefnx {statistics} {[@dots{}] =} evfit (@var{x}, @var{alpha}, @var{censor}, @var{freq}) ## @deftypefnx {statistics} {[@dots{}] =} evfit (@var{x}, @var{alpha}, @var{censor}, @var{freq}, @var{options}) ## ## Estimate parameters and confidence intervals for the extreme value distribution. ## ## @code{@var{paramhat} = evfit (@var{x})} returns the maximum likelihood ## estimates of the parameters of the extreme value distribution (also known as ## the Gumbel or the type I generalized extreme value distribution) given the ## data in @var{x}. @qcode{@var{paramhat}(1)} is the location parameter, ## @var{mu}, and @qcode{@var{paramhat}(2)} is the scale parameter, @var{sigma}. ## ## @code{[@var{paramhat}, @var{paramci}] = evfit (@var{x})} returns the 95% ## confidence intervals for the parameter estimates. ## ## @code{[@dots{}] = evfit (@var{x}, @var{alpha})} also returns the ## @qcode{100 * (1 - @var{alpha})} percent confidence intervals for the ## parameter estimates. By default, the optional argument @var{alpha} is ## 0.05 corresponding to 95% confidence intervals. Pass in @qcode{[]} for ## @var{alpha} to use the default values. ## ## @code{[@dots{}] = evfit (@var{x}, @var{alpha}, @var{censor})} accepts a ## boolean vector, @var{censor}, of the same size as @var{x} with @qcode{1}s for ## observations that are right-censored and @qcode{0}s for observations that are ## observed exactly. By default, or if left empty, ## @qcode{@var{censor} = zeros (size (@var{x}))}. ## ## @code{[@dots{}] = evfit (@var{x}, @var{alpha}, @var{censor}, @var{freq})} ## accepts a frequency vector, @var{freq}, of the same size as @var{x}. ## @var{freq} typically contains integer frequencies for the corresponding ## elements in @var{x}, but it can contain any non-integer non-negative values. ## By default, or if left empty, @qcode{@var{freq} = ones (size (@var{x}))}. ## ## @code{[@dots{}] = evfit (@dots{}, @var{options})} specifies control ## parameters for the iterative algorithm used to compute the maximum likelihood ## estimates. @var{options} is a structure with the following field and its ## default value: ## @itemize ## @item @qcode{@var{options}.TolX = 1e-6} ## @end itemize ## ## The Gumbel distribution is used to model the distribution of the maximum (or ## the minimum) of a number of samples of various distributions. This version ## is suitable for modeling minima. For modeling maxima, use the alternative ## Gumbel fitting function, @code{gumbelfit}. ## ## Further information about the Gumbel distribution can be found at ## @url{https://en.wikipedia.org/wiki/Gumbel_distribution} ## ## @seealso{evcdf, evinv, evpdf, evrnd, evlike, evstat, gumbelfit} ## @end deftypefn function [paramhat, paramci] = evfit (x, alpha, censor, freq, options) ## Check X for being a double precision vector if (! isvector (x) || ! isa (x, "double")) error ("evfit: X must be a double-precision vector."); endif ## Check that X does not contain missing values (NaNs) if (any (isnan (x))) error ("evfit: X must NOT contain missing values (NaNs)."); endif ## Check alpha if (nargin > 1) if (! isscalar (alpha) || ! isreal (alpha) || alpha <= 0 || alpha >= 1) error ("evfit: wrong value for ALPHA."); endif else alpha = 0.05; endif ## Check censor vector if (nargin > 2) if (! isempty (censor) && ! all (size (censor) == size (x))) error ("evfit: X and CENSOR vectors mismatch."); endif else censor = zeros (size (x)); endif ## Check frequency vector if (nargin > 3) if (! isempty (freq) && ! all (size (freq) == size (x))) error ("evfit: X and FREQ vectors mismatch."); endif ## Remove elements with zero frequency (if applicable) rm = find (freq == 0); if (length (rm) > 0) x(rm) = []; censor(rm) = []; freq(rm) = []; endif else freq = ones (size (x)); endif ## Get options structure or add defaults if (nargin > 4) if (! isstruct (options) || ! isfield (options, "TolX")) error (strcat (["evfit: 'options' 5th argument must be a structure"], ... [" with 'TolX' field present."])); endif else options.TolX = 1e-6; endif ## If X is a column vector, make X, CENSOR, and FREQ row vectors if (size (x, 1) > 1) x = x(:)'; censor = censor(:)'; freq = freq(:)'; endif ## Censor x and get number of samples sample_size = sum (freq); censored_sample_size = sum (freq .* censor); uncensored_sample_size = sample_size - censored_sample_size; x_range = range (x); x_max = max (x); ## Check cases that cannot make a fit. ## 1. All observations are censored if (sample_size == 0 || uncensored_sample_size == 0 || ! isfinite (x_range)) paramhat = NaN (1, 2); paramci = NaN (2, 2); return endif ## 2. Constant x in X if (censored_sample_size == 0 && x_range == 0) paramhat = [x(1), 0]; if (sample_size == 1) paramci = [-Inf, 0; Inf, Inf]; else paramci = [paramhat, paramhat]; endif return elseif (censored_sample_size == 0 && x_range != 0) ## Data can fit, so preprocess them to make likelihood eqn more stable. ## Shift x to max(x) == 0, min(x) = -1. x_0 = (x - x_max) ./ x_range; ## Get a rough initial estimate for scale parameter initial_sigma_parm = (sqrt (6) * std (x_0)) / pi; uncensored_weights = sum (freq .* x_0) ./ sample_size; endif ## 3. All uncensored observations are equal and greater than all censored ones uncensored_x_range = range (x(censor == 0)); uncensored_x = x(censor == 0); if (censored_sample_size > 0 && uncensored_x_range == 0 ... && uncensored_x(1) >= x_max) paramhat = [uncensored_x(1), 0]; if uncensored_sample_size == 1 paramci = [-Inf, 0; Inf, Inf]; else paramci = [paramhat; paramhat]; end return else ## Data can fit, so preprocess them to make likelihood eqn more stable. ## Shift x to max(x) == 0, min(x) = -1. x_0 = (x - x_max) ./ x_range; ## Get a rough initial estimate for scale parameter if (uncensored_x_range > 0) [F_y, y] = ecdf (x_0, "censoring", censor', "frequency", freq'); pmid = (F_y(1:(end-1)) + F_y(2:end)) / 2; linefit = polyfit (log (- log (1 - pmid)), y(2:end), 1); initial_sigma_parm = linefit(1); else initial_sigma_parm = 1; endif uncensored_weights = sum (freq .* x_0 .* (1 - censor)) ./ ... uncensored_sample_size; endif ## Find lower and upper boundaries for bracketing the likelihood equation for ## the extreme value scale parameter if (evscale_lkeq (initial_sigma_parm, x_0, freq, uncensored_weights) > 0) upper = initial_sigma_parm; lower = 0.5 * upper; while (evscale_lkeq (lower, x_0, freq, uncensored_weights) > 0) upper = lower; lower = 0.5 * upper; if (lower <= realmin ("double")) error ("evfit: no solution for maximum likelihood estimates."); endif endwhile boundaries = [lower, upper]; else lower = initial_sigma_parm; upper = 2 * lower; while (evscale_lkeq (upper, x_0, freq, uncensored_weights) < 0) lower = upper; upper = 2 * lower; if (upper > realmax ("double")) error ("evfit: no solution for maximum likelihood estimates."); endif endwhile boundaries = [lower, upper]; endif ## Compute maximum likelihood for scale parameter as the root of the equation ## Custom code for finding the value within the boundaries [lower, upper] that ## evscale_lkeq function returns zero ## First check that there is a root within the boundaries new_lo = boundaries(1); new_up = boundaries(2); v_lower = evscale_lkeq (new_lo, x_0, freq, uncensored_weights); v_upper = evscale_lkeq (new_up, x_0, freq, uncensored_weights); if (! (sign (v_lower) * sign (v_upper) <= 0)) error ("evfit: no solution for maximum likelihood estimates."); endif ## Get a value at mid boundary range old_sigma = new_lo; new_sigma = (new_lo + new_up) / 2; new_fzero = evscale_lkeq (new_sigma, x_0, freq, uncensored_weights); ## Start searching cur_iter = 0; max_iter = 1e+3; while (cur_iter < max_iter && abs (old_sigma - new_sigma) > options.TolX) cur_iter++; if (new_fzero < 0) old_sigma = new_sigma; new_lo = new_sigma; new_sigma = (new_lo + new_up) / 2; new_fzero = evscale_lkeq (new_sigma, x_0, freq, uncensored_weights); else old_sigma = new_sigma; new_up = new_sigma; new_sigma = (new_lo + new_up) / 2; new_fzero = evscale_lkeq (new_sigma, x_0, freq, uncensored_weights); endif endwhile ## Check for maximum number of iterations if (cur_iter == max_iter) warning (strcat (["evfit: maximum number of function "], ... [" evaluations (1e+4) has been reached."])); endif ## Compute MU muhat = new_sigma .* log (sum (freq .* exp (x_0 ./ new_sigma)) ./ ... uncensored_sample_size); ## Transform MU and SIGMA back to original location and scale paramhat = [(x_range*muhat)+x_max, x_range*new_sigma]; ## Compute the CI for MU and SIGMA if (nargout == 2) probs = [alpha/2; 1-alpha/2]; [~, acov] = evlike (paramhat, x, censor, freq); transfhat = [paramhat(1), log(paramhat(2))]; se = sqrt (diag (acov))'; se(2) = se(2) ./ paramhat(2); paramci = norminv ([probs, probs], [transfhat; transfhat], [se; se]); paramci(:,2) = exp (paramci(:,2)); endif endfunction ## Likelihood equation for the extreme value scale parameter. function v = evscale_lkeq (sigma, x, freq, x_weighted_uncensored) freq = freq .* exp (x ./ sigma); v = sigma + x_weighted_uncensored - sum (x .* freq) / sum (freq); endfunction %!demo %! ## Sample 3 populations from different extreme value distibutions %! rand ("seed", 1); # for reproducibility %! r1 = evrnd (2, 5, 400, 1); %! rand ("seed", 12); # for reproducibility %! r2 = evrnd (-5, 3, 400, 1); %! rand ("seed", 13); # for reproducibility %! r3 = evrnd (14, 8, 400, 1); %! r = [r1, r2, r3]; %! %! ## Plot them normalized and fix their colors %! hist (r, 25, 0.4); %! h = findobj (gca, "Type", "patch"); %! set (h(1), "facecolor", "c"); %! set (h(2), "facecolor", "g"); %! set (h(3), "facecolor", "r"); %! ylim ([0, 0.28]) %! xlim ([-30, 30]); %! hold on %! %! ## Estimate their MU and SIGMA parameters %! mu_sigmaA = evfit (r(:,1)); %! mu_sigmaB = evfit (r(:,2)); %! mu_sigmaC = evfit (r(:,3)); %! %! ## Plot their estimated PDFs %! x = [min(r(:)):max(r(:))]; %! y = evpdf (x, mu_sigmaA(1), mu_sigmaA(2)); %! plot (x, y, "-pr"); %! y = evpdf (x, mu_sigmaB(1), mu_sigmaB(2)); %! plot (x, y, "-sg"); %! y = evpdf (x, mu_sigmaC(1), mu_sigmaC(2)); %! plot (x, y, "-^c"); %! legend ({"Normalized HIST of sample 1 with μ=2 and σ=5", ... %! "Normalized HIST of sample 2 with μ=-5 and σ=3", ... %! "Normalized HIST of sample 3 with μ=14 and σ=8", ... %! sprintf("PDF for sample 1 with estimated μ=%0.2f and σ=%0.2f", ... %! mu_sigmaA(1), mu_sigmaA(2)), ... %! sprintf("PDF for sample 2 with estimated μ=%0.2f and σ=%0.2f", ... %! mu_sigmaB(1), mu_sigmaB(2)), ... %! sprintf("PDF for sample 3 with estimated μ=%0.2f and σ=%0.2f", ... %! mu_sigmaC(1), mu_sigmaC(2))}) %! title ("Three population samples from different extreme value distibutions") %! hold off ## Test output %!test %! x = 1:50; %! [paramhat, paramci] = evfit (x); %! paramhat_out = [32.6811, 13.0509]; %! paramci_out = [28.8504, 10.5294; 36.5118, 16.1763]; %! assert (paramhat, paramhat_out, 1e-4); %! assert (paramci, paramci_out, 1e-4); %!test %! x = 1:50; %! [paramhat, paramci] = evfit (x, 0.01); %! paramci_out = [27.6468, 9.8426; 37.7155, 17.3051]; %! assert (paramci, paramci_out, 1e-4); ## Test input validation %!error evfit (ones (2,5)); %!error evfit (single (ones (1,5))); %!error evfit ([1, 2, 3, 4, NaN]); %!error evfit ([1, 2, 3, 4, 5], 1.2); %!error ... %! evfit ([1, 2, 3, 4, 5], 0.05, [1 1 0]); %!error ... %! evfit ([1, 2, 3, 4, 5], 0.05, [], [1 1 0]); %!error evfit ([1, 2, 3, 4, 5], 0.05, [], [], 2); statistics-release-1.6.3/inst/dist_fit/evlike.m000066400000000000000000000135761456127120000215620ustar00rootroot00000000000000## Copyright (C) 2022-2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{nlogL} =} evlike (@var{params}, @var{x}) ## @deftypefnx {statistics} {[@var{nlogL}, @var{acov}] =} evlike (@var{params}, @var{x}) ## @deftypefnx {statistics} {[@dots{}] =} evlike (@var{params}, @var{x}, @var{censor}) ## @deftypefnx {statistics} {[@dots{}] =} evlike (@var{params}, @var{x}, @var{censor}, @var{freq}) ## ## Negative log-likelihood for the extreme value distribution. ## ## @code{@var{nlogL} = evlike (@var{params}, @var{x})} returns the negative ## log likelihood of the data in @var{x} corresponding to the extreme value ## distribution (also known as the Gumbel or the type I generalized extreme ## value distribution) with (1) location parameter @var{mu} and (2) scale ## parameter @var{sigma} given in the two-element vector @var{params}. ## ## @code{[@var{nlogL}, @var{acov}] = evlike (@var{params}, @var{x})} also ## returns the inverse of Fisher's information matrix, @var{acov}. If the input ## parameter values in @var{params} are the maximum likelihood estimates, the ## diagonal elements of @var{acov} are their asymptotic variances. ## ## @code{[@dots{}] = evlike (@var{params}, @var{x}, @var{censor})} accepts a ## boolean vector, @var{censor}, of the same size as @var{x} with @qcode{1}s for ## observations that are right-censored and @qcode{0}s for observations that are ## observed exactly. By default, or if left empty, ## @qcode{@var{censor} = zeros (size (@var{x}))}. ## ## @code{[@dots{}] = evlike (@var{params}, @var{x}, @var{censor}, @var{freq})} ## accepts a frequency vector, @var{freq}, of the same size as @var{x}. ## @var{freq} typically contains integer frequencies for the corresponding ## elements in @var{x}, but it can contain any non-integer non-negative values. ## By default, or if left empty, @qcode{@var{freq} = ones (size (@var{x}))}. ## ## The Gumbel distribution is used to model the distribution of the maximum (or ## the minimum) of a number of samples of various distributions. This version ## is suitable for modeling minima. For modeling maxima, use the alternative ## Gumbel likelihood function, @code{gumbellike}. ## ## Further information about the Gumbel distribution can be found at ## @url{https://en.wikipedia.org/wiki/Gumbel_distribution} ## ## @seealso{evcdf, evinv, evpdf, evrnd, evfit, evstat, gumbellike} ## @end deftypefn function [nlogL, acov] = evlike (params, x, censor, freq) ## Check input arguments and add defaults if (nargin < 2) error ("evlike: function called with too few input arguments."); endif if (numel (params) != 2) error ("evlike: wrong parameters length."); endif if (! isvector (x)) error ("evlike: X must be a vector."); endif if (nargin < 3 || isempty (censor)) censor = zeros (size (x)); elseif (! isequal (size (x), size (censor))) error ("evlike: X and CENSOR vectors mismatch."); endif if (nargin < 4 || isempty (freq)) freq = ones (size (x)); elseif (isequal (size (x), size (freq))) nulls = find (freq == 0); if (numel (nulls) > 0) x(nulls) = []; censor(nulls) = []; freq(nulls) = []; endif else error ("evlike: X and FREQ vectors mismatch."); endif ## Get mu and sigma values mu = params(1); sigma = params(2); ## sigma must be positive, otherwise make it NaN if (sigma <= 0) sigma = NaN; endif ## Compute the individual log-likelihood terms z = (x - mu) ./ sigma; expz = exp (z); L = (z - log (sigma)) .* (1 - censor) - expz; ## Force a log(0)==-Inf for X from extreme right tail L(z == Inf) = -Inf; ## Neg-log-likelihood is the sum of the individual contributions nlogL = -sum (freq .* L); ## Compute the negative hessian at the parameter values. ## Invert to get the observed information matrix. if (nargout == 2) unc = (1 - censor); nH11 = sum(freq .* expz); nH12 = sum(freq .* ((z + 1) .* expz - unc)); nH22 = sum(freq .* (z .* (z+2) .* expz - ((2 .* z + 1) .* unc))); acov = (sigma .^ 2) * ... [nH22 -nH12; -nH12 nH11] / (nH11 * nH22 - nH12 * nH12); endif endfunction ## Test output %!test %! x = 1:50; %! [nlogL, acov] = evlike ([2.3, 1.2], x); %! avar_out = [-1.2778e-13, 3.1859e-15; 3.1859e-15, -7.9430e-17]; %! assert (nlogL, 3.242264755689906e+17, 1e-14); %! assert (acov, avar_out, 1e-3); %!test %! x = 1:50; %! [nlogL, acov] = evlike ([2.3, 1.2], x * 0.5); %! avar_out = [-7.6094e-05, 3.9819e-06; 3.9819e-06, -2.0836e-07]; %! assert (nlogL, 481898704.0472211, 1e-6); %! assert (acov, avar_out, 1e-3); %!test %! x = 1:50; %! [nlogL, acov] = evlike ([21, 15], x); %! avar_out = [11.73913876598908, -5.9546128523121216; ... %! -5.954612852312121, 3.708060045170236]; %! assert (nlogL, 223.7612479380652, 1e-13); %! assert (acov, avar_out, 1e-14); ## Test input validation %!error evlike ([12, 15]) %!error evlike ([12, 15, 3], [1:50]) %!error evlike ([12, 3], ones (10, 2)) %!error ... %! evlike ([12, 15], [1:50], [1, 2, 3]) %!error ... %! evlike ([12, 15], [1:50], [], [1, 2, 3]) statistics-release-1.6.3/inst/dist_fit/expfit.m000066400000000000000000000315151456127120000215730ustar00rootroot00000000000000## Copyright (C) 2021 Nicholas R. Jankowski ## Copyright (C) 2023 Andreas Bertsatos ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{muhat} =} expfit (@var{x}) ## @deftypefnx {statistics} {[@var{muhat}, @var{muci}] =} expfit (@var{x}) ## @deftypefnx {statistics} {[@var{muhat}, @var{muci}] =} expfit (@var{x}, @var{alpha}) ## @deftypefnx {statistics} {[@dots{}] =} expfit (@var{x}, @var{alpha}, @var{censor}) ## @deftypefnx {statistics} {[@dots{}] =} expfit (@var{x}, @var{alpha}, @var{censor}, @var{freq}) ## ## Estimate mean and confidence intervals for the exponential distribution. ## ## @code{@var{muhat} = expfit (@var{x})} returns the maximum likelihood estimate ## of the mean parameter, @var{muhat}, of the exponential distribution given the ## data in @var{x}. @var{x} is expected to be a non-negative vector. If @var{x} ## is an array, the mean will be computed for each column of @var{x}. If any ## elements of @var{x} are NaN, that vector's mean will be returned as NaN. ## ## @code{[@var{muhat}, @var{muci}] = expfit (@var{x})} returns the 95% ## confidence intervals for the parameter estimate. If @var{x} is a vector, ## @var{muci} is a two element column vector. If @var{x} is an array, each ## column of data will have a confidence interval returned as a two-row array. ## ## @code{[@dots{}] = evfit (@var{x}, @var{alpha})} also returns the ## @qcode{100 * (1 - @var{alpha})} percent confidence intervals for the ## parameter estimates. By default, the optional argument @var{alpha} is ## 0.05 corresponding to 95% confidence intervals. Pass in @qcode{[]} for ## @var{alpha} to use the default values. Any invalid values for @var{alpha} ## will return NaN for both CI bounds. ## ## @code{[@dots{}] = expfit (@var{x}, @var{alpha}, @var{censor})} accepts a ## logical or numeric array, @var{censor}, of the same size as @var{x} with ## @qcode{1}s for observations that are right-censored and @qcode{0}s for ## observations that are observed exactly. Any non-zero elements are regarded ## as @qcode{1}s. By default, or if left empty, ## @qcode{@var{censor} = zeros (size (@var{x}))}. ## ## @code{[@dots{}] = expfit (@var{x}, @var{alpha}, @var{censor}, @var{freq})} ## accepts a frequency array, @var{freq}, of the same size as @var{x}. ## @var{freq} typically contains integer frequencies for the corresponding ## elements in @var{x}, but it can contain any non-integer non-negative values. ## By default, or if left empty, @qcode{@var{freq} = ones (size (@var{x}))}. ## ## Matlab incompatibility: Matlab's @code{expfit} produces unpredictable results ## for some cases with higher dimensions (specifically 1 x m x n x ... arrays). ## Octave's implementation allows for @math{nxD} arrays, consistently performing ## calculations on individual column vectors. Additionally, @var{censor} and ## @var{freq} can be used with arrays of any size, whereas Matlab only allows ## their use when @var{x} is a vector. ## ## A common alternative parameterization of the exponential distribution is to ## use the parameter @math{λ} defined as the mean number of events in an ## interval as opposed to the parameter @math{μ}, which is the mean wait time ## for an event to occur. @math{λ} and @math{μ} are reciprocals, ## i.e. @math{μ = 1 / λ}. ## ## Further information about the exponential distribution can be found at ## @url{https://en.wikipedia.org/wiki/Exponential_distribution} ## ## @seealso{expcdf, expinv, explpdf, exprnd, explike, expstat} ## @end deftypefn function [muhat, muci] = expfit (x, alpha = 0.05, censor = [], freq = []) ## Check arguments if (nargin == 0 || nargin > 4 || nargout > 2) print_usage (); endif if (! (isnumeric (x) || islogical (x))) x = double (x); endif ## Guarantee working with column vectors if (isvector (x)) x = x(:); endif if (any (x(:) < 0)) error ("expfit: X cannot be negative."); endif sz_s = size (x); if (isempty (alpha)) alpha = 0.05; elseif (! (isscalar (alpha))) error ("expfit: ALPHA must be a scalar quantity."); endif if (isempty (censor) && isempty (freq)) ## Simple case without freq or censor, shortcut other validations muhat = mean (x, 1); if (nargout == 2) X = sum (x, 1); muci = [2*X./chi2inv(1 - alpha / 2, 2 * sz_s(1));... 2*X./chi2inv(alpha / 2, 2 * sz_s(1))]; endif else ## Input validation for censor and freq if (isempty (censor)) ## Expand to full censor with values that don't affect results censor = zeros (sz_s); elseif (! (isnumeric (censor) || islogical (censor))) ## Check for incorrect freq type error ("expfit: CENSOR must be a numeric or logical array.") elseif (isvector (censor)) ## Guarantee working with a column vector censor = censor(:); endif if (isempty (freq)) ## Expand to full censor with values that don't affect results freq = ones (sz_s); elseif (! (isnumeric (freq) || islogical (freq))) ## Check for incorrect freq type error ("expfit: FREQ must be a numeric or logical array.") elseif (isvector (freq)) ## Guarantee working with a column vector freq = freq(:); endif ## Check that size of censor and freq match x if (! (isequal (size (censor), sz_s))) error("expfit: X and CENSOR vectors mismatch."); elseif (! isequal (size (freq), sz_s)) error("expfit: X and FREQ vectors mismatch."); endif ## Trivial case where censor and freq have no effect if (all (censor(:) == 0 & freq(:) == 1)) muhat = mean (x, 1); if (nargout == 2) X = sum (x, 1); muci = [2*X./chi2inv(1 - alpha / 2, 2 * sz_s(1));... 2*X./chi2inv(alpha / 2, 2 * sz_s(1))]; endif ## No censoring, just adjust sample counts for freq elseif (all (censor(:) == 0)) X = sum (x.*freq, 1); n = sum (freq, 1); muhat = X ./ n; if (nargout == 2) muci = [2*X./chi2inv(1 - alpha / 2, 2 * n);... 2*X./chi2inv(alpha / 2, 2 * n)]; endif ## Censoring, but no sample counts adjustment elseif (all (freq(:) == 1)) censor = logical(censor); # convert any numeric censor'x to 0s and 1s X = sum (x, 1); r = sz_s(1) - sum (censor, 1); muhat = X ./ r; if (nargout == 2) muci = [2*X./chi2inv(1 - alpha / 2, 2 * r);... 2*X./chi2inv(alpha / 2, 2 * r)]; endif ## Both censoring and sample count adjustment else censor = logical (censor); # convert any numeric censor'x to 0s and 1s X = sum (x .* freq , 1); r = sum (freq .* (! censor), 1); muhat = X ./ r; if (nargout == 2) muci = [2*X./chi2inv(1 - alpha / 2, 2 * r);... 2*X./chi2inv(alpha / 2, 2 * r)]; endif endif ## compatibility check, NaN for columns where all censor's or freq's remove ## all samples null_columns = all (censor) | ! all (freq); muhat(null_columns) = NaN; if (nargout == 2) muci(:,null_columns) = NaN; endif endif endfunction %!demo %! ## Sample 3 populations from 3 different exponential distibutions %! rande ("seed", 1); # for reproducibility %! r1 = exprnd (2, 4000, 1); %! rande ("seed", 2); # for reproducibility %! r2 = exprnd (5, 4000, 1); %! rande ("seed", 3); # for reproducibility %! r3 = exprnd (12, 4000, 1); %! r = [r1, r2, r3]; %! %! ## Plot them normalized and fix their colors %! hist (r, 48, 0.52); %! h = findobj (gca, "Type", "patch"); %! set (h(1), "facecolor", "c"); %! set (h(2), "facecolor", "g"); %! set (h(3), "facecolor", "r"); %! hold on %! %! ## Estimate their mu parameter %! muhat = expfit (r); %! %! ## Plot their estimated PDFs %! x = [0:max(r(:))]; %! y = exppdf (x, muhat(1)); %! plot (x, y, "-pr"); %! y = exppdf (x, muhat(2)); %! plot (x, y, "-sg"); %! y = exppdf (x, muhat(3)); %! plot (x, y, "-^c"); %! ylim ([0, 0.6]) %! xlim ([0, 40]) %! legend ({"Normalized HIST of sample 1 with μ=2", ... %! "Normalized HIST of sample 2 with μ=5", ... %! "Normalized HIST of sample 3 with μ=12", ... %! sprintf("PDF for sample 1 with estimated μ=%0.2f", muhat(1)), ... %! sprintf("PDF for sample 2 with estimated μ=%0.2f", muhat(2)), ... %! sprintf("PDF for sample 3 with estimated μ=%0.2f", muhat(3))}) %! title ("Three population samples from different exponential distibutions") %! hold off ## Tests for mean %!assert (expfit (1), 1) %!assert (expfit (1:3), 2) %!assert (expfit ([1:3]'), 2) %!assert (expfit (1:3, []), 2) %!assert (expfit (1:3, [], [], []), 2) %!assert (expfit (magic (3)), [5 5 5]) %!assert (expfit (cat (3, magic (3), 2*magic (3))), cat (3,[5 5 5], [10 10 10])) %!assert (expfit (1:3, 0.1, [0 0 0], [1 1 1]), 2) %!assert (expfit ([1:3]', 0.1, [0 0 0]', [1 1 1]'), 2) %!assert (expfit (1:3, 0.1, [0 0 0]', [1 1 1]'), 2) %!assert (expfit (1:3, 0.1, [1 0 0], [1 1 1]), 3) %!assert (expfit (1:3, 0.1, [0 0 0], [4 1 1]), 1.5) %!assert (expfit (1:3, 0.1, [1 0 0], [4 1 1]), 4.5) %!assert (expfit (1:3, 0.1, [1 0 1], [4 1 1]), 9) %!assert (expfit (1:3, 0.1, [], [-1 1 1]), 4) %!assert (expfit (1:3, 0.1, [], [0.5 1 1]), 2.2) %!assert (expfit (1:3, 0.1, [1 1 1]), NaN) %!assert (expfit (1:3, 0.1, [], [0 0 0]), NaN) %!assert (expfit (reshape (1:9, [3 3])), [2 5 8]) %!assert (expfit (reshape (1:9, [3 3]), [], eye(3)), [3 7.5 12]) %!assert (expfit (reshape (1:9, [3 3]), [], 2*eye(3)), [3 7.5 12]) %!assert (expfit (reshape (1:9, [3 3]), [], [], [2 2 2; 1 1 1; 1 1 1]), ... %! [1.75 4.75 7.75]) %!assert (expfit (reshape (1:9, [3 3]), [], [], [2 2 2; 1 1 1; 1 1 1]), ... %! [1.75 4.75 7.75]) %!assert (expfit (reshape (1:9, [3 3]), [], eye(3), [2 2 2; 1 1 1; 1 1 1]), ... %! [3.5 19/3 31/3]) ## Tests for confidence intervals %!assert ([~,muci] = expfit (1:3, 0), [0; Inf]) %!assert ([~,muci] = expfit (1:3, 2), [Inf; 0]) %!assert ([~,muci] = expfit (1:3, 0.1, [1 1 1]), [NaN; NaN]) %!assert ([~,muci] = expfit (1:3, 0.1, [], [0 0 0]), [NaN; NaN]) %!assert ([~,muci] = expfit (1:3, -1), [NaN; NaN]) %!assert ([~,muci] = expfit (1:3, 5), [NaN; NaN]) #!assert ([~,muci] = expfit ([1:3;1:3], -1), NaN(2, 3)] #!assert ([~,muci] = expfit ([1:3;1:3], 5), NaN(2, 3)] %!assert ([~,muci] = expfit (1:3), [0.830485728373393; 9.698190330474096], ... %! 1000*eps) %!assert ([~,muci] = expfit (1:3, 0.1), ... %! [0.953017262058213; 7.337731146400207], 1000*eps) %!assert ([~,muci] = expfit ([1:3;2:4]), ... %! [0.538440777613095, 0.897401296021825, 1.256361814430554; ... %! 12.385982973214016, 20.643304955356694, 28.900626937499371], ... %! 1000*eps) %!assert ([~,muci] = expfit ([1:3;2:4], [], [1 1 1; 0 0 0]), ... %! 100*[0.008132550920455, 0.013554251534091, 0.018975952147727; ... %! 1.184936706156216, 1.974894510260360, 2.764852314364504], ... %! 1000*eps) %!assert ([~,muci] = expfit ([1:3;2:4], [], [], [3 3 3; 1 1 1]), ... %! [0.570302756652583, 1.026544961974649, 1.482787167296715; ... %! 4.587722594914109, 8.257900670845396, 11.928078746776684], ... %! 1000*eps) %!assert ([~,muci] = expfit ([1:3;2:4], [], [0 0 0; 1 1 1], [3 3 3; 1 1 1]), ... %! [0.692071440311161, 1.245728592560089, 1.799385744809018; ... %! 8.081825275395081, 14.547285495711145, 21.012745716027212], ... %! 1000*eps) %!test %! x = reshape (1:8, [4 2]); %! x(4) = NaN; %! [muhat,muci] = expfit (x); %! assert ({muhat, muci}, {[NaN, 6.5], ... %! [NaN, 2.965574334593430;NaN, 23.856157493553368]}, 1000*eps); %!test %! x = magic (3); %! censor = [0 1 0; 0 1 0; 0 1 0]; %! freq = [1 1 0; 1 1 0; 1 1 0]; %! [muhat,muci] = expfit (x, [], censor, freq); %! assert ({muhat, muci}, {[5 NaN NaN], ... %! [[2.076214320933482; 24.245475826185242],NaN(2)]}, 1000*eps); ## Test input validation %!error expfit () %!error expfit (1,2,3,4,5) %!error [a b censor] = expfit (1) %!error expfit (1, [1 2]) %!error expfit ([-1 2 3 4 5]) %!error expfit ([1:5], [], "test") %!error expfit ([1:5], [], [], "test") %!error expfit ([1:5], [], [0 0 0 0]) %!error expfit ([1:5], [], [], [1 1 1 1]) statistics-release-1.6.3/inst/dist_fit/explike.m000066400000000000000000000123301456127120000217270ustar00rootroot00000000000000## Copyright (C) 2021 Nir Krakauer ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{nlogL} =} explike (@var{mu}, @var{x}) ## @deftypefnx {statistics} {[@var{nlogL}, @var{avar}] =} explike (@var{mu}, @var{x}) ## @deftypefnx {statistics} {[@dots{}] =} explike (@var{mu}, @var{x}, @var{censor}) ## @deftypefnx {statistics} {[@dots{}] =} explike (@var{mu}, @var{x}, @var{censor}, @var{freq}) ## ## Negative log-likelihood for the exponential distribution. ## ## @code{@var{nlogL} = explike (@var{mu}, @var{x})} returns the negative ## log likelihood of the data in @var{x} corresponding to the exponential ## distribution with mean parameter @var{mu}. @var{x} must be a vector of ## non-negative values, otherwise @qcode{NaN} is returned. ## ## @code{[@var{nlogL}, @var{avar}] = explike (@var{mu}, @var{x})} also ## returns the inverse of Fisher's information matrix, @var{avar}. If the input ## mean parameter, @var{mu}, is the maximum likelihood estimate, @var{avar} is ## its asymptotic variance. ## ## @code{[@dots{}] = explike (@var{mu}, @var{x}, @var{censor})} accepts a ## boolean vector, @var{censor}, of the same size as @var{x} with @qcode{1}s for ## observations that are right-censored and @qcode{0}s for observations that are ## observed exactly. By default, or if left empty, ## @qcode{@var{censor} = zeros (size (@var{x}))}. ## ## @code{[@dots{}] = explike (@var{mu}, @var{x}, @var{censor}, @var{freq})} ## accepts a frequency vector, @var{freq}, of the same size as @var{x}. ## @var{freq} typically contains integer frequencies for the corresponding ## elements in @var{x}, but it can contain any non-integer non-negative values. ## By default, or if left empty, @qcode{@var{freq} = ones (size (@var{x}))}. ## ## A common alternative parameterization of the exponential distribution is to ## use the parameter @math{λ} defined as the mean number of events in an ## interval as opposed to the parameter @math{μ}, which is the mean wait time ## for an event to occur. @math{λ} and @math{μ} are reciprocals, ## i.e. @math{μ = 1 / λ}. ## ## Further information about the exponential distribution can be found at ## @url{https://en.wikipedia.org/wiki/Exponential_distribution} ## ## @seealso{expcdf, expinv, exppdf, exprnd, expfit, expstat} ## @end deftypefn function [nlogL, avar] = explike (mu, x, censor, freq) ## Check input arguments if (nargin < 2) error ("explike: function called with too few input arguments."); endif if (! isvector (x)) error ("explike: X must be a vector."); endif if (numel (mu) != 1) error ("explike: MU must be a scalar."); endif ## Return NaNs for non-positive MU or negative values in X if (mu <= 0 || any (x(:) < 0)) nlogL = NaN; if (nargout > 1) avar = NaN; endif return endif if (nargin < 3 || isempty (censor)) censor = zeros (size (x)); elseif (! isequal (size (x), size (censor))) error ("explike: X and CENSOR vectors mismatch."); endif if (nargin < 4 || isempty (freq)) freq = ones (size (x)); elseif (isequal (size (x), size (freq))) nulls = find (freq == 0); if (numel (nulls) > 0) x(nulls) = []; censor(nulls) = []; freq(nulls) = []; endif else error ("explike: X and FREQ vectors mismatch."); endif ## Start processing numx = numel (x); sumz = sum (x .* freq) / mu; numc = numx - sum (freq .* censor); ## Calculate negative log likelihood nlogL = sumz + numc * log (mu); ## Optionally calculate the inverse (reciprocal) of the second derivative ## of the negative log likelihood with respect to parameter if (nargout > 1) avar = (mu ^ 2) ./ (2 * sumz - numc); endif endfunction %!test %! x = 12; %! beta = 5; %! [L, V] = explike (beta, x); %! expected_L = 4.0094; %! expected_V = 6.5789; %! assert (L, expected_L, 0.001); %! assert (V, expected_V, 0.001); %!test %! x = 1:5; %! beta = 2; %! [L, V] = explike (beta, x); %! expected_L = 10.9657; %! expected_V = 0.4; %! assert (L, expected_L, 0.001); %! assert (V, expected_V, 0.001); ## Test input validation %!error explike () %!error explike (2) %!error explike ([12, 3], [1:50]) %!error explike (3, ones (10, 2)) %!error ... %! explike (3, [1:50], [1, 2, 3]) %!error ... %! explike (3, [1:50], [], [1, 2, 3]) statistics-release-1.6.3/inst/dist_fit/gamfit.m000066400000000000000000000371251456127120000215460ustar00rootroot00000000000000## Copyright (C) 2019 Nir Krakauer ## Copyright (C) 2023 Andreas Bertsatos ## Based on previous work by Martijn van Oosterhout ## originally granted to the public domain. ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{paramhat} =} gamfit (@var{x}) ## @deftypefnx {statistics} {[@var{paramhat}, @var{paramci}] =} gamfit (@var{x}) ## @deftypefnx {statistics} {[@var{paramhat}, @var{paramci}] =} gamfit (@var{x}, @var{alpha}) ## @deftypefnx {statistics} {[@dots{}] =} gamfit (@var{x}, @var{alpha}, @var{censor}) ## @deftypefnx {statistics} {[@dots{}] =} gamfit (@var{x}, @var{alpha}, @var{censor}, @var{freq}) ## @deftypefnx {statistics} {[@dots{}] =} gamfit (@var{x}, @var{alpha}, @var{censor}, @var{freq}, @var{options}) ## ## Estimate parameters and confidence intervals for the Gamma distribution. ## ## @code{@var{paramhat} = gamfit (@var{x})} returns the maximum likelihood ## estimates of the parameters of the Gamma distribution given the data in ## @var{x}. @qcode{@var{paramhat}(1)} is the shape parameter, @var{k}, and ## @qcode{@var{paramhat}(2)} is the scale parameter, @var{theta}. ## ## @code{[@var{paramhat}, @var{paramci}] = gamfit (@var{x})} returns the 95% ## confidence intervals for the parameter estimates. ## ## @code{[@dots{}] = gamfit (@var{x}, @var{alpha})} also returns the ## @qcode{100 * (1 - @var{alpha})} percent confidence intervals for the ## parameter estimates. By default, the optional argument @var{alpha} is ## 0.05 corresponding to 95% confidence intervals. Pass in @qcode{[]} for ## @var{alpha} to use the default values. ## ## @code{[@dots{}] = gamfit (@var{x}, @var{alpha}, @var{censor})} accepts a ## boolean vector, @var{censor}, of the same size as @var{x} with @qcode{1}s for ## observations that are right-censored and @qcode{0}s for observations that are ## observed exactly. By default, or if left empty, ## @qcode{@var{censor} = zeros (size (@var{x}))}. ## ## @code{[@dots{}] = gamfit (@var{x}, @var{alpha}, @var{censor}, @var{freq})} ## accepts a frequency vector, @var{freq}, of the same size as @var{x}. ## @var{freq} typically contains integer frequencies for the corresponding ## elements in @var{x}, but it can contain any non-integer non-negative values. ## By default, or if left empty, @qcode{@var{freq} = ones (size (@var{x}))}. ## ## @code{[@dots{}] = gamfit (@dots{}, @var{options})} specifies control ## parameters for the iterative algorithm used to compute the maximum likelihood ## estimates. @var{options} is a structure with the following field and its ## default value: ## @itemize ## @item @qcode{@var{options}.Display = "off"} ## @item @qcode{@var{options}.MaxFunEvals = 1000} ## @item @qcode{@var{options}.MaxIter = 500} ## @item @qcode{@var{options}.TolX = 1e-6} ## @end itemize ## ## There are two equivalent parameterizations in common use: ## @enumerate ## @item With a shape parameter @math{k} and a scale parameter @math{θ}, which ## is used by @code{gamcdf}. ## @item With a shape parameter @math{α = k} and an inverse scale parameter ## @math{β = 1 / θ}, called a rate parameter. ## @end enumerate ## ## Further information about the Gamma distribution can be found at ## @url{https://en.wikipedia.org/wiki/Gamma_distribution} ## ## @seealso{gamcdf, gampdf, gaminv, gamrnd, gamlike} ## @end deftypefn function [paramhat, paramci] = gamfit (x, alpha, censor, freq, options) ## Check input arguments if (! isvector (x)) error ("gamfit: X must be a vector."); endif ## Check alpha if (nargin < 2 || isempty (alpha)) alpha = 0.05; else if (! isscalar (alpha) || ! isreal (alpha) || alpha <= 0 || alpha >= 1) error ("gamfit: wrong value for ALPHA."); endif endif ## Check censor vector if (nargin < 3 || isempty (censor)) censor = zeros (size (x)); elseif (! isequal (size (x), size (censor))) error ("gamfit: X and CENSOR vectors mismatch."); endif ## Check frequency vector if (nargin < 4 || isempty (freq)) freq = ones (size (x)); elseif (! isequal (size (x), size (freq))) error ("gamfit: X and FREQ vectors mismatch."); endif ## Get options structure or add defaults if (nargin < 5) options.Display = "off"; options.MaxFunEvals = 400; options.MaxIter = 200; options.TolX = 1e-6; else if (! isstruct (options) || ! isfield (options, "Display") || ! isfield (options, "MaxFunEvals") || ! isfield (options, "MaxIter") || ! isfield (options, "TolX")) error (strcat (["gamfit: 'options' 5th argument must be a"], ... [" structure with 'Display', 'MaxFunEvals',"], ... [" 'MaxIter', and 'TolX' fields present."])); endif endif ## Get sample size and data type cls = class (x); szx = sum (freq); ncen = sum (freq .* censor); nunc = szx - ncen; ## Check for illegal value in X if (ncen == 0 && any (x < 0)) error ("gamfit: X cannot contain negative values."); endif if (ncen > 0 && any (x <= 0)) error ("gamfit: X must contain positive values."); endif ## Handle ill-conditioned cases: no data or all censored if (szx == 0 || nunc == 0 || any (! isfinite (x))) paramhat = nan (1, 2, cls); paramci = nan (2, cls); return endif ## Check for identical data in X if (! isscalar (x) && max (abs (diff (x)) ./ x(2:end)) <= sqrt (eps)) paramhat = cast ([Inf, 0], cls); paramci = cast ([Inf, 0; Inf, 0], cls); return endif ## When CENSOR and FREQ are default if (all (censor == 0) && all (freq == 1)) ## Optimize with respect to log(k), since both K and THETA must be positive meanx = mean (x); x0 = 0; ## Minimize negative log-likelihood to estimate parameters f = @(logk) gamfit_search (logk, meanx, x); [logk, ~, err, output] = fminsearch (f, x0, options); ## Inverse log(k) k = exp (logk); theta = meanx / k; paramhat = [k, theta]; ## Handle errors if (err == 0) if (output.funcCount >= options.MaxFunEvals) warning (strcat (["gamfit: maximum number of function"], ... [" evaluations are exceeded."])); elseif (output.iterations >= options.MaxIter) warning ("gamfit: maximum number of iterations are exceeded."); endif elseif (err < 0) error ("gamfit: NoSolution."); endif endif ## No censoring if (all (censor == 0)) ## Scale data to allow parameter estimation ## for extremely large or small values scale = sum (freq .* x) / szx; ## Check for all data being ~zero if (scale < realmin (cls)) paramhat = cast ([NaN, 0], cls); paramci = cast ([NaN, 0; NaN, 0], cls); return endif scaledx = x / scale; ## Use Method of Moments for initial estimates meansqx = sum (freq .* (scaledx - 1) .^ 2) / szx; theta = meansqx * szx / (szx - 1); k = 1 / theta; ## Ensure that MLEs is possible, otherwise return initial estimates if (any (scaledx == 0)) paramhat = [k, theta*scale]; paramci = nan (2, cls); warning ("gamfit: X contains zeros."); return ## Compute MLEs else ## Bracket the root of the scale parameter likelihood equation sumlogx = sum (freq .* log (scaledx)); bracket = sumlogx / szx; if (lkeqn (k, bracket) > 0) upper = k; lower = 0.5 * upper; while (lkeqn (lower, bracket) > 0) upper = lower; lower = 0.5 * upper; if (lower < realmin (cls)) error ("gamfit: no solution"); endif endwhile else lower = k; upper = 2 * lower; while (lkeqn (upper, bracket) < 0) lower = upper; upper = 2 * lower; if (upper > realmax (cls)) error ("gamfit: no solution"); endif endwhile endif bounds = [lower upper]; ## Find the root of the likelihood equation. opts = optimset ("fzero"); opts = optimset (opts, "Display", "off"); f = @(k) lkeqn (k, bracket); [k, lkeqnval, err] = fzero (f, bounds, opts); ## Rescale THETA paramhat = [k, (1/k)*scale]; endif ## With censoring else ## Get uncensored data notc = ! censor; xunc = x(notc); freq_notc = freq(notc); ## Ensure that MLEs is possible and get initial estimates xuncbar = sum (freq_notc .* xunc) / nunc; s2unc = sum (freq_notc .* (xunc - xuncbar) .^ 2) / nunc; if s2unc <= 100.*eps(xuncbar.^2) ## When all uncensored observations are equal and greater than all ## the censored observations, the likelihood surface becomes infinite if (max (xunc) == max (x)) paramhat = cast ([Inf, 0], cls); if (nunc > 1) paramci = cast ([Inf, 0; Inf, 0], cls); else paramci = cast ([0, 0; Inf, Inf], cls); endif return endif ## Set some default parameter estimates. x0 = [2, xuncbar./2]; else ## Fit a Weibull distribution and equate the parameter estimates ## into a Gamma distribution wblphat = wblfit (x, alpha, censor, freq); [m, v] = wblstat (wblphat(1), wblphat(2)); x0 = [m.*m./v, v./m]; endif ## Minimize negative log-likelihood to estimate parameters f = @(params) gamlike (params, x, censor, freq); [paramhat, ~, err, output] = fminsearch (f, x0, options); ## Force positive parameter values paramhat = abs (paramhat); ## Handle errors if (err == 0) if (output.funcCount >= options.MaxFunEvals) warning (strcat (["gamfit: maximum number of function"], ... [" evaluations are exceeded."])); elseif (output.iterations >= options.MaxIter) warning ("gamfit: maximum number of iterations are exceeded."); endif elseif (err < 0) error ("gamfit: no solution."); endif endif ## Compute CIs using a log normal approximation for parameters. if (nargout > 1) ## Compute asymptotic covariance [~, acov] = gamlike (paramhat, x, censor, freq); ## Get standard errors stderr = sqrt (diag (acov))'; stderr = stderr ./ paramhat; ## Apply log transform phatlog = log (paramhat); ## Compute normal quantiles z = probit (alpha / 2); ## Compute CI paramci = [phatlog; phatlog] + [stderr; stderr] .* [z, z; -z, -z]; ## Inverse log transform paramci = exp (paramci); endif endfunction ## Helper function so we only have to minimize for one variable. function nlogL = gamfit_search (logk, meanx, x) k = exp (logk); theta = meanx / k; nlogL = gamlike ([k, theta], x); endfunction ## Helper function for MLE with no censoring function v = lkeqn (k, bracket) v = -bracket - log (k) + psi (k); endfunction %!demo %! ## Sample 3 populations from different Gamma distibutions %! randg ("seed", 5); # for reproducibility %! r1 = gamrnd (1, 2, 2000, 1); %! randg ("seed", 2); # for reproducibility %! r2 = gamrnd (2, 2, 2000, 1); %! randg ("seed", 7); # for reproducibility %! r3 = gamrnd (7.5, 1, 2000, 1); %! r = [r1, r2, r3]; %! %! ## Plot them normalized and fix their colors %! hist (r, 75, 4); %! h = findobj (gca, "Type", "patch"); %! set (h(1), "facecolor", "c"); %! set (h(2), "facecolor", "g"); %! set (h(3), "facecolor", "r"); %! ylim ([0, 0.62]); %! xlim ([0, 12]); %! hold on %! %! ## Estimate their α and β parameters %! k_thetaA = gamfit (r(:,1)); %! k_thetaB = gamfit (r(:,2)); %! k_thetaC = gamfit (r(:,3)); %! %! ## Plot their estimated PDFs %! x = [0.01,0.1:0.2:18]; %! y = gampdf (x, k_thetaA(1), k_thetaA(2)); %! plot (x, y, "-pr"); %! y = gampdf (x, k_thetaB(1), k_thetaB(2)); %! plot (x, y, "-sg"); %! y = gampdf (x, k_thetaC(1), k_thetaC(2)); %! plot (x, y, "-^c"); %! hold off %! legend ({"Normalized HIST of sample 1 with k=1 and θ=2", ... %! "Normalized HIST of sample 2 with k=2 and θ=2", ... %! "Normalized HIST of sample 3 with k=7.5 and θ=1", ... %! sprintf("PDF for sample 1 with estimated k=%0.2f and θ=%0.2f", ... %! k_thetaA(1), k_thetaA(2)), ... %! sprintf("PDF for sample 2 with estimated k=%0.2f and θ=%0.2f", ... %! k_thetaB(1), k_thetaB(2)), ... %! sprintf("PDF for sample 3 with estimated k=%0.2f and θ=%0.2f", ... %! k_thetaC(1), k_thetaC(2))}) %! title ("Three population samples from different Gamma distibutions") %! hold off ## Test output %!shared x %! x = [1.2 1.6 1.7 1.8 1.9 2.0 2.2 2.6 3.0 3.5 4.0 4.8 5.6 6.6 7.6]; %!test %! [paramhat, paramci] = gamfit (x); %! assert (paramhat, [3.4248, 0.9752], 1e-4); %! assert (paramci, [1.7287, 0.4670; 6.7852, 2.0366], 1e-4); %!test %! [paramhat, paramci] = gamfit (x, 0.01); %! assert (paramhat, [3.4248, 0.9752], 1e-4); %! assert (paramci, [1.3945, 0.3705; 8.4113, 2.5668], 1e-4); %!test %! freq = [1 1 1 1 2 1 1 1 1 2 1 1 1 1 2]; %! [paramhat, paramci] = gamfit (x, [], [], freq); %! assert (paramhat, [3.3025, 1.0615], 1e-4); %! assert (paramci, [1.7710, 0.5415; 6.1584, 2.0806], 1e-4); %!test %! [paramhat, paramci] = gamfit (x, [], [], [1:15]); %! assert (paramhat, [4.4484, 0.9689], 1e-4); %! assert (paramci, [3.4848, 0.7482; 5.6785, 1.2546], 1e-4); %!test %! [paramhat, paramci] = gamfit (x, 0.01, [], [1:15]); %! assert (paramhat, [4.4484, 0.9689], 1e-4); %! assert (paramci, [3.2275, 0.6899; 6.1312, 1.3608], 1e-4); %!test %! cens = [0 0 0 0 1 0 0 0 0 0 0 0 0 0 0]; %! [paramhat, paramci] = gamfit (x, [], cens, [1:15]); %! assert (paramhat, [4.7537, 0.9308], 1e-4); %! assert (paramci, [3.7123, 0.7162; 6.0872, 1.2097], 1e-4); %!test %! cens = [0 0 0 0 1 0 0 0 0 0 0 0 0 0 0]; %! freq = [1 1 1 1 2 1 1 1 1 2 1 1 1 1 2]; %! [paramhat, paramci] = gamfit (x, [], cens, freq); %! assert (paramhat, [3.4736, 1.0847], 1e-4); %! assert (paramci, [1.8286, 0.5359; 6.5982, 2.1956], 1e-4); ## Test edge cases %!test %! [paramhat, paramci] = gamfit ([1 1 1 1 1 1]); %! assert (paramhat, [Inf, 0]); %! assert (paramci, [Inf, 0; Inf, 0]); %!test %! [paramhat, paramci] = gamfit ([1 1 1 1 1 1], [], [1 1 1 1 1 1]); %! assert (paramhat, [NaN, NaN]); %! assert (paramci, [NaN, NaN; NaN, NaN]); %!test %! [paramhat, paramci] = gamfit ([1 1 1 1 1 1], [], [], [1 1 1 1 1 1]); %! assert (paramhat, [Inf, 0]); %! assert (paramci, [Inf, 0; Inf, 0]); ## Test class of input preserved %!assert (class (gamfit (single (x))), "single") ## Test input validation %!error gamfit (ones (2)) %!error gamfit (x, 1) %!error gamfit (x, -1) %!error gamfit (x, {0.05}) %!error gamfit (x, "k") %!error gamfit (x, i) %!error gamfit (x, [0.01 0.02]) %!error gamfit (x, [], [1 1]) %!error gamfit (x, [], [], [1 1]) %!error gamfit ([1 2 3 -4]) %!error gamfit ([1 2 0], [], [1 0 0]) statistics-release-1.6.3/inst/dist_fit/gamlike.m000066400000000000000000000244071456127120000217070ustar00rootroot00000000000000## Copyright (C) 2022-2023 Andreas Bertsatos ## Based on previous work by Martijn van Oosterhout ## originally granted to the public domain. ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{nlogL} =} gamlike (@var{params}, @var{x}) ## @deftypefnx {statistics} {[@var{nlogL}, @var{acov}] =} gamlike (@var{params}, @var{x}) ## @deftypefnx {statistics} {[@dots{}] =} gamlike (@var{params}, @var{x}, @var{censor}) ## @deftypefnx {statistics} {[@dots{}] =} gamlike (@var{params}, @var{x}, @var{censor}, @var{freq}) ## ## Negative log-likelihood for the Gamma distribution. ## ## @code{@var{nlogL} = gamlike (@var{params}, @var{x})} returns the negative ## log likelihood of the data in @var{x} corresponding to the Gamma distribution ## with (1) shape parameter @var{k} and (2) scale parameter @var{theta} given in ## the two-element vector @var{params}. ## ## @code{[@var{nlogL}, @var{acov}] = gamlike (@var{params}, @var{x})} also ## returns the inverse of Fisher's information matrix, @var{acov}. If the input ## parameter values in @var{params} are the maximum likelihood estimates, the ## diagonal elements of @var{acov} are their asymptotic variances. ## ## @code{[@dots{}] = gamlike (@var{params}, @var{x}, @var{censor})} accepts a ## boolean vector, @var{censor}, of the same size as @var{x} with @qcode{1}s for ## observations that are right-censored and @qcode{0}s for observations that are ## observed exactly. By default, or if left empty, ## @qcode{@var{censor} = zeros (size (@var{x}))}. ## ## @code{[@dots{}] = gamlike (@var{params}, @var{x}, @var{censor}, @var{freq})} ## accepts a frequency vector, @var{freq}, of the same size as @var{x}. ## @var{freq} typically contains integer frequencies for the corresponding ## elements in @var{x}, but it can contain any non-integer non-negative values. ## By default, or if left empty, @qcode{@var{freq} = ones (size (@var{x}))}. ## ## There are two equivalent parameterizations in common use: ## @enumerate ## @item With a shape parameter @math{k} and a scale parameter @math{θ}, which ## is used by @code{gamcdf}. ## @item With a shape parameter @math{α = k} and an inverse scale parameter ## @math{β = 1 / θ}, called a rate parameter. ## @end enumerate ## ## Further information about the Gamma distribution can be found at ## @url{https://en.wikipedia.org/wiki/Gamma_distribution} ## ## @seealso{gamcdf, gampdf, gaminv, gamrnd, gamfit} ## @end deftypefn function [nlogL, acov] = gamlike (params, x, censor, freq) ## Check input arguments and add defaults if (nargin < 2) error ("gamlike: function called with too few input arguments."); endif if (numel (params) != 2) error ("gamlike: wrong parameters length."); endif if (! isvector (x)) error ("gamlike: X must be a vector."); endif if (nargin < 3 || isempty (censor)) censor = zeros (size (x)); elseif (! isequal (size (x), size (censor))) error ("gamlike: X and CENSOR vectors mismatch."); endif if (nargin < 4 || isempty (freq)) freq = ones (size (x)); elseif (isequal (size (x), size (freq))) nulls = find (freq == 0); if (numel (nulls) > 0) x(nulls) = []; censor(nulls) = []; freq(nulls) = []; endif else error ("gamlike: X and FREQ vectors mismatch."); endif ## Get K and THETA values k = params(1); t = params(2); ## Parameters K and THETA must be positive, otherwise make them NaN k(k <= 0) = NaN; t(t <= 0) = NaN; ## Data in X must be positive, otherwise make it NaN x(x <= 0) = NaN; ## Compute the individual log-likelihood terms z = x ./ t; L = (k - 1) .* log (z) - z - gammaln (k) - log (t); n_censored = sum (freq .* censor); if (n_censored > 0) z_censored = z(logical (censor)); Scen = gammainc (z_censored, k, "upper"); L(logical (censor)) = log (Scen); endif ## Force a log(0)==-Inf for X from extreme right tail L(z == Inf) = -Inf; ## Neg-log-likelihood is the sum of the individual contributions nlogL = -sum (freq .* L); ## Compute the negative hessian at the parameter values. ## Invert to get the observed information matrix. if (nargout == 2) ## Calculate all data dL11 = -psi (1, k) * ones (size (z), "like", z); dL12 = -(1 ./ t) * ones (size (z), "like", z); dL22 = -(2 .* z - k) ./ (t .^ 2); ## Calculate censored data if (n_censored > 0) ## Compute derivatives [y, dy, d2y] = dgammainc (z_censored, k); dlnS = dy ./ y; d2lnS = d2y ./ y - dlnS.*dlnS; #[dlnS,d2lnS] = dlngamsf(z_censored,k); logzcen = log(z_censored); tmp = exp(k .* logzcen - z_censored - gammaln(k) - log(t)) ./ Scen; dL11(logical (censor)) = d2lnS; dL12(logical (censor)) = tmp .* (logzcen - dlnS - psi(0,k)); dL22(logical (censor)) = tmp .* ((z_censored-1-k)./t - tmp); endif nH11 = -sum(freq .* dL11); nH12 = -sum(freq .* dL12); nH22 = -sum(freq .* dL22); nH = [nH11 nH12; nH12 nH22]; if (any (isnan (nH(:)))) acov = nan (2, "like", nH); else acov = inv (nH); endif endif endfunction ## Compute the incomplete Gamma function with its 1st and 2nd derivatives function [y, dy, d2y] = dgammainc (x, k) ## Initialize return variables y = nan (size (x)); dy = y; d2y = y; ## Use approximation for K > 2^20 ulim = 2^20; is_lim = find (k > ulim); if (! isempty (is_lim)) x(is_lim) = max (ulim - 1/3 + sqrt (ulim ./ k(is_lim)) .* ... (x(is_lim) - (k(is_lim) - 1/3)), 0); k(is_lim) = ulim; endif ## For x < k+1 is_lo = find(x < k + 1 & x != 0); if (! isempty (is_lo)) x_lo = x(is_lo); k_lo = k(is_lo); k_1 = k_lo; step = 1; d1st = 0; d2st = 0; stsum = step; d1sum = d1st; d2sum = d2st; while norm (step, "inf") >= 100 * eps (norm (stsum, "inf")) k_1 += 1; step = step .* x_lo ./ k_1; d1st = (d1st .* x_lo - step) ./ k_1; d2st = (d2st .* x_lo - 2 .* d1st) ./ k_1; stsum = stsum + step; d1sum = d1sum + d1st; d2sum = d2sum + d2st; endwhile fklo = exp (-x_lo + k_lo .* log (x_lo) - gammaln (k_lo + 1)); y_lo = fklo .* stsum; ## Fix very small k y_lo(x_lo > 0 & y_lo > 1) = 1; ## Compute 1st derivative dlogfklo = (log (x_lo) - psi (k_lo + 1)); d1fklo = fklo .* dlogfklo; d1y_lo = d1fklo .* stsum + fklo .* d1sum; ## Compute 2nd derivative d2fklo = d1fklo .* dlogfklo - fklo .* psi (1, k_lo + 1); d2y_lo = d2fklo .* stsum + 2 .* d1fklo .* d1sum + fklo .* d2sum; ## Considering the upper tail y(is_lo) = 1 - y_lo; dy(is_lo) = -d1y_lo; d2y(is_lo) = -d2y_lo; endif ## For x >= k+1 is_hi = find(x >= k+1); if (! isempty (is_hi)) x_hi = x(is_hi); k_hi = k(is_hi); zc = 0; k0 = 0; k1 = k_hi; x0 = 1; x1 = x_hi; d1k0 = 0; d1k1 = 1; d1x0 = 0; d1x1 = 0; d2k0 = 0; d2k1 = 0; d2x0 = 0; d2x2 = 0; kx = k_hi ./ x_hi; d1kx = 1 ./ x_hi; d2kx = 0; start = 1; while norm (d2kx - start, "Inf") > 100 * eps (norm (d2kx, "Inf")) rescale = 1 ./ x1; zc += 1; n_k = zc - k_hi; d2k0 = (d2k1 + d2k0 .* n_k - 2 .* d1k0) .* rescale; d2x0 = (d2x2 + d2x0 .* n_k - 2 .* d1x0) .* rescale; d1k0 = (d1k1 + d1k0 .* n_k - k0) .* rescale; d1x0 = (d1x1 + d1x0 .* n_k - x0) .* rescale; k0 = (k1 + k0 .* n_k) .* rescale; x0 = 1 + (x0 .* n_k) .* rescale; nrescale = zc .* rescale; d2k1 = d2k0 .* x_hi + d2k1 .* nrescale; d2x2 = d2x0 .* x_hi + d2x2 .* nrescale; d1k1 = d1k0 .* x_hi + d1k1 .* nrescale; d1x1 = d1x0 .* x_hi + d1x1 .* nrescale; k1 = k0 .* x_hi + k1 .* nrescale; x1 = x0 .* x_hi + zc; start = d2kx; kx = k1 ./ x1; d1kx = (d1k1 - kx .* d1x1) ./ x1; d2kx = (d2k1 - d1kx .* d1x1 - kx .* d2x2 - d1kx .* d1x1) ./ x1; endwhile fkhi = exp (-x_hi + k_hi .* log (x_hi) - gammaln (k_hi + 1)); y_hi = fkhi .* kx; ## Compute 1st derivative dlogfkhi = (log (x_hi) - psi (k_hi + 1)); d1fkhi = fkhi .* dlogfkhi; d1y_hi = d1fkhi .* kx + fkhi .* d1kx; ## Compute 2nd derivative d2fkhi = d1fkhi .* dlogfkhi - fkhi .* psi (1, k_hi + 1); d2y_hi = d2fkhi .* kx + 2 .* d1fkhi .* d1kx + fkhi .* d2kx; ## Considering the upper tail y(is_hi) = y_hi; dy(is_hi) = d1y_hi; d2y(is_hi) = d2y_hi; endif ## Handle x == 0 is_x0 = find (x == 0); if (! isempty (is_x0)) ## Considering the upper tail y(is_x0) = 1; dy(is_x0) = 0; d2y(is_x0) = 0; endif ## Handle k == 0 is_k0 = find (k == 0); if (! isempty (is_k0)) is_k0x0 = find (k == 0 & x == 0); ## Considering the upper tail y(is_k0) = 0; dy(is_k0x0) = Inf; d2y(is_k0x0) = -Inf; endif endfunction ## Test output %!test %! [nlogL, acov] = gamlike([2, 3], [2, 3, 4, 5, 6, 7, 8, 9]); %! assert (nlogL, 19.4426, 1e-4); %! assert (acov, [2.7819, -5.0073; -5.0073, 9.6882], 1e-4); %!test %! [nlogL, acov] = gamlike([2, 3], [5:45]); %! assert (nlogL, 305.8070, 1e-4); %! assert (acov, [0.0423, -0.0087; -0.0087, 0.0167], 1e-4); %!test %! [nlogL, acov] = gamlike([2, 13], [5:45]); %! assert (nlogL, 163.2261, 1e-4); %! assert (acov, [0.2362, -1.6631; -1.6631, 13.9440], 1e-4); ## Test input validation %!error ... %! gamlike ([12, 15]) %!error gamlike ([12, 15, 3], [1:50]) %!error gamlike ([12, 3], ones (10, 2)) %!error ... %! gamlike ([12, 15], [1:50], [1, 2, 3]) %!error ... %! gamlike ([12, 15], [1:50], [], [1, 2, 3]) statistics-release-1.6.3/inst/dist_fit/geofit.m000066400000000000000000000132241456127120000215460ustar00rootroot00000000000000## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{pshat} =} geofit (@var{x}) ## @deftypefnx {statistics} {[@var{pshat}, @var{psci}] =} geofit (@var{x}) ## @deftypefnx {statistics} {[@var{pshat}, @var{psci}] =} geofit (@var{x}, @var{alpha}) ## @deftypefnx {statistics} {[@var{pshat}, @var{psci}] =} geofit (@var{x}, @var{alpha}, @var{freq}) ## ## Estimate parameter and confidence intervals for the geometric distribution. ## ## @code{@var{pshat} = geofit (@var{x})} returns the maximum likelihood estimate ## (MLE) of the probability of success for the geometric distribution. @var{x} ## must be a vector. ## ## @code{[@var{pshat}, @var{psci}] = geofit (@var{x}, @var{alpha})} also returns ## the @qcode{100 * (1 - @var{alpha})} percent confidence intervals of the ## estimated parameter. By default, the optional argument @var{alpha} is 0.05 ## corresponding to 95% confidence intervals. Pass in @qcode{[]} for ## @var{alpha} to use the default values. ## ## @code{[@dots{}] = geofit (@var{x}, @var{alpha}, @var{freq})} accepts a ## frequency vector, @var{freq}, of the same size as @var{x}. @var{freq} ## typically contains integer frequencies for the corresponding elements in ## @var{x}, but it can contain any non-integer non-negative values. By default, ## or if left empty, @qcode{@var{freq} = ones (size (@var{x}))}. ## ## The geometric distribution models the number of failures (@var{x}) of a ## Bernoulli trial with probability @var{ps} before the first success. ## ## Further information about the geometric distribution can be found at ## @url{https://en.wikipedia.org/wiki/Geometric_distribution} ## ## @seealso{geocdf, geoinv, geopdf, geornd, geostat} ## @end deftypefn function [pshat, psci] = geofit (x, alpha, freq) ## Check input arguments if (nargin < 1) error ("geofit: function called with too few input arguments."); endif ## Check data inX if (any (x < 0)) error ("geofit: X cannot have negative values."); endif if (! isvector (x)) error ("geofit: X must be a vector."); endif ## Check ALPHA if (nargin < 2 || isempty (alpha)) alpha = 0.05; elseif (! isscalar (alpha) || ! isreal (alpha) || alpha <= 0 || alpha >= 1) error ("geofit: wrong value for ALPHA."); endif ## Check frequency vector if (nargin < 3 || isempty (freq)) freq = ones (size (x)); elseif (! isequal (size (x), size (freq))) error ("geofit: X and FREQ vector mismatch."); endif ## Expand frequency and censor vectors (if necessary) if (! all (freq == 1)) xf = []; for i = 1:numel (freq) xf = [xf, repmat(x(i), 1, freq(i))]; endfor x = xf; freq = ones (size (x)); endif ## Compute PS estimate pshat = 1 ./ (1 + mean (x)); ## Compute confidence interval of PS if (nargout > 1) sz = numel (x); serr = pshat .* sqrt ((1 - pshat) ./ sz); psci = norminv ([alpha/2; 1-alpha/2], [pshat; pshat], [serr; serr]); endif endfunction %!demo %! ## Sample 2 populations from different geometric distibutions %! rande ("seed", 1); # for reproducibility %! r1 = geornd (0.15, 1000, 1); %! rande ("seed", 2); # for reproducibility %! r2 = geornd (0.5, 1000, 1); %! r = [r1, r2]; %! %! ## Plot them normalized and fix their colors %! hist (r, 0:0.5:20.5, 1); %! h = findobj (gca, "Type", "patch"); %! set (h(1), "facecolor", "c"); %! set (h(2), "facecolor", "g"); %! hold on %! %! ## Estimate their probability of success %! pshatA = geofit (r(:,1)); %! pshatB = geofit (r(:,2)); %! %! ## Plot their estimated PDFs %! x = [0:15]; %! y = geopdf (x, pshatA); %! plot (x, y, "-pg"); %! y = geopdf (x, pshatB); %! plot (x, y, "-sc"); %! xlim ([0, 15]) %! ylim ([0, 0.6]) %! legend ({"Normalized HIST of sample 1 with ps=0.15", ... %! "Normalized HIST of sample 2 with ps=0.50", ... %! sprintf("PDF for sample 1 with estimated ps=%0.2f", ... %! mean (pshatA)), ... %! sprintf("PDF for sample 2 with estimated ps=%0.2f", ... %! mean (pshatB))}) %! title ("Two population samples from different geometric distibutions") %! hold off ## Test output %!test %! x = 0:5; %! [pshat, psci] = geofit (x); %! assert (pshat, 0.2857, 1e-4); %! assert (psci, [0.092499; 0.478929], 1e-5); %!test %! x = 0:5; %! [pshat, psci] = geofit (x, [], [1 1 1 1 1 1]); %! assert (pshat, 0.2857, 1e-4); %! assert (psci, [0.092499; 0.478929], 1e-5); %!assert (geofit ([1 1 2 3]), geofit ([1 2 3], [] ,[2 1 1])) ## Test input validation %!error geofit () %!error geofit (-1, [1 2 3 3]) %!error geofit (1, 0) %!error geofit (1, 1.2) %!error geofit (1, [0.02 0.05]) %!error ... %! geofit ([1.5, 0.2], [], [0, 0, 0, 0, 0]) %!error ... %! geofit ([1.5, 0.2], [], [1, 1, 1]) statistics-release-1.6.3/inst/dist_fit/gevfit.m000066400000000000000000000241301456127120000215530ustar00rootroot00000000000000## Copyright (C) 2012-2021 Nir Krakauer ## Copyright (C) 2022-2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{paramhat} =} gevfit (@var{x}) ## @deftypefnx {statistics} {[@var{paramhat}, @var{paramci}] =} gevfit (@var{x}) ## @deftypefnx {statistics} {[@var{paramhat}, @var{paramci}] =} gevfit (@var{x}, @var{alpha}) ## @deftypefnx {statistics} {[@dots{}] =} gevfit (@var{x}, @var{alpha}, @var{options}) ## ## Estimate parameters and confidence intervals for the generalized extreme ## value (GEV) distribution. ## ## @subheading Arguments ## ## @code{@var{paramhat} = gevfit (@var{x})} returns the maximum likelihood ## estimates of the parameters of the GEV distribution given the data in ## @var{x}. @qcode{@var{paramhat}(1)} is the shape parameter, @var{k}, and ## @qcode{@var{paramhat}(2)} is the scale parameter, @var{sigma}, and ## @qcode{@var{paramhat}(3)} is the location parameter, @var{mu}. ## ## @code{[@var{paramhat}, @var{paramci}] = gevfit (@var{x})} returns the 95% ## confidence intervals for the parameter estimates. ## ## @code{[@dots{}] = gevfit (@var{x}, @var{alpha})} also returns the ## @qcode{100 * (1 - @var{alpha})} percent confidence intervals for the ## parameter estimates. By default, the optional argument @var{alpha} is ## 0.05 corresponding to 95% confidence intervals. Pass in @qcode{[]} for ## @var{alpha} to use the default values. ## ## @code{[@dots{}] = gevfit (@dots{}, @var{options})} specifies control ## parameters for the iterative algorithm used to compute the maximum likelihood ## estimates. @var{options} is a structure with the following field and its ## default value: ## @itemize ## @item @qcode{@var{options}.Display = "off"} ## @item @qcode{@var{options}.MaxFunEvals = 1000} ## @item @qcode{@var{options}.MaxIter = 500} ## @item @qcode{@var{options}.TolX = 1e-6} ## @end itemize ## ## When @qcode{@var{k} < 0}, the GEV is the type III extreme value distribution. ## When @qcode{@var{k} > 0}, the GEV distribution is the type II, or Frechet, ## extreme value distribution. If @var{W} has a Weibull distribution as ## computed by the @code{wblcdf} function, then @qcode{-@var{W}} has a type III ## extreme value distribution and @qcode{1/@var{W}} has a type II extreme value ## distribution. In the limit as @var{k} approaches @qcode{0}, the GEV is the ## mirror image of the type I extreme value distribution as computed by the ## @code{evcdf} function. ## ## The mean of the GEV distribution is not finite when @qcode{@var{k} >= 1}, and ## the variance is not finite when @qcode{@var{k} >= 1/2}. The GEV distribution ## has positive density only for values of @var{x} such that ## @qcode{@var{k} * (@var{x} - @var{mu}) / @var{sigma} > -1}. ## ## Further information about the generalized extreme value distribution can be ## found at ## @url{https://en.wikipedia.org/wiki/Generalized_extreme_value_distribution} ## ## @subheading References ## @enumerate ## @item ## Rolf-Dieter Reiss and Michael Thomas. @cite{Statistical Analysis of Extreme ## Values with Applications to Insurance, Finance, Hydrology and Other Fields}. ## Chapter 1, pages 16-17, Springer, 2007. ## @end enumerate ## ## @seealso{gevcdf, gevinv, gevpdf, gevrnd, gevlike, gevstat} ## @end deftypefn function [paramhat, paramci] = gevfit (x, alpha, options) ## Check X is vector if (! isvector (x)) error ("gevfit: X must be a vector."); endif ## Force to column vector x = x(:); ## Get X type and convert to double for computation is_type = class (x); if (strcmpi (is_type, "single")) x = double (x); endif ## Check that X is not constant and does not contain NaNs sample_size = length (x); if (sample_size == 0 || any (isnan (x))) paramhat = NaN (1,3, is_type); paramci = NaN (2,3, is_type); warning ("gevfit: X contains NaNs."); return elseif (numel (unique (x)) == 1) paramhat = cast ([0, 0, unique(x)], is_type); if (length (x) == 1) paramci = cast ([-Inf, 0, -Inf; Inf, Inf, Inf], is_type); else paramci = [paramhat; paramhat]; endif warning ("gevfit: X is a constant vector."); return endif ## Check ALPHA if (nargin < 2 || isempty (alpha)) alpha = 0.05; else if (! isscalar (alpha) || ! isreal (alpha) || alpha <= 0 || alpha >= 1) error ("gevfit: wrong value for ALPHA."); endif endif ## Get options structure or add defaults if (nargin < 3) options.Display = "off"; options.MaxFunEvals = 400; options.MaxIter = 200; options.TolX = 1e-6; else if (! isstruct (options) || ! isfield (options, "Display") || ! isfield (options, "MaxFunEvals") || ! isfield (options, "MaxIter") || ! isfield (options, "TolX")) error (strcat (["gevfit: 'options' 5th argument must be a"], ... [" structure with 'Display', 'MaxFunEvals',"], ... [" 'MaxIter', and 'TolX' fields present."])); endif endif ## Compute initial parameters if not parsed as an input argument F = (0.5:1:(sample_size - 0.5))' ./ sample_size; k_0 = fminsearch (@(k) 1 - corr (x, gevinv (F, k, 1, 0)), 0); paramguess = [k_0, polyfit(gevinv(F,k_0,1,0),x',1)]; ## Check if x support initial parameters or fall back to unbounded evfit if (k_0 < 0 && (max (x) > - paramguess(2) / k_0 + paramguess(3)) || ... k_0 > 0 && (min (x) < - paramguess(2) / k_0 + paramguess(3))) paramguess = [evfit(x), 0]; paramguess = flip (paramguess); endif ## Minimize the negative log-likelihood according to initial parameters paramguess(2) = log (paramguess(2)); fhandle = @(paramguess) nll (paramguess, x); [paramhat, ~, exitflag, output] = fminsearch (fhandle, paramguess, options); paramhat(2) = exp (paramhat(2)); ## Display errors and warnings if any if (exitflag == 0) if (output.funcCount >= output.iterations) warning ("gevfit: maximum number of evaluations reached"); else warning ("gevfit: reached iteration limit"); endif elseif (exitflag == -1) error ("gevfit: No solution"); endif ## Return a row vector for Matlab compatibility paramhat = paramhat(:)'; ## Check for second output argument if (nargout > 1) [~, acov] = gevlike (paramhat, x); param_se = sqrt (diag (acov))'; if (any (iscomplex (param_se))) warning (["gevfit: Fisher information matrix not positive definite;", ... " parameter optimization likely did not converge"]); paramci = NaN (2, 3, is_type); else p_vals = [alpha/2; 1-alpha/2]; k_ci = norminv (p_vals, paramhat(1), param_se(1)); s_ci = exp (norminv (p_vals, log (paramhat(2)), param_se(2) ./ paramhat(2))); m_ci = norminv (p_vals, paramhat(3), param_se(3)); paramci = [k_ci, s_ci, m_ci]; endif endif endfunction ## Negative log-likelihood for the GEV (log(sigma) parameterization) function out = nll (parms, x) k_0 = parms(1); log_sigma = parms(2); sigma = exp (log_sigma); mu = parms(3); n = numel (x); z = (x - mu) ./ sigma; if abs(k_0) > eps u = 1 + k_0.*z; if min(u) > 0 lnu = log1p (k_0 .* z); out = n * log_sigma + sum (exp ((-1 / k_0) * lnu)) + ... (1 + 1 / k_0) * sum (lnu); else out = Inf; endif else out = n * log_sigma + sum (exp (-z) + z); endif endfunction %!demo %! ## Sample 2 populations from 2 different exponential distibutions %! rand ("seed", 1); # for reproducibility %! r1 = gevrnd (-0.5, 1, 2, 5000, 1); %! rand ("seed", 2); # for reproducibility %! r2 = gevrnd (0, 1, -4, 5000, 1); %! r = [r1, r2]; %! %! ## Plot them normalized and fix their colors %! hist (r, 50, 5); %! h = findobj (gca, "Type", "patch"); %! set (h(1), "facecolor", "c"); %! set (h(2), "facecolor", "g"); %! hold on %! %! ## Estimate their k, sigma, and mu parameters %! k_sigma_muA = gevfit (r(:,1)); %! k_sigma_muB = gevfit (r(:,2)); %! %! ## Plot their estimated PDFs %! x = [-10:0.5:20]; %! y = gevpdf (x, k_sigma_muA(1), k_sigma_muA(2), k_sigma_muA(3)); %! plot (x, y, "-pr"); %! y = gevpdf (x, k_sigma_muB(1), k_sigma_muB(2), k_sigma_muB(3)); %! plot (x, y, "-sg"); %! ylim ([0, 0.7]) %! xlim ([-7, 5]) %! legend ({"Normalized HIST of sample 1 with ξ=-0.5, σ=1, μ=2", ... %! "Normalized HIST of sample 2 with ξ=0, σ=1, μ=-4", %! sprintf("PDF for sample 1 with estimated ξ=%0.2f, σ=%0.2f, μ=%0.2f", ... %! k_sigma_muA(1), k_sigma_muA(2), k_sigma_muA(3)), ... %! sprintf("PDF for sample 3 with estimated ξ=%0.2f, σ=%0.2f, μ=%0.2f", ... %! k_sigma_muB(1), k_sigma_muB(2), k_sigma_muB(3))}) %! title ("Two population samples from different exponential distibutions") %! hold off ## Test output %!test %! x = 1:50; %! [pfit, pci] = gevfit (x); %! pfit_out = [-0.4407, 15.1923, 21.5309]; %! pci_out = [-0.7532, 11.5878, 16.5686; -0.1282, 19.9183, 26.4926]; %! assert (pfit, pfit_out, 1e-3); %! assert (pci, pci_out, 1e-3); %!test %! x = 1:2:50; %! [pfit, pci] = gevfit (x); %! pfit_out = [-0.4434, 15.2024, 21.0532]; %! pci_out = [-0.8904, 10.3439, 14.0168; 0.0035, 22.3429, 28.0896]; %! assert (pfit, pfit_out, 1e-3); %! assert (pci, pci_out, 1e-3); ## Test input validation %!error gevfit (ones (2,5)); %!error gevfit ([1, 2, 3, 4, 5], 1.2); %!error gevfit ([1, 2, 3, 4, 5], 0); %!error gevfit ([1, 2, 3, 4, 5], "alpha"); statistics-release-1.6.3/inst/dist_fit/gevfit_lmom.m000066400000000000000000000066701456127120000226100ustar00rootroot00000000000000## Copyright (C) 2012 Nir Krakauer ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {[@var{paramhat}, @var{paramci}] =} gevfit_lmom (@var{data}) ## ## Find an estimator (@var{paramhat}) of the generalized extreme value (GEV) ## distribution fitting @var{data} using the method of L-moments. ## ## @subheading Arguments ## ## @itemize @bullet ## @item ## @var{data} is the vector of given values. ## @end itemize ## ## @subheading Return values ## ## @itemize @bullet ## @item ## @var{parmhat} is the 3-parameter maximum-likelihood parameter vector ## [@var{k}; @var{sigma}; @var{mu}], where @var{k} is the shape parameter of the ## GEV distribution, @var{sigma} is the scale parameter of the GEV distribution, ## and @var{mu} is the location parameter of the GEV distribution. ## @item ## @var{paramci} has the approximate 95% confidence intervals of the parameter ## values (currently not implemented). ## ## @end itemize ## ## @subheading Examples ## ## @example ## @group ## data = gevrnd (0.1, 1, 0, 100, 1); ## [pfit, pci] = gevfit_lmom (data); ## p1 = gevcdf (data,pfit(1),pfit(2),pfit(3)); ## [f, x] = ecdf (data); ## plot(data, p1, 's', x, f) ## @end group ## @end example ## @seealso{gevfit} ## @subheading References ## ## @enumerate ## @item ## Ailliot, P.; Thompson, C. & Thomson, P. Mixed methods for fitting the GEV ## distribution, Water Resources Research, 2011, 47, W05551 ## ## @end enumerate ## @end deftypefn function [paramhat, paramci] = gevfit_lmom (data) # Check arguments if (nargin < 1) print_usage; endif # find the L-moments data = sort (data(:))'; n = numel(data); L1 = mean(data); L2 = sum(data .* (2*(1:n) - n - 1)) / (2*nchoosek(n, 2)); # or mean(triu(data' - data, 1, 'pack')) / 2; b = bincoeff((1:n) - 1, 2); L3 = sum(data .* (b - 2 * ((1:n) - 1) .* (n - (1:n)) + fliplr(b))) / (3*nchoosek(n, 3)); #match the moments to the GEV distribution #first find k based on L3/L2 f = @(k) (L3/L2 + 3)/2 - limdiv((1 - 3^(k)), (1 - 2^(k))); k = fzero(f, 0); #next find sigma and mu given k if abs(k) < 1E-8 sigma = L2 / log(2); eg = 0.57721566490153286; %Euler-Mascheroni constant mu = L1 - sigma * eg; else sigma = -k*L2 / (gamma(1 - k) * (1 - 2^(k))); mu = L1 - sigma * ((gamma(1 - k) - 1) / k); endif paramhat = [k; sigma; mu]; if nargout > 1 paramci = NaN; endif endfunction #internal function to accurately evaluate (1 - 3^k)/(1 - 2^k) in the limit as k --> 0 function c = limdiv(a, b) # c = ifelse (abs(b) < 1E-8, log(3)/log(2), a ./ b); if abs(b) < 1E-8 c = log(3)/log(2); else c = a / b; endif endfunction %!xtest <31070> %! data = 1:50; %! [pfit, pci] = gevfit_lmom (data); %! expected_p = [-0.28 15.01 20.22]'; %! assert (pfit, expected_p, 0.1); statistics-release-1.6.3/inst/dist_fit/gevlike.m000066400000000000000000000301711456127120000217170ustar00rootroot00000000000000## Copyright (C) 2012 Nir Krakauer ## Copyright (C) 2022 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{nlogL} =} gevlike (@var{params}, @var{x}) ## @deftypefnx {statistics} {[@var{nlogL}, @var{acov}] =} gevlike (@var{params}, @var{x}) ## ## Negative log-likelihood for the generalized extreme value (GEV) distribution. ## ## @code{@var{nlogL} = gevlike (@var{params}, @var{x})} returns the negative ## log likelihood of the data in @var{x} corresponding to the GEV distribution ## with (1) shape parameter @var{k}, (2) scale parameter @var{sigma}, and (3) ## location parameter @var{mu} given in the three-element vector @var{params}. ## ## @code{[@var{nlogL}, @var{acov}] = gevlike (@var{params}, @var{x})} also ## returns the inverse of Fisher's information matrix, @var{acov}. If the input ## parameter values in @var{params} are the maximum likelihood estimates, the ## diagonal elements of @var{acov} are their asymptotic variances. ## ## When @qcode{@var{k} < 0}, the GEV is the type III extreme value distribution. ## When @qcode{@var{k} > 0}, the GEV distribution is the type II, or Frechet, ## extreme value distribution. If @var{W} has a Weibull distribution as ## computed by the @code{wblcdf} function, then @qcode{-@var{W}} has a type III ## extreme value distribution and @qcode{1/@var{W}} has a type II extreme value ## distribution. In the limit as @var{k} approaches @qcode{0}, the GEV is the ## mirror image of the type I extreme value distribution as computed by the ## @code{evcdf} function. ## ## The mean of the GEV distribution is not finite when @qcode{@var{k} >= 1}, and ## the variance is not finite when @qcode{@var{k} >= 1/2}. The GEV distribution ## has positive density only for values of @var{x} such that ## @qcode{@var{k} * (@var{x} - @var{mu}) / @var{sigma} > -1}. ## ## Further information about the generalized extreme value distribution can be ## found at ## @url{https://en.wikipedia.org/wiki/Generalized_extreme_value_distribution} ## ## @subheading References ## @enumerate ## @item ## Rolf-Dieter Reiss and Michael Thomas. @cite{Statistical Analysis of Extreme ## Values with Applications to Insurance, Finance, Hydrology and Other Fields}. ## Chapter 1, pages 16-17, Springer, 2007. ## @end enumerate ## ## @seealso{gevcdf, gevinv, gevpdf, gevrnd, gevfit, gevstat} ## @end deftypefn function [nlogL, acov] = gevlike (params, x) ## Check input arguments if (nargin < 2) error ("gevlike: function called with too few input arguments."); endif if (! isvector (x)) error ("gevlike: X must be a vector."); endif if (length (params) != 3) error ("gevlike: PARAMS must be a three-element vector."); endif k = params(1); sigma = params(2); mu = params(3); ## Calculate negative log likelihood [nll, k_terms] = gevnll (x, k, sigma, mu); nlogL = sum (nll(:)); ## Optionally calculate the first and second derivatives of the negative log ## likelihood with respect to parameters if (nargout > 1) [Grad, kk_terms] = gevgrad (x, k, sigma, mu, k_terms); FIM = gevfim (x, k, sigma, mu, k_terms, kk_terms); acov = inv (FIM); endif endfunction ## Internal function to calculate negative log likelihood for gevlike function [nlogL, k_terms] = gevnll (x, k, sigma, mu) k_terms = []; a = (x - mu) ./ sigma; if (all (k == 0)) nlogL = exp(-a) + a + log(sigma); else aa = k .* a; ## Use a series expansion to find the log likelihood more accurately ## when k is small if (min (abs (aa)) < 1E-3 && max (abs (aa)) < 0.5) k_terms = 1; sgn = 1; i = 0; while 1 sgn = -sgn; i++; newterm = (sgn / (i + 1)) * (aa .^ i); k_terms = k_terms + newterm; if (max (abs (newterm)) <= eps) break endif endwhile nlogL = exp (-a .* k_terms) + a .* (k + 1) .* k_terms + log (sigma); else b = 1 + aa; nlogL = b .^ (-1 ./ k) + (1 + 1 ./ k) .* log (b) + log (sigma); nlogL(b <= 0) = Inf; endif endif endfunction ## Calculate the gradient of the negative log likelihood of x x with respect ## to the parameters of the generalized extreme value distribution for gevlike function [G, kk_terms] = gevgrad (x, k, sigma, mu, k_terms) kk_terms = []; G = ones(3, 1); ## Use the expressions for first derivatives that are the limits as k --> 0 if (k == 0) a = (x - mu) ./ sigma; f = exp(-a) - 1; ## k g = a .* (1 + a .* f / 2); G(1) = sum(g(:)); ## sigma g = (a .* f + 1) ./ sigma; G(2) = sum(g(:)); ## mu g = f ./ sigma; G(3) = sum(g(:)); return endif a = (x - mu) ./ sigma; b = 1 + k .* a; ## Negative log likelihood is locally infinite if (any (b <= 0)) G(:) = 0; return endif ## k c = log(b); d = 1 ./ k + 1; ## Use a series expansion to find the gradient more accurately when k is small if (nargin > 4 && ! isempty (k_terms)) aa = k .* a; f = exp (-a .* k_terms); kk_terms = 0.5; sgn = 1; i = 0; while 1 sgn = -sgn; i++; newterm = (sgn * (i + 1) / (i + 2)) * (aa .^ i); kk_terms = kk_terms + newterm; if (max (abs (newterm)) <= eps) break endif endwhile g = a .* ((a .* kk_terms) .* (f - 1 - k) + k_terms); else g = (c ./ k - a ./ b) ./ (k .* b .^ (1/k)) - c ./ (k .^ 2) + a .* d ./ b; endif G(1) = sum(g(:)); ## sigma ## Use a series expansion to find the gradient more accurately when k is small if nargin > 4 && ~isempty(k_terms) g = (1 - a .* (a .* k .* kk_terms - k_terms) .* (f - k - 1)) ./ sigma; else g = (a .* b .^ (-d) - (k + 1) .* a ./ b + 1) ./ sigma; endif G(2) = sum(g(:)); ## mu ## Use a series expansion to find the gradient more accurately when k is small if (nargin > 4 && ! isempty (k_terms)) g = - (a .* k .* kk_terms - k_terms) .* (f - k - 1) ./ sigma; else g = (b .^ (-d) - (k + 1) ./ b) ./ sigma; end G(3) = sum(g(:)); endfunction ## Internal function to calculate the Fisher information matrix for gevlike function ACOV = gevfim (x, k, sigma, mu, k_terms, kk_terms) ACOV = ones(3); ## Use the expressions for second derivatives that are the limits as k --> 0 if (k == 0) ## k, k a = (x - mu) ./ sigma; f = exp(-a); der = (a .^ 2) .* (a .* (a/4 - 2/3) .* f + 2/3 * a - 1); ACOV(1, 1) = sum(der(:)); ## sigma, sigma der = (sigma .^ -2) .* (a .* ((a - 2) .* f + 2) - 1); ACOV(2, 2) = sum(der(:)); ## mu, mu der = (sigma .^ -2) .* f; ACOV(3, 3) = sum(der(:)); ## k, sigma der = (-a ./ sigma) .* (a .* (1 - a/2) .* f - a + 1); ACOV(1, 2) = ACOV(2, 1) = sum(der(:)); ## k, mu der = (-1 ./ sigma) .* (a .* (1 - a/2) .* f - a + 1); ACOV(1, 3) = ACOV(3, 1) = sum(der(:)); ## sigma, mu der = (1 + (a - 1) .* f) ./ (sigma .^ 2); ACOV(2, 3) = ACOV(3, 2) = sum(der(:)); return endif ## General case z = 1 + k .* (x - mu) ./ sigma; ## k, k a = (x - mu) ./ sigma; b = k .* a + 1; c = log(b); d = 1 ./ k + 1; ## Use a series expansion to find the derivatives more accurately ## when k is small if (nargin > 5 && ! isempty (kk_terms)) aa = k .* a; f = exp (-a .* k_terms); kkk_terms = 2/3; sgn = 1; i = 0; while 1 sgn = -sgn; i++; newterm = (sgn * (i + 1) * (i + 2) / (i + 3)) * (aa .^ i); kkk_terms = kkk_terms + newterm; if (max (abs (newterm)) <= eps) break endif endwhile der = (a .^ 2) .* (a .* (a .* kk_terms .^ 2 - kkk_terms) .* ... f + a .* (1 + k) .* kkk_terms - 2 * kk_terms); else der = ((((c ./ k.^2) - (a ./ (k .* b))) .^ 2) ./ (b .^ (1 ./ k))) + ... ((-2*c ./ k.^3) + (2*a ./ (k.^2 .* b)) + ((a ./ b) .^ 2 ./ k)) ./ ... (b .^ (1 ./ k)) + 2*c ./ k.^3 - (2*a ./ (k.^2 .* b)) - (d .* (a ./ b) .^ 2); endif der(z <= 0) = 0; # no probability mass in this region ACOV(1, 1) = sum (der(:)); ## sigma, sigma ## Use a series expansion to find the derivatives more accurately ## when k is small if (nargin > 5 && ! isempty (kk_terms)) der = ((-2*a .* k_terms + 4 * a .^ 2 .* k .* kk_terms - a .^ 3 .* ... (k .^ 2) .* kkk_terms) .* (f - k - 1) + f .* ((a .* ... (k_terms - a .* k .* kk_terms)) .^ 2) - 1) ./ (sigma .^ 2); else der = (sigma .^ -2) .* (-2 * a .* b .^ (-d) + d .* k .* a .^ 2 .* ... (b .^ (-d-1)) + 2 .* d .* k .* a ./ b - d .* (k .* a ./ b) .^ 2 - 1); end der(z <= 0) = 0; # no probability mass in this region ACOV(2, 2) = sum (der(:)); ## mu, mu ## Use a series expansion to find the derivatives more accurately ## when k is small if (nargin > 5 && ! isempty (kk_terms)) der = (f .* (a .* k .* kk_terms - k_terms) .^ 2 - a .* k .^ 2 .* ... kkk_terms .* (f - k - 1)) ./ (sigma .^ 2); else der = (d .* (sigma .^ -2)) .* (k .* (b .^ (-d-1)) - (k ./ b) .^ 2); endif der(z <= 0) = 0; # no probability mass in this region ACOV(3, 3) = sum (der(:)); ## k, mu ## Use a series expansion to find the derivatives more accurately ## when k is small if (nargin > 5 && ! isempty (kk_terms)) der = 2 * a .* kk_terms .* (f - 1 - k) - a .^ 2 .* k_terms .* ... kk_terms .* f + k_terms; der = -der ./ sigma; else der = ((b .^ (-d)) .* (c ./ k - a ./ b) ./ k - a .* (b .^ (-d-1)) + ... ((1 ./ k) - d) ./ b + a .* k .* d ./ (b .^ 2)) ./ sigma; endif der(z <= 0) = 0; # no probability mass in this region ACOV(1, 3) = ACOV(3, 1) = sum (der(:)); ## k, sigma der = a .* der; der(z <= 0) = 0; # no probability mass in this region ACOV(1, 2) = ACOV(2, 1) = sum (der(:)); ## sigma, mu ## Use a series expansion to find the derivatives more accurately ## when k is small if (nargin > 5 && ! isempty (kk_terms)) der = ((-k_terms + 3 * a .* k .* kk_terms - (a .* k) .^ 2 .* ... kkk_terms) .* (f - k - 1) + a .* (k_terms - a .* k .* ... kk_terms) .^ 2 .* f) ./ (sigma .^ 2); else der = (-(b .^ (-d)) + a .* k .* d .* (b .^ (-d-1)) + ... (d .* k ./ b) - a .* (k./b).^2 .* d) ./ (sigma .^ 2); end der(z <= 0) = 0; # no probability mass in this region ACOV(2, 3) = ACOV(3, 2) = sum (der(:)); endfunction ## Test output %!test %! x = 1; %! k = 0.2; %! sigma = 0.3; %! mu = 0.5; %! [L, C] = gevlike ([k sigma mu], x); %! expected_L = 0.75942; %! expected_C = [-0.12547 1.77884 1.06731; 1.77884 16.40761 8.48877; 1.06731 8.48877 0.27979]; %! assert (L, expected_L, 0.001); %! assert (C, inv (expected_C), 0.001); %!test %! x = 1; %! k = 0; %! sigma = 0.3; %! mu = 0.5; %! [L, C] = gevlike ([k sigma mu], x); %! expected_L = 0.65157; %! expected_C = [0.090036 3.41229 2.047337; 3.412229 24.760027 12.510190; 2.047337 12.510190 2.098618]; %! assert (L, expected_L, 0.001); %! assert (C, inv (expected_C), 0.001); %!test %! x = -5:-1; %! k = -0.2; %! sigma = 0.3; %! mu = 0.5; %! [L, C] = gevlike ([k sigma mu], x); %! expected_L = 3786.4; %! expected_C = [1.6802e-07, 4.6110e-06, 8.7297e-05; ... %! 4.6110e-06, 7.5693e-06, 1.2034e-05; ... %! 8.7297e-05, 1.2034e-05, -0.0019125]; %! assert (L, expected_L, -0.001); %! assert (C, expected_C, -0.001); ## Test input validation %!error gevlike (3.25) %!error gevlike ([1, 2, 3], ones (2)) %!error ... %! gevlike ([1, 2], [1, 3, 5, 7]) %!error ... %! gevlike ([1, 2, 3, 4], [1, 3, 5, 7]) statistics-release-1.6.3/inst/dist_fit/gpfit.m000066400000000000000000000253111456127120000214020ustar00rootroot00000000000000## Copyright (C) 2022-2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## Octave is free software; you can redistribute it and/or modify it ## under the terms of the GNU General Public License as published by ## the Free Software Foundation; either version 3 of the License, or (at ## your option) any later version. ## ## Octave is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with Octave; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{paramhat} =} gpfit (@var{x}) ## @deftypefnx {statistics} {[@var{paramhat}, @var{paramci}] =} gpfit (@var{x}) ## @deftypefnx {statistics} {[@var{paramhat}, @var{paramci}] =} gpfit (@var{x}, @var{alpha}) ## @deftypefnx {statistics} {[@dots{}] =} gpfit (@var{x}, @var{alpha}, @var{options}) ## ## Estimate parameters and confidence intervals for the generalized Pareto ## distribution. ## ## @code{@var{paramhat} = gpfit (@var{x})} returns the maximum likelihood ## estimates of the parameters of the generalized Pareto distribution given the ## data in @var{x}. @qcode{@var{paramhat}(1)} is the shape parameter, @var{k}, ## and @qcode{@var{paramhat}(2)} is the scale parameter, @var{sigma}. Other ## functions for the generalized Pareto, such as @code{gpcdf}, allow a location ## parameter, @var{mu}. However, @code{gpfit} does not estimate a location ## parameter, and it must be assumed known, and subtracted from @var{x} before ## calling @code{gpfit}. ## ## @code{[@var{paramhat}, @var{paramci}] = gpfit (@var{x})} returns the 95% ## confidence intervals for the parameter estimates. ## ## @code{[@dots{}] = gpfit (@var{x}, @var{alpha})} also returns the ## @qcode{100 * (1 - @var{alpha})} percent confidence intervals for the ## parameter estimates. By default, the optional argument @var{alpha} is ## 0.05 corresponding to 95% confidence intervals. Pass in @qcode{[]} for ## @var{alpha} to use the default values. ## ## @code{[@dots{}] = gpfit (@var{x}, @var{alpha}, @var{options})} specifies ## control parameters for the iterative algorithm used to compute ML estimates ## with the @code{fminsearch} function. @var{options} is a structure with the ## following fields and their default values: ## @itemize ## @item @qcode{@var{options}.Display = "off"} ## @item @qcode{@var{options}.MaxFunEvals = 400} ## @item @qcode{@var{options}.MaxIter = 200} ## @item @qcode{@var{options}.TolBnd = 1e-6} ## @item @qcode{@var{options}.TolFun = 1e-6} ## @item @qcode{@var{options}.TolX = 1e-6} ## @end itemize ## ## When @qcode{@var{k} = 0} and @qcode{@var{mu} = 0}, the Generalized Pareto CDF ## is equivalent to the exponential distribution. When @qcode{@var{k} > 0} and ## @code{@var{mu} = @var{k} / @var{k}} the Generalized Pareto is equivalent to ## the Pareto distribution. The mean of the Generalized Pareto is not finite ## when @qcode{@var{k} >= 1} and the variance is not finite when ## @qcode{@var{k} >= 1/2}. When @qcode{@var{k} >= 0}, the Generalized Pareto ## has positive density for @qcode{@var{x} > @var{mu}}, or, when ## @qcode{@var{mu} < 0}, for ## @qcode{0 <= (@var{x} - @var{mu}) / @var{sigma} <= -1 / @var{k}}. ## ## Further information about the generalized Pareto distribution can be found at ## @url{https://en.wikipedia.org/wiki/Generalized_Pareto_distribution} ## ## @seealso{gpcdf, gpinv, gppdf, gprnd, gplike, gpstat} ## @end deftypefn function [paramhat, paramci] = gpfit (x, alpha, options) ## Check input arguments, X must be a vector of positive values if (! isvector (x)) error ("gpfit: X must be a vector."); endif if (any (x <= 0)) error ("gpfit: X must contain only positive values."); endif ## Add default value for alpha if not supplied if (nargin < 2 || isempty (alpha)) alpha = 0.05; endif ## Check for valid value of alpha if (! isscalar (alpha) || ! isnumeric (alpha) || alpha <= 0 || alpha >= 1) error ("gpfit: wrong value for alpha."); endif ## Add default values to OPTIONS structure if (nargin < 3 || isempty (options)) options.Display = "off"; options.MaxFunEvals = 400; options.MaxIter = 200; options.TolBnd = 1e-6; options.TolFun = 1e-6; options.TolX = 1e-6; elseif (! isstruct (options)) error ("gpfit: OPTIONS must be a structure for 'fminsearch' function."); endif if (! isfield (options, "Display")) options.Display = "off"; endif if (! isfield (options, "MaxFunEvals")) options.MaxFunEvals = 400; endif if (! isfield (options, "MaxIter")) options.MaxIter = 200; endif if (! isfield (options, "TolBnd")) options.TolBnd = 1e-6; endif if (! isfield (options, "TolFun")) options.TolFun = 1e-6; endif if (! isfield (options, "TolX")) options.TolX = 1e-6; endif ## Get class of X is_class = class (x); if (strcmp (is_class, "single")) x = double (x); endif ## Remove NaN from data and make a warning if (any (isnan (x))) x(isnan(x)) = []; warning ("gpfit: X contains NaN values, which are ignored."); endif ## Remove Inf from data and make a warning if (any (isinf (x))) x(isnan(x)) = []; warning ("gpfit: X contains Inf values, which are ignored."); endif ## Get sample size, max and range of X x_max = max (x); x_size = length (x); x_range = range (x); ## Check for appropriate sample size or all observations being equal if (x_size == 0) paramhat = NaN (1, 2, is_class); paramci = NaN (2, 2, is_class); warning ("gpfit: X contains no data."); return elseif (x_range < realmin (is_class)) if (x_max <= sqrt (realmax (is_class))) paramhat = cast ([NaN 0], is_class); endif paramci = [paramhat; paramhat]; warning ("gpfit: X contains constant data."); return endif ## Make an initial guess x_mean = mean (x); x_var = var (x); k0 = -0.5 .* (x_mean .^ 2 ./ x_var - 1); s0 = 0.5 .* x_mean .* (x_mean .^ 2 ./ x_var + 1); ## If initial guess fails, start with an exponential fit if (k0 < 0 && (x_max >= -s0 / k0)) k0 = 0; s0 = x_mean; endif paramhat = [k0, log(s0)]; ## Maximize the log-likelihood with respect to shape and log_scale. [paramhat, ~, err, output] = fminsearch (@negloglike, paramhat, options, x); paramhat(2) = exp (paramhat(2)); ## Check output of fminsearch and produce warnings or errors if applicable if (err == 0) if (output.funcCount >= options.MaxFunEvals) warning ("gpfit: reached evaluation limit."); else warning ("gpfit: reached iteration limit."); endif elseif (err < 0) error ("gpfit: no solution."); endif ## Check if converged to boundaries if ((paramhat(1) < 0) && (x_max > -paramhat(2)/paramhat(1) - options.TolBnd)) warning ("gpfit: converged to boundary 1."); reachedBnd = true; elseif (paramhat(1) <= -1 / 2) warning ("gpfit: converged to boundary 2."); reachedBnd = true; else reachedBnd = false; endif ## If second output argument is requested if (nargout > 1) if (! reachedBnd) probs = [alpha/2; 1-alpha/2]; [~, acov] = gplike (paramhat, x); se = sqrt (diag (acov))'; ## Compute the CI for shape using a normal distribution for khat. kci = norminv (probs, paramhat(1), se(1)); ## Compute the CI for scale using a normal approximation for ## log(sigmahat), and transform back to the original scale. lnsigci = norminv (probs, log (paramhat(2)), se(2) ./ paramhat(2)); paramci = [kci, exp(lnsigci)]; else paramci = [NaN, NaN; NaN, NaN]; endif endif ## Preserve class tu output arguments paramhat = cast (paramhat, is_class); if (nargout > 1) paramci = cast (paramci, is_class); endif endfunction ## Negative log-likelihood for the GP function nll = negloglike (paramhat, data) shape = paramhat(1); log_scale = paramhat(2); scale = exp (log_scale); sample_size = numel (data); z = data ./ scale; if (abs (shape) > eps) if (shape > 0 || max (z) < -1 / shape) nll = sample_size * log_scale + (1 + 1/shape) * sum (log1p (shape .* z)); else nll = Inf; endif else nll = sample_size * log_scale + sum (z); endif endfunction %!demo %! ## Sample 2 populations from different generalized Pareto distibutions %! ## Assume location parameter is known %! mu = 0; %! rand ("seed", 5); # for reproducibility %! r1 = gprnd (1, 2, mu, 20000, 1); %! rand ("seed", 2); # for reproducibility %! r2 = gprnd (3, 1, mu, 20000, 1); %! r = [r1, r2]; %! %! ## Plot them normalized and fix their colors %! hist (r, [0.1:0.2:100], 5); %! h = findobj (gca, "Type", "patch"); %! set (h(1), "facecolor", "r"); %! set (h(2), "facecolor", "c"); %! ylim ([0, 1]); %! xlim ([0, 5]); %! hold on %! %! ## Estimate their α and β parameters %! k_sigmaA = gpfit (r(:,1)); %! k_sigmaB = gpfit (r(:,2)); %! %! ## Plot their estimated PDFs %! x = [0.01, 0.1:0.2:18]; %! y = gppdf (x, k_sigmaA(1), k_sigmaA(2), mu); %! plot (x, y, "-pc"); %! y = gppdf (x, k_sigmaB(1), k_sigmaB(2), mu); %! plot (x, y, "-sr"); %! hold off %! legend ({"Normalized HIST of sample 1 with k=1 and σ=2", ... %! "Normalized HIST of sample 2 with k=2 and σ=2", ... %! sprintf("PDF for sample 1 with estimated k=%0.2f and σ=%0.2f", ... %! k_sigmaA(1), k_sigmaA(2)), ... %! sprintf("PDF for sample 3 with estimated k=%0.2f and σ=%0.2f", ... %! k_sigmaB(1), k_sigmaB(2))}) %! title ("Three population samples from different generalized Pareto distibutions") %! text (2, 0.7, "Known location parameter μ = 0") %! hold off ## Test output %!test %! shape = 5; scale = 2; %! x = gprnd (shape, scale, 0, 1, 100000); %! [hat, ci] = gpfit (x); %! assert (hat, [shape, scale], 1e-1); %! assert (ci, [shape, scale; shape, scale], 2e-1); %!test %! shape = 1; scale = 1; %! x = gprnd (shape, scale, 0, 1, 100000); %! [hat, ci] = gpfit (x); %! assert (hat, [shape, scale], 1e-1); %! assert (ci, [shape, scale; shape, scale], 1e-1); %!test %! shape = 3; scale = 2; %! x = gprnd (shape, scale, 0, 1, 100000); %! [hat, ci] = gpfit (x); %! assert (hat, [shape, scale], 1e-1); %! assert (ci, [shape, scale; shape, scale], 1e-1); ## Test input validation %!error gpfit (ones (2)) %!error gpfit ([-1, 2]) %!error gpfit ([0, 1, 2]) %!error gpfit ([1, 2], 0) %!error gpfit ([1, 2], 1.2) %!error ... %! gpfit ([1:10], 0.05, 5) statistics-release-1.6.3/inst/dist_fit/gplike.m000066400000000000000000000120731456127120000215450ustar00rootroot00000000000000## Copyright (C) 2022-2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## Octave is free software; you can redistribute it and/or modify it ## under the terms of the GNU General Public License as published by ## the Free Software Foundation; either version 3 of the License, or (at ## your option) any later version. ## ## Octave is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with Octave; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{nlogL} =} gplike (@var{params}, @var{x}) ## @deftypefnx {statistics} {[@var{nlogL}, @var{acov}] =} gplike (@var{params}, @var{x}) ## ## Negative log-likelihood for the generalized Pareto distribution. ## ## @code{@var{nlogL} = gplike (@var{params}, @var{x})} returns the negative ## log-likelihood of the data in @var{x} corresponding to the generalized Pareto ## distribution with (1) shape parameter @var{k} and (2) scale parameter ## @var{sigma} given in the two-element vector @var{params}. @code{gplike} ## does not allow a location parameter and it must be assumed known, and ## subtracted from @var{x} before calling @code{gplike}. ## ## @code{[@var{nlogL}, @var{acov}] = gplike (@var{params}, @var{x})} returns ## the inverse of Fisher's information matrix, @var{acov}. If the input ## parameter values in @var{params} are the maximum likelihood estimates, the ## diagonal elements of @var{acov} are their asymptotic variances. @var{acov} ## is based on the observed Fisher's information, not the expected information. ## ## When @qcode{@var{k} = 0} and @qcode{@var{mu} = 0}, the Generalized Pareto CDF ## is equivalent to the exponential distribution. When @qcode{@var{k} > 0} and ## @code{@var{mu} = @var{k} / @var{k}} the Generalized Pareto is equivalent to ## the Pareto distribution. The mean of the Generalized Pareto is not finite ## when @qcode{@var{k} >= 1} and the variance is not finite when ## @qcode{@var{k} >= 1/2}. When @qcode{@var{k} >= 0}, the Generalized Pareto ## has positive density for @qcode{@var{x} > @var{mu}}, or, when ## @qcode{@var{mu} < 0}, for ## @qcode{0 <= (@var{x} - @var{mu}) / @var{sigma} <= -1 / @var{k}}. ## ## Further information about the generalized Pareto distribution can be found at ## @url{https://en.wikipedia.org/wiki/Generalized_Pareto_distribution} ## ## @seealso{gpcdf, gpinv, gppdf, gprnd, gpfit, gpstat} ## @end deftypefn function [nlogL, acov] = gplike (params, x) ## Check input arguments if (nargin < 2) error ("gplike: function called with too few input arguments."); endif if (! isvector (x)) error ("gplike: X must be a vector."); endif if (numel (params) != 2) error ("gplike: PARAMS must be a two-element vector."); endif ## Get SHAPE and SCALE parameters shape = params(1); scale = params(2); ## Get sample size and scale x sz = numel (x); z = x ./ scale; ## For SHAPE > 0 if (abs (shape) > eps) if (shape > 0 || max (z) < -1 / shape) sumLn = sum (log1p (shape .* z)); nlogL = sz * log (scale) + (1 + 1 / shape) .* sumLn; if (nargout > 1) v = z ./ (1 + shape .* z); sumv = sum (v); sumvsq = sum (v .^ 2); nH11 = 2 * sumLn ./ shape ^ 3 - ... 2 * sumv ./ shape ^ 2 - (1 + 1 / shape) .* sumvsq; nH12 = (-sumv + (shape + 1) .* sumvsq) ./ scale; nH22 = (-sz + 2 * (shape + 1) .* sumv - ... shape * (shape + 1) .* sumvsq) ./ scale ^ 2; acov = [nH22, -nH12; -nH12, nH11] / (nH11 * nH22 - nH12 * nH12); endif else ## The support of the GP when shape<0 is 0 < y < abs(scale/shape) nlogL = Inf; if (nargout > 1) acov = [NaN NaN; NaN NaN]; endif endif else # For shape = 0 ## Handle limit explicitly to prevent (1/0) * log(1) == Inf*0 == NaN. nlogL = sz*log(scale) + sum(z); if (nargout > 1) sumz = sum (z); sumzsq = sum (z .^ 2); sumzcb = sum (z .^ 3); nH11 = (2 / 3) * sumzcb - sumzsq; nH12 = (-sumz + sumzsq) ./ scale; nH22 = (-sz + 2 * sumz) ./ scale ^ 2; acov = [nH22, -nH12; -nH12, nH11] / (nH11 * nH22 - nH12 * nH12); endif endif endfunction ## Test output %!assert (gplike ([2, 3], 4), 3.047536764863501, 1e-14) %!assert (gplike ([1, 2], 4), 2.890371757896165, 1e-14) %!assert (gplike ([2, 3], [1:10]), 32.57864322725392, 1e-14) %!assert (gplike ([1, 2], [1:10]), 31.65666282460443, 1e-14) %!assert (gplike ([1, NaN], [1:10]), NaN) ## Test input validation %!error gplike () %!error gplike (1) %!error gplike ([1, 2], []) %!error gplike ([1, 2], ones (2)) %!error gplike (2, [1:10]) statistics-release-1.6.3/inst/dist_fit/gumbelfit.m000066400000000000000000000211161456127120000222460ustar00rootroot00000000000000## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{paramhat} =} gumbelfit (@var{x}) ## @deftypefnx {statistics} {[@var{paramhat}, @var{paramci}] =} gumbelfit (@var{x}) ## @deftypefnx {statistics} {[@var{paramhat}, @var{paramci}] =} gumbelfit (@var{x}, @var{alpha}) ## @deftypefnx {statistics} {[@dots{}] =} gumbelfit (@var{x}, @var{alpha}, @var{censor}) ## @deftypefnx {statistics} {[@dots{}] =} gumbelfit (@var{x}, @var{alpha}, @var{censor}, @var{freq}) ## @deftypefnx {statistics} {[@dots{}] =} gumbelfit (@var{x}, @var{alpha}, @var{censor}, @var{freq}, @var{options}) ## ## Estimate parameters and confidence intervals for Gumbel distribution. ## ## @code{@var{paramhat} = gumbelfit (@var{x})} returns the maximum likelihood ## estimates of the parameters of the Gumbel distribution (also known as ## the extreme value or the type I generalized extreme value distribution) given ## in @var{x}. @qcode{@var{paramhat}(1)} is the location parameter, @var{mu}, ## and @qcode{@var{paramhat}(2)} is the scale parameter, @var{beta}. ## ## @code{[@var{paramhat}, @var{paramci}] = gumbelfit (@var{x})} returns the 95% ## confidence intervals for the parameter estimates. ## ## @code{[@dots{}] = gumbelfit (@var{x}, @var{alpha})} also returns the ## @qcode{100 * (1 - @var{alpha})} percent confidence intervals for the ## parameter estimates. By default, the optional argument @var{alpha} is ## 0.05 corresponding to 95% confidence intervals. Pass in @qcode{[]} for ## @var{alpha} to use the default values. ## ## @code{[@dots{}] = gumbelfit (@var{x}, @var{alpha}, @var{censor})} accepts a ## boolean vector, @var{censor}, of the same size as @var{x} with @qcode{1}s for ## observations that are right-censored and @qcode{0}s for observations that are ## observed exactly. By default, or if left empty, ## @qcode{@var{censor} = zeros (size (@var{x}))}. ## ## @code{[@dots{}] = gumbelfit (@var{x}, @var{alpha}, @var{censor}, @var{freq})} ## accepts a frequency vector, @var{freq}, of the same size as @var{x}. ## @var{freq} typically contains integer frequencies for the corresponding ## elements in @var{x}, but it can contain any non-integer non-negative values. ## By default, or if left empty, @qcode{@var{freq} = ones (size (@var{x}))}. ## ## @code{[@dots{}] = gumbelfit (@dots{}, @var{options})} specifies control ## parameters for the iterative algorithm used to compute the maximum likelihood ## estimates. @var{options} is a structure with the following field and its ## default value: ## @itemize ## @item @qcode{@var{options}.TolX = 1e-6} ## @end itemize ## ## The Gumbel distribution is used to model the distribution of the maximum (or ## the minimum) of a number of samples of various distributions. This version ## is suitable for modeling maxima. For modeling minima, use the alternative ## extreme value fitting function, @code{evfit}. ## ## Further information about the Gumbel distribution can be found at ## @url{https://en.wikipedia.org/wiki/Gumbel_distribution} ## ## @seealso{gumbelcdf, gumbelinv, gumbelpdf, gumbelrnd, gumbellike, gumbelstat, ## evfit} ## @end deftypefn function [paramhat, paramci] = gumbelfit (x, alpha, censor, freq, options) ## Check X for being a double precision vector if (! isvector (x) || ! isa (x, "double")) error ("gumbelfit: X must be a double-precision vector."); endif ## Check that X does not contain missing values (NaNs) if (any (isnan (x))) error ("gumbelfit: X must NOT contain missing values (NaNs)."); endif ## Check alpha if (nargin > 1) if (! isscalar (alpha) || ! isreal (alpha) || alpha <= 0 || alpha >= 1) error ("gumbelfit: wrong value for ALPHA."); endif else alpha = 0.05; endif ## Check censor vector if (nargin > 2) if (! isempty (censor) && ! all (size (censor) == size (x))) error ("gumbelfit: X and CENSOR vectors mismatch."); endif else censor = zeros (size (x)); endif ## Check frequency vector if (nargin > 3) if (! isempty (freq) && ! all (size (freq) == size (x))) error ("gumbelfit: X and FREQ vectors mismatch."); endif ## Remove elements with zero frequency (if applicable) rm = find (freq == 0); if (length (rm) > 0) x(rm) = []; censor(rm) = []; freq(rm) = []; endif else freq = ones (size (x)); endif ## Get options structure or add defaults if (nargin > 4) if (! isstruct (options) || ! isfield (options, "TolX")) error (strcat (["gumbelfit: 'options' 5th argument must be a"], ... [" structure with 'TolX' field present."])); endif else options.TolX = 1e-6; endif ## If X is a column vector, make X, CENSOR, and FREQ row vectors if (size (x, 1) > 1) x = x(:)'; censor = censor(:)'; freq = freq(:)'; endif ## Call evfit to do the actual computation on the negative X try [paramhat, paramci] = evfit (-x, alpha, censor, freq, options); catch error ("gumbelfit: no solution for maximum likelihood estimates."); end_try_catch ## Flip sign on estimated parameter MU paramhat(1) = -paramhat(1); ## Flip sign on confidence intervals of parameter MU paramci(:,1) = -flip (paramci(:,1)); endfunction %!demo %! ## Sample 3 populations from different Gumbel distibutions %! rand ("seed", 1); # for reproducibility %! r1 = gumbelrnd (2, 5, 400, 1); %! rand ("seed", 11); # for reproducibility %! r2 = gumbelrnd (-5, 3, 400, 1); %! rand ("seed", 16); # for reproducibility %! r3 = gumbelrnd (14, 8, 400, 1); %! r = [r1, r2, r3]; %! %! ## Plot them normalized and fix their colors %! hist (r, 25, 0.32); %! h = findobj (gca, "Type", "patch"); %! set (h(1), "facecolor", "c"); %! set (h(2), "facecolor", "g"); %! set (h(3), "facecolor", "r"); %! ylim ([0, 0.28]) %! xlim ([-11, 50]); %! hold on %! %! ## Estimate their MU and BETA parameters %! mu_betaA = gumbelfit (r(:,1)); %! mu_betaB = gumbelfit (r(:,2)); %! mu_betaC = gumbelfit (r(:,3)); %! %! ## Plot their estimated PDFs %! x = [min(r(:)):max(r(:))]; %! y = gumbelpdf (x, mu_betaA(1), mu_betaA(2)); %! plot (x, y, "-pr"); %! y = gumbelpdf (x, mu_betaB(1), mu_betaB(2)); %! plot (x, y, "-sg"); %! y = gumbelpdf (x, mu_betaC(1), mu_betaC(2)); %! plot (x, y, "-^c"); %! legend ({"Normalized HIST of sample 1 with μ=2 and β=5", ... %! "Normalized HIST of sample 2 with μ=-5 and β=3", ... %! "Normalized HIST of sample 3 with μ=14 and β=8", ... %! sprintf("PDF for sample 1 with estimated μ=%0.2f and β=%0.2f", ... %! mu_betaA(1), mu_betaA(2)), ... %! sprintf("PDF for sample 2 with estimated μ=%0.2f and β=%0.2f", ... %! mu_betaB(1), mu_betaB(2)), ... %! sprintf("PDF for sample 3 with estimated μ=%0.2f and β=%0.2f", ... %! mu_betaC(1), mu_betaC(2))}) %! title ("Three population samples from different Gumbel distibutions") %! hold off ## Test output %!test %! x = 1:50; %! [paramhat, paramci] = gumbelfit (x); %! paramhat_out = [18.3188, 13.0509]; %! paramci_out = [14.4882, 10.5294; 22.1495, 16.1763]; %! assert (paramhat, paramhat_out, 1e-4); %! assert (paramci, paramci_out, 1e-4); %!test %! x = 1:50; %! [paramhat, paramci] = gumbelfit (x, 0.01); %! paramci_out = [13.2845, 9.8426; 23.3532, 17.3051]; %! assert (paramci, paramci_out, 1e-4); ## Test input validation %!error gumbelfit (ones (2,5)); %!error ... %! gumbelfit (single (ones (1,5))); %!error ... %! gumbelfit ([1, 2, 3, 4, NaN]); %!error gumbelfit ([1, 2, 3, 4, 5], 1.2); %!error ... %! gumbelfit ([1, 2, 3, 4, 5], 0.05, [1 1 0]); %!error ... %! gumbelfit ([1, 2, 3, 4, 5], 0.05, [], [1 1 0]); %!error ... %! gumbelfit ([1, 2, 3, 4, 5], 0.05, [], [], 2); statistics-release-1.6.3/inst/dist_fit/gumbellike.m000066400000000000000000000137071456127120000224170ustar00rootroot00000000000000## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{nlogL} =} gumbellike (@var{params}, @var{x}) ## @deftypefnx {statistics} {[@var{nlogL}, @var{avar}] =} gumbellike (@var{params}, @var{x}) ## @deftypefnx {statistics} {[@dots{}] =} gumbellike (@var{params}, @var{x}, @var{censor}) ## @deftypefnx {statistics} {[@dots{}] =} gumbellike (@var{params}, @var{x}, @var{censor}, @var{freq}) ## ## Negative log-likelihood for the extreme value distribution. ## ## @code{@var{nlogL} = gumbellike (@var{params}, @var{x})} returns the negative ## log likelihood of the data in @var{x} corresponding to the Gumbel ## distribution (also known as the extreme value or the type I generalized ## extreme value distribution) with (1) location parameter @var{mu} and (2) ## scale parameter @var{beta} given in the two-element vector @var{params}. ## ## @code{[@var{nlogL}, @var{acov}] = gumbellike (@var{params}, @var{x})} also ## returns the inverse of Fisher's information matrix, @var{acov}. If the input ## parameter values in @var{params} are the maximum likelihood estimates, the ## diagonal elements of @var{acov} are their asymptotic variances. ## ## @code{[@dots{}] = gumbellike (@var{params}, @var{x}, @var{censor})} accepts a ## boolean vector, @var{censor}, of the same size as @var{x} with @qcode{1}s for ## observations that are right-censored and @qcode{0}s for observations that are ## observed exactly. By default, or if left empty, ## @qcode{@var{censor} = zeros (size (@var{x}))}. ## ## @code{[@dots{}] = gumbellike (@var{params}, @var{x}, @var{censor}, @var{freq})} ## accepts a frequency vector, @var{freq}, of the same size as @var{x}. ## @var{freq} typically contains integer frequencies for the corresponding ## elements in @var{x}, but it can contain any non-integer non-negative values. ## By default, or if left empty, @qcode{@var{freq} = ones (size (@var{x}))}. ## ## The Gumbel distribution is used to model the distribution of the maximum (or ## the minimum) of a number of samples of various distributions. This version ## is suitable for modeling maxima. For modeling minima, use the alternative ## extreme value likelihood function, @code{evlike}. ## ## Further information about the Gumbel distribution can be found at ## @url{https://en.wikipedia.org/wiki/Gumbel_distribution} ## ## @seealso{gumbelcdf, gumbelinv, gumbelpdf, gumbelrnd, gumbelfit, gumbelstat, ## evlike} ## @end deftypefn function [nlogL, avar] = gumbellike (params, x, censor, freq) ## Check input arguments and add defaults if (nargin < 2) error ("gumbellike: too few input arguments."); endif if (numel (params) != 2) error ("gumbellike: wrong parameters length."); endif if (! isvector (x)) error ("gumbellike: X must be a vector."); endif if (nargin < 3 || isempty (censor)) censor = zeros (size (x)); elseif (! isequal (size (x), size (censor))) error ("gumbellike: X and CENSOR vectors mismatch."); endif if (nargin < 4 || isempty (freq)) freq = ones (size (x)); elseif (isequal (size (x), size (freq))) nulls = find (freq == 0); if (numel (nulls) > 0) x(nulls) = []; censor(nulls) = []; freq(nulls) = []; endif else error ("gumbellike: X and FREQ vectors mismatch."); endif ## Get mu and sigma values mu = params(1); sigma = params(2); ## sigma must be positive, otherwise make it NaN if (sigma <= 0) sigma = NaN; endif ## Compute the individual log-likelihood terms. Force a log(0)==-Inf for ## x from extreme right tail, instead of getting exp(Inf-Inf)==NaN. z = (x - mu) ./ sigma; expz = exp (z); L = (z - log (sigma)) .* (1 - censor) - expz; L(z == Inf) = -Inf; ## Neg-log-like is the sum of the individual contributions nlogL = -sum (freq .* L); ## Compute the negative hessian at the parameter values. ## Invert to get the observed information matrix. if (nargout == 2) unc = (1-censor); nH11 = sum(freq .* expz); nH12 = sum(freq .* ((z + 1) .* expz - unc)); nH22 = sum(freq .* (z .* (z+2) .* expz - ((2 .* z + 1) .* unc))); avar = (sigma .^ 2) * ... [nH22 -nH12; -nH12 nH11] / (nH11 * nH22 - nH12 * nH12); endif endfunction ## Test output %!test %! x = 1:50; %! [nlogL, avar] = gumbellike ([2.3, 1.2], x); %! avar_out = [-1.2778e-13, 3.1859e-15; 3.1859e-15, -7.9430e-17]; %! assert (nlogL, 3.242264755689906e+17, 1e-14); %! assert (avar, avar_out, 1e-3); %!test %! x = 1:50; %! [nlogL, avar] = gumbellike ([2.3, 1.2], x * 0.5); %! avar_out = [-7.6094e-05, 3.9819e-06; 3.9819e-06, -2.0836e-07]; %! assert (nlogL, 481898704.0472211, 1e-6); %! assert (avar, avar_out, 1e-3); %!test %! x = 1:50; %! [nlogL, avar] = gumbellike ([21, 15], x); %! avar_out = [11.73913876598908, -5.9546128523121216; ... %! -5.954612852312121, 3.708060045170236]; %! assert (nlogL, 223.7612479380652, 1e-13); %! assert (avar, avar_out, 1e-14); ## Test input validation %!error gumbellike ([12, 15]); %!error gumbellike ([12, 15, 3], [1:50]); %!error gumbellike ([12, 3], ones (10, 2)); %!error gumbellike ([12, 15], [1:50], [1, 2, 3]); %!error gumbellike ([12, 15], [1:50], [], [1, 2, 3]); statistics-release-1.6.3/inst/dist_fit/hnfit.m000066400000000000000000000140711456127120000214020ustar00rootroot00000000000000## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{paramhat} =} hnfit (@var{x}, @var{mu}) ## @deftypefnx {statistics} {[@var{paramhat}, @var{paramci}] =} hnfit (@var{x}, @var{mu}) ## @deftypefnx {statistics} {[@var{paramhat}, @var{paramci}] =} hnfit (@var{x}, @var{mu}, @var{alpha}) ## ## Estimate parameters and confidence intervals for the half-normal distribution. ## ## @code{@var{paramhat} = hnfit (@var{x})} returns the maximum likelihood ## estimates of the parameters of the half-normal distribution given the data in ## vector @var{x}. @qcode{@var{paramhat}(1)} is the location parameter, ## @var{mu}, and @qcode{@var{paramhat}(2)} is the scale parameter, @var{sigma}. ## Although @var{mu} is returned in the estimated @var{paramhat}, @code{hnfit} ## does not estimate the location parameter @var{mu}, and it must be assumed to ## be known. ## ## @code{[@var{paramhat}, @var{paramci}] = hnfit (@var{x}, @var{mu})} returns ## the 95% confidence intervals for the estimated scale parameter @var{sigma}. ## The first colummn of @var{paramci} includes the location parameter @var{mu} ## without any confidence bounds. ## ## @code{[@dots{}] = hnfit (@var{x}, @var{alpha})} also returns the ## @qcode{100 * (1 - @var{alpha})} percent confidence intervals of the estimated ## scale parameter. By default, the optional argument @var{alpha} is 0.05 ## corresponding to 95% confidence intervals. ## ## The half-normal CDF is only defined for @qcode{@var{x} >= @var{mu}}. ## ## Further information about the half-normal distribution can be found at ## @url{https://en.wikipedia.org/wiki/Half-normal_distribution} ## ## @seealso{hncdf, hninv, hnpdf, hnrnd, hnlike} ## @end deftypefn function [paramhat, paramci] = hnfit (x, mu, alpha) ## Check for valid number of input arguments if (nargin < 2) error ("hnfit: function called with too few input arguments."); endif ## Check X for being a vector if (isempty (x)) phat = nan (1, 2, class (x)); pci = nan (2, 2, class (x)); return elseif (! isvector (x) || ! isreal (x)) error ("hnfit: X must be a vector of real values."); endif ## Check for MU being a scalar real value if (! isscalar (mu) || ! isreal (mu)) error ("hnfit: MU must be a real scalar value."); endif ## Check X >= MU if (any (x < mu)) error ("hnfit: X cannot contain values less than MU."); endif ## Parse ALPHA argument or add default if (nargin > 2) if (! isscalar (alpha) || ! isreal (alpha) || alpha <= 0 || alpha >= 1) error ("hnfit: wrong value for ALPHA."); endif else alpha = 0.05; endif ## Estimate parameters sz = numel (x); x = x - mu; sigmahat = sqrt (sum (x .* x) ./ sz); paramhat = [mu, sigmahat]; ## Compute confidence intervals if (nargout == 2) chi2cr = chi2inv ([alpha/2, 1-alpha/2], sz); shatlo = sigmahat * sqrt (sz / chi2inv (1 - alpha / 2, sz)); shathi = sigmahat * sqrt (sz / chi2inv (alpha / 2, sz)); paramci = [mu, shatlo; mu, shathi]; endif endfunction %!demo %! ## Sample 2 populations from different half-normal distibutions %! rand ("seed", 1); # for reproducibility %! r1 = hnrnd (0, 5, 5000, 1); %! rand ("seed", 2); # for reproducibility %! r2 = hnrnd (0, 2, 5000, 1); %! r = [r1, r2]; %! %! ## Plot them normalized and fix their colors %! hist (r, [0.5:20], 1); %! h = findobj (gca, "Type", "patch"); %! set (h(1), "facecolor", "c"); %! set (h(2), "facecolor", "g"); %! hold on %! %! ## Estimate their shape parameters %! mu_sigmaA = hnfit (r(:,1), 0); %! mu_sigmaB = hnfit (r(:,2), 0); %! %! ## Plot their estimated PDFs %! x = [0:0.2:10]; %! y = hnpdf (x, mu_sigmaA(1), mu_sigmaA(2)); %! plot (x, y, "-pr"); %! y = hnpdf (x, mu_sigmaB(1), mu_sigmaB(2)); %! plot (x, y, "-sg"); %! xlim ([0, 10]) %! ylim ([0, 0.5]) %! legend ({"Normalized HIST of sample 1 with μ=0 and σ=5", ... %! "Normalized HIST of sample 2 with μ=0 and σ=2", ... %! sprintf("PDF for sample 1 with estimated μ=%0.2f and σ=%0.2f", ... %! mu_sigmaA(1), mu_sigmaA(2)), ... %! sprintf("PDF for sample 2 with estimated μ=%0.2f and σ=%0.2f", ... %! mu_sigmaB(1), mu_sigmaB(2))}) %! title ("Two population samples from different half-normal distibutions") %! hold off ## Test output %!test %! x = 1:20; %! [paramhat, paramci] = hnfit (x, 0); %! assert (paramhat, [0, 11.9791], 1e-4); %! assert (paramci, [0, 9.1648; 0, 17.2987], 1e-4); %!test %! x = 1:20; %! [paramhat, paramci] = hnfit (x, 0, 0.01); %! assert (paramci, [0, 8.4709; 0, 19.6487], 1e-4); ## Test input validation %!error hnfit () %!error hnfit (1) %!error hnfit ([0.2, 0.5+i], 0); %!error hnfit (ones (2,2) * 0.5, 0); %!error ... %! hnfit ([0.5, 1.2], [0, 1]); %!error ... %! hnfit ([0.5, 1.2], 5+i); %!error ... %! hnfit ([1:5], 2); %!error hnfit ([0.01:0.1:0.99], 0, 1.2); %!error hnfit ([0.01:0.1:0.99], 0, i); %!error hnfit ([0.01:0.1:0.99], 0, -1); %!error hnfit ([0.01:0.1:0.99], 0, [0.05, 0.01]); statistics-release-1.6.3/inst/dist_fit/hnlike.m000066400000000000000000000067121456127120000215470ustar00rootroot00000000000000## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{nlogL} =} hnlike (@var{params}, @var{x}) ## @deftypefnx {statistics} {[@var{nlogL}, @var{acov}] =} hnlike (@var{params}, @var{x}) ## ## Negative log-likelihood for the half-normal distribution. ## ## @code{@var{nlogL} = hnlike (@var{params}, @var{x})} returns the negative ## log likelihood of the data in @var{x} corresponding to the half-normal ## distribution with (1) location parameter @var{mu} and (2) scale parameter ## @var{sigma} given in the two-element vector @var{params}. ## ## @code{[@var{nlogL}, @var{acov}] = hnlike (@var{params}, @var{x})} returns ## the inverse of Fisher's information matrix, @var{acov}. If the input ## parameter values in @var{params} are the maximum likelihood estimates, the ## diagonal elements of @var{params} are their asymptotic variances. ## ## The half-normal CDF is only defined for @qcode{@var{x} >= @var{mu}}. ## ## Further information about the half-normal distribution can be found at ## @url{https://en.wikipedia.org/wiki/Half-normal_distribution} ## ## @seealso{hncdf, hninv, hnpdf, hnrnd, hnfit} ## @end deftypefn function [nlogL, acov] = hnlike (params, x) ## Check input arguments and add defaults if (nargin < 2) error ("hnlike: function called with too few input arguments."); endif if (numel (params) != 2) error ("hnlike: wrong parameters length."); endif ## Check X for being a vector if (isempty (x)) phat = nan (1, 2, class (x)); pci = nan (2, 2, class (x)); return elseif (! isvector (x) || ! isreal (x)) error ("hnlike: X must be a vector of real values."); endif ## Get MU and SIGMA parameters mu = params(1); sigma = params(2); ## Force X to column vector x = x(:); ## Return NaN for out of range parameters or data. sigma(sigma <= 0) = NaN; x(x < mu) = NaN; z = (x - mu) ./ sigma; ## Sum up the individual log-likelihood terms nlogL = -sum (-0.5 .* z .* z - log (sqrt (pi ./ 2) .* sigma)); ## Compute asymptotic covariance (if requested) if (nargout == 2) nH = -sum (1 - 3 .* z .* z); avar = (sigma .^ 2) ./ nH; acov = [0, 0; 0, avar]; endif endfunction ## Test output %!test %! x = 1:20; %! paramhat = hnfit (x, 0); %! [nlogL, acov] = hnlike (paramhat, x); %! assert (nlogL, 64.179177404891300, 1e-14); ## Test input validation %!error ... %! hnlike ([12, 15]); %!error hnlike ([12, 15, 3], [1:50]); %!error hnlike ([3], [1:50]); %!error ... %! hnlike ([0, 3], ones (2)); %!error ... %! hnlike ([0, 3], [1, 2, 3, 4, 5+i]); statistics-release-1.6.3/inst/dist_fit/invgfit.m000066400000000000000000000230361456127120000217410ustar00rootroot00000000000000## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{paramhat} =} invgfit (@var{x}) ## @deftypefnx {statistics} {[@var{paramhat}, @var{paramci}] =} invgfit (@var{x}) ## @deftypefnx {statistics} {[@var{paramhat}, @var{paramci}] =} invgfit (@var{x}, @var{alpha}) ## @deftypefnx {statistics} {[@dots{}] =} invgfit (@var{x}, @var{alpha}, @var{censor}) ## @deftypefnx {statistics} {[@dots{}] =} invgfit (@var{x}, @var{alpha}, @var{censor}, @var{freq}) ## @deftypefnx {statistics} {[@dots{}] =} invgfit (@var{x}, @var{alpha}, @var{censor}, @var{freq}, @var{options}) ## ## Estimate mean and confidence intervals for the inverse Gaussian distribution. ## ## @code{@var{mu0} = invgfit (@var{x})} returns the maximum likelihood ## estimates of the parameters of the inverse Gaussian distribution given the ## data in @var{x}. @qcode{@var{paramhat}(1)} is the scale parameter, @var{mu}, ## and @qcode{@var{paramhat}(2)} is the shape parameter, @var{lambda}. ## ## @code{[@var{paramhat}, @var{paramci}] = invgfit (@var{x})} returns the 95% ## confidence intervals for the parameter estimates. ## ## @code{[@dots{}] = invgfit (@var{x}, @var{alpha})} also returns the ## @qcode{100 * (1 - @var{alpha})} percent confidence intervals for the ## parameter estimates. By default, the optional argument @var{alpha} is ## 0.05 corresponding to 95% confidence intervals. Pass in @qcode{[]} for ## @var{alpha} to use the default values. ## ## @code{[@dots{}] = invgfit (@var{x}, @var{alpha}, @var{censor})} accepts a ## boolean vector, @var{censor}, of the same size as @var{x} with @qcode{1}s for ## observations that are right-censored and @qcode{0}s for observations that are ## observed exactly. By default, or if left empty, ## @qcode{@var{censor} = zeros (size (@var{x}))}. ## ## @code{[@dots{}] = invgfit (@var{x}, @var{alpha}, @var{censor}, @var{freq})} ## accepts a frequency vector, @var{freq}, of the same size as @var{x}. ## @var{freq} typically contains integer frequencies for the corresponding ## elements in @var{x}, but it can contain any non-integer non-negative values. ## By default, or if left empty, @qcode{@var{freq} = ones (size (@var{x}))}. ## ## @code{[@dots{}] = invgfit (@dots{}, @var{options})} specifies control ## parameters for the iterative algorithm used to compute ML estimates with the ## @code{fminsearch} function. @var{options} is a structure with the following ## fields and their default values: ## @itemize ## @item @qcode{@var{options}.Display = "off"} ## @item @qcode{@var{options}.MaxFunEvals = 400} ## @item @qcode{@var{options}.MaxIter = 200} ## @item @qcode{@var{options}.TolX = 1e-6} ## @end itemize ## ## Further information about the inverse Gaussian distribution can be found at ## @url{https://en.wikipedia.org/wiki/Inverse_Gaussian_distribution} ## ## @seealso{invgcdf, invginv, invgpdf, invgrnd, invglike} ## @end deftypefn function [paramhat, paramci] = invgfit (x, alpha, censor, freq, options) ## Check input arguments if (! isvector (x)) error ("invgfit: X must be a vector."); elseif (any (x <= 0)) error ("invgfit: X must contain only positive values."); endif ## Check alpha if (nargin < 2 || isempty (alpha)) alpha = 0.05; else if (! isscalar (alpha) || ! isreal (alpha) || alpha <= 0 || alpha >= 1) error ("invgfit: wrong value for ALPHA."); endif endif ## Check censor vector if (nargin < 3 || isempty (censor)) censor = zeros (size (x)); elseif (! isequal (size (x), size (censor))) error ("invgfit: X and CENSOR vectors mismatch."); endif ## Check frequency vector if (nargin < 4 || isempty (freq)) freq = ones (size (x)); elseif (! isequal (size (x), size (freq))) error ("invgfit: X and FREQ vectors mismatch."); endif ## Get options structure or add defaults if (nargin < 5) options.Display = "off"; options.MaxFunEvals = 400; options.MaxIter = 200; options.TolX = 1e-6; else if (! isstruct (options) || ! isfield (options, "Display") || ! isfield (options, "MaxFunEvals") || ! isfield (options, "MaxIter") || ! isfield (options, "TolX")) error (strcat (["invgfit: 'options' 5th argument must be a"], ... [" structure with 'Display', 'MaxFunEvals',"], ... [" 'MaxIter', and 'TolX' fields present."])); endif endif n_censored = sum (freq .* censor); ## Compute parameters for uncensored data if (n_censored == 0) ## Expand frequency vector (MATLAB does not do this, in R2018 at least) xf = []; for i = 1:numel (freq) xf = [xf, repmat(x(i), 1, freq(i))]; endfor xbar = mean (xf); paramhat = [xbar, (1 ./ mean (1 ./ x - 1 ./ xbar))]; else ## Use MLEs of the uncensored data as initial searching values x_uncensored = x(censor == 0); mu0 = mean (x_uncensored); lambda0 = 1 ./ mean (1 ./ x_uncensored - 1 ./ mu0); x0 = [mu0, lambda0]; ## Minimize negative log-likelihood to estimate parameters f = @(params) invglike (params, x, censor, freq); [paramhat, ~, err, output] = fminsearch (f, x0, options); ## Force positive parameter values paramhat = abs (paramhat); ## Handle errors if (err == 0) if (output.funcCount >= options.MaxFunEvals) msg = "invgfit: maximum number of function evaluations are exceeded."; warning (msg); elseif (output.iterations >= options.MaxIter) warning ("invgfit: maximum number of iterations are exceeded."); endif elseif (err < 0) error ("invgfit: no solution."); endif endif ## Compute CIs using a log normal approximation for parameters. if (nargout > 1) ## Compute asymptotic covariance [~, acov] = invglike (paramhat, x, censor, freq); ## Get standard errors stderr = sqrt (diag (acov))'; ## Get normal quantiles probs = [alpha/2; 1-alpha/2]; ## Compute CI paramci = norminv([probs, probs], [paramhat; paramhat], [stderr; stderr]); endif endfunction %!demo %! ## Sample 3 populations from different inverse Gaussian distibutions %! rand ("seed", 5); randn ("seed", 5); # for reproducibility %! r1 = invgrnd (1, 0.2, 2000, 1); %! rand ("seed", 2); randn ("seed", 2); # for reproducibility %! r2 = invgrnd (1, 3, 2000, 1); %! rand ("seed", 7); randn ("seed", 7); # for reproducibility %! r3 = invgrnd (3, 1, 2000, 1); %! r = [r1, r2, r3]; %! %! ## Plot them normalized and fix their colors %! hist (r, [0.1:0.1:3.2], 9); %! h = findobj (gca, "Type", "patch"); %! set (h(1), "facecolor", "c"); %! set (h(2), "facecolor", "g"); %! set (h(3), "facecolor", "r"); %! ylim ([0, 3]); %! xlim ([0, 3]); %! hold on %! %! ## Estimate their MU and LAMBDA parameters %! mu_lambdaA = invgfit (r(:,1)); %! mu_lambdaB = invgfit (r(:,2)); %! mu_lambdaC = invgfit (r(:,3)); %! %! ## Plot their estimated PDFs %! x = [0:0.1:3]; %! y = invgpdf (x, mu_lambdaA(1), mu_lambdaA(2)); %! plot (x, y, "-pr"); %! y = invgpdf (x, mu_lambdaB(1), mu_lambdaB(2)); %! plot (x, y, "-sg"); %! y = invgpdf (x, mu_lambdaC(1), mu_lambdaC(2)); %! plot (x, y, "-^c"); %! hold off %! legend ({"Normalized HIST of sample 1 with μ=1 and λ=0.5", ... %! "Normalized HIST of sample 2 with μ=2 and λ=0.3", ... %! "Normalized HIST of sample 3 with μ=4 and λ=0.5", ... %! sprintf("PDF for sample 1 with estimated μ=%0.2f and λ=%0.2f", ... %! mu_lambdaA(1), mu_lambdaA(2)), ... %! sprintf("PDF for sample 2 with estimated μ=%0.2f and λ=%0.2f", ... %! mu_lambdaB(1), mu_lambdaB(2)), ... %! sprintf("PDF for sample 3 with estimated μ=%0.2f and λ=%0.2f", ... %! mu_lambdaC(1), mu_lambdaC(2))}) %! title ("Three population samples from different inverse Gaussian distibutions") %! hold off ## Test output %!test %! paramhat = invgfit ([1:50]); %! paramhat_out = [25.5, 19.6973]; %! assert (paramhat, paramhat_out, 1e-4); %!test %! paramhat = invgfit ([1:5]); %! paramhat_out = [3, 8.1081]; %! assert (paramhat, paramhat_out, 1e-4); ## Test input validation %!error invgfit (ones (2,5)); %!error invgfit ([-1 2 3 4]); %!error invgfit ([1, 2, 3, 4, 5], 1.2); %!error invgfit ([1, 2, 3, 4, 5], 0); %!error invgfit ([1, 2, 3, 4, 5], "alpha"); %!error ... %! invgfit ([1, 2, 3, 4, 5], 0.05, [1 1 0]); %!error ... %! invgfit ([1, 2, 3, 4, 5], [], [1 1 0 1 1]'); %!error ... %! invgfit ([1, 2, 3, 4, 5], 0.05, zeros (1,5), [1 1 0]); %!error ... %! invgfit ([1, 2, 3, 4, 5], [], [], [1 1 0 1 1]'); %!error ... %! invgfit ([1, 2, 3, 4, 5], 0.05, [], [], 2); statistics-release-1.6.3/inst/dist_fit/invglike.m000066400000000000000000000163331456127120000221050ustar00rootroot00000000000000## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{nlogL} =} invglike (@var{params}, @var{x}) ## @deftypefnx {statistics} {[@var{nlogL}, @var{acov}] =} invglike (@var{params}, @var{x}) ## @deftypefnx {statistics} {[@dots{}] =} invglike (@var{params}, @var{x}, @var{censor}) ## @deftypefnx {statistics} {[@dots{}] =} invglike (@var{params}, @var{x}, @var{censor}, @var{freq}) ## ## Negative log-likelihood for the inverse Gaussian distribution. ## ## @code{@var{nlogL} = invglike (@var{params}, @var{x})} returns the negative ## log likelihood of the data in @var{x} corresponding to the inverse Gaussian ## distribution with (1) scale parameter @var{mu} and (2) shape parameter ## @var{lambda} given in the two-element vector @var{params}. ## ## @code{[@var{nlogL}, @var{acov}] = invglike (@var{params}, @var{x})} also ## returns the inverse of Fisher's information matrix, @var{acov}. If the input ## parameter values in @var{params} are the maximum likelihood estimates, the ## diagonal elements of @var{params} are their asymptotic variances. ## ## @code{[@dots{}] = invglike (@var{params}, @var{x}, @var{censor})} accepts a ## boolean vector, @var{censor}, of the same size as @var{x} with @qcode{1}s for ## observations that are right-censored and @qcode{0}s for observations that are ## observed exactly. By default, or if left empty, ## @qcode{@var{censor} = zeros (size (@var{x}))}. ## ## @code{[@dots{}] = invglike (@var{params}, @var{x}, @var{censor}, @var{freq})} ## accepts a frequency vector, @var{freq}, of the same size as @var{x}. ## @var{freq} typically contains integer frequencies for the corresponding ## elements in @var{x}, but it can contain any non-integer non-negative values. ## By default, or if left empty, @qcode{@var{freq} = ones (size (@var{x}))}. ## ## Further information about the inverse Gaussian distribution can be found at ## @url{https://en.wikipedia.org/wiki/Inverse_Gaussian_distribution} ## ## @seealso{invgcdf, invginv, invgpdf, invgrnd, invgfit} ## @end deftypefn function [nlogL, acov] = invglike (params, x, censor, freq) ## Check input arguments if (nargin < 2) error ("invglike: function called with too few input arguments."); endif if (! isvector (x)) error ("invglike: X must be a vector."); endif if (any (x < 0)) error ("invglike: X must have positive values."); endif if (length (params) != 2) error ("invglike: PARAMS must be a two-element vector."); endif ## Check censor vector if (nargin < 3 || isempty (censor)) censor = zeros (size (x)); elseif (! isequal (size (x), size (censor))) error ("invglike: X and CENSOR vector mismatch."); endif ## Check frequency vector if (nargin < 4 || isempty (freq)) freq = ones (size (x)); elseif (! isequal (size (x), size (freq))) error ("invglike: X and FREQ vector mismatch."); endif ## Get parameters mu = params(1); lambda = params(2); L = 0.5 .* log (lambda ./ (2 * pi)) - 1.5 .* log (x) ... -lambda .* (x ./ mu - 1) .^ 2 ./ (2 .* x); n_censored = sum (freq .* censor); ## Handle censored data if (n_censored > 0) censored = (censor == 1); x_censored = x(censored); sqrt_lx = sqrt (lambda ./ x_censored); z_censored = -(x_censored ./ mu - 1) .* sqrt_lx; w_censored = -(x_censored ./ mu + 1) .* sqrt_lx; Fz = 0.5 .* erfc (-z_censored ./ sqrt (2)); Fw = 0.5 .* erfc (-w_censored ./ sqrt (2)); S_censored = Fz - exp (2 .* lambda ./ mu) .* Fw; L(censored) = log (S_censored); endif ## Sum up the neg log likelihood nlogL = -sum (freq .* L); ## Compute asymptotic covariance if (nargout > 1) ## Compute first order central differences of the log-likelihood gradient dp = 0.0001 .* max (abs (params), 1); ngrad_p1 = invg_grad (params + [dp(1), 0], x, censor, freq); ngrad_m1 = invg_grad (params - [dp(1), 0], x, censor, freq); ngrad_p2 = invg_grad (params + [0, dp(2)], x, censor, freq); ngrad_m2 = invg_grad (params - [0, dp(2)], x, censor, freq); ## Compute negative Hessian by normalizing the differences by the increment nH = [(ngrad_p1(:) - ngrad_m1(:))./(2 * dp(1)), ... (ngrad_p2(:) - ngrad_m2(:))./(2 * dp(2))]; ## Force neg Hessian being symmetric nH = 0.5 .* (nH + nH'); ## Check neg Hessian is positive definite [R, p] = chol (nH); if (p > 0) warning ("invglike: non positive definite Hessian matrix."); acov = NaN (2); return endif ## ACOV estimate is the negative inverse of the Hessian. Rinv = inv (R); acov = Rinv * Rinv; endif endfunction ## Helper function for computing negative gradient function ngrad = invg_grad (params, x, censor, freq) mu = params(1); lambda = params(2); dL1 = lambda .* (x - mu) ./ mu .^ 3; dL2 = 1 ./ (2 .* lambda) - (x ./ mu - 1) .^ 2 ./ (2 .* x); n_censored = sum (freq .* censor); if (n_censored > 0) censored = (censor == 1); x_censored = x(censored); sqrt_lx = sqrt (lambda ./ x_censored); exp_lmu = exp (2 .* lambda ./ mu); z_censored = -(x_censored ./ mu - 1) .* sqrt_lx; w_censored = -(x_censored ./ mu + 1) .* sqrt_lx; Fw = 0.5 .* erfc (-w_censored ./ sqrt (2)); fz = exp (-0.5 .* z_censored .^ 2) ./ sqrt (2 .* pi); fw = exp (-0.5 .* w_censored .^ 2) ./ sqrt (2 .* pi); dS1cen = (fz - exp_lmu .* fw) .* (x_censored ./ mu .^ 2) .* sqrt_lx ... + 2 .* Fw .* exp_lmu .* lambda ./ mu .^ 2; dS2cen = 0.5 .* (fz .* z_censored - exp_lmu .* fw .* w_censored) ... ./ lambda - 2 .* Fw .* exp_lmu ./ mu; dL1(censored) = dS1cen ./ Scen; dL2(censored) = dS2cen ./ Scen; endif ngrad = -[sum(freq .* dL1), sum(freq .* dL2)]; endfunction ## Test results %!test %! nlogL = invglike ([25.5, 19.6973], [1:50]); %! assert (nlogL, 219.1516, 1e-4); %!test %! nlogL = invglike ([3, 8.1081], [1:5]); %! assert (nlogL, 9.0438, 1e-4); ## Test input validation %!error invglike (3.25) %!error invglike ([5, 0.2], ones (2)) %!error invglike ([5, 0.2], [-1, 3]) %!error ... %! invglike ([1, 0.2, 3], [1, 3, 5, 7]) %!error ... %! invglike ([1.5, 0.2], [1:5], [0, 0, 0]) %!error ... %! invglike ([1.5, 0.2], [1:5], [0, 0, 0, 0, 0], [1, 1, 1]) %!error ... %! invglike ([1.5, 0.2], [1:5], [], [1, 1, 1]) statistics-release-1.6.3/inst/dist_fit/logifit.m000066400000000000000000000226501456127120000217310ustar00rootroot00000000000000## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{paramhat} =} logifit (@var{x}) ## @deftypefnx {statistics} {[@var{paramhat}, @var{paramci}] =} logifit (@var{x}) ## @deftypefnx {statistics} {[@var{paramhat}, @var{paramci}] =} logifit (@var{x}, @var{alpha}) ## @deftypefnx {statistics} {[@dots{}] =} logifit (@var{x}, @var{alpha}, @var{censor}) ## @deftypefnx {statistics} {[@dots{}] =} logifit (@var{x}, @var{alpha}, @var{censor}, @var{freq}) ## @deftypefnx {statistics} {[@dots{}] =} logifit (@var{x}, @var{alpha}, @var{censor}, @var{freq}, @var{options}) ## ## Estimate mean and confidence intervals for the logistic distribution. ## ## @code{@var{mu0} = logifit (@var{x})} returns the maximum likelihood ## estimates of the parameters of the logistic distribution given the data in ## @var{x}. @qcode{@var{paramhat}(1)} is the scale parameter, @var{mu}, and ## @qcode{@var{paramhat}(2)} is the shape parameter, @var{s}. ## ## @code{[@var{paramhat}, @var{paramci}] = logifit (@var{x})} returns the 95% ## confidence intervals for the parameter estimates. ## ## @code{[@dots{}] = logifit (@var{x}, @var{alpha})} also returns the ## @qcode{100 * (1 - @var{alpha})} percent confidence intervals for the ## parameter estimates. By default, the optional argument @var{alpha} is ## 0.05 corresponding to 95% confidence intervals. Pass in @qcode{[]} for ## @var{alpha} to use the default values. ## ## @code{[@dots{}] = logifit (@var{x}, @var{alpha}, @var{censor})} accepts a ## boolean vector, @var{censor}, of the same size as @var{x} with @qcode{1}s for ## observations that are right-censored and @qcode{0}s for observations that are ## observed exactly. By default, or if left empty, ## @qcode{@var{censor} = zeros (size (@var{x}))}. ## ## @code{[@dots{}] = logifit (@var{x}, @var{alpha}, @var{censor}, @var{freq})} ## accepts a frequency vector, @var{freq}, of the same size as @var{x}. ## @var{freq} typically contains integer frequencies for the corresponding ## elements in @var{x}, but it can contain any non-integer non-negative values. ## By default, or if left empty, @qcode{@var{freq} = ones (size (@var{x}))}. ## ## @code{[@dots{}] = logifit (@dots{}, @var{options})} specifies control ## parameters for the iterative algorithm used to compute ML estimates with the ## @code{fminsearch} function. @var{options} is a structure with the following ## fields and their default values: ## @itemize ## @item @qcode{@var{options}.Display = "off"} ## @item @qcode{@var{options}.MaxFunEvals = 400} ## @item @qcode{@var{options}.MaxIter = 200} ## @item @qcode{@var{options}.TolX = 1e-6} ## @end itemize ## ## Further information about the logistic distribution can be found at ## @url{https://en.wikipedia.org/wiki/Logistic_distribution} ## ## @seealso{logicdf, logiinv, logipdf, logirnd, logilike} ## @end deftypefn function [paramhat, paramci] = logifit (x, alpha, censor, freq, options) ## Check input arguments if (! isvector (x)) error ("logifit: X must be a vector."); endif ## Check alpha if (nargin < 2 || isempty (alpha)) alpha = 0.05; else if (! isscalar (alpha) || ! isreal (alpha) || alpha <= 0 || alpha >= 1) error ("logifit: wrong value for ALPHA."); endif endif ## Check censor vector if (nargin < 3 || isempty (censor)) censor = zeros (size (x)); elseif (! isequal (size (x), size (censor))) error ("logifit: X and CENSOR vectors mismatch."); endif ## Check frequency vector if (nargin < 4 || isempty (freq)) freq = ones (size (x)); elseif (! isequal (size (x), size (freq))) error ("logifit: X and FREQ vectors mismatch."); endif ## Get options structure or add defaults if (nargin < 5) options.Display = "off"; options.MaxFunEvals = 400; options.MaxIter = 200; options.TolX = 1e-6; else if (! isstruct (options) || ! isfield (options, "Display") || ! isfield (options, "MaxFunEvals") || ! isfield (options, "MaxIter") || ! isfield (options, "TolX")) error (strcat (["logifit: 'options' 5th argument must be a"], ... [" structure with 'Display', 'MaxFunEvals',"], ... [" 'MaxIter', and 'TolX' fields present."])); endif endif ## Expand frequency and censor vectors (if necessary) if (! all (freq == 1)) xf = []; cf = []; for i = 1:numel (freq) xf = [xf, repmat(x(i), 1, freq(i))]; cf = [cf, repmat(censor(i), 1, freq(i))]; endfor x = xf; freq = ones (size (x)); censor = cf; endif ## Use MLEs of the uncensored data as initial searching values x_uncensored = x(censor == 0); mu0 = mean (x_uncensored); s0 = std (x_uncensored) .* sqrt (3) ./ pi; x0 = [mu0, s0]; ## Minimize negative log-likelihood to estimate parameters f = @(params) logilike (params, x, censor, freq); [paramhat, ~, err, output] = fminsearch (f, x0, options); ## Handle errors if (err == 0) if (output.funcCount >= options.MaxFunEvals) msg = "logifit: maximum number of function evaluations are exceeded."; warning (msg); elseif (output.iterations >= options.MaxIter) warning ("logifit: maximum number of iterations are exceeded."); endif elseif (err < 0) error ("logifit: no solution."); endif ## Compute CIs using a log normal approximation for parameters. if (nargout > 1) ## Compute asymptotic covariance [~, acov] = logilike (paramhat, x, censor, freq); ## Get standard errors se = sqrt (diag (acov))'; ## Get normal quantiles probs = [alpha/2; 1-alpha/2]; ## Compute muci using a normal approximation paramci(:,1) = norminv (probs, paramhat(1), se(1)); ## Compute sci using a normal approximation for log (s) and transform back paramci(:,2) = exp (norminv (probs, log (paramhat(2)), log (se(2)))); endif endfunction %!demo %! ## Sample 3 populations from different logistic distibutions %! rand ("seed", 5) # for reproducibility %! r1 = logirnd (2, 1, 2000, 1); %! rand ("seed", 2) # for reproducibility %! r2 = logirnd (5, 2, 2000, 1); %! rand ("seed", 7) # for reproducibility %! r3 = logirnd (9, 4, 2000, 1); %! r = [r1, r2, r3]; %! %! ## Plot them normalized and fix their colors %! hist (r, [-6:20], 1); %! h = findobj (gca, "Type", "patch"); %! set (h(1), "facecolor", "c"); %! set (h(2), "facecolor", "g"); %! set (h(3), "facecolor", "r"); %! ylim ([0, 0.3]); %! xlim ([-5, 20]); %! hold on %! %! ## Estimate their MU and LAMBDA parameters %! mu_sA = logifit (r(:,1)); %! mu_sB = logifit (r(:,2)); %! mu_sC = logifit (r(:,3)); %! %! ## Plot their estimated PDFs %! x = [-5:0.5:20]; %! y = logipdf (x, mu_sA(1), mu_sA(2)); %! plot (x, y, "-pr"); %! y = logipdf (x, mu_sB(1), mu_sB(2)); %! plot (x, y, "-sg"); %! y = logipdf (x, mu_sC(1), mu_sC(2)); %! plot (x, y, "-^c"); %! hold off %! legend ({"Normalized HIST of sample 1 with μ=1 and s=0.5", ... %! "Normalized HIST of sample 2 with μ=2 and s=0.3", ... %! "Normalized HIST of sample 3 with μ=4 and s=0.5", ... %! sprintf("PDF for sample 1 with estimated μ=%0.2f and s=%0.2f", ... %! mu_sA(1), mu_sA(2)), ... %! sprintf("PDF for sample 2 with estimated μ=%0.2f and s=%0.2f", ... %! mu_sB(1), mu_sB(2)), ... %! sprintf("PDF for sample 3 with estimated μ=%0.2f and s=%0.2f", ... %! mu_sC(1), mu_sC(2))}) %! title ("Three population samples from different logistic distibutions") %! hold off ## Test output %!test %! paramhat = logifit ([1:50]); %! paramhat_out = [25.5, 8.7724]; %! assert (paramhat, paramhat_out, 1e-4); %!test %! paramhat = logifit ([1:5]); %! paramhat_out = [3, 0.8645]; %! assert (paramhat, paramhat_out, 1e-4); %!test %! paramhat = logifit ([1:6], [], [], [1 1 1 1 1 0]); %! paramhat_out = [3, 0.8645]; %! assert (paramhat, paramhat_out, 1e-4); %!test %! paramhat = logifit ([1:5], [], [], [1 1 1 1 2]); %! paramhat_out = logifit ([1:5, 5]); %! assert (paramhat, paramhat_out, 1e-4); ## Test input validation %!error logifit (ones (2,5)); %!error logifit ([1, 2, 3, 4, 5], 1.2); %!error logifit ([1, 2, 3, 4, 5], 0); %!error logifit ([1, 2, 3, 4, 5], "alpha"); %!error ... %! logifit ([1, 2, 3, 4, 5], 0.05, [1 1 0]); %!error ... %! logifit ([1, 2, 3, 4, 5], [], [1 1 0 1 1]'); %!error ... %! logifit ([1, 2, 3, 4, 5], 0.05, zeros (1,5), [1 1 0]); %!error ... %! logifit ([1, 2, 3, 4, 5], [], [], [1 1 0 1 1]'); %!error ... %! logifit ([1, 2, 3, 4, 5], 0.05, [], [], 2); statistics-release-1.6.3/inst/dist_fit/logilike.m000066400000000000000000000147731456127120000221020ustar00rootroot00000000000000## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{nlogL} =} logilike (@var{params}, @var{x}) ## @deftypefnx {statistics} {[@var{nlogL}, @var{acov}] =} logilike (@var{params}, @var{x}) ## @deftypefnx {statistics} {[@dots{}] =} logilike (@var{params}, @var{x}, @var{censor}) ## @deftypefnx {statistics} {[@dots{}] =} logilike (@var{params}, @var{x}, @var{censor}, @var{freq}) ## ## Negative log-likelihood for the logistic distribution. ## ## @code{@var{nlogL} = logilike (@var{params}, @var{x})} returns the negative ## log likelihood of the data in @var{x} corresponding to the logistic ## distribution with (1) location parameter @var{mu} and (2) scale parameter ## @var{s} given in the two-element vector @var{params}. ## ## @code{[@var{nlogL}, @var{acov}] = logilike (@var{params}, @var{x})} also ## returns the inverse of Fisher's information matrix, @var{acov}. If the input ## parameter values in @var{params} are the maximum likelihood estimates, the ## diagonal elements of @var{params} are their asymptotic variances. ## ## @code{[@dots{}] = logilike (@var{params}, @var{x}, @var{censor})} accepts a ## boolean vector, @var{censor}, of the same size as @var{x} with @qcode{1}s for ## observations that are right-censored and @qcode{0}s for observations that are ## observed exactly. By default, or if left empty, ## @qcode{@var{censor} = zeros (size (@var{x}))}. ## ## @code{[@dots{}] = logilike (@var{params}, @var{x}, @var{censor}, @var{freq})} ## accepts a frequency vector, @var{freq}, of the same size as @var{x}. ## @var{freq} typically contains integer frequencies for the corresponding ## elements in @var{x}, but it can contain any non-integer non-negative values. ## By default, or if left empty, @qcode{@var{freq} = ones (size (@var{x}))}. ## ## Further information about the logistic distribution can be found at ## @url{https://en.wikipedia.org/wiki/Logistic_distribution} ## ## @seealso{logicdf, logiinv, logipdf, logirnd, logifit} ## @end deftypefn function [nlogL, acov] = logilike (params, x, censor, freq) ## Check input arguments if (nargin < 2) error ("logilike: function called with too few input arguments."); endif if (! isvector (x)) error ("logilike: X must be a vector."); endif if (length (params) != 2) error ("logilike: PARAMS must be a two-element vector."); endif ## Check censor vector if (nargin < 3 || isempty (censor)) censor = zeros (size (x)); elseif (! isequal (size (x), size (censor))) error ("logilike: X and CENSOR vector mismatch."); endif ## Check frequency vector if (nargin < 4 || isempty (freq)) freq = ones (size (x)); elseif (! isequal (size (x), size (freq))) error ("logilike: X and FREQ vector mismatch."); endif ## Expand frequency and censor vectors (if necessary) if (! all (freq == 1)) xf = []; cf = []; for i = 1:numel (freq) xf = [xf, repmat(x(i), 1, freq(i))]; cf = [cf, repmat(censor(i), 1, freq(i))]; endfor x = xf; freq = ones (size (x)); censor = cf; endif ## Get parameters mu = params(1); s = params(2); z = (x - mu) ./ s; logclogitz = log (1 ./ (1 + exp (z))); k = (z > 700); if (any (k)) logclogitz(k) = z(k); endif L = z + 2 .* logclogitz - log (s); n_censored = sum (freq .* censor); ## Handle censored data if (n_censored > 0) censored = (censor == 1); L(censored) = logclogitz(censored); endif ## Sum up the neg log likelihood if (s < 0) nlogL = Inf; else nlogL = -sum (freq .* L); endif ## Compute asymptotic covariance if (nargout > 1) ## Compute first order central differences of the log-likelihood gradient dp = 0.0001 .* max (abs (params), 1); ngrad_p1 = logi_grad (params + [dp(1), 0], x, censor, freq); ngrad_m1 = logi_grad (params - [dp(1), 0], x, censor, freq); ngrad_p2 = logi_grad (params + [0, dp(2)], x, censor, freq); ngrad_m2 = logi_grad (params - [0, dp(2)], x, censor, freq); ## Compute negative Hessian by normalizing the differences by the increment nH = [(ngrad_p1(:) - ngrad_m1(:))./(2 * dp(1)), ... (ngrad_p2(:) - ngrad_m2(:))./(2 * dp(2))]; ## Force neg Hessian being symmetric nH = 0.5 .* (nH + nH'); ## Check neg Hessian is positive definite [R, p] = chol (nH); if (p > 0) warning ("logilike: non positive definite Hessian matrix."); acov = NaN (2); return endif ## ACOV estimate is the negative inverse of the Hessian. Rinv = inv (R); acov = Rinv * Rinv; endif endfunction ## Helper function for computing negative gradient function ngrad = logi_grad (params, x, censor, freq) mu = params(1); s = params(2); z = (x - mu) ./ s; logitz = 1 ./ (1 + exp (-z)); dL1 = (2 .* logitz - 1) ./ sigma; dL2 = z .* dL1 - 1 ./ sigma; n_censored = sum (freq .* censor); if (n_censored > 0) censored = (censor == 1); dL1(censored) = logitz(censored) ./ sigma; dL2(censored) = z(censored) .* dL1(censored); endif ngrad = -[sum(freq .* dL1), sum(freq .* dL2)]; endfunction ## Test results %!test %! nlogL = logilike ([25.5, 8.7725], [1:50]); %! assert (nlogL, 206.6769, 1e-4); %!test %! nlogL = logilike ([3, 0.8645], [1:5]); %! assert (nlogL, 9.0699, 1e-4); ## Test input validation %!error logilike (3.25) %!error logilike ([5, 0.2], ones (2)) %!error ... %! logilike ([1, 0.2, 3], [1, 3, 5, 7]) %!error ... %! logilike ([1.5, 0.2], [1:5], [0, 0, 0]) %!error ... %! logilike ([1.5, 0.2], [1:5], [0, 0, 0, 0, 0], [1, 1, 1]) %!error ... %! logilike ([1.5, 0.2], [1:5], [], [1, 1, 1]) statistics-release-1.6.3/inst/dist_fit/loglfit.m000066400000000000000000000236231456127120000217350ustar00rootroot00000000000000## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{paramhat} =} loglfit (@var{x}) ## @deftypefnx {statistics} {[@var{paramhat}, @var{paramci}] =} loglfit (@var{x}) ## @deftypefnx {statistics} {[@var{paramhat}, @var{paramci}] =} loglfit (@var{x}, @var{alpha}) ## @deftypefnx {statistics} {[@dots{}] =} loglfit (@var{x}, @var{alpha}, @var{censor}) ## @deftypefnx {statistics} {[@dots{}] =} loglfit (@var{x}, @var{alpha}, @var{censor}, @var{freq}) ## @deftypefnx {statistics} {[@dots{}] =} loglfit (@var{x}, @var{alpha}, @var{censor}, @var{freq}, @var{options}) ## ## Estimate mean and confidence intervals for the log-logistic distribution. ## ## @code{@var{mu0} = loglfit (@var{x})} returns the maximum likelihood ## estimates of the parameters of the log-logistic distribution given the data ## in @var{x}. @qcode{@var{paramhat}(1)} is the scale parameter, @var{a}, and ## @qcode{@var{paramhat}(2)} is the shape parameter, @var{b}. ## ## @code{[@var{paramhat}, @var{paramci}] = loglfit (@var{x})} returns the 95% ## confidence intervals for the parameter estimates. ## ## @code{[@dots{}] = loglfit (@var{x}, @var{alpha})} also returns the ## @qcode{100 * (1 - @var{alpha})} percent confidence intervals for the ## parameter estimates. By default, the optional argument @var{alpha} is ## 0.05 corresponding to 95% confidence intervals. Pass in @qcode{[]} for ## @var{alpha} to use the default values. ## ## @code{[@dots{}] = loglfit (@var{x}, @var{alpha}, @var{censor})} accepts a ## boolean vector, @var{censor}, of the same size as @var{x} with @qcode{1}s for ## observations that are right-censored and @qcode{0}s for observations that are ## observed exactly. By default, or if left empty, ## @qcode{@var{censor} = zeros (size (@var{x}))}. ## ## @code{[@dots{}] = loglfit (@var{x}, @var{alpha}, @var{censor}, @var{freq})} ## accepts a frequency vector, @var{freq}, of the same size as @var{x}. ## @var{freq} typically contains integer frequencies for the corresponding ## elements in @var{x}, but it can contain any non-integer non-negative values. ## By default, or if left empty, @qcode{@var{freq} = ones (size (@var{x}))}. ## ## @code{[@dots{}] = loglfit (@dots{}, @var{options})} specifies control ## parameters for the iterative algorithm used to compute ML estimates with the ## @code{fminsearch} function. @var{options} is a structure with the following ## fields and their default values: ## @itemize ## @item @qcode{@var{options}.Display = "off"} ## @item @qcode{@var{options}.MaxFunEvals = 400} ## @item @qcode{@var{options}.MaxIter = 200} ## @item @qcode{@var{options}.TolX = 1e-6} ## @end itemize ## ## Further information about the log-logistic distribution can be found at ## @url{https://en.wikipedia.org/wiki/Log-logistic_distribution} ## ## MATLAB compatibility: MATLAB uses an alternative parameterization given by ## the pair @math{μ, s}, i.e. @var{mu} and @var{s}, in analogy with the logistic ## distribution. Their relation to the @var{a} and @var{b} parameters is given ## below: ## ## @itemize ## @item @qcode{@var{a} = exp (@var{mu})} ## @item @qcode{@var{b} = 1 / @var{s}} ## @end itemize ## ## @seealso{loglcdf, loglinv, loglpdf, loglrnd, logllike} ## @end deftypefn function [paramhat, paramci] = loglfit (x, alpha, censor, freq, options) ## Check input arguments if (! isvector (x)) error ("loglfit: X must be a vector."); endif ## Check alpha if (nargin < 2 || isempty (alpha)) alpha = 0.05; else if (! isscalar (alpha) || ! isreal (alpha) || alpha <= 0 || alpha >= 1) error ("loglfit: wrong value for ALPHA."); endif endif ## Check censor vector if (nargin < 3 || isempty (censor)) censor = zeros (size (x)); elseif (! isequal (size (x), size (censor))) error ("loglfit: X and CENSOR vectors mismatch."); endif ## Check frequency vector if (nargin < 4 || isempty (freq)) freq = ones (size (x)); elseif (! isequal (size (x), size (freq))) error ("loglfit: X and FREQ vectors mismatch."); endif ## Get options structure or add defaults if (nargin < 5) options.Display = "off"; options.MaxFunEvals = 400; options.MaxIter = 200; options.TolX = 1e-6; else if (! isstruct (options) || ! isfield (options, "Display") || ! isfield (options, "MaxFunEvals") || ! isfield (options, "MaxIter") || ! isfield (options, "TolX")) error (strcat (["loglfit: 'options' 5th argument must be a"], ... [" structure with 'Display', 'MaxFunEvals',"], ... [" 'MaxIter', and 'TolX' fields present."])); endif endif ## Expand frequency and censor vectors (if necessary) if (! all (freq == 1)) xf = []; cf = []; for i = 1:numel (freq) xf = [xf, repmat(x(i), 1, freq(i))]; cf = [cf, repmat(censor(i), 1, freq(i))]; endfor x = xf; freq = ones (size (x)); censor = cf; endif ## Use MLEs of the uncensored data as initial searching values x_uncensored = x(censor == 0); a0 = mean (x_uncensored); b0 = 1 ./ (std (x_uncensored) .* sqrt (3) ./ pi); x0 = [a0, b0]; ## Minimize negative log-likelihood to estimate parameters f = @(params) logllike (params, x, censor, freq); [paramhat, ~, err, output] = fminsearch (f, x0, options); ## Force positive parameter values paramhat = abs (paramhat); ## Handle errors if (err == 0) if (output.funcCount >= options.MaxFunEvals) msg = "loglfit: maximum number of function evaluations are exceeded."; warning (msg); elseif (output.iterations >= options.MaxIter) warning ("loglfit: maximum number of iterations are exceeded."); endif elseif (err < 0) error ("loglfit: no solution."); endif ## Compute CIs using a log normal approximation for parameters. if (nargout > 1) ## Compute asymptotic covariance [~, acov] = logllike (paramhat, x, censor, freq); ## Get standard errors se = sqrt (diag (acov))'; ## Get normal quantiles probs = [alpha/2; 1-alpha/2]; ## Compute muci using a normal approximation paramci(:,1) = norminv (probs, paramhat(1), se(1)); ## Compute sci using a normal approximation for log (s) and transform back paramci(:,2) = exp (norminv (probs, log (paramhat(2)), log (se(2)))); endif endfunction %!demo %! ## Sample 3 populations from different log-logistic distibutions %! rand ("seed", 5) # for reproducibility %! r1 = loglrnd (1, 1, 2000, 1); %! rand ("seed", 2) # for reproducibility %! r2 = loglrnd (1, 2, 2000, 1); %! rand ("seed", 7) # for reproducibility %! r3 = loglrnd (1, 8, 2000, 1); %! r = [r1, r2, r3]; %! %! ## Plot them normalized and fix their colors %! hist (r, [0.05:0.1:2.5], 10); %! h = findobj (gca, "Type", "patch"); %! set (h(1), "facecolor", "c"); %! set (h(2), "facecolor", "g"); %! set (h(3), "facecolor", "r"); %! ylim ([0, 3.5]); %! xlim ([0, 2.0]); %! hold on %! %! ## Estimate their MU and LAMBDA parameters %! a_bA = loglfit (r(:,1)); %! a_bB = loglfit (r(:,2)); %! a_bC = loglfit (r(:,3)); %! %! ## Plot their estimated PDFs %! x = [0.01:0.1:2.01]; %! y = loglpdf (x, a_bA(1), a_bA(2)); %! plot (x, y, "-pr"); %! y = loglpdf (x, a_bB(1), a_bB(2)); %! plot (x, y, "-sg"); %! y = loglpdf (x, a_bC(1), a_bC(2)); %! plot (x, y, "-^c"); %! legend ({"Normalized HIST of sample 1 with α=1 and β=1", ... %! "Normalized HIST of sample 2 with α=1 and β=2", ... %! "Normalized HIST of sample 3 with α=1 and β=8", ... %! sprintf("PDF for sample 1 with estimated α=%0.2f and β=%0.2f", ... %! a_bA(1), a_bA(2)), ... %! sprintf("PDF for sample 2 with estimated α=%0.2f and β=%0.2f", ... %! a_bB(1), a_bB(2)), ... %! sprintf("PDF for sample 3 with estimated α=%0.2f and β=%0.2f", ... %! a_bC(1), a_bC(2))}) %! title ("Three population samples from different log-logistic distibutions") %! hold off ## Test output %!test %! paramhat = loglfit ([1:50]); %! paramhat_out = [exp(3.097175), 1/0.468525]; %! assert (paramhat, paramhat_out, 1e-4); %!test %! paramhat = loglfit ([1:5]); %! paramhat_out = [exp(1.01124), 1/0.336449]; %! assert (paramhat, paramhat_out, 1e-4); %!test %! paramhat = loglfit ([1:6], [], [], [1 1 1 1 1 0]); %! paramhat_out = [exp(1.01124), 1/0.336449]; %! assert (paramhat, paramhat_out, 1e-4); %!test %! paramhat = loglfit ([1:5], [], [], [1 1 1 1 2]); %! paramhat_out = loglfit ([1:5, 5]); %! assert (paramhat, paramhat_out, 1e-4); ## Test input validation %!error loglfit (ones (2,5)); %!error loglfit ([1, 2, 3, 4, 5], 1.2); %!error loglfit ([1, 2, 3, 4, 5], 0); %!error loglfit ([1, 2, 3, 4, 5], "alpha"); %!error ... %! loglfit ([1, 2, 3, 4, 5], 0.05, [1 1 0]); %!error ... %! loglfit ([1, 2, 3, 4, 5], [], [1 1 0 1 1]'); %!error ... %! loglfit ([1, 2, 3, 4, 5], 0.05, zeros (1,5), [1 1 0]); %!error ... %! loglfit ([1, 2, 3, 4, 5], [], [], [1 1 0 1 1]'); %!error ... %! loglfit ([1, 2, 3, 4, 5], 0.05, [], [], 2); statistics-release-1.6.3/inst/dist_fit/logllike.m000066400000000000000000000155611456127120000221010ustar00rootroot00000000000000## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{nlogL} =} logllike (@var{params}, @var{x}) ## @deftypefnx {statistics} {[@var{nlogL}, @var{acov}] =} logllike (@var{params}, @var{x}) ## @deftypefnx {statistics} {[@dots{}] =} logllike (@var{params}, @var{x}, @var{censor}) ## @deftypefnx {statistics} {[@dots{}] =} logllike (@var{params}, @var{x}, @var{censor}, @var{freq}) ## ## Negative log-likelihood for the log-logistic distribution. ## ## @code{@var{nlogL} = logllike (@var{params}, @var{x})} returns the negative ## log likelihood of the data in @var{x} corresponding to the log-logistic ## distribution with (1) scale parameter @var{a} and (2) shape parameter @var{b} ## given in the two-element vector @var{params}. ## ## @code{[@var{nlogL}, @var{acov}] = logllike (@var{params}, @var{x})} also ## returns the inverse of Fisher's information matrix, @var{acov}. If the input ## parameter values in @var{params} are the maximum likelihood estimates, the ## diagonal elements of @var{params} are their asymptotic variances. ## ## @code{[@dots{}] = logllike (@var{params}, @var{x}, @var{censor})} accepts a ## boolean vector, @var{censor}, of the same size as @var{x} with @qcode{1}s for ## observations that are right-censored and @qcode{0}s for observations that are ## observed exactly. By default, or if left empty, ## @qcode{@var{censor} = zeros (size (@var{x}))}. ## ## @code{[@dots{}] = logllike (@var{params}, @var{x}, @var{censor}, @var{freq})} ## accepts a frequency vector, @var{freq}, of the same size as @var{x}. ## @var{freq} typically contains integer frequencies for the corresponding ## elements in @var{x}, but it can contain any non-integer non-negative values. ## By default, or if left empty, @qcode{@var{freq} = ones (size (@var{x}))}. ## ## Further information about the log-logistic distribution can be found at ## @url{https://en.wikipedia.org/wiki/Log-logistic_distribution} ## ## MATLAB compatibility: MATLAB uses an alternative parameterization given by ## the pair @math{μ, s}, i.e. @var{mu} and @var{s}, in analogy with the logistic ## distribution. Their relation to the @var{a} and @var{b} parameters is given ## below: ## ## @itemize ## @item @qcode{@var{a} = exp (@var{mu})} ## @item @qcode{@var{b} = 1 / @var{s}} ## @end itemize ## ## @seealso{loglcdf, loglinv, loglpdf, loglrnd, loglfit} ## @end deftypefn function [nlogL, acov] = logllike (params, x, censor, freq) ## Check input arguments if (nargin < 2) error ("logllike: function called with too few input arguments."); endif if (! isvector (x)) error ("logllike: X must be a vector."); endif if (length (params) != 2) error ("logllike: PARAMS must be a two-element vector."); endif ## Check censor vector if (nargin < 3 || isempty (censor)) censor = zeros (size (x)); elseif (! isequal (size (x), size (censor))) error ("logllike: X and CENSOR vector mismatch."); endif ## Check frequency vector if (nargin < 4 || isempty (freq)) freq = ones (size (x)); elseif (! isequal (size (x), size (freq))) error ("logllike: X and FREQ vector mismatch."); endif ## Expand frequency and censor vectors (if necessary) if (! all (freq == 1)) xf = []; cf = []; for i = 1:numel (freq) xf = [xf, repmat(x(i), 1, freq(i))]; cf = [cf, repmat(censor(i), 1, freq(i))]; endfor x = xf; freq = ones (size (x)); censor = cf; endif ## Get parameters a = params(1); b = params(2); z = (log (x) - log (a)) .* b; logclogitz = log (1 ./ (1 + exp (z))); k = (z > 700); if (any (k)) logclogitz(k) = z(k); endif L = z + 2 .* logclogitz - log (1 / b) - log (x); n_censored = sum (freq .* censor); ## Handle censored data if (n_censored > 0) censored = (censor == 1); L(censored) = logclogitz(censored); endif ## Sum up the neg log likelihood nlogL = -sum (freq .* L); ## Compute asymptotic covariance if (nargout > 1) ## Compute first order central differences of the log-likelihood gradient dp = 0.0001 .* max (abs (params), 1); ngrad_p1 = logl_grad (params + [dp(1), 0], x, censor, freq); ngrad_m1 = logl_grad (params - [dp(1), 0], x, censor, freq); ngrad_p2 = logl_grad (params + [0, dp(2)], x, censor, freq); ngrad_m2 = logl_grad (params - [0, dp(2)], x, censor, freq); ## Compute negative Hessian by normalizing the differences by the increment nH = [(ngrad_p1(:) - ngrad_m1(:))./(2 * dp(1)), ... (ngrad_p2(:) - ngrad_m2(:))./(2 * dp(2))]; ## Force neg Hessian being symmetric nH = 0.5 .* (nH + nH'); ## Check neg Hessian is positive definite [R, p] = chol (nH); if (p > 0) warning ("logllike: non positive definite Hessian matrix."); acov = NaN (2); return endif ## ACOV estimate is the negative inverse of the Hessian. Rinv = inv (R); acov = Rinv * Rinv; endif endfunction ## Helper function for computing negative gradient function ngrad = logl_grad (params, x, censor, freq) a = params(1); b = params(2); z = (log (x) - log (a)) .* b; logitz = 1 ./ (1 + exp (-z)); dL1 = (2 .* logitz - 1) .* b; dL2 = z .* dL1 - b; n_censored = sum (freq .* censor); if (n_censored > 0) censored = (censor == 1); dL1(censored) = logitz(censored) .* b; dL2(censored) = z(censored) .* dL1(censored); endif ngrad = -[sum(freq .* dL1), sum(freq .* dL2)]; endfunction ## Test output %!test %! nlogL = logllike ([exp(3.09717), 1/0.468525], [1:50]); %! assert (nlogL, 211.2965, 1e-4); %!test %! nlogL = logllike ([exp(1.01124), 1/0.336449], [1:5]); %! assert (nlogL, 9.2206, 1e-4); ## Test input validation %!error logllike (3.25) %!error logllike ([5, 0.2], ones (2)) %!error ... %! logllike ([1, 0.2, 3], [1, 3, 5, 7]) %!error ... %! logllike ([1.5, 0.2], [1:5], [0, 0, 0]) %!error ... %! logllike ([1.5, 0.2], [1:5], [0, 0, 0, 0, 0], [1, 1, 1]) %!error ... %! logllike ([1.5, 0.2], [1:5], [], [1, 1, 1]) statistics-release-1.6.3/inst/dist_fit/lognfit.m000066400000000000000000000210741456127120000217350ustar00rootroot00000000000000## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## Octave is free software; you can redistribute it and/or modify it ## under the terms of the GNU General Public License as published by ## the Free Software Foundation; either version 3 of the License, or (at ## your option) any later version. ## ## Octave is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with Octave; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{paramhat} =} lognfit (@var{x}) ## @deftypefnx {statistics} {[@var{paramhat}, @var{paramci}] =} lognfit (@var{x}) ## @deftypefnx {statistics} {[@var{paramhat}, @var{paramci}] =} lognfit (@var{x}, @var{alpha}) ## @deftypefnx {statistics} {[@dots{}] =} lognfit (@var{x}, @var{alpha}, @var{censor}) ## @deftypefnx {statistics} {[@dots{}] =} lognfit (@var{x}, @var{alpha}, @var{censor}, @var{freq}) ## @deftypefnx {statistics} {[@dots{}] =} lognfit (@var{x}, @var{alpha}, @var{censor}, @var{freq}, @var{options}) ## ## Estimate parameters and confidence intervals for the log-normal distribution. ## ## @code{@var{paramhat} = lognfit (@var{x})} returns the maximum likelihood ## estimates of the parameters of the log-normal distribution given the data in ## vector @var{x}. @qcode{@var{paramhat}([1, 2])} corresponds to the mean and ## standard deviation, respectively, of the associated normal distribution. ## ## If a random variable follows this distribution, its logarithm is normally ## distributed with mean @var{mu} and standard deviation @var{sigma}. ## ## @code{[@var{paramhat}, @var{paramci}] = lognfit (@var{x})} returns the 95% ## confidence intervals for the parameter estimates. ## ## @code{[@dots{}] = lognfit (@var{x}, @var{alpha})} also returns the ## @qcode{100 * (1 - @var{alpha})} percent confidence intervals for the ## parameter estimates. By default, the optional argument @var{alpha} is ## 0.05 corresponding to 95% confidence intervals. Pass in @qcode{[]} for ## @var{alpha} to use the default values. ## ## @code{[@dots{}] = lognfit (@var{x}, @var{alpha}, @var{censor})} accepts a ## boolean vector, @var{censor}, of the same size as @var{x} with @qcode{1}s for ## observations that are right-censored and @qcode{0}s for observations that are ## observed exactly. By default, or if left empty, ## @qcode{@var{censor} = zeros (size (@var{x}))}. ## ## @code{[@dots{}] = lognfit (@var{x}, @var{alpha}, @var{censor}, @var{freq})} ## accepts a frequency vector, @var{freq}, of the same size as @var{x}. ## @var{freq} typically contains integer frequencies for the corresponding ## elements in @var{x}, but it can contain any non-integer non-negative values. ## By default, or if left empty, @qcode{@var{freq} = ones (size (@var{x}))}. ## ## @code{[@dots{}] = lognfit (@dots{}, @var{options})} specifies control ## parameters for the iterative algorithm used to compute ML estimates with the ## @code{fminsearch} function. @var{options} is a structure with the following ## fields and their default values: ## @itemize ## @item @qcode{@var{options}.Display = "off"} ## @item @qcode{@var{options}.MaxFunEvals = 400} ## @item @qcode{@var{options}.MaxIter = 200} ## @item @qcode{@var{options}.TolX = 1e-6} ## @end itemize ## ## With no censor, the estimate of the standard deviation, ## @qcode{@var{paramhat}(2)}, is the square root of the unbiased estimate of the ## variance of @qcode{log (@var{x})}. With censored data, the maximum ## likelihood estimate is returned. ## ## Further information about the log-normal distribution can be found at ## @url{https://en.wikipedia.org/wiki/Log-normal_distribution} ## ## @seealso{logncdf, logninv, lognpdf, lognrnd, lognlike, lognstat} ## @end deftypefn function [paramhat, paramci] = lognfit (x, alpha, censor, freq, options) ## Check X for valid data if (! isvector (x) || ! isnumeric (x) || any (x <= 0)) error ("lognfit: X must be a numeric vector of positive values."); endif ## Check alpha if (nargin < 2 || isempty (alpha)) alpha = 0.05; else if (! isscalar (alpha) || ! isreal (alpha) || alpha <= 0 || alpha >= 1) error ("lognfit: wrong value for ALPHA."); endif endif ## Check censor vector if (nargin < 3 || isempty (censor)) censor = []; elseif (! isequal (size (x), size (censor))) error ("lognfit: X and CENSOR vectors mismatch."); endif ## Check frequency vector if (nargin < 4 || isempty (freq)) freq = []; elseif (! isequal (size(x), size(freq))) error ("lognfit: X and FREQ vectors mismatch."); endif ## Check options structure or add defaults if (nargin > 4 && ! isempty (options)) if (! isstruct (options) || ! isfield (options, "Display") || ! isfield (options, "MaxFunEvals") || ! isfield (options, "MaxIter") || ! isfield (options, "TolX")) error (strcat (["lognfit: 'options' 5th argument must be a"], ... [" structure with 'Display', 'MaxFunEvals',"], ... [" 'MaxIter', and 'TolX' fields present."])); endif else options = []; endif ## Fit a normal distribution to the logged data if (nargout <= 1) [muhat, sigmahat] = normfit (log (x), alpha, censor, freq, options); paramhat = [muhat, sigmahat]; else [muhat, sigmahat, muci, sigmaci] = normfit (log (x), alpha, ... censor, freq, options); paramhat = [muhat, sigmahat]; paramci = [muci, sigmaci]; endif endfunction %!demo %! ## Sample 3 populations from 3 different log-normal distibutions %! randn ("seed", 1); # for reproducibility %! r1 = lognrnd (0, 0.25, 1000, 1); %! randn ("seed", 2); # for reproducibility %! r2 = lognrnd (0, 0.5, 1000, 1); %! randn ("seed", 3); # for reproducibility %! r3 = lognrnd (0, 1, 1000, 1); %! r = [r1, r2, r3]; %! %! ## Plot them normalized and fix their colors %! hist (r, 30, 2); %! h = findobj (gca, "Type", "patch"); %! set (h(1), "facecolor", "c"); %! set (h(2), "facecolor", "g"); %! set (h(3), "facecolor", "r"); %! hold on %! %! ## Estimate their mu and sigma parameters %! mu_sigmaA = lognfit (r(:,1)); %! mu_sigmaB = lognfit (r(:,2)); %! mu_sigmaC = lognfit (r(:,3)); %! %! ## Plot their estimated PDFs %! x = [0:0.1:6]; %! y = lognpdf (x, mu_sigmaA(1), mu_sigmaA(2)); %! plot (x, y, "-pr"); %! y = lognpdf (x, mu_sigmaB(1), mu_sigmaB(2)); %! plot (x, y, "-sg"); %! y = lognpdf (x, mu_sigmaC(1), mu_sigmaC(2)); %! plot (x, y, "-^c"); %! ylim ([0, 2]) %! xlim ([0, 6]) %! hold off %! legend ({"Normalized HIST of sample 1 with mu=0, σ=0.25", ... %! "Normalized HIST of sample 2 with mu=0, σ=0.5", ... %! "Normalized HIST of sample 3 with mu=0, σ=1", ... %! sprintf("PDF for sample 1 with estimated mu=%0.2f and σ=%0.2f", ... %! mu_sigmaA(1), mu_sigmaA(2)), ... %! sprintf("PDF for sample 2 with estimated mu=%0.2f and σ=%0.2f", ... %! mu_sigmaB(1), mu_sigmaB(2)), ... %! sprintf("PDF for sample 3 with estimated mu=%0.2f and σ=%0.2f", ... %! mu_sigmaC(1), mu_sigmaC(2))}, "location", "northeast") %! title ("Three population samples from different log-normal distibutions") %! hold off ## Test output %!test %! x = lognrnd (3, 5, [1000, 1]); %! [paramhat, paramci] = lognfit (x, 0.01); %! assert (paramci(1,1) < 3); %! assert (paramci(1,2) > 3); %! assert (paramci(2,1) < 5); %! assert (paramci(2,2) > 5); ## Test input validation %!error ... %! lognfit (ones (20,3)) %!error ... %! lognfit ({1, 2, 3, 4, 5}) %!error ... %! lognfit ([-1, 2, 3, 4, 5]) %!error lognfit (ones (20,1), 0) %!error lognfit (ones (20,1), -0.3) %!error lognfit (ones (20,1), 1.2) %!error lognfit (ones (20,1), [0.05, 0.1]) %!error lognfit (ones (20,1), 0.02+i) %!error ... %! lognfit (ones (20,1), [], zeros(15,1)) %!error ... %! lognfit (ones (20,1), [], zeros(20,1), ones(25,1)) %!error lognfit (ones (20,1), [], zeros(20,1), ones(20,1), "options") statistics-release-1.6.3/inst/dist_fit/lognlike.m000066400000000000000000000141411456127120000220740ustar00rootroot00000000000000## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## Octave is free software; you can redistribute it and/or modify it ## under the terms of the GNU General Public License as published by ## the Free Software Foundation; either version 3 of the License, or (at ## your option) any later version. ## ## Octave is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with Octave; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{nlogL} =} lognlike (@var{params}, @var{x}) ## @deftypefnx {statistics} {[@var{nlogL}, @var{avar}] =} lognlike (@var{params}, @var{x}) ## @deftypefnx {statistics} {[@dots{}] =} lognlike (@var{params}, @var{x}, @var{censor}) ## @deftypefnx {statistics} {[@dots{}] =} lognlike (@var{params}, @var{x}, @var{censor}, @var{freq}) ## ## Negative log-likelihood for the log-normal distribution. ## ## @code{@var{nlogL} = lognlike (@var{params}, @var{x})} returns the negative ## log-likelihood of the data in @var{x} corresponding to the log-normal ## distribution with (1) location parameter @var{mu} and (2) scale parameter ## @var{sigma} given in the two-element vector @var{params}, which correspond to ## the mean and standard deviation of the associated normal distribution. ## Missing values, @qcode{NaNs}, are ignored. Negative values of @var{x} are ## treated as missing values. ## ## If a random variable follows this distribution, its logarithm is normally ## distributed with mean @var{mu} and standard deviation @var{sigma}. ## ## @code{[@var{nlogL}, @var{avar}] = lognlike (@var{params}, @var{x})} ## returns the inverse of Fisher's information matrix, @var{avar}. If the input ## parameter values in @var{params} are the maximum likelihood estimates, the ## diagonal elements of @var{avar} are their asymptotic variances. @var{avar} ## is based on the observed Fisher's information, not the expected information. ## ## @code{[@dots{}] = lognlike (@var{params}, @var{x}, @var{censor})} accepts a ## boolean vector, @var{censor}, of the same size as @var{x} with @qcode{1}s for ## observations that are right-censored and @qcode{0}s for observations that are ## observed exactly. By default, or if left empty, ## @qcode{@var{censor} = zeros (size (@var{x}))}. ## ## @code{[@dots{}] = lognlike (@var{params}, @var{x}, @var{censor}, @var{freq})} ## accepts a frequency vector, @var{freq}, of the same size as @var{x}. ## @var{freq} typically contains integer frequencies for the corresponding ## elements in @var{x}, but it can contain any non-integer non-negative values. ## By default, or if left empty, @qcode{@var{freq} = ones (size (@var{x}))}. ## ## Further information about the log-normal distribution can be found at ## @url{https://en.wikipedia.org/wiki/Log-normal_distribution} ## ## @seealso{logncdf, logninv, lognpdf, lognrnd, lognfit, lognstat} ## @end deftypefn function [nlogL, avar] = lognlike (params, x, censor, freq) ## Check input arguments if (nargin < 2) error ("lognlike: function called with too few input arguments."); endif if (! isvector (x)) error ("lognlike: X must be a vector."); endif if (numel (params) != 2) error ("lognlike: PARAMS must be a two-element vector."); endif if (nargin < 3 || isempty (censor)) censor = []; elseif (! isequal (size (x), size (censor))) error ("lognlike: X and CENSOR vectors mismatch."); endif if nargin < 4 || isempty (freq) freq = []; elseif (isequal (size (x), size (freq))) nulls = find (freq == 0); if (numel (nulls) > 0) x(nulls) = []; if (numel (censor) == numel (freq)) censor(nulls) = []; endif freq(nulls) = []; endif else error ("lognlike: X and FREQ vectors mismatch."); endif ## Treat negative data in X as missing values x(x < 0) = NaN; ## Calculate on log data logx = log(x); if (nargout <= 1) nlogL = normlike (params, logx, censor, freq); else [nlogL, avar] = normlike (params, logx, censor, freq); endif ## Compute censored and frequency if (isempty (freq)) freq = 1; endif if (isempty (censor)) censor = 0; endif nlogL = nlogL + sum (freq .* logx .* (1 - censor)); endfunction ## Test output %!test %! x = 1:50; %! [nlogL, avar] = lognlike ([0, 0.25], x); %! avar_out = [-5.4749e-03, 2.8308e-04; 2.8308e-04, -1.1916e-05]; %! assert (nlogL, 3962.330333301793, 1e-10); %! assert (avar, avar_out, 1e-7); %!test %! x = 1:50; %! [nlogL, avar] = lognlike ([0, 0.25], x * 0.5); %! avar_out = [-7.6229e-03, 4.8722e-04; 4.8722e-04, -2.6754e-05]; %! assert (nlogL, 2473.183051225747, 1e-10); %! assert (avar, avar_out, 1e-7); %!test %! x = 1:50; %! [nlogL, avar] = lognlike ([0, 0.5], x); %! avar_out = [-2.1152e-02, 2.2017e-03; 2.2017e-03, -1.8535e-04]; %! assert (nlogL, 1119.072424020455, 1e-12); %! assert (avar, avar_out, 1e-6); %!test %! x = 1:50; %! censor = ones (1, 50); %! censor([2, 4, 6, 8, 12, 14]) = 0; %! [nlogL, avar] = lognlike ([0, 0.5], x, censor); %! avar_out = [-1.9823e-02, 2.0370e-03; 2.0370e-03, -1.6618e-04]; %! assert (nlogL, 1091.746371145497, 1e-12); %! assert (avar, avar_out, 1e-6); %!test %! x = 1:50; %! censor = ones (1, 50); %! censor([2, 4, 6, 8, 12, 14]) = 0; %! [nlogL, avar] = lognlike ([0, 1], x, censor); %! avar_out = [-6.8634e-02, 1.3968e-02; 1.3968e-02, -2.1664e-03]; %! assert (nlogL, 349.3969104144271, 1e-12); %! assert (avar, avar_out, 1e-6); ## Test input validation %!error ... %! lognlike ([12, 15]); %!error lognlike ([12, 15], ones (2)); %!error ... %! lognlike ([12, 15, 3], [1:50]); %!error ... %! lognlike ([12, 15], [1:50], [1, 2, 3]); %!error ... %! lognlike ([12, 15], [1:50], [], [1, 2, 3]); statistics-release-1.6.3/inst/dist_fit/nakafit.m000066400000000000000000000217151456127120000217120ustar00rootroot00000000000000## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{paramhat} =} nakafit (@var{x}) ## @deftypefnx {statistics} {[@var{paramhat}, @var{paramci}] =} nakafit (@var{x}) ## @deftypefnx {statistics} {[@var{paramhat}, @var{paramci}] =} nakafit (@var{x}, @var{alpha}) ## @deftypefnx {statistics} {[@dots{}] =} nakafit (@var{x}, @var{alpha}, @var{censor}) ## @deftypefnx {statistics} {[@dots{}] =} nakafit (@var{x}, @var{alpha}, @var{censor}, @var{freq}) ## @deftypefnx {statistics} {[@dots{}] =} nakafit (@var{x}, @var{alpha}, @var{censor}, @var{freq}, @var{options}) ## ## Estimate mean and confidence intervals for the Nakagami distribution. ## ## @code{@var{mu0} = nakafit (@var{x})} returns the maximum likelihood ## estimates of the parameters of the Nakagami distribution given the data in ## @var{x}. @qcode{@var{paramhat}(1)} is the scale parameter, @var{mu}, and ## @qcode{@var{paramhat}(2)} is the shape parameter, @var{omega}. ## ## @code{[@var{paramhat}, @var{paramci}] = nakafit (@var{x})} returns the 95% ## confidence intervals for the parameter estimates. ## ## @code{[@dots{}] = nakafit (@var{x}, @var{alpha})} also returns the ## @qcode{100 * (1 - @var{alpha})} percent confidence intervals for the ## parameter estimates. By default, the optional argument @var{alpha} is ## 0.05 corresponding to 95% confidence intervals. Pass in @qcode{[]} for ## @var{alpha} to use the default values. ## ## @code{[@dots{}] = nakafit (@var{x}, @var{alpha}, @var{censor})} accepts a ## boolean vector, @var{censor}, of the same size as @var{x} with @qcode{1}s for ## observations that are right-censored and @qcode{0}s for observations that are ## observed exactly. By default, or if left empty, ## @qcode{@var{censor} = zeros (size (@var{x}))}. ## ## @code{[@dots{}] = nakafit (@var{x}, @var{alpha}, @var{censor}, @var{freq})} ## accepts a frequency vector, @var{freq}, of the same size as @var{x}. ## @var{freq} typically contains integer frequencies for the corresponding ## elements in @var{x}, but it can contain any non-integer non-negative values. ## By default, or if left empty, @qcode{@var{freq} = ones (size (@var{x}))}. ## ## @code{[@dots{}] = nakafit (@dots{}, @var{options})} specifies control ## parameters for the iterative algorithm used to compute ML estimates with the ## @code{fminsearch} function. @var{options} is a structure with the following ## fields and their default values: ## @itemize ## @item @qcode{@var{options}.Display = "off"} ## @item @qcode{@var{options}.MaxFunEvals = 400} ## @item @qcode{@var{options}.MaxIter = 200} ## @item @qcode{@var{options}.TolX = 1e-6} ## @end itemize ## ## Further information about the Nakagami distribution can be found at ## @url{https://en.wikipedia.org/wiki/Nakagami_distribution} ## ## @seealso{nakacdf, nakainv, nakapdf, nakarnd, nakalike} ## @end deftypefn function [paramhat, paramci] = nakafit (x, alpha, censor, freq, options) ## Check input arguments if (! isvector (x)) error ("nakafit: X must be a vector."); endif ## Check alpha if (nargin < 2 || isempty (alpha)) alpha = 0.05; else if (! isscalar (alpha) || ! isreal (alpha) || alpha <= 0 || alpha >= 1) error ("nakafit: wrong value for ALPHA."); endif endif ## Check censor vector if (nargin < 3 || isempty (censor)) censor = zeros (size (x)); elseif (! isequal (size (x), size (censor))) error ("nakafit: X and CENSOR vectors mismatch."); endif ## Check frequency vector if (nargin < 4 || isempty (freq)) freq = ones (size (x)); elseif (! isequal (size (x), size (freq))) error ("nakafit: X and FREQ vectors mismatch."); endif ## Get options structure or add defaults if (nargin < 5) options.Display = "off"; options.MaxFunEvals = 400; options.MaxIter = 200; options.TolX = 1e-6; else if (! isstruct (options) || ! isfield (options, "Display") || ! isfield (options, "MaxFunEvals") || ! isfield (options, "MaxIter") || ! isfield (options, "TolX")) error (strcat (["nakafit: 'options' 5th argument must be a"], ... [" structure with 'Display', 'MaxFunEvals',"], ... [" 'MaxIter', and 'TolX' fields present."])); endif endif ## Expand frequency and censor vectors (if necessary) if (! all (freq == 1)) xf = []; cf = []; for i = 1:numel (freq) xf = [xf, repmat(x(i), 1, freq(i))]; cf = [cf, repmat(censor(i), 1, freq(i))]; endfor x = xf; freq = ones (size (x)); censor = cf; endif ## Get parameter estimates from the Gamma distribution paramhat = gamfit (x .^ 2, alpha, censor, freq, options); ## Transform back to Nakagami parameters paramhat(2) = paramhat(1) .* paramhat(2); ## Compute CIs using a log normal approximation for parameters. if (nargout > 1) ## Compute asymptotic covariance [~, acov] = nakalike (paramhat, x, censor, freq); ## Get standard errors se = sqrt (diag (acov))'; ## Get normal quantiles probs = [alpha/2; 1-alpha/2]; ## Compute muci using a normal approximation paramci(:,1) = norminv (probs, paramhat(1), se(1)); ## Compute omegaci using a normal approximation for log (omega) paramci(:,2) = exp (norminv (probs, log (paramhat(2)), log (se(2)))); endif endfunction %!demo %! ## Sample 3 populations from different Nakagami distibutions %! randg ("seed", 5) # for reproducibility %! r1 = nakarnd (0.5, 1, 2000, 1); %! randg ("seed", 2) # for reproducibility %! r2 = nakarnd (5, 1, 2000, 1); %! randg ("seed", 7) # for reproducibility %! r3 = nakarnd (2, 2, 2000, 1); %! r = [r1, r2, r3]; %! %! ## Plot them normalized and fix their colors %! hist (r, [0.05:0.1:3.5], 10); %! h = findobj (gca, "Type", "patch"); %! set (h(1), "facecolor", "c"); %! set (h(2), "facecolor", "g"); %! set (h(3), "facecolor", "r"); %! ylim ([0, 2.5]); %! xlim ([0, 3.0]); %! hold on %! %! ## Estimate their MU and LAMBDA parameters %! mu_omegaA = nakafit (r(:,1)); %! mu_omegaB = nakafit (r(:,2)); %! mu_omegaC = nakafit (r(:,3)); %! %! ## Plot their estimated PDFs %! x = [0.01:0.1:3.01]; %! y = nakapdf (x, mu_omegaA(1), mu_omegaA(2)); %! plot (x, y, "-pr"); %! y = nakapdf (x, mu_omegaB(1), mu_omegaB(2)); %! plot (x, y, "-sg"); %! y = nakapdf (x, mu_omegaC(1), mu_omegaC(2)); %! plot (x, y, "-^c"); %! legend ({"Normalized HIST of sample 1 with μ=0.5 and ω=1", ... %! "Normalized HIST of sample 2 with μ=5 and ω=1", ... %! "Normalized HIST of sample 3 with μ=2 and ω=2", ... %! sprintf("PDF for sample 1 with estimated μ=%0.2f and ω=%0.2f", ... %! mu_omegaA(1), mu_omegaA(2)), ... %! sprintf("PDF for sample 2 with estimated μ=%0.2f and ω=%0.2f", ... %! mu_omegaB(1), mu_omegaB(2)), ... %! sprintf("PDF for sample 3 with estimated μ=%0.2f and ω=%0.2f", ... %! mu_omegaC(1), mu_omegaC(2))}) %! title ("Three population samples from different Nakagami distibutions") %! hold off ## Test output %!test %! paramhat = nakafit ([1:50]); %! paramhat_out = [0.7355, 858.5]; %! assert (paramhat, paramhat_out, 1e-4); %!test %! paramhat = nakafit ([1:5]); %! paramhat_out = [1.1740, 11]; %! assert (paramhat, paramhat_out, 1e-4); %!test %! paramhat = nakafit ([1:6], [], [], [1 1 1 1 1 0]); %! paramhat_out = [1.1740, 11]; %! assert (paramhat, paramhat_out, 1e-4); %!test %! paramhat = nakafit ([1:5], [], [], [1 1 1 1 2]); %! paramhat_out = nakafit ([1:5, 5]); %! assert (paramhat, paramhat_out, 1e-4); ## Test input validation %!error nakafit (ones (2,5)); %!error nakafit ([1, 2, 3, 4, 5], 1.2); %!error nakafit ([1, 2, 3, 4, 5], 0); %!error nakafit ([1, 2, 3, 4, 5], "alpha"); %!error ... %! nakafit ([1, 2, 3, 4, 5], 0.05, [1 1 0]); %!error ... %! nakafit ([1, 2, 3, 4, 5], [], [1 1 0 1 1]'); %!error ... %! nakafit ([1, 2, 3, 4, 5], 0.05, zeros (1,5), [1 1 0]); %!error ... %! nakafit ([1, 2, 3, 4, 5], [], [], [1 1 0 1 1]'); %!error ... %! nakafit ([1, 2, 3, 4, 5], 0.05, [], [], 2); statistics-release-1.6.3/inst/dist_fit/nakalike.m000066400000000000000000000247061456127120000220570ustar00rootroot00000000000000## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{nlogL} =} nakalike (@var{params}, @var{x}) ## @deftypefnx {statistics} {[@var{nlogL}, @var{acov}] =} nakalike (@var{params}, @var{x}) ## @deftypefnx {statistics} {[@dots{}] =} nakalike (@var{params}, @var{x}, @var{censor}) ## @deftypefnx {statistics} {[@dots{}] =} nakalike (@var{params}, @var{x}, @var{censor}, @var{freq}) ## ## Negative log-likelihood for the Nakagami distribution. ## ## @code{@var{nlogL} = nakalike (@var{params}, @var{x})} returns the negative ## log likelihood of the data in @var{x} corresponding to the Nakagami ## distribution with (1) scale parameter @var{mu} and (2) shape parameter ## @var{omega} given in the two-element vector @var{params}. ## ## @code{[@var{nlogL}, @var{acov}] = nakalike (@var{params}, @var{x})} also ## returns the inverse of Fisher's information matrix, @var{acov}. If the input ## parameter values in @var{params} are the maximum likelihood estimates, the ## diagonal elements of @var{params} are their asymptotic variances. ## ## @code{[@dots{}] = nakalike (@var{params}, @var{x}, @var{censor})} accepts a ## boolean vector, @var{censor}, of the same size as @var{x} with @qcode{1}s for ## observations that are right-censored and @qcode{0}s for observations that are ## observed exactly. By default, or if left empty, ## @qcode{@var{censor} = zeros (size (@var{x}))}. ## ## @code{[@dots{}] = nakalike (@var{params}, @var{x}, @var{censor}, @var{freq})} ## accepts a frequency vector, @var{freq}, of the same size as @var{x}. ## @var{freq} typically contains integer frequencies for the corresponding ## elements in @var{x}, but it can contain any non-integer non-negative values. ## By default, or if left empty, @qcode{@var{freq} = ones (size (@var{x}))}. ## ## Further information about the Nakagami distribution can be found at ## @url{https://en.wikipedia.org/wiki/Nakagami_distribution} ## ## @seealso{nakacdf, nakainv, nakapdf, nakarnd, nakafit} ## @end deftypefn function [nlogL, acov] = nakalike (params, x, censor, freq) ## Check input arguments if (nargin < 2) error ("nakalike: function called with too few input arguments."); endif if (! isvector (x)) error ("nakalike: X must be a vector."); endif if (length (params) != 2) error ("nakalike: PARAMS must be a two-element vector."); endif ## Check censor vector if (nargin < 3 || isempty (censor)) censor = zeros (size (x)); elseif (! isequal (size (x), size (censor))) error ("nakalike: X and CENSOR vector mismatch."); endif ## Check frequency vector if (nargin < 4 || isempty (freq)) freq = ones (size (x)); elseif (! isequal (size (x), size (freq))) error ("nakalike: X and FREQ vector mismatch."); endif ## Expand frequency and censor vectors (if necessary) if (! all (freq == 1)) xf = []; cf = []; for i = 1:numel (freq) xf = [xf, repmat(x(i), 1, freq(i))]; cf = [cf, repmat(censor(i), 1, freq(i))]; endfor x = xf; freq = ones (size (x)); censor = cf; endif ## Get parameters mu = params(1); omega = params(2); log_a = gammaln (mu); log_b = log (omega / mu); z = x .^ 2 ./ (omega / mu); log_z = log (z); L = (mu - 1) .* log_z - z - log_a - log_b + log (2 .* x); ## Handle censored data n_censored = sum (freq .* censor); if (n_censored > 0) censored = (censor == 1); z_censored = z(censored); [S, dS] = dgammainc (z_censored, mu); L(censored) = log (S); endif ## Sum up the neg log likelihood nlogL = -sum (freq .* L); ## Compute asymptotic covariance if (nargout > 1) ## Compute first order central differences of the log-likelihood gradient dp = 0.0001 .* max (abs (params), 1); ngrad_p1 = logl_grad (params + [dp(1), 0], x, censor, freq); ngrad_m1 = logl_grad (params - [dp(1), 0], x, censor, freq); ngrad_p2 = logl_grad (params + [0, dp(2)], x, censor, freq); ngrad_m2 = logl_grad (params - [0, dp(2)], x, censor, freq); ## Compute negative Hessian by normalizing the differences by the increment nH = [(ngrad_p1(:) - ngrad_m1(:))./(2 * dp(1)), ... (ngrad_p2(:) - ngrad_m2(:))./(2 * dp(2))]; ## Force neg Hessian being symmetric nH = 0.5 .* (nH + nH'); ## Check neg Hessian is positive definite [R, p] = chol (nH); if (p > 0) warning ("nakalike: non positive definite Hessian matrix."); acov = NaN (2); return endif ## ACOV estimate is the negative inverse of the Hessian. Rinv = inv (R); acov = Rinv * Rinv; endif endfunction ## Helper function for computing negative gradient function ngrad = logl_grad (params, x, censor, freq) mu = params(1); omega = params(2); ## Transform to Gamma parameters log_a = gammaln (mu); log_b = log (omega / mu); z = x .^ 2 ./ (omega / mu); log_z = log (z); dL1 = log_z - psi (mu); dL2 = (z - mu) ./ (omega / mu); ## Handle censored data n_censored = sum (freq .* censor); if (n_censored > 0) censored = (censor == 1); z_censored = z(censored); [S, dS] = dgammainc (z_censored, a); dL1(censored) = dS ./ S; tmp = mu .* log_z(censored) - log_b - z_censored - log_a; dL2(censored) = exp (tmp) ./ S; endif ngrad = -[sum(freq .* dL1), sum(freq .* dL2)]; ## Transform back to Nakagami parameters ngrad = ngrad * [1, 0; (-omega ./ (mu .^ 2)), (1 ./ mu)]; endfunction ## Compute the incomplete Gamma function with its 1st and 2nd derivatives function [y, dy, d2y] = dgammainc (x, k) ## Initialize return variables y = nan (size (x)); dy = y; d2y = y; ## Use approximation for K > 2^20 ulim = 2^20; is_lim = find (k > ulim); if (! isempty (is_lim)) x(is_lim) = max (ulim - 1/3 + sqrt (ulim ./ k(is_lim)) .* ... (x(is_lim) - (k(is_lim) - 1/3)), 0); k(is_lim) = ulim; endif ## For x < k+1 is_lo = find(x < k + 1 & x != 0); if (! isempty (is_lo)) x_lo = x(is_lo); k_lo = k(is_lo); k_1 = k_lo; step = 1; d1st = 0; d2st = 0; stsum = step; d1sum = d1st; d2sum = d2st; while norm (step, "inf") >= 100 * eps (norm (stsum, "inf")) k_1 += 1; step = step .* x_lo ./ k_1; d1st = (d1st .* x_lo - step) ./ k_1; d2st = (d2st .* x_lo - 2 .* d1st) ./ k_1; stsum = stsum + step; d1sum = d1sum + d1st; d2sum = d2sum + d2st; endwhile fklo = exp (-x_lo + k_lo .* log (x_lo) - gammaln (k_lo + 1)); y_lo = fklo .* stsum; ## Fix very small k y_lo(x_lo > 0 & y_lo > 1) = 1; ## Compute 1st derivative dlogfklo = (log (x_lo) - psi (k_lo + 1)); d1fklo = fklo .* dlogfklo; d1y_lo = d1fklo .* stsum + fklo .* d1sum; ## Compute 2nd derivative d2fklo = d1fklo .* dlogfklo - fklo .* psi (1, k_lo + 1); d2y_lo = d2fklo .* stsum + 2 .* d1fklo .* d1sum + fklo .* d2sum; ## Considering the upper tail y(is_lo) = 1 - y_lo; dy(is_lo) = -d1y_lo; d2y(is_lo) = -d2y_lo; endif ## For x >= k+1 is_hi = find(x >= k+1); if (! isempty (is_hi)) x_hi = x(is_hi); k_hi = k(is_hi); zc = 0; k0 = 0; k1 = k_hi; x0 = 1; x1 = x_hi; d1k0 = 0; d1k1 = 1; d1x0 = 0; d1x1 = 0; d2k0 = 0; d2k1 = 0; d2x0 = 0; d2x2 = 0; kx = k_hi ./ x_hi; d1kx = 1 ./ x_hi; d2kx = 0; start = 1; while norm (d2kx - start, "Inf") > 100 * eps (norm (d2kx, "Inf")) rescale = 1 ./ x1; zc += 1; n_k = zc - k_hi; d2k0 = (d2k1 + d2k0 .* n_k - 2 .* d1k0) .* rescale; d2x0 = (d2x2 + d2x0 .* n_k - 2 .* d1x0) .* rescale; d1k0 = (d1k1 + d1k0 .* n_k - k0) .* rescale; d1x0 = (d1x1 + d1x0 .* n_k - x0) .* rescale; k0 = (k1 + k0 .* n_k) .* rescale; x0 = 1 + (x0 .* n_k) .* rescale; nrescale = zc .* rescale; d2k1 = d2k0 .* x_hi + d2k1 .* nrescale; d2x2 = d2x0 .* x_hi + d2x2 .* nrescale; d1k1 = d1k0 .* x_hi + d1k1 .* nrescale; d1x1 = d1x0 .* x_hi + d1x1 .* nrescale; k1 = k0 .* x_hi + k1 .* nrescale; x1 = x0 .* x_hi + zc; start = d2kx; kx = k1 ./ x1; d1kx = (d1k1 - kx .* d1x1) ./ x1; d2kx = (d2k1 - d1kx .* d1x1 - kx .* d2x2 - d1kx .* d1x1) ./ x1; endwhile fkhi = exp (-x_hi + k_hi .* log (x_hi) - gammaln (k_hi + 1)); y_hi = fkhi .* kx; ## Compute 1st derivative dlogfkhi = (log (x_hi) - psi (k_hi + 1)); d1fkhi = fkhi .* dlogfkhi; d1y_hi = d1fkhi .* kx + fkhi .* d1kx; ## Compute 2nd derivative d2fkhi = d1fkhi .* dlogfkhi - fkhi .* psi (1, k_hi + 1); d2y_hi = d2fkhi .* kx + 2 .* d1fkhi .* d1kx + fkhi .* d2kx; ## Considering the upper tail y(is_hi) = y_hi; dy(is_hi) = d1y_hi; d2y(is_hi) = d2y_hi; endif ## Handle x == 0 is_x0 = find (x == 0); if (! isempty (is_x0)) ## Considering the upper tail y(is_x0) = 1; dy(is_x0) = 0; d2y(is_x0) = 0; endif ## Handle k == 0 is_k0 = find (k == 0); if (! isempty (is_k0)) is_k0x0 = find (k == 0 & x == 0); ## Considering the upper tail y(is_k0) = 0; dy(is_k0x0) = Inf; d2y(is_k0x0) = -Inf; endif endfunction ## Test output %!test %! nlogL = nakalike ([0.735504, 858.5], [1:50]); %! assert (nlogL, 202.8689, 1e-4); %!test %! nlogL = nakalike ([1.17404, 11], [1:5]); %! assert (nlogL, 8.6976, 1e-4); ## Test input validation %!error nakalike (3.25) %!error nakalike ([5, 0.2], ones (2)) %!error ... %! nakalike ([1, 0.2, 3], [1, 3, 5, 7]) %!error ... %! nakalike ([1.5, 0.2], [1:5], [0, 0, 0]) %!error ... %! nakalike ([1.5, 0.2], [1:5], [0, 0, 0, 0, 0], [1, 1, 1]) %!error ... %! nakalike ([1.5, 0.2], [1:5], [], [1, 1, 1]) statistics-release-1.6.3/inst/dist_fit/nbinfit.m000066400000000000000000000227571456127120000217350ustar00rootroot00000000000000## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{paramhat} =} nbinfit (@var{x}) ## @deftypefnx {statistics} {[@var{paramhat}, @var{paramci}] =} nbinfit (@var{x}) ## @deftypefnx {statistics} {[@var{paramhat}, @var{paramci}] =} nbinfit (@var{x}, @var{alpha}) ## @deftypefnx {statistics} {[@var{paramhat}, @var{paramci}] =} nbinfit (@var{x}, @var{alpha}, @var{options}) ## ## Estimate parameter and confidence intervals for the negative binomial ## distribution. ## ## @code{@var{paramhat} = nbinfit (@var{x})} returns the maximum likelihood ## estimates of the parameters of the negative binomial distribution given the ## data in vector @var{x}. @qcode{@var{paramhat}(1)} is the number of successes ## until the experiment is stopped, @var{r}, and @qcode{@var{paramhat}(1)} is ## the probability of success in each experiment, @var{ps}. ## ## @code{[@var{paramhat}, @var{paramci}] = nbinfit (@var{x})} returns the 95% ## confidence intervals for the parameter estimates. ## ## @code{[@var{paramhat}, @var{paramci}] = nbinfit (@var{x}, @var{alpha})} also ## returns the @qcode{100 * (1 - @var{alpha})} percent confidence intervals of ## the estimated parameter. By default, the optional argument @var{alpha} is ## 0.05 corresponding to 95% confidence intervals. ## ## @code{[@var{paramhat}, @var{paramci}] = nbinfit (@var{x}, @var{alpha}, ## @var{options})} specifies control parameters for the iterative algorithm used ## to compute ML estimates with the @code{fminsearch} function. @var{options} ## is a structure with the following fields and their default values: ## @itemize ## @item @qcode{@var{options}.Display = "off"} ## @item @qcode{@var{options}.MaxFunEvals = 400} ## @item @qcode{@var{options}.MaxIter = 200} ## @item @qcode{@var{options}.TolX = 1e-6} ## @end itemize ## ## When @var{r} is an integer, the negative binomial distribution is also known ## as the Pascal distribution and it models the number of failures in @var{x} ## before a specified number of successes is reached in a series of independent, ## identical trials. Its parameters are the probability of success in a single ## trial, @var{ps}, and the number of successes, @var{r}. A special case of the ## negative binomial distribution, when @qcode{@var{r} = 1}, is the geometric ## distribution, which models the number of failures before the first success. ## ## @var{r} can also have non-integer positive values, in which form the negative ## binomial distribution, also known as the Polya distribution, has no ## interpretation in terms of repeated trials, but, like the Poisson ## distribution, it is useful in modeling count data. The negative binomial ## distribution is more general than the Poisson distribution because it has a ## variance that is greater than its mean, making it suitable for count data ## that do not meet the assumptions of the Poisson distribution. In the limit, ## as @var{r} increases to infinity, the negative binomial distribution ## approaches the Poisson distribution. ## ## Further information about the negative binomial distribution can be found at ## @url{https://en.wikipedia.org/wiki/Negative_binomial_distribution} ## ## @seealso{nbincdf, nbininv, nbinpdf, nbinrnd, nbinlike, nbinstat} ## @end deftypefn function [paramhat, paramci] = nbinfit (x, alpha, options) ## Check data in X if (any (x < 0)) error ("nbinfit: X cannot have negative values."); endif if (! isvector (x)) error ("nbinfit: X must be a vector."); endif if (any (x < 0) || any (x != round (x)) || any (isinf (x))) error ("nbinfit: X must be a non-negative integer."); endif ## Check ALPHA if (nargin < 2 || isempty (alpha)) alpha = 0.05; elseif (! isscalar (alpha) || ! isreal (alpha) || alpha <= 0 || alpha >= 1) error ("nbinfit: wrong value for ALPHA."); endif ## Get options structure or add defaults if (nargin < 3) options.Display = "off"; options.MaxFunEvals = 400; options.MaxIter = 200; options.TolX = 1e-6; else if (! isstruct (options) || ! isfield (options, "Display") || ! isfield (options, "MaxFunEvals") || ! isfield (options, "MaxIter") || ! isfield (options, "TolX")) error (strcat (["nbinfit: 'options' 3rd argument must be a"], ... [" structure with 'Display', 'MaxFunEvals',"], ... [" 'MaxIter', and 'TolX' fields present."])); endif endif ## Ensure that a negative binomial fit is valid. xbar = mean (x); varx = var (x); if (varx <= xbar) paramhat = cast ([Inf, 1.0], class (x)); paramci = cast ([Inf, 1; Inf, 1], class (x)); fprintf ("warning: nbinfit: mean exceeds variance.\n"); return endif ## Use Method of Moments estimates as starting point for MLEs. rhat = (xbar .* xbar) ./ (varx - xbar); ## Minimize negative log-likelihood to estimate parameters by parameterizing ## with mu=r(1-p)/p, so it becomes 1-parameter search for rhat. f = @(rhat) nbinfit_search (rhat, x, numel (x), sum (x), options.TolX); [rhat, ~, err, output] = fminsearch (f, rhat, options); ## Handle errors if (err == 0) if (output.funcCount >= options.MaxFunEvals) warning (strcat (["nbinfit: maximum number of function"], ... [" evaluations are exceeded."])); elseif (output.iterations >= options.MaxIter) warning ("nbinfit: maximum number of iterations are exceeded."); endif elseif (err < 0) error ("nbinfit: NoSolution."); endif ## Compute parameter estimates pshat = rhat ./ (xbar + rhat); paramhat = [rhat, pshat]; ## Compute confidence interval if (nargout > 1) [~, avar] = nbinlike (paramhat, x); ## Get standard errors sigma = sqrt (diag (avar)); ## Get normal quantiles probs = [alpha/2; 1-alpha/2]; ## Compute paramci using a normal approximation paramci = norminv ([probs, probs], [paramhat; paramhat], [sigma'; sigma']); ## Restrict CI to valid values: r >= 0, 0 <= ps <= 1 paramci(paramci < 0) = 0; if (paramci(2,2) > 1) paramci(2,2) = 1; endif endif endfunction ## Helper function for minimizing the negative log-likelihood function nll = nbinfit_search (r, x, nx, sx, tol) if (r < tol) nll = Inf; else xbar = sx / nx; nll = -sum (gammaln (r +x )) + nx * gammaln (r) ... -nx * r * log (r / (xbar + r)) - sx * log (xbar / (xbar + r)); endif endfunction %!demo %! ## Sample 2 populations from different negative binomial distibutions %! randp ("seed", 5); randg ("seed", 5); # for reproducibility %! r1 = nbinrnd (2, 0.15, 5000, 1); %! randp ("seed", 8); randg ("seed", 8); # for reproducibility %! r2 = nbinrnd (5, 0.2, 5000, 1); %! r = [r1, r2]; %! %! ## Plot them normalized and fix their colors %! hist (r, [0:51], 1); %! h = findobj (gca, "Type", "patch"); %! set (h(1), "facecolor", "c"); %! set (h(2), "facecolor", "g"); %! hold on %! %! ## Estimate their probability of success %! r_psA = nbinfit (r(:,1)); %! r_psB = nbinfit (r(:,2)); %! %! ## Plot their estimated PDFs %! x = [0:40]; %! y = nbinpdf (x, r_psA(1), r_psA(2)); %! plot (x, y, "-pg"); %! x = [min(r(:,2)):max(r(:,2))]; %! y = nbinpdf (x, r_psB(1), r_psB(2)); %! plot (x, y, "-sc"); %! ylim ([0, 0.1]) %! xlim ([0, 50]) %! legend ({"Normalized HIST of sample 1 with r=2 and ps=0.15", ... %! "Normalized HIST of sample 2 with r=5 and ps=0.2", ... %! sprintf("PDF for sample 1 with estimated r=%0.2f and ps=%0.2f", ... %! r_psA(1), r_psA(2)), ... %! sprintf("PDF for sample 2 with estimated r=%0.2f and ps=%0.2f", ... %! r_psB(1), r_psB(2))}) %! title ("Two population samples from negative different binomial distibutions") %! hold off ## Test output %!test %! [paramhat, paramci] = nbinfit ([1:50]); %! assert (paramhat, [2.420857, 0.086704], 1e-6); %! assert (paramci(:,1), [1.382702; 3.459012], 1e-6); %! assert (paramci(:,2), [0.049676; 0.123732], 1e-6); %!test %! [paramhat, paramci] = nbinfit ([1:20]); %! assert (paramhat, [3.588233, 0.254697], 1e-6); %! assert (paramci(:,1), [0.451693; 6.724774], 1e-6); %! assert (paramci(:,2), [0.081143; 0.428251], 1e-6); %!test %! [paramhat, paramci] = nbinfit ([1:10]); %! assert (paramhat, [8.8067, 0.6156], 1e-4); %! assert (paramci(:,1), [0; 30.7068], 1e-4); %! assert (paramci(:,2), [0.0217; 1], 1e-4); ## Test input validation %!error nbinfit ([-1 2 3 3]) %!error nbinfit (ones (2)) %!error nbinfit ([1 2 1.2 3]) %!error nbinfit ([1 2 3], 0) %!error nbinfit ([1 2 3], 1.2) %!error nbinfit ([1 2 3], [0.02 0.05]) %!error ... %! nbinfit ([1, 2, 3, 4, 5], 0.05, 2); statistics-release-1.6.3/inst/dist_fit/nbinlike.m000066400000000000000000000127261456127120000220720ustar00rootroot00000000000000## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{nlogL} =} nbinlike (@var{params}, @var{x}) ## @deftypefnx {statistics} {[@var{nlogL}, @var{avar}] =} nbinlike (@var{params}, @var{x}) ## ## Negative log-likelihood for the negative binomial distribution. ## ## @code{@var{nlogL} = nbinlike (@var{params}, @var{x})} returns the negative ## log likelihood of the negative binomial distribution with (1) parameter ## @var{r} and (2) parameter @var{ps}, given in the two-element vector ## @var{params}, where @var{r} is the number of successes until the experiment ## is stopped and @var{ps} is the probability of success in each experiment, ## given the number of failures in @var{x}. ## ## @code{[@var{nlogL}, @var{avar}] = nbinlike (@var{params}, @var{x})} also ## returns the inverse of Fisher's information matrix, @var{avar}. If the input ## parameter values in @var{params} are the maximum likelihood estimates, the ## diagonal elements of @var{params} are their asymptotic variances. ## ## When @var{r} is an integer, the negative binomial distribution is also known ## as the Pascal distribution and it models the number of failures in @var{x} ## before a specified number of successes is reached in a series of independent, ## identical trials. Its parameters are the probability of success in a single ## trial, @var{ps}, and the number of successes, @var{r}. A special case of the ## negative binomial distribution, when @qcode{@var{r} = 1}, is the geometric ## distribution, which models the number of failures before the first success. ## ## @var{r} can also have non-integer positive values, in which form the negative ## binomial distribution, also known as the Polya distribution, has no ## interpretation in terms of repeated trials, but, like the Poisson ## distribution, it is useful in modeling count data. The negative binomial ## distribution is more general than the Poisson distribution because it has a ## variance that is greater than its mean, making it suitable for count data ## that do not meet the assumptions of the Poisson distribution. In the limit, ## as @var{r} increases to infinity, the negative binomial distribution ## approaches the Poisson distribution. ## ## Further information about the negative binomial distribution can be found at ## @url{https://en.wikipedia.org/wiki/Negative_binomial_distribution} ## ## @seealso{nbincdf, nbininv, nbinpdf, nbinrnd, nbinfit, nbinstat} ## @end deftypefn function [nlogL, avar] = nbinlike (params, x) ## Check input arguments if (nargin < 2) error ("nbinlike: function called with too few input arguments."); endif if (! isvector (x)) error ("nbinlike: X must be a vector."); endif if (any (x < 0)) error ("nbinlike: X cannot have negative values."); endif if (any (x != fix (x))) error ("nbinlike: number of failures, X, must be integers."); endif if (length (params) != 2) error ("nbinlike: PARAMS must be a two-element vector."); endif if (params(1) <= 0) error (strcat (["nbinlike: number of successes, PARAMS(1), must be"], ... [" a real positive value."])); endif if (params(2) < 0 || params(2) > 1) error (strcat (["nbinlike: probability of success, PARAMS(2), must be"], ... [" in the range [0,1]."])); endif ## Compute negative log-likelihood and asymptotic variance r = params(1); ps = params(2); nx = numel (x); glnr = gammaln (r + x) - gammaln (x + 1) - gammaln (r); sumx = sum (x); nlogL = -(sum (glnr) + nx * r * log (ps)) - sumx * log (1 - ps); if (nargout == 2) dL11 = sum (psi (1, r + x) - psi (1, r)); dL12 = nx ./ ps; dL22 = -nx .*r ./ ps .^ 2 - sumx ./ (1 - ps) .^ 2; nH = -[dL11, dL12; dL12, dL22]; if (any (isnan (nH(:)))) avar = [NaN, NaN; NaN, NaN]; else avar = inv (nH); end endif endfunction ## Test output %!assert (nbinlike ([2.42086, 0.0867043],[1:50]), 205.5942, 1e-4) %!assert (nbinlike ([3.58823, 0.254697], [1:20]), 63.6435, 1e-4) %!assert (nbinlike ([8.80671, 0.615565], [1:10]), 24.7410, 1e-4) %!assert (nbinlike ([22.1756, 0.831306], [1:8]), 17.9528, 1e-4) ## Test input validation %!error nbinlike (3.25) %!error nbinlike ([5, 0.2], ones (2)) %!error nbinlike ([5, 0.2], [-1, 3]) %!error ... %! nbinlike ([1, 0.2, 3], [1, 3, 5, 7]) %!error nbinlike ([-5, 0.2], [1:15]) %!error nbinlike ([0, 0.2], [1:15]) %!error nbinlike ([5, 1.2], [3, 5]) %!error nbinlike ([5, -0.2], [3, 5]) statistics-release-1.6.3/inst/dist_fit/normfit.m000066400000000000000000000407661456127120000217620ustar00rootroot00000000000000## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## Octave is free software; you can redistribute it and/or modify it ## under the terms of the GNU General Public License as published by ## the Free Software Foundation; either version 3 of the License, or (at ## your option) any later version. ## ## Octave is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with Octave; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{muhat} =} normfit (@var{x}) ## @deftypefnx {statistics} {[@var{muhat}, @var{sigmahat}] =} normfit (@var{x}) ## @deftypefnx {statistics} {[@var{muhat}, @var{sigmahat}, @var{muci}] =} normfit (@var{x}) ## @deftypefnx {statistics} {[@var{muhat}, @var{sigmahat}, @var{muci}, @var{sigmaci}] =} normfit (@var{x}) ## @deftypefnx {statistics} {[@dots{}] =} normfit (@var{x}, @var{alpha}) ## @deftypefnx {statistics} {[@dots{}] =} normfit (@var{x}, @var{alpha}, @var{censor}) ## @deftypefnx {statistics} {[@dots{}] =} normfit (@var{x}, @var{alpha}, @var{censor}, @var{freq}) ## @deftypefnx {statistics} {[@dots{}] =} normfit (@var{x}, @var{alpha}, @var{censor}, @var{freq}, @var{options}) ## ## Estimate parameters and confidence intervals for the normal distribution. ## ## @code{[@var{muhat}, @var{sigmahat}] = normfit (@var{x})} estimates the ## parameters of the normal distribution given the data in @var{x}. @var{muhat} ## is an estimate of the mean, and @var{sigmahat} is an estimate of the standard ## deviation. ## ## @code{[@var{muhat}, @var{sigmahat}, @var{muci}, @var{sigmaci}] = normfit ## (@var{x})} returns the 95% confidence intervals for the mean and standard ## deviation estimates in the arrays @var{muci} and @var{sigmaci}, respectively. ## ## @itemize ## @item ## @var{x} can be a vector or a matrix. When @var{x} is a matrix, the parameter ## estimates and their confidence intervals are computed for each column. In ## this case, @code{normfit} supports only 2 input arguments, @var{x} and ## @var{alpha}. Optional arguments @var{censor}, @var{freq}, and @var{options} ## can be used only when @var{x} is a vector. ## ## @item ## @var{alpha} is a scalar value in the range @math{(0,1)} specifying the ## confidence level for the confidence intervals calculated as ## @math{100x(1 – alpha)%}. By default, the optional argument @var{alpha} is ## 0.05 corresponding to 95% confidence intervals. Pass in @qcode{[]} for ## @var{alpha} to use the default values. ## ## @item ## @var{censor} is a logical vector of the same length as @var{x} specifying ## whether each value in @var{x} is right-censored or not. 1 indicates ## observations that are right-censored and 0 indicates observations that are ## fully observed. With censoring, @var{muhat} and @var{sigmahat} are the ## maximum likelihood estimates (MLEs). If empty, the default is an array of ## 0s, meaning that all observations are fully observed. ## ## @item ## @var{freq} is a vector of the same length as @var{x} and it typically ## contains non-negative integer counts of the corresponding elements in ## @var{x}. If empty, the default is an array of 1s, meaning one observation ## per element of @var{x}. To obtain the weighted MLEs for a data set with ## censoring, specify weights of observations, normalized to the number of ## observations in @var{x}. However, when there is no censored data (default), ## the returned estimate for standard deviation is not exactly the WMLE. To ## compute the weighted MLE, multiply the value returned in @var{sigmahat} by ## @code{(SUM (@var{freq}) - 1) / SUM (@var{freq})}. This correction is needed ## because @code{normfit} normally computes @var{sigmahat} using an unbiased ## variance estimator when there is no censored data. When there is censoring ## in the data, the correction is not needed, since @code{normfit} does not use ## the unbiased variance estimator in that case. ## ## @item ## @var{options} is a structure with the control parameters for ## @code{fminsearch} which is used internally to compute MLEs for censored data. ## By default, it uses the following options: ## @itemize ## @item @qcode{@var{options}.Display = "off"} ## @item @qcode{@var{options}.MaxFunEvals = 400} ## @item @qcode{@var{options}.MaxIter = 200} ## @item @qcode{@var{options}.TolX = 1e-6} ## @end itemize ## @end itemize ## ## @seealso{normcdf, norminv, normpdf, normrnd, normlike, normstat} ## @end deftypefn function [muhat, sigmahat, muci, sigmaci] = normfit (x, alpha, censor, freq, options) ## Check for valid number of input arguments narginchk (1, 5); ## Check X for being a vector or a matrix if (ndims (x) != 2) error ("normfit: X must not be a multi-dimensional array."); endif if (! isvector (x)) if (nargin < 3) [n, ncols] = size (x); else error ("normfit: matrix data acceptable only under 2-arg syntax."); endif else n = numel (x); ncols = 1; endif ## Check alpha if (nargin < 2 || isempty (alpha)) alpha = 0.05; else if (! isscalar (alpha) || ! isreal (alpha) || alpha <= 0 || alpha >= 1) error ("normfit: wrong value for ALPHA."); endif endif ## Check censor vector if (nargin < 3 || isempty (censor)) censor = 0; elseif (! isequal (size (x), size (censor))) error ("normfit: X and CENSOR vectors mismatch."); endif ## Check frequency vector if (nargin < 4 || isempty (freq)) freq = 1; elseif (isequal (size (x), size (freq))) n = sum (freq); is_zero = find (freq == 0); if (numel (is_zero) > 0) x(is_zero) = []; if (numel (censor) == numel (freq)) censor(is_zero) = []; endif freq(is_zero) = []; end else error ("normfit: X and FREQ vectors mismatch."); endif ## Check options structure or add defaults if (nargin > 4 && ! isempty (options)) if (! isstruct (options) || ! isfield (options, "Display") || ! isfield (options, "MaxFunEvals") || ! isfield (options, "MaxIter") || ! isfield (options, "TolX")) error (strcat (["normfit: 'options' 5th argument must be a"], ... [" structure with 'Display', 'MaxFunEvals',"], ... [" 'MaxIter', and 'TolX' fields present."])); endif else options.Display = "off"; options.MaxFunEvals = 400; options.MaxIter = 200; options.TolX = 1e-6; endif ## Get number of censored and uncensored elements n_censored = sum(freq.*censor); % a scalar in all cases n_uncensored = n - n_censored; % a scalar in all cases ## Compute total sum in X totalsum = sum(freq.*x); ## Check cases that cannot make a fit. ## 1. Handle Infs and NaNs if (! isfinite (totalsum)) muhat = totalsum; sigmahat = NaN ("like", x); muci = NaN (2, 1, "like", x); sigmaci = NaN (2, 1, "like", x); return endif ## 2. All observations are censored or empty data if (n == 0 || n_uncensored == 0) muhat = NaN (1, ncols,'like',x); sigmahat = NaN (1, ncols,'like',x); muci = NaN (2, ncols,'like',x); sigmaci = NaN (2, ncols,'like',x); return endif ## 3. No censored values, compute parameter estimates explicitly. if (n_censored == 0) muhat = totalsum ./ n; if (n > 1) if numel(muhat) == 1 # X is a vector xc = x - muhat; else # X is a matrix xc = x - repmat (muhat, [n, 1]); endif sigmahat = sqrt (sum (conj (xc) .* xc .* freq) ./ (n - 1)); else sigmahat = zeros (1, ncols, "like", x); endif if (nargout > 2) if (n > 1) paramhat = [muhat; sigmahat]; ci = norm_ci (paramhat, [], alpha, x, [], freq); muci = ci(:,:,1); sigmaci = ci(:,:,2); else muci = [-Inf; Inf] * ones (1, ncols, "like", x); sigmaci = [0; Inf] * ones (1, ncols, "like", x); endif endif return endif ## 4. Αll uncensored observations equal and greater than all the ## censored observations x_uncensored = x(censor == 0); range_x_uncensored = range (x_uncensored); if (range_x_uncensored < realmin (class (x))) if (x_uncensored(1) == max (x)) muhat = x_uncensored(1); sigmahat = zeros ('like',x); if (n_uncensored > 1) muci = [muhat; muhat]; sigmaci = zeros (2, 1, "like", x); else muci = cast ([-Inf; Inf], "like", x); sigmaci = cast ([0; Inf], "like", x); endif return endif endif ## Get an initial estimate for parameters using the "least squares" method if (range_x_uncensored > 0) if (numel (freq) == numel (x)) [p,q] = ecdf (x, "censoring", censor, "frequency", freq); else [p,q] = ecdf (x, "censoring", censor); endif pmid = (p(1:(end-1)) + p(2:end)) / 2; linefit = polyfit (-sqrt (2) * erfcinv (2 * pmid), q(2:end), 1); paramhat = linefit ([2 1]); else # only one uncensored element in X paramhat = [x_uncensored(1) 1]; endif ## Optimize the parameters as doubles, regardless of input data type paramhat = cast (paramhat, "double"); ## Search for parameter that minimize the negative log likelihood function [paramhat, ~, err, output] = fminsearch ... (@(ph) norm_nlogl (ph, x, censor, freq), paramhat, options); ## Handle errors if (err == 0) if (output.funcCount >= options.MaxFunEvals) warning ("normfit: maximum number of function evaluations are exceeded."); elseif (output.iterations >= options.MaxIter) warning ("normfit: maximum number of iterations are exceeded."); endif elseif (err < 0) error ("normfit: NoSolution."); endif ## Make sure the outputs match the input data type muhat = cast (paramhat(1), "like", x); sigmahat = cast (paramhat(2), "like", x); if (nargout > 2) paramhat = paramhat(:); if (numel (freq) == numel (x)) [~, avar] = normlike (paramhat, x, censor, freq); else [~, avar] = normlike (paramhat, x, censor); endif ci = norm_ci (paramhat, avar, alpha, x, censor, freq); muci = ci(:,:,1); sigmaci = ci(:,:,2); endif endfunction ## Negative log-likelihood function and gradient for normal distribution. function [nlogL, avar] = norm_nlogl (params, x, censor, freq) ## Get mu and sigma values mu = params(1); sigma = params(2); ## Compute the individual log-likelihood terms. Force a log(0)==-Inf for ## data from extreme right tail, instead of getting exp(Inf-Inf)==NaN. z = (x - mu) ./ sigma; L = -0.5 .* z .^ 2 - log (sqrt (2 .* pi) .* sigma); if (any (censor)) censored = censor == 1; z_censor = z(censored); S_censor = 0.5 * erfc (z_censor / sqrt (2)); L(censored) = log (S_censor); endif ## Neg-log-like is the sum of the individual contributions nlogL = -sum (freq .* L); ## Compute the negative hessian at the parameter values. ## Invert to get the observed information matrix. if (nargout == 2) dL11 = -ones (size (z), class (z)); dL12 = -2 .* z; dL22 = 1 - 3 .* z .^ 2; if (any (censor)) dlogScen = exp (-0.5 .* z_censor .^ 2) ./ (sqrt (2 * pi) .* S_censor); d2logScen = dlogScen .* (dlogScen - z_censor); dL11(censored) = -d2logScen; dL12(censored) = -dlogScen - z_censor .* d2logScen; dL22(censored) = -z_censor .* (2 .* dlogScen + z_censor .* d2logScen); endif nH11 = -sum (freq .* dL11); nH12 = -sum (freq .* dL12); nH22 = -sum (freq .* dL22); avar = (sigma .^ 2) * [nH22, -nH12; -nH12, nH11] / ... (nH11 * nH22 - nH12 * nH12); endif endfunction ## Confidence intervals for normal distribution function ci = norm_ci (paramhat, cv, alpha, x, censor, freq) ## Check for missing input arguments if (nargin < 6 || isempty (freq)) freq = ones (size (x)); endif if (nargin < 5 || isempty (censor)) censor = false (size (x)); endif if (isvector (paramhat)) paramhat = paramhat(:); endif muhat = paramhat(1,:); sigmahat = paramhat(2,:); ## Get number of elements if (isempty (freq) || isequal (freq, 1)) if (isvector (x)) n = length (x); else n = size (x, 1); endif else n = sum (freq); endif ## Get number of censored and uncensored elements n_censored = sum (freq .* censor); n_uncensored = n - n_censored; ## Just in case if (any (censor) && (n == 0 || n_uncensored == 0 || ! isfinite (paramhat(1)))) ## X is a vector muci = NaN(2,1); sigmaci = NaN(2,1); ci = cast (cat (3, muci, sigmaci), "like", x); return endif ## Get confidence intervals for each parameter if ((isempty (censor) || ! any (censor(:))) && ! isequal (cv,zeros(2,2))) ## Use exact formulas tcrit = tinv ([alpha/2, 1-alpha/2], n-1); muci = [muhat+tcrit(1)*sigmahat/sqrt(n); muhat+tcrit(2)*sigmahat/sqrt(n)]; chi2crit = chi2inv ([alpha/2, 1-alpha/2], n-1); sigmaci = [sigmahat*sqrt((n-1)./chi2crit(2)); ... sigmahat*sqrt((n-1)./chi2crit(1))]; else ## Use normal approximation probs = [alpha/2; 1-alpha/2]; se = sqrt (diag (cv))'; z = norminv (probs); ## Compute the CI for mu using a normal distribution for muhat. muci = muhat + se(1) .* z; ## Compute the CI for sigma using a normal approximation for ## log(sigmahat), and transform back to the original scale. logsigci = log (sigmahat) + (se(2) ./ sigmahat) .* z; sigmaci = exp (logsigci); endif ## Return as a single array ci = cat (3, muci, sigmaci); endfunction %!demo %! ## Sample 3 populations from 3 different normal distibutions %! randn ("seed", 1); # for reproducibility %! r1 = normrnd (2, 5, 5000, 1); %! randn ("seed", 2); # for reproducibility %! r2 = normrnd (5, 2, 5000, 1); %! randn ("seed", 3); # for reproducibility %! r3 = normrnd (9, 4, 5000, 1); %! r = [r1, r2, r3]; %! %! ## Plot them normalized and fix their colors %! hist (r, 15, 0.4); %! h = findobj (gca, "Type", "patch"); %! set (h(1), "facecolor", "c"); %! set (h(2), "facecolor", "g"); %! set (h(3), "facecolor", "r"); %! hold on %! %! ## Estimate their mu and sigma parameters %! [muhat, sigmahat] = normfit (r); %! %! ## Plot their estimated PDFs %! x = [min(r(:)):max(r(:))]; %! y = normpdf (x, muhat(1), sigmahat(1)); %! plot (x, y, "-pr"); %! y = normpdf (x, muhat(2), sigmahat(2)); %! plot (x, y, "-sg"); %! y = normpdf (x, muhat(3), sigmahat(3)); %! plot (x, y, "-^c"); %! ylim ([0, 0.5]) %! xlim ([-20, 20]) %! hold off %! legend ({"Normalized HIST of sample 1 with mu=2, σ=5", ... %! "Normalized HIST of sample 2 with mu=5, σ=2", ... %! "Normalized HIST of sample 3 with mu=9, σ=4", ... %! sprintf("PDF for sample 1 with estimated mu=%0.2f and σ=%0.2f", ... %! muhat(1), sigmahat(1)), ... %! sprintf("PDF for sample 2 with estimated mu=%0.2f and σ=%0.2f", ... %! muhat(2), sigmahat(2)), ... %! sprintf("PDF for sample 3 with estimated mu=%0.2f and σ=%0.2f", ... %! muhat(3), sigmahat(3))}, "location", "northwest") %! title ("Three population samples from different normal distibutions") %! hold off ## Test output %!test %! load lightbulb %! idx = find (lightbulb(:,2) == 0); %! censoring = lightbulb(idx,3) == 1; %! [muHat, sigmaHat] = normfit (lightbulb(idx,1), [], censoring); %! assert (muHat, 9496.59586737857, 1e-11); %! assert (sigmaHat, 3064.021012796456, 2e-12); %!test %! randn ("seed", 234); %! x = normrnd (3, 5, [1000, 1]); %! [muHat, sigmaHat, muCI, sigmaCI] = normfit (x, 0.01); %! assert (muCI(1) < 3); %! assert (muCI(2) > 3); %! assert (sigmaCI(1) < 5); %! assert (sigmaCI(2) > 5); ## Test input validation %!error ... %! normfit (ones (3,3,3)) %!error ... %! normfit (ones (20,3), [], zeros (20,1)) %!error normfit (ones (20,1), 0) %!error normfit (ones (20,1), -0.3) %!error normfit (ones (20,1), 1.2) %!error normfit (ones (20,1), [0.05 0.1]) %!error normfit (ones (20,1), 0.02+i) %!error ... %! normfit (ones (20,1), [], zeros(15,1)) %!error ... %! normfit (ones (20,1), [], zeros(20,1), ones(25,1)) %!error normfit (ones (20,1), [], zeros(20,1), ones(20,1), "options") statistics-release-1.6.3/inst/dist_fit/normlike.m000066400000000000000000000146511456127120000221160ustar00rootroot00000000000000## Copyright (C) 2022 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## Octave is free software; you can redistribute it and/or modify it ## under the terms of the GNU General Public License as published by ## the Free Software Foundation; either version 3 of the License, or (at ## your option) any later version. ## ## Octave is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with Octave; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{nlogL} =} normlike (@var{params}, @var{x}) ## @deftypefnx {statistics} {[@var{nlogL}, @var{avar}] =} normlike (@var{params}, @var{x}) ## @deftypefnx {statistics} {[@dots{}] =} normlike (@var{params}, @var{x}, @var{censor}) ## @deftypefnx {statistics} {[@dots{}] =} normlike (@var{params}, @var{x}, @var{censor}, @var{freq}) ## ## Negative log-likelihood for the normal distribution. ## ## @code{@var{nlogL} = normlike (@var{params}, @var{x})} returns the negative ## log-likelihood for the normal distribution, evaluated at parameters ## @var{params(1)} = mean and @var{params(2)} = standard deviation, given ## @var{x}. @var{nlogL} is a scalar. ## ## @code{[@var{nlogL}, @var{avar}] = normlike (@var{params}, @var{x})} ## returns the inverse of Fisher's information matrix, @var{avar}. If the input ## parameter values in @var{params} are the maximum likelihood estimates, the ## diagonal elements of @var{avar} are their asymptotic variances. @var{avar} ## is based on the observed Fisher's information, not the expected information. ## ## @code{[@dots{}] = normlike (@var{params}, @var{x}, @var{censor})} accepts ## a boolean vector of the same size as @var{x} that is 1 for observations ## that are right-censored and 0 for observations that are observed exactly. ## ## @code{[@dots{}] = normlike (@var{params}, @var{x}, @var{censor}, ## @var{freq})} accepts a frequency vector of the same size as @var{x}. ## @var{freq} typically contains integer frequencies for the corresponding ## elements in @var{x}, but it may contain any non-integer non-negative ## values. Pass in [] for @var{censor} to use its default value. ## ## @seealso{normcdf, norminv, normpdf, normrnd, normfit, normstat} ## @end deftypefn function [nlogL, avar] = normlike (params, x, censor, freq) ## Check input arguments if (nargin < 2) error ("normlike: too few input arguments."); endif if (! isvector (x)) error ("normlike: X must be a vector."); endif if (numel (params) != 2) error ("normlike: PARAMS must be a two-element vector."); endif if (nargin < 3 || isempty (censor)) censor = zeros (size (x)); elseif (! isequal (size (x), size (censor))) error ("normlike: X and CENSOR vectors mismatch."); endif if nargin < 4 || isempty (freq) freq = ones (size (x)); elseif (isequal (size (x), size (freq))) nulls = find (freq == 0); if (numel (nulls) > 0) x(nulls) = []; censor(nulls) = []; freq(nulls) = []; endif else error ("normlike: X and FREQ vectors mismatch."); endif ## Get mu and sigma values mu = params(1); sigma = params(2); ## sigma must be positive, otherwise make it NaN if (sigma <= 0) sigma = NaN; endif ## Compute the individual log-likelihood terms. Force a log(0)==-Inf for ## x from extreme right tail, instead of getting exp(Inf-Inf)==NaN. z = (x - mu) ./ sigma; L = -0.5 .* z .^ 2 - log (sqrt (2 .* pi) .* sigma); if (any (censor)) censored = censor == 1; z_censor = z(censored); S_censor = 0.5 * erfc (z_censor / sqrt (2)); L(censored) = log (S_censor); endif ## Neg-log-like is the sum of the individual contributions nlogL = -sum (freq .* L); ## Compute the negative hessian at the parameter values. ## Invert to get the observed information matrix. if (nargout == 2) dL11 = -ones (size (z), class (z)); dL12 = -2 .* z; dL22 = 1 - 3 .* z .^ 2; if (any (censor)) dlogScen = exp (-0.5 .* z_censor .^ 2) ./ (sqrt (2 * pi) .* S_censor); d2logScen = dlogScen .* (dlogScen - z_censor); dL11(censored) = -d2logScen; dL12(censored) = -dlogScen - z_censor .* d2logScen; dL22(censored) = -z_censor .* (2 .* dlogScen + z_censor .* d2logScen); endif nH11 = -sum (freq .* dL11); nH12 = -sum (freq .* dL12); nH22 = -sum (freq .* dL22); avar = (sigma .^ 2) * [nH22, -nH12; -nH12, nH11] / ... (nH11 * nH22 - nH12 * nH12); endif endfunction ## Test input validation %!error normlike ([12, 15]); %!error normlike ([12, 15], ones (2)); %!error ... %! normlike ([12, 15, 3], [1:50]); %!error ... %! normlike ([12, 15], [1:50], [1, 2, 3]); %!error ... %! normlike ([12, 15], [1:50], [], [1, 2, 3]); ## Results compared with Matlab %!test %! x = 1:50; %! [nlogL, avar] = normlike ([2.3, 1.2], x); %! avar_out = [7.5767e-01, -1.8850e-02; -1.8850e-02, 4.8750e-04]; %! assert (nlogL, 13014.95883783327, 1e-10); %! assert (avar, avar_out, 1e-4); %!test %! x = 1:50; %! [nlogL, avar] = normlike ([2.3, 1.2], x * 0.5); %! avar_out = [3.0501e-01, -1.5859e-02; -1.5859e-02, 9.1057e-04]; %! assert (nlogL, 2854.802587833265, 1e-10); %! assert (avar, avar_out, 1e-4); %!test %! x = 1:50; %! [nlogL, avar] = normlike ([21, 15], x); %! avar_out = [5.460474308300396, -1.600790513833993; ... %! -1.600790513833993, 2.667984189723321]; %! assert (nlogL, 206.738325604233, 1e-12); %! assert (avar, avar_out, 1e-14); %!test %! x = 1:50; %! censor = ones (1, 50); %! censor([2, 4, 6, 8, 12, 14]) = 0; %! [nlogL, avar] = normlike ([2.3, 1.2], x, censor); %! avar_out = [3.0501e-01, -1.5859e-02; -1.5859e-02, 9.1057e-04]; %! assert (nlogL, Inf); %! assert (avar, [NaN, NaN; NaN, NaN]); %!test %! x = 1:50; %! censor = ones (1, 50); %! censor([2, 4, 6, 8, 12, 14]) = 0; %! [nlogL, avar] = normlike ([21, 15], x, censor); %! avar_out = [24.4824488866131, -10.6649544179636; ... %! -10.6649544179636, 6.22827849965737]; %! assert (nlogL, 86.9254371829733, 1e-12); %! assert (avar, avar_out, 8e-14); statistics-release-1.6.3/inst/dist_fit/poissfit.m000066400000000000000000000144561456127120000221410ustar00rootroot00000000000000## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{lambdahat} =} poissfit (@var{x}) ## @deftypefnx {statistics} {[@var{lambdahat}, @var{lambdaci}] =} poissfit (@var{x}) ## @deftypefnx {statistics} {[@var{lambdahat}, @var{lambdaci}] =} poissfit (@var{x}, @var{alpha}) ## @deftypefnx {statistics} {[@var{lambdahat}, @var{lambdaci}] =} poissfit (@var{x}, @var{alpha}, @var{freq}) ## ## Estimate parameter and confidence intervals for the Poisson distribution. ## ## @code{@var{lambdahat} = poissfit (@var{x})} returns the maximum likelihood ## estimate of the rate parameter, @var{lambda}, of the Poisson distribution ## given the data in @var{x}. @var{x} must be a vector of non-negative values. ## ## @code{[@var{lambdahat}, @var{lambdaci}] = poissfit (@var{x})} returns the 95% ## confidence intervals for the parameter estimate. ## ## @code{[@var{lambdahat}, @var{lambdaci}] = poissfit (@var{x}, @var{alpha})} ## also returns the @qcode{100 * (1 - @var{alpha})} percent confidence intervals ## of the estimated parameter. By default, the optional argument @var{alpha} is ## 0.05 corresponding to 95% confidence intervals. Pass in @qcode{[]} for ## @var{alpha} to use the default values. ## ## @code{[@dots{}] = poissfit (@var{x}, @var{alpha}, @var{freq})} accepts a ## frequency vector or matrix, @var{freq}, of the same size as @var{x}. ## @var{freq} typically contains integer frequencies for the corresponding ## elements in @var{x}. @var{freq} cannot contain negative values. ## ## Further information about the Poisson distribution can be found at ## @url{https://en.wikipedia.org/wiki/Poisson_distribution} ## ## @seealso{poisscdf, poissinv, poisspdf, poissrnd, poisslike, poisstat} ## @end deftypefn function [lambdahat, lambdaci] = poissfit (x, alpha, freq=[]) ## Check input arguments if (any (x < 0)) error ("poissfit: X cannot have negative values."); endif if (nargin < 2 || isempty (alpha)) alpha = 0.05; elseif (! isscalar (alpha) || ! isreal (alpha) || alpha <= 0 || alpha >= 1) error ("poissfit: wrong value for ALPHA."); endif if (isempty (freq)) freq = ones (size (x)); elseif (! isequal (size (x), size (freq))) error ("poissfit: X and FREQ vectors mismatch."); elseif (any (freq < 0)) error ("poissfit: FREQ must not contain negative values."); endif if (isvector (x)) x = x(:); freq = freq(:); endif ## Compute lambdahat n = sum (freq, 1); lambdahat = double (sum (x .* freq) ./ n); ## Compute confidence intervals lambdasum = n .* lambdahat; ## Select elements for exact method or normal approximation k = (lambdasum < 100); if (any (k)) # exact method lb(k) = chi2inv (alpha / 2, 2 * lambdasum(k)) / 2; ub(k) = chi2inv (1 - alpha / 2, 2 * (lambdasum(k) + 1)) / 2; endif k = ! k; if (any (k)) # normal approximation lb(k) = norminv (alpha / 2, lambdasum(k), sqrt (lambdasum(k))); ub(k) = norminv (1 - alpha / 2, lambdasum(k), sqrt (lambdasum(k))); endif lambdaci = [lb; ub] / n; endfunction %!demo %! ## Sample 3 populations from 3 different Poisson distibutions %! randp ("seed", 2); # for reproducibility %! r1 = poissrnd (1, 1000, 1); %! randp ("seed", 2); # for reproducibility %! r2 = poissrnd (4, 1000, 1); %! randp ("seed", 3); # for reproducibility %! r3 = poissrnd (10, 1000, 1); %! r = [r1, r2, r3]; %! %! ## Plot them normalized and fix their colors %! hist (r, [0:20], 1); %! h = findobj (gca, "Type", "patch"); %! set (h(1), "facecolor", "c"); %! set (h(2), "facecolor", "g"); %! set (h(3), "facecolor", "r"); %! hold on %! %! ## Estimate their lambda parameter %! lambdahat = poissfit (r); %! %! ## Plot their estimated PDFs %! x = [0:20]; %! y = poisspdf (x, lambdahat(1)); %! plot (x, y, "-pr"); %! y = poisspdf (x, lambdahat(2)); %! plot (x, y, "-sg"); %! y = poisspdf (x, lambdahat(3)); %! plot (x, y, "-^c"); %! xlim ([0, 20]) %! ylim ([0, 0.4]) %! legend ({"Normalized HIST of sample 1 with λ=1", ... %! "Normalized HIST of sample 2 with λ=4", ... %! "Normalized HIST of sample 3 with λ=10", ... %! sprintf("PDF for sample 1 with estimated λ=%0.2f", ... %! lambdahat(1)), ... %! sprintf("PDF for sample 2 with estimated λ=%0.2f", ... %! lambdahat(2)), ... %! sprintf("PDF for sample 3 with estimated λ=%0.2f", ... %! lambdahat(3))}) %! title ("Three population samples from different Poisson distibutions") %! hold off ## Test output %!test %! x = [1 3 2 4 5 4 3 4]; %! [lhat, lci] = poissfit (x); %! assert (lhat, 3.25) %! assert (lci, [2.123007901949543; 4.762003010390628], 1e-14) %!test %! x = [1 3 2 4 5 4 3 4]; %! [lhat, lci] = poissfit (x, 0.01); %! assert (lhat, 3.25) %! assert (lci, [1.842572740234582; 5.281369033298528], 1e-14) %!test %! x = [1 2 3 4 5]; %! f = [1 1 2 3 1]; %! [lhat, lci] = poissfit (x, [], f); %! assert (lhat, 3.25) %! assert (lci, [2.123007901949543; 4.762003010390628], 1e-14) %!test %! x = [1 2 3 4 5]; %! f = [1 1 2 3 1]; %! [lhat, lci] = poissfit (x, 0.01, f); %! assert (lhat, 3.25) %! assert (lci, [1.842572740234582; 5.281369033298528], 1e-14) ## Test input validation %!error poissfit ([1 2 -1 3]) %!error poissfit ([1 2 3], 0) %!error poissfit ([1 2 3], 1.2) %!error poissfit ([1 2 3], [0.02 0.05]) %!error %! poissfit ([1 2 3], [], [1 5]) %!error %! poissfit ([1 2 3], [], [1 5 -1]) statistics-release-1.6.3/inst/dist_fit/poisslike.m000066400000000000000000000076071456127120000223030ustar00rootroot00000000000000## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{nlogL} =} poisslike (@var{lambda}, @var{x}) ## @deftypefnx {statistics} {[@var{nlogL}, @var{avar}] =} poisslike (@var{lambda}, @var{x}) ## @deftypefnx {statistics} {[@dots{}] =} poisslike (@var{lambda}, @var{x}, @var{freq}) ## ## Negative log-likelihood for the Poisson distribution. ## ## @code{@var{nlogL} = poisslike (@var{lambda}, @var{x})} returns the negative ## log likelihood of the data in @var{x} corresponding to the Poisson ## distribution with rate parameter @var{lambda}. @var{x} must be a vector of ## non-negative values. ## ## @code{[@var{nlogL}, @var{avar}] = poisslike (@var{lambda}, @var{x})} also ## returns the inverse of Fisher's information matrix, @var{avar}. If the input ## rate parameter, @var{lambda}, is the maximum likelihood estimate, @var{avar} ## is its asymptotic variance. ## ## @code{[@dots{}] = poisslike (@var{lambda}, @var{x}, @var{freq})} accepts a ## frequency vector, @var{freq}, of the same size as @var{x}. @var{freq} ## typically contains integer frequencies for the corresponding elements in ## @var{x}, but it can contain any non-integer non-negative values. By default, ## or if left empty, @qcode{@var{freq} = ones (size (@var{x}))}. ## ## Further information about the Poisson distribution can be found at ## @url{https://en.wikipedia.org/wiki/Poisson_distribution} ## ## @seealso{poisscdf, poissinv, poisspdf, poissrnd, poissfit, poisstat} ## @end deftypefn function [nlogL, avar] = poisslike (lambda, x, freq=[]) ## Check input arguments if (nargin < 2) error ("poisslike: function called with too few input arguments."); endif if (! isscalar (lambda) || ! isnumeric (lambda) || lambda <= 0) error ("poisslike: LAMBDA must be a positive scalar."); endif if (! isvector (x) || any (x < 0)) error ("poisslike: X must be a vector of non-negative values."); endif if (isempty (freq)) freq = ones (size (x)); elseif (! isequal (size (x), size (freq))) error ("poisslike: X and FREQ vectors mismatch."); elseif (any (freq < 0)) error ("poisslike: FREQ must not contain negative values."); endif ## Compute negative log-likelihood and asymptotic covariance n = sum (freq); nlogL = - sum (freq .* log (poisspdf (x, lambda))); avar = lambda / n; endfunction ## Test output %!test %! x = [1 3 2 4 5 4 3 4]; %! [nlogL, avar] = poisslike (3.25, x); %! assert (nlogL, 13.9533, 1e-4) %!test %! x = [1 2 3 4 5]; %! f = [1 1 2 3 1]; %! [nlogL, avar] = poisslike (3.25, x, f); %! assert (nlogL, 13.9533, 1e-4) ## Test input validation %!error poisslike (1) %!error poisslike ([1 2 3], [1 2]) %!error ... %! poisslike (3.25, ones (10, 2)) %!error ... %! poisslike (3.25, [1 2 3 -4 5]) %!error ... %! poisslike (3.25, ones (10, 1), ones (8,1)) %!error ... %! poisslike (3.25, ones (1, 8), [1 1 1 1 1 1 1 -1]) statistics-release-1.6.3/inst/dist_fit/raylfit.m000066400000000000000000000170161456127120000217460ustar00rootroot00000000000000## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{sigmaA} =} raylfit (@var{x}) ## @deftypefnx {statistics} {[@var{sigmaA}, @var{sigmaci}] =} raylfit (@var{x}) ## @deftypefnx {statistics} {[@var{sigmaA}, @var{sigmaci}] =} raylfit (@var{x}, @var{alpha}) ## @deftypefnx {statistics} {[@var{sigmaA}, @var{sigmaci}] =} raylfit (@var{x}, @var{alpha}, @var{censor}) ## @deftypefnx {statistics} {[@var{sigmaA}, @var{sigmaci}] =} raylfit (@var{x}, @var{alpha}, @var{censor}, @var{freq}) ## ## Estimate parameter and confidence intervals for the Rayleigh distribution. ## ## @code{@var{sigmaA} = raylfit (@var{x})} returns the maximum likelihood ## estimate of the rate parameter, @var{lambda}, of the Rayleigh distribution ## given the data in @var{x}. @var{x} must be a vector of non-negative values. ## ## @code{[@var{sigmaA}, @var{sigmaci}] = raylfit (@var{x})} returns the 95% ## confidence intervals for the parameter estimate. ## ## @code{[@var{sigmaA}, @var{sigmaci}] = raylfit (@var{x}, @var{alpha})} ## also returns the @qcode{100 * (1 - @var{alpha})} percent confidence intervals ## of the estimated parameter. By default, the optional argument @var{alpha} is ## 0.05 corresponding to 95% confidence intervals. Pass in @qcode{[]} for ## @var{alpha} to use the default values. ## ## @code{[@dots{}] = raylfit (@var{x}, @var{alpha}, @var{censor})} accepts a ## boolean vector, @var{censor}, of the same size as @var{x} with @qcode{1}s for ## observations that are right-censored and @qcode{0}s for observations that are ## observed exactly. By default, or if left empty, ## @qcode{@var{censor} = zeros (size (@var{x}))}. ## ## @code{[@dots{}] = raylfit (@var{x}, @var{alpha}, @var{censor}, @var{freq})} ## accepts a frequency vector or matrix, @var{freq}, of the same size as @var{x}. ## @var{freq} typically contains integer frequencies for the corresponding ## elements in @var{x}. @var{freq} cannot contain negative values. ## ## Further information about the Rayleigh distribution can be found at ## @url{https://en.wikipedia.org/wiki/Rayleigh_distribution} ## ## @seealso{raylcdf, raylinv, raylpdf, raylrnd, rayllike, raylstat} ## @end deftypefn function [sigmaA, sigmaci] = raylfit (x, alpha, censor, freq) ## Check input arguments if (any (x < 0)) error ("raylfit: X cannot have negative values."); endif if (! isvector (x)) error ("raylfit: X must be a vector."); endif ## Check alpha if (nargin < 2 || isempty (alpha)) alpha = 0.05; elseif (! isscalar (alpha) || ! isreal (alpha) || alpha <= 0 || alpha >= 1) error ("raylfit: wrong value for ALPHA."); endif ## Check censor vector if (nargin < 3 || isempty (censor)) censor = zeros (size (x)); elseif (! isequal (size (x), size (censor))) error ("raylfit: X and CENSOR vectors mismatch."); endif ## Check frequency vector if (nargin < 4 || isempty (freq)) freq = ones (size (x)); elseif (! isequal (size (x), size (freq))) error ("raylfit: X and FREQ vectors mismatch."); elseif (any (freq < 0)) error ("raylfit: FREQ must not contain negative values."); endif ## Remove any censored data censored = censor == 1; freq(censored) = []; x(censored) = []; ## Expand frequency vector (if necessary) if (! all (freq == 1)) xf = []; for i = 1:numel (freq) xf = [xf, repmat(x(i), 1, freq(i))]; endfor x = xf; endif ## Compute sigmaA sigmaA = sqrt (0.5 * mean (x .^ 2)); ## Compute confidence intervals (based on chi-squared) if (nargout > 1) sx = 2 * numel (x); ci = [1-alpha/2; alpha/2]; sigmaci = sqrt (sx * sigmaA .^ 2 ./ chi2inv (ci, sx)); endif endfunction %!demo %! ## Sample 3 populations from 3 different Rayleigh distibutions %! rand ("seed", 2); # for reproducibility %! r1 = raylrnd (1, 1000, 1); %! rand ("seed", 2); # for reproducibility %! r2 = raylrnd (2, 1000, 1); %! rand ("seed", 3); # for reproducibility %! r3 = raylrnd (4, 1000, 1); %! r = [r1, r2, r3]; %! %! ## Plot them normalized and fix their colors %! hist (r, [0.5:0.5:10.5], 2); %! h = findobj (gca, "Type", "patch"); %! set (h(1), "facecolor", "c"); %! set (h(2), "facecolor", "g"); %! set (h(3), "facecolor", "r"); %! hold on %! %! ## Estimate their lambda parameter %! sigmaA = raylfit (r(:,1)); %! sigmaB = raylfit (r(:,2)); %! sigmaC = raylfit (r(:,3)); %! %! ## Plot their estimated PDFs %! x = [0:0.1:10]; %! y = raylpdf (x, sigmaA); %! plot (x, y, "-pr"); %! y = raylpdf (x, sigmaB); %! plot (x, y, "-sg"); %! y = raylpdf (x, sigmaC); %! plot (x, y, "-^c"); %! xlim ([0, 10]) %! ylim ([0, 0.7]) %! legend ({"Normalized HIST of sample 1 with σ=1", ... %! "Normalized HIST of sample 2 with σ=2", ... %! "Normalized HIST of sample 3 with σ=4", ... %! sprintf("PDF for sample 1 with estimated σ=%0.2f", ... %! sigmaA), ... %! sprintf("PDF for sample 2 with estimated σ=%0.2f", ... %! sigmaB), ... %! sprintf("PDF for sample 3 with estimated σ=%0.2f", ... %! sigmaC)}) %! title ("Three population samples from different Rayleigh distibutions") %! hold off ## Test output %!test %! x = [1 3 2 4 5 4 3 4]; %! [shat, sci] = raylfit (x); %! assert (shat, 2.4495, 1e-4) %! assert (sci, [1.8243; 3.7279], 1e-4) %!test %! x = [1 3 2 4 5 4 3 4]; %! [shat, sci] = raylfit (x, 0.01); %! assert (shat, 2.4495, 1e-4) %! assert (sci, [1.6738; 4.3208], 1e-4) %!test %! x = [1 2 3 4 5]; %! f = [1 1 2 3 1]; %! [shat, sci] = raylfit (x, [], [], f); %! assert (shat, 2.4495, 1e-4) %! assert (sci, [1.8243; 3.7279], 1e-4) %!test %! x = [1 2 3 4 5]; %! f = [1 1 2 3 1]; %! [shat, sci] = raylfit (x, 0.01, [], f); %! assert (shat, 2.4495, 1e-4) %! assert (sci, [1.6738; 4.3208], 1e-4) %!test %! x = [1 2 3 4 5 6]; %! c = [0 0 0 0 0 1]; %! f = [1 1 2 3 1 1]; %! [shat, sci] = raylfit (x, 0.01, c, f); %! assert (shat, 2.4495, 1e-4) %! assert (sci, [1.6738; 4.3208], 1e-4) ## Test input validation %!error raylfit (ones (2,5)); %!error raylfit ([1 2 -1 3]) %!error raylfit ([1 2 3], 0) %!error raylfit ([1 2 3], 1.2) %!error raylfit ([1 2 3], [0.02 0.05]) %!error ... %! raylfit ([1, 2, 3, 4, 5], 0.05, [1 1 0]); %!error ... %! raylfit ([1, 2, 3, 4, 5], [], [1 1 0 1 1]'); %!error ... %! raylfit ([1, 2, 3, 4, 5], 0.05, zeros (1,5), [1 1 0]); %!error ... %! raylfit ([1, 2, 3, 4, 5], [], [], [1 1 0 1 1]'); %!error %! raylfit ([1 2 3], [], [], [1 5]) %!error %! raylfit ([1 2 3], [], [], [1 5 -1]) statistics-release-1.6.3/inst/dist_fit/rayllike.m000066400000000000000000000117451456127120000221130ustar00rootroot00000000000000## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{nlogL} =} rayllike (@var{sigma}, @var{x}) ## @deftypefnx {statistics} {[@var{nlogL}, @var{acov}] =} rayllike (@var{sigma}, @var{x}) ## @deftypefnx {statistics} {[@dots{}] =} rayllike (@var{sigma}, @var{x}, @var{freq}) ## ## Negative log-likelihood for the Rayleigh distribution. ## ## @code{@var{nlogL} = rayllike (@var{sigma}, @var{x})} returns the negative ## log likelihood of the data in @var{x} corresponding to the Rayleigh ## distribution with rate parameter @var{sigma}. @var{x} must be a vector of ## non-negative values. ## ## @code{[@var{nlogL}, @var{acov}] = rayllike (@var{sigma}, @var{x})} also ## returns the inverse of Fisher's information matrix, @var{acov}. If the input ## rate parameter, @var{sigma}, is the maximum likelihood estimate, @var{acov} ## is its asymptotic variance. ## ## @code{[@dots{}] = rayllike (@var{sigma}, @var{x}, @var{freq})} accepts a ## frequency vector, @var{freq}, of the same size as @var{x}. @var{freq} ## typically contains integer frequencies for the corresponding elements in ## @var{x}, but it can contain any non-integer non-negative values. By default, ## or if left empty, @qcode{@var{freq} = ones (size (@var{x}))}. ## ## Further information about the Rayleigh distribution can be found at ## @url{https://en.wikipedia.org/wiki/Rayleigh_distribution} ## ## @seealso{raylcdf, raylinv, raylpdf, raylrnd, raylfit, raylstat} ## @end deftypefn function [nlogL, acov] = rayllike (sigma, x, censor, freq) ## Check input arguments if (nargin < 2) error ("rayllike: function called with too few input arguments."); endif if (! isscalar (sigma) || ! isnumeric (sigma) || sigma <= 0) error ("rayllike: SIGMA must be a positive scalar."); endif if (! isvector (x) || any (x < 0)) error ("rayllike: X must be a vector of non-negative values."); endif ## Check censor vector if (nargin < 3 || isempty (censor)) censor = zeros (size (x)); elseif (! isequal (size (x), size (censor))) error ("rayllike: X and CENSOR vectors mismatch."); endif ## Check frequency vector if (nargin < 4 || isempty (freq)) freq = ones (size (x)); elseif (! isequal (size (x), size (freq))) error ("rayllike: X and FREQ vectors mismatch."); elseif (any (freq < 0)) error ("rayllike: FREQ must not contain negative values."); endif ## Compute negative log-likelihood and asymptotic covariance zsq = (x / sigma) .^ 2; logfz = -2 * log (sigma) - zsq / 2 + log (x); dlogfz = (zsq - 2) / sigma; logS = - zsq / 2; dlogS = - zsq * 3 / sigma ^ 2; logfz(censor == 1) = logS(censor == 1); nlogL = - sum (freq .* logfz); if (nargout > 1) d2 = (2 - 3 * zsq) / sigma ^ 2; d2(censor == 1) = - 3 * zsq(censor == 1) / sigma ^ 2; acov = - sum (freq .* d2); endif endfunction ## Test output %!test %! x = [1 3 2 4 5 4 3 4]; %! [nlogL, acov] = rayllike (3.25, x); %! assert (nlogL, 14.7442, 1e-4) %!test %! x = [1 2 3 4 5]; %! f = [1 1 2 3 1]; %! [nlogL, acov] = rayllike (3.25, x, [], f); %! assert (nlogL, 14.7442, 1e-4) %!test %! x = [1 2 3 4 5 6]; %! f = [1 1 2 3 1 0]; %! [nlogL, acov] = rayllike (3.25, x, [], f); %! assert (nlogL, 14.7442, 1e-4) %!test %! x = [1 2 3 4 5 6]; %! c = [0 0 0 0 0 1]; %! f = [1 1 2 3 1 0]; %! [nlogL, acov] = rayllike (3.25, x, c, f); %! assert (nlogL, 14.7442, 1e-4) ## Test input validation %!error rayllike (1) %!error rayllike ([1 2 3], [1 2]) %!error ... %! rayllike (3.25, ones (10, 2)) %!error ... %! rayllike (3.25, [1 2 3 -4 5]) %!error ... %! rayllike (3.25, [1, 2, 3, 4, 5], [1 1 0]); %!error ... %! rayllike (3.25, [1, 2, 3, 4, 5], [1 1 0 1 1]'); %!error ... %! rayllike (3.25, [1, 2, 3, 4, 5], zeros (1,5), [1 1 0]); %!error ... %! rayllike (3.25, [1, 2, 3, 4, 5], [], [1 1 0 1 1]'); %!error ... %! rayllike (3.25, ones (1, 8), [], [1 1 1 1 1 1 1 -1]) statistics-release-1.6.3/inst/dist_fit/ricefit.m000066400000000000000000000254661456127120000217310ustar00rootroot00000000000000## Copyright (C) 2024 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{paramhat} =} ricefit (@var{x}) ## @deftypefnx {statistics} {[@var{paramhat}, @var{paramci}] =} ricefit (@var{x}) ## @deftypefnx {statistics} {[@var{paramhat}, @var{paramci}] =} ricefit (@var{x}, @var{alpha}) ## @deftypefnx {statistics} {[@dots{}] =} ricefit (@var{x}, @var{alpha}, @var{censor}) ## @deftypefnx {statistics} {[@dots{}] =} ricefit (@var{x}, @var{alpha}, @var{censor}, @var{freq}) ## @deftypefnx {statistics} {[@dots{}] =} ricefit (@var{x}, @var{alpha}, @var{censor}, @var{freq}, @var{options}) ## ## Estimate parameters and confidence intervals for the Gamma distribution. ## ## @code{@var{paramhat} = ricefit (@var{x})} returns the maximum likelihood ## estimates of the parameters of the Rician distribution given the data in ## @var{x}. @qcode{@var{paramhat}(1)} is the non-centrality (distance) ## parameter, @var{nu}, and @qcode{@var{paramhat}(2)} is the scale parameter, ## @var{sigma}. ## ## @code{[@var{paramhat}, @var{paramci}] = ricefit (@var{x})} returns the 95% ## confidence intervals for the parameter estimates. ## ## @code{[@dots{}] = ricefit (@var{x}, @var{alpha})} also returns the ## @qcode{100 * (1 - @var{alpha})} percent confidence intervals for the ## parameter estimates. By default, the optional argument @var{alpha} is ## 0.05 corresponding to 95% confidence intervals. Pass in @qcode{[]} for ## @var{alpha} to use the default values. ## ## @code{[@dots{}] = ricefit (@var{x}, @var{alpha}, @var{censor})} accepts a ## boolean vector, @var{censor}, of the same size as @var{x} with @qcode{1}s for ## observations that are right-censored and @qcode{0}s for observations that are ## observed exactly. By default, or if left empty, ## @qcode{@var{censor} = zeros (size (@var{x}))}. ## ## @code{[@dots{}] = ricefit (@var{x}, @var{alpha}, @var{censor}, @var{freq})} ## accepts a frequency vector, @var{freq}, of the same size as @var{x}. ## @var{freq} typically contains integer frequencies for the corresponding ## elements in @var{x}, but it can contain any non-integer non-negative values. ## By default, or if left empty, @qcode{@var{freq} = ones (size (@var{x}))}. ## ## @code{[@dots{}] = ricefit (@dots{}, @var{options})} specifies control ## parameters for the iterative algorithm used to compute the maximum likelihood ## estimates. @var{options} is a structure with the following field and its ## default value: ## @itemize ## @item @qcode{@var{options}.Display = "off"} ## @item @qcode{@var{options}.MaxFunEvals = 1000} ## @item @qcode{@var{options}.MaxIter = 500} ## @item @qcode{@var{options}.TolX = 1e-6} ## @end itemize ## ## Further information about the Rician distribution can be found at ## @url{https://en.wikipedia.org/wiki/Rice_distribution} ## ## @seealso{ricecdf, ricepdf, riceinv, ricernd, ricelike, ricestat} ## @end deftypefn function [paramhat, paramci] = ricefit (x, alpha, censor, freq, options) ## Check input arguments if (! isvector (x)) error ("ricefit: X must be a vector."); endif ## Check alpha if (nargin < 2 || isempty (alpha)) alpha = 0.05; else if (! isscalar (alpha) || ! isreal (alpha) || alpha <= 0 || alpha >= 1) error ("ricefit: wrong value for ALPHA."); endif endif ## Check censor vector if (nargin < 3 || isempty (censor)) censor = zeros (size (x)); elseif (! isequal (size (x), size (censor))) error ("ricefit: X and CENSOR vectors mismatch."); endif ## Check frequency vector if (nargin < 4 || isempty (freq)) freq = ones (size (x)); elseif (! isequal (size (x), size (freq))) error ("ricefit: X and FREQ vectors mismatch."); endif ## Get options structure or add defaults if (nargin < 5) options.Display = "off"; options.MaxFunEvals = 400; options.MaxIter = 200; options.TolX = 1e-6; else if (! isstruct (options) || ! isfield (options, "Display") || ! isfield (options, "MaxFunEvals") || ! isfield (options, "MaxIter") || ! isfield (options, "TolX")) error (strcat (["ricefit: 'options' 5th argument must be a"], ... [" structure with 'Display', 'MaxFunEvals',"], ... [" 'MaxIter', and 'TolX' fields present."])); endif endif ## Get sample size and data type cls = class (x); szx = sum (freq); ncen = sum (freq .* censor); nunc = szx - ncen; ## Check for illegal value in X if (any (x <= 0)) error ("ricefit: X must contain positive values."); endif ## Handle ill-conditioned cases: no data or all censored if (szx == 0 || nunc == 0 || any (! isfinite (x))) paramhat = nan (1, 2, cls); paramci = nan (2, cls); return endif ## Check for identical data in X if (! isscalar (x) && max (abs (diff (x)) ./ x(2:end)) <= sqrt (eps)) paramhat = cast ([Inf, 0], cls); paramci = cast ([Inf, 0; Inf, 0], cls); return endif ## Use 2nd and 4th Moment Estimators of uncensored data as starting point xsq_uncensored = x(censor == 0) .^ 2; meanxsq = mean (xsq_uncensored); meanx_4 = mean (xsq_uncensored .^ 2); if (meanxsq ^ 2 < meanx_4 && meanx_4 < 2 * meanxsq ^ 2) nu_4 = 2 * meanxsq ^ 2 - meanx_4; nusq = sqrt (nu_4); sigmasq = 0.5 * (meanxsq - nusq); params = cast ([sqrt(nusq), sqrt(sigmasq)], cls); else params = cast ([1, 1], cls); endif ## Minimize negative log-likelihood to estimate parameters f = @(params) ricelike (params, x, censor, freq); [paramhat, ~, err, output] = fminsearch (f, params, options); ## Force positive parameter values paramhat = abs (paramhat); ## Handle errors if (err == 0) if (output.funcCount >= options.MaxFunEvals) warning (strcat (["ricefit: maximum number of function"], ... [" evaluations are exceeded."])); elseif (output.iterations >= options.MaxIter) warning ("ricefit: maximum number of iterations are exceeded."); endif elseif (err < 0) error ("ricefit: no solution."); endif ## Compute CIs using a log normal approximation for parameters. if (nargout > 1) ## Compute asymptotic covariance [~, acov] = ricelike (paramhat, x, censor, freq); ## Get standard errors stderr = sqrt (diag (acov))'; stderr = stderr ./ paramhat; ## Apply log transform phatlog = log (paramhat); ## Compute normal quantiles z = probit (alpha / 2); ## Compute CI paramci = [phatlog; phatlog] + [stderr; stderr] .* [z, z; -z, -z]; ## Inverse log transform paramci = exp (paramci); endif endfunction %!demo %! ## Sample 3 populations from different Gamma distibutions %! randg ("seed", 5); # for reproducibility %! randp ("seed", 6); %! r1 = ricernd (1, 2, 3000, 1); %! randg ("seed", 2); # for reproducibility %! randp ("seed", 8); %! r2 = ricernd (2, 4, 3000, 1); %! randg ("seed", 7); # for reproducibility %! randp ("seed", 9); %! r3 = ricernd (7.5, 1, 3000, 1); %! r = [r1, r2, r3]; %! %! ## Plot them normalized and fix their colors %! hist (r, 75, 4); %! h = findobj (gca, "Type", "patch"); %! set (h(1), "facecolor", "c"); %! set (h(2), "facecolor", "g"); %! set (h(3), "facecolor", "r"); %! ylim ([0, 0.7]); %! xlim ([0, 12]); %! hold on %! %! ## Estimate their α and β parameters %! nu_sigmaA = ricefit (r(:,1)); %! nu_sigmaB = ricefit (r(:,2)); %! nu_sigmaC = ricefit (r(:,3)); %! %! ## Plot their estimated PDFs %! x = [0.01,0.1:0.2:18]; %! y = ricepdf (x, nu_sigmaA(1), nu_sigmaA(2)); %! plot (x, y, "-pr"); %! y = ricepdf (x, nu_sigmaB(1), nu_sigmaB(2)); %! plot (x, y, "-sg"); %! y = ricepdf (x, nu_sigmaC(1), nu_sigmaC(2)); %! plot (x, y, "-^c"); %! hold off %! legend ({"Normalized HIST of sample 1 with k=1 and θ=2", ... %! "Normalized HIST of sample 2 with k=2 and θ=4", ... %! "Normalized HIST of sample 3 with k=7.5 and θ=1", ... %! sprintf("PDF for sample 1 with estimated k=%0.2f and θ=%0.2f", ... %! nu_sigmaA(1), nu_sigmaA(2)), ... %! sprintf("PDF for sample 2 with estimated k=%0.2f and θ=%0.2f", ... %! nu_sigmaB(1), nu_sigmaB(2)), ... %! sprintf("PDF for sample 3 with estimated k=%0.2f and θ=%0.2f", ... %! nu_sigmaC(1), nu_sigmaC(2))}) %! title ("Three population samples from different Rician distibutions") %! hold off ## Test output %!test %! [paramhat, paramci] = ricefit ([1:50]); %! assert (paramhat, [15.3057, 17.6668], 1e-4); %! assert (paramci, [9.5468, 11.7802; 24.5383, 26.4952], 1e-4); %!test %! [paramhat, paramci] = ricefit ([1:50], 0.01); %! assert (paramhat, [15.3057, 17.6668], 1e-4); %! assert (paramci, [8.2309, 10.3717; 28.4615, 30.0934], 1e-4); %!test %! [paramhat, paramci] = ricefit ([1:5]); %! assert (paramhat, [2.3123, 1.6812], 1e-4); %! assert (paramci, [1.0819, 0.6376; 4.9424, 4.4331], 1e-4); %!test %! [paramhat, paramci] = ricefit ([1:5], 0.01); %! assert (paramhat, [2.3123, 1.6812], 1e-4); %! assert (paramci, [0.8521, 0.4702; 6.2747, 6.0120], 1e-4); %!test %! freq = [1 1 1 1 5]; %! [paramhat, paramci] = ricefit ([1:5], [], [], freq); %! assert (paramhat, [3.5181, 1.5565], 1e-4); %! assert (paramci, [2.5893, 0.9049; 4.7801, 2.6772], 1e-4); %!test %! censor = [1 0 0 0 0]; %! [paramhat, paramci] = ricefit ([1:5], [], censor); %! assert (paramhat, [3.2978, 1.1527], 1e-4); %! assert (paramci, [2.3192, 0.5476; 4.6895, 2.4261], 1e-4); ## Test class of input preserved %!assert (class (ricefit (single ([1:50]))), "single") ## Test input validation %!error ricefit (ones (2)) %!error ricefit ([1:50], 1) %!error ricefit ([1:50], -1) %!error ricefit ([1:50], {0.05}) %!error ricefit ([1:50], "k") %!error ricefit ([1:50], i) %!error ricefit ([1:50], [0.01 0.02]) %!error ricefit ([1:50], [], [1 1]) %!error ricefit ([1:50], [], [], [1 1]) %!error ricefit ([1 2 3 -4]) %!error ricefit ([1 2 0], [], [1 0 0]) statistics-release-1.6.3/inst/dist_fit/ricelike.m000066400000000000000000000202741456127120000220630ustar00rootroot00000000000000## Copyright (C) 2024 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{nlogL} =} ricelike (@var{params}, @var{x}) ## @deftypefnx {statistics} {[@var{nlogL}, @var{acov}] =} ricelike (@var{params}, @var{x}) ## @deftypefnx {statistics} {[@dots{}] =} ricelike (@var{params}, @var{x}, @var{censor}) ## @deftypefnx {statistics} {[@dots{}] =} ricelike (@var{params}, @var{x}, @var{censor}, @var{freq}) ## ## Negative log-likelihood for the Rician distribution. ## ## @code{@var{nlogL} = ricelike (@var{params}, @var{x})} returns the negative ## log likelihood of the data in @var{x} corresponding to the Rician ## distribution with (1) non-centrality (distance) parameter @var{nu} and (2) ## scale parameter @var{sigma} given in the two-element vector @var{params}. ## ## @code{[@var{nlogL}, @var{acov}] = ricelike (@var{params}, @var{x})} also ## returns the inverse of Fisher's information matrix, @var{acov}. If the input ## parameter values in @var{params} are the maximum likelihood estimates, the ## diagonal elements of @var{params} are their asymptotic variances. ## ## @code{[@dots{}] = ricelike (@var{params}, @var{x}, @var{censor})} accepts a ## boolean vector, @var{censor}, of the same size as @var{x} with @qcode{1}s for ## observations that are right-censored and @qcode{0}s for observations that are ## observed exactly. By default, or if left empty, ## @qcode{@var{censor} = zeros (size (@var{x}))}. ## ## @code{[@dots{}] = ricelike (@var{params}, @var{x}, @var{censor}, @var{freq})} ## accepts a frequency vector, @var{freq}, of the same size as @var{x}. ## @var{freq} typically contains integer frequencies for the corresponding ## elements in @var{x}, but it can contain any non-integer non-negative values. ## By default, or if left empty, @qcode{@var{freq} = ones (size (@var{x}))}. ## ## Further information about the Rician distribution can be found at ## @url{https://en.wikipedia.org/wiki/Rice_distribution} ## ## @seealso{ricecdf, riceinv, ricepdf, ricernd, ricefit, ricestat} ## @end deftypefn function [nlogL, acov] = ricelike (params, x, censor, freq) ## Check input arguments if (nargin < 2) error ("ricelike: function called with too few input arguments."); endif if (! isvector (x)) error ("ricelike: X must be a vector."); endif if (length (params) != 2) error ("ricelike: PARAMS must be a two-element vector."); endif ## Check censor vector if (nargin < 3 || isempty (censor)) censor = zeros (size (x)); elseif (! isequal (size (x), size (censor))) error ("ricelike: X and CENSOR vector mismatch."); endif ## Check frequency vector if (nargin < 4 || isempty (freq)) freq = ones (size (x)); elseif (! isequal (size (x), size (freq))) error ("ricelike: X and FREQ vector mismatch."); endif ## Expand frequency and censor vectors (if necessary) if (! all (freq == 1)) xf = []; cf = []; for i = 1:numel (freq) xf = [xf, repmat(x(i), 1, freq(i))]; cf = [cf, repmat(censor(i), 1, freq(i))]; endfor x = xf; freq = ones (size (x)); censor = cf; endif ## Get parameters nu = params(1); sigma = params(2); theta = nu ./ sigma; xsigma = x ./ sigma; xstheta = xsigma.*theta; I_0 = besseli (0, xstheta, 1); XNS = (xsigma .^ 2 + theta .^ 2) ./ 2; ## Compute log likelihood L = -XNS + log (I_0) + log (xsigma ./ sigma) + xstheta; ## Handle censored data n_censored = sum (freq .* censor); if (n_censored > 0) censored = (censor == 1); xsigma_censored = xsigma(censored); Q = marcumQ1 (theta, xsigma_censored); L(censored) = log (Q); endif ## Sum up the neg log likelihood nlogL = -sum (freq .* L); ## Compute asymptotic covariance if (nargout > 1) ## Compute first order central differences of the log-likelihood gradient dp = 0.0001 .* max (abs (params), 1); ngrad_p1 = rice_grad (params + [dp(1), 0], x, censor, freq); ngrad_m1 = rice_grad (params - [dp(1), 0], x, censor, freq); ngrad_p2 = rice_grad (params + [0, dp(2)], x, censor, freq); ngrad_m2 = rice_grad (params - [0, dp(2)], x, censor, freq); ## Compute negative Hessian by normalizing the differences by the increment nH = [(ngrad_p1(:) - ngrad_m1(:))./(2 * dp(1)), ... (ngrad_p2(:) - ngrad_m2(:))./(2 * dp(2))]; ## Force neg Hessian being symmetric nH = 0.5 .* (nH + nH'); ## Check neg Hessian is positive definite [R, p] = chol (nH); if (p > 0) warning ("ricelike: non positive definite Hessian matrix."); acov = NaN (2); return endif ## ACOV estimate is the negative inverse of the Hessian. Rinv = inv (R); acov = Rinv * Rinv; endif endfunction ## Helper function for computing negative gradient function ngrad = rice_grad (params, x, censor, freq) ## Get parameters nu = params(1); sigma = params(2); theta = nu ./ sigma; xsigma = x ./ sigma; xstheta = xsigma.*theta; I_0 = besseli (0, xstheta, 1); XNS = (xsigma .^ 2 + theta .^ 2) ./ 2; ## Compute derivatives I_1 = besseli(1, xstheta, 1); dII = I_1 ./ I_0; dL1 = (-theta + dII .* xsigma) ./ sigma; dL2 = -2 * (1 - XNS + dII .* xstheta) ./ sigma; ## Handle censored data n_censored = sum (freq .* censor); if (n_censored > 0) censored = (censor == 1); xsigma_censored = xsigma(censored); Q = marcumQ1 (theta, xsigma_censored); expt = exp (-XNS(censored) + xstheta(censored)); dQdtheta = xsigma_censored .* I_1(censored) .* expt; dQdz = -xsigma_censored .* I_0(censored) .* expt; dtheta1 = 1 ./ sigma; dtheta2 = -theta ./ sigma; dz2 = -xsigma_censored ./ sigma; dL1(censored) = dQdtheta .* dtheta1 ./ Q; dL2(censored) = (dQdtheta .* dtheta2 + dQdz .* dz2) ./ Q; endif ## Compute gradient ngrad = -[sum(freq .* dL1) sum(freq .* dL2)]; endfunction ## Marcum's "Q" function of order 1 function Q = marcumQ1 (a, b) ## Prepare output matrix if (isa (a, "single") || isa (b, "single")) Q = NaN (size (b), "single"); else Q = NaN (size (b)); endif ## Force marginal cases Q(a != Inf & b == 0) = 1; Q(a != Inf & b == Inf) = 0; Q(a == Inf & b != Inf) = 1; z = isnan (Q) & a == 0 & b != Inf; if (any(z)) Q(z) = exp ((-b(z) .^ 2) ./ 2); end ## Compute the remaining cases z = isnan (Q) & ! isnan (a) & ! isnan (b); if (any(z(:))) aa = (a(z) .^ 2) ./ 2; bb = (b(z) .^ 2) ./ 2; eA = exp (-aa); eB = bb .* exp (-bb); h = eA; d = eB .* h; s = d; j = (d > s.*eps(class(d))); k = 1; while (any (j)) eA = aa .* eA ./ k; h = h + eA; eB = bb .* eB ./ (k + 1); d = eB .* h; s(j) = s (j) + d(j); j = (d > s .* eps (class (d))); k = k + 1; endwhile Q(z) = 1 - s; endif endfunction ## Test output %!test %! nlogL = ricelike ([15.3057344, 17.6668458], [1:50]); %! assert (nlogL, 204.5230311010569, 1e-12); %!test %! nlogL = ricelike ([2.312346885, 1.681228265], [1:5]); %! assert (nlogL, 8.65562164930058, 1e-12); ## Test input validation %!error ricelike (3.25) %!error ricelike ([5, 0.2], ones (2)) %!error ... %! ricelike ([1, 0.2, 3], [1, 3, 5, 7]) %!error ... %! ricelike ([1.5, 0.2], [1:5], [0, 0, 0]) %!error ... %! ricelike ([1.5, 0.2], [1:5], [0, 0, 0, 0, 0], [1, 1, 1]) %!error ... %! ricelike ([1.5, 0.2], [1:5], [], [1, 1, 1]) statistics-release-1.6.3/inst/dist_fit/unidfit.m000066400000000000000000000125161456127120000217360ustar00rootroot00000000000000## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{Nhat} =} unidfit (@var{x}) ## @deftypefnx {statistics} {[@var{Nhat}, @var{Nci}] =} unidfit (@var{x}) ## @deftypefnx {statistics} {[@var{Nhat}, @var{Nci}] =} unidfit (@var{x}, @var{alpha}) ## @deftypefnx {statistics} {[@var{Nhat}, @var{Nci}] =} unidfit (@var{x}, @var{alpha}, @var{freq}) ## ## Estimate parameter and confidence intervals for the discrete uniform distribution. ## ## @code{@var{Nhat} = unidfit (@var{x})} returns the maximum likelihood estimate ## (MLE) of the maximum observable value for the discrete uniform distribution. ## @var{x} must be a vector. ## ## @code{[@var{Nhat}, @var{Nci}] = unidfit (@var{x}, @var{alpha})} also ## returns the @qcode{100 * (1 - @var{alpha})} percent confidence intervals of ## the estimated parameter. By default, the optional argument @var{alpha} is ## 0.05 corresponding to 95% confidence intervals. Pass in @qcode{[]} for ## @var{alpha} to use the default values. ## ## @code{[@dots{}] = unidfit (@var{x}, @var{alpha}, @var{freq})} accepts a ## frequency vector, @var{freq}, of the same size as @var{x}. @var{freq} ## typically contains integer frequencies for the corresponding elements in ## @var{x}, but it can contain any non-integer non-negative values. By default, ## or if left empty, @qcode{@var{freq} = ones (size (@var{x}))}. ## ## Further information about the discrete uniform distribution can be found at ## @url{https://en.wikipedia.org/wiki/Discrete_uniform_distribution} ## ## @seealso{unidcdf, unidinv, unidpdf, unidrnd, unidstat} ## @end deftypefn function [Nhat, Nci] = unidfit (x, alpha, freq) ## Check input arguments if (nargin < 1) error ("unidfit: function called with too few input arguments."); endif ## Check data inX if (any (x < 0)) error ("unidfit: X cannot have negative values."); endif if (! isvector (x)) error ("unidfit: X must be a vector."); endif ## Check ALPHA if (nargin < 2 || isempty (alpha)) alpha = 0.05; elseif (! isscalar (alpha) || ! isreal (alpha) || alpha <= 0 || alpha >= 1) error ("unidfit: wrong value for ALPHA."); endif ## Check frequency vector if (nargin < 3 || isempty (freq)) freq = ones (size (x)); elseif (! isequal (size (x), size (freq))) error ("unidfit: X and FREQ vector mismatch."); endif ## Expand frequency and censor vectors (if necessary) if (! all (freq == 1)) xf = []; for i = 1:numel (freq) xf = [xf, repmat(x(i), 1, freq(i))]; endfor x = xf; freq = ones (size (x)); endif ## Compute N estimate Nhat = max (x); ## Compute confidence interval of N if (nargout > 1) Nci = [Nhat; ceil(Nhat ./ alpha .^ (1 ./ numel (x)))]; endif endfunction %!demo %! ## Sample 2 populations from different discrete uniform distibutions %! rand ("seed", 1); # for reproducibility %! r1 = unidrnd (5, 1000, 1); %! rand ("seed", 2); # for reproducibility %! r2 = unidrnd (9, 1000, 1); %! r = [r1, r2]; %! %! ## Plot them normalized and fix their colors %! hist (r, 0:0.5:20.5, 1); %! h = findobj (gca, "Type", "patch"); %! set (h(1), "facecolor", "c"); %! set (h(2), "facecolor", "g"); %! hold on %! %! ## Estimate their probability of success %! NhatA = unidfit (r(:,1)); %! NhatB = unidfit (r(:,2)); %! %! ## Plot their estimated PDFs %! x = [0:10]; %! y = unidpdf (x, NhatA); %! plot (x, y, "-pg"); %! y = unidpdf (x, NhatB); %! plot (x, y, "-sc"); %! xlim ([0, 10]) %! ylim ([0, 0.4]) %! legend ({"Normalized HIST of sample 1 with N=5", ... %! "Normalized HIST of sample 2 with N=9", ... %! sprintf("PDF for sample 1 with estimated N=%0.2f", NhatA), ... %! sprintf("PDF for sample 2 with estimated N=%0.2f", NhatB)}) %! title ("Two population samples from different discrete uniform distibutions") %! hold off ## Test output %!test %! x = 0:5; %! [Nhat, Nci] = unidfit (x); %! assert (Nhat, 5); %! assert (Nci, [5; 9]); %!test %! x = 0:5; %! [Nhat, Nci] = unidfit (x, [], [1 1 1 1 1 1]); %! assert (Nhat, 5); %! assert (Nci, [5; 9]); %!assert (unidfit ([1 1 2 3]), unidfit ([1 2 3], [] ,[2 1 1])) ## Test input validation %!error unidfit () %!error unidfit (-1, [1 2 3 3]) %!error unidfit (1, 0) %!error unidfit (1, 1.2) %!error unidfit (1, [0.02 0.05]) %!error ... %! unidfit ([1.5, 0.2], [], [0, 0, 0, 0, 0]) %!error ... %! unidfit ([1.5, 0.2], [], [1, 1, 1]) statistics-release-1.6.3/inst/dist_fit/unifit.m000066400000000000000000000133101456127120000215630ustar00rootroot00000000000000## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{paramhat} =} unifit (@var{x}) ## @deftypefnx {statistics} {[@var{paramhat}, @var{paramci}] =} unifit (@var{x}) ## @deftypefnx {statistics} {[@var{paramhat}, @var{paramci}] =} unifit (@var{x}, @var{alpha}) ## @deftypefnx {statistics} {[@var{paramhat}, @var{paramci}] =} unifit (@var{x}, @var{alpha}, @var{freq}) ## ## Estimate parameter and confidence intervals for the continuous uniform distribution. ## ## @code{@var{paramhat} = unifit (@var{x})} returns the maximum likelihood ## estimate (MLE) of the parameters @var{a} and @var{b} of the continuous ## uniform distribution given the data in @var{x}. @var{x} must be a vector. ## ## @code{[@var{paramhat}, @var{paramci}] = unifit (@var{x}, @var{alpha})} also ## returns the @qcode{100 * (1 - @var{alpha})} percent confidence intervals of ## the estimated parameter. By default, the optional argument @var{alpha} is ## 0.05 corresponding to 95% confidence intervals. Pass in @qcode{[]} for ## @var{alpha} to use the default values. ## ## @code{[@dots{}] = unifit (@var{x}, @var{alpha}, @var{freq})} accepts a ## frequency vector, @var{freq}, of the same size as @var{x}. @var{freq} ## typically contains integer frequencies for the corresponding elements in ## @var{x}, but it can contain any non-integer non-negative values. By default, ## or if left empty, @qcode{@var{freq} = ones (size (@var{x}))}. ## ## Further information about the continuous uniform distribution can be found at ## @url{https://en.wikipedia.org/wiki/Discrete_uniform_distribution} ## ## @seealso{unifcdf, unifinv, unifpdf, unifrnd, unifstat} ## @end deftypefn function [paramhat, paramci] = unifit (x, alpha, freq) ## Check input arguments if (nargin < 1) error ("unifit: function called with too few input arguments."); endif ## Check data inX if (any (x < 0)) error ("unifit: X cannot have negative values."); endif if (! isvector (x)) error ("unifit: X must be a vector."); endif ## Check ALPHA if (nargin < 2 || isempty (alpha)) alpha = 0.05; elseif (! isscalar (alpha) || ! isreal (alpha) || alpha <= 0 || alpha >= 1) error ("unifit: wrong value for ALPHA."); endif ## Check frequency vector if (nargin < 3 || isempty (freq)) freq = ones (size (x)); elseif (! isequal (size (x), size (freq))) error ("unifit: X and FREQ vector mismatch."); endif ## Expand frequency and censor vectors (if necessary) if (! all (freq == 1)) xf = []; for i = 1:numel (freq) xf = [xf, repmat(x(i), 1, freq(i))]; endfor x = xf; freq = ones (size (x)); endif ## Compute A and B estimates ahat = min (x); bhat = max(x); paramhat = [ahat, bhat]; ## Compute confidence interval of A and B if (nargout > 1) tmp = (bhat - ahat) ./ alpha .^ (1 ./ numel (x)); paramci = [bhat-tmp, ahat+tmp; ahat, bhat]; endif endfunction %!demo %! ## Sample 2 populations from different continuous uniform distibutions %! rand ("seed", 5); # for reproducibility %! r1 = unifrnd (2, 5, 2000, 1); %! rand ("seed", 6); # for reproducibility %! r2 = unifrnd (3, 9, 2000, 1); %! r = [r1, r2]; %! %! ## Plot them normalized and fix their colors %! hist (r, 0:0.5:10, 2); %! h = findobj (gca, "Type", "patch"); %! set (h(1), "facecolor", "c"); %! set (h(2), "facecolor", "g"); %! hold on %! %! ## Estimate their probability of success %! a_bA = unifit (r(:,1)); %! a_bB = unifit (r(:,2)); %! %! ## Plot their estimated PDFs %! x = [0:10]; %! y = unifpdf (x, a_bA(1), a_bA(2)); %! plot (x, y, "-pg"); %! y = unifpdf (x, a_bB(1), a_bB(2)); %! plot (x, y, "-sc"); %! xlim ([1, 10]) %! ylim ([0, 0.5]) %! legend ({"Normalized HIST of sample 1 with a=2 and b=5", ... %! "Normalized HIST of sample 2 with a=3 and b=9", ... %! sprintf("PDF for sample 1 with estimated a=%0.2f and b=%0.2f", ... %! a_bA(1), a_bA(2)), ... %! sprintf("PDF for sample 2 with estimated a=%0.2f and b=%0.2f", ... %! a_bB(1), a_bB(2))}) %! title ("Two population samples from different continuous uniform distibutions") %! hold off ## Test output %!test %! x = 0:5; %! [paramhat, paramci] = unifit (x); %! assert (paramhat, [0, 5]); %! assert (paramci, [-3.2377, 8.2377; 0, 5], 1e-4); %!test %! x = 0:5; %! [paramhat, paramci] = unifit (x, [], [1 1 1 1 1 1]); %! assert (paramhat, [0, 5]); %! assert (paramci, [-3.2377, 8.2377; 0, 5], 1e-4); %!assert (unifit ([1 1 2 3]), unifit ([1 2 3], [] ,[2 1 1])) ## Test input validation %!error unifit () %!error unifit (-1, [1 2 3 3]) %!error unifit (1, 0) %!error unifit (1, 1.2) %!error unifit (1, [0.02 0.05]) %!error ... %! unifit ([1.5, 0.2], [], [0, 0, 0, 0, 0]) %!error ... %! unifit ([1.5, 0.2], [], [1, 1, 1]) statistics-release-1.6.3/inst/dist_fit/wblfit.m000066400000000000000000000177001456127120000215630ustar00rootroot00000000000000## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{paramhat} =} wblfit (@var{x}) ## @deftypefnx {statistics} {[@var{paramhat}, @var{paramci}] =} wblfit (@var{x}) ## @deftypefnx {statistics} {[@var{paramhat}, @var{paramci}] =} wblfit (@var{x}, @var{alpha}) ## @deftypefnx {statistics} {[@dots{}] =} wblfit (@var{x}, @var{alpha}, @var{censor}) ## @deftypefnx {statistics} {[@dots{}] =} wblfit (@var{x}, @var{alpha}, @var{censor}, @var{freq}) ## @deftypefnx {statistics} {[@dots{}] =} wblfit (@var{x}, @var{alpha}, @var{censor}, @var{freq}, @var{options}) ## ## Estimate mean and confidence intervals for the Weibull distribution. ## ## @code{@var{muhat} = wblfit (@var{x})} returns the maximum likelihood ## estimates of the parameters of the Weibull distribution given the data in ## @var{x}. @qcode{@var{paramhat}(1)} is the scale parameter, @math{lambda}, ## and @qcode{@var{paramhat}(2)} is the shape parameter, @math{k}. ## ## @code{[@var{paramhat}, @var{paramci}] = wblfit (@var{x})} returns the 95% ## confidence intervals for the parameter estimates. ## ## @code{[@dots{}] = wblfit (@var{x}, @var{alpha})} also returns the ## @qcode{100 * (1 - @var{alpha})} percent confidence intervals for the ## parameter estimates. By default, the optional argument @var{alpha} is ## 0.05 corresponding to 95% confidence intervals. Pass in @qcode{[]} for ## @var{alpha} to use the default values. ## ## @code{[@dots{}] = wblfit (@var{x}, @var{alpha}, @var{censor})} accepts a ## boolean vector, @var{censor}, of the same size as @var{x} with @qcode{1}s for ## observations that are right-censored and @qcode{0}s for observations that are ## observed exactly. By default, or if left empty, ## @qcode{@var{censor} = zeros (size (@var{x}))}. ## ## @code{[@dots{}] = wblfit (@var{x}, @var{alpha}, @var{censor}, @var{freq})} ## accepts a frequency vector, @var{freq}, of the same size as @var{x}. ## @var{freq} typically contains integer frequencies for the corresponding ## elements in @var{x}, but it can contain any non-integer non-negative values. ## By default, or if left empty, @qcode{@var{freq} = ones (size (@var{x}))}. ## ## @code{[@dots{}] = evfit (@dots{}, @var{options})} specifies control ## parameters for the iterative algorithm used to compute ML estimates with the ## @code{fminsearch} function. @var{options} is a structure with the following ## fields and their default values: ## @itemize ## @item @qcode{@var{options}.Display = "off"} ## @item @qcode{@var{options}.TolX = 1e-6} ## @end itemize ## ## Further information about the Weibull distribution can be found at ## @url{https://en.wikipedia.org/wiki/Weibull_distribution} ## ## @seealso{wblcdf, wblinv, wblpdf, wblrnd, wbllike, wblstat} ## @end deftypefn function [paramhat, paramci] = wblfit (x, alpha, censor, freq, options) ## Check input arguments if (! isvector (x)) error ("wblfit: X must be a vector."); elseif (any (x <= 0)) error ("wblfit: X must contain only positive values."); endif ## Check alpha if (nargin < 2 || isempty (alpha)) alpha = 0.05; else if (! isscalar (alpha) || ! isreal (alpha) || alpha <= 0 || alpha >= 1) error ("wblfit: wrong value for ALPHA."); endif endif ## Check censor vector if (nargin < 3 || isempty (censor)) censor = zeros (size (x)); elseif (! isequal (size (x), size (censor))) error ("wblfit: X and CENSOR vectors mismatch."); endif ## Check frequency vector if (nargin < 4 || isempty (freq)) freq = ones (size (x)); elseif (! isequal (size (x), size (freq))) error ("wblfit: X and FREQ vectors mismatch."); endif ## Get options structure or add defaults if (nargin < 5) options.Display = "off"; options.TolX = 1e-6; else if (! isstruct (options) || ! isfield (options, "Display") || ... ! isfield (options, "TolX")) error (strcat (["wblfit: 'options' 5th argument must be a structure"], ... [" with 'Display' and 'TolX' fields present."])); endif endif ## Fit an extreme value distribution to the logged data, then transform to ## the Weibull parameter scales. [paramhatEV, paramciEV] = evfit (log (x), alpha, censor, freq, options); paramhat = [exp(paramhatEV(1)), 1./paramhatEV(2)]; if (nargout > 1) paramci = [exp(paramciEV(:,1)) 1./paramciEV([2 1],2)]; endif endfunction %!demo %! ## Sample 3 populations from 3 different Weibull distibutions %! rande ("seed", 1); # for reproducibility %! r1 = wblrnd(2, 4, 2000, 1); %! rande ("seed", 2); # for reproducibility %! r2 = wblrnd(5, 2, 2000, 1); %! rande ("seed", 5); # for reproducibility %! r3 = wblrnd(1, 5, 2000, 1); %! r = [r1, r2, r3]; %! %! ## Plot them normalized and fix their colors %! hist (r, 30, [2.5 2.1 3.2]); %! h = findobj (gca, "Type", "patch"); %! set (h(1), "facecolor", "c"); %! set (h(2), "facecolor", "g"); %! set (h(3), "facecolor", "r"); %! ylim ([0, 2]); %! xlim ([0, 10]); %! hold on %! %! ## Estimate their lambda parameter %! lambda_kA = wblfit (r(:,1)); %! lambda_kB = wblfit (r(:,2)); %! lambda_kC = wblfit (r(:,3)); %! %! ## Plot their estimated PDFs %! x = [0:0.1:15]; %! y = wblpdf (x, lambda_kA(1), lambda_kA(2)); %! plot (x, y, "-pr"); %! y = wblpdf (x, lambda_kB(1), lambda_kB(2)); %! plot (x, y, "-sg"); %! y = wblpdf (x, lambda_kC(1), lambda_kC(2)); %! plot (x, y, "-^c"); %! hold off %! legend ({"Normalized HIST of sample 1 with λ=2 and k=4", ... %! "Normalized HIST of sample 2 with λ=5 and k=2", ... %! "Normalized HIST of sample 3 with λ=1 and k=5", ... %! sprintf("PDF for sample 1 with estimated λ=%0.2f and k=%0.2f", ... %! lambda_kA(1), lambda_kA(2)), ... %! sprintf("PDF for sample 2 with estimated λ=%0.2f and k=%0.2f", ... %! lambda_kB(1), lambda_kB(2)), ... %! sprintf("PDF for sample 3 with estimated λ=%0.2f and k=%0.2f", ... %! lambda_kC(1), lambda_kC(2))}) %! title ("Three population samples from different Weibull distibutions") %! hold off ## Test results %!test %! x = 1:50; %! [paramhat, paramci] = wblfit (x); %! paramhat_out = [28.3636, 1.7130]; %! paramci_out = [23.9531, 1.3551; 33.5861, 2.1655]; %! assert (paramhat, paramhat_out, 1e-4); %! assert (paramci, paramci_out, 1e-4); %!test %! x = 1:50; %! [paramhat, paramci] = wblfit (x, 0.01); %! paramci_out = [22.7143, 1.2589; 35.4179, 2.3310]; %! assert (paramci, paramci_out, 1e-4); ## Test input validation %!error wblfit (ones (2,5)); %!error wblfit ([-1 2 3 4]); %!error wblfit ([1, 2, 3, 4, 5], 1.2); %!error wblfit ([1, 2, 3, 4, 5], 0); %!error wblfit ([1, 2, 3, 4, 5], "alpha"); %!error ... %! wblfit ([1, 2, 3, 4, 5], 0.05, [1 1 0]); %!error ... %! wblfit ([1, 2, 3, 4, 5], [], [1 1 0 1 1]'); %!error ... %! wblfit ([1, 2, 3, 4, 5], 0.05, zeros (1,5), [1 1 0]); %!error ... %! wblfit ([1, 2, 3, 4, 5], [], [], [1 1 0 1 1]'); %!error ... %! wblfit ([1, 2, 3, 4, 5], 0.05, [], [], 2); statistics-release-1.6.3/inst/dist_fit/wbllike.m000066400000000000000000000127371456127120000217320ustar00rootroot00000000000000## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR l PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{nlogL} =} wbllike (@var{params}, @var{x}) ## @deftypefnx {statistics} {[@var{nlogL}, @var{acov}] =} wbllike (@var{params}, @var{x}) ## @deftypefnx {statistics} {[@dots{}] =} wbllike (@var{params}, @var{x}, @var{alpha}, @var{censor}) ## @deftypefnx {statistics} {[@dots{}] =} wbllike (@var{params}, @var{x}, @var{alpha}, @var{censor}, @var{freq}) ## ## Negative log-likelihood for the Weibull distribution. ## ## @code{@var{nlogL} = wbllike (@var{params}, @var{data})} returns the negative ## log-likelihood of the data in @var{x} corresponding to the Weibull ## distribution with (1) scale parameter @var{lambda} and (2) shape parameter ## @var{k} given in the two-element vector @var{params}. ## ## @code{[@var{nlogL}, @var{acov}] = wbllike (@var{params}, @var{data})} also ## returns the inverse of Fisher's information matrix, @var{acov}. If the input ## parameter values in @var{params} are the maximum likelihood estimates, the ## diagonal elements of @var{acov} are their asymptotic variances. @var{acov} ## is based on the observed Fisher's information, not the expected information. ## ## @code{[@dots{}] = wbllike (@var{params}, @var{data}, @var{censor})} accepts a ## boolean vector, @var{censor}, of the same size as @var{x} with @qcode{1}s for ## observations that are right-censored and @qcode{0}s for observations that are ## observed exactly. By default, or if left empty, ## @qcode{@var{censor} = zeros (size (@var{x}))}. ## ## @code{[@dots{}] = wbllike (@var{params}, @var{data}, @var{censor}, ## @var{freq})} accepts a frequency vector, @var{freq}, of the same size as ## @var{x}. @var{freq} typically contains integer frequencies for the ## corresponding elements in @var{x}, but may contain any non-integer ## non-negative values. By default, or if left empty, ## @qcode{@var{freq} = ones (size (@var{x}))}. ## ## Further information about the Weibull distribution can be found at ## @url{https://en.wikipedia.org/wiki/Weibull_distribution} ## ## @seealso{wblcdf, wblinv, wblpdf, wblrnd, wblfit, wblstat} ## @end deftypefn function [nlogL, acov] = wbllike (params, x, censor, freq) ## Check input arguments and add defaults if (nargin < 2) error ("wbllike: too few input arguments."); endif if (numel (params) != 2) error ("wbllike: wrong parameters length."); endif if (! isvector (x)) error ("wbllike: X must be a vector."); endif if (nargin < 3 || isempty (censor)) censor = zeros (size (x)); elseif (! isequal (size (x), size (censor))) error ("wbllike: X and CENSOR vectors mismatch."); endif if (nargin < 4 || isempty (freq)) freq = ones (size (x)); elseif (isequal (size (x), size (freq))) nulls = find (freq == 0); if (numel (nulls) > 0) x(nulls) = []; censor(nulls) = []; freq(nulls) = []; endif else error ("wbllike: X and FREQ vectors mismatch."); endif ## Get lambda and k parameter values l = params(1); k = params(2); ## Force NaNs for out of range parameters or x. l(l <= 0) = NaN; k(k <= 0) = NaN; x(x < 0) = NaN; ## Compute the individual log-likelihood terms z = x ./ l; logz = log (z); expz = exp (k .* logz); ilogL = ((k - 1) .* logz + log (k ./ l)) .* (1 - censor) - expz; ilogL(z == Inf) = -Inf; ## Sum up the individual log-likelihood contributions nlogL = -sum (freq .* ilogL); ## Compute the negative hessian and invert to get the information matrix. if (nargout > 1) ucen = (1 - censor); nH11 = sum (freq .* (k .* ((1 + k) .* expz - ucen))) ./ l .^ 2; nH12 = -sum(freq .* (((1 + k .* logz) .* expz - ucen))) ./ l; nH22 = sum(freq .* ((logz .^ 2) .* expz + ucen ./ k .^ 2)); acov = [nH22, -nH12; -nH12, nH11] / (nH11 * nH22 - nH12 * nH12); endif endfunction ## Results compared with Matlab %!test %! x = 1:50; %! [nlogL, acov] = wbllike ([2.3, 1.2], x); %! avar_out = [0.0250, 0.0062; 0.0062, 0.0017]; %! assert (nlogL, 945.9589180651594, 1e-12); %! assert (acov, avar_out, 1e-4); %!test %! x = 1:50; %! [nlogL, acov] = wbllike ([2.3, 1.2], x * 0.5); %! avar_out = [-0.3238, -0.1112; -0.1112, -0.0376]; %! assert (nlogL, 424.9879809704742, 6e-14); %! assert (acov, avar_out, 1e-4); %!test %! x = 1:50; %! [nlogL, acov] = wbllike ([21, 15], x); %! avar_out = [-0.00001236, -0.00001166; -0.00001166, -0.00001009]; %! assert (nlogL, 1635190.328991511, 1e-8); %! assert (acov, avar_out, 1e-8); ## Test input validation %!error wbllike ([12, 15]); %!error wbllike ([12, 15, 3], [1:50]); %!error wbllike ([12, 3], ones (10, 2)); %!error wbllike ([12, 15], [1:50], [1, 2, 3]); %!error wbllike ([12, 15], [1:50], [], [1, 2, 3]); statistics-release-1.6.3/inst/dist_fun/000077500000000000000000000000001456127120000201175ustar00rootroot00000000000000statistics-release-1.6.3/inst/dist_fun/betacdf.m000066400000000000000000000143641456127120000216750ustar00rootroot00000000000000## Copyright (C) 2012 Rik Wehbring ## Copyright (C) 1995-2016 Kurt Hornik ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{p} =} betacdf (@var{x}, @var{a}, @var{b}) ## @deftypefnx {statistics} {@var{p} =} betacdf (@var{x}, @var{a}, @var{b}, @qcode{"upper"}) ## ## Beta cumulative distribution function (CDF). ## ## For each element of @var{x}, compute the cumulative distribution function of ## the Beta distribution with shape parameters @var{a} and @var{b}. The size of ## @var{p} is the common size of @var{x}, @var{a}, and @var{b}. A scalar input ## functions as a constant matrix of the same size as the other inputs. ## ## @code{@var{p} = betacdf (@var{x}, @var{a}, @var{b}, "upper")} computes the ## upper tail probability of the Beta distribution with parameters @var{a} and ## @var{b}, at the values in @var{x}. ## ## Further information about the Beta distribution can be found at ## @url{https://en.wikipedia.org/wiki/Beta_distribution} ## ## @seealso{betainv, betapdf, betarnd, betafit, betalike, betastat} ## @end deftypefn function p = betacdf (x, a, b, uflag) ## Check for valid number of input arguments if (nargin < 3) error ("betacdf: function called with too few input arguments."); endif ## Check for valid "upper" flag if (nargin > 3) if (! strcmpi (uflag, "upper")) error ("betacdf: invalid argument for upper tail."); else uflag = true; endif else uflag = false; endif ## Check for common size of X, A, and B if (! isscalar (x) || ! isscalar (a) || ! isscalar (b)) [err, x, a, b] = common_size (x, a, b); if (err > 0) error ("betacdf: X, A, and B must be of common size or scalars."); endif endif ## Check for X, A, and B being reals if (iscomplex (x) || iscomplex (a) || iscomplex (b)) error ("betacdf: X, A, and B must not be complex."); endif ## Check for class type if (isa (x, "single") || isa (a, "single") || isa (b, "single")) is_type = "single"; else is_type = "double"; endif ## Find valid values in parameters and data okPARAM = (0 < a & a < Inf) & (0 < b & b < Inf); okDATA = (okPARAM & (0 <= x & x <= 1)); all_OK = all (okDATA(:)); ## Force NaNs for out of range parameters. ## Fill in edges cases when X is outside 0 or 1. if (! all_OK) p = NaN (size (okDATA), is_type); if (uflag) p(okPARAM & x <= 0) = 1; p(okPARAM & x >= 1) = 0; else p(okPARAM & x < 0) = 0; p(okPARAM & x > 1) = 1; endif ## Remove the out of range/edge cases. Return, if there's nothing left. if (any (okDATA(:))) if (numel (x) > 1) x = x(okDATA); endif if (numel (a) > 1) a = a(okDATA); endif if (numel (b) > 1) b = b(okDATA); endif else return; endif endif ## Call betainc for the actual work if (uflag) pk = betainc (x, a, b, "upper"); else pk = betainc (x, a, b); endif ## Relocate the values to the correct places if necessary. if all_OK p = pk; else p(okDATA) = pk; endif endfunction %!demo %! ## Plot various CDFs from the Beta distribution %! x = 0:0.005:1; %! p1 = betacdf (x, 0.5, 0.5); %! p2 = betacdf (x, 5, 1); %! p3 = betacdf (x, 1, 3); %! p4 = betacdf (x, 2, 2); %! p5 = betacdf (x, 2, 5); %! plot (x, p1, "-b", x, p2, "-g", x, p3, "-r", x, p4, "-c", x, p5, "-m") %! grid on %! legend ({"α = β = 0.5", "α = 5, β = 1", "α = 1, β = 3", ... %! "α = 2, β = 2", "α = 2, β = 5"}, "location", "northwest") %! title ("Beta CDF") %! xlabel ("values in x") %! ylabel ("probability") ## Test output %!shared x, y, x1, x2 %! x = [-1 0 0.5 1 2]; %! y = [0 0 0.75 1 1]; %!assert (betacdf (x, ones (1, 5), 2 * ones (1, 5)), y) %!assert (betacdf (x, 1, 2 * ones (1, 5)), y) %!assert (betacdf (x, ones (1, 5), 2), y) %!assert (betacdf (x, [0 1 NaN 1 1], 2), [NaN 0 NaN 1 1]) %!assert (betacdf (x, 1, 2 * [0 1 NaN 1 1]), [NaN 0 NaN 1 1]) %!assert (betacdf ([x(1:2) NaN x(4:5)], 1, 2), [y(1:2) NaN y(4:5)]) %! x1 = [0.1:0.2:0.9]; %!assert (betacdf (x1, 2, 2), [0.028, 0.216, 0.5, 0.784, 0.972], 1e-14); %!assert (betacdf (x1, 2, 2, "upper"), 1 - [0.028, 0.216, 0.5, 0.784, 0.972],... %! 1e-14); %! x2 = [1, 2, 3]; %!assert (betacdf (0.5, x2, x2), [0.5, 0.5, 0.5], 1e-14); %!assert (betacdf ([x, NaN], 1, 2), [y, NaN]) ## Test class of input preserved %!assert (betacdf (single ([x, NaN]), 1, 2), single ([y, NaN])) %!assert (betacdf ([x, NaN], single (1), 2), single ([y, NaN])) %!assert (betacdf ([x, NaN], 1, single (2)), single ([y, NaN])) ## Test input validation %!error betacdf () %!error betacdf (1) %!error betacdf (1, 2) %!error betacdf (1, 2, 3, 4, 5) %!error betacdf (1, 2, 3, "tail") %!error betacdf (1, 2, 3, 4) %!error ... %! betacdf (ones (3), ones (2), ones (2)) %!error ... %! betacdf (ones (2), ones (3), ones (2)) %!error ... %! betacdf (ones (2), ones (2), ones (3)) %!error betacdf (i, 2, 2) %!error betacdf (2, i, 2) %!error betacdf (2, 2, i) statistics-release-1.6.3/inst/dist_fun/betainv.m000066400000000000000000000132671456127120000217360ustar00rootroot00000000000000## Copyright (C) 2012 Rik Wehbring ## Copyright (C) 1995-2016 Kurt Hornik ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{x} =} betainv (@var{p}, @var{a}, @var{b}) ## ## Inverse of the Beta distribution (iCDF). ## ## For each element of @var{p}, compute the quantile (the inverse of the CDF) ## of the Beta distribution with shape parameters @var{a} and @var{b}. The size ## of @var{x} is the common size of @var{x}, @var{a}, and @var{b}. A scalar ## input functions as a constant matrix of the same size as the other inputs. ## ## Further information about the Beta distribution can be found at ## @url{https://en.wikipedia.org/wiki/Beta_distribution} ## ## @seealso{betacdf, betapdf, betarnd, betafit, betalike, betastat} ## @end deftypefn function x = betainv (p, a, b) ## Check for valid number of input arguments if (nargin < 3) error ("betainv: function called with too few input arguments."); endif ## Check for common size of P, A, and B if (! isscalar (p) || ! isscalar (a) || ! isscalar (b)) [retval, p, a, b] = common_size (p, a, b); if (retval > 0) error ("betainv: P, A, and B must be of common size or scalars."); endif endif ## Check for P, A, and B being reals if (iscomplex (p) || iscomplex (a) || iscomplex (b)) error ("betainv: P, A, and B must not be complex."); endif ## Check for class type if (isa (p, "single") || isa (a, "single") || isa (b, "single")) x = zeros (size (p), "single"); else x = zeros (size (p)); endif k = (p < 0) | (p > 1) | !(a > 0) | !(b > 0) | isnan (p); x(k) = NaN; k = (p == 1) & (a > 0) & (b > 0); x(k) = 1; k = find ((p > 0) & (p < 1) & (a > 0) & (b > 0)); if (! isempty (k)) if (! isscalar (a) || ! isscalar (b)) a = a(k); b = b(k); y = a ./ (a + b); else y = a / (a + b) * ones (size (k)); endif p = p(k); if (isa (y, "single")) myeps = eps ("single"); else myeps = eps; endif l = find (y < myeps); if (any (l)) y(l) = sqrt (myeps) * ones (length (l), 1); endif l = find (y > 1 - myeps); if (any (l)) y(l) = 1 - sqrt (myeps) * ones (length (l), 1); endif y_new = y; loopcnt = 0; do y_old = y_new; h = (betacdf (y_old, a, b) - p) ./ betapdf (y_old, a, b); y_new = y_old - h; ind = find (y_new <= myeps); if (any (ind)) y_new(ind) = y_old(ind) / 10; endif ind = find (y_new >= 1 - myeps); if (any (ind)) y_new(ind) = 1 - (1 - y_old(ind)) / 10; endif h = y_old - y_new; until (max (abs (h)) < sqrt (myeps) || ++loopcnt == 40) if (loopcnt == 40) warning ("betainv: calculation failed to converge for some values."); endif x(k) = y_new; endif endfunction %!demo %! ## Plot various iCDFs from the Beta distribution %! p = 0.001:0.001:0.999; %! x1 = betainv (p, 0.5, 0.5); %! x2 = betainv (p, 5, 1); %! x3 = betainv (p, 1, 3); %! x4 = betainv (p, 2, 2); %! x5 = betainv (p, 2, 5); %! plot (p, x1, "-b", p, x2, "-g", p, x3, "-r", p, x4, "-c", p, x5, "-m") %! grid on %! legend ({"α = β = 0.5", "α = 5, β = 1", "α = 1, β = 3", ... %! "α = 2, β = 2", "α = 2, β = 5"}, "location", "southeast") %! title ("Beta iCDF") %! xlabel ("probability") %! ylabel ("values in x") ## Test output %!shared p %! p = [-1 0 0.75 1 2]; %!assert (betainv (p, ones (1,5), 2*ones (1,5)), [NaN 0 0.5 1 NaN], eps) %!assert (betainv (p, 1, 2*ones (1,5)), [NaN 0 0.5 1 NaN], eps) %!assert (betainv (p, ones (1,5), 2), [NaN 0 0.5 1 NaN], eps) %!assert (betainv (p, [1 0 NaN 1 1], 2), [NaN NaN NaN 1 NaN]) %!assert (betainv (p, 1, 2*[1 0 NaN 1 1]), [NaN NaN NaN 1 NaN]) %!assert (betainv ([p(1:2) NaN p(4:5)], 1, 2), [NaN 0 NaN 1 NaN]) ## Test class of input preserved %!assert (betainv ([p, NaN], 1, 2), [NaN 0 0.5 1 NaN NaN], eps) %!assert (betainv (single ([p, NaN]), 1, 2), single ([NaN 0 0.5 1 NaN NaN])) %!assert (betainv ([p, NaN], single (1), 2), single ([NaN 0 0.5 1 NaN NaN]), eps("single")) %!assert (betainv ([p, NaN], 1, single (2)), single ([NaN 0 0.5 1 NaN NaN]), eps("single")) ## Test input validation %!error betainv () %!error betainv (1) %!error betainv (1,2) %!error betainv (1,2,3,4) %!error ... %! betainv (ones (3), ones (2), ones (2)) %!error ... %! betainv (ones (2), ones (3), ones (2)) %!error ... %! betainv (ones (2), ones (2), ones (3)) %!error betainv (i, 2, 2) %!error betainv (2, i, 2) %!error betainv (2, 2, i) statistics-release-1.6.3/inst/dist_fun/betapdf.m000066400000000000000000000133741456127120000217120ustar00rootroot00000000000000## Copyright (C) 2010 Christos Dimitrakakis ## Copyright (C) 2012 Rik Wehbring ## Copyright (C) 1995-2016 Kurt Hornik ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{y} =} betapdf (@var{x}, @var{a}, @var{b}) ## ## Beta probability density function (PDF). ## ## For each element of @var{x}, compute the probability density function (PDF) ## of the Beta distribution with shape parameters @var{a} and @var{b}. The size ## of @var{y} is the common size of @var{x}, @var{a}, and @var{b}. A scalar ## input functions as a constant matrix of the same size as the other inputs. ## ## Further information about the Beta distribution can be found at ## @url{https://en.wikipedia.org/wiki/Beta_distribution} ## ## @seealso{betacdf, betainv, betarnd, betafit, betalike, betastat} ## @end deftypefn function y = betapdf (x, a, b) ## Check for valid number of input arguments if (nargin < 3) error ("betapdf: function called with too few input arguments."); endif ## Check for common size of X, A, and B if (! isscalar (x) || ! isscalar (a) || ! isscalar (b)) [retval, x, a, b] = common_size (x, a, b); if (retval > 0) error ("betapdf: X, A, and B must be of common size or scalars."); endif endif ## Check for X, A, and B being reals if (iscomplex (x) || iscomplex (a) || iscomplex (b)) error ("betapdf: X, A, and B must not be complex."); endif ## Check for class type if (isa (x, "single") || isa (a, "single") || isa (b, "single")); y = zeros (size (x), "single"); else y = zeros (size (x)); endif k = !(a > 0) | !(b > 0) | isnan (x); y(k) = NaN; k = (x > 0) & (x < 1) & (a > 0) & (b > 0) & ((a != 1) | (b != 1)); if (isscalar (a) && isscalar (b)) y(k) = exp ((a - 1) * log (x(k)) + (b - 1) * log (1 - x(k)) + gammaln (a + b) - gammaln (a) - gammaln (b)); else y(k) = exp ((a(k) - 1) .* log (x(k)) + (b(k) - 1) .* log (1 - x(k)) + gammaln (a(k) + b(k)) - gammaln (a(k)) - gammaln (b(k))); endif ## Most important special cases when the density is finite. k = (x == 0) & (a == 1) & (b > 0) & (b != 1); if (isscalar (a) && isscalar (b)) y(k) = exp (gammaln (a + b) - gammaln (a) - gammaln (b)); else y(k) = exp (gammaln (a(k) + b(k)) - gammaln (a(k)) - gammaln (b(k))); endif k = (x == 1) & (b == 1) & (a > 0) & (a != 1); if (isscalar (a) && isscalar (b)) y(k) = exp (gammaln (a + b) - gammaln (a) - gammaln (b)); else y(k) = exp (gammaln (a(k) + b(k)) - gammaln (a(k)) - gammaln (b(k))); endif k = (x >= 0) & (x <= 1) & (a == 1) & (b == 1); y(k) = 1; ## Other special case when the density at the boundary is infinite. k = (x == 0) & (a < 1); y(k) = Inf; k = (x == 1) & (b < 1); y(k) = Inf; endfunction %!demo %! ## Plot various PDFs from the Beta distribution %! x = 0.001:0.001:0.999; %! y1 = betapdf (x, 0.5, 0.5); %! y2 = betapdf (x, 5, 1); %! y3 = betapdf (x, 1, 3); %! y4 = betapdf (x, 2, 2); %! y5 = betapdf (x, 2, 5); %! plot (x, y1, "-b", x, y2, "-g", x, y3, "-r", x, y4, "-c", x, y5, "-m") %! grid on %! ylim ([0, 2.5]) %! legend ({"α = β = 0.5", "α = 5, β = 1", "α = 1, β = 3", ... %! "α = 2, β = 2", "α = 2, β = 5"}, "location", "north") %! title ("Beta PDF") %! xlabel ("values in x") %! ylabel ("density") ## Test output %!shared x, y %! x = [-1 0 0.5 1 2]; %! y = [0 2 1 0 0]; %!assert (betapdf (x, ones (1, 5), 2 * ones (1, 5)), y) %!assert (betapdf (x, 1, 2 * ones (1, 5)), y) %!assert (betapdf (x, ones (1, 5), 2), y) %!assert (betapdf (x, [0 NaN 1 1 1], 2), [NaN NaN y(3:5)]) %!assert (betapdf (x, 1, 2 * [0 NaN 1 1 1]), [NaN NaN y(3:5)]) %!assert (betapdf ([x, NaN], 1, 2), [y, NaN]) ## Test class of input preserved %!assert (betapdf (single ([x, NaN]), 1, 2), single ([y, NaN])) %!assert (betapdf ([x, NaN], single (1), 2), single ([y, NaN])) %!assert (betapdf ([x, NaN], 1, single (2)), single ([y, NaN])) ## Beta (1/2,1/2) == arcsine distribution %!test %! x = rand (10,1); %! y = 1 ./ (pi * sqrt (x .* (1 - x))); %! assert (betapdf (x, 1/2, 1/2), y, 100 * eps); ## Test large input values to betapdf %!assert (betapdf (0.5, 1000, 1000), 35.678, 1e-3) ## Test input validation %!error betapdf () %!error betapdf (1) %!error betapdf (1,2) %!error betapdf (1,2,3,4) %!error ... %! betapdf (ones (3), ones (2), ones (2)) %!error ... %! betapdf (ones (2), ones (3), ones (2)) %!error ... %! betapdf (ones (2), ones (2), ones (3)) %!error betapdf (i, 2, 2) %!error betapdf (2, i, 2) %!error betapdf (2, 2, i) statistics-release-1.6.3/inst/dist_fun/betarnd.m000066400000000000000000000162031456127120000217160ustar00rootroot00000000000000## Copyright (C) 2012 Rik Wehbring ## Copyright (C) 1995-2016 Kurt Hornik ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{r} =} betarnd (@var{a}, @var{b}) ## @deftypefnx {statistics} {@var{r} =} betarnd (@var{a}, @var{b}, @var{rows}) ## @deftypefnx {statistics} {@var{r} =} betarnd (@var{a}, @var{b}, @var{rows}, @var{cols}, @dots{}) ## @deftypefnx {statistics} {@var{r} =} betarnd (@var{a}, @var{b}, [@var{sz}]) ## ## Random arrays from the Beta distribution. ## ## @code{@var{r} = betarnd (@var{a}, @var{b})} returns an array of random ## numbers chosen from the Beta distribution with shape parameters @var{a} and ## @var{b}. The size of @var{r} is the common size of @var{a} and @var{b}. ## A scalar input functions as a constant matrix of the same size as the other ## inputs. ## ## When called with a single size argument, @code{betarnd} returns a square ## matrix with the dimension specified. When called with more than one scalar ## argument, the first two arguments are taken as the number of rows and columns ## and any further arguments specify additional matrix dimensions. The size may ## also be specified with a row vector of dimensions, @var{sz}. ## ## Further information about the Beta distribution can be found at ## @url{https://en.wikipedia.org/wiki/Beta_distribution} ## ## @seealso{betacdf, betainv, betapdf, betafit, betalike, betastat} ## @end deftypefn function r = betarnd (a, b, varargin) ## Check for valid number of input arguments if (nargin < 2) error ("betarnd: function called with too few input arguments."); endif ## Check for common size of A and B if (! isscalar (a) || ! isscalar (b)) [retval, a, b] = common_size (a, b); if (retval > 0) error ("betarnd: A and B must be of common size or scalars."); endif endif ## Check for A and B being reals if (iscomplex (a) || iscomplex (b)) error ("betarnd: A and B must not be complex."); endif ## Parse and check SIZE arguments if (nargin == 2) sz = size (a); elseif (nargin == 3) if (isscalar (varargin{1}) && varargin{1} >= 0 ... && varargin{1} == fix (varargin{1})) sz = [varargin{1}, varargin{1}]; elseif (isrow (varargin{1}) && all (varargin{1} >= 0) ... && all (varargin{1} == fix (varargin{1}))) sz = varargin{1}; elseif error (strcat (["betarnd: SZ must be a scalar or a row vector"], ... [" of non-negative integers."])); endif elseif (nargin > 3) posint = cellfun (@(x) (! isscalar (x) || x < 0 || x != fix (x)), varargin); if (any (posint)) error ("betarnd: dimensions must be non-negative integers."); endif sz = [varargin{:}]; endif ## Check that parameters match requested dimensions in size if (! isscalar (a) && ! isequal (size (a), sz)) error ("betarnd: A and B must be scalars or of size SZ."); endif ## Check for class type if (isa (a, "single") || isa (b, "single")) cls = "single"; else cls = "double"; endif ## Generate random sample from Beta distribution if (isscalar (a) && isscalar (b)) if ((a > 0) && (a < Inf) && (b > 0) && (b < Inf)) tmpr = randg (a, sz, cls); r = tmpr ./ (tmpr + randg (b, sz, cls)); else r = NaN (sz, cls); endif else r = NaN (sz, cls); k = (a > 0) & (a < Inf) & (b > 0) & (b < Inf); tmpr = randg (a(k), cls); r(k) = tmpr ./ (tmpr + randg (b(k), cls)); endif endfunction ## Test output %!assert (size (betarnd (2, 1/2)), [1 1]) %!assert (size (betarnd (2 * ones (2, 1), 1/2)), [2, 1]) %!assert (size (betarnd (2 * ones (2, 2), 1/2)), [2, 2]) %!assert (size (betarnd (2, 1/2 * ones (2, 1))), [2, 1]) %!assert (size (betarnd (1, 1/2 * ones (2, 2))), [2, 2]) %!assert (size (betarnd (ones (2, 1), 1)), [2, 1]) %!assert (size (betarnd (ones (2, 2), 1)), [2, 2]) %!assert (size (betarnd (2, 1/2, 3)), [3, 3]) %!assert (size (betarnd (1, 1, [4, 1])), [4, 1]) %!assert (size (betarnd (1, 1, 4, 1)), [4, 1]) %!assert (size (betarnd (1, 1, 4, 1, 5)), [4, 1, 5]) %!assert (size (betarnd (1, 1, 0, 1)), [0, 1]) %!assert (size (betarnd (1, 1, 1, 0)), [1, 0]) %!assert (size (betarnd (1, 1, 1, 2, 0, 5)), [1, 2, 0, 5]) ## Test class of input preserved %!assert (class (betarnd (1, 1)), "double") %!assert (class (betarnd (1, single (0))), "single") %!assert (class (betarnd (1, single ([0, 0]))), "single") %!assert (class (betarnd (1, single (1), 2)), "single") %!assert (class (betarnd (1, single ([1, 1]), 1, 2)), "single") %!assert (class (betarnd (single (1), 1, 2)), "single") %!assert (class (betarnd (single ([1, 1]), 1, 1, 2)), "single") ## Test input validation %!error betarnd () %!error betarnd (1) %!error ... %! betarnd (ones (3), ones (2)) %!error ... %! betarnd (ones (2), ones (3)) %!error betarnd (i, 2) %!error betarnd (1, i) %!error ... %! betarnd (1, 1/2, -1) %!error ... %! betarnd (1, 1/2, 1.2) %!error ... %! betarnd (1, 1/2, ones (2)) %!error ... %! betarnd (1, 1/2, [2 -1 2]) %!error ... %! betarnd (1, 1/2, [2 0 2.5]) %!error ... %! betarnd (1, 1/2, 2, -1, 5) %!error ... %! betarnd (1, 1/2, 2, 1.5, 5) %!error ... %! betarnd (2, 1/2 * ones (2), 3) %!error ... %! betarnd (2, 1/2 * ones (2), [3, 2]) %!error ... %! betarnd (2, 1/2 * ones (2), 3, 2) %!error ... %! betarnd (2 * ones (2), 1/2, 3) %!error ... %! betarnd (2 * ones (2), 1/2, [3, 2]) %!error ... %! betarnd (2 * ones (2), 1/2, 3, 2) statistics-release-1.6.3/inst/dist_fun/binocdf.m000066400000000000000000000142621456127120000217060ustar00rootroot00000000000000## Copyright (C) 2012 Rik Wehbring ## Copyright (C) 1995-2016 Kurt Hornik ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{p} =} binocdf (@var{x}, @var{n}, @var{ps}) ## @deftypefnx {statistics} {@var{p} =} binocdf (@var{x}, @var{n}, @var{ps}, @qcode{"upper"}) ## ## Binomial cumulative distribution function (CDF). ## ## For each element of @var{x}, compute the cumulative distribution function ## (CDF) of the binomial distribution with parameters @var{n} and @var{ps}, ## where @var{n} is the number of trials and @var{ps} is the probability of ## success. The size of @var{p} is the common size of @var{x}, @var{n}, and ## @var{ps}. A scalar input functions as a constant matrix of the same size as ## the other inputs. ## ## @code{@var{p} = binocdf (@var{x}, @var{n}, @var{ps}, "upper")} computes the ## upper tail probability of the binomial distribution with parameters ## @var{n} and @var{ps}, at the values in @var{x}. ## ## Further information about the binomial distribution can be found at ## @url{https://en.wikipedia.org/wiki/Binomial_distribution} ## ## @seealso{binoinv, binopdf, binornd, binofit, binolike, binostat, binotest} ## @end deftypefn function p = binocdf (x, n, ps, uflag) ## Check for valid number of input arguments if (nargin < 3) error ("binocdf: function called with too few input arguments."); endif ## Check for valid "upper" flag if (nargin == 4) if (strcmp (uflag, "upper")) uflag = true; else error ("binocdf: invalid argument for upper tail."); endif else uflag = false; endif ## Check for common size of X, N, and PS if (! isscalar (x) || ! isscalar (n) || ! isscalar (ps)) [retval, x, n, ps] = common_size (x, n, ps); if (retval > 0) error ("binocdf: X, N, and PS must be of common size or scalars."); endif endif ## Check for X, N, and PS being reals if (iscomplex (x) || iscomplex (n) || iscomplex (ps)) error ("binocdf: X, N, and PS must not be complex."); endif ## Check for class type if (isa (x, "single") || isa (n, "single") || isa (ps, "single")); p = nan (size (x), "single"); else p = nan (size (x)); endif k = (x >= n) & (n >= 0) & (n == fix (n) & (ps >= 0) & (ps <= 1)); p(k) = !uflag; k = (x < 0) & (n >= 0) & (n == fix (n) & (ps >= 0) & (ps <= 1)); p(k) = uflag; k = (x >= 0) & (x < n) & (n == fix (n)) & (ps >= 0) & (ps <= 1); tmp = floor (x(k)); if (! uflag) if (isscalar (n) && isscalar (ps)) p(k) = betainc (1 - ps, n - tmp, tmp + 1); else p(k) = betainc (1 - ps(k), n(k) - tmp, tmp + 1); endif else if (isscalar (n) && isscalar (ps)); p(k) = betainc (ps, tmp + 1, n - tmp); else p(k) = betainc (ps(k), tmp + 1, n(k) - tmp); endif endif endfunction %!demo %! ## Plot various CDFs from the binomial distribution %! x = 0:40; %! p1 = binocdf (x, 20, 0.5); %! p2 = binocdf (x, 20, 0.7); %! p3 = binocdf (x, 40, 0.5); %! plot (x, p1, "*b", x, p2, "*g", x, p3, "*r") %! grid on %! legend ({"n = 20, ps = 0.5", "n = 20, ps = 0.7", ... %! "n = 40, ps = 0.5"}, "location", "southeast") %! title ("Binomial CDF") %! xlabel ("values in x (number of successes)") %! ylabel ("probability") ## Test output %!shared x, p, p1 %! x = [-1 0 1 2 3]; %! p = [0 1/4 3/4 1 1]; %! p1 = 1 - p; %!assert (binocdf (x, 2 * ones (1, 5), 0.5 * ones (1, 5)), p, eps) %!assert (binocdf (x, 2, 0.5 * ones (1, 5)), p, eps) %!assert (binocdf (x, 2 * ones (1, 5), 0.5), p, eps) %!assert (binocdf (x, 2 * [0 -1 NaN 1.1 1], 0.5), [0 NaN NaN NaN 1]) %!assert (binocdf (x, 2, 0.5 * [0 -1 NaN 3 1]), [0 NaN NaN NaN 1]) %!assert (binocdf ([x(1:2) NaN x(4:5)], 2, 0.5), [p(1:2) NaN p(4:5)], eps) %!assert (binocdf (99, 100, 0.1, "upper"), 1e-100, 1e-112); %!assert (binocdf (x, 2 * ones (1, 5), 0.5*ones (1,5), "upper"), p1, eps) %!assert (binocdf (x, 2, 0.5 * ones (1, 5), "upper"), p1, eps) %!assert (binocdf (x, 2 * ones (1, 5), 0.5, "upper"), p1, eps) %!assert (binocdf (x, 2 * [0 -1 NaN 1.1 1], 0.5, "upper"), [1 NaN NaN NaN 0]) %!assert (binocdf (x, 2, 0.5 * [0 -1 NaN 3 1], "upper"), [1 NaN NaN NaN 0]) %!assert (binocdf ([x(1:2) NaN x(4:5)], 2, 0.5, "upper"), [p1(1:2) NaN p1(4:5)]) %!assert (binocdf ([x, NaN], 2, 0.5), [p, NaN], eps) ## Test class of input preserved %!assert (binocdf (single ([x, NaN]), 2, 0.5), single ([p, NaN])) %!assert (binocdf ([x, NaN], single (2), 0.5), single ([p, NaN])) %!assert (binocdf ([x, NaN], 2, single (0.5)), single ([p, NaN])) ## Test input validation %!error binocdf () %!error binocdf (1) %!error binocdf (1, 2) %!error binocdf (1, 2, 3, 4, 5) %!error binocdf (1, 2, 3, "tail") %!error binocdf (1, 2, 3, 4) %!error ... %! binocdf (ones (3), ones (2), ones (2)) %!error ... %! binocdf (ones (2), ones (3), ones (2)) %!error ... %! binocdf (ones (2), ones (2), ones (3)) %!error binocdf (i, 2, 2) %!error binocdf (2, i, 2) %!error binocdf (2, 2, i) statistics-release-1.6.3/inst/dist_fun/binoinv.m000066400000000000000000000203071456127120000217430ustar00rootroot00000000000000## Copyright (C) 2016-2017 Lachlan Andrew ## Copyright (C) 2012-2016 Rik Wehbring ## Copyright (C) 1995-2012 Kurt Hornik ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{x} =} binoinv (@var{p}, @var{n}, @var{ps}) ## ## Inverse of the Binomial cumulative distribution function (iCDF). ## ## For each element of @var{p}, compute the quantile (the inverse of the CDF) of ## the binomial distribution with parameters @var{n} and @var{ps}, where @var{n} ## is the number of trials and @var{ps} is the probability of success. The size ## of @var{x} is the common size of @var{p}, @var{n}, and @var{ps}. A scalar ## input functions as a constant matrix of the same size as the other inputs. ## ## Further information about the binomial distribution can be found at ## @url{https://en.wikipedia.org/wiki/Binomial_distribution} ## ## @seealso{binocdf, binopdf, binornd, binofit, binolike, binostat, binotest} ## @end deftypefn function x = binoinv (p, n, ps) ## Check for valid number of input arguments if (nargin < 3) error ("binoinv: function called with too few input arguments."); endif ## Check for common size of P, N, and PS if (! isscalar (n) || ! isscalar (ps)) [retval, p, n, ps] = common_size (p, n, ps); if (retval > 0) error ("binoinv: P, N, and PS must be of common size or scalars."); endif endif ## Check for P, N, and PS being reals if (iscomplex (p) || iscomplex (n) || iscomplex (ps)) error ("binoinv: P, N, and PS must not be complex."); endif ## Check for class type if (isa (p, "single") || isa (n, "single") || isa (ps, "single")); x = zeros (size (p), "single"); else x = zeros (size (p)); endif k = (! (p >= 0) | ! (p <= 1) | ! (n >= 0) | (n != fix (n)) | ! (ps >= 0) | ... ! (ps <= 1)); x(k) = NaN; k = find ((p >= 0) & (p <= 1) & (n >= 0) & (n == fix (n) ... & (ps >= 0) & (ps <= 1))); if (! isempty (k)) p = p(k); if (isscalar (n) && isscalar (ps)) [x(k), unfinished] = scalar_binoinv (p(:), n, ps); k = k(unfinished); if (! isempty (k)) x(k) = bin_search_binoinv (p(k), n, ps); endif else [x(k), unfinished] = vector_binoinv (p(:), n(:), ps(:)); k = k(unfinished); if (! isempty (k)) x(k) = bin_search_binoinv (p(k), n(k), ps(k)); endif endif endif endfunction ## Core algorithm to calculate the inverse binomial, for n and ps real scalars ## and x a column vector, and for which the output is not NaN or Inf. ## Compute CDF in batches of doubling size until CDF > p, or answer > 500 ## Return the locations of unfinished cases in k. function [m, k] = scalar_binoinv (p, n, ps) k = 1:length (p); m = zeros (size (p)); prev_limit = 0; limit = 10; cdf = 0; v = 0; do cdf = binocdf (prev_limit:limit-1, n, ps); r = bsxfun (@le, p(k), cdf); [v, m(k)] = max (r, [], 2); # find first instance of p <= cdf m(k) += prev_limit - 1; k = k(v == 0); prev_limit = limit; limit += limit; until (isempty (k) || limit >= 1000) endfunction ## Core algorithm to calculate the inverse binomial, for n, ps, and x column ## vectors, and for which the output is not NaN or Inf. ## Compute CDF in batches of doubling size until CDF > p, or answer > 500 ## Return the locations of unfinished cases in k. ## Calculates CDF by summing PDF, which is faster than calls to binocdf. function [m, k] = vector_binoinv (p, n, ps) k = 1:length(p); m = zeros (size (p)); prev_limit = 0; limit = 10; cdf = 0; v = 0; do xx = repmat (prev_limit:limit-1, [length(k), 1]); nn = kron (ones (1, limit-prev_limit), n(k)); pp = kron (ones (1, limit-prev_limit), ps(k)); pdf = binopdf (xx, nn, pp); pdf(:,1) += cdf(v==0, end); cdf = cumsum (pdf, 2); r = bsxfun (@le, p(k), cdf); [v, m(k)] = max (r, [], 2); # find first instance of p <= cdf m(k) += prev_limit - 1; k = k(v == 0); prev_limit = limit; limit += min (limit, max (1e4/numel (k), 10)); # limit memory use until (isempty (k) || limit >= 1000) endfunction ## Vectorized binary search. ## Can handle vectors n and ps, and is faster than the scalar case when the ## answer is large. ## Could be optimized to call binocdf only for a subset of the p at each stage, ## but care must be taken to handle both scalar and vector n, ps. Bookkeeping ## may cost more than the extra computations. function m = bin_search_binoinv (p, n, ps) k = 1:length (p); lower = zeros (size (p)); limit = 500; # lower bound on point at which prev phase finished while (any (k) && limit < 1e100) cdf = binocdf (limit, n, ps); k = (p > cdf); lower(k) = limit; limit += limit; endwhile upper = max (2*lower, 1); k = find (lower != limit/2); # elements for which above loop finished for i = 1:ceil (log2 (max (lower))) mid = (upper + lower)/2; cdf = binocdf (floor(mid(:)), n, ps); r = (p <= cdf); upper(r) = mid(r); lower(! r) = mid(! r); endfor m = ceil (lower); m(p > binocdf (m(:), n, ps)) += 1; # fix off-by-one errors from binary search endfunction %!demo %! ## Plot various iCDFs from the binomial distribution %! p = 0.001:0.001:0.999; %! x1 = binoinv (p, 20, 0.5); %! x2 = binoinv (p, 20, 0.7); %! x3 = binoinv (p, 40, 0.5); %! plot (p, x1, "-b", p, x2, "-g", p, x3, "-r") %! grid on %! legend ({"n = 20, ps = 0.5", "n = 20, ps = 0.7", ... %! "n = 40, ps = 0.5"}, "location", "southeast") %! title ("Binomial iCDF") %! xlabel ("probability") %! ylabel ("values in x (number of successes)") ## Test output %!shared p %! p = [-1 0 0.5 1 2]; %!assert (binoinv (p, 2*ones (1,5), 0.5*ones (1,5)), [NaN 0 1 2 NaN]) %!assert (binoinv (p, 2, 0.5*ones (1,5)), [NaN 0 1 2 NaN]) %!assert (binoinv (p, 2*ones (1,5), 0.5), [NaN 0 1 2 NaN]) %!assert (binoinv (p, 2*[0 -1 NaN 1.1 1], 0.5), [NaN NaN NaN NaN NaN]) %!assert (binoinv (p, 2, 0.5*[0 -1 NaN 3 1]), [NaN NaN NaN NaN NaN]) %!assert (binoinv ([p(1:2) NaN p(4:5)], 2, 0.5), [NaN 0 NaN 2 NaN]) ## Test class of input preserved %!assert (binoinv ([p, NaN], 2, 0.5), [NaN 0 1 2 NaN NaN]) %!assert (binoinv (single ([p, NaN]), 2, 0.5), single ([NaN 0 1 2 NaN NaN])) %!assert (binoinv ([p, NaN], single (2), 0.5), single ([NaN 0 1 2 NaN NaN])) %!assert (binoinv ([p, NaN], 2, single (0.5)), single ([NaN 0 1 2 NaN NaN])) ## Test accuracy, to within +/- 1 since it is a discrete distribution %!shared x, tol %! x = magic (3) + 1; %! tol = 1; %!assert (binoinv (binocdf (1:10, 11, 0.1), 11, 0.1), 1:10, tol) %!assert (binoinv (binocdf (1:10, 2*(1:10), 0.1), 2*(1:10), 0.1), 1:10, tol) %!assert (binoinv (binocdf (x, 2*x, 1./x), 2*x, 1./x), x, tol) ## Test input validation %!error binoinv () %!error binoinv (1) %!error binoinv (1,2) %!error binoinv (1,2,3,4) %!error ... %! binoinv (ones (3), ones (2), ones (2)) %!error ... %! binoinv (ones (2), ones (3), ones (2)) %!error ... %! binoinv (ones (2), ones (2), ones (3)) %!error binoinv (i, 2, 2) %!error binoinv (2, i, 2) %!error binoinv (2, 2, i) statistics-release-1.6.3/inst/dist_fun/binopdf.m000066400000000000000000000272311456127120000217230ustar00rootroot00000000000000## Copyright (C) 2012 Rik Wehbring ## Copyright (C) 1995-2016 Kurt Hornik ## Copyright (C) 2021 Nicholas R. Jankowski ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{y} =} binopdf (@var{x}, @var{n}, @var{ps}) ## ## Binomial probability density function (PDF). ## ## For each element of @var{x}, compute the probability density function (PDF) ## of the binomial distribution with parameters @var{n} and @var{ps}, where ## @var{n} is the number of trials and @var{ps} is the probability of success. ## The size of @var{y} is the common size of @var{x}, @var{n}, and @var{ps}. A ## scalar input functions as a constant matrix of the same size as the other ## inputs. ## ## Matlab incompatibility: Octave's @code{binopdf} does not allow complex ## input values. Matlab 2021b returns values for complex inputs despite the ## documentation indicates integer and real value inputs are required. ## ## Further information about the binomial distribution can be found at ## @url{https://en.wikipedia.org/wiki/Binomial_distribution} ## ## @seealso{binocdf, binoinv, binornd, binofit, binolike, binostat, binotest} ## @end deftypefn function y = binopdf (x, n, ps) ## Check for valid number of input arguments if (nargin < 3) error ("binopdf: function called with too few input arguments."); endif ## Check for common size of X, N, and PS if (! isscalar (x) || ! isscalar (n) || ! isscalar (ps)) [retval, x, n, ps] = common_size (x, n, ps); if (retval > 0) error ("binopdf: X, N, and PS must be of common size or scalars."); endif endif ## Check for X, N, and PS being reals if (iscomplex (x) || iscomplex (n) || iscomplex (ps)) error ("binopdf: X, N, and PS must not be complex."); endif sz_x = size (x); # save original size for reshape later x = x(:); n = n(:); ps = ps(:); # columns for easier vectorization ## Initialize output, preserve class of output if any are singles if (isa (x, "single") || isa (n, "single") || isa (ps, "single")); y = zeros (numel (x), 1, "single"); else y = zeros (numel (x), 1); endif ## k - index of array locations needing calculation k = (x == fix (x)) & (n == fix (n)) & (n >= 0) & (ps >= 0) & (ps <= 1) ... & (x >= 0) & (x <= n); nx = n - x; q = 1 - ps; ## Catch special cases ahead of calculations: ## Matlab incompatibility: Matlab 2021b returns values for complex inputs ## despite documentation indicating integer and real value inputs required. ## Octave chooses to return an NaN instead. catch_special = (iscomplex (x) | iscomplex (n) | iscomplex (ps)); k(catch_special) = false; y(catch_special) = NaN; ## x = 0 and x = n cases where ps != 0 or 1, respectivly ## remove them from k, use alternate calculation to avoid /0 catch_special = (x == 0)& (! catch_special); k(catch_special) = false; y(catch_special) = exp (n(catch_special) .* log (q(catch_special))); catch_special = (nx == 0) & (! catch_special); k(catch_special) = false; y(catch_special) = exp (n(catch_special) .* log (ps(catch_special))); ## Perform Loader pdf calculation on non-trivial elements if (any (k)) y(k) = loader_expansion (x(k), n(k), ps(k), nx(k), q(k)); endif ## Trivial case special outputs: ksp = ((ps == 0) & (x == 0)) | (ps == 1) & (x == n); y(ksp) = 1; ## Input NaN, n not pos int, or ps outside [0,1], ## set output to NaN (overrides 0 or 1) ksp = (n ~= fix (n)) | (n < 0) | (ps < 0) | (ps > 1) | isnan (x) ... | isnan (n) | isnan (ps); y(ksp) = NaN; y = reshape (y, sz_x); ## restore output to input shape endfunction function y = loader_expansion (x, n, ps, nx, q) ## Precalculated constants, d_n from n = 0 to 30 ## extended from Loader using octave symbolic vpa ## out to n = 30 d_n = [ 0.08106146679532725821967026359438236013860, 0.04134069595540929409382208140711750802535, 0.02767792568499833914878929274624466659538, 0.02079067210376509311152277176784865633309, 0.01664469118982119216319486537359339114739, 0.01387612882307074799874572702376290856175, 0.01189670994589177009505572411765943862013, 0.01041126526197209649747856713253462919952, 0.00925546218271273291772863663310013611743, 0.00833056343336287125646931865962855220929, 0.00757367548795184079497202421159508389293, 0.00694284010720952986566415266347536265992, 0.00640899418800420706843963108297831257520, 0.00595137011275884773562441604646945832642, 0.00555473355196280137103868995979228464907, 0.00520765591960964044071799685790189865099, 0.00490139594843473786071681819096755442865, 0.00462915374933402859242721316419232323878, 0.00438556024923232426828773634861946570116, 0.00416631969199692245746292338221831613633, 0.00396795421864085961728763680734281467287, 0.00378761806844443457786667706893349200129, 0.00362296022468309470738119836390285473489, 0.00347202138297876696294511542270952959204, 0.00333315563672809287580701911737271025035, 0.00320497022805503801118415655381541759643, 0.00308627868260877706325624133564397946129, 0.00297606398355040882602116255686080370692, 0.00287344936235246638755235148906672207372, 0.00277767492975269360359490376220667282839 ]; stored_dn = numel(d_n); ## Indices for precalculated vs to-be-calculated values n_precalc = (n > 0) & (n < stored_dn); x_precalc = (x > 0) & (x < stored_dn); nx_precalc = (nx > 0) & (nx < stored_dn); [delta_n, delta_x, delta_nx] = deal (zeros (size (x))); ## Fetch precalculated values delta_n(n_precalc) = d_n(n(n_precalc)); delta_x(x_precalc) = d_n(x(x_precalc)); delta_nx(nx_precalc) = d_n(nx(nx_precalc)); ## Calculate any other d(n) values delta_n(!n_precalc) = delta_fn (n(!n_precalc)); delta_x(!x_precalc) = delta_fn (x(!x_precalc)); delta_nx(!nx_precalc) = delta_fn (nx(!nx_precalc)); ## Calculate exp(log(pdf)); y = exp ((delta_n - delta_x - delta_nx - ... deviance (x, n .* ps) - ... deviance (nx, n .* q)) - ... 0.5 * (log(2*pi) + log (x) + log (1-x./n))); endfunction function y = delta_fn (n) ## Stirling formula error term approximations based on Loader paper. ## exact expression, n^n overflows to Inf for n > ~145: ## = log (n!*exp(n)/(sqrt(2pi*n)*n^n)); ## ## Rewritten to avoid overflow out to n> 1e305. accurate to ~10^-12 ## = n + gammaln (n+1) - (n+0.5) * log(n) - log(2*pi)/2; ## ## Approximated as: ## accurate to ~10^-16 for n=30. underflow to 0 at n~10^309 ## = 1/(12n)-1/(360n^3)+1/(1260n^5)- 1/(1680n.^7)+1/(1188n^9) + O(n^-11); ## ## Factored to reduced operation count. Used by Loader and in R: ## 25% faster than unfactored form. ## =(1/12-(1/360-(1/1260-(1/1680-(1/1188)/n^2)/n^2)/n^2)/n^2)/n; nn = n.^2; y = (0.08333333333333333333333333333333333333333 - ... (0.00277777777777777777777777777777777777778 - ... (0.00079365079365079365079365079365079365079 - ... (0.00059523809523809523809523809523809523810 - ... (0.00084175084175084175084175084175084175084)./nn)./nn)./nn)./nn)./n; endfunction function D = deviance (x, np) ## requires equal length column inputs epsilon = x ./ np; v = (epsilon - 1) ./ (epsilon + 1); vtest = abs (v) < 0.1; if (any (vtest)) ## For abs(v) < 0.1, do taylor expansion for higher precision. Expansion ## term: v^(2j+1)/(2j+1). For abs(v)< 0.1, term drops slowest for max ## abs(v) = 0.1. (n+1)th term is <= 10. jmax = 12; two_jpone = 2 * [1:jmax] + 1; # sum term 2*j+1 (row vector expansion) D = zeros (numel (epsilon), 1); ## D = (x-np)*v + 2*x*sum_over_j(v^2j+1 / 2j+1) D(vtest) = (x(vtest) - np(vtest)) .* v(vtest) + 2 .* x(vtest) .* ... sum (v(vtest).^(two_jpone) ./ two_jpone, 2); D(! vtest) = x(! vtest) .* (log (epsilon(! vtest)) - 1) + np(! vtest); else D = x.* (log (epsilon) - 1) + np; endif endfunction %!demo %! ## Plot various PDFs from the binomial distribution %! x = 0:40; %! y1 = binopdf (x, 20, 0.5); %! y2 = binopdf (x, 20, 0.7); %! y3 = binopdf (x, 40, 0.5); %! plot (x, y1, "*b", x, y2, "*g", x, y3, "*r") %! grid on %! ylim ([0, 0.25]) %! legend ({"n = 20, ps = 0.5", "n = 20, ps = 0.7", ... %! "n = 40, ps = 0.5"}, "location", "northeast") %! title ("Binomial PDF") %! xlabel ("values in x (number of successes)") %! ylabel ("density") ## Test output %!shared x, y %! x = [-1 0 1 2 3]; %! y = [0 1/4 1/2 1/4 0]; %!assert (binopdf (x, 2 * ones (1, 5), 0.5 * ones (1, 5)), y, eps) %!assert (binopdf (x, 2, 0.5 * ones (1, 5)), y, eps) %!assert (binopdf (x, 2 * ones (1, 5), 0.5), y, eps) %!assert (binopdf (x, 2 * [0 -1 NaN 1.1 1], 0.5), [0 NaN NaN NaN 0]) %!assert (binopdf (x, 2, 0.5 * [0 -1 NaN 3 1]), [0 NaN NaN NaN 0]) %!assert (binopdf ([x, NaN], 2, 0.5), [y, NaN], eps) %!assert (binopdf (cat (3, x, x), 2, 0.5), cat (3, y, y), eps) ## Test Special input values %!assert (binopdf (1, 1, 1), 1) %!assert (binopdf (0, 3, 0), 1) %!assert (binopdf (2, 2, 1), 1) %!assert (binopdf (1, 2, 1), 0) %!assert (binopdf (0, 1.1, 0), NaN) %!assert (binopdf (1, 2, -1), NaN) %!assert (binopdf (1, 2, 1.5), NaN) ## Test empty inputs %!assert (binopdf ([], 1, 1), []) %!assert (binopdf (1, [], 1), []) %!assert (binopdf (1, 1, []), []) %!assert (binopdf (ones (1, 0), 2, .5), ones(1, 0)) %!assert (binopdf (ones (0, 1), 2, .5), ones(0, 1)) %!assert (binopdf (ones (0, 1, 2), 2, .5), ones(0, 1, 2)) %!assert (binopdf (1, ones (0, 1, 2), .5), ones(0, 1, 2)) %!assert (binopdf (1, 2, ones (0, 1, 2)), ones(0, 1, 2)) %!assert (binopdf (ones (1, 0, 2), 2, .5), ones(1, 0, 2)) %!assert (binopdf (ones (1, 2, 0), 2, .5), ones(1, 2, 0)) %!assert (binopdf (ones (0, 1, 2), NaN, .5), ones(0, 1, 2)) %!assert (binopdf (ones (0, 1, 2), 2, NaN), ones(0, 1, 2)) ## Test class of input preserved %!assert (binopdf (single ([x, NaN]), 2, 0.5), single ([y, NaN])) %!assert (binopdf ([x, NaN], single (2), 0.5), single ([y, NaN])) %!assert (binopdf ([x, NaN], 2, single (0.5)), single ([y, NaN])) ## Test input validation %!error binopdf () %!error binopdf (1) %!error binopdf (1, 2) %!error binopdf (1, 2, 3, 4) %!error ... %! binopdf (ones (3), ones (2), ones (2)) %!error ... %! binopdf (ones (2), ones (3), ones (2)) %!error ... %! binopdf (ones (2), ones (2), ones (3)) %!error binopdf (i, 2, 2) %!error binopdf (2, i, 2) %!error binopdf (2, 2, i) statistics-release-1.6.3/inst/dist_fun/binornd.m000066400000000000000000000173641456127120000217430ustar00rootroot00000000000000## Copyright (C) 2015 Michael Leitner ## Copyright (C) 2012 Rik Wehbring ## Copyright (C) 1995-2016 Kurt Hornik ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{r} =} binornd (@var{n}, @var{ps}) ## @deftypefnx {statistics} {@var{r} =} binornd (@var{n}, @var{ps}, @var{rows}) ## @deftypefnx {statistics} {@var{r} =} binornd (@var{n}, @var{ps}, @var{rows}, @var{cols}, @dots{}) ## @deftypefnx {statistics} {@var{r} =} binornd (@var{n}, @var{ps}, [@var{sz}]) ## ## Random arrays from the Binomial distribution. ## ## @code{@var{r} = binornd (@var{n}, @var{ps})} returns a matrix of random ## samples from the binomial distribution with parameters @var{n} and @var{ps}, ## where @var{n} is the number of trials and @var{ps} is the probability of ## success. The size of @var{r} is the common size of @var{n} and @var{ps}. ## A scalar input functions as a constant matrix of the same size as the other ## inputs. ## ## When called with a single size argument, @code{binornd} returns a square ## matrix with the dimension specified. When called with more than one scalar ## argument, the first two arguments are taken as the number of rows and columns ## and any further arguments specify additional matrix dimensions. The size may ## also be specified with a row vector of dimensions, @var{sz}. ## ## Further information about the binomial distribution can be found at ## @url{https://en.wikipedia.org/wiki/Binomial_distribution} ## ## @seealso{binocdf, binoinv, binopdf, binofit, binolike, binostat, binotest} ## @end deftypefn function r = binornd (n, ps, varargin) ## Check for valid number of input arguments if (nargin < 2) error ("binornd: function called with too few input arguments."); endif ## Check for common size of N and PS if (! isscalar (n) || ! isscalar (ps)) [retval, n, ps] = common_size (n, ps); if (retval > 0) error ("binornd: N and PS must be of common size or scalars."); endif endif ## Check for N and PS being reals if (iscomplex (n) || iscomplex (ps)) error ("binornd: N and PS must not be complex."); endif ## Parse and check SIZE arguments if (nargin == 2) sz = size (n); elseif (nargin == 3) if (isscalar (varargin{1}) && varargin{1} >= 0 ... && varargin{1} == fix (varargin{1})) sz = [varargin{1}, varargin{1}]; elseif (isrow (varargin{1}) && all (varargin{1} >= 0) ... && all (varargin{1} == fix (varargin{1}))) sz = varargin{1}; elseif error (strcat (["binornd: SZ must be a scalar or a row vector"], ... [" of non-negative integers."])); endif elseif (nargin > 3) posint = cellfun (@(x) (! isscalar (x) || x < 0 || x != fix (x)), varargin); if (any (posint)) error ("binornd: dimensions must be non-negative integers."); endif sz = [varargin{:}]; endif ## Check that parameters match requested dimensions in size if (! isscalar (n) && ! isequal (size (n), sz)) error ("binornd: N and PS must be scalars or of size SZ."); endif ## Check for class type if (isa (n, "single") || isa (ps, "single")) cls = "single"; else cls = "double"; endif ## Generate random sample from binomial distribution if (isscalar (n) && isscalar (ps)) if ((n > 0) && (n < Inf) && (n == fix (n)) && (ps >= 0) && (ps <= 1)) nel = prod (sz); tmp = rand (n, nel); r = sum (tmp < ps, 1); r = reshape (r, sz); if (strcmp (cls, "single")) r = single (r); endif elseif ((n == 0) && (ps >= 0) && (ps <= 1)) r = zeros (sz, cls); else r = NaN (sz, cls); endif else r = zeros (sz, cls); k = !(n >= 0) | !(n < Inf) | !(n == fix (n)) | !(ps >= 0) | !(ps <= 1); r(k) = NaN; k = (n > 0) & (n < Inf) & (n == fix (n)) & (ps >= 0) & (ps <= 1); if (any (k(:))) L = sum (k(:)); ind = repelems ((1 : L), [(1 : L); n(k)(:)'])'; p_ext = ps(k)(ind)(:); r(k) = accumarray (ind, rand (sum(n(k)(:)), 1) < p_ext); endif endif endfunction ## Test output %!assert (size (binornd (2, 1/2)), [1 1]) %!assert (size (binornd (2 * ones (2, 1), 1/2)), [2, 1]) %!assert (size (binornd (2 * ones (2, 2), 1/2)), [2, 2]) %!assert (size (binornd (2, 1/2 * ones (2, 1))), [2, 1]) %!assert (size (binornd (1, 1/2 * ones (2, 2))), [2, 2]) %!assert (size (binornd (ones (2, 1), 1)), [2, 1]) %!assert (size (binornd (ones (2, 2), 1)), [2, 2]) %!assert (size (binornd (2, 1/2, 3)), [3, 3]) %!assert (size (binornd (1, 1, [4, 1])), [4, 1]) %!assert (size (binornd (1, 1, 4, 1)), [4, 1]) %!assert (size (binornd (1, 1, 4, 1, 5)), [4, 1, 5]) %!assert (size (binornd (1, 1, 0, 1)), [0, 1]) %!assert (size (binornd (1, 1, 1, 0)), [1, 0]) %!assert (size (binornd (1, 1, 1, 2, 0, 5)), [1, 2, 0, 5]) ## Test class of input preserved %!assert (class (binornd (1, 1)), "double") %!assert (class (binornd (1, single (0))), "single") %!assert (class (binornd (1, single ([0, 0]))), "single") %!assert (class (binornd (1, single (1), 2)), "single") %!assert (class (binornd (1, single ([1, 1]), 1, 2)), "single") %!assert (class (binornd (single (1), 1, 2)), "single") %!assert (class (binornd (single ([1, 1]), 1, 1, 2)), "single") ## Test input validation %!error binornd () %!error binornd (1) %!error ... %! binornd (ones (3), ones (2)) %!error ... %! binornd (ones (2), ones (3)) %!error binornd (i, 2) %!error binornd (1, i) %!error ... %! binornd (1, 1/2, -1) %!error ... %! binornd (1, 1/2, 1.2) %!error ... %! binornd (1, 1/2, ones (2)) %!error ... %! binornd (1, 1/2, [2 -1 2]) %!error ... %! binornd (1, 1/2, [2 0 2.5]) %!error ... %! binornd (1, 1/2, 2, -1, 5) %!error ... %! binornd (1, 1/2, 2, 1.5, 5) %!error ... %! binornd (2, 1/2 * ones (2), 3) %!error ... %! binornd (2, 1/2 * ones (2), [3, 2]) %!error ... %! binornd (2, 1/2 * ones (2), 3, 2) %!error ... %! binornd (2 * ones (2), 1/2, 3) %!error ... %! binornd (2 * ones (2), 1/2, [3, 2]) %!error ... %! binornd (2 * ones (2), 1/2, 3, 2) statistics-release-1.6.3/inst/dist_fun/bisacdf.m000066400000000000000000000154331456127120000216760ustar00rootroot00000000000000## Copyright (C) 1995-2015 Kurt Hornik ## Copyright (C) 2016 Dag Lyberg ## Copyright (C) 2018 John Donoghue ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## Octave is free software; you can redistribute it and/or modify it ## under the terms of the GNU General Public License as published by ## the Free Software Foundation; either version 3 of the License, or (at ## your option) any later version. ## ## Octave is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with Octave; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{p} =} bisacdf (@var{x}, @var{beta}, @var{gamma}) ## @deftypefnx {statistics} {@var{p} =} bisacdf (@var{x}, @var{beta}, @var{gamma}, @qcode{"upper"}) ## ## Birnbaum-Saunders cumulative distribution function (CDF). ## ## For each element of @var{x}, compute the cumulative distribution function ## (CDF) of the Birnbaum-Saunders distribution with scale parameter @var{beta} ## and shape parameter @var{gamma}. The size of @var{p} is the common size of ## @var{x}, @var{beta} and @var{gamma}. A scalar input functions as a constant ## matrix of the same size as the other inputs. ## ## @code{@var{p} = bisacdf (@var{x}, @var{beta}, @var{gamma}, "upper")} ## computes the upper tail probability of the Birnbaum-Saunders distribution ## with parameters @var{beta} and @var{gamma}, at the values in @var{x}. ## ## Further information about the Birnbaum-Saunders distribution can be found at ## @url{https://en.wikipedia.org/wiki/Birnbaum%E2%80%93Saunders_distribution} ## ## @seealso{bisainv, bisapdf, bisarnd, bisafit, bisalike, bisastat} ## @end deftypefn function p = bisacdf (x, beta, gamma, uflag) ## Check for valid number of input arguments if (nargin < 3) error ("bisacdf: function called with too few input arguments."); endif ## Check for valid "upper" flag if (nargin > 3) if (! strcmpi (uflag, "upper")) error ("bisacdf: invalid argument for upper tail."); else uflag = true; endif else uflag = false; endif ## Check for common size of X, BETA, and GAMMA if (! isscalar (x) || ! isscalar (beta) || ! isscalar (gamma)) [retval, x, beta, gamma] = common_size (x, beta, gamma); if (retval > 0) error (strcat (["bisacdf: X, BETA, and GAMMA must be of"], ... [" common size or scalars."])); endif endif ## Check for X, BETA, and GAMMA being reals if (iscomplex (x) || iscomplex (beta) || iscomplex (gamma)) error ("bisacdf: X, BETA, and GAMMA must not be complex."); endif ## Check for class type if (isa (x, "single") || isa (beta, "single") || isa (gamma, "single")) p = zeros (size (x), "single"); else p = zeros (size (x)); endif ## Force NaNs for out of range parameters. k = isnan (x) | ! (beta > 0) | ! (beta < Inf) ... | ! (gamma > 0) | ! (gamma < Inf); p(k) = NaN; ## Find valid values in parameters and data k = (x > 0) & (x <= Inf) & (beta > 0) & (beta < Inf) ... & (gamma > 0) & (gamma < Inf); xk = x(k); ## Compute Birnbaum-Saunders CDF if (isscalar (beta) && isscalar (gamma)) if (uflag) z = (-sqrt (xk ./ beta) + sqrt (beta ./ xk)) ./ gamma; else z = (sqrt (xk ./ beta) - sqrt (beta ./ xk)) ./ gamma; endif p(k) = 0.5 * erfc (-z ./ sqrt (2)); else if (uflag) z = (-sqrt (xk ./ beta(k)) + sqrt (beta(k) ./ xk)) ./ gamma(k); else z = (sqrt (xk ./ beta(k)) - sqrt (beta(k) ./ xk)) ./ gamma(k); endif p(k) = 0.5 * erfc (-z ./ sqrt (2)); endif endfunction %!demo %! ## Plot various CDFs from the Birnbaum-Saunders distribution %! x = 0.01:0.01:4; %! p1 = bisacdf (x, 1, 0.5); %! p2 = bisacdf (x, 1, 1); %! p3 = bisacdf (x, 1, 2); %! p4 = bisacdf (x, 1, 5); %! p5 = bisacdf (x, 1, 10); %! plot (x, p1, "-b", x, p2, "-g", x, p3, "-r", x, p4, "-c", x, p5, "-m") %! grid on %! legend ({"β = 1, γ = 0.5", "β = 1, γ = 1", "β = 1, γ = 2", ... %! "β = 1, γ = 5", "β = 1, γ = 10"}, "location", "southeast") %! title ("Birnbaum-Saunders CDF") %! xlabel ("values in x") %! ylabel ("probability") %!demo %! ## Plot various CDFs from the Birnbaum-Saunders distribution %! x = 0.01:0.01:6; %! p1 = bisacdf (x, 1, 0.3); %! p2 = bisacdf (x, 2, 0.3); %! p3 = bisacdf (x, 1, 0.5); %! p4 = bisacdf (x, 3, 0.5); %! p5 = bisacdf (x, 5, 0.5); %! plot (x, p1, "-b", x, p2, "-g", x, p3, "-r", x, p4, "-c", x, p5, "-m") %! grid on %! legend ({"β = 1, γ = 0.3", "β = 2, γ = 0.3", "β = 1, γ = 0.5", ... %! "β = 3, γ = 0.5", "β = 5, γ = 0.5"}, "location", "southeast") %! title ("Birnbaum-Saunders CDF") %! xlabel ("values in x") %! ylabel ("probability") ## Test output %!shared x, y %! x = [-1, 0, 1, 2, Inf]; %! y = [0, 0, 1/2, 0.76024993890652337, 1]; %!assert (bisacdf (x, ones (1,5), ones (1,5)), y, eps) %!assert (bisacdf (x, 1, 1), y, eps) %!assert (bisacdf (x, 1, ones (1,5)), y, eps) %!assert (bisacdf (x, ones (1,5), 1), y, eps) %!assert (bisacdf (x, 1, 1), y, eps) %!assert (bisacdf (x, 1, [1, 1, NaN, 1, 1]), [y(1:2), NaN, y(4:5)], eps) %!assert (bisacdf (x, [1, 1, NaN, 1, 1], 1), [y(1:2), NaN, y(4:5)], eps) %!assert (bisacdf ([x, NaN], 1, 1), [y, NaN], eps) ## Test class of input preserved %!assert (bisacdf (single ([x, NaN]), 1, 1), single ([y, NaN]), eps ("single")) %!assert (bisacdf ([x, NaN], 1, single (1)), single ([y, NaN]), eps ("single")) %!assert (bisacdf ([x, NaN], single (1), 1), single ([y, NaN]), eps ("single")) ## Test input validation %!error bisacdf () %!error bisacdf (1) %!error bisacdf (1, 2) %!error ... %! bisacdf (1, 2, 3, 4, 5) %!error bisacdf (1, 2, 3, "tail") %!error bisacdf (1, 2, 3, 4) %!error ... %! bisacdf (ones (3), ones (2), ones(2)) %!error ... %! bisacdf (ones (2), ones (3), ones(2)) %!error ... %! bisacdf (ones (2), ones (2), ones(3)) %!error bisacdf (i, 4, 3) %!error bisacdf (1, i, 3) %!error bisacdf (1, 4, i) statistics-release-1.6.3/inst/dist_fun/bisainv.m000066400000000000000000000141131456127120000217300ustar00rootroot00000000000000## Copyright (C) 1995-2015 Kurt Hornik ## Copyright (C) 2016 Dag Lyberg ## Copyright (C) 2018 John Donoghue ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## Octave is free software; you can redistribute it and/or modify it ## under the terms of the GNU General Public License as published by ## the Free Software Foundation; either version 3 of the License, or (at ## your option) any later version. ## ## Octave is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with Octave; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{x} =} bisainv (@var{p}, @var{beta}, @var{gamma}) ## ## Inverse of the Birnbaum-Saunders cumulative distribution function (iCDF). ## ## For each element of @var{p}, compute the quantile (the inverse of the CDF) of ## the Birnbaum-Saunders distribution with scale parameter @var{beta} and shape ## parameter @var{gamma}. The size of @var{x} is the common size of @var{p}, ## @var{beta}, and @var{gamma}. A scalar input functions as a constant matrix ## of the same size as the other inputs. ## ## Further information about the Birnbaum-Saunders distribution can be found at ## @url{https://en.wikipedia.org/wiki/Birnbaum%E2%80%93Saunders_distribution} ## ## @seealso{bisainv, bisapdf, bisarnd, bisafit, bisalike, bisastat} ## @end deftypefn function x = bisainv (p, beta, gamma) ## Check for valid number of input arguments if (nargin < 3) error ("bisainv: function called with too few input arguments."); endif ## Check for common size of X, BETA, and GAMMA if (! isscalar (p) || ! isscalar (beta) || ! isscalar (gamma)) [retval, p, beta, gamma] = common_size (p, beta, gamma); if (retval > 0) error (strcat (["bisainv: P, BETA, and GAMMA must be of"], ... [" common size or scalars."])); endif endif ## Check for X, BETA, and GAMMA being reals if (iscomplex (p) || iscomplex (beta) || iscomplex (gamma)) error ("bisainv: P, BETA, and GAMMA must not be complex."); endif ## Check for class type if (isa (p, "single") || isa (beta, "single") || isa (gamma, "single")) x = zeros (size (p), "single"); else x = zeros (size (p)); endif ## Force NaNs for out of range parameters kn = isnan (p) | (p < 0) | (p > 1) | ! (beta > 0) | ! (beta < Inf) ... | ! (gamma > 0) | ! (gamma < Inf); x(kn) = NaN; ## Find valid values in parameters kv = (beta > 0) & (beta < Inf) & (gamma > 0) & (gamma < Inf); ## Handle edge cases k0 = (p == 0) & kv; x(k0) = 0; k1 = (p == 1) & kv; x(k1) = Inf; ## Handle all other valid cases k = (p > 0) & (p < 1) & kv; if (isscalar (beta) && isscalar (gamma)) z = -sqrt (2) .* erfcinv (2 .* p(k)) .* gamma; x(k) = 0.25 .* beta .* (z + sqrt (4 + z .^ 2)) .^ 2; else z = -sqrt (2) .* erfcinv (2 .* p(k)) .* gamma(k); x(k) = 0.25 .* beta(k) .* (z + sqrt (4 + z .^ 2)) .^ 2; endif endfunction %!demo %! ## Plot various iCDFs from the Birnbaum-Saunders distribution %! p = 0.001:0.001:0.999; %! x1 = bisainv (p, 1, 0.5); %! x2 = bisainv (p, 1, 1); %! x3 = bisainv (p, 1, 2); %! x4 = bisainv (p, 1, 5); %! x5 = bisainv (p, 1, 10); %! plot (p, x1, "-b", p, x2, "-g", p, x3, "-r", p, x4, "-c", p, x5, "-m") %! grid on %! ylim ([0, 10]) %! legend ({"β = 1, γ = 0.5", "β = 1, γ = 1", "β = 1, γ = 2", ... %! "β = 1, γ = 5", "β = 1, γ = 10"}, "location", "northwest") %! title ("Birnbaum-Saunders iCDF") %! xlabel ("probability") %! ylabel ("values in x") %!demo %! ## Plot various iCDFs from the Birnbaum-Saunders distribution %! p = 0.001:0.001:0.999; %! x1 = bisainv (p, 1, 0.3); %! x2 = bisainv (p, 2, 0.3); %! x3 = bisainv (p, 1, 0.5); %! x4 = bisainv (p, 3, 0.5); %! x5 = bisainv (p, 5, 0.5); %! plot (p, x1, "-b", p, x2, "-g", p, x3, "-r", p, x4, "-c", p, x5, "-m") %! grid on %! ylim ([0, 10]) %! legend ({"β = 1, γ = 0.3", "β = 2, γ = 0.3", "β = 1, γ = 0.5", ... %! "β = 3, γ = 0.5", "β = 5, γ = 0.5"}, "location", "northwest") %! title ("Birnbaum-Saunders iCDF") %! xlabel ("probability") %! ylabel ("values in x") ## Test output %!shared p, y, f %! f = @(p,b,c) (b * (c * norminv (p) + sqrt (4 + (c * norminv(p))^2))^2) / 4; %! p = [-1, 0, 1/4, 1/2, 1, 2]; %! y = [NaN, 0, f(1/4, 1, 1), 1, Inf, NaN]; %!assert (bisainv (p, ones (1,6), ones (1,6)), y) %!assert (bisainv (p, 1, ones (1,6)), y) %!assert (bisainv (p, ones (1,6), 1), y) %!assert (bisainv (p, 1, 1), y) %!assert (bisainv (p, 1, [1, 1, 1, NaN, 1, 1]), [y(1:3), NaN, y(5:6)]) %!assert (bisainv (p, [1, 1, 1, NaN, 1, 1], 1), [y(1:3), NaN, y(5:6)]) %!assert (bisainv ([p, NaN], 1, 1), [y, NaN]) ## Test class of input preserved %!assert (bisainv (single ([p, NaN]), 1, 1), single ([y, NaN]), eps ("single")) %!assert (bisainv ([p, NaN], 1, single (1)), single ([y, NaN]), eps ("single")) %!assert (bisainv ([p, NaN], single (1), 1), single ([y, NaN]), eps ("single")) ## Test input validation %!error bisainv () %!error bisainv (1) %!error bisainv (1, 2) %!error bisainv (1, 2, 3, 4) %!error ... %! bisainv (ones (3), ones (2), ones(2)) %!error ... %! bisainv (ones (2), ones (3), ones(2)) %!error ... %! bisainv (ones (2), ones (2), ones(3)) %!error bisainv (i, 4, 3) %!error bisainv (1, i, 3) %!error bisainv (1, 4, i) statistics-release-1.6.3/inst/dist_fun/bisapdf.m000066400000000000000000000140321456127120000217050ustar00rootroot00000000000000## Copyright (C) 2018 John Donoghue ## Copyright (C) 2016 Dag Lyberg ## Copyright (C) 1995-2015 Kurt Hornik ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## Octave is free software; you can redistribute it and/or modify it ## under the terms of the GNU General Public License as published by ## the Free Software Foundation; either version 3 of the License, or (at ## your option) any later version. ## ## Octave is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with Octave; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{y} =} bisapdf (@var{x}, @var{beta}, @var{gamma}) ## ## Birnbaum-Saunders probability density function (PDF). ## ## For each element of @var{x}, compute the probability density function (PDF) ## of the Birnbaum-Saunders distribution with scale parameter @var{beta} and ## shape parameter @var{gamma}. The size of @var{y} is the common size of ## @var{x}, @var{beta}, and @var{gamma}. A scalar input functions as a constant ## matrix of the same size as the other inputs. ## ## Further information about the Birnbaum-Saunders distribution can be found at ## @url{https://en.wikipedia.org/wiki/Birnbaum%E2%80%93Saunders_distribution} ## ## @seealso{bisacdf, bisapdf, bisarnd, bisafit, bisalike, bisastat} ## @end deftypefn function y = bisapdf (x, beta, gamma) ## Check for valid number of input arguments if (nargin < 3) error ("bisapdf: function called with too few input arguments."); endif ## Check for common size of X, BETA and GAMMA if (! isscalar (x) || ! isscalar (beta) || ! isscalar (gamma)) [retval, x, beta, gamma] = common_size (x, beta, gamma); if (retval > 0) error (strcat (["bisapdf: X, BETA, and GAMMA must be of"], ... [" common size or scalars."])); endif endif ## Check for X, BETA and GAMMA being reals if (iscomplex (x) || iscomplex (beta) || iscomplex (gamma)) error ("bisapdf: X, BETA, and GAMMA must not be complex."); endif ## Check for class type if (isa (x, "single") || isa (beta, "single") || isa (gamma, "single")) y = zeros (size (x), "single"); else y = zeros (size (x)); endif ## Force NaNs for out of range parameters. k = isnan (x) | ! (beta > 0) | ! (beta < Inf) ... | ! (gamma > 0) | ! (gamma < Inf); y(k) = NaN; ## Find valid values in parameters and data k = (x > 0) & (x < Inf) & (beta > 0) & (beta < Inf) ... & (gamma > 0) & (gamma < Inf); xk = x(k); if (isscalar (beta) && isscalar (gamma)) z = (sqrt (xk ./ beta) - sqrt (beta ./ xk)) ./ gamma; w = (sqrt (xk ./ beta) + sqrt (beta ./ xk)) ./ gamma; y(k) = (exp (-0.5 .* z .^ 2) ./ sqrt (2 .* pi)) .* w ./ (2.*xk); else z = (sqrt (xk ./ beta(k)) - sqrt (beta(k) ./ xk)) ./ gamma(k); w = (sqrt (xk ./ beta(k)) + sqrt (beta(k) ./ xk)) ./ gamma(k); y(k) = (exp (-0.5 .* z .^ 2) ./ sqrt (2 .* pi)) .* w ./ (2 .* xk); endif endfunction %!demo %! ## Plot various PDFs from the Birnbaum-Saunders distribution %! x = 0.01:0.01:4; %! y1 = bisapdf (x, 1, 0.5); %! y2 = bisapdf (x, 1, 1); %! y3 = bisapdf (x, 1, 2); %! y4 = bisapdf (x, 1, 5); %! y5 = bisapdf (x, 1, 10); %! plot (x, y1, "-b", x, y2, "-g", x, y3, "-r", x, y4, "-c", x, y5, "-m") %! grid on %! ylim ([0, 1.5]) %! legend ({"β = 1 ,γ = 0.5", "β = 1, γ = 1", "β = 1, γ = 2", ... %! "β = 1, γ = 5", "β = 1, γ = 10"}, "location", "northeast") %! title ("Birnbaum-Saunders PDF") %! xlabel ("values in x") %! ylabel ("density") %!demo %! ## Plot various PDFs from the Birnbaum-Saunders distribution %! x = 0.01:0.01:6; %! y1 = bisapdf (x, 1, 0.3); %! y2 = bisapdf (x, 2, 0.3); %! y3 = bisapdf (x, 1, 0.5); %! y4 = bisapdf (x, 3, 0.5); %! y5 = bisapdf (x, 5, 0.5); %! plot (x, y1, "-b", x, y2, "-g", x, y3, "-r", x, y4, "-c", x, y5, "-m") %! grid on %! ylim ([0, 1.5]) %! legend ({"β = 1, γ = 0.3", "β = 2, γ = 0.3", "β = 1, γ = 0.5", ... %! "β = 3, γ = 0.5", "β = 5, γ = 0.5"}, "location", "northeast") %! title ("Birnbaum-Saunders CDF") %! xlabel ("values in x") %! ylabel ("density") ## Test output %!shared x, y %! x = [-1, 0, 1, 2, Inf]; %! y = [0, 0, 0.3989422804014327, 0.1647717335503959, 0]; %!assert (bisapdf (x, ones (1,5), ones (1,5)), y, eps) %!assert (bisapdf (x, 1, 1), y, eps) %!assert (bisapdf (x, 1, ones (1,5)), y, eps) %!assert (bisapdf (x, ones (1,5), 1), y, eps) %!assert (bisapdf (x, 1, [1, 1, NaN, 1, 1]), [y(1:2), NaN, y(4:5)], eps) %!assert (bisapdf (x, [1, 1, NaN, 1, 1], 1), [y(1:2), NaN, y(4:5)], eps) %!assert (bisapdf ([x, NaN], 1, 1), [y, NaN], eps) ## Test class of input preserved %!assert (bisapdf (single ([x, NaN]), 1, 1), single ([y, NaN]), eps ("single")) %!assert (bisapdf ([x, NaN], 1, single (1)), single ([y, NaN]), eps ("single")) %!assert (bisapdf ([x, NaN], single (1), 1), single ([y, NaN]), eps ("single")) ## Test input validation %!error bisapdf () %!error bisapdf (1) %!error bisapdf (1, 2) %!error bisapdf (1, 2, 3, 4) %!error ... %! bisapdf (ones (3), ones (2), ones(2)) %!error ... %! bisapdf (ones (2), ones (3), ones(2)) %!error ... %! bisapdf (ones (2), ones (2), ones(3)) %!error bisapdf (i, 4, 3) %!error bisapdf (1, i, 3) %!error bisapdf (1, 4, i) statistics-release-1.6.3/inst/dist_fun/bisarnd.m000066400000000000000000000157071456127120000217310ustar00rootroot00000000000000## Copyright (C) 2018 John Donoghue ## Copyright (C) 2016 Dag Lyberg ## Copyright (C) 1995-2015 Kurt Hornik ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## Octave is free software; you can redistribute it and/or modify it ## under the terms of the GNU General Public License as published by ## the Free Software Foundation; either version 3 of the License, or (at ## your option) any later version. ## ## Octave is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with Octave; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{r} =} bisarnd (@var{beta}, @var{gamma}) ## @deftypefnx {statistics} {@var{r} =} bisarnd (@var{beta}, @var{gamma}, @var{rows}) ## @deftypefnx {statistics} {@var{r} =} bisarnd (@var{beta}, @var{gamma}, @var{rows}, @var{cols}, @dots{}) ## @deftypefnx {statistics} {@var{r} =} bisarnd (@var{beta}, @var{gamma}, [@var{sz}]) ## ## Random arrays from the Birnbaum-Saunders distribution. ## ## @code{@var{r} = bisarnd (@var{beta}, @var{gamma})} returns an array of ## random numbers chosen from the Birnbaum-Saunders distribution with scale ## parameter @var{beta} and shape parameter @var{gamma}. The size of @var{r} is ## the common size of @var{beta} and @var{gamma}. A scalar input functions as a ## constant matrix of the same size as the other inputs. ## ## When called with a single size argument, @code{bisarnd} returns a square ## matrix with the dimension specified. When called with more than one scalar ## argument, the first two arguments are taken as the number of rows and columns ## and any further arguments specify additional matrix dimensions. The size may ## also be specified with a row vector of dimensions, @var{sz}. ## ## Further information about the Birnbaum-Saunders distribution can be found at ## @url{https://en.wikipedia.org/wiki/Birnbaum%E2%80%93Saunders_distribution} ## ## @seealso{bisacdf, bisainv, bisapdf, bisafit, bisalike, bisastat} ## @end deftypefn function r = bisarnd (beta, gamma, varargin) ## Check for valid number of input arguments if (nargin < 2) error ("bisarnd: function called with too few input arguments."); endif ## Check for common size of BETA and GAMMA if (! isscalar (beta) || ! isscalar (gamma)) [retval, beta, gamma] = common_size (beta, gamma); if (retval > 0) error ("bisarnd: BETA and GAMMA must be of common size or scalars."); endif endif ## Check for BETA and GAMMA being reals if (iscomplex (beta) || iscomplex (gamma)) error ("bisarnd: BETA and GAMMA must not be complex."); endif ## Parse and check SIZE arguments if (nargin == 2) sz = size (beta); elseif (nargin == 3) if (isscalar (varargin{1}) && varargin{1} >= 0 ... && varargin{1} == fix (varargin{1})) sz = [varargin{1}, varargin{1}]; elseif (isrow (varargin{1}) && all (varargin{1} >= 0) ... && all (varargin{1} == fix (varargin{1}))) sz = varargin{1}; elseif error (strcat (["bisarnd: SZ must be a scalar or a row vector"], ... [" of non-negative integers."])); endif elseif (nargin > 3) posint = cellfun (@(x) (! isscalar (x) || x < 0 || x != fix (x)), varargin); if (any (posint)) error ("bisarnd: dimensions must be non-negative integers."); endif sz = [varargin{:}]; endif ## Check that parameters match requested dimensions in size if (! isscalar (beta) && ! isequal (size (beta), sz)) error ("bisarnd: BETA and GAMMA must be scalars or of size SZ."); endif ## Check for class type if (isa (beta, "single") || isa (gamma, "single")) cls = "single"; else cls = "double"; endif ## Generate random sample from Birnbaum-Saunders distribution if (isscalar (beta) && isscalar (gamma)) if ((beta > 0) && (beta < Inf) && (gamma > 0) && (gamma < Inf)) r = rand (sz, cls); y = gamma * norminv (r); r = beta * (y + sqrt (4 + y .^ 2)) .^ 2 / 4; else r = NaN (sz, cls); endif else r = NaN (sz, cls); k = (beta > 0) & (beta < Inf) & (gamma > 0) & (gamma < Inf); r(k) = rand (sum (k(:)),1); y = gamma(k) .* norminv (r(k)); r(k) = beta(k) .* (y + sqrt (4 + y.^2)).^2 / 4; endif endfunction ## Test output %!assert (size (bisarnd (1, 1)), [1 1]) %!assert (size (bisarnd (1, ones (2,1))), [2, 1]) %!assert (size (bisarnd (1, ones (2,2))), [2, 2]) %!assert (size (bisarnd (ones (2,1), 1)), [2, 1]) %!assert (size (bisarnd (ones (2,2), 1)), [2, 2]) %!assert (size (bisarnd (1, 1, 3)), [3, 3]) %!assert (size (bisarnd (1, 1, [4, 1])), [4, 1]) %!assert (size (bisarnd (1, 1, 4, 1)), [4, 1]) %!assert (size (bisarnd (1, 1, 4, 1, 5)), [4, 1, 5]) %!assert (size (bisarnd (1, 1, 0, 1)), [0, 1]) %!assert (size (bisarnd (1, 1, 1, 0)), [1, 0]) %!assert (size (bisarnd (1, 1, 1, 2, 0, 5)), [1, 2, 0, 5]) ## Test class of input preserved %!assert (class (bisarnd (1, 1)), "double") %!assert (class (bisarnd (1, single (1))), "single") %!assert (class (bisarnd (1, single ([1, 1]))), "single") %!assert (class (bisarnd (single (1), 1)), "single") %!assert (class (bisarnd (single ([1, 1]), 1)), "single") ## Test input validation %!error bisarnd () %!error bisarnd (1) %!error ... %! bisarnd (ones (3), ones (2)) %!error ... %! bisarnd (ones (2), ones (3)) %!error bisarnd (i, 2, 3) %!error bisarnd (1, i, 3) %!error ... %! bisarnd (1, 2, -1) %!error ... %! bisarnd (1, 2, 1.2) %!error ... %! bisarnd (1, 2, ones (2)) %!error ... %! bisarnd (1, 2, [2 -1 2]) %!error ... %! bisarnd (1, 2, [2 0 2.5]) %!error ... %! bisarnd (1, 2, 2, -1, 5) %!error ... %! bisarnd (1, 2, 2, 1.5, 5) %!error ... %! bisarnd (2, ones (2), 3) %!error ... %! bisarnd (2, ones (2), [3, 2]) %!error ... %! bisarnd (2, ones (2), 3, 2) statistics-release-1.6.3/inst/dist_fun/burrcdf.m000066400000000000000000000151201456127120000217230ustar00rootroot00000000000000## Copyright (C) 1995-2015 Kurt Hornik ## Copyright (C) 2016 Dag Lyberg ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## Octave is free software; you can redistribute it and/or modify it ## under the terms of the GNU General Public License as published by ## the Free Software Foundation; either version 3 of the License, or (at ## your option) any later version. ## ## Octave is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with Octave; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{p} =} burrcdf (@var{x}, @var{lambda}, @var{c}, @var{k}) ## @deftypefnx {statistics} {@var{p} =} burrcdf (@var{x}, @var{lambda}, @var{c}, @var{k}, @qcode{"upper"}) ## ## Burr type XII cumulative distribution function (CDF). ## ## For each element of @var{x}, compute the cumulative distribution function ## (CDF) of the Burr type XII distribution with scale parameter @var{lambda}, ## first shape parameter @var{c}, and second shape parameter @var{k}. The size ## of @var{p} is the common size of @var{x}, @var{lambda}, @var{c}, and @var{k}. ## A scalar input functions as a constant matrix of the same size as the other ## inputs. ## ## @code{@var{p} = burrcdf (@var{x}, @var{beta}, @var{gamma}, "upper")} ## computes the upper tail probability of the Birnbaum-Saunders distribution ## with parameters @var{beta} and @var{gamma}, at the values in @var{x}. ## ## Further information about the Burr distribution can be found at ## @url{https://en.wikipedia.org/wiki/Burr_distribution} ## ## @seealso{burrinv, burrpdf, burrrnd, burrfit, burrlike} ## @end deftypefn function p = burrcdf (x, lambda, c, k, uflag) ## Check for valid number of input arguments if (nargin < 4) error ("burrcdf: function called with too few input arguments."); endif ## Check for valid "upper" flag if (nargin > 4) if (! strcmpi (uflag, "upper")) error ("burrcdf: invalid argument for upper tail."); else uflag = true; endif else uflag = false; endif ## Check for common size of X, LANBDA, C and K if (! isscalar (x) || ! isscalar (lambda) || ! isscalar (c) || ! isscalar (k)) [retval, x, lambda, c, k] = common_size (x, lambda, c, k); if (retval > 0) error ("burrcdf: X, LAMBDA, C, and K must be of common size or scalars."); endif endif ## Check for X, LANBDA, C and K being reals if (iscomplex (x) || iscomplex (lambda) || iscomplex (c) || iscomplex (k)) error ("burrcdf: X, LAMBDA, C, and K must not be complex."); endif ## Check for class type if (isa (x, "single") || isa (lambda, "single") || isa (c, "single") ... || isa (k, "single")) p = zeros (size (x), "single"); else p = zeros (size (x)); endif ## Force NaNs for out of range parameters j = isnan (x) | ! (lambda > 0) | ! (c > 0) | ! (k > 0); p(j) = NaN; ## Find valid values in parameters and data j = (x > 0) & (lambda > 0) & (lambda < Inf) & (c > 0) & (c < Inf) ... & (k > 0) & (k < Inf); ## Compute Burr CDF if (isscalar (lambda) && isscalar(c) && isscalar(k)) if (uflag) p(j) = (1 + (x(j) / lambda) .^ c) .^ (-k); else p(j) = 1 - (1 + (x(j) / lambda) .^ c) .^ (-k); endif else if (uflag) p(j) = (1 + (x(j) ./ lambda(j)) .^ c(j)) .^ (-k(j)); else p(j) = 1 - (1 + (x(j) ./ lambda(j)) .^ c(j)) .^ (-k(j)); endif endif endfunction %!demo %! ## Plot various CDFs from the Burr type XII distribution %! x = 0.001:0.001:5; %! p1 = burrcdf (x, 1, 1, 1); %! p2 = burrcdf (x, 1, 1, 2); %! p3 = burrcdf (x, 1, 1, 3); %! p4 = burrcdf (x, 1, 2, 1); %! p5 = burrcdf (x, 1, 3, 1); %! p6 = burrcdf (x, 1, 0.5, 2); %! plot (x, p1, "-b", x, p2, "-g", x, p3, "-r", ... %! x, p4, "-c", x, p5, "-m", x, p6, "-k") %! grid on %! legend ({"λ = 1, c = 1, k = 1", "λ = 1, c = 1, k = 2", ... %! "λ = 1, c = 1, k = 3", "λ = 1, c = 2, k = 1", ... %! "λ = 1, c = 3, k = 1", "λ = 1, c = 0.5, k = 2"}, ... %! "location", "southeast") %! title ("Burr type XII CDF") %! xlabel ("values in x") %! ylabel ("probability") ## Test output %!shared x, y %! x = [-1, 0, 1, 2, Inf]; %! y = [0, 0, 1/2, 2/3, 1]; %!assert (burrcdf (x, ones(1,5), ones (1,5), ones (1,5)), y, eps) %!assert (burrcdf (x, 1, 1, 1), y, eps) %!assert (burrcdf (x, [1, 1, NaN, 1, 1], 1, 1), [y(1:2), NaN, y(4:5)], eps) %!assert (burrcdf (x, 1, [1, 1, NaN, 1, 1], 1), [y(1:2), NaN, y(4:5)], eps) %!assert (burrcdf (x, 1, 1, [1, 1, NaN, 1, 1]), [y(1:2), NaN, y(4:5)], eps) %!assert (burrcdf ([x, NaN], 1, 1, 1), [y, NaN], eps) ## Test class of input preserved %!assert (burrcdf (single ([x, NaN]), 1, 1, 1), single ([y, NaN]), eps("single")) %!assert (burrcdf ([x, NaN], single (1), 1, 1), single ([y, NaN]), eps("single")) %!assert (burrcdf ([x, NaN], 1, single (1), 1), single ([y, NaN]), eps("single")) %!assert (burrcdf ([x, NaN], 1, 1, single (1)), single ([y, NaN]), eps("single")) ## Test input validation %!error burrcdf () %!error burrcdf (1) %!error burrcdf (1, 2) %!error burrcdf (1, 2, 3) %!error ... %! burrcdf (1, 2, 3, 4, 5, 6) %!error burrcdf (1, 2, 3, 4, "tail") %!error burrcdf (1, 2, 3, 4, 5) %!error ... %! burrcdf (ones (3), ones (2), ones(2), ones(2)) %!error ... %! burrcdf (ones (2), ones (3), ones(2), ones(2)) %!error ... %! burrcdf (ones (2), ones (2), ones(3), ones(2)) %!error ... %! burrcdf (ones (2), ones (2), ones(2), ones(3)) %!error burrcdf (i, 2, 3, 4) %!error burrcdf (1, i, 3, 4) %!error burrcdf (1, 2, i, 4) %!error burrcdf (1, 2, 3, i) statistics-release-1.6.3/inst/dist_fun/burrinv.m000066400000000000000000000136601456127120000217720ustar00rootroot00000000000000## Copyright (C) 1995-2015 Kurt Hornik ## Copyright (C) 2016 Dag Lyberg ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## Octave is free software; you can redistribute it and/or modify it ## under the terms of the GNU General Public License as published by ## the Free Software Foundation; either version 3 of the License, or (at ## your option) any later version. ## ## Octave is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with Octave; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{x} =} burrinv (@var{p}, @var{lambda}, @var{c}, @var{k}) ## ## Inverse of the Burr type XII cumulative distribution function (iCDF). ## ## For each element of @var{p}, compute the quantile (the inverse of the CDF) of ## the Burr type XII distribution with scale parameter @var{lambda}, first shape ## parameter @var{c}, and second shape parameter @var{k}. The size of @var{x} ## is the common size of @var{p}, @var{lambda}, @var{c}, and @var{k}. A scalar ## input functions as a constant matrix of the same size as the other inputs. ## ## Further information about the Burr distribution can be found at ## @url{https://en.wikipedia.org/wiki/Burr_distribution} ## ## @seealso{burrcdf, burrpdf, burrrnd, burrfit, burrlike} ## @end deftypefn function x = burrinv (p, lambda, c, k) ## Check for valid number of input arguments if (nargin < 4) error ("burrinv: function called with too few input arguments."); endif ## Check for common size of P, LANBDA, C and K if (! isscalar (p) || ! isscalar (lambda) || ! isscalar (c) || ! isscalar (k)) [retval, p, lambda, c, k] = common_size (p, lambda, c, k); if (retval > 0) error ("burrinv: P, LAMBDA, C, and K must be of common size or scalars."); endif endif ## Check for P, LANBDA, C and K being reals if (iscomplex (p) || iscomplex (lambda) || iscomplex (c) || iscomplex (k)) error ("burrinv: P, LAMBDA, C, and K must not be complex."); endif ## Check for class type if (isa (p, "single") || isa (lambda, "single") || isa (c, "single") ... || isa (k, "single")) x = zeros (size (p), "single"); else x = zeros (size (p)); endif ## Force NaNs for out of range parameters j = isnan (p) | (p < 0) | (p > 1) | ! (lambda > 0) | ! (c > 0) | ! (k > 0); x(j) = NaN; ## Handle edge cases j = (p == 1) & (lambda > 0) & (lambda < Inf) & (c > 0) & (c < Inf) ... & (k > 0) & (k < Inf); x(j) = Inf; ## Handle all other valid cases j = (0 < p) & (p < 1) & (0 < lambda) & (lambda < Inf) & (0 < c) & (c < Inf) ... & (0 < k) & (k < Inf); if (isscalar (lambda) && isscalar(c) && isscalar(k)) x(j) = ((1 - p(j) / lambda).^(-1 / k) - 1).^(1 / c) ; else x(j) = ((1 - p(j) ./ lambda(j)).^(-1 ./ k(j)) - 1).^(1 ./ c(j)) ; endif endfunction %!demo %! ## Plot various iCDFs from the Burr type XII distribution %! p = 0.001:0.001:0.999; %! x1 = burrinv (p, 1, 1, 1); %! x2 = burrinv (p, 1, 1, 2); %! x3 = burrinv (p, 1, 1, 3); %! x4 = burrinv (p, 1, 2, 1); %! x5 = burrinv (p, 1, 3, 1); %! x6 = burrinv (p, 1, 0.5, 2); %! plot (p, x1, "-b", p, x2, "-g", p, x3, "-r", ... %! p, x4, "-c", p, x5, "-m", p, x6, "-k") %! grid on %! ylim ([0, 5]) %! legend ({"λ = 1, c = 1, k = 1", "λ = 1, c = 1, k = 2", ... %! "λ = 1, c = 1, k = 3", "λ = 1, c = 2, k = 1", ... %! "λ = 1, c = 3, k = 1", "λ = 1, c = 0.5, k = 2"}, ... %! "location", "northwest") %! title ("Burr type XII iCDF") %! xlabel ("probability") %! ylabel ("values in x") ## Test output %!shared p, y %! p = [-Inf, -1, 0, 1/2, 1, 2, Inf]; %! y = [NaN, NaN, 0, 1 , Inf, NaN, NaN]; %!assert (burrinv (p, ones (1,7), ones (1,7), ones(1,7)), y, eps) %!assert (burrinv (p, 1, 1, 1), y, eps) %!assert (burrinv (p, [1, 1, 1, NaN, 1, 1, 1], 1, 1), [y(1:3), NaN, y(5:7)], eps) %!assert (burrinv (p, 1, [1, 1, 1, NaN, 1, 1, 1], 1), [y(1:3), NaN, y(5:7)], eps) %!assert (burrinv (p, 1, 1, [1, 1, 1, NaN, 1, 1, 1]), [y(1:3), NaN, y(5:7)], eps) %!assert (burrinv ([p, NaN], 1, 1, 1), [y, NaN], eps) ## Test class of input preserved %!assert (burrinv (single ([p, NaN]), 1, 1, 1), single ([y, NaN]), eps("single")) %!assert (burrinv ([p, NaN], single (1), 1, 1), single ([y, NaN]), eps("single")) %!assert (burrinv ([p, NaN], 1, single (1), 1), single ([y, NaN]), eps("single")) %!assert (burrinv ([p, NaN], 1, 1, single (1)), single ([y, NaN]), eps("single")) ## Test input validation %!error burrinv () %!error burrinv (1) %!error burrinv (1, 2) %!error burrinv (1, 2, 3) %!error ... %! burrinv (1, 2, 3, 4, 5) %!error ... %! burrinv (ones (3), ones (2), ones(2), ones(2)) %!error ... %! burrinv (ones (2), ones (3), ones(2), ones(2)) %!error ... %! burrinv (ones (2), ones (2), ones(3), ones(2)) %!error ... %! burrinv (ones (2), ones (2), ones(2), ones(3)) %!error burrinv (i, 2, 3, 4) %!error burrinv (1, i, 3, 4) %!error burrinv (1, 2, i, 4) %!error burrinv (1, 2, 3, i) statistics-release-1.6.3/inst/dist_fun/burrpdf.m000066400000000000000000000134031456127120000217420ustar00rootroot00000000000000## Copyright (C) 1995-2015 Kurt Hornik ## Copyright (C) 2016 Dag Lyberg ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## Octave is free software; you can redistribute it and/or modify it ## under the terms of the GNU General Public License as published by ## the Free Software Foundation; either version 3 of the License, or (at ## your option) any later version. ## ## Octave is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with Octave; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{y} =} burrpdf (@var{x}, @var{lambda}, @var{c}, @var{k}) ## ## Burr type XII probability density function (PDF). ## ## For each element of @var{x}, compute the probability density function (PDF) ## of the Burr type XII distribution with with scale parameter @var{lambda}, ## first shape parameter @var{c}, and second shape parameter @var{k}. The size ## of @var{y} is the common size of @var{x}, @var{lambda}, @var{c}, and @var{k}. ## A scalar input functions as a constant matrix of the same size as the other ## inputs. ## ## Further information about the Burr distribution can be found at ## @url{https://en.wikipedia.org/wiki/Burr_distribution} ## ## @seealso{burrcdf, burrinv, burrrnd, burrfit, burrlike} ## @end deftypefn function y = burrpdf (x, lambda, c, k) ## Check for valid number of input arguments if (nargin < 4) error ("burrpdf: function called with too few input arguments."); endif ## Check for common size of X, LANBDA, C and K if (! isscalar (x) || ! isscalar (lambda) || ! isscalar (c) || ! isscalar (k)) [retval, x, lambda, c, k] = common_size (x, lambda, c, k); if (retval > 0) error ("burrpdf: X, LAMBDA, C, and K must be of common size or scalars."); endif endif ## Check for X, LANBDA, C and K being reals if (iscomplex (x) || iscomplex (lambda) || iscomplex (c) || iscomplex (k)) error ("burrpdf: X, LAMBDA, C, and K must not be complex."); endif ## Check for class type if (isa (x, "single") || isa (lambda, "single") ... || isa (c, "single") || isa (k, "single")) y = zeros (size (x), "single"); else y = zeros (size (x)); endif ## Force NaNs for out of range parameters j = isnan (x) | ! (lambda > 0) | ! (c > 0) | ! (k > 0); y(j) = NaN; ## Find valid values in parameters and data j = (x > 0) & (0 < lambda) & (lambda < Inf) & (0 < c) & (c < Inf) ... & (0 < k) & (k < Inf); ## Compute Burr PDF if (isscalar (lambda) && isscalar (c) && isscalar(k)) y(j) = (c * k / lambda) .* (x(j) / lambda) .^ (c - 1) ./ ... (1 + (x(j) / lambda) .^ c) .^ (k + 1); else y(j) = (c(j) .* k(j) ./ lambda(j) ) .* x(j).^(c(j) - 1) ./ ... (1 + (x(j) ./ lambda(j) ) .^ c(j)) .^ (k(j) + 1); endif endfunction %!demo %! ## Plot various PDFs from the Burr type XII distribution %! x = 0.001:0.001:3; %! y1 = burrpdf (x, 1, 1, 1); %! y2 = burrpdf (x, 1, 1, 2); %! y3 = burrpdf (x, 1, 1, 3); %! y4 = burrpdf (x, 1, 2, 1); %! y5 = burrpdf (x, 1, 3, 1); %! y6 = burrpdf (x, 1, 0.5, 2); %! plot (x, y1, "-b", x, y2, "-g", x, y3, "-r", ... %! x, y4, "-c", x, y5, "-m", x, y6, "-k") %! grid on %! ylim ([0, 2]) %! legend ({"λ = 1, c = 1, k = 1", "λ = 1, c = 1, k = 2", ... %! "λ = 1, c = 1, k = 3", "λ = 1, c = 2, k = 1", ... %! "λ = 1, c = 3, k = 1", "λ = 1, c = 0.5, k = 2"}, ... %! "location", "northeast") %! title ("Burr type XII PDF") %! xlabel ("values in x") %! ylabel ("density") ## Test output %!shared x, y %! x = [-1, 0, 1, 2, Inf]; %! y = [0, 0, 1/4, 1/9, 0]; %!assert (burrpdf (x, ones(1,5), ones (1,5), ones (1,5)), y) %!assert (burrpdf (x, 1, 1, 1), y) %!assert (burrpdf (x, [1, 1, NaN, 1, 1], 1, 1), [y(1:2), NaN, y(4:5)]) %!assert (burrpdf (x, 1, [1, 1, NaN, 1, 1], 1), [y(1:2), NaN, y(4:5)]) %!assert (burrpdf (x, 1, 1, [1, 1, NaN, 1, 1]), [y(1:2), NaN, y(4:5)]) %!assert (burrpdf ([x, NaN], 1, 1, 1), [y, NaN]) ## Test class of input preserved %!assert (burrpdf (single ([x, NaN]), 1, 1, 1), single ([y, NaN])) %!assert (burrpdf ([x, NaN], single (1), 1, 1), single ([y, NaN])) %!assert (burrpdf ([x, NaN], 1, single (1), 1), single ([y, NaN])) %!assert (burrpdf ([x, NaN], 1, 1, single (1)), single ([y, NaN])) ## Test input validation %!error burrpdf () %!error burrpdf (1) %!error burrpdf (1, 2) %!error burrpdf (1, 2, 3) %!error ... %! burrpdf (1, 2, 3, 4, 5) %!error ... %! burrpdf (ones (3), ones (2), ones(2), ones(2)) %!error ... %! burrpdf (ones (2), ones (3), ones(2), ones(2)) %!error ... %! burrpdf (ones (2), ones (2), ones(3), ones(2)) %!error ... %! burrpdf (ones (2), ones (2), ones(2), ones(3)) %!error burrpdf (i, 2, 3, 4) %!error burrpdf (1, i, 3, 4) %!error burrpdf (1, 2, i, 4) %!error burrpdf (1, 2, 3, i) statistics-release-1.6.3/inst/dist_fun/burrrnd.m000066400000000000000000000160561456127120000217630ustar00rootroot00000000000000## Copyright (C) 1995-2015 Kurt Hornik ## Copyright (C) 2016 Dag Lyberg ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## Octave is free software; you can redistribute it and/or modify it ## under the terms of the GNU General Public License as published by ## the Free Software Foundation; either version 3 of the License, or (at ## your option) any later version. ## ## Octave is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR LAMBDA PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with Octave; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{r} =} burrrnd (@var{lambda}, @var{c}, @var{k}) ## @deftypefnx {statistics} {@var{r} =} burrrnd (@var{lambda}, @var{c}, @var{k}, @var{rows}) ## @deftypefnx {statistics} {@var{r} =} burrrnd (@var{lambda}, @var{c}, @var{k}, @var{rows}, @var{cols}, @dots{}) ## @deftypefnx {statistics} {@var{r} =} burrrnd (@var{lambda}, @var{c}, @var{k}, [@var{sz}]) ## ## Random arrays from the Burr type XII distribution. ## ## @code{@var{r} = burrrnd (@var{lambda}, @var{c}, @var{k})} returns an array of ## random numbers chosen from the Burr type XII distribution with scale ## parameter @var{lambda}, first shape parameter @var{c}, and second shape ## parameter@var{c}, and @var{k}. The size of @var{r} is the common size of ## @var{lambda}, @var{c}, and @var{k}. LAMBDA scalar input functions as a ## constant matrix of the same size as the other inputs. ## ## When called with a single size argument, @code{burrrnd} returns a square ## matrix with the dimension specified. When called with more than one scalar ## argument, the first two arguments are taken as the number of rows and columns ## and any further arguments specify additional matrix dimensions. The size may ## also be specified with a row vector of dimensions, @var{sz}. ## ## Further information about the Burr distribution can be found at ## @url{https://en.wikipedia.org/wiki/Burr_distribution} ## ## @seealso{burrcdf, burrinv, burrpdf, burrfit, burrlike} ## @end deftypefn function r = burrrnd (lambda, c, k, varargin) ## Check for valid number of input arguments if (nargin < 3) error ("burrrnd: function called with too few input arguments."); endif ## Check for common size of LAMBDA, C, and K if (! isscalar (lambda) || ! isscalar (c) || ! isscalar (k)) [retval, lambda, c, k] = common_size (lambda, c, k); if (retval > 0) error ("burrrnd: LAMBDA, C, and K must be of common size or scalars."); endif endif ## Check for LAMBDA, C, and K being reals if (iscomplex (lambda) || iscomplex (c) || iscomplex (k)) error ("burrrnd: LAMBDA, C, and K must not be complex."); endif ## Parse and check SIZE arguments if (nargin == 3) sz = size (lambda); elseif (nargin == 4) if (isscalar (varargin{1}) && varargin{1} >= 0 ... && varargin{1} == fix (varargin{1})) sz = [varargin{1}, varargin{1}]; elseif (isrow (varargin{1}) && all (varargin{1} >= 0) ... && all (varargin{1} == fix (varargin{1}))) sz = varargin{1}; elseif error (strcat (["burrrnd: SZ must be a scalar or a row vector"], ... [" of non-negative integers."])); endif elseif (nargin > 4) posint = cellfun (@(x) (! isscalar (x) || x < 0 || x != fix (x)), varargin); if (any (posint)) error ("burrrnd: dimensions must be non-negative integers."); endif sz = [varargin{:}]; endif ## Check that parameters match requested dimensions in size if (! isscalar (lambda) && ! isequal (size (lambda), sz)) error ("burrrnd: LAMBDA, C, and K must be scalars or of size SZ."); endif ## Check for class type if (isa (lambda, "single") || isa (c, "single") || isa (k, "single")) cls = "single"; else cls = "double"; endif ## Generate random sample from Burr type XII distribution lambda(lambda <= 0) = NaN; c(c <= 0) = NaN; k(k <= 0) = NaN; r = lambda .* (((1 - rand (sz, cls)) .^ (-(1./k))) - 1) .^ (1./c); endfunction ## Test output %!assert (size (burrrnd (1, 1, 1)), [1 1]) %!assert (size (burrrnd (ones (2,1), 1, 1)), [2, 1]) %!assert (size (burrrnd (ones (2,2), 1, 1)), [2, 2]) %!assert (size (burrrnd (1, ones (2,1), 1)), [2, 1]) %!assert (size (burrrnd (1, ones (2,2), 1)), [2, 2]) %!assert (size (burrrnd (1, 1, ones (2,1))), [2, 1]) %!assert (size (burrrnd (1, 1, ones (2,2))), [2, 2]) %!assert (size (burrrnd (1, 1, 1, 3)), [3, 3]) %!assert (size (burrrnd (1, 1, 1, [4 1])), [4, 1]) %!assert (size (burrrnd (1, 1, 1, 4, 1)), [4, 1]) ## Test class of input preserved %!assert (class (burrrnd (1,1,1)), "double") %!assert (class (burrrnd (single (1),1,1)), "single") %!assert (class (burrrnd (single ([1 1]),1,1)), "single") %!assert (class (burrrnd (1,single (1),1)), "single") %!assert (class (burrrnd (1,single ([1 1]),1)), "single") %!assert (class (burrrnd (1,1,single (1))), "single") %!assert (class (burrrnd (1,1,single ([1 1]))), "single") ## Test input validation %!error burrrnd () %!error burrrnd (1) %!error burrrnd (1, 2) %!error ... %! burrrnd (ones (3), ones (2), ones (2)) %!error ... %! burrrnd (ones (2), ones (3), ones (2)) %!error ... %! burrrnd (ones (2), ones (2), ones (3)) %!error burrrnd (i, 2, 3) %!error burrrnd (1, i, 3) %!error burrrnd (1, 2, i) %!error ... %! burrrnd (1, 2, 3, -1) %!error ... %! burrrnd (1, 2, 3, 1.2) %!error ... %! burrrnd (1, 2, 3, ones (2)) %!error ... %! burrrnd (1, 2, 3, [2 -1 2]) %!error ... %! burrrnd (1, 2, 3, [2 0 2.5]) %!error ... %! burrrnd (1, 2, 3, 2, -1, 5) %!error ... %! burrrnd (1, 2, 3, 2, 1.5, 5) %!error ... %! burrrnd (2, ones (2), 2, 3) %!error ... %! burrrnd (2, ones (2), 2, [3, 2]) %!error ... %! burrrnd (2, ones (2), 2, 3, 2) statistics-release-1.6.3/inst/dist_fun/bvncdf.m000066400000000000000000000204351456127120000215430ustar00rootroot00000000000000## Copyright (C) 2022-2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{p} =} bvncdf (@var{x}, @var{mu}, @var{sigma}) ## @deftypefnx {statistics} {@var{p} =} bvncdf (@var{x}, [], @var{sigma}) ## ## Bivariate normal cumulative distribution function (CDF). ## ## @code{@var{p} = bvncdf (@var{x}, @var{mu}, @var{sigma})} will compute the ## bivariate normal cumulative distribution function of @var{x} given a mean ## parameter @var{mu} and a scale parameter @var{sigma}. ## ## @itemize ## @item @var{x} must be an @math{Nx2} matrix with each variable as a column ## vector. ## @item @var{mu} can be either a scalar (common mean) or a two-element row ## vector (each element corresponds to a variable). If empty, a zero mean is ## assumed. ## @item @var{sigma} can be a scalar (common variance) or a @math{2x2} ## covariance matrix, which must be positive definite. ## @end itemize ## ## @seealso{mvncdf} ## @end deftypefn ## Code adapted from Thomas H. Jørgensen's work in BVNcdf.m function retrieved ## from https://www.tjeconomics.com/code/ function p = bvncdf (x, mu, sigma) ## Check input arguments and add defaults if (size (x, 2) != 2) error (strcat (["bvncdf: X must be an Nx2 matrix with each variable"], ... [" as a column vector."])); endif if (isempty (mu)) mu = [0, 0]; elseif (isscalar (mu)) mu = [mu, mu]; elseif (numel (mu) == 2); mu = mu(:)'; else error ("bvncdf: MU must be a scalar or a two-element vector."); endif if (numel (sigma) == 1) sigma = sigma * ones (2,2); sigma(1,1) = 1; sigma(2,2) = 1; elseif (numel (sigma) != 4) error (strcat (["bvncdf: the covariance matrix must be either a"], ... [" scalar or a 2x2 matrix."])); endif ## Test for symmetric positive definite covariance matrix [~, err] = chol (sigma); if (err != 0) error (strcat (["bvncdf: the covariance matrix is not positive"], ... [" definite and/or symmetric."])); endif dh = (x(:,1) - mu(:,1)) / sqrt (sigma(1,1)); dk = (x(:,2) - mu(:,2)) / sqrt (sigma(2,2)); r = sigma(1,2) / sqrt (sigma(1,1) * sigma(2,2)); p = NaN (size (dh)); p(dh == Inf & dk == Inf) = 1; p(dk == Inf) = 0.5 * erfc (- dh(dk == Inf) / sqrt (2)); p(dh == Inf) = 0.5 * erfc (- dh(dk == Inf) / sqrt (2)); p(dh == -Inf | dk == -Inf) = 0; ind = (dh > -Inf & dh < Inf & dk > -Inf & dk < Inf); ## For p(x1 < dh, x2 < dk, r) if (sum (ind) > 0) p(ind) = calculate_bvncdf (-dh(ind), -dk(ind), r); endif endfunction function p = calculate_bvncdf (dh,dk,r) if (abs (r) < 0.3) lg = 3; ## Gauss Legendre points and weights, n = 6 w = [0.1713244923791705, 0.3607615730481384, 0.4679139345726904]; x = [0.9324695142031522, 0.6612093864662647, 0.2386191860831970]; elseif (abs (r) < 0.75) lg = 6; ## Gauss Legendre points and weights, n = 12 w = [.04717533638651177, 0.1069393259953183, 0.1600783285433464, ... 0.2031674267230659, 0.2334925365383547, 0.2491470458134029]; x = [0.9815606342467191, 0.9041172563704750, 0.7699026741943050, ... 0.5873179542866171, 0.3678314989981802, 0.1252334085114692]; else lg = 10; ## Gauss Legendre points and weights, n = 20 w = [.01761400713915212, .04060142980038694, .06267204833410906, ... .08327674157670475, 0.1019301198172404, 0.1181945319615184, ... 0.1316886384491766, 0.1420961093183821, 0.1491729864726037, ... 0.1527533871307259]; x = [0.9931285991850949, 0.9639719272779138, 0.9122344282513259, ... 0.8391169718222188, 0.7463319064601508, 0.6360536807265150, ... 0.5108670019508271, 0.3737060887154196, 0.2277858511416451, ... 0.07652652113349733]; endif dim1 = ones (size (dh, 1), 1); dim2 = ones (1, lg); hk = dh .* dk; bvn = dim1 * 0; phi_dh = 0.5 * erfc (dh / sqrt (2)); phi_dk = 0.5 * erfc (dk / sqrt (2)); if (abs(r) < 0.925) hs = (dh .* dh + dk .* dk) / 2; asr = asin (r); sn1 = sin (asr * (1 - x) / 2); sn2 = sin (asr * (1 + x) / 2); bvn = sum ((dim1 * w) .* exp (((dim1 * sn1) .* (hk * dim2) - ... hs * dim2) ./ (1 - dim1 * (sn1 .^ 2))) + ... (dim1 * w) .* exp (((dim1 * sn2) .* (hk * dim2) - ... hs * dim2) ./ (1 - dim1 * (sn2 .^ 2))), 2) * ... asr / (4 * pi) + phi_dh .* phi_dk; else twopi = 2 * pi; if r < 0 dk = -dk; hk = -hk; endif if abs(r) < 1 as = (1 - r) * (1 + r); a = sqrt (as); bs = (dh - dk) .^ 2; c = (4 - hk) / 8; d = (12 - hk) / 16; asr = - (bs ./ as + hk) / 2; ind = asr > -100; bvn(ind) = a * exp (asr(ind)) .* (1 - (c(ind) .* (bs(ind) - as)) ... .* (1 - d(ind) .* bs(ind) / 5) /3 ... + (c(ind) .* d(ind)) .* as .^ 2 / 5 ); ind = hk > -100; b = sqrt (bs); phi_ba = 0.5 * erfc ((b/a) / sqrt (2)); sp = sqrt (twopi) * phi_ba; bvn(ind) = bvn(ind) - (exp (-hk(ind) / 2) .* sp(ind)) ... .* b(ind) .* (1 - c(ind) .* bs(ind) ... .* (1 - d(ind) .* bs(ind) / 5) /3); a = a/2; for is = -1:2:1 xs = (a + a * is * x) .^ 2; rs = sqrt (1 - xs); asr1 = - ((bs * dim2) ./ (dim1 * xs) + hk * dim2) / 2; ind1 = (asr1 > -100); sp1 = (1 + (c * dim2) .* (dim1 * xs) .* ... (1 + (d * dim2) .* (dim1 * xs))); ep1 = exp (- (hk * dim2) .* (1 - dim1 * rs) ./ ... (2 * (1 + dim1 * rs))) ./ (dim1 * rs); bvn = bvn + sum (a .* (dim1 * w) .* exp (asr1 .* ind1) ... .* (ep1 .* ind1 - sp1 .* ind1), 2); endfor bvn = -bvn/twopi; endif if (r > 0) tmp = max (dh, dk); bvn = bvn + 0.5 * erfc (tmp / sqrt(2)); elseif (r < 0) phi_dh = 0.5 * erfc (dh / sqrt (2)); phi_dk = 0.5 * erfc (dk / sqrt (2)); bvn = - bvn + max (0, phi_dh - phi_dk); endif endif p = max (0, min (1, bvn)); endfunction %!demo %! mu = [1, -1]; %! sigma = [0.9, 0.4; 0.4, 0.3]; %! [X1, X2] = meshgrid (linspace (-1, 3, 25)', linspace (-3, 1, 25)'); %! x = [X1(:), X2(:)]; %! p = bvncdf (x, mu, sigma); %! Z = reshape (p, 25, 25); %! surf (X1, X2, Z); %! title ("Bivariate Normal Distribution"); %! ylabel "X1" %! xlabel "X2" ## Test output %!test %! mu = [1, -1]; %! sigma = [0.9, 0.4; 0.4, 0.3]; %! [X1,X2] = meshgrid (linspace (-1, 3, 25)', linspace (-3, 1, 25)'); %! x = [X1(:), X2(:)]; %! p = bvncdf (x, mu, sigma); %! p_out = [0.00011878988774500, 0.00034404112322371, ... %! 0.00087682502191813, 0.00195221905058185, ... %! 0.00378235566873474, 0.00638175749734415, ... %! 0.00943764224329656, 0.01239164888125426, ... %! 0.01472750274376648, 0.01623228313374828]'; %! assert (p([1:10]), p_out, 1e-16); %!test %! mu = [1, -1]; %! sigma = [0.9, 0.4; 0.4, 0.3]; %! [X1,X2] = meshgrid (linspace (-1, 3, 25)', linspace (-3, 1, 25)'); %! x = [X1(:), X2(:)]; %! p = bvncdf (x, mu, sigma); %! p_out = [0.8180695783608276, 0.8854485749482751, ... %! 0.9308108777385832, 0.9579855743025508, ... %! 0.9722897881414742, 0.9788150170059926, ... %! 0.9813597788804785, 0.9821977956568989, ... %! 0.9824283794464095, 0.9824809345614861]'; %! assert (p([616:625]), p_out, 2e-16); %!error bvncdf (randn (25,3), [], [1, 1; 1, 1]); %!error bvncdf (randn (25,2), [], [1, 1; 1, 1]); %!error bvncdf (randn (25,2), [], ones (3, 2)); statistics-release-1.6.3/inst/dist_fun/bvtcdf.m000066400000000000000000000143621456127120000215530ustar00rootroot00000000000000## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{p} =} bvtcdf (@var{x}, @var{rho}, @var{df}) ## @deftypefnx {statistics} {@var{p} =} bvtcdf (@var{x}, @var{rho}, @var{df}, @var{Tol}) ## ## Bivariate Student's t cumulative distribution function (CDF). ## ## @code{@var{p} = bvtcdf (@var{x}, @var{rho}, @var{df})} will compute the ## bivariate student's t cumulative distribution function of @var{x}, which must ## be an @math{Nx2} matrix, given a correlation coefficient @var{rho}, which ## must be a scalar, and @var{df} degrees of freedom, which can be a scalar or a ## vector of positive numbers commensurate with @var{x}. ## ## @var{Tol} is the tolerance for numerical integration and by default ## @code{@var{Tol} = 1e-8}. ## ## @seealso{mvtcdf} ## @end deftypefn function p = bvtcdf (x, rho, df, TolFun) narginchk (3,4); if (nargin < 4) TolFun = 1e-8; endif if (isa (x, "single") || isa (rho, "single") || isa (df, "single")) is_type = "single"; else is_type = "double"; endif if (abs (rho) < 1) largeNu = 1e4; general = ! (fix (df) == df & df < largeNu); if (isscalar (df)) if general p = generalDF (x, rho, repmat (df, size (x, 1), 1), TolFun); else p = integerDF (x, rho, df); endif else p = zeros (size (x, 1), 1, is_type); ## For large of non-integer df if (any (general)) p(general) = generalDF (x(general,:), rho, df(general), TolFun); endif ## For small integer df for i = find (! general(:)') p(i) = integerDF (x(i,:), rho, df(i)); endfor endif elseif (rho == 1) p = tcdf (min (x, [], 2), df); p(any (isnan( x), 2)) = NaN; else p = tcdf (x(:,1), df) - tcdf (-x(:,2), df); endif endfunction ## CDF for the bivariate t with integer degrees of freedom function p = integerDF (x, rho, df) x1 = x(:,1); x2 = x(:,2); tau = 1 - rho .^ 2; x1rx2 = x1 - rho * x2; x2rx1 = x2 - rho * x1; sx1r2 = sign (x1rx2); sx2r1 = sign (x2rx1); dfx1s = df + x1 .^ 2; dfx2s = df + x2 .^ 2; tdfx1x2 = tau * dfx2s ./ x1rx2 .^ 2; tdfx2x1 = tau * dfx1s ./ x2rx1 .^ 2; x_tdf12 = 1 ./ (1 + tdfx1x2); x_tdf21 = 1 ./ (1 + tdfx2x1); y_tdf12 = 1 ./ (1 + 1 ./ tdfx1x2); y_tdf21 = 1 ./ (1 + 1 ./ tdfx2x1); sqrtDF = sqrt (df); halfDF = df/2; if (fix (halfDF) == halfDF) # for even DF p1 = atan2 (sqrt (tau), -rho) ./ (2 * pi); c1 = x1 ./ (4 * sqrt (dfx1s)); c2 = x2 ./ (4 * sqrt (dfx2s)); beta12 = 2 *atan2 (sqrt (x_tdf12), sqrt (y_tdf12)) / pi; beta21 = 2 *atan2 (sqrt (x_tdf21), sqrt (y_tdf21)) / pi; p2 = (1 + sx1r2 .* beta12) .* c2 + (1 + sx2r1 .* beta21) .* c1; betaT12 = 2 * sqrt (x_tdf12 .* y_tdf12) / pi; betaT21 = 2 * sqrt (x_tdf21 .* y_tdf21) / pi; for j = 2:halfDF fact = df * (j - 1.5) / (j - 1); c2 = c2 .* fact ./ dfx2s; c1 = c1 .* fact ./ dfx1s; beta12 = beta12 + betaT12; beta21 = beta21 + betaT21; p2 = p2 + (1 + sx1r2 .* beta12) .* c2 + (1 + sx2r1 .* beta21) .* c1; fact = 2 * (j - 1) / (2 * (j - 1) + 1); betaT12 = fact * betaT12 .* y_tdf12; betaT21 = fact * betaT21 .* y_tdf21; endfor else # for odd DF x1x2p = x1.*x2; x1x2s = x1 + x2; t1 = sqrt (x1 .^ 2 - 2 * rho * x1x2p + x2 .^ 2 + tau * df); t2 = x1x2p + rho * df; t3 = x1x2p - df; p1 = atan2 (sqrtDF .* (-x1x2s .* t2 - t3 .* t1), ... t3 .* t2 - df .* x1x2s .* t1) ./ (2 * pi); p1 = p1 + (p1 < 0); p2 = 0; if (df > 1) c1 = sqrtDF .* x1 ./ (2 * pi .* dfx1s); c2 = sqrtDF .* x2 ./ (2 * pi .* dfx2s); betaT12 = sqrt (x_tdf12); betaT21 = sqrt (x_tdf21); beta12 = betaT12; beta21 = betaT21; p2 = (1 + sx1r2 .* beta12) .* c2 + (1 + sx2r1 .* beta21) .* c1; for j = 2:(halfDF - 0.5) fact = df * (j - 1) / (j - 0.5); c2 = fact * c2 ./ dfx2s; c1 = fact * c1 ./ dfx1s; fact = 1 - 0.5 / (j - 1); betaT12 = fact * betaT12 .* y_tdf12; betaT21 = fact * betaT21 .* y_tdf21; beta12 = beta12 + betaT12; beta21 = beta21 + betaT21; p2 = p2 + (1 + sx1r2 .* beta12) .* c2 + (1 + sx2r1 .* beta21) .* c1; endfor endif endif p = p1 + p2; ## Fix limit cases large = 1e10; p(x1 < -large | x2 < -large) = 0; p(x1 > large) = tcdf (x2(x1 > large), df); p(x2 > large) = tcdf (x1(x2 > large), df); endfunction ## CDF for the bivariate t with arbitrary degrees of freedom. function p = generalDF (x, rho, df, TolFun) n = size (x, 1); if (rho >= 0) p1 = tcdf (min (x, [], 2), df); p1(any (isnan (x), 2)) = NaN; else p1 = tcdf (x(:,1), df) - tcdf (-x(:,2), df); p1(p1 < 0) = 0; endif lo = asin (rho); hi = (sign (rho) + (rho == 0)) .* pi ./ 2; p2 = zeros (size (p1), class (rho)); for i = 1:n b1 = x(i,1); b2 = x(i,2); v = df(i); if (isfinite (b1) && isfinite (b2)) p2(i) = quadgk (@bvtIntegrand, lo, hi, "AbsTol", TolFun, "RelTol", 0); endif endfor p = p1 - p2 ./ (2 .* pi); function integrand = bvtIntegrand (theta) st = sin (theta); ct2 = cos (theta).^2; integrand = (1 ./ (1 + ((b1 * st - b2) .^ 2 ./ ct2 + b1 .^ 2) / v)) ... .^ (v / 2); endfunction endfunction ## Test output %!test %! x = [1, 2]; %! rho = [1, 0.5; 0.5, 1]; %! df = 4; %! assert (bvtcdf(x, rho(2), df), mvtcdf(x, rho, df), 1e-14); %!test %! x = [3, 2;2, 4;1, 5]; %! rho = [1, 0.5; 0.5, 1]; %! df = 4; %! assert (bvtcdf(x, rho(2), df), mvtcdf(x, rho, df), 1e-14); statistics-release-1.6.3/inst/dist_fun/cauchycdf.m000066400000000000000000000133361456127120000222340ustar00rootroot00000000000000## Copyright (C) 2012 Rik Wehbring ## Copyright (C) 1995-2016 Kurt Hornik ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{p} =} cauchycdf (@var{x}, @var{x0}, @var{gamma}) ## @deftypefnx {statistics} {@var{p} =} cauchycdf (@var{x}, @var{x0}, @var{gamma}, @qcode{"upper"}) ## ## Cauchy cumulative distribution function (CDF). ## ## For each element of @var{x}, compute the cumulative distribution function ## (CDF) of the Cauchy distribution with location parameter @var{x0} and scale ## parameter @var{gamma}. The size of @var{p} is the common size of @var{x}, ## @var{x0}, and @var{gamma}. A scalar input functions as a constant matrix of ## the same size as the other inputs. ## ## @code{@var{p} = cauchycdf (@var{x}, @var{x0}, @var{gamma}, "upper")} computes ## the upper tail probability of the Cauchy distribution with parameters ## @var{x0} and @var{gamma}, at the values in @var{x}. ## ## Further information about the Cauchy distribution can be found at ## @url{https://en.wikipedia.org/wiki/Cauchy_distribution} ## ## @seealso{cauchyinv, cauchypdf, cauchyrnd} ## @end deftypefn function p = cauchycdf (x, x0, gamma, uflag) ## Check for valid number of input arguments if (nargin < 3) error ("cauchycdf: function called with too few input arguments."); endif ## Check for valid "upper" flag if (nargin > 3) if (! strcmpi (uflag, "upper")) error ("cauchycdf: invalid argument for upper tail."); else uflag = true; endif else uflag = false; endif ## Check for common size of X, X0, and GAMMA if (! isscalar (x) || ! isscalar (x0) || ! isscalar (gamma)) [retval, x, x0, gamma] = common_size (x, x0, gamma); if (retval > 0) error (strcat (["cauchycdf: X, X0, and GAMMA must be of"], ... [" common size or scalars."])); endif endif ## Check for X, X0, and GAMMA being reals if (iscomplex (x) || iscomplex (x0) || iscomplex (gamma)) error ("cauchycdf: X, X0, and GAMMA must not be complex."); endif ## Check for class type if (isa (x, "single") || isa (x0, "single") || isa (gamma, "single")); p = NaN (size (x), "single"); else p = NaN (size (x)); endif ## Find valid values in parameters and data k = ! isinf (x0) & (gamma > 0) & (gamma < Inf); ## Compute Cauchy CDF if (isscalar (x0) && isscalar (gamma)) if (uflag) p = 0.5 + atan ((-x(k) + x0) / gamma) / pi; else p = 0.5 + atan ((x(k) - x0) / gamma) / pi; endif else if (uflag) p(k) = 0.5 + atan ((-x(k) + x0(k)) ./ gamma(k)) / pi; else p(k) = 0.5 + atan ((x(k) - x0(k)) ./ gamma(k)) / pi; endif endif endfunction %!demo %! ## Plot various CDFs from the Cauchy distribution %! x = -5:0.01:5; %! p1 = cauchycdf (x, 0, 0.5); %! p2 = cauchycdf (x, 0, 1); %! p3 = cauchycdf (x, 0, 2); %! p4 = cauchycdf (x, -2, 1); %! plot (x, p1, "-b", x, p2, "-g", x, p3, "-r", x, p4, "-c") %! grid on %! xlim ([-5, 5]) %! legend ({"x0 = 0, γ = 0.5", "x0 = 0, γ = 1", ... %! "x0 = 0, γ = 2", "x0 = -2, γ = 1"}, "location", "southeast") %! title ("Cauchy CDF") %! xlabel ("values in x") %! ylabel ("probability") ## Test output %!shared x, y %! x = [-1 0 0.5 1 2]; %! y = 1/pi * atan ((x-1) / 2) + 1/2; %!assert (cauchycdf (x, ones (1,5), 2*ones (1,5)), y) %!assert (cauchycdf (x, 1, 2*ones (1,5)), y) %!assert (cauchycdf (x, ones (1,5), 2), y) %!assert (cauchycdf (x, [-Inf 1 NaN 1 Inf], 2), [NaN y(2) NaN y(4) NaN]) %!assert (cauchycdf (x, 1, 2*[0 1 NaN 1 Inf]), [NaN y(2) NaN y(4) NaN]) %!assert (cauchycdf ([x(1:2) NaN x(4:5)], 1, 2), [y(1:2) NaN y(4:5)]) %!assert (cauchycdf ([x, NaN], 1, 2), [y, NaN]) ## Test class of input preserved %!assert (cauchycdf (single ([x, NaN]), 1, 2), single ([y, NaN]), eps ("single")) %!assert (cauchycdf ([x, NaN], single (1), 2), single ([y, NaN]), eps ("single")) %!assert (cauchycdf ([x, NaN], 1, single (2)), single ([y, NaN]), eps ("single")) ## Test input validation %!error cauchycdf () %!error cauchycdf (1) %!error ... %! cauchycdf (1, 2) %!error ... %! cauchycdf (1, 2, 3, 4, 5) %!error cauchycdf (1, 2, 3, "tail") %!error cauchycdf (1, 2, 3, 4) %!error ... %! cauchycdf (ones (3), ones (2), ones (2)) %!error ... %! cauchycdf (ones (2), ones (3), ones (2)) %!error ... %! cauchycdf (ones (2), ones (2), ones (3)) %!error cauchycdf (i, 2, 2) %!error cauchycdf (2, i, 2) %!error cauchycdf (2, 2, i) statistics-release-1.6.3/inst/dist_fun/cauchyinv.m000066400000000000000000000121471456127120000222730ustar00rootroot00000000000000## Copyright (C) 2012 Rik Wehbring ## Copyright (C) 1995-2016 Kurt Hornik ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{x} =} cauchyinv (@var{p}, @var{x0}, @var{gamma}) ## ## Inverse of the Cauchy cumulative distribution function (iCDF). ## ## For each element of @var{p}, compute the quantile (the inverse of the CDF) of ## the Cauchy distribution with location parameter @var{x0} and scale parameter ## @var{gamma}. The size of @var{x} is the common size of @var{p}, @var{x0}, ## and @var{gamma}. A scalar input functions as a constant matrix of the same ## size as the other inputs. ## ## Further information about the Cauchy distribution can be found at ## @url{https://en.wikipedia.org/wiki/Cauchy_distribution} ## ## @seealso{cauchycdf, cauchypdf, cauchyrnd} ## @end deftypefn function x = cauchyinv (p, x0, gamma) ## Check for valid number of input arguments if (nargin < 3) error ("cauchyinv: function called with too few input arguments."); endif ## Check for common size of P, X0, and GAMMA if (! isscalar (p) || ! isscalar (x0) || ! isscalar (gamma)) [retval, p, x0, gamma] = common_size (p, x0, gamma); if (retval > 0) error (strcat (["cauchyinv: P, X0, and GAMMA must be of"], ... [" common size or scalars."])); endif endif ## Check for P, X0, and GAMMA being reals if (iscomplex (p) || iscomplex (x0) || iscomplex (gamma)) error ("cauchyinv: P, X0, and GAMMA must not be complex."); endif ## Check for class type if (isa (p, "single") || isa (x0, "single") || isa (gamma, "single")) x = NaN (size (p), "single"); else x = NaN (size (p)); endif ## Find valid values in parameters ok = ! isinf (x0) & (gamma > 0) & (gamma < Inf); ## Handle edge cases k0 = (p == 0) & ok; x(k0) = -Inf; k1 = (p == 1) & ok; x(k1) = Inf; ## Handle all other valid cases k = (p > 0) & (p < 1) & ok; if (isscalar (x0) && isscalar (gamma)) x(k) = x0 - gamma * cot (pi * p(k)); else x(k) = x0(k) - gamma(k) .* cot (pi * p(k)); endif endfunction %!demo %! ## Plot various iCDFs from the Cauchy distribution %! p = 0.001:0.001:0.999; %! x1 = cauchyinv (p, 0, 0.5); %! x2 = cauchyinv (p, 0, 1); %! x3 = cauchyinv (p, 0, 2); %! x4 = cauchyinv (p, -2, 1); %! plot (p, x1, "-b", p, x2, "-g", p, x3, "-r", p, x4, "-c") %! grid on %! ylim ([-5, 5]) %! legend ({"x0 = 0, γ = 0.5", "x0 = 0, γ = 1", ... %! "x0 = 0, γ = 2", "x0 = -2, γ = 1"}, "location", "northwest") %! title ("Cauchy iCDF") %! xlabel ("probability") %! ylabel ("values in x") ## Test output %!shared p %! p = [-1 0 0.5 1 2]; %!assert (cauchyinv (p, ones (1,5), 2 * ones (1,5)), [NaN -Inf 1 Inf NaN], eps) %!assert (cauchyinv (p, 1, 2 * ones (1,5)), [NaN -Inf 1 Inf NaN], eps) %!assert (cauchyinv (p, ones (1,5), 2), [NaN -Inf 1 Inf NaN], eps) %!assert (cauchyinv (p, [1 -Inf NaN Inf 1], 2), [NaN NaN NaN NaN NaN]) %!assert (cauchyinv (p, 1, 2 * [1 0 NaN Inf 1]), [NaN NaN NaN NaN NaN]) %!assert (cauchyinv ([p(1:2) NaN p(4:5)], 1, 2), [NaN -Inf NaN Inf NaN]) %!assert (cauchyinv ([p, NaN], 1, 2), [NaN -Inf 1 Inf NaN NaN], eps) ## Test class of input preserved %!assert (cauchyinv (single ([p, NaN]), 1, 2), ... %! single ([NaN -Inf 1 Inf NaN NaN]), eps ("single")) %!assert (cauchyinv ([p, NaN], single (1), 2), ... %! single ([NaN -Inf 1 Inf NaN NaN]), eps ("single")) %!assert (cauchyinv ([p, NaN], 1, single (2)), ... %! single ([NaN -Inf 1 Inf NaN NaN]), eps ("single")) ## Test input validation %!error cauchyinv () %!error cauchyinv (1) %!error ... %! cauchyinv (1, 2) %!error cauchyinv (1, 2, 3, 4) %!error ... %! cauchyinv (ones (3), ones (2), ones(2)) %!error ... %! cauchyinv (ones (2), ones (3), ones(2)) %!error ... %! cauchyinv (ones (2), ones (2), ones(3)) %!error cauchyinv (i, 4, 3) %!error cauchyinv (1, i, 3) %!error cauchyinv (1, 4, i) statistics-release-1.6.3/inst/dist_fun/cauchypdf.m000066400000000000000000000116621456127120000222510ustar00rootroot00000000000000## Copyright (C) 2012 Rik Wehbring ## Copyright (C) 1995-2016 Kurt Hornik ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{y} =} cauchypdf (@var{x}, @var{x0}, @var{gamma}) ## ## Cauchy probability density function (PDF). ## ## For each element of @var{x}, compute the probability density function (PDF) ## of the Cauchy distribution with location parameter @var{x0} and scale ## parameter @var{gamma}. The size of @var{y} is the common size of @var{x}, ## @var{x0}, and @var{gamma}. A scalar input functions as a constant matrix of ## the same size as the other inputs. ## ## Further information about the Cauchy distribution can be found at ## @url{https://en.wikipedia.org/wiki/Cauchy_distribution} ## ## @seealso{cauchycdf, cauchypdf, cauchyrnd} ## @end deftypefn function y = cauchypdf (x, x0, gamma) ## Check for valid number of input arguments if (nargin < 3) error ("cauchypdf: function called with too few input arguments."); endif ## Check for common size of X, X0, and GAMMA if (! isscalar (x) || ! isscalar (x0) || ! isscalar (gamma)) [retval, x, x0, gamma] = common_size (x, x0, gamma); if (retval > 0) error (strcat (["cauchypdf: X, X0, and GAMMA must be of"], ... [" common size or scalars."])); endif endif ## Check for X, X0, and GAMMA being reals if (iscomplex (x) || iscomplex (x0) || iscomplex (gamma)) error ("cauchypdf: X, X0, and GAMMA must not be complex."); endif ## Check for class type if (isa (x, "single") || isa (x0, "single") || isa (gamma, "single")) y = NaN (size (x), "single"); else y = NaN (size (x)); endif ## Find valid values in parameters k = ! isinf (x0) & (gamma > 0) & (gamma < Inf); if (isscalar (x0) && isscalar (gamma)) y(k) = ((1 ./ (1 + ((x(k) - x0) / gamma) .^ 2)) / pi / gamma); else y(k) = ((1 ./ (1 + ((x(k) - x0(k)) ./ gamma(k)) .^ 2)) / pi ./ gamma(k)); endif endfunction %!demo %! ## Plot various PDFs from the Cauchy distribution %! x = -5:0.01:5; %! y1 = cauchypdf (x, 0, 0.5); %! y2 = cauchypdf (x, 0, 1); %! y3 = cauchypdf (x, 0, 2); %! y4 = cauchypdf (x, -2, 1); %! plot (x, y1, "-b", x, y2, "-g", x, y3, "-r", x, y4, "-c") %! grid on %! xlim ([-5, 5]) %! ylim ([0, 0.7]) %! legend ({"x0 = 0, γ = 0.5", "x0 = 0, γ = 1", ... %! "x0 = 0, γ = 2", "x0 = -2, γ = 1"}, "location", "northeast") %! title ("Cauchy PDF") %! xlabel ("values in x") %! ylabel ("density") ## Test output %!shared x, y %! x = [-1 0 0.5 1 2]; %! y = 1/pi * ( 2 ./ ((x-1).^2 + 2^2) ); %!assert (cauchypdf (x, ones (1,5), 2*ones (1,5)), y) %!assert (cauchypdf (x, 1, 2*ones (1,5)), y) %!assert (cauchypdf (x, ones (1,5), 2), y) %!assert (cauchypdf (x, [-Inf 1 NaN 1 Inf], 2), [NaN y(2) NaN y(4) NaN]) %!assert (cauchypdf (x, 1, 2*[0 1 NaN 1 Inf]), [NaN y(2) NaN y(4) NaN]) %!assert (cauchypdf ([x, NaN], 1, 2), [y, NaN]) ## Test class of input preserved %!assert (cauchypdf (single ([x, NaN]), 1, 2), single ([y, NaN]), eps ("single")) %!assert (cauchypdf ([x, NaN], single (1), 2), single ([y, NaN]), eps ("single")) %!assert (cauchypdf ([x, NaN], 1, single (2)), single ([y, NaN]), eps ("single")) ## Cauchy (0,1) == Student's T distribution with 1 DOF %!test %! x = rand (10, 1); %! assert (cauchypdf (x, 0, 1), tpdf (x, 1), eps); ## Test input validation %!error cauchypdf () %!error cauchypdf (1) %!error ... %! cauchypdf (1, 2) %!error cauchypdf (1, 2, 3, 4) %!error ... %! cauchypdf (ones (3), ones (2), ones(2)) %!error ... %! cauchypdf (ones (2), ones (3), ones(2)) %!error ... %! cauchypdf (ones (2), ones (2), ones(3)) %!error cauchypdf (i, 4, 3) %!error cauchypdf (1, i, 3) %!error cauchypdf (1, 4, i) statistics-release-1.6.3/inst/dist_fun/cauchyrnd.m000066400000000000000000000155171456127120000222660ustar00rootroot00000000000000## Copyright (C) 2012 Rik Wehbring ## Copyright (C) 1995-2016 Kurt Hornik ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{r} =} cauchyrnd (@var{x0}, @var{gamma}) ## @deftypefnx {statistics} {@var{r} =} cauchyrnd (@var{x0}, @var{gamma}, @var{rows}) ## @deftypefnx {statistics} {@var{r} =} cauchyrnd (@var{x0}, @var{gamma}, @var{rows}, @var{cols}, @dots{}) ## @deftypefnx {statistics} {@var{r} =} cauchyrnd (@var{x0}, @var{gamma}, [@var{sz}]) ## ## Random arrays from the Cauchy distribution. ## ## @code{@var{r} = cauchyrnd (@var{x0}, @var{gamma})} returns an array of ## random numbers chosen from the Cauchy distribution with location parameter ## @var{x0} and scale parameter @var{gamma}. The size of @var{r} is the common ## size of @var{x0} and @var{gamma}. A scalar input functions as a constant ## matrix of the same size as the other inputs. ## ## When called with a single size argument, @code{cauchyrnd} returns a square ## matrix with the dimension specified. When called with more than one scalar ## argument, the first two arguments are taken as the number of rows and columns ## and any further arguments specify additional matrix dimensions. The size may ## also be specified with a row vector of dimensions, @var{sz}. ## ## Further information about the Cauchy distribution can be found at ## @url{https://en.wikipedia.org/wiki/Cauchy_distribution} ## ## @seealso{cauchycdf, cauchyinv, cauchypdf} ## @end deftypefn function r = cauchyrnd (x0, gamma, varargin) ## Check for valid number of input arguments if (nargin < 2) error ("cauchyrnd: function called with too few input arguments."); endif ## Check for common size of X0 and GAMMA if (! isscalar (x0) || ! isscalar (gamma)) [retval, x0, gamma] = common_size (x0, gamma); if (retval > 0) error (strcat (["cauchyrnd: X0 and GAMMA must be of common"], ... [" size or scalars."])); endif endif ## Check for X0 and GAMMA being reals if (iscomplex (x0) || iscomplex (gamma)) error ("cauchyrnd: X0 and GAMMA must not be complex."); endif ## Parse and check SIZE arguments if (nargin == 2) sz = size (x0); elseif (nargin == 3) if (isscalar (varargin{1}) && varargin{1} >= 0 ... && varargin{1} == fix (varargin{1})) sz = [varargin{1}, varargin{1}]; elseif (isrow (varargin{1}) && all (varargin{1} >= 0) ... && all (varargin{1} == fix (varargin{1}))) sz = varargin{1}; elseif error (strcat (["cauchyrnd: SZ must be a scalar or a row vector"], ... [" of non-negative integers."])); endif elseif (nargin > 3) posint = cellfun (@(x) (! isscalar (x) || x < 0 || x != fix (x)), varargin); if (any (posint)) error ("cauchyrnd: dimensions must be non-negative integers."); endif sz = [varargin{:}]; endif ## Check that parameters match requested dimensions in size if (! isscalar (x0) && ! isequal (size (x0), sz)) error ("cauchyrnd: X0 and GAMMA must be scalars or of size SZ."); endif ## Check for class type if (isa (x0, "single") || isa (gamma, "single")) cls = "single"; else cls = "double"; endif ## Generate random sample from Cauchy distribution if (isscalar (x0) && isscalar (gamma)) if (! isinf (x0) && (gamma > 0) && (gamma < Inf)) r = x0 - cot (pi * rand (sz, cls)) * gamma; else r = NaN (sz, cls); endif else r = NaN (sz, cls); k = ! isinf (x0) & (gamma > 0) & (gamma < Inf); r(k) = x0(k)(:) - cot (pi * rand (sum (k(:)), 1, cls)) .* gamma(k)(:); endif endfunction ## Test output %!assert (size (cauchyrnd (1, 1)), [1 1]) %!assert (size (cauchyrnd (1, ones (2,1))), [2, 1]) %!assert (size (cauchyrnd (1, ones (2,2))), [2, 2]) %!assert (size (cauchyrnd (ones (2,1), 1)), [2, 1]) %!assert (size (cauchyrnd (ones (2,2), 1)), [2, 2]) %!assert (size (cauchyrnd (1, 1, 3)), [3, 3]) %!assert (size (cauchyrnd (1, 1, [4, 1])), [4, 1]) %!assert (size (cauchyrnd (1, 1, 4, 1)), [4, 1]) %!assert (size (cauchyrnd (1, 1, 4, 1, 5)), [4, 1, 5]) %!assert (size (cauchyrnd (1, 1, 0, 1)), [0, 1]) %!assert (size (cauchyrnd (1, 1, 1, 0)), [1, 0]) %!assert (size (cauchyrnd (1, 1, 1, 2, 0, 5)), [1, 2, 0, 5]) ## Test class of input preserved %!assert (class (cauchyrnd (1, 1)), "double") %!assert (class (cauchyrnd (1, single (1))), "single") %!assert (class (cauchyrnd (1, single ([1, 1]))), "single") %!assert (class (cauchyrnd (single (1), 1)), "single") %!assert (class (cauchyrnd (single ([1, 1]), 1)), "single") ## Test input validation %!error cauchyrnd () %!error cauchyrnd (1) %!error ... %! cauchyrnd (ones (3), ones (2)) %!error ... %! cauchyrnd (ones (2), ones (3)) %!error cauchyrnd (i, 2, 3) %!error cauchyrnd (1, i, 3) %!error ... %! cauchyrnd (1, 2, -1) %!error ... %! cauchyrnd (1, 2, 1.2) %!error ... %! cauchyrnd (1, 2, ones (2)) %!error ... %! cauchyrnd (1, 2, [2 -1 2]) %!error ... %! cauchyrnd (1, 2, [2 0 2.5]) %!error ... %! cauchyrnd (1, 2, 2, -1, 5) %!error ... %! cauchyrnd (1, 2, 2, 1.5, 5) %!error ... %! cauchyrnd (2, ones (2), 3) %!error ... %! cauchyrnd (2, ones (2), [3, 2]) %!error ... %! cauchyrnd (2, ones (2), 3, 2) statistics-release-1.6.3/inst/dist_fun/chi2cdf.m000066400000000000000000000112721456127120000216020ustar00rootroot00000000000000## Copyright (C) 2012 Rik Wehbring ## Copyright (C) 1995-2016 Kurt Hornik ## Copyright (C) 2022-2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{p} =} chi2cdf (@var{x}, @var{df}) ## @deftypefnx {statistics} {@var{p} =} chi2cdf (@var{x}, @var{df}, @qcode{"upper"}) ## ## Chi-squared cumulative distribution function (CDF). ## ## For each element of @var{x}, compute the cumulative distribution function ## (CDF) of the chi-squared distribution with @var{df} degrees of freedom. The ## chi-squared density function with @var{df} degrees of freedom is the same as ## a gamma density function with parameters @qcode{@var{df}/2} and @qcode{2}. ## ## The size of @var{p} is the common size of @var{x} and @var{df}. A scalar ## input functions as a constant matrix of the same size as the other input. ## ## @code{@var{p} = chi2cdf (@var{x}, @var{df}, "upper")} computes the upper tail ## probability of the chi-squared distribution with @var{df} degrees of freedom, ## at the values in @var{x}. ## ## Further information about the chi-squared distribution can be found at ## @url{https://en.wikipedia.org/wiki/Chi-squared_distribution} ## ## @seealso{chi2inv, chi2pdf, chi2rnd, chi2stat} ## @end deftypefn function p = chi2cdf (x, df, uflag) ## Check for valid number of input arguments if (nargin < 2) error ("chi2cdf: function called with too few input arguments."); endif ## Check for valid "upper" flag if (nargin > 2 && ! strcmpi (uflag, "upper")) error ("chi2cdf: invalid argument for upper tail."); else uflag = []; endif ## Check for common size of X and DF if (! isscalar (x) || ! isscalar (df)) [err, x, df] = common_size (x, df); if (err > 0) error ("chi2cdf: X and DF must be of common size or scalars."); endif endif ## Check for X and DF being reals if (iscomplex (x) || iscomplex (df)) error ("chi2cdf: X and DF must not be complex."); endif ## Compute chi-squared CDF p = gamcdf (x, df/2, 2, uflag); endfunction %!demo %! ## Plot various CDFs from the chi-squared distribution %! x = 0:0.01:8; %! p1 = chi2cdf (x, 1); %! p2 = chi2cdf (x, 2); %! p3 = chi2cdf (x, 3); %! p4 = chi2cdf (x, 4); %! p5 = chi2cdf (x, 6); %! p6 = chi2cdf (x, 9); %! plot (x, p1, "-b", x, p2, "-g", x, p3, "-r", ... %! x, p4, "-c", x, p5, "-m", x, p6, "-y") %! grid on %! xlim ([0, 8]) %! legend ({"df = 1", "df = 2", "df = 3", ... %! "df = 4", "df = 6", "df = 9"}, "location", "southeast") %! title ("Chi-squared CDF") %! xlabel ("values in x") %! ylabel ("probability") ## Test output %!shared x, p, u %! x = [-1, 0, 0.5, 1, 2]; %! p = [0, (1 - exp (-x(2:end) / 2))]; %! u = [1, 0, NaN, 0.3934693402873666, 0.6321205588285577]; %!assert (chi2cdf (x, 2 * ones (1,5)), p, eps) %!assert (chi2cdf (x, 2), p, eps) %!assert (chi2cdf (x, 2 * [1, 0, NaN, 1, 1]), [p(1), 1, NaN, p(4:5)], eps) %!assert (chi2cdf (x, 2 * [1, 0, NaN, 1, 1], "upper"), ... %! [p(1), 1, NaN, u(4:5)], eps) %!assert (chi2cdf ([x(1:2), NaN, x(4:5)], 2), [p(1:2), NaN, p(4:5)], eps) ## Test class of input preserved %!assert (chi2cdf ([x, NaN], 2), [p, NaN], eps) %!assert (chi2cdf (single ([x, NaN]), 2), single ([p, NaN]), eps ("single")) %!assert (chi2cdf ([x, NaN], single (2)), single ([p, NaN]), eps ("single")) ## Test input validation %!error chi2cdf () %!error chi2cdf (1) %!error chi2cdf (1, 2, 3, 4) %!error chi2cdf (1, 2, 3) %!error chi2cdf (1, 2, "uper") %!error ... %! chi2cdf (ones (3), ones (2)) %!error ... %! chi2cdf (ones (2), ones (3)) %!error chi2cdf (i, 2) %!error chi2cdf (2, i) statistics-release-1.6.3/inst/dist_fun/chi2inv.m000066400000000000000000000073771456127120000216550ustar00rootroot00000000000000## Copyright (C) 2012 Rik Wehbring ## Copyright (C) 1995-2016 Kurt Hornik ## Copyright (C) 2023 Andreas Bertsatos ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{x} =} chi2inv (@var{p}, @var{df}) ## ## Inverse of the chi-squared cumulative distribution function (iCDF). ## ## For each element of @var{p}, compute the quantile (the inverse of the CDF) of ## the chi-squared distribution with @var{df} degrees of freedom. The size of ## @var{x} is the common size of @var{p} and @var{df}. A scalar input functions ## as a constant matrix of the same size as the other inputs. ## ## Further information about the chi-squared distribution can be found at ## @url{https://en.wikipedia.org/wiki/Chi-squared_distribution} ## ## @seealso{chi2cdf, chi2pdf, chi2rnd, chi2stat} ## @end deftypefn function x = chi2inv (p, df) ## Check for valid number of input arguments if (nargin < 2) error ("chi2inv: function called with too few input arguments."); endif ## Check for common size of P and DF if (! isscalar (p) || ! isscalar (df)) [retval, p, df] = common_size (p, df); if (retval > 0) error ("chi2inv: P and DF must be of common size or scalars."); endif endif ## Check for P and DF being reals if (iscomplex (p) || iscomplex (df)) error ("chi2inv: P and DF must not be complex."); endif ## Compute chi-squared iCDF x = gaminv (p, df/2, 2); endfunction %!demo %! ## Plot various iCDFs from the chi-squared distribution %! p = 0.001:0.001:0.999; %! x1 = chi2inv (p, 1); %! x2 = chi2inv (p, 2); %! x3 = chi2inv (p, 3); %! x4 = chi2inv (p, 4); %! x5 = chi2inv (p, 6); %! x6 = chi2inv (p, 9); %! plot (p, x1, "-b", p, x2, "-g", p, x3, "-r", ... %! p, x4, "-c", p, x5, "-m", p, x6, "-y") %! grid on %! ylim ([0, 8]) %! legend ({"df = 1", "df = 2", "df = 3", ... %! "df = 4", "df = 6", "df = 9"}, "location", "northwest") %! title ("Chi-squared iCDF") %! xlabel ("probability") %! ylabel ("values in x") ## Test output %!shared p %! p = [-1 0 0.3934693402873666 1 2]; %!assert (chi2inv (p, 2*ones (1,5)), [NaN 0 1 Inf NaN], 5*eps) %!assert (chi2inv (p, 2), [NaN 0 1 Inf NaN], 5*eps) %!assert (chi2inv (p, 2*[0 1 NaN 1 1]), [NaN 0 NaN Inf NaN], 5*eps) %!assert (chi2inv ([p(1:2) NaN p(4:5)], 2), [NaN 0 NaN Inf NaN], 5*eps) ## Test class of input preserved %!assert (chi2inv ([p, NaN], 2), [NaN 0 1 Inf NaN NaN], 5*eps) %!assert (chi2inv (single ([p, NaN]), 2), single ([NaN 0 1 Inf NaN NaN]), 5*eps ("single")) %!assert (chi2inv ([p, NaN], single (2)), single ([NaN 0 1 Inf NaN NaN]), 5*eps ("single")) ## Test input validation %!error chi2inv () %!error chi2inv (1) %!error chi2inv (1,2,3) %!error ... %! chi2inv (ones (3), ones (2)) %!error ... %! chi2inv (ones (2), ones (3)) %!error chi2inv (i, 2) %!error chi2inv (2, i) statistics-release-1.6.3/inst/dist_fun/chi2pdf.m000066400000000000000000000070721456127120000216220ustar00rootroot00000000000000## Copyright (C) 2012 Rik Wehbring ## Copyright (C) 1995-2016 Kurt Hornik ## Copyright (C) 2023 Andreas Bertsatos ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{y} =} chi2pdf (@var{x}, @var{df}) ## ## Chi-squared probability density function (PDF). ## ## For each element of @var{x}, compute the probability density function (PDF) ## of the chi-squared distribution with @var{df} degrees of freedom. The size ## of @var{y} is the common size of @var{x} and @var{df}. A scalar input ## functions as a constant matrix of the same size as the other inputs. ## ## Further information about the chi-squared distribution can be found at ## @url{https://en.wikipedia.org/wiki/Chi-squared_distribution} ## ## @seealso{chi2cdf, chi2pdf, chi2rnd, chi2stat} ## @end deftypefn function y = chi2pdf (x, df) ## Check for valid number of input arguments if (nargin < 2) error ("chi2pdf: function called with too few input arguments."); endif ## Check for common size of X and DF if (! isscalar (x) || ! isscalar (df)) [retval, x, df] = common_size (x, df); if (retval > 0) error ("chi2pdf: X and DF must be of common size or scalars."); endif endif ## Check for X and DF being reals if (iscomplex (x) || iscomplex (df)) error ("chi2pdf: X and DF must not be complex."); endif ## Compute chi-squared PDF y = gampdf (x, df/2, 2); endfunction %!demo %! ## Plot various PDFs from the chi-squared distribution %! x = 0:0.01:8; %! y1 = chi2pdf (x, 1); %! y2 = chi2pdf (x, 2); %! y3 = chi2pdf (x, 3); %! y4 = chi2pdf (x, 4); %! y5 = chi2pdf (x, 6); %! y6 = chi2pdf (x, 9); %! plot (x, y1, "-b", x, y2, "-g", x, y3, "-r", ... %! x, y4, "-c", x, y5, "-m", x, y6, "-y") %! grid on %! xlim ([0, 8]) %! ylim ([0, 0.5]) %! legend ({"df = 1", "df = 2", "df = 3", ... %! "df = 4", "df = 6", "df = 9"}, "location", "northeast") %! title ("Chi-squared PDF") %! xlabel ("values in x") %! ylabel ("density") ## Test output %!shared x, y %! x = [-1 0 0.5 1 Inf]; %! y = [0, 1/2 * exp(-x(2:5)/2)]; %!assert (chi2pdf (x, 2*ones (1,5)), y) %!assert (chi2pdf (x, 2), y) %!assert (chi2pdf (x, 2*[1 0 NaN 1 1]), [y(1) NaN NaN y(4:5)]) %!assert (chi2pdf ([x, NaN], 2), [y, NaN]) ## Test class of input preserved %!assert (chi2pdf (single ([x, NaN]), 2), single ([y, NaN])) %!assert (chi2pdf ([x, NaN], single (2)), single ([y, NaN])) ## Test input validation %!error chi2pdf () %!error chi2pdf (1) %!error chi2pdf (1,2,3) %!error ... %! chi2pdf (ones (3), ones (2)) %!error ... %! chi2pdf (ones (2), ones (3)) %!error chi2pdf (i, 2) %!error chi2pdf (2, i) statistics-release-1.6.3/inst/dist_fun/chi2rnd.m000066400000000000000000000127261456127120000216360ustar00rootroot00000000000000## Copyright (C) 2012 Rik Wehbring ## Copyright (C) 1995-2016 Kurt Hornik ## Copyright (C) 2023 Andreas Bertsatos ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{r} =} chi2rnd (@var{df}) ## @deftypefnx {statistics} {@var{r} =} chi2rnd (@var{df}, @var{rows}) ## @deftypefnx {statistics} {@var{r} =} chi2rnd (@var{df}, @var{rows}, @var{cols}, @dots{}) ## @deftypefnx {statistics} {@var{r} =} chi2rnd (@var{df}, [@var{sz}]) ## ## Random arrays from the chi-squared distribution. ## ## @code{@var{r} = chi2rnd (@var{df})} returns an array of random numbers chosen ## from the chi-squared distribution with @var{df} degrees of freedom. The size ## of @var{r} is the size of @var{df}. ## ## When called with a single size argument, @code{chi2rnd} returns a square ## matrix with the dimension specified. When called with more than one scalar ## argument, the first two arguments are taken as the number of rows and columns ## and any further arguments specify additional matrix dimensions. The size may ## also be specified with a row vector of dimensions, @var{sz}. ## ## Further information about the chi-squared distribution can be found at ## @url{https://en.wikipedia.org/wiki/Chi-squared_distribution} ## ## @seealso{chi2cdf, chi2inv, chi2pdf, chi2stat} ## @end deftypefn function r = chi2rnd (df, varargin) ## Check for valid number of input arguments if (nargin < 1) error ("chi2rnd: function called with too few input arguments."); endif ## Check for DF being reals if (iscomplex (df)) error ("chi2rnd: DF must not be complex."); endif ## Parse and check SIZE arguments if (nargin == 1) sz = size (df); elseif (nargin == 2) if (isscalar (varargin{1}) && varargin{1} >= 0 ... && varargin{1} == fix (varargin{1})) sz = [varargin{1}, varargin{1}]; elseif (isrow (varargin{1}) && all (varargin{1} >= 0) ... && all (varargin{1} == fix (varargin{1}))) sz = varargin{1}; elseif error (strcat (["chi2rnd: SZ must be a scalar or a row vector"], ... [" of non-negative integers."])); endif elseif (nargin > 2) posint = cellfun (@(x) (! isscalar (x) || x < 0 || x != fix (x)), varargin); if (any (posint)) error ("chi2rnd: dimensions must be non-negative integers."); endif sz = [varargin{:}]; endif ## Check that parameter match requested dimensions in size if (! isscalar (df) && ! isequal (size (df), sz)) error ("chi2rnd: DF must be scalar or of size SZ."); endif ## Check for class type if (isa (df, "single")) cls = "single"; else cls = "double"; endif ## Generate random sample from chi-squared distribution if (isscalar (df)) if ((df > 0) && (df < Inf)) r = 2 * randg (df/2, sz, cls); else r = NaN (sz, cls); endif else r = NaN (sz, cls); k = (df > 0) | (df < Inf); r(k) = 2 * randg (df(k)/2, cls); endif endfunction ## Test output %!assert (size (chi2rnd (2)), [1, 1]) %!assert (size (chi2rnd (ones (2,1))), [2, 1]) %!assert (size (chi2rnd (ones (2,2))), [2, 2]) %!assert (size (chi2rnd (1, 3)), [3, 3]) %!assert (size (chi2rnd (1, [4 1])), [4, 1]) %!assert (size (chi2rnd (1, 4, 1)), [4, 1]) %!assert (size (chi2rnd (1, 4, 1)), [4, 1]) %!assert (size (chi2rnd (1, 4, 1, 5)), [4, 1, 5]) %!assert (size (chi2rnd (1, 0, 1)), [0, 1]) %!assert (size (chi2rnd (1, 1, 0)), [1, 0]) %!assert (size (chi2rnd (1, 1, 2, 0, 5)), [1, 2, 0, 5]) ## Test class of input preserved %!assert (class (chi2rnd (2)), "double") %!assert (class (chi2rnd (single (2))), "single") %!assert (class (chi2rnd (single ([2 2]))), "single") ## Test input validation %!error chi2rnd () %!error chi2rnd (i) %!error ... %! chi2rnd (1, -1) %!error ... %! chi2rnd (1, 1.2) %!error ... %! chi2rnd (1, ones (2)) %!error ... %! chi2rnd (1, [2 -1 2]) %!error ... %! chi2rnd (1, [2 0 2.5]) %!error ... %! chi2rnd (ones (2), ones (2)) %!error ... %! chi2rnd (1, 2, -1, 5) %!error ... %! chi2rnd (1, 2, 1.5, 5) %!error chi2rnd (ones (2,2), 3) %!error chi2rnd (ones (2,2), [3, 2]) %!error chi2rnd (ones (2,2), 2, 3) statistics-release-1.6.3/inst/dist_fun/copulacdf.m000066400000000000000000000224661456127120000222470ustar00rootroot00000000000000## Copyright (C) 2008 Arno Onken ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{p} =} copulacdf (@var{family}, @var{x}, @var{theta}) ## @deftypefnx {statistics} {@var{p} =} copulacdf ('t', @var{x}, @var{theta}, @var{df}) ## ## Copula family cumulative distribution functions (CDF). ## ## @subheading Arguments ## ## @itemize @bullet ## @item ## @var{family} is the copula family name. Currently, @var{family} can ## be @code{'Gaussian'} for the Gaussian family, @code{'t'} for the ## Student's t family, @code{'Clayton'} for the Clayton family, ## @code{'Gumbel'} for the Gumbel-Hougaard family, @code{'Frank'} for ## the Frank family, @code{'AMH'} for the Ali-Mikhail-Haq family, or ## @code{'FGM'} for the Farlie-Gumbel-Morgenstern family. ## ## @item ## @var{x} is the support where each row corresponds to an observation. ## ## @item ## @var{theta} is the parameter of the copula. For the Gaussian and ## Student's t copula, @var{theta} must be a correlation matrix. For ## bivariate copulas @var{theta} can also be a correlation coefficient. ## For the Clayton family, the Gumbel-Hougaard family, the Frank family, ## and the Ali-Mikhail-Haq family, @var{theta} must be a vector with the ## same number of elements as observations in @var{x} or be scalar. For ## the Farlie-Gumbel-Morgenstern family, @var{theta} must be a matrix of ## coefficients for the Farlie-Gumbel-Morgenstern polynomial where each ## row corresponds to one set of coefficients for an observation in ## @var{x}. A single row is expanded. The coefficients are in binary ## order. ## ## @item ## @var{df} is the degrees of freedom for the Student's t family. ## @var{df} must be a vector with the same number of elements as ## observations in @var{x} or be scalar. ## @end itemize ## ## @subheading Return values ## ## @itemize @bullet ## @item ## @var{p} is the cumulative distribution of the copula at each row of ## @var{x} and corresponding parameter @var{theta}. ## @end itemize ## ## @subheading Examples ## ## @example ## @group ## x = [0.2:0.2:0.6; 0.2:0.2:0.6]; ## theta = [1; 2]; ## p = copulacdf ("Clayton", x, theta) ## @end group ## ## @group ## x = [0.2:0.2:0.6; 0.2:0.1:0.4]; ## theta = [0.2, 0.1, 0.1, 0.05]; ## p = copulacdf ("FGM", x, theta) ## @end group ## @end example ## ## @subheading References ## ## @enumerate ## @item ## Roger B. Nelsen. @cite{An Introduction to Copulas}. Springer, ## New York, second edition, 2006. ## @end enumerate ## ## @seealso{copulapdf, copularnd} ## @end deftypefn function p = copulacdf (family, x, theta, df) ## Check arguments if (nargin != 3 && (nargin != 4 || ! strcmpi (family, "t"))) print_usage (); endif if (! ischar (family)) error (strcar (["copulacdf: family must be one of 'Gaussian',"], ... [" 't', 'Clayton', 'Gumbel', 'Frank', 'AMH', and 'FGM'."])); endif if (! isempty (x) && ! ismatrix (x)) error ("copulacdf: X must be a numeric matrix."); endif [n, d] = size (x); lower_family = lower (family); ## Check family and copula parameters switch (lower_family) case {"gaussian", "t"} ## Family with a covariance matrix if (d == 2 && isscalar (theta)) ## Expand a scalar to a correlation matrix theta = [1, theta; theta, 1]; endif if (any (size (theta) != [d, d]) || any (diag (theta) != 1) || ... any (any (theta != theta')) || min (eig (theta)) <= 0) error ("copulacdf: THETA must be a correlation matrix."); endif if (nargin == 4) ## Student's t family if (! isscalar (df) && (! isvector (df) || length (df) != n)) error (strcar (["copulacdf: DF must be a vector with the same"], ... [" number of rows as X or be scalar."])); endif df = df(:); endif case {"clayton", "gumbel", "frank", "amh"} ## Archimedian one parameter family if (! isvector (theta) || (! isscalar (theta) && length (theta) != n)) error (strcat (["copulacdf: THETA must be a vector with the same"], ... [" number of rows as X or be scalar."])); endif theta = theta(:); if (n > 1 && isscalar (theta)) theta = repmat (theta, n, 1); endif case {"fgm"} ## Exponential number of parameters if (! ismatrix (theta) || size (theta, 2) != (2 .^ d - d - 1) || ... (size (theta, 1) != 1 && size (theta, 1) != n)) error (strcat (["copulacdf: THETA must be a row vector of length"], ... [" 2^d-d-1 or a matrix of size N x (2^d-d-1)."])); endif if (n > 1 && size (theta, 1) == 1) theta = repmat (theta, n, 1); endif otherwise error ("copulacdf: unknown copula family '%s'.", family); endswitch if (n == 0) ## Input is empty p = zeros (0, 1); else ## Truncate input to unit hypercube x(x < 0) = 0; x(x > 1) = 1; ## Compute the cumulative distribution function according to family switch (lower_family) case {"gaussian"} ## The Gaussian family p = mvncdf (norminv (x), zeros (1, d), theta); ## No parameter bounds check k = []; case {"t"} ## The Student's t family p = mvtcdf (tinv (x, df), theta, df); ## No parameter bounds check k = []; case {"clayton"} ## The Clayton family p = exp (-log (max (sum (x .^ (repmat (-theta, 1, d)), 2) ... - d + 1, 0)) ./ theta); ## Product copula at columns where theta == 0 k = find (theta == 0); if (any (k)) p(k) = prod (x(k, :), 2); endif ## Check bounds if (d > 2) k = find (! (theta >= 0) | ! (theta < inf)); else k = find (! (theta >= -1) | ! (theta < inf)); endif case {"gumbel"} ## The Gumbel-Hougaard family p = exp (-(sum ((-log (x)) .^ repmat (theta, 1, d), 2)) ... .^ (1 ./ theta)); ## Check bounds k = find (! (theta >= 1) | ! (theta < inf)); case {"frank"} ## The Frank family p = -log (1 + (prod (expm1 (repmat (-theta, 1, d) .* x), 2)) ./ ... (expm1 (-theta) .^ (d - 1))) ./ theta; ## Product copula at columns where theta == 0 k = find (theta == 0); if (any (k)) p(k) = prod (x(k, :), 2); endif ## Check bounds if (d > 2) k = find (! (theta > 0) | ! (theta < inf)); else k = find (! (theta > -inf) | ! (theta < inf)); endif case {"amh"} ## The Ali-Mikhail-Haq family p = (theta - 1) ./ (theta - prod ((1 + repmat (theta, 1, d) ... .* (x - 1)) ./ x, 2)); ## Check bounds if (d > 2) k = find (! (theta >= 0) | ! (theta < 1)); else k = find (! (theta >= -1) | ! (theta < 1)); endif case {"fgm"} ## The Farlie-Gumbel-Morgenstern family ## All binary combinations bcomb = logical (floor (mod (((0:(2 .^ d - 1))' * 2 .^ ... ((1 - d):0)), 2))); ecomb = ones (size (bcomb)); ecomb(bcomb) = -1; ## Summation over all combinations of order >= 2 bcomb = bcomb(sum (bcomb, 2) >= 2, end:-1:1); ## Linear constraints matrix ac = zeros (size (ecomb, 1), size (bcomb, 1)); ## Matrix to compute p ap = zeros (size (x, 1), size (bcomb, 1)); for i = 1:size (bcomb, 1) ac(:, i) = -prod (ecomb(:, bcomb(i, :)), 2); ap(:, i) = prod (1 - x(:, bcomb(i, :)), 2); endfor p = prod (x, 2) .* (1 + sum (ap .* theta, 2)); ## Check linear constraints k = false (n, 1); for i = 1:n k(i) = any (ac * theta(i, :)' > 1); endfor endswitch ## Out of bounds parameters if (any (k)) p(k) = NaN; endif endif endfunction ## Test output %!test %! x = [0.2:0.2:0.6; 0.2:0.2:0.6]; %! theta = [1; 2]; %! p = copulacdf ("Clayton", x, theta); %! expected_p = [0.1395; 0.1767]; %! assert (p, expected_p, 0.001); %!test %! x = [0.2:0.2:0.6; 0.2:0.2:0.6]; %! p = copulacdf ("Gumbel", x, 2); %! expected_p = [0.1464; 0.1464]; %! assert (p, expected_p, 0.001); %!test %! x = [0.2:0.2:0.6; 0.2:0.2:0.6]; %! theta = [1; 2]; %! p = copulacdf ("Frank", x, theta); %! expected_p = [0.0699; 0.0930]; %! assert (p, expected_p, 0.001); %!test %! x = [0.2:0.2:0.6; 0.2:0.2:0.6]; %! theta = [0.3; 0.7]; %! p = copulacdf ("AMH", x, theta); %! expected_p = [0.0629; 0.0959]; %! assert (p, expected_p, 0.001); %!test %! x = [0.2:0.2:0.6; 0.2:0.1:0.4]; %! theta = [0.2, 0.1, 0.1, 0.05]; %! p = copulacdf ("FGM", x, theta); %! expected_p = [0.0558; 0.0293]; %! assert (p, expected_p, 0.001); statistics-release-1.6.3/inst/dist_fun/copulapdf.m000066400000000000000000000143241456127120000222560ustar00rootroot00000000000000## Copyright (C) 2008 Arno Onken ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{y} =} copulapdf (@var{family}, @var{x}, @var{theta}) ## ## Copula family probability density functions (PDF). ## ## @subheading Arguments ## ## @itemize @bullet ## @item ## @var{family} is the copula family name. Currently, @var{family} can ## be @code{'Clayton'} for the Clayton family, @code{'Gumbel'} for the ## Gumbel-Hougaard family, @code{'Frank'} for the Frank family, or ## @code{'AMH'} for the Ali-Mikhail-Haq family. ## ## @item ## @var{x} is the support where each row corresponds to an observation. ## ## @item ## @var{theta} is the parameter of the copula. The elements of ## @var{theta} must be greater than or equal to @code{-1} for the ## Clayton family, greater than or equal to @code{1} for the ## Gumbel-Hougaard family, arbitrary for the Frank family, and greater ## than or equal to @code{-1} and lower than @code{1} for the ## Ali-Mikhail-Haq family. Moreover, @var{theta} must be non-negative ## for dimensions greater than @code{2}. @var{theta} must be a column ## vector with the same number of rows as @var{x} or be scalar. ## @end itemize ## ## @subheading Return values ## ## @itemize @bullet ## @item ## @var{y} is the probability density of the copula at each row of ## @var{x} and corresponding parameter @var{theta}. ## @end itemize ## ## @subheading Examples ## ## @example ## @group ## x = [0.2:0.2:0.6; 0.2:0.2:0.6]; ## theta = [1; 2]; ## y = copulapdf ("Clayton", x, theta) ## @end group ## ## @group ## y = copulapdf ("Gumbel", x, 2) ## @end group ## @end example ## ## @subheading References ## ## @enumerate ## @item ## Roger B. Nelsen. @cite{An Introduction to Copulas}. Springer, ## New York, second edition, 2006. ## @end enumerate ## ## @seealso{copulacdf, copularnd} ## @end deftypefn function y = copulapdf (family, x, theta) ## Check arguments if (nargin != 3) print_usage (); endif if (! ischar (family)) error (strcat (["copulapdf: family must be one of 'Clayton',"], ... [" 'Gumbel', 'Frank', and 'AMH'."])); endif if (! isempty (x) && ! ismatrix (x)) error ("copulapdf: X must be a numeric matrix."); endif [n, d] = size (x); if (! isvector (theta) || (! isscalar (theta) && size (theta, 1) != n)) error (strcat (["copulapdf: THETA must be a column vector with the"], ... [" same number of rows as X or be scalar."])); endif if (n == 0) ## Input is empty y = zeros (0, 1); else if (n > 1 && isscalar (theta)) theta = repmat (theta, n, 1); endif ## Truncate input to unit hypercube x(x < 0) = 0; x(x > 1) = 1; ## Compute the cumulative distribution function according to family lowerarg = lower (family); if (strcmp (lowerarg, "clayton")) ## The Clayton family log_cdf = -log (max (sum (x .^ (repmat (-theta, 1, d)), 2) ... - d + 1, 0)) ./ theta; y = prod (repmat (theta, 1, d) .* repmat (0:(d - 1), n, 1) + 1, 2) ... .* exp ((1 + theta .* d) .* log_cdf - ... (theta + 1) .* sum (log (x), 2)); ## Product copula at columns where theta == 0 k = find (theta == 0); if (any (k)) y(k) = 1; endif ## Check theta if (d > 2) k = find (! (theta >= 0) | ! (theta < inf)); else k = find (! (theta >= -1) | ! (theta < inf)); endif elseif (strcmp (lowerarg, "gumbel")) ## The Gumbel-Hougaard family g = sum ((-log (x)) .^ repmat (theta, 1, d), 2); c = exp (-g .^ (1 ./ theta)); y = ((prod (-log (x), 2)) .^ (theta - 1)) ./ prod (x, 2) .* c .* ... (g .^ (2 ./ theta - 2) + (theta - 1) .* g .^ (1 ./ theta - 2)); ## Check theta k = find (! (theta >= 1) | ! (theta < inf)); elseif (strcmp (lowerarg, "frank")) ## The Frank family if (d != 2) error ("copulapdf: Frank copula PDF implemented as bivariate only."); endif y = (theta .* exp (theta .* (1 + sum (x, 2))) .* (exp (theta) - 1)) ./ ... (exp (theta) - exp (theta + theta .* x(:, 1)) + ... exp (theta .* sum (x, 2)) - exp (theta + theta .* x(:, 2))) .^ 2; ## Product copula at columns where theta == 0 k = find (theta == 0); if (any (k)) y(k) = 1; endif ## Check theta k = find (! (theta > -inf) | ! (theta < inf)); elseif (strcmp (lowerarg, "amh")) ## The Ali-Mikhail-Haq family if (d != 2) error (strcat (["copulapdf: Ali-Mikhail-Haq copula PDF"], ... [" implemented as bivariate only."])); endif z = theta .* prod (x - 1, 2) - 1; y = (theta .* (1 - sum (x, 2) - prod (x, 2) - z) - 1) ./ (z .^ 3); ## Check theta k = find (! (theta >= -1) | ! (theta < 1)); else error ("copulapdf: unknown copula family '%s'.", family); endif if (any (k)) y(k) = NaN; endif endif endfunction ## Test output %!test %! x = [0.2:0.2:0.6; 0.2:0.2:0.6]; %! theta = [1; 2]; %! y = copulapdf ("Clayton", x, theta); %! expected_p = [0.9872; 0.7295]; %! assert (y, expected_p, 0.001); %!test %! x = [0.2:0.2:0.6; 0.2:0.2:0.6]; %! y = copulapdf ("Gumbel", x, 2); %! expected_p = [0.9468; 0.9468]; %! assert (y, expected_p, 0.001); %!test %! x = [0.2, 0.6; 0.2, 0.6]; %! theta = [1; 2]; %! y = copulapdf ("Frank", x, theta); %! expected_p = [0.9378; 0.8678]; %! assert (y, expected_p, 0.001); %!test %! x = [0.2, 0.6; 0.2, 0.6]; %! theta = [0.3; 0.7]; %! y = copulapdf ("AMH", x, theta); %! expected_p = [0.9540; 0.8577]; %! assert (y, expected_p, 0.001); statistics-release-1.6.3/inst/dist_fun/copularnd.m000066400000000000000000000200141456127120000222610ustar00rootroot00000000000000## Copyright (C) 2012 Arno Onken ## ## This program is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {@var{r} =} copularnd (@var{family}, @var{theta}, @var{n}) ## @deftypefnx {Function File} {@var{r} =} copularnd (@var{family}, @var{theta}, @var{n}, @var{d}) ## @deftypefnx {Function File} {@var{r} =} copularnd ('t', @var{theta}, @var{df}, @var{n}) ## ## Random arrays from the copula family distributions. ## ## @subheading Arguments ## ## @itemize @bullet ## @item ## @var{family} is the copula family name. Currently, @var{family} can be ## @code{'Gaussian'} for the Gaussian family, @code{'t'} for the Student's t ## family, or @code{'Clayton'} for the Clayton family. ## ## @item ## @var{theta} is the parameter of the copula. For the Gaussian and Student's t ## copula, @var{theta} must be a correlation matrix. For bivariate copulas ## @var{theta} can also be a correlation coefficient. For the Clayton family, ## @var{theta} must be a vector with the same number of elements as samples to ## be generated or be scalar. ## ## @item ## @var{df} is the degrees of freedom for the Student's t family. @var{df} must ## be a vector with the same number of elements as samples to be generated or ## be scalar. ## ## @item ## @var{n} is the number of rows of the matrix to be generated. @var{n} must be ## a non-negative integer and corresponds to the number of samples to be ## generated. ## ## @item ## @var{d} is the number of columns of the matrix to be generated. @var{d} must ## be a positive integer and corresponds to the dimension of the copula. ## @end itemize ## ## @subheading Return values ## ## @itemize @bullet ## @item ## @var{r} is a matrix of random samples from the copula with @var{n} samples ## of distribution dimension @var{d}. ## @end itemize ## ## @subheading Examples ## ## @example ## @group ## theta = 0.5; ## r = copularnd ("Gaussian", theta); ## @end group ## ## @group ## theta = 0.5; ## df = 2; ## r = copularnd ("t", theta, df); ## @end group ## ## @group ## theta = 0.5; ## n = 2; ## r = copularnd ("Clayton", theta, n); ## @end group ## @end example ## ## @subheading References ## ## @enumerate ## @item ## Roger B. Nelsen. @cite{An Introduction to Copulas}. Springer, New York, ## second edition, 2006. ## @end enumerate ## @end deftypefn function r = copularnd (family, theta, df, n) ## Check arguments if (nargin < 2) print_usage (); endif if (! ischar (family)) error ("copularnd: family must be one of 'Gaussian', 't', and 'Clayton'."); endif lower_family = lower (family); ## Check family and copula parameters switch (lower_family) case {"gaussian"} ## Gaussian family if (isscalar (theta)) ## Expand a scalar to a correlation matrix theta = [1, theta; theta, 1]; endif if (! ismatrix (theta) || any (diag (theta) != 1) || ... any (any (theta != theta')) || min (eig (theta)) <= 0) error ("copularnd: THETA must be a correlation matrix."); endif if (nargin > 3) d = n; if (! isscalar (d) || d != size (theta, 1)) error ("copularnd: D must correspond to dimension of theta."); endif else d = size (theta, 1); endif if (nargin < 3) n = 1; else n = df; if (! isscalar (n) || (n < 0) || round (n) != n) error ("copularnd: N must be a non-negative integer."); endif endif case {"t"} ## Student's t family if (nargin < 3) print_usage (); endif if (isscalar (theta)) ## Expand a scalar to a correlation matrix theta = [1, theta; theta, 1]; endif if (! ismatrix (theta) || any (diag (theta) != 1) || ... any (any (theta != theta')) || min (eig (theta)) <= 0) error ("copularnd: THETA must be a correlation matrix."); endif if (! isscalar (df) && (! isvector (df) || length (df) != n)) error (strcat (["copularnd: DF must be a vector with the same"], ... [" number of rows as r or be scalar."])); endif df = df(:); if (nargin < 4) n = 1; else if (! isscalar (n) || (n < 0) || round (n) != n) error ("copularnd: N must be a non-negative integer."); endif endif case {"clayton"} ## Archimedian one parameter family if (nargin < 4) ## Default is bivariate d = 2; else d = n; if (! isscalar (d) || (d < 2) || round (d) != d) error ("copularnd: D must be an integer greater than 1."); endif endif if (nargin < 3) ## Default is one sample n = 1; else n = df; if (! isscalar (n) || (n < 0) || round (n) != n) error ("copularnd: N must be a non-negative integer."); endif endif if (! isvector (theta) || (! isscalar (theta) && size (theta, 1) != n)) error (strcaty (["copularnd: THETA must be a column vector with"], ... [" the number of rows equal to N or be scalar."])); endif if (n > 1 && isscalar (theta)) theta = repmat (theta, n, 1); endif otherwise error ("copularnd: unknown copula family '%s'.", family); endswitch if (n == 0) ## Input is empty r = zeros (0, d); else ## Draw random samples according to family switch (lower_family) case {"gaussian"} ## The Gaussian family r = normcdf (mvnrnd (zeros (1, d), theta, n), 0, 1); ## No parameter bounds check k = []; case {"t"} ## The Student's t family r = tcdf (mvtrnd (theta, df, n), df); ## No parameter bounds check k = []; case {"clayton"} ## The Clayton family u = rand (n, d); if (d == 2) r = zeros (n, 2); ## Conditional distribution method for the bivariate case which also ## works for theta < 0 r(:, 1) = u(:, 1); r(:, 2) = (1 + u(:, 1) .^ (-theta) .* (u(:, 2) .^ ... (-theta ./ (1 + theta)) - 1)) .^ (-1 ./ theta); else ## Apply the algorithm by Marshall and Olkin: ## Frailty distribution for Clayton copula is gamma y = randg (1 ./ theta, n, 1); r = (1 - log (u) ./ repmat (y, 1, d)) .^ (-1 ./ repmat (theta, 1, d)); endif k = find (theta == 0); if (any (k)) ## Product copula at columns k r(k, :) = u(k, :); endif ## Continue argument check if (d == 2) k = find (! (theta >= -1) | ! (theta < inf)); else k = find (! (theta >= 0) | ! (theta < inf)); endif endswitch ## Out of bounds parameters if (any (k)) r(k, :) = NaN; endif endif endfunction ## Test output %!test %! theta = 0.5; %! r = copularnd ("Gaussian", theta); %! assert (size (r), [1, 2]); %! assert (all ((r >= 0) & (r <= 1))); %!test %! theta = 0.5; %! df = 2; %! r = copularnd ("t", theta, df); %! assert (size (r), [1, 2]); %! assert (all ((r >= 0) & (r <= 1))); %!test %! theta = 0.5; %! r = copularnd ("Clayton", theta); %! assert (size (r), [1, 2]); %! assert (all ((r >= 0) & (r <= 1))); %!test %! theta = 0.5; %! n = 2; %! r = copularnd ("Clayton", theta, n); %! assert (size (r), [n, 2]); %! assert (all ((r >= 0) & (r <= 1))); %!test %! theta = [1; 2]; %! n = 2; %! d = 3; %! r = copularnd ("Clayton", theta, n, d); %! assert (size (r), [n, d]); %! assert (all ((r >= 0) & (r <= 1))); statistics-release-1.6.3/inst/dist_fun/evcdf.m000066400000000000000000000215271456127120000213730ustar00rootroot00000000000000## Copyright (C) 2022-2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{p} =} evcdf (@var{x}) ## @deftypefnx {statistics} {@var{p} =} evcdf (@var{x}, @var{mu}) ## @deftypefnx {statistics} {@var{p} =} evcdf (@var{x}, @var{mu}, @var{sigma}) ## @deftypefnx {statistics} {@var{p} =} evcdf (@dots{}, @qcode{"upper"}) ## @deftypefnx {statistics} {[@var{p}, @var{plo}, @var{pup}] =} evcdf (@var{x}, @var{mu}, @var{sigma}, @var{pcov}) ## @deftypefnx {statistics} {[@var{p}, @var{plo}, @var{pup}] =} evcdf (@var{x}, @var{mu}, @var{sigma}, @var{pcov}, @var{alpha}) ## @deftypefnx {statistics} {[@var{p}, @var{plo}, @var{pup}] =} evcdf (@dots{}, @qcode{"upper"}) ## ## Extreme value cumulative distribution function (CDF). ## ## For each element of @var{x}, compute the cumulative distribution function ## (CDF) of the extreme value distribution (also known as the Gumbel or the type ## I generalized extreme value distribution) at the values in @var{x} with ## location parameter @var{mu} and scale parameter @var{sigma}. The size of ## @var{p} is the common size of @var{x}, @var{mu} and @var{sigma}. A scalar ## input functions as a constant matrix of the same size as the other inputs. ## ## Default values are @var{mu} = 0 and @var{sigma} = 1. ## ## When called with three output arguments, i.e. @qcode{[@var{p}, @var{plo}, ## @var{pup}]}, @code{evcdf} computes the confidence bounds for @var{p} when the ## input parameters @var{mu} and @var{sigma} are estimates. In such case, ## @var{pcov}, a @math{2x2} matrix containing the covariance matrix of the ## estimated parameters, is necessary. Optionally, @var{alpha}, which has a ## default value of 0.05, specifies the @qcode{100 * (1 - @var{alpha})} percent ## confidence bounds. @var{plo} and @var{pup} are arrays of the same size as ## @var{p} containing the lower and upper confidence bounds. ## ## @code{[@dots{}] = evcdf (@dots{}, "upper")} computes the upper tail ## probability of the extreme value distribution with parameters @var{x0} and ## @var{gamma}, at the values in @var{x}. ## ## The Gumbel distribution is used to model the distribution of the maximum (or ## the minimum) of a number of samples of various distributions. This version ## is suitable for modeling minima. For modeling maxima, use the alternative ## Gumbel CDF, @code{gumbelcdf}. ## ## Further information about the Gumbel distribution can be found at ## @url{https://en.wikipedia.org/wiki/Gumbel_distribution} ## ## @seealso{evinv, evpdf, evrnd, evfit, evlike, evstat, gumbelcdf} ## @end deftypefn function [varargout] = evcdf (x, varargin) ## Check for valid number of input arguments if (nargin < 1 || nargin > 6) error ("evcdf: invalid number of input arguments."); endif ## Check for 'upper' flag if (nargin > 1 && strcmpi (varargin{end}, "upper")) uflag = true; varargin(end) = []; elseif (nargin > 1 && ischar (varargin{end}) && ... ! strcmpi (varargin{end}, "upper")) error ("evcdf: invalid argument for upper tail."); else uflag = false; endif ## Get extra arguments (if they exist) or add defaults if (numel (varargin) > 0) mu = varargin{1}; else mu = 0; endif if (numel (varargin) > 1) sigma = varargin{2}; else sigma = 1; endif if (numel (varargin) > 2) pcov = varargin{3}; ## Check for valid covariance matrix 2x2 if (! isequal (size (pcov), [2, 2])) error ("evcdf: invalid size of covariance matrix."); endif else ## Check that cov matrix is provided if 3 output arguments are requested if (nargout > 1) error ("evcdf: covariance matrix is required for confidence bounds."); endif pcov = []; endif if (numel (varargin) > 3) alpha = varargin{4}; ## Check for valid alpha value if (! isnumeric (alpha) || numel (alpha) !=1 || alpha <= 0 || alpha >= 1) error ("evcdf: invalid value for alpha."); endif else alpha = 0.05; endif ## Check for common size of X, MU, and SIGMA if (! isscalar (x) || ! isscalar (mu) || ! isscalar (sigma)) [err, x, mu, sigma] = common_size (x, mu, sigma); if (err > 0) error ("evcdf: X, MU, and SIGMA must be of common size or scalars."); endif endif ## Check for X, MU, and SIGMA being reals if (iscomplex (x) || iscomplex (mu) || iscomplex (sigma)) error ("evcdf: X, MU, and SIGMA must not be complex."); endif ## Return NaNs for out of range parameters. sigma(sigma <= 0) = NaN; ## Compute extreme value cdf z = (x - mu) ./ sigma; if (uflag) p = exp (-exp (z)); else p = -expm1 (-exp (z)); endif ## Check for appropriate class if (isa (x, "single") || isa (mu, "single") || isa (sigma, "single")); is_class = "single"; else is_class = "double"; endif ## Prepare output varargout{1} = cast (p, is_class); if (nargout > 1) plo = NaN (size (z), is_class); pup = NaN (size (z), is_class); endif ## Check sigma if (isscalar (sigma)) if (sigma > 0) sigma_p = true (size (z)); else if (nargout == 3) varargout{2} = plo; varargout{3} = pup; endif return; endif else sigma_p = sigma > 0; endif ## Compute confidence bounds (if requested) if (nargout >= 2) zvar = (pcov(1,1) + 2 * pcov(1,2) * z(sigma_p) + ... pcov(2,2) * z(sigma_p) .^ 2) ./ (sigma .^ 2); if (any (zvar < 0)) error ("evcdf: bad covariance matrix."); endif normz = -probit (alpha / 2); halfwidth = normz * sqrt (zvar); zlo = z(sigma_p) - halfwidth; zup = z(sigma_p) + halfwidth; if (uflag) plo(sigma_p) = exp (-exp (zup)); pup(sigma_p) = exp (-exp (zlo)); else plo(sigma_p) = -expm1 (-exp (zlo)); pup(sigma_p) = -expm1 (-exp (zup)); endif varargout{2} = plo; varargout{3} = pup; endif endfunction %!demo %! ## Plot various CDFs from the extreme value distribution %! x = -10:0.01:10; %! p1 = evcdf (x, 0.5, 2); %! p2 = evcdf (x, 1.0, 2); %! p3 = evcdf (x, 1.5, 3); %! p4 = evcdf (x, 3.0, 4); %! plot (x, p1, "-b", x, p2, "-g", x, p3, "-r", x, p4, "-c") %! grid on %! legend ({"μ = 0.5, σ = 2", "μ = 1.0, σ = 2", ... %! "μ = 1.5, σ = 3", "μ = 3.0, σ = 4"}, "location", "southeast") %! title ("Extreme value CDF") %! xlabel ("values in x") %! ylabel ("probability") ## Test output %!shared x, y %! x = [-Inf, 1, 2, Inf]; %! y = [0, 0.6321, 0.9340, 1]; %!assert (evcdf (x, ones (1,4), ones (1,4)), y, 1e-4) %!assert (evcdf (x, 1, ones (1,4)), y, 1e-4) %!assert (evcdf (x, ones (1,4), 1), y, 1e-4) %!assert (evcdf (x, [0, -Inf, NaN, Inf], 1), [0, 1, NaN, NaN], 1e-4) %!assert (evcdf (x, 1, [Inf, NaN, -1, 0]), [NaN, NaN, NaN, NaN], 1e-4) %!assert (evcdf ([x(1:2), NaN, x(4)], 1, 1), [y(1:2), NaN, y(4)], 1e-4) %!assert (evcdf (x, "upper"), [1, 0.0660, 0.0006, 0], 1e-4) ## Test class of input preserved %!assert (evcdf ([x, NaN], 1, 1), [y, NaN], 1e-4) %!assert (evcdf (single ([x, NaN]), 1, 1), single ([y, NaN]), 1e-4) %!assert (evcdf ([x, NaN], single (1), 1), single ([y, NaN]), 1e-4) %!assert (evcdf ([x, NaN], 1, single (1)), single ([y, NaN]), 1e-4) ## Test input validation %!error evcdf () %!error evcdf (1,2,3,4,5,6,7) %!error evcdf (1, 2, 3, 4, "uper") %!error ... %! evcdf (ones (3), ones (2), ones (2)) %!error evcdf (2, 3, 4, [1, 2]) %!error ... %! [p, plo, pup] = evcdf (1, 2, 3) %!error [p, plo, pup] = ... %! evcdf (1, 2, 3, [1, 0; 0, 1], 0) %!error [p, plo, pup] = ... %! evcdf (1, 2, 3, [1, 0; 0, 1], 1.22) %!error [p, plo, pup] = ... %! evcdf (1, 2, 3, [1, 0; 0, 1], "alpha", "upper") %!error evcdf (i, 2, 2) %!error evcdf (2, i, 2) %!error evcdf (2, 2, i) %!error ... %! [p, plo, pup] = evcdf (1, 2, 3, [1, 0; 0, -inf], 0.04) statistics-release-1.6.3/inst/dist_fun/evinv.m000066400000000000000000000165731456127120000214400ustar00rootroot00000000000000## Copyright (C) 2022-2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{x} =} evinv (@var{p}) ## @deftypefnx {statistics} {@var{x} =} evinv (@var{p}, @var{mu}) ## @deftypefnx {statistics} {@var{x} =} evinv (@var{p}, @var{mu}, @var{sigma}) ## @deftypefnx {statistics} {[@var{x}, @var{xlo}, @var{xup}] =} evinv (@var{p}, @var{mu}, @var{sigma}, @var{pcov}) ## @deftypefnx {statistics} {[@var{x}, @var{xlo}, @var{xup}] =} evinv (@var{p}, @var{mu}, @var{sigma}, @var{pcov}, @var{alpha}) ## ## Inverse of the extreme value cumulative distribution function (iCDF). ## ## For each element of @var{p}, compute the quantile (the inverse of the CDF) of ## the extreme value distribution (also known as the Gumbel or the type I ## generalized extreme value distribution) with location parameter @var{mu} and ## scale parameter @var{sigma}. The size of @var{x} is the common size of ## @var{p}, @var{mu} and @var{sigma}. A scalar input functions as a constant ## matrix of the same size as the other inputs. ## ## Default values are @var{mu} = 0 and @var{sigma} = 1. ## ## When called with three output arguments, i.e. @qcode{[@var{x}, @var{xlo}, ## @var{xup}]}, @code{evinv} computes the confidence bounds for @var{x} when the ## input parameters @var{mu} and @var{sigma} are estimates. In such case, ## @var{pcov}, a @math{2x2} matrix containing the covariance matrix of the ## estimated parameters, is necessary. Optionally, @var{alpha}, which has a ## default value of 0.05, specifies the @qcode{100 * (1 - @var{alpha})} percent ## confidence bounds. @var{xlo} and @var{xup} are arrays of the same size as ## @var{x} containing the lower and upper confidence bounds. ## ## The Gumbel distribution is used to model the distribution of the maximum (or ## the minimum) of a number of samples of various distributions. This version ## is suitable for modeling minima. For modeling maxima, use the alternative ## Gumbel iCDF, @code{gumbelinv}. ## ## Further information about the Gumbel distribution can be found at ## @url{https://en.wikipedia.org/wiki/Gumbel_distribution} ## ## @seealso{evcdf, evpdf, evrnd, evfit, evlike, evstat, gumbelinv} ## @end deftypefn function [x, xlo, xup] = evinv (p, mu, sigma, pcov, alpha) ## Check for valid number of input arguments if (nargin < 1 || nargin > 5) error ("evinv: invalid number of input arguments."); endif ## Add defaults (if missing input arguments) if (nargin < 2) mu = 0; endif if (nargin < 3) sigma = 1; endif ## Check if PCOV is provided when confidence bounds are requested if (nargout > 2) if (nargin < 4) error ("evinv: covariance matrix is required for confidence bounds."); endif ## Check for valid covariance matrix 2x2 if (! isequal (size (pcov), [2, 2])) error ("evinv: invalid size of covariance matrix."); endif ## Check for valid alpha value if (nargin < 5) alpha = 0.05; elseif (! isnumeric (alpha) || numel (alpha) !=1 || alpha <= 0 || alpha >= 1) error ("evinv: invalid value for alpha."); endif endif ## Check for common size of P, MU, and SIGMA if (! isscalar (p) || ! isscalar (mu) || ! isscalar (sigma)) [err, p, mu, sigma] = common_size (p, mu, sigma); if (err > 0) error ("evinv: P, MU, and SIGMA must be of common size or scalars."); endif endif ## Check for P, MU, and SIGMA being reals if (iscomplex (p) || iscomplex (mu) || iscomplex (sigma)) error ("evinv: P, MU, and SIGMA must not be complex."); endif ## Check for appropriate class if (isa (p, "single") || isa (mu, "single") || isa (sigma, "single")); is_class = "single"; else is_class = "double"; endif ## Compute inverse of type 1 extreme value cdf k = (eps <= p & p < 1); if (all (k(:))) q = log (-log (1 - p)); else q = zeros (size (p), is_class); q(k) = log (-log (1 - p(k))); ## Return -Inf for p = 0 and Inf for p = 1 q(p < eps) = -Inf; q(p == 1) = Inf; ## Return NaN for out of range values of P q(p < 0 | 1 < p | isnan (p)) = NaN; endif ## Return NaN for out of range values of SIGMA sigma(sigma <= 0) = NaN; x = sigma .* q + mu; ## Compute confidence bounds if requested. if (nargout >= 2) xvar = pcov(1,1) + 2 * pcov(1,2) * q + pcov(2,2) * q .^ 2; if (any (xvar < 0)) error ("evinv: bad covariance matrix."); endif z = -norminv (alpha / 2); halfwidth = z * sqrt (xvar); xlo = x - halfwidth; xup = x + halfwidth; endif endfunction %!demo %! ## Plot various iCDFs from the extreme value distribution %! p = 0.001:0.001:0.999; %! x1 = evinv (p, 0.5, 2); %! x2 = evinv (p, 1.0, 2); %! x3 = evinv (p, 1.5, 3); %! x4 = evinv (p, 3.0, 4); %! plot (p, x1, "-b", p, x2, "-g", p, x3, "-r", p, x4, "-c") %! grid on %! ylim ([-10, 10]) %! legend ({"μ = 0.5, σ = 2", "μ = 1.0, σ = 2", ... %! "μ = 1.5, σ = 3", "μ = 3.0, σ = 4"}, "location", "northwest") %! title ("Extreme value iCDF") %! xlabel ("probability") %! ylabel ("values in x") ## Test output %!shared p, x %! p = [0, 0.05, 0.5 0.95]; %! x = [-Inf, -2.9702, -0.3665, 1.0972]; %!assert (evinv (p), x, 1e-4) %!assert (evinv (p, zeros (1,4), ones (1,4)), x, 1e-4) %!assert (evinv (p, 0, ones (1,4)), x, 1e-4) %!assert (evinv (p, zeros (1,4), 1), x, 1e-4) %!assert (evinv (p, [0, -Inf, NaN, Inf], 1), [-Inf, -Inf, NaN, Inf], 1e-4) %!assert (evinv (p, 0, [Inf, NaN, -1, 0]), [-Inf, NaN, NaN, NaN], 1e-4) %!assert (evinv ([p(1:2), NaN, p(4)], 0, 1), [x(1:2), NaN, x(4)], 1e-4) ## Test class of input preserved %!assert (evinv ([p, NaN], 0, 1), [x, NaN], 1e-4) %!assert (evinv (single ([p, NaN]), 0, 1), single ([x, NaN]), 1e-4) %!assert (evinv ([p, NaN], single (0), 1), single ([x, NaN]), 1e-4) %!assert (evinv ([p, NaN], 0, single (1)), single ([x, NaN]), 1e-4) ## Test input validation %!error evinv () %!error evinv (1,2,3,4,5,6) %!error ... %! evinv (ones (3), ones (2), ones (2)) %!error ... %! [p, plo, pup] = evinv (2, 3, 4, [1, 2]) %!error ... %! [p, plo, pup] = evinv (1, 2, 3) %!error [p, plo, pup] = ... %! evinv (1, 2, 3, [1, 0; 0, 1], 0) %!error [p, plo, pup] = ... %! evinv (1, 2, 3, [1, 0; 0, 1], 1.22) %!error evinv (i, 2, 2) %!error evinv (2, i, 2) %!error evinv (2, 2, i) %!error ... %! [p, plo, pup] = evinv (1, 2, 3, [-1, -10; -Inf, -Inf], 0.04) statistics-release-1.6.3/inst/dist_fun/evpdf.m000066400000000000000000000103731456127120000214050ustar00rootroot00000000000000## Copyright (C) 2022-2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{y} =} evpdf (@var{x}) ## @deftypefnx {statistics} {@var{y} =} evpdf (@var{x}, @var{mu}) ## @deftypefnx {statistics} {@var{y} =} evpdf (@var{x}, @var{mu}, @var{sigma}) ## ## Extreme value probability density function (PDF). ## ## For each element of @var{x}, compute the probability density function (PDF) ## of the extreme value distribution (also known as the Gumbel or the type I ## generalized extreme value distribution) with location parameter @var{mu} and ## scale parameter @var{sigma}. The size of @var{y} is the common size of ## @var{x}, @var{mu} and @var{sigma}. A scalar input functions as a constant ## matrix of the same size as the other inputs. ## ## Default values are @var{mu} = 0 and @var{sigma} = 1. ## ## The Gumbel distribution is used to model the distribution of the maximum (or ## the minimum) of a number of samples of various distributions. This version ## is suitable for modeling minima. For modeling maxima, use the alternative ## Gumbel iCDF, @code{gumbelinv}. ## ## Further information about the Gumbel distribution can be found at ## @url{https://en.wikipedia.org/wiki/Gumbel_distribution} ## ## @seealso{evcdf, evinv, evrnd, evfit, evlike, evstat, gumbelpdf} ## @end deftypefn function y = evpdf (x, mu, sigma) ## Check for valid number of input arguments if (nargin < 1) error ("evpdf: function called with too few input arguments."); endif ## Add defaults (if missing input arguments) if (nargin < 2) mu = 0; endif if (nargin < 3) sigma = 1; endif ## Check for common size of X, MU, and SIGMA if (! isscalar (x) || ! isscalar (mu) || ! isscalar (sigma)) [err, x, mu, sigma] = common_size (x, mu, sigma); if (err > 0) error ("evpdf: X, MU, and SIGMA must be of common size or scalars."); endif endif ## Check for X, MU, and SIGMA being reals if (iscomplex (x) || iscomplex (mu) || iscomplex (sigma)) error ("evpdf: X, MU, and SIGMA must not be complex."); endif ## Return NaNs for out of range parameters sigma(sigma <= 0) = NaN; ## Compute pdf of type 1 extreme value distribution z = (x - mu) ./ sigma; y = exp (z - exp (z)) ./ sigma; ## Force 0 for extreme right tail, instead of getting exp (Inf - Inf) = NaN y(z == Inf) = 0; endfunction %!demo %! ## Plot various PDFs from the Extreme value distribution %! x = -10:0.001:10; %! y1 = evpdf (x, 0.5, 2); %! y2 = evpdf (x, 1.0, 2); %! y3 = evpdf (x, 1.5, 3); %! y4 = evpdf (x, 3.0, 4); %! plot (x, y1, "-b", x, y2, "-g", x, y3, "-r", x, y4, "-c") %! grid on %! ylim ([0, 0.2]) %! legend ({"μ = 0.5, σ = 2", "μ = 1.0, σ = 2", ... %! "μ = 1.5, σ = 3", "μ = 3.0, σ = 4"}, "location", "northeast") %! title ("Extreme value PDF") %! xlabel ("values in x") %! ylabel ("density") ## Test output %!shared x, y0, y1 %! x = [-5, 0, 1, 2, 3]; %! y0 = [0.0067, 0.3679, 0.1794, 0.0046, 0]; %! y1 = [0.0025, 0.2546, 0.3679, 0.1794, 0.0046]; %!assert (evpdf (x), y0, 1e-4) %!assert (evpdf (x, zeros (1,5), ones (1,5)), y0, 1e-4) %!assert (evpdf (x, ones (1,5), ones (1,5)), y1, 1e-4) ## Test input validation %!error evpdf () %!error ... %! evpdf (ones (3), ones (2), ones (2)) %!error evpdf (i, 2, 2) %!error evpdf (2, i, 2) %!error evpdf (2, 2, i) statistics-release-1.6.3/inst/dist_fun/evrnd.m000066400000000000000000000152321456127120000214160ustar00rootroot00000000000000## Copyright (C) 2022-2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{r} =} evrnd (@var{mu}, @var{sigma}) ## @deftypefnx {statistics} {@var{r} =} evrnd (@var{mu}, @var{sigma}, @var{rows}) ## @deftypefnx {statistics} {@var{r} =} evrnd (@var{mu}, @var{sigma}, @var{rows}, @var{cols}, @dots{}) ## @deftypefnx {statistics} {@var{r} =} evrnd (@var{mu}, @var{sigma}, [@var{sz}]) ## ## Random arrays from the extreme value distribution. ## ## @code{@var{r} = evrnd (@var{mu}, @var{sigma})} returns an array of random ## numbers chosen from the extreme value distribution (also known as the Gumbel ## or the type I generalized extreme value distribution) with location ## parameter @var{mu} and scale parameter @var{sigma}. The size of @var{r} is ## the common size of @var{mu} and @var{sigma}. A scalar input functions as a ## constant matrix of the same size as the other inputs. ## ## When called with a single size argument, @code{evrnd} returns a square ## matrix with the dimension specified. When called with more than one scalar ## argument, the first two arguments are taken as the number of rows and columns ## and any further arguments specify additional matrix dimensions. The size may ## also be specified with a row vector of dimensions, @var{sz}. ## ## The Gumbel distribution is used to model the distribution of the maximum (or ## the minimum) of a number of samples of various distributions. This version ## is suitable for modeling minima. For modeling maxima, use the alternative ## Gumbel iCDF, @code{gumbelinv}. ## ## Further information about the Gumbel distribution can be found at ## @url{https://en.wikipedia.org/wiki/Gumbel_distribution} ## ## @seealso{evcdf, evinv, evpdf, evfit, evlike, evstat} ## @end deftypefn function r = evrnd (mu, sigma, varargin) ## Check for valid number of input arguments if (nargin < 2) error ("evrnd: function called with too few input arguments."); endif ## Check for common size of MU and SIGMA if (! isscalar (mu) || ! isscalar (sigma)) [retval, mu, sigma] = common_size (mu, sigma); if (retval > 0) error ("evrnd: MU and SIGMA must be of common size or scalars."); endif endif ## Check for MU and SIGMA being reals if (iscomplex (mu) || iscomplex (sigma)) error ("evrnd: MU and SIGMA must not be complex."); endif ## Parse and check SIZE arguments if (nargin == 2) sz = size (mu); elseif (nargin == 3) if (isscalar (varargin{1}) && varargin{1} >= 0 ... && varargin{1} == fix (varargin{1})) sz = [varargin{1}, varargin{1}]; elseif (isrow (varargin{1}) && all (varargin{1} >= 0) ... && all (varargin{1} == fix (varargin{1}))) sz = varargin{1}; elseif error (strcat (["evrnd: SZ must be a scalar or a row vector"], ... [" of non-negative integers."])); endif elseif (nargin > 3) posint = cellfun (@(x) (! isscalar (x) || x < 0 || x != fix (x)), varargin); if (any (posint)) error ("evrnd: dimensions must be non-negative integers."); endif sz = [varargin{:}]; endif ## Check that parameters match requested dimensions in size if (! isscalar (mu) && ! isequal (size (mu), sz)) error ("evrnd: MU and SIGMA must be scalars or of size SZ."); endif ## Check for class type if (isa (mu, "single") || isa (sigma, "single")) cls = "single"; else cls = "double"; endif ## Return NaNs for out of range values of SIGMA sigma(sigma < 0) = NaN; ## Generate uniform random values, and apply the extreme value inverse CDF. r = log (-log (rand (sz, cls))) .* sigma + mu; endfunction ## Test output %!assert (size (evrnd (1, 1)), [1 1]) %!assert (size (evrnd (1, ones (2,1))), [2, 1]) %!assert (size (evrnd (1, ones (2,2))), [2, 2]) %!assert (size (evrnd (ones (2,1), 1)), [2, 1]) %!assert (size (evrnd (ones (2,2), 1)), [2, 2]) %!assert (size (evrnd (1, 1, 3)), [3, 3]) %!assert (size (evrnd (1, 1, [4, 1])), [4, 1]) %!assert (size (evrnd (1, 1, 4, 1)), [4, 1]) %!assert (size (evrnd (1, 1, 4, 1, 5)), [4, 1, 5]) %!assert (size (evrnd (1, 1, 0, 1)), [0, 1]) %!assert (size (evrnd (1, 1, 1, 0)), [1, 0]) %!assert (size (evrnd (1, 1, 1, 2, 0, 5)), [1, 2, 0, 5]) ## Test class of input preserved %!assert (class (evrnd (1, 1)), "double") %!assert (class (evrnd (1, single (1))), "single") %!assert (class (evrnd (1, single ([1, 1]))), "single") %!assert (class (evrnd (single (1), 1)), "single") %!assert (class (evrnd (single ([1, 1]), 1)), "single") ## Test input validation %!error evrnd () %!error evrnd (1) %!error ... %! evrnd (ones (3), ones (2)) %!error ... %! evrnd (ones (2), ones (3)) %!error evrnd (i, 2, 3) %!error evrnd (1, i, 3) %!error ... %! evrnd (1, 2, -1) %!error ... %! evrnd (1, 2, 1.2) %!error ... %! evrnd (1, 2, ones (2)) %!error ... %! evrnd (1, 2, [2 -1 2]) %!error ... %! evrnd (1, 2, [2 0 2.5]) %!error ... %! evrnd (1, 2, 2, -1, 5) %!error ... %! evrnd (1, 2, 2, 1.5, 5) %!error ... %! evrnd (2, ones (2), 3) %!error ... %! evrnd (2, ones (2), [3, 2]) %!error ... %! evrnd (2, ones (2), 3, 2) statistics-release-1.6.3/inst/dist_fun/expcdf.m000066400000000000000000000201701456127120000215460ustar00rootroot00000000000000## Copyright (C) 2012 Rik Wehbring ## Copyright (C) 1995-2016 Kurt Hornik ## Copyright (C) 2023 Andreas Bertsatos ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{p} =} expcdf (@var{x}) ## @deftypefnx {statistics} {@var{p} =} expcdf (@var{x}, @var{mu}) ## @deftypefnx {statistics} {@var{p} =} expcdf (@dots{}, @qcode{"upper"}) ## @deftypefnx {statistics} {[@var{p}, @var{plo}, @var{pup}] =} expcdf (@var{x}, @var{mu}, @var{pcov}) ## @deftypefnx {statistics} {[@var{p}, @var{plo}, @var{pup}] =} expcdf (@var{x}, @var{mu}, @var{pcov}, @var{alpha}) ## @deftypefnx {statistics} {[@var{p}, @var{plo}, @var{pup}] =} expcdf (@dots{}, @qcode{"upper"}) ## ## Exponential cumulative distribution function (CDF). ## ## For each element of @var{x}, compute the cumulative distribution function ## (CDF) of the exponential distribution with mean parameter @var{mu}. The size ## of @var{p} is the common size of @var{x} and @var{mu}. A scalar input ## functions as a constant matrix of the same size as the other inputs. ## ## Default value is @var{mu} = 1. ## ## A common alternative parameterization of the exponential distribution is to ## use the parameter @math{λ} defined as the mean number of events in an ## interval as opposed to the parameter @math{μ}, which is the mean wait time ## for an event to occur. @math{λ} and @math{μ} are reciprocals, ## i.e. @math{μ = 1 / λ}. ## ## When called with three output arguments, i.e. @qcode{[@var{p}, @var{plo}, ## @var{pup}]}, @code{expcdf} computes the confidence bounds for @var{p} when ## the input parameter @var{mu} is an estimate. In such case, @var{pcov}, a ## scalar value with the variance of the estimated parameter @var{mu}, is ## necessary. Optionally, @var{alpha}, which has a default value of 0.05, ## specifies the @qcode{100 * (1 - @var{alpha})} percent confidence bounds. ## @var{plo} and @var{pup} are arrays of the same size as @var{p} containing the ## lower and upper confidence bounds. ## ## @code{[@dots{}] = expcdf (@dots{}, "upper")} computes the upper tail ## probability of the exponential distribution with parameter @var{mu}, at the ## values in @var{x}. ## ## Further information about the exponential distribution can be found at ## @url{https://en.wikipedia.org/wiki/Exponential_distribution} ## ## @seealso{expinv, exppdf, exprnd, expfit, explike, expstat} ## @end deftypefn function [varargout] = expcdf (x, varargin) ## Check for valid number of input arguments if (nargin < 1 || nargin > 5) error ("expcdf: invalid number of input arguments."); endif ## Check for "upper" flag if (nargin > 1 && strcmpi (varargin{end}, "upper")) uflag = true; varargin(end) = []; elseif (nargin > 1 && ischar (varargin{end}) && ... ! strcmpi (varargin{end}, "upper")) error ("expcdf: invalid argument for upper tail."); else uflag = false; endif ## Get extra arguments (if they exist) or add defaults if (numel (varargin) > 0) mu = varargin{1}; else mu = 1; endif if (numel (varargin) > 1) pcov = varargin{2}; ## Check for variance being a scalar if (! isscalar (pcov)) error ("expcdf: invalid size of variance, PCOV must be a scalar."); endif if (pcov < 0) error ("expcdf: variance, PCOV, cannot be negative."); endif else ## Check that cov matrix is provided if 3 output arguments are requested if (nargout > 1) error ("expcdf: variance, PCOV, is required for confidence bounds."); endif pcov = []; endif if (numel (varargin) > 2) alpha = varargin{3}; ## Check for valid alpha value if (! isnumeric (alpha) || numel (alpha) != 1 || alpha <= 0 || alpha >= 1) error ("expcdf: invalid value for alpha."); endif else alpha = 0.05; endif ## Check for common size of X and MU if (! isscalar (x) || ! isscalar (mu)) [err, x, mu] = common_size (x, mu); if (err > 0) error ("expcdf: X and MU must be of common size or scalars."); endif endif ## Check for X and MU being reals if (iscomplex (x) || iscomplex (mu)) error ("expcdf: X and MU must not be complex."); endif ## Check for appropriate class if (isa (x, "single") || isa (mu, "single")); is_class = "single"; else is_class = "double"; endif ## Return NaNs for out of range parameters. mu(mu <= 0) = NaN; ## Compute P value for exponential cdf z = x ./ mu; ## Force 0 for negative X z(z < 0) = 0; ## Check uflag if (uflag) p = exp (-z); else p = -expm1 (-z); endif ## Prepare output varargout{1} = cast (p, is_class); if (nargout > 1) plo = NaN (size (z), is_class); pup = NaN (size (z), is_class); endif ## Compute confidence bounds (if requested) if (nargout >= 2) ## Convert to log scale log_z = log (z); norm_z = -probit (alpha / 2); halfwidth = norm_z * sqrt (pcov ./ (mu .^ 2)); zlo = log_z - halfwidth; zup = log_z + halfwidth; ## Convert to original scale if (uflag) plo = exp (-exp (zup)); pup = exp (-exp (zlo)); else plo = - expm1 (-exp (zlo)); pup = - expm1 (-exp (zup)); endif ## Prepare output varargout{2} = plo; varargout{3} = pup; endif endfunction %!demo %! ## Plot various CDFs from the exponential distribution %! x = 0:0.01:5; %! p1 = expcdf (x, 2/3); %! p2 = expcdf (x, 1.0); %! p3 = expcdf (x, 2.0); %! plot (x, p1, "-b", x, p2, "-g", x, p3, "-r") %! grid on %! legend ({"μ = 2/3", "μ = 1", "μ = 2"}, "location", "southeast") %! title ("Exponential CDF") %! xlabel ("values in x") %! ylabel ("probability") ## Test output %!shared x, p %! x = [-1 0 0.5 1 Inf]; %! p = [0, 1 - exp(-x(2:end)/2)]; %!assert (expcdf (x, 2*ones (1,5)), p) %!assert (expcdf (x, 2), p) %!assert (expcdf (x, 2*[1 0 NaN 1 1]), [0 NaN NaN p(4:5)]) ## Test class of input preserved %!assert (expcdf ([x, NaN], 2), [p, NaN]) %!assert (expcdf (single ([x, NaN]), 2), single ([p, NaN])) %!assert (expcdf ([x, NaN], single (2)), single ([p, NaN])) ## Test values against MATLAB output %!test %! [p, plo, pup] = expcdf (1, 2, 3); %! assert (p, 0.39346934028737, 1e-14); %! assert (plo, 0.08751307220484, 1e-14); %! assert (pup, 0.93476821257933, 1e-14); %!test %! [p, plo, pup] = expcdf (1, 2, 2, 0.1); %! assert (p, 0.39346934028737, 1e-14); %! assert (plo, 0.14466318041675, 1e-14); %! assert (pup, 0.79808291849140, 1e-14); %!test %! [p, plo, pup] = expcdf (1, 2, 2, 0.1, "upper"); %! assert (p, 0.60653065971263, 1e-14); %! assert (plo, 0.20191708150860, 1e-14); %! assert (pup, 0.85533681958325, 1e-14); ## Test input validation %!error expcdf () %!error expcdf (1, 2 ,3 ,4 ,5, 6) %!error expcdf (1, 2, 3, 4, "uper") %!error ... %! expcdf (ones (3), ones (2)) %!error ... %! expcdf (2, 3, [1, 2]) %!error ... %! [p, plo, pup] = expcdf (1, 2) %!error [p, plo, pup] = ... %! expcdf (1, 2, 3, 0) %!error [p, plo, pup] = ... %! expcdf (1, 2, 3, 1.22) %!error [p, plo, pup] = ... %! expcdf (1, 2, 3, "alpha", "upper") %!error expcdf (i, 2) %!error expcdf (2, i) %!error ... %! [p, plo, pup] = expcdf (1, 2, -1, 0.04) statistics-release-1.6.3/inst/dist_fun/expinv.m000066400000000000000000000157561456127120000216240ustar00rootroot00000000000000## Copyright (C) 2012 Rik Wehbring ## Copyright (C) 1995-2016 Kurt Hornik ## Copyright (C) 2023 Andreas Bertsatos ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{x} =} expinv (@var{p}) ## @deftypefnx {statistics} {@var{x} =} expinv (@var{p}, @var{mu}) ## @deftypefnx {statistics} {[@var{x}, @var{xlo}, @var{xup}] =} expinv (@var{p}, @var{mu}, @var{pcov}) ## @deftypefnx {statistics} {[@var{x}, @var{xlo}, @var{xup}] =} expinv (@var{p}, @var{mu}, @var{pcov}, @var{alpha}) ## ## Inverse of the exponential cumulative distribution function (iCDF). ## ## For each element of @var{p}, compute the quantile (the inverse of the CDF) of ## the exponential distribution with mean @var{mu}. The size of @var{x} is the ## common size of @var{p} and @var{mu}. A scalar input functions as a constant ## matrix of the same size as the other inputs. ## ## Default value is @var{mu} = 1. ## ## A common alternative parameterization of the exponential distribution is to ## use the parameter @math{λ} defined as the mean number of events in an ## interval as opposed to the parameter @math{μ}, which is the mean wait time ## for an event to occur. @math{λ} and @math{μ} are reciprocals, ## i.e. @math{μ = 1 / λ}. ## ## When called with three output arguments, i.e. @qcode{[@var{x}, @var{xlo}, ## @var{xup}]}, @code{expinv} computes the confidence bounds for @var{x} when ## the input parameter @var{mu} is an estimate. In such case, @var{pcov}, a ## scalar value with the variance of the estimated parameter @var{mu}, is ## necessary. Optionally, @var{alpha}, which has a default value of 0.05, ## specifies the @qcode{100 * (1 - @var{alpha})} percent confidence bounds. ## @var{xlo} and @var{xup} are arrays of the same size as @var{x} containing the ## lower and upper confidence bounds. ## ## Further information about the exponential distribution can be found at ## @url{https://en.wikipedia.org/wiki/Exponential_distribution} ## ## @seealso{expcdf, exppdf, exprnd, expfit, explike, expstat} ## @end deftypefn function [varargout] = expinv (p, varargin) ## Check for valid number of input arguments if (nargin < 1 || nargin > 4) error ("expinv: invalid number of input arguments."); endif ## Get extra arguments (if they exist) or add defaults if (numel (varargin) > 0) mu = varargin{1}; else mu = 1; endif if (numel (varargin) > 1) pcov = varargin{2}; ## Check for variance being a scalar if (! isscalar (pcov)) error ("expinv: invalid size of variance, PCOV must be a scalar."); endif if (pcov < 0) error ("expinv: variance, PCOV, cannot be negative."); endif else ## Check that cov matrix is provided if 3 output arguments are requested if (nargout > 1) error ("expinv: variance, PCOV, is required for confidence bounds."); endif pcov = []; endif if (numel (varargin) > 2) alpha = varargin{3}; ## Check for valid alpha value if (! isnumeric (alpha) || numel (alpha) != 1 || alpha <= 0 || alpha >= 1) error ("expinv: invalid value for alpha."); endif else alpha = 0.05; endif ## Check for common size of P and MU if (! isscalar (p) || ! isscalar (mu)) [retval, p, mu] = common_size (p, mu); if (retval > 0) error ("expinv: P and MU must be of common size or scalars."); endif endif ## Check for P and MU being reals if (iscomplex (p) || iscomplex (mu)) error ("expinv: P and MU must not be complex."); endif ## Check for appropriate class if (isa (p, "single") || isa (mu, "single")); is_class = "single"; else is_class = "double"; endif ## Create output matrix if (isa (p, "single") || isa (mu, "single")) x = NaN (size (p), "single"); else x = NaN (size (p)); endif ## Handle edge cases k = (p == 1) & (mu > 0); x(k) = Inf; ## Handle valid cases k = (p >= 0) & (p < 1) & (mu > 0); if (isscalar (mu)) x(k) = - mu * log (1 - p(k)); else x(k) = - mu(k) .* log (1 - p(k)); endif ## Prepare output varargout{1} = cast (x, is_class); if (nargout > 1) xlo = NaN (size (z), is_class); xup = NaN (size (z), is_class); endif ## Compute confidence bounds (if requested) if (nargout >= 2) ## Convert to log scale log_x = log (x); z = -probit (alpha / 2); halfwidth = z * sqrt (pcov ./ (mu.^2)); ## Convert to original scale xlo = exp (log_x - halfwidth); xup = exp (log_x + halfwidth); ## Prepare output varargout{2} = plo; varargout{3} = pup; endif endfunction %!demo %! ## Plot various iCDFs from the exponential distribution %! p = 0.001:0.001:0.999; %! x1 = expinv (p, 2/3); %! x2 = expinv (p, 1.0); %! x3 = expinv (p, 2.0); %! plot (p, x1, "-b", p, x2, "-g", p, x3, "-r") %! grid on %! ylim ([0, 5]) %! legend ({"μ = 2/3", "μ = 1", "μ = 2"}, "location", "northwest") %! title ("Exponential iCDF") %! xlabel ("probability") %! ylabel ("values in x") ## Test output %!shared p %! p = [-1 0 0.3934693402873666 1 2]; %!assert (expinv (p, 2*ones (1,5)), [NaN 0 1 Inf NaN], eps) %!assert (expinv (p, 2), [NaN 0 1 Inf NaN], eps) %!assert (expinv (p, 2*[1 0 NaN 1 1]), [NaN NaN NaN Inf NaN], eps) %!assert (expinv ([p(1:2) NaN p(4:5)], 2), [NaN 0 NaN Inf NaN], eps) ## Test class of input preserved %!assert (expinv ([p, NaN], 2), [NaN 0 1 Inf NaN NaN], eps) %!assert (expinv (single ([p, NaN]), 2), single ([NaN 0 1 Inf NaN NaN]), eps) %!assert (expinv ([p, NaN], single (2)), single ([NaN 0 1 Inf NaN NaN]), eps) ## Test input validation %!error expinv () %!error expinv (1, 2 ,3 ,4 ,5) %!error ... %! expinv (ones (3), ones (2)) %!error ... %! expinv (2, 3, [1, 2]) %!error ... %! [x, xlo, xup] = expinv (1, 2) %!error [x, xlo, xup] = ... %! expinv (1, 2, 3, 0) %!error [x, xlo, xup] = ... %! expinv (1, 2, 3, 1.22) %!error [x, xlo, xup] = ... %! expinv (1, 2, 3, [0.05, 0.1]) %!error expinv (i, 2) %!error expinv (2, i) %!error ... %! [x, xlo, xup] = expinv (1, 2, -1, 0.04) statistics-release-1.6.3/inst/dist_fun/exppdf.m000066400000000000000000000100551456127120000215640ustar00rootroot00000000000000## Copyright (C) 2012 Rik Wehbring ## Copyright (C) 1995-2016 Kurt Hornik ## Copyright (C) 2023 Andreas Bertsatos ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{y} =} exppdf (@var{x}) ## @deftypefnx {statistics} {@var{y} =} exppdf (@var{x}, @var{mu}) ## ## Exponential probability density function (PDF). ## ## For each element of @var{x}, compute the probability density function (PDF) ## of the exponential distribution with mean parameter @var{mu}. The size of ## @var{y} is the common size of @var{x} and @var{mu}. A scalar input functions ## as a constant matrix of the same size as the other inputs. ## ## Default value for @var{mu} = 1. ## ## A common alternative parameterization of the exponential distribution is to ## use the parameter @math{λ} defined as the mean number of events in an ## interval as opposed to the parameter @math{μ}, which is the mean wait time ## for an event to occur. @math{λ} and @math{μ} are reciprocals, ## i.e. @math{μ = 1 / λ}. ## ## Further information about the exponential distribution can be found at ## @url{https://en.wikipedia.org/wiki/Exponential_distribution} ## ## @seealso{expcdf, expinv, exprnd, expfit, explike, expstat} ## @end deftypefn function y = exppdf (x, mu) ## Check for valid number of input arguments if (nargin < 1) error ("exppdf: function called with too few input arguments."); endif ## Add defaults (if missing input arguments) if (nargin < 2) mu = 0; endif ## Check for common size of X and MU if (! isscalar (x) || ! isscalar (mu)) [retval, x, mu] = common_size (x, mu); if (retval > 0) error ("exppdf: X and MU must be of common size or scalars."); endif endif ## Check for X and MU being reals if (iscomplex (x) || iscomplex (mu)) error ("exppdf: X and MU must not be complex."); endif ## Check for appropriate class if (isa (x, "single") || isa (mu, "single")) y = zeros (size (x), "single"); else y = zeros (size (x)); endif k = isnan (x) | !(mu > 0); y(k) = NaN; k = (x >= 0) & (x < Inf) & (mu > 0); if (isscalar (mu)) y(k) = exp (-x(k) / mu) / mu; else y(k) = exp (-x(k) ./ mu(k)) ./ mu(k); endif endfunction %!demo %! ## Plot various PDFs from the exponential distribution %! x = 0:0.01:5; %! y1 = exppdf (x, 2/3); %! y2 = exppdf (x, 1.0); %! y3 = exppdf (x, 2.0); %! plot (x, y1, "-b", x, y2, "-g", x, y3, "-r") %! grid on %! ylim ([0, 1.5]) %! legend ({"μ = 2/3", "μ = 1", "μ = 2"}, "location", "northeast") %! title ("Exponential PDF") %! xlabel ("values in x") %! ylabel ("density") ## Test output %!shared x,y %! x = [-1 0 0.5 1 Inf]; %! y = gampdf (x, 1, 2); %!assert (exppdf (x, 2*ones (1,5)), y) %!assert (exppdf (x, 2*[1 0 NaN 1 1]), [y(1) NaN NaN y(4:5)]) %!assert (exppdf ([x, NaN], 2), [y, NaN]) ## Test class of input preserved %!assert (exppdf (single ([x, NaN]), 2), single ([y, NaN])) %!assert (exppdf ([x, NaN], single (2)), single ([y, NaN])) ## Test input validation %!error exppdf () %!error exppdf (1,2,3) %!error ... %! exppdf (ones (3), ones (2)) %!error ... %! exppdf (ones (2), ones (3)) %!error exppdf (i, 2) %!error exppdf (2, i) statistics-release-1.6.3/inst/dist_fun/exprnd.m000066400000000000000000000133771456127120000216100ustar00rootroot00000000000000## Copyright (C) 2012 Rik Wehbring ## Copyright (C) 1995-2016 Kurt Hornik ## Copyright (C) 2023 Andreas Bertsatos ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{r} =} exprnd (@var{mu}) ## @deftypefnx {statistics} {@var{r} =} exprnd (@var{mu}, @var{rows}) ## @deftypefnx {statistics} {@var{r} =} exprnd (@var{mu}, @var{rows}, @var{cols}, @dots{}) ## @deftypefnx {statistics} {@var{r} =} exprnd (@var{mu}, [@var{sz}]) ## ## Random arrays from the exponential distribution. ## ## @code{@var{r} = exprnd (@var{mu})} returns an array of random numbers chosen ## from the exponential distribution with mean parameter @var{mu}. The size of ## @var{r} is the size of @var{mu}. ## ## When called with a single size argument, @code{exprnd} returns a square ## matrix with the dimension specified. When called with more than one scalar ## argument, the first two arguments are taken as the number of rows and columns ## and any further arguments specify additional matrix dimensions. The size may ## also be specified with a row vector of dimensions, @var{sz}. ## ## A common alternative parameterization of the exponential distribution is to ## use the parameter @math{λ} defined as the mean number of events in an ## interval as opposed to the parameter @math{μ}, which is the mean wait time ## for an event to occur. @math{λ} and @math{μ} are reciprocals, ## i.e. @math{μ = 1 / λ}. ## ## Further information about the exponential distribution can be found at ## @url{https://en.wikipedia.org/wiki/Exponential_distribution} ## ## @seealso{expcdf, expinv, exppdf, expfit, explike, expstat} ## @end deftypefn function r = exprnd (mu, varargin) ## Check for valid number of input arguments if (nargin < 1) error ("exprnd: function called with too few input arguments."); endif ## Check for MU being real if (iscomplex (mu)) error ("exprnd: MU must not be complex."); endif ## Parse and check SIZE arguments if (nargin == 1) sz = size (mu); elseif (nargin == 2) if (isscalar (varargin{1}) && varargin{1} >= 0 ... && varargin{1} == fix (varargin{1})) sz = [varargin{1}, varargin{1}]; elseif (isrow (varargin{1}) && all (varargin{1} >= 0) ... && all (varargin{1} == fix (varargin{1}))) sz = varargin{1}; elseif error (strcat (["exprnd: SZ must be a scalar or a row vector"], ... [" of non-negative integers."])); endif elseif (nargin > 2) posint = cellfun (@(x) (! isscalar (x) || x < 0 || x != fix (x)), varargin); if (any (posint)) error ("exprnd: dimensions must be non-negative integers."); endif sz = [varargin{:}]; endif ## Check that parameters match requested dimensions in size if (! isscalar (mu) && ! isequal (size (mu), sz)) error ("exprnd: MU must be scalar or of size SZ."); endif ## Check for class type if (isa (mu, "single")) cls = "single"; else cls = "double"; endif ## Generate random sample from exponential distribution if (isscalar (mu)) if ((mu > 0) && (mu < Inf)) r = rande (sz, cls) * mu; else r = NaN (sz, cls); endif else r = NaN (sz, cls); k = (mu > 0) & (mu < Inf); r(k) = rande (sum (k(:)), 1, cls) .* mu(k)(:); endif endfunction ## Test output %!assert (size (exprnd (2)), [1, 1]) %!assert (size (exprnd (ones (2,1))), [2, 1]) %!assert (size (exprnd (ones (2,2))), [2, 2]) %!assert (size (exprnd (1, 3)), [3, 3]) %!assert (size (exprnd (1, [4 1])), [4, 1]) %!assert (size (exprnd (1, 4, 1)), [4, 1]) %!assert (size (exprnd (1, 4, 1)), [4, 1]) %!assert (size (exprnd (1, 4, 1, 5)), [4, 1, 5]) %!assert (size (exprnd (1, 0, 1)), [0, 1]) %!assert (size (exprnd (1, 1, 0)), [1, 0]) %!assert (size (exprnd (1, 1, 2, 0, 5)), [1, 2, 0, 5]) ## Test class of input preserved %!assert (class (exprnd (2)), "double") %!assert (class (exprnd (single (2))), "single") %!assert (class (exprnd (single ([2 2]))), "single") ## Test input validation %!error exprnd () %!error exprnd (i) %!error ... %! exprnd (1, -1) %!error ... %! exprnd (1, 1.2) %!error ... %! exprnd (1, ones (2)) %!error ... %! exprnd (1, [2 -1 2]) %!error ... %! exprnd (1, [2 0 2.5]) %!error ... %! exprnd (ones (2), ones (2)) %!error ... %! exprnd (1, 2, -1, 5) %!error ... %! exprnd (1, 2, 1.5, 5) %!error exprnd (ones (2,2), 3) %!error exprnd (ones (2,2), [3, 2]) %!error exprnd (ones (2,2), 2, 3) statistics-release-1.6.3/inst/dist_fun/fcdf.m000066400000000000000000000150221456127120000211770ustar00rootroot00000000000000## Copyright (C) 2012 Rik Wehbring ## Copyright (C) 1995-2016 Kurt Hornik ## Copyright (C) 2022-2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{p} =} fcdf (@var{x}, @var{df1}, @var{df2}) ## @deftypefnx {statistics} {@var{p} =} fcdf (@var{x}, @var{df1}, @var{df2}, @qcode{"upper"}) ## ## @math{F}-cumulative distribution function (CDF). ## ## For each element of @var{x}, compute the cumulative distribution function ## (CDF) of the @math{F}-distribution with @var{df1} and @var{df2} degrees of ## freedom. The size of @var{p} is the common size of @var{x}, @var{df1}, and ## @var{df2}. A scalar input functions as a constant matrix of the same size as ## the other inputs. ## ## @code{@var{p} = fcdf (@var{x}, @var{df1}, @var{df2}, "upper")} computes the ## upper tail probability of the @math{F}-distribution with @var{df1} and ## @var{df2} degrees of freedom, at the values in @var{x}. ## ## Further information about the @math{F}-distribution can be found at ## @url{https://en.wikipedia.org/wiki/F-distribution} ## ## @seealso{finv, fpdf, frnd, fstat} ## @end deftypefn function p = fcdf (x, df1, df2, uflag) ## Check for valid number of input arguments if (nargin < 3) error ("fcdf: function called with too few input arguments."); endif ## Check for "upper" flag if (nargin > 3 && strcmpi (uflag, "upper")) notnan = ! isnan (x); x(notnan) = 1 ./ max (0, x(notnan)); tmp=df1; df1=df2; df2=tmp; elseif (nargin > 3 && ! strcmpi (uflag, "upper")) error ("fcdf: invalid argument for upper tail."); endif ## Check for common size of X, DF1, and DF2 if (! isscalar (x) || ! isscalar (df1) || ! isscalar (df2)) [err, x, df1, df2] = common_size (x, df1, df2); if (err > 0) error ("fcdf: X, DF1, and DF2 must be of common size or scalars."); endif endif ## Check for X, DF1, and DF2 being reals if (iscomplex (x) || iscomplex (df1) || iscomplex (df2)) error ("fcdf: X, DF1, and DF2 must not be complex."); endif ## Check for class type if (isa (x, "single") || isa (df1, "single") || isa (df2, "single")) p = zeros (size (x), "single"); else p = zeros (size (x)); endif ## Check X for NaNs while DFs <= 0 and make P = NaNs make_nan = (df1 <= 0 | df2 <= 0 | isnan(x) | isnan(df1) | isnan(df2)); p(make_nan) = NaN; ## Check remaining valid X for Inf values and make P = 1 is_inf = (x == Inf) & ! make_nan; if any (is_inf(:)) p(is_inf) = 1; make_nan = (make_nan | is_inf); endif ## Compute P when X > 0. k = find(x > 0 & ! make_nan & isfinite(df1) & isfinite(df2)); if (any (k)) k1 = (df2(k) <= x(k) .* df1(k)); if (any (k1)) kk = k(k1); xx = df2(kk) ./ (df2(kk) + x(kk) .* df1(kk)); p(kk) = betainc (xx, df2(kk)/2, df1(kk)/2, "upper"); end if (any (! k1)) kk = k(! k1); num = df1(kk) .* x(kk); xx = num ./ (num + df2(kk)); p(kk) = betainc (xx, df1(kk)/2, df2(kk)/2, "lower"); endif endif if any(~isfinite(df1(:)) | ~isfinite(df2(:))) k = find (x > 0 & ! make_nan & isfinite (df1) & ! isfinite (df2) & df2 > 0); if (any (k)) p(k) = gammainc (df1(k) .* x(k) ./ 2, df1(k) ./ 2, "lower"); end k = find (x > 0 & ! make_nan & ! isfinite (df1) & df1 > 0 & isfinite (df2)); if (any (k)) p(k) = gammainc (df2(k) ./ x(k) ./ 2, df2(k) ./ 2, "upper"); end k = find (x > 0 & ! make_nan & ! isfinite (df1) & df1 > 0 & ... ! isfinite (df2) & df2 > 0); if (any (k)) if (nargin >= 4 && x(k) == 1) p(k) = 0; else p(k) = (x(k)>=1); end endif endif endfunction %!demo %! ## Plot various CDFs from the F distribution %! x = 0.01:0.01:4; %! p1 = fcdf (x, 1, 2); %! p2 = fcdf (x, 2, 1); %! p3 = fcdf (x, 5, 2); %! p4 = fcdf (x, 10, 1); %! p5 = fcdf (x, 100, 100); %! plot (x, p1, "-b", x, p2, "-g", x, p3, "-r", x, p4, "-c", x, p5, "-m") %! grid on %! legend ({"df1 = 1, df2 = 2", "df1 = 2, df2 = 1", ... %! "df1 = 5, df2 = 2", "df1 = 10, df2 = 1", ... %! "df1 = 100, df2 = 100"}, "location", "southeast") %! title ("F CDF") %! xlabel ("values in x") %! ylabel ("probability") ## Test output %!shared x, y %! x = [-1, 0, 0.5, 1, 2, Inf]; %! y = [0, 0, 1/3, 1/2, 2/3, 1]; %!assert (fcdf (x, 2*ones (1,6), 2*ones (1,6)), y, eps) %!assert (fcdf (x, 2, 2*ones (1,6)), y, eps) %!assert (fcdf (x, 2*ones (1,6), 2), y, eps) %!assert (fcdf (x, [0 NaN Inf 2 2 2], 2), [NaN NaN 0.1353352832366127 y(4:6)], eps) %!assert (fcdf (x, 2, [0 NaN Inf 2 2 2]), [NaN NaN 0.3934693402873666 y(4:6)], eps) %!assert (fcdf ([x(1:2) NaN x(4:6)], 2, 2), [y(1:2) NaN y(4:6)], eps) ## Test class of input preserved %!assert (fcdf ([x, NaN], 2, 2), [y, NaN], eps) %!assert (fcdf (single ([x, NaN]), 2, 2), single ([y, NaN]), eps ("single")) %!assert (fcdf ([x, NaN], single (2), 2), single ([y, NaN]), eps ("single")) %!assert (fcdf ([x, NaN], 2, single (2)), single ([y, NaN]), eps ("single")) ## Test input validation %!error fcdf () %!error fcdf (1) %!error fcdf (1, 2) %!error fcdf (1, 2, 3, 4) %!error fcdf (1, 2, 3, "tail") %!error ... %! fcdf (ones (3), ones (2), ones (2)) %!error ... %! fcdf (ones (2), ones (3), ones (2)) %!error ... %! fcdf (ones (2), ones (2), ones (3)) %!error fcdf (i, 2, 2) %!error fcdf (2, i, 2) %!error fcdf (2, 2, i) statistics-release-1.6.3/inst/dist_fun/finv.m000066400000000000000000000112131456127120000212350ustar00rootroot00000000000000## Copyright (C) 2012 Rik Wehbring ## Copyright (C) 1995-2016 Kurt Hornik ## Copyright (C) 2023 Andreas Bertsatos ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{x} =} finv (@var{p}, @var{df1}, @var{df2}) ## ## Inverse of the @math{F}-cumulative distribution function (iCDF). ## ## For each element of @var{p}, compute the quantile (the inverse of the CDF) of ## the @math{F}-distribution with @var{df1} and @var{df2} degrees of freedom. ## The size of @var{x} is the common size of @var{p}, @var{df1}, and @var{df2}. ## A scalar input functions as a constant matrix of the same size as the other ## inputs. ## ## Further information about the @math{F}-distribution can be found at ## @url{https://en.wikipedia.org/wiki/F-distribution} ## ## @seealso{fcdf, fpdf, frnd, fstat} ## @end deftypefn function x = finv (p, df1, df2) ## Check for valid number of input arguments if (nargin < 3) error ("finv: function called with too few input arguments."); endif ## Check for common size of P, DF1, and DF2 if (! isscalar (p) || ! isscalar (df1) || ! isscalar (df2)) [retval, p, df1, df2] = common_size (p, df1, df2); if (retval > 0) error ("finv: P, DF1, and DF2 must be of common size or scalars."); endif endif ## Check for P, DF1, and DF2 being reals if (iscomplex (p) || iscomplex (df1) || iscomplex (df2)) error ("finv: P, DF1, and DF2 must not be complex."); endif ## Check for class type if (isa (p, "single") || isa (df1, "single") || isa (df2, "single")) x = NaN (size (p), "single"); else x = NaN (size (p)); endif k = (p == 1) & (df1 > 0) & (df1 < Inf) & (df2 > 0) & (df2 < Inf); x(k) = Inf; k = (p >= 0) & (p < 1) & (df1 > 0) & (df1 < Inf) & (df2 > 0) & (df2 < Inf); if (isscalar (df1) && isscalar (df2)) x(k) = ((1 ./ betainv (1 - p(k), df2/2, df1/2) - 1) * df2 / df1); else x(k) = ((1 ./ betainv (1 - p(k), df2(k)/2, df1(k)/2) - 1) .* df2(k) ./ df1(k)); endif endfunction %!demo %! ## Plot various iCDFs from the F distribution %! p = 0.001:0.001:0.999; %! x1 = finv (p, 1, 1); %! x2 = finv (p, 2, 1); %! x3 = finv (p, 5, 2); %! x4 = finv (p, 10, 1); %! x5 = finv (p, 100, 100); %! plot (p, x1, "-b", p, x2, "-g", p, x3, "-r", p, x4, "-c", p, x5, "-m") %! grid on %! ylim ([0, 4]) %! legend ({"df1 = 1, df2 = 2", "df1 = 2, df2 = 1", ... %! "df1 = 5, df2 = 2", "df1 = 10, df2 = 1", ... %! "df1 = 100, df2 = 100"}, "location", "northwest") %! title ("F iCDF") %! xlabel ("probability") %! ylabel ("values in x") ## Test output %!shared p %! p = [-1 0 0.5 1 2]; %!assert (finv (p, 2*ones (1,5), 2*ones (1,5)), [NaN 0 1 Inf NaN]) %!assert (finv (p, 2, 2*ones (1,5)), [NaN 0 1 Inf NaN]) %!assert (finv (p, 2*ones (1,5), 2), [NaN 0 1 Inf NaN]) %!assert (finv (p, [2 -Inf NaN Inf 2], 2), [NaN NaN NaN NaN NaN]) %!assert (finv (p, 2, [2 -Inf NaN Inf 2]), [NaN NaN NaN NaN NaN]) %!assert (finv ([p(1:2) NaN p(4:5)], 2, 2), [NaN 0 NaN Inf NaN]) ## Test class of input preserved %!assert (finv ([p, NaN], 2, 2), [NaN 0 1 Inf NaN NaN]) %!assert (finv (single ([p, NaN]), 2, 2), single ([NaN 0 1 Inf NaN NaN])) %!assert (finv ([p, NaN], single (2), 2), single ([NaN 0 1 Inf NaN NaN])) %!assert (finv ([p, NaN], 2, single (2)), single ([NaN 0 1 Inf NaN NaN])) ## Test input validation %!error finv () %!error finv (1) %!error finv (1,2) %!error ... %! finv (ones (3), ones (2), ones (2)) %!error ... %! finv (ones (2), ones (3), ones (2)) %!error ... %! finv (ones (2), ones (2), ones (3)) %!error finv (i, 2, 2) %!error finv (2, i, 2) %!error finv (2, 2, i) statistics-release-1.6.3/inst/dist_fun/fpdf.m000066400000000000000000000117531456127120000212230ustar00rootroot00000000000000## Copyright (C) 2012 Rik Wehbring ## Copyright (C) 1995-2016 Kurt Hornik ## Copyright (C) 2023 Andreas Bertsatos ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{y} =} fpdf (@var{x}, @var{df1}, @var{df2}) ## ## @math{F}-probability density function (PDF). ## ## For each element of @var{x}, compute the probability density function (PDF) ## of the @math{F}-distribution with @var{df1} and @var{df2} degrees of freedom. ## The size of @var{y} is the common size of @var{x}, @var{df1}, and @var{df2}. ## A scalar input functions as a constant matrix of the same size as the other ## inputs. ## ## Further information about the @math{F}-distribution can be found at ## @url{https://en.wikipedia.org/wiki/F-distribution} ## ## @seealso{fcdf, finv, frnd, fstat} ## @end deftypefn function y = fpdf (x, df1, df2) ## Check for valid number of input arguments if (nargin < 3) error ("fpdf: function called with too few input arguments."); endif ## Check for common size of X, DF1, and DF2 if (! isscalar (x) ||! isscalar (df1) || ! isscalar (df2)) [retval, x, df1, df2] = common_size (x, df1, df2); if (retval > 0) error ("fpdf: X, DF1, and DF2 must be of common size or scalars."); endif endif ## Check for X, DF1, and DF2 being reals if (iscomplex (x) || iscomplex (df1) || iscomplex (df2)) error ("fpdf: X, DF1, and DF2 must not be complex."); endif ## Check for class type if (isa (x, "single") || isa (df1, "single") || isa (df2, "single")) y = zeros (size (x), "single"); else y = zeros (size (x)); endif k = isnan (x) | !(df1 > 0) | !(df1 < Inf) | !(df2 > 0) | !(df2 < Inf); y(k) = NaN; k = (x > 0) & (x < Inf) & (df1 > 0) & (df1 < Inf) & (df2 > 0) & (df2 < Inf); if (isscalar (df1) && isscalar (df2)) tmp = df1 / df2 * x(k); y(k) = (exp ((df1/2 - 1) * log (tmp) ... - ((df1 + df2) / 2) * log (1 + tmp)) ... * (df1 / df2) ./ beta (df1/2, df2/2)); else tmp = df1(k) .* x(k) ./ df2(k); y(k) = (exp ((df1(k)/2 - 1) .* log (tmp) ... - ((df1(k) + df2(k)) / 2) .* log (1 + tmp)) ... .* (df1(k) ./ df2(k)) ./ beta (df1(k)/2, df2(k)/2)); endif endfunction %!demo %! ## Plot various PDFs from the F distribution %! x = 0.01:0.01:4; %! y1 = fpdf (x, 1, 1); %! y2 = fpdf (x, 2, 1); %! y3 = fpdf (x, 5, 2); %! y4 = fpdf (x, 10, 1); %! y5 = fpdf (x, 100, 100); %! plot (x, y1, "-b", x, y2, "-g", x, y3, "-r", x, y4, "-c", x, y5, "-m") %! grid on %! ylim ([0, 2.5]) %! legend ({"df1 = 1, df2 = 2", "df1 = 2, df2 = 1", ... %! "df1 = 5, df2 = 2", "df1 = 10, df2 = 1", ... %! "df1 = 100, df2 = 100"}, "location", "northeast") %! title ("F PDF") %! xlabel ("values in x") %! ylabel ("density") ## Test output %!shared x, y %! x = [-1 0 0.5 1 2]; %! y = [0 0 4/9 1/4 1/9]; %!assert (fpdf (x, 2*ones (1,5), 2*ones (1,5)), y, eps) %!assert (fpdf (x, 2, 2*ones (1,5)), y, eps) %!assert (fpdf (x, 2*ones (1,5), 2), y, eps) %!assert (fpdf (x, [0 NaN Inf 2 2], 2), [NaN NaN NaN y(4:5)], eps) %!assert (fpdf (x, 2, [0 NaN Inf 2 2]), [NaN NaN NaN y(4:5)], eps) %!assert (fpdf ([x, NaN], 2, 2), [y, NaN], eps) %!test #F (x, 1, df1) == T distribution (sqrt (x), df1) / sqrt (x) %! xr = rand (10,1); %! xr = xr(x > 0.1 & x < 0.9); %! yr = tpdf (sqrt (xr), 2) ./ sqrt (xr); %! assert (fpdf (xr, 1, 2), yr, 5*eps); ## Test class of input preserved %!assert (fpdf (single ([x, NaN]), 2, 2), single ([y, NaN]), eps ("single")) %!assert (fpdf ([x, NaN], single (2), 2), single ([y, NaN]), eps ("single")) %!assert (fpdf ([x, NaN], 2, single (2)), single ([y, NaN]), eps ("single")) ## Test input validation %!error fpdf () %!error fpdf (1) %!error fpdf (1,2) %!error ... %! fpdf (ones (3), ones (2), ones (2)) %!error ... %! fpdf (ones (2), ones (3), ones (2)) %!error ... %! fpdf (ones (2), ones (2), ones (3)) %!error fpdf (i, 2, 2) %!error fpdf (2, i, 2) %!error fpdf (2, 2, i) statistics-release-1.6.3/inst/dist_fun/frnd.m000066400000000000000000000146411456127120000212340ustar00rootroot00000000000000## Copyright (C) 2012 Rik Wehbring ## Copyright (C) 1995-2016 Kurt Hornik ## Copyright (C) 2023 Andreas Bertsatos ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{r} =} frnd (@var{df1}, @var{df2}) ## @deftypefnx {statistics} {@var{r} =} frnd (@var{df1}, @var{df2}, @var{rows}) ## @deftypefnx {statistics} {@var{r} =} frnd (@var{df1}, @var{df2}, @var{rows}, @var{cols}, @dots{}) ## @deftypefnx {statistics} {@var{r} =} frnd (@var{df1}, @var{df2}, [@var{sz}]) ## ## Random arrays from the @math{F}-distribution. ## ## @code{@var{r} = frnd (@var{df1}, @var{df2})} returns an array of random ## numbers chosen from the @math{F}-distribution with @var{df1} and @var{df2} ## degrees of freedom. The size of @var{r} is the common size of @var{df1} and ## @var{df2}. A scalar input functions as a constant matrix of the same size as ## the other inputs. ## ## When called with a single size argument, @code{frnd} returns a square ## matrix with the dimension specified. When called with more than one scalar ## argument, the first two arguments are taken as the number of rows and columns ## and any further arguments specify additional matrix dimensions. The size may ## also be specified with a row vector of dimensions, @var{sz}. ## ## Further information about the @math{F}-distribution can be found at ## @url{https://en.wikipedia.org/wiki/F-distribution} ## ## @seealso{fcdf, finv, fpdf, fstat} ## @end deftypefn function r = frnd (df1, df2, varargin) ## Check for valid number of input arguments if (nargin < 2) error ("frnd: function called with too few input arguments."); endif ## Check for common size of DF1 and DF2 if (! isscalar (df1) || ! isscalar (df2)) [retval, df1, df2] = common_size (df1, df2); if (retval > 0) error ("frnd: DF1 and DF2 must be of common size or scalars."); endif endif ## Check for DF1 and DF2 being reals if (iscomplex (df1) || iscomplex (df2)) error ("frnd: DF1 and DF2 must not be complex."); endif ## Parse and check SIZE arguments if (nargin == 2) sz = size (df1); elseif (nargin == 3) if (isscalar (varargin{1}) && varargin{1} >= 0 ... && varargin{1} == fix (varargin{1})) sz = [varargin{1}, varargin{1}]; elseif (isrow (varargin{1}) && all (varargin{1} >= 0) ... && all (varargin{1} == fix (varargin{1}))) sz = varargin{1}; elseif error (strcat (["frnd: SZ must be a scalar or a row vector"], ... [" of non-negative integers."])); endif elseif (nargin > 3) posint = cellfun (@(x) (! isscalar (x) || x < 0 || x != fix (x)), varargin); if (any (posint)) error ("frnd: dimensions must be non-negative integers."); endif sz = [varargin{:}]; endif ## Check that parameters match requested dimensions in size if (! isscalar (df1) && ! isequal (size (df1), sz)) error ("frnd: DF1 and DF2 must be scalars or of size SZ."); endif ## Check for class type if (isa (df1, "single") || isa (df2, "single")) cls = "single"; else cls = "double"; endif ## Generate random sample from F distribution if (isscalar (df1) && isscalar (df2)) if ((df1 > 0) && (df1 < Inf) && (df2 > 0) && (df2 < Inf)) r = df2/df1 * randg (df1/2, sz, cls) ./ randg (df2/2, sz, cls); else r = NaN (sz, cls); endif else r = NaN (sz, cls); k = (df1 > 0) & (df1 < Inf) & (df2 > 0) & (df2 < Inf); r(k) = df2(k) ./ df1(k) .* randg (df1(k)/2, cls) ./ randg (df2(k)/2, cls); endif endfunction ## Test output %!assert (size (frnd (1, 1)), [1 1]) %!assert (size (frnd (1, ones (2,1))), [2, 1]) %!assert (size (frnd (1, ones (2,2))), [2, 2]) %!assert (size (frnd (ones (2,1), 1)), [2, 1]) %!assert (size (frnd (ones (2,2), 1)), [2, 2]) %!assert (size (frnd (1, 1, 3)), [3, 3]) %!assert (size (frnd (1, 1, [4, 1])), [4, 1]) %!assert (size (frnd (1, 1, 4, 1)), [4, 1]) %!assert (size (frnd (1, 1, 4, 1, 5)), [4, 1, 5]) %!assert (size (frnd (1, 1, 0, 1)), [0, 1]) %!assert (size (frnd (1, 1, 1, 0)), [1, 0]) %!assert (size (frnd (1, 1, 1, 2, 0, 5)), [1, 2, 0, 5]) ## Test class of input preserved %!assert (class (frnd (1, 1)), "double") %!assert (class (frnd (1, single (1))), "single") %!assert (class (frnd (1, single ([1, 1]))), "single") %!assert (class (frnd (single (1), 1)), "single") %!assert (class (frnd (single ([1, 1]), 1)), "single") ## Test input validation %!error frnd () %!error frnd (1) %!error ... %! frnd (ones (3), ones (2)) %!error ... %! frnd (ones (2), ones (3)) %!error frnd (i, 2, 3) %!error frnd (1, i, 3) %!error ... %! frnd (1, 2, -1) %!error ... %! frnd (1, 2, 1.2) %!error ... %! frnd (1, 2, ones (2)) %!error ... %! frnd (1, 2, [2 -1 2]) %!error ... %! frnd (1, 2, [2 0 2.5]) %!error ... %! frnd (1, 2, 2, -1, 5) %!error ... %! frnd (1, 2, 2, 1.5, 5) %!error ... %! frnd (2, ones (2), 3) %!error ... %! frnd (2, ones (2), [3, 2]) %!error ... %! frnd (2, ones (2), 3, 2) statistics-release-1.6.3/inst/dist_fun/gamcdf.m000066400000000000000000000306161456127120000215240ustar00rootroot00000000000000## Copyright (C) 2012 Rik Wehbring ## Copyright (C) 1995-2016 Kurt Hornik ## Copyright (C) 2022-2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{p} =} gamcdf (@var{x}, @var{k}) ## @deftypefnx {statistics} {@var{p} =} gamcdf (@var{x}, @var{k}, @var{theta}) ## @deftypefnx {statistics} {@var{p} =} gamcdf (@dots{}, @qcode{"upper"}) ## @deftypefnx {statistics} {[@var{p}, @var{plo}, @var{pup}] =} evcdf (@var{x}, @var{k}, @var{theta}, @var{pcov}) ## @deftypefnx {statistics} {[@var{p}, @var{plo}, @var{pup}] =} evcdf (@var{x}, @var{k}, @var{theta}, @var{pcov}, @var{alpha}) ## @deftypefnx {statistics} {[@var{p}, @var{plo}, @var{pup}] =} evcdf (@dots{}, @qcode{"upper"}) ## ## Gamma cumulative distribution function (CDF). ## ## For each element of @var{x}, compute the cumulative distribution function ## (CDF) of the Gamma distribution with shape parameter @var{k} and scale ## parameter @var{theta}. When called with only one parameter, then @var{theta} ## defaults to 1. The size of @var{p} is the common size of @var{x}, @var{k}, ## and @var{theta}. A scalar input functions as a constant matrix of the same ## size as the other inputs. ## ## When called with three output arguments, i.e. @qcode{[@var{p}, @var{plo}, ## @var{pup}]}, @code{gamcdf} computes the confidence bounds for @var{p} when ## the input parameters @var{k} and @var{theta} are estimates. In such case, ## @var{pcov}, a @math{2x2} matrix containing the covariance matrix of the ## estimated parameters, is necessary. Optionally, @var{alpha}, which has a ## default value of 0.05, specifies the @qcode{100 * (1 - @var{alpha})} percent ## confidence bounds. @var{plo} and @var{pup} are arrays of the same size as ## @var{p} containing the lower and upper confidence bounds. ## ## @code{[@dots{}] = gamcdf (@dots{}, "upper")} computes the upper tail ## probability of the Gamma distribution with parameters @var{k} and ## @var{theta}, at the values in @var{x}. ## ## There are two equivalent parameterizations in common use: ## @enumerate ## @item With a shape parameter @math{k} and a scale parameter @math{θ}, which ## is used by @code{gamcdf}. ## @item With a shape parameter @math{α = k} and an inverse scale parameter ## @math{β = 1 / θ}, called a rate parameter. ## @end enumerate ## ## Further information about the Gamma distribution can be found at ## @url{https://en.wikipedia.org/wiki/Gamma_distribution} ## ## @seealso{gaminv, gampdf, gamrnd, gamfit, gamlike, gamstat} ## @end deftypefn function [varargout] = gamcdf (x, varargin) ## Check for valid number of input arguments if (nargin < 2 || nargin > 6) error ("gamcdf: invalid number of input arguments."); endif ## Check for "upper" flag if (nargin > 2 && strcmpi (varargin{end}, "upper")) uflag = true; varargin(end) = []; elseif (nargin > 2 && ischar (varargin{end}) && ... ! strcmpi (varargin{end}, "upper")) error ("gamcdf: invalid argument for upper tail."); elseif (nargin > 2 && isempty (varargin{end})) uflag = false; varargin(end) = []; else uflag = false; endif ## Get extra arguments (if they exist) or add defaults k = varargin{1}; if (numel (varargin) > 1) theta = varargin{2}; else theta = 1; endif if (numel (varargin) > 2) pcov = varargin{3}; ## Check for valid covariance matrix 2x2 if (! isequal (size (pcov), [2, 2])) error ("gamcdf: invalid size of covariance matrix."); endif else ## Check that cov matrix is provided if 3 output arguments are requested if (nargout > 1) error ("gamcdf: covariance matrix is required for confidence bounds."); endif pcov = []; endif if (numel (varargin) > 3) alpha = varargin{4}; ## Check for valid alpha value if (! isnumeric (alpha) || numel (alpha) !=1 || alpha <= 0 || alpha >= 1) error ("gamcdf: invalid value for alpha."); endif else alpha = 0.05; endif ## Check for common size of X, K, and THETA if (! isscalar (x) || ! isscalar (k) || ! isscalar (theta)) [err, x, k, theta] = common_size (x, k, theta); if (err > 0) error ("gamcdf: X, K, and THETA must be of common size or scalars."); endif endif ## Check for X, K, and THETA being reals if (iscomplex (x) || iscomplex (k) || iscomplex (theta)) error ("gamcdf: X, K, and THETA must not be complex."); endif ## Prepare parameters so that gammainc returns NaN for out of range parameters k(k < 0) = NaN; theta(theta < 0) = NaN; ## Prepare data so that gammainc returns 0 for negative X x(x < 0) = 0; ## Compute gammainc z = x ./ theta; if (uflag) p = gammainc (z, k, "upper"); ## Fix NaNs to gammainc output when k == NaN p(isnan (k)) = NaN; else p = gammainc (z, k); ## Fix NaNs to gammainc output when k == NaN p(isnan (k)) = NaN; endif ## Check for appropriate class if (isa (x, "single") || isa (k, "single") || isa (theta, "single")); is_class = "single"; else is_class = "double"; endif ## Prepare output varargout{1} = cast (p, is_class); if (nargout > 1) plo = NaN (size (z), is_class); pup = NaN (size (z), is_class); endif ## Compute confidence bounds (if requested) if (nargout >= 2) ## Approximate the variance of p on the logit scale logitp = log (p ./ (1 - p)); dp = 1 ./ (p .* (1 - p)); dk = dgammainc (z, k) .* dp; dt = -exp (k .* log (z) - z - gammaln (k) - log (theta)) .* dp; varLogitp = pcov(1,1) .* dk .^ 2 + 2 .* pcov(1,2) .* dk .* dt + ... pcov(2,2) .* dt .^ 2; if (any (varLogitp(:) < 0)) error ("gamcdf: bad covariance matrix."); endif ## Use k normal approximation on the logit scale, then transform back to ## the original CDF scale halfwidth = -norminv (alpha / 2) * sqrt (varLogitp); explogitplo = exp (logitp - halfwidth); explogitpup = exp (logitp + halfwidth); plo = explogitplo ./ (1 + explogitplo); pup = explogitpup ./ (1 + explogitpup); varargout{2} = plo; varargout{3} = pup; endif endfunction ## Compute 1st derivative of the incomplete Gamma function function dy = dgammainc (x, k) ## Initialize return variables dy = nan (size (x)); ## Use approximation for K > 2^20 ulim = 2^20; is_lim = find (k > ulim); if (! isempty (is_lim)) x(is_lim) = max (ulim - 1/3 + sqrt (ulim ./ k(is_lim)) .* ... (x(is_lim) - (k(is_lim) - 1/3)), 0); k(is_lim) = ulim; endif ## For x < k+1 is_lo = find (x < k + 1 & x != 0); if (! isempty (is_lo)) x_lo = x(is_lo); k_lo = k(is_lo); k_1 = k_lo; step = 1; d1st = 0; stsum = step; d1sum = d1st; while norm (step, "inf") >= 100 * eps (norm (stsum, "inf")) k_1 += 1; step = step .* x_lo ./ k_1; d1st = (d1st .* x_lo - step) ./ k_1; stsum = stsum + step; d1sum = d1sum + d1st; endwhile fklo = exp (-x_lo + k_lo .* log (x_lo) - gammaln (k_lo + 1)); ## Compute 1st derivative dlogfklo = (log (x_lo) - psi (k_lo + 1)); d1fklo = fklo .* dlogfklo; d1y_lo = d1fklo .* stsum + fklo .* d1sum; dy(is_lo) = d1y_lo; endif ## For x >= k+1 is_hi = find (x >= k+1); if (! isempty (is_hi)) x_hi = x(is_hi); k_hi = k(is_hi); zc = 0; k0 = 0; k1 = k_hi; x0 = 1; x1 = x_hi; d1k0 = 0; d1k1 = 1; d1x0 = 0; d1x1 = 0; kx = k_hi ./ x_hi; d1kx = 1 ./ x_hi; d2kx = 0; start = 1; while norm (d2kx - start, "Inf") > 100 * eps (norm (d2kx, "Inf")) rescale = 1 ./ x1; zc += 1; n_k = zc - k_hi; d1k0 = (d1k1 + d1k0 .* n_k - k0) .* rescale; d1x0 = (d1x1 + d1x0 .* n_k - x0) .* rescale; k0 = (k1 + k0 .* n_k) .* rescale; x0 = 1 + (x0 .* n_k) .* rescale; nrescale = zc .* rescale; d1k1 = d1k0 .* x_hi + d1k1 .* nrescale; d1x1 = d1x0 .* x_hi + d1x1 .* nrescale; k1 = k0 .* x_hi + k1 .* nrescale; x1 = x0 .* x_hi + zc; start = d2kx; kx = k1 ./ x1; d1kx = (d1k1 - kx .* d1x1) ./ x1; endwhile fkhi = exp (-x_hi + k_hi .* log (x_hi) - gammaln (k_hi+1)); ## Compute 1st derivative dlogfkhi = (log (x_hi) - psi (k_hi + 1)); d1fkhi = fkhi .* dlogfkhi; d1y_hi = d1fkhi .* kx + fkhi .* d1kx; dy(is_hi) = -d1y_hi; endif ## Handle x == 0 is_x0 = find (x == 0); if (! isempty (is_x0)) dy(is_x0) = 0; endif ## Handle k == 0 is_k0 = find (k == 0); if (! isempty (is_k0)) is_k0x0 = find (k == 0 & x == 0); dy(is_k0x0) = -Inf; endif endfunction %!demo %! ## Plot various CDFs from the Gamma distribution %! x = 0:0.01:20; %! p1 = gamcdf (x, 1, 2); %! p2 = gamcdf (x, 2, 2); %! p3 = gamcdf (x, 3, 2); %! p4 = gamcdf (x, 5, 1); %! p5 = gamcdf (x, 9, 0.5); %! p6 = gamcdf (x, 7.5, 1); %! p7 = gamcdf (x, 0.5, 1); %! plot (x, p1, "-r", x, p2, "-g", x, p3, "-y", x, p4, "-m", ... %! x, p5, "-k", x, p6, "-b", x, p7, "-c") %! grid on %! legend ({"α = 1, θ = 2", "α = 2, θ = 2", "α = 3, θ = 2", ... %! "α = 5, θ = 1", "α = 9, θ = 0.5", "α = 7.5, θ = 1", ... %! "α = 0.5, θ = 1"}, "location", "southeast") %! title ("Gamma CDF") %! xlabel ("values in x") %! ylabel ("probability") ## Test output %!shared x, y, u %! x = [-1, 0, 0.5, 1, 2, Inf]; %! y = [0, gammainc(x(2:end), 1)]; %! u = [0, NaN, NaN, 1, 0.1353352832366127, 0]; %!assert (gamcdf (x, ones (1,6), ones (1,6)), y, eps) %!assert (gamcdf (x, ones (1,6), ones (1,6), []), y, eps) %!assert (gamcdf (x, 1, ones (1,6)), y, eps) %!assert (gamcdf (x, ones (1,6), 1), y, eps) %!assert (gamcdf (x, [0, -Inf, NaN, Inf, 1, 1], 1), [1, NaN, NaN, 0, y(5:6)], eps) %!assert (gamcdf (x, [0, -Inf, NaN, Inf, 1, 1], 1, "upper"), u, eps) %!assert (gamcdf (x, 1, [0, -Inf, NaN, Inf, 1, 1]), [NaN, NaN, NaN, 0, y(5:6)], eps) %!assert (gamcdf ([x(1:2), NaN, x(4:6)], 1, 1), [y(1:2), NaN, y(4:6)], eps) ## Test class of input preserved %!assert (gamcdf ([x, NaN], 1, 1), [y, NaN]) %!assert (gamcdf (single ([x, NaN]), 1, 1), single ([y, NaN]), eps ("single")) %!assert (gamcdf ([x, NaN], single (1), 1), single ([y, NaN]), eps ("single")) %!assert (gamcdf ([x, NaN], 1, single (1)), single ([y, NaN]), eps ("single")) ## Test input validation %!error gamcdf () %!error gamcdf (1) %!error gamcdf (1, 2, 3, 4, 5, 6, 7) %!error gamcdf (1, 2, 3, "uper") %!error gamcdf (1, 2, 3, 4, 5, "uper") %!error gamcdf (2, 3, 4, [1, 2]) %!error ... %! [p, plo, pup] = gamcdf (1, 2, 3) %!error ... %! [p, plo, pup] = gamcdf (1, 2, 3, "upper") %!error [p, plo, pup] = ... %! gamcdf (1, 2, 3, [1, 0; 0, 1], 0) %!error [p, plo, pup] = ... %! gamcdf (1, 2, 3, [1, 0; 0, 1], 1.22) %!error [p, plo, pup] = ... %! gamcdf (1, 2, 3, [1, 0; 0, 1], "alpha", "upper") %!error ... %! gamcdf (ones (3), ones (2), ones (2)) %!error ... %! gamcdf (ones (2), ones (3), ones (2)) %!error ... %! gamcdf (ones (2), ones (2), ones (3)) %!error gamcdf (i, 2, 2) %!error gamcdf (2, i, 2) %!error gamcdf (2, 2, i) %!error ... %! [p, plo, pup] = gamcdf (1, 2, 3, [1, 0; 0, -inf], 0.04) statistics-release-1.6.3/inst/dist_fun/gaminv.m000066400000000000000000000146041456127120000215630ustar00rootroot00000000000000## Copyright (C) 2012 Rik Wehbring ## Copyright (C) 1995-2016 Kurt Hornik ## Copyright (C) 2022-2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{x} =} gaminv (@var{p}, @var{k}, @var{theta}) ## ## Inverse of the Gamma cumulative distribution function (iCDF). ## ## For each element of @var{p}, compute the quantile (the inverse of the CDF) of ## the Gamma distribution with shape parameter @var{k} and scale parameter ## @var{theta}. The size of @var{x} is the common size of @var{p}, @var{k}, ## and @var{theta}. A scalar input functions as a constant matrix of the same ## size as the other inputs. ## ## There are two equivalent parameterizations in common use: ## @enumerate ## @item With a shape parameter @math{k} and a scale parameter @math{θ}, which ## is used by @code{gaminv}. ## @item With a shape parameter @math{α = k} and an inverse scale parameter ## @math{β = 1 / θ}, called a rate parameter. ## @end enumerate ## ## Further information about the Gamma distribution can be found at ## @url{https://en.wikipedia.org/wiki/Gamma_distribution} ## ## @seealso{gamcdf, gampdf, gamrnd, gamfit, gamlike, gamstat} ## @end deftypefn function x = gaminv (p, k, theta) ## Check for valid number of input arguments if (nargin < 3) error ("gaminv: function called with too few input arguments."); endif ## Check for common size of P, K, and THETA if (! isscalar (p) || ! isscalar (k) || ! isscalar (theta)) [retval, p, k, theta] = common_size (p, k, theta); if (retval > 0) error ("gaminv: P, K, and THETA must be of common size or scalars."); endif endif ## Check for P, K, and THETA being reals if (iscomplex (p) || iscomplex (k) || iscomplex (theta)) error ("gaminv: P, K, and THETA must not be complex."); endif ## Check for class type if (isa (p, "single") || isa (k, "single") || isa (theta, "single")) x = zeros (size (p), "single"); else x = zeros (size (p)); endif ## Force NaNs for out of range parameters is_nan = ((p < 0) | (p > 1) | isnan (p) ... | ! (k > 0) | ! (k < Inf) | ! (theta > 0) | ! (theta < Inf)); x(is_nan) = NaN; ## Handle edge cases is_inf = (p == 1) & (k > 0) & (k < Inf) & (theta > 0) & (theta < Inf); x(is_inf) = Inf; ## Handle all other valid cases is_valid = find ((p > 0) & (p < 1) & (k > 0) & ... (k < Inf) & (theta > 0) & (theta < Inf)); if (! isempty (is_valid)) if (! isscalar (k) || ! isscalar (theta)) k = k(is_valid); theta = theta(is_valid); y = k .* theta; else y = k * theta * ones (size (is_valid)); endif p = p(is_valid); ## Call GAMMAINCINV to find k root of GAMMAINC q = gammaincinv (p, k); tol = sqrt (eps (ones (1, 1, class(q)))); check_cdf = ((abs (gammainc (q, k) - p) ./ p) > tol); ## Check for any cdf being far off from tolerance if (any (check_cdf(:))) warning ("gaminv: calculation failed to converge for some values."); endif x(is_valid) = q .* theta; endif endfunction %!demo %! ## Plot various iCDFs from the Gamma distribution %! p = 0.001:0.001:0.999; %! x1 = gaminv (p, 1, 2); %! x2 = gaminv (p, 2, 2); %! x3 = gaminv (p, 3, 2); %! x4 = gaminv (p, 5, 1); %! x5 = gaminv (p, 9, 0.5); %! x6 = gaminv (p, 7.5, 1); %! x7 = gaminv (p, 0.5, 1); %! plot (p, x1, "-r", p, x2, "-g", p, x3, "-y", p, x4, "-m", ... %! p, x5, "-k", p, x6, "-b", p, x7, "-c") %! ylim ([0, 20]) %! grid on %! legend ({"α = 1, θ = 2", "α = 2, θ = 2", "α = 3, θ = 2", ... %! "α = 5, θ = 1", "α = 9, θ = 0.5", "α = 7.5, θ = 1", ... %! "α = 0.5, θ = 1"}, "location", "northwest") %! title ("Gamma iCDF") %! xlabel ("probability") %! ylabel ("x") ## Test output %!shared p %! p = [-1 0 0.63212055882855778 1 2]; %!assert (gaminv (p, ones (1,5), ones (1,5)), [NaN 0 1 Inf NaN], eps) %!assert (gaminv (p, 1, ones (1,5)), [NaN 0 1 Inf NaN], eps) %!assert (gaminv (p, ones (1,5), 1), [NaN 0 1 Inf NaN], eps) %!assert (gaminv (p, [1 -Inf NaN Inf 1], 1), [NaN NaN NaN NaN NaN]) %!assert (gaminv (p, 1, [1 -Inf NaN Inf 1]), [NaN NaN NaN NaN NaN]) %!assert (gaminv ([p(1:2) NaN p(4:5)], 1, 1), [NaN 0 NaN Inf NaN]) %!assert (gaminv ([p(1:2) NaN p(4:5)], 1, 1), [NaN 0 NaN Inf NaN]) ## Test for accuracy when p is small. Results compared to Matlab %!assert (gaminv (1e-16, 1, 1), 1e-16, eps) %!assert (gaminv (1e-16, 1, 2), 2e-16, eps) %!assert (gaminv (1e-20, 3, 5), 1.957434012161815e-06, eps) %!assert (gaminv (1e-15, 1, 1), 1e-15, eps) %!assert (gaminv (1e-35, 1, 1), 1e-35, eps) ## Test class of input preserved %!assert (gaminv ([p, NaN], 1, 1), [NaN 0 1 Inf NaN NaN], eps) %!assert (gaminv (single ([p, NaN]), 1, 1), single ([NaN 0 1 Inf NaN NaN]), ... %! eps ("single")) %!assert (gaminv ([p, NaN], single (1), 1), single ([NaN 0 1 Inf NaN NaN]), ... %! eps ("single")) %!assert (gaminv ([p, NaN], 1, single (1)), single ([NaN 0 1 Inf NaN NaN]), ... %! eps ("single")) ## Test input validation %!error gaminv () %!error gaminv (1) %!error gaminv (1,2) %!error ... %! gaminv (ones (3), ones (2), ones (2)) %!error ... %! gaminv (ones (2), ones (3), ones (2)) %!error ... %! gaminv (ones (2), ones (2), ones (3)) %!error gaminv (i, 2, 2) %!error gaminv (2, i, 2) %!error gaminv (2, 2, i) statistics-release-1.6.3/inst/dist_fun/gampdf.m000066400000000000000000000126341456127120000215410ustar00rootroot00000000000000## Copyright (C) 2012 Rik Wehbring ## Copyright (C) 1995-2016 Kurt Hornik ## Copyright (C) 2023 Andreas Bertsatos ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{y} =} gampdf (@var{x}, @var{k}, @var{theta}) ## ## Gamma probability density function (PDF). ## ## For each element of @var{x}, compute the probability density function (PDF) ## of the Gamma distribution with shape parameter @var{k} and scale parameter ## @var{theta}. The size of @var{y} is the common size of @var{x}, @var{k} and ## @var{theta}. A scalar input functions as a constant matrix of the same size ## as the other inputs. ## ## There are two equivalent parameterizations in common use: ## @enumerate ## @item With a shape parameter @math{k} and a scale parameter @math{θ}, which ## is used by @code{gampdf}. ## @item With a shape parameter @math{α = k} and an inverse scale parameter ## @math{β = 1 / θ}, called a rate parameter. ## @end enumerate ## ## Further information about the Gamma distribution can be found at ## @url{https://en.wikipedia.org/wiki/Gamma_distribution} ## ## @seealso{gamcdf, gaminv, gamrnd, gamfit, gamlike, gamstat} ## @end deftypefn function y = gampdf (x, k, theta) ## Check for valid number of input arguments if (nargin < 3) error ("gampdf: function called with too few input arguments."); endif ## Check for common size of X, K, and THETA if (! isscalar (k) || ! isscalar (theta)) [retval, x, k, theta] = common_size (x, k, theta); if (retval > 0) error ("gampdf: X, K, and THETA must be of common size or scalars."); endif endif ## Check for X, K, and THETA being reals if (iscomplex (x) || iscomplex (k) || iscomplex (theta)) error ("gampdf: X, K, and THETA must not be complex."); endif ## Check for class type if (isa (x, "single") || isa (k, "single") || isa (theta, "single")) y = zeros (size (x), "single"); else y = zeros (size (x)); endif ## Force NaNs for out of range parameters is_nan = ! (k > 0) | ! (theta > 0) | isnan (x); y(is_nan) = NaN; ## Handle all other valid cases v = (x >= 0) & (k > 0) & (k <= 1) & (theta > 0); if (isscalar (k) && isscalar (theta)) y(v) = (x(v) .^ (k - 1)) ... .* exp (- x(v) / theta) / gamma (k) / (theta ^ k); else y(v) = (x(v) .^ (k(v) - 1)) ... .* exp (- x(v) ./ theta(v)) ./ gamma (k(v)) ./ (theta(v) .^ k(v)); endif v = (x >= 0) & (k > 1) & (theta > 0); if (isscalar (k) && isscalar (theta)) y(v) = exp (- k * log (theta) + (k-1) * log (x(v)) - x(v) / theta - gammaln (k)); else y(v) = exp (- k(v) .* log (theta(v)) + (k(v)-1) .* log (x(v)) - x(v) ./ theta(v) - gammaln (k(v))); endif endfunction %!demo %! ## Plot various PDFs from the Gamma distribution %! x = 0:0.01:20; %! y1 = gampdf (x, 1, 2); %! y2 = gampdf (x, 2, 2); %! y3 = gampdf (x, 3, 2); %! y4 = gampdf (x, 5, 1); %! y5 = gampdf (x, 9, 0.5); %! y6 = gampdf (x, 7.5, 1); %! y7 = gampdf (x, 0.5, 1); %! plot (x, y1, "-r", x, y2, "-g", x, y3, "-y", x, y4, "-m", ... %! x, y5, "-k", x, y6, "-b", x, y7, "-c") %! grid on %! ylim ([0,0.5]) %! legend ({"α = 1, θ = 2", "α = 2, θ = 2", "α = 3, θ = 2", ... %! "α = 5, θ = 1", "α = 9, θ = 0.5", "α = 7.5, θ = 1", ... %! "α = 0.5, θ = 1"}, "location", "northeast") %! title ("Gamma PDF") %! xlabel ("values in x") %! ylabel ("density") ## Test output %!shared x, y %! x = [-1 0 0.5 1 Inf]; %! y = [0 exp(-x(2:end))]; %!assert (gampdf (x, ones (1,5), ones (1,5)), y) %!assert (gampdf (x, 1, ones (1,5)), y) %!assert (gampdf (x, ones (1,5), 1), y) %!assert (gampdf (x, [0 -Inf NaN Inf 1], 1), [NaN NaN NaN NaN y(5)]) %!assert (gampdf (x, 1, [0 -Inf NaN Inf 1]), [NaN NaN NaN 0 y(5)]) %!assert (gampdf ([x, NaN], 1, 1), [y, NaN]) ## Test class of input preserved %!assert (gampdf (single ([x, NaN]), 1, 1), single ([y, NaN])) %!assert (gampdf ([x, NaN], single (1), 1), single ([y, NaN])) %!assert (gampdf ([x, NaN], 1, single (1)), single ([y, NaN])) ## Test input validation %!error gampdf () %!error gampdf (1) %!error gampdf (1,2) %!error ... %! gampdf (ones (3), ones (2), ones (2)) %!error ... %! gampdf (ones (2), ones (3), ones (2)) %!error ... %! gampdf (ones (2), ones (2), ones (3)) %!error gampdf (i, 2, 2) %!error gampdf (2, i, 2) %!error gampdf (2, 2, i) statistics-release-1.6.3/inst/dist_fun/gamrnd.m000066400000000000000000000155211456127120000215510ustar00rootroot00000000000000## Copyright (C) 2012 Rik Wehbring ## Copyright (C) 1995-2016 Kurt Hornik ## Copyright (C) 2023 Andreas Bertsatos ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{r} =} gamrnd (@var{k}, @var{theta}) ## @deftypefnx {statistics} {@var{r} =} gamrnd (@var{k}, @var{theta}, @var{rows}) ## @deftypefnx {statistics} {@var{r} =} gamrnd (@var{k}, @var{theta}, @var{rows}, @var{cols}, @dots{}) ## @deftypefnx {statistics} {@var{r} =} gamrnd (@var{k}, @var{theta}, [@var{sz}]) ## ## Random arrays from the Gamma distribution. ## ## @code{@var{r} = gamrnd (@var{k}, @var{theta})} returns an array of random ## numbers chosen from the Gamma distribution with shape parameter @var{k} and ## scale parameter @var{theta}. The size of @var{r} is the common size of ## @var{k} and @var{theta}. A scalar input functions as a constant matrix of ## the same size as the other inputs. ## ## When called with a single size argument, @code{gamrnd} returns a square ## matrix with the dimension specified. When called with more than one scalar ## argument, the first two arguments are taken as the number of rows and columns ## and any further arguments specify additional matrix dimensions. The size may ## also be specified with a row vector of dimensions, @var{sz}. ## ## There are two equivalent parameterizations in common use: ## @enumerate ## @item With a shape parameter @math{k} and a scale parameter @math{θ}, which ## is used by @code{gamrnd}. ## @item With a shape parameter @math{α = k} and an inverse scale parameter ## @math{β = 1 / θ}, called a rate parameter. ## @end enumerate ## ## Further information about the Gamma distribution can be found at ## @url{https://en.wikipedia.org/wiki/Gamma_distribution} ## ## @seealso{gamcdf, gaminv, gampdf, gamfit, gamlike, gamstat} ## @end deftypefn function r = gamrnd (k, theta, varargin) ## Check for valid number of input arguments if (nargin < 2) error ("gamrnd: function called with too few input arguments."); endif ## Check for common size of K and THETA if (! isscalar (k) || ! isscalar (theta)) [retval, k, theta] = common_size (k, theta); if (retval > 0) error ("gamrnd: K and THETA must be of common size or scalars."); endif endif ## Check for K and THETA being reals if (iscomplex (k) || iscomplex (theta)) error ("gamrnd: K and THETA must not be complex."); endif ## Parse and check SIZE arguments if (nargin == 2) sz = size (k); elseif (nargin == 3) if (isscalar (varargin{1}) && varargin{1} >= 0 ... && varargin{1} == fix (varargin{1})) sz = [varargin{1}, varargin{1}]; elseif (isrow (varargin{1}) && all (varargin{1} >= 0) ... && all (varargin{1} == fix (varargin{1}))) sz = varargin{1}; elseif error (strcat (["gamrnd: SZ must be a scalar or a row vector"], ... [" of non-negative integers."])); endif elseif (nargin > 3) posint = cellfun (@(x) (! isscalar (x) || x < 0 || x != fix (x)), varargin); if (any (posint)) error ("gamrnd: dimensions must be non-negative integers."); endif sz = [varargin{:}]; endif ## Check that parameters match requested dimensions in size if (! isscalar (k) && ! isequal (size (k), sz)) error ("gamrnd: K and THETA must be scalars or of size SZ."); endif ## Check for class type if (isa (k, "single") || isa (theta, "single")) cls = "single"; else cls = "double"; endif ## Generate random sample from Gamma distribution if (isscalar (k) && isscalar (theta)) if ((k > 0) && (k < Inf) && (theta > 0) && (theta < Inf)) r = theta * randg (k, sz, cls); else r = NaN (sz, cls); endif else r = NaN (sz, cls); valid = (k > 0) & (k < Inf) & (theta > 0) & (theta < Inf); r(valid) = theta(valid) .* randg (k(valid), cls); endif endfunction ## Test output %!assert (size (gamrnd (1, 1)), [1 1]) %!assert (size (gamrnd (1, ones (2,1))), [2, 1]) %!assert (size (gamrnd (1, ones (2,2))), [2, 2]) %!assert (size (gamrnd (ones (2,1), 1)), [2, 1]) %!assert (size (gamrnd (ones (2,2), 1)), [2, 2]) %!assert (size (gamrnd (1, 1, 3)), [3, 3]) %!assert (size (gamrnd (1, 1, [4, 1])), [4, 1]) %!assert (size (gamrnd (1, 1, 4, 1)), [4, 1]) %!assert (size (gamrnd (1, 1, 4, 1, 5)), [4, 1, 5]) %!assert (size (gamrnd (1, 1, 0, 1)), [0, 1]) %!assert (size (gamrnd (1, 1, 1, 0)), [1, 0]) %!assert (size (gamrnd (1, 1, 1, 2, 0, 5)), [1, 2, 0, 5]) ## Test class of input preserved %!assert (class (gamrnd (1, 1)), "double") %!assert (class (gamrnd (1, single (1))), "single") %!assert (class (gamrnd (1, single ([1, 1]))), "single") %!assert (class (gamrnd (single (1), 1)), "single") %!assert (class (gamrnd (single ([1, 1]), 1)), "single") ## Test input validation %!error gamrnd () %!error gamrnd (1) %!error ... %! gamrnd (ones (3), ones (2)) %!error ... %! gamrnd (ones (2), ones (3)) %!error gamrnd (i, 2, 3) %!error gamrnd (1, i, 3) %!error ... %! gamrnd (1, 2, -1) %!error ... %! gamrnd (1, 2, 1.2) %!error ... %! gamrnd (1, 2, ones (2)) %!error ... %! gamrnd (1, 2, [2 -1 2]) %!error ... %! gamrnd (1, 2, [2 0 2.5]) %!error ... %! gamrnd (1, 2, 2, -1, 5) %!error ... %! gamrnd (1, 2, 2, 1.5, 5) %!error ... %! gamrnd (2, ones (2), 3) %!error ... %! gamrnd (2, ones (2), [3, 2]) %!error ... %! gamrnd (2, ones (2), 3, 2) statistics-release-1.6.3/inst/dist_fun/geocdf.m000066400000000000000000000123661456127120000215340ustar00rootroot00000000000000## Copyright (C) 2012 Rik Wehbring ## Copyright (C) 1995-2016 Kurt Hornik ## Copyright (C) 2023 Andreas Bertsatos ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{p} =} geocdf (@var{x}, @var{ps}) ## @deftypefnx {statistics} {@var{p} =} geocdf (@var{x}, @var{ps}, @qcode{"upper"}) ## ## Geometric cumulative distribution function (CDF). ## ## For each element of @var{x}, compute the cumulative distribution function ## (CDF) of the geometric distribution with probability of success parameter ## @var{ps}. The size of @var{p} is the common size of @var{x} and @var{ps}. ## A scalar input functions as a constant matrix of the same size as the other ## inputs. ## ## @code{@var{p} = geocdf (@var{x}, @var{ps}, "upper")} computes the upper tail ## probability of the geometric distribution with parameter @var{ps}, at the ## values in @var{x}. ## ## The geometric distribution models the number of failures (@var{x}) of a ## Bernoulli trial with probability @var{ps} before the first success. ## ## Further information about the geometric distribution can be found at ## @url{https://en.wikipedia.org/wiki/Geometric_distribution} ## ## @seealso{geoinv, geopdf, geornd, geofit, geostat} ## @end deftypefn function p = geocdf (x, ps, uflag) ## Check for valid number of input arguments if (nargin < 2) error ("geocdf: function called with too few input arguments."); endif ## Check for common size of X and PS if (! isscalar (x) || ! isscalar (ps)) [retval, x, ps] = common_size (x, ps); if (retval > 0) error ("geocdf: X and PS must be of common size or scalars."); endif endif ## Check for X and PS being reals if (iscomplex (x) || iscomplex (ps)) error ("geocdf: X and PS must not be complex."); endif ## Check for class type if (isa (x, "single") || isa (ps, "single")) p = zeros (size (x), "single"); else p = zeros (size (x)); endif ## Return NaN for out of range parameters k = isnan (x) | ! (ps >= 0) | ! (ps <= 1); p(k) = NaN; ## Return 1 for valid range parameters when X = Inf k = (x == Inf) & (ps >= 0) & (ps <= 1); p(k) = 1; ## Return 0 for X < 0 x(x < 0) = -1; ## Check for "upper" flag if (nargin > 2 && strcmpi (uflag, "upper")) uflag = true; elseif (nargin > 2 && ! strcmpi (uflag, "upper")) error ("geocdf: invalid argument for upper tail."); else uflag = false; endif ## Get valid instances k = (x >= 0) & (x < Inf) & (x == fix (x)) & (ps > 0) & (ps <= 1); ## Compute CDF if (uflag) if (any (k)) p(k) = betainc (ps(k), 1, (x(k)) + 1, "upper"); endif else if (isscalar (ps)) p(k) = 1 - ((1 - ps) .^ (x(k) + 1)); else p(k) = 1 - ((1 - ps(k)) .^ (x(k) + 1)); endif endif endfunction %!demo %! ## Plot various CDFs from the geometric distribution %! x = 0:10; %! p1 = geocdf (x, 0.2); %! p2 = geocdf (x, 0.5); %! p3 = geocdf (x, 0.7); %! plot (x, p1, "*b", x, p2, "*g", x, p3, "*r") %! grid on %! xlim ([0, 10]) %! legend ({"ps = 0.2", "ps = 0.5", "ps = 0.7"}, "location", "southeast") %! title ("Geometric CDF") %! xlabel ("values in x (number of failures)") %! ylabel ("probability") ## Test output %!test %! p = geocdf ([1, 2, 3, 4], 0.25); %! assert (p(1), 0.4375000000, 1e-14); %! assert (p(2), 0.5781250000, 1e-14); %! assert (p(3), 0.6835937500, 1e-14); %! assert (p(4), 0.7626953125, 1e-14); %!test %! p = geocdf ([1, 2, 3, 4], 0.25, "upper"); %! assert (p(1), 0.5625000000, 1e-14); %! assert (p(2), 0.4218750000, 1e-14); %! assert (p(3), 0.3164062500, 1e-14); %! assert (p(4), 0.2373046875, 1e-14); %!shared x, p %! x = [-1 0 1 Inf]; %! p = [0 0.5 0.75 1]; %!assert (geocdf (x, 0.5*ones (1,4)), p) %!assert (geocdf (x, 0.5), p) %!assert (geocdf (x, 0.5*[-1 NaN 4 1]), [NaN NaN NaN p(4)]) %!assert (geocdf ([x(1:2) NaN x(4)], 0.5), [p(1:2) NaN p(4)]) ## Test class of input preserved %!assert (geocdf ([x, NaN], 0.5), [p, NaN]) %!assert (geocdf (single ([x, NaN]), 0.5), single ([p, NaN])) %!assert (geocdf ([x, NaN], single (0.5)), single ([p, NaN])) ## Test input validation %!error geocdf () %!error geocdf (1) %!error ... %! geocdf (ones (3), ones (2)) %!error ... %! geocdf (ones (2), ones (3)) %!error geocdf (i, 2) %!error geocdf (2, i) %!error geocdf (2, 3, "tail") %!error geocdf (2, 3, 5) statistics-release-1.6.3/inst/dist_fun/geoinv.m000066400000000000000000000100431456127120000215620ustar00rootroot00000000000000## Copyright (C) 2012 Rik Wehbring ## Copyright (C) 1995-2016 Kurt Hornik ## Copyright (C) 2023 Andreas Bertsatos ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{x} =} geoinv (@var{p}, @var{ps}) ## ## Inverse of the geometric cumulative distribution function (iCDF). ## ## For each element of @var{p}, compute the quantile (the inverse of the CDF) of ## the geometric distribution with probability of success parameter @var{ps}. ## The size of @var{x} is the common size of @var{p} and @var{ps}. A scalar ## input functions as a constant matrix of the same size as the other inputs. ## ## The geometric distribution models the number of failures (@var{p}) of a ## Bernoulli trial with probability @var{ps} before the first success. ## ## Further information about the geometric distribution can be found at ## @url{https://en.wikipedia.org/wiki/Geometric_distribution} ## ## @seealso{geocdf, geopdf, geornd, geofit, geostat} ## @end deftypefn function x = geoinv (p, ps) ## Check for valid number of input arguments if (nargin < 2) error ("geoinv: function called with too few input arguments."); endif ## Check for common size of P and PS if (! isscalar (ps) || ! isscalar (ps)) [retval, p, ps] = common_size (p, ps); if (retval > 0) error ("geoinv: P and PS must be of common size or scalars."); endif endif ## Check for P and PS being reals if (iscomplex (p) || iscomplex (ps)) error ("geoinv: P and PS must not be complex."); endif ## Check for class type if (isa (p, "single") || isa (ps, "single")) x = NaN (size (p), "single"); else x = NaN (size (p)); endif ## Handle edge cases k = (p == 1) & (ps >= 0) & (ps <= 1); x(k) = Inf; ## Get valid instances k = (p >= 0) & (p < 1) & (ps > 0) & (ps <= 1); ## Compute iCDF if (isscalar (ps)) x(k) = max (ceil (log (1 - p(k)) / log (1 - ps)) - 1, 0); else x(k) = max (ceil (log (1 - p(k)) ./ log (1 - ps(k))) - 1, 0); endif endfunction %!demo %! ## Plot various iCDFs from the geometric distribution %! p = 0.001:0.001:0.999; %! x1 = geoinv (p, 0.2); %! x2 = geoinv (p, 0.5); %! x3 = geoinv (p, 0.7); %! plot (p, x1, "-b", p, x2, "-g", p, x3, "-r") %! grid on %! ylim ([0, 10]) %! legend ({"ps = 0.2", "ps = 0.5", "ps = 0.7"}, "location", "northwest") %! title ("Geometric iCDF") %! xlabel ("probability") %! ylabel ("values in x (number of failures)") ## Test output %!shared p %! p = [-1 0 0.75 1 2]; %!assert (geoinv (p, 0.5*ones (1,5)), [NaN 0 1 Inf NaN]) %!assert (geoinv (p, 0.5), [NaN 0 1 Inf NaN]) %!assert (geoinv (p, 0.5*[1 -1 NaN 4 1]), [NaN NaN NaN NaN NaN]) %!assert (geoinv ([p(1:2) NaN p(4:5)], 0.5), [NaN 0 NaN Inf NaN]) ## Test class of input preserved %!assert (geoinv ([p, NaN], 0.5), [NaN 0 1 Inf NaN NaN]) %!assert (geoinv (single ([p, NaN]), 0.5), single ([NaN 0 1 Inf NaN NaN])) %!assert (geoinv ([p, NaN], single (0.5)), single ([NaN 0 1 Inf NaN NaN])) ## Test input validation %!error geoinv () %!error geoinv (1) %!error ... %! geoinv (ones (3), ones (2)) %!error ... %! geoinv (ones (2), ones (3)) %!error ... %! geoinv (i, 2) %!error ... %! geoinv (2, i) statistics-release-1.6.3/inst/dist_fun/geopdf.m000066400000000000000000000072151456127120000215460ustar00rootroot00000000000000## Copyright (C) 2012 Rik Wehbring ## Copyright (C) 1995-2016 Kurt Hornik ## Copyright (C) 2023 Andreas Bertsatos ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{y} =} geopdf (@var{x}, @var{ps}) ## ## Geometric probability density function (PDF). ## ## For each element of @var{x}, compute the probability density function (PDF) ## of the geometric distribution with probability of success parameter @var{ps}. ## The size of @var{y} is the common size of @var{x} and @var{ps}. A scalar ## input functions as a constant matrix of the same size as the other inputs. ## ## The geometric distribution models the number of failures (@var{x}) of a ## Bernoulli trial with probability @var{ps} before the first success. ## ## Further information about the geometric distribution can be found at ## @url{https://en.wikipedia.org/wiki/Geometric_distribution} ## ## @seealso{geocdf, geoinv, geornd, geofit, geostat} ## @end deftypefn function y = geopdf (x, ps) ## Check for valid number of input arguments if (nargin < 2) error ("geopdf: function called with too few input arguments."); endif ## Check for common size of X and PS if (! isscalar (x) || ! isscalar (ps)) [retval, x, ps] = common_size (x, ps); if (retval > 0) error ("geopdf: X and PS must be of common size or scalars."); endif endif ## Check for X and PS being reals if (iscomplex (x) || iscomplex (ps)) error ("geopdf: X and PS must not be complex."); endif ## Check for class type if (isa (x, "single") || isa (ps, "single")) y = zeros (size (x), "single"); else y = zeros (size (x)); endif ## Return NaN for out of range parameters k = isnan (x) | (x == Inf) | !(ps >= 0) | !(ps <= 1); y(k) = NaN; ## Get valid instances k = (x >= 0) & (x < Inf) & (x == fix (x)) & (ps > 0) & (ps <= 1); ## Compute CDF if (isscalar (ps)) y(k) = ps * ((1 - ps) .^ x(k)); else y(k) = ps(k) .* ((1 - ps(k)) .^ x(k)); endif endfunction %!demo %! ## Plot various PDFs from the geometric distribution %! x = 0:10; %! y1 = geopdf (x, 0.2); %! y2 = geopdf (x, 0.5); %! y3 = geopdf (x, 0.7); %! plot (x, y1, "*b", x, y2, "*g", x, y3, "*r") %! grid on %! ylim ([0, 0.8]) %! legend ({"ps = 0.2", "ps = 0.5", "ps = 0.7"}, "location", "northeast") %! title ("Geometric PDF") %! xlabel ("values in x (number of failures)") %! ylabel ("density") ## Test output %!shared x, y %! x = [-1 0 1 Inf]; %! y = [0, 1/2, 1/4, NaN]; %!assert (geopdf (x, 0.5*ones (1,4)), y) %!assert (geopdf (x, 0.5), y) %!assert (geopdf (x, 0.5*[-1 NaN 4 1]), [NaN NaN NaN y(4)]) %!assert (geopdf ([x, NaN], 0.5), [y, NaN]) ## Test class of input preserved %!assert (geopdf (single ([x, NaN]), 0.5), single ([y, NaN]), 5*eps ("single")) %!assert (geopdf ([x, NaN], single (0.5)), single ([y, NaN]), 5*eps ("single")) ## Test input validation %!error geopdf () %!error geopdf (1) %!error geopdf (1,2,3) %!error geopdf (ones (3), ones (2)) %!error geopdf (ones (2), ones (3)) %!error geopdf (i, 2) %!error geopdf (2, i) statistics-release-1.6.3/inst/dist_fun/geornd.m000066400000000000000000000132061456127120000215550ustar00rootroot00000000000000## Copyright (C) 2012 Rik Wehbring ## Copyright (C) 1995-2016 Kurt Hornik ## Copyright (C) 2023 Andreas Bertsatos ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{r} =} geornd (@var{ps}) ## @deftypefnx {statistics} {@var{r} =} geornd (@var{ps}, @var{rows}) ## @deftypefnx {statistics} {@var{r} =} geornd (@var{ps}, @var{rows}, @var{cols}, @dots{}) ## @deftypefnx {statistics} {@var{r} =} geornd (@var{ps}, [@var{sz}]) ## ## Random arrays from the geometric distribution. ## ## @code{@var{r} = geornd (@var{ps})} returns an array of random numbers chosen ## from the Birnbaum-Saunders distribution with probability of success parameter ## @var{ps}. The size of @var{r} is the size of @var{ps}. ## ## When called with a single size argument, @code{geornd} returns a square ## matrix with the dimension specified. When called with more than one scalar ## argument, the first two arguments are taken as the number of rows and columns ## and any further arguments specify additional matrix dimensions. The size may ## also be specified with a row vector of dimensions, @var{sz}. ## ## The geometric distribution models the number of failures (@var{x}) of a ## Bernoulli trial with probability @var{ps} before the first success. ## ## Further information about the geometric distribution can be found at ## @url{https://en.wikipedia.org/wiki/Geometric_distribution} ## ## @seealso{geocdf, geoinv, geopdf, geofit, geostat} ## @end deftypefn function r = geornd (ps, varargin) ## Check for valid number of input arguments if (nargin < 1) error ("geornd: function called with too few input arguments."); endif ## Check for PS being reals if (iscomplex (ps)) error ("geornd: PS must not be complex."); endif ## Parse and check SIZE arguments if (nargin == 1) sz = size (ps); elseif (nargin == 2) if (isscalar (varargin{1}) && varargin{1} >= 0 ... && varargin{1} == fix (varargin{1})) sz = [varargin{1}, varargin{1}]; elseif (isrow (varargin{1}) && all (varargin{1} >= 0) ... && all (varargin{1} == fix (varargin{1}))) sz = varargin{1}; elseif error (strcat (["geornd: SZ must be a scalar or a row vector"], ... [" of non-negative integers."])); endif elseif (nargin > 2) posint = cellfun (@(x) (! isscalar (x) || x < 0 || x != fix (x)), varargin); if (any (posint)) error ("geornd: dimensions must be non-negative integers."); endif sz = [varargin{:}]; endif ## Check that parameter match requested dimensions in size if (! isscalar (ps) && ! isequal (size (ps), sz)) error ("geornd: PS must be scalar or of size SZ."); endif ## Check for class type if (isa (ps, "single")) cls = "single"; else cls = "double"; endif ## Generate random sample from geometric distribution if (isscalar (ps)) if (ps > 0 && ps < 1); r = floor (- rande (sz, cls) ./ log (1 - ps)); elseif (ps == 0) r = Inf (sz, cls); elseif (ps == 1) r = zeros (sz, cls); elseif (ps < 0 || ps > 1) r = NaN (sz, cls); endif else r = floor (- rande (sz, cls) ./ log (1 - ps)); k = ! (ps >= 0) | ! (ps <= 1); r(k) = NaN; k = (ps == 0); r(k) = Inf; endif endfunction ## Test output %!assert (size (geornd (0.5)), [1, 1]) %!assert (size (geornd (0.5*ones (2,1))), [2, 1]) %!assert (size (geornd (0.5*ones (2,2))), [2, 2]) %!assert (size (geornd (0.5, 3)), [3, 3]) %!assert (size (geornd (0.5, [4 1])), [4, 1]) %!assert (size (geornd (0.5, 4, 1)), [4, 1]) ## Test class of input preserved %!assert (class (geornd (0.5)), "double") %!assert (class (geornd (single (0.5))), "single") %!assert (class (geornd (single ([0.5 0.5]))), "single") %!assert (class (geornd (single (0))), "single") %!assert (class (geornd (single (1))), "single") ## Test input validation %!error geornd () %!error geornd (i) %!error ... %! geornd (1, -1) %!error ... %! geornd (1, 1.2) %!error ... %! geornd (1, ones (2)) %!error ... %! geornd (1, [2 -1 2]) %!error ... %! geornd (1, [2 0 2.5]) %!error ... %! geornd (ones (2), ones (2)) %!error ... %! geornd (1, 2, -1, 5) %!error ... %! geornd (1, 2, 1.5, 5) %!error geornd (ones (2,2), 3) %!error geornd (ones (2,2), [3, 2]) %!error geornd (ones (2,2), 2, 3) statistics-release-1.6.3/inst/dist_fun/gevcdf.m000066400000000000000000000170401456127120000215350ustar00rootroot00000000000000## Copyright (C) 2012 Nir Krakauer ## Copyright (C) 2023 Andreas Bertsatos ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{p} =} gevcdf (@var{x}, @var{k}, @var{sigma}, @var{mu}) ## @deftypefnx {statistics} {@var{p} =} gevcdf (@var{x}, @var{k}, @var{sigma}, @var{mu}, @qcode{"upper"}) ## ## Generalized extreme value (GEV) cumulative distribution function (CDF). ## ## For each element of @var{x}, compute the cumulative distribution function ## (CDF) of the GEV distribution with shape parameter @var{k}, scale parameter ## @var{sigma}, and location parameter @var{mu}. The size of @var{p} is the ## common size of @var{x}, @var{k}, @var{sigma}, and @var{mu}. A scalar input ## functions as a constant matrix of the same size as the other inputs. ## ## @code{[@dots{}] = gevcdf (@var{x}, @var{k}, @var{sigma}, @var{mu}, "upper")} ## computes the upper tail probability of the GEV distribution with parameters ## @var{k}, @var{sigma}, and @var{mu}, at the values in @var{x}. ## ## When @qcode{@var{k} < 0}, the GEV is the type III extreme value distribution. ## When @qcode{@var{k} > 0}, the GEV distribution is the type II, or Frechet, ## extreme value distribution. If @var{W} has a Weibull distribution as ## computed by the @code{wblcdf} function, then @qcode{-@var{W}} has a type III ## extreme value distribution and @qcode{1/@var{W}} has a type II extreme value ## distribution. In the limit as @var{k} approaches @qcode{0}, the GEV is the ## mirror image of the type I extreme value distribution as computed by the ## @code{evcdf} function. ## ## The mean of the GEV distribution is not finite when @qcode{@var{k} >= 1}, and ## the variance is not finite when @qcode{@var{k} >= 1/2}. The GEV distribution ## has positive density only for values of @var{x} such that ## @qcode{@var{k} * (@var{x} - @var{mu}) / @var{sigma} > -1}. ## ## Further information about the generalized extreme value distribution can be ## found at ## @url{https://en.wikipedia.org/wiki/Generalized_extreme_value_distribution} ## ## @subheading References ## @enumerate ## @item ## Rolf-Dieter Reiss and Michael Thomas. @cite{Statistical Analysis of Extreme ## Values with Applications to Insurance, Finance, Hydrology and Other Fields}. ## Chapter 1, pages 16-17, Springer, 2007. ## @end enumerate ## ## @seealso{gevinv, gevpdf, gevrnd, gevfit, gevlike, gevstat} ## @end deftypefn function p = gevcdf (x, k, sigma, mu, uflag) ## Check for valid number of input arguments if (nargin < 4) error ("gevcdf: function called with too few input arguments."); endif ## Check for valid "upper" flag if (nargin > 4) if (! strcmpi (uflag, "upper")) error ("gevcdf: invalid argument for upper tail."); else uflag = true; endif else uflag = false; endif ## Check for common size of X, K, SIGMA, and MU if (! isscalar (x) || ! isscalar (k) || ! isscalar (sigma) || ! isscalar (mu)) [err, x, k, sigma, mu] = common_size (x, k, sigma, mu); if (err > 0) error ("gevcdf: X, K, SIGMA, and MU must be of common size or scalars."); endif endif ## Check for X, K, SIGMA, and MU being reals if (iscomplex (x) || iscomplex (k) || iscomplex (sigma) || iscomplex (mu)) error ("gevcdf: X, K, SIGMA, and MU must not be complex."); endif ## Check for class type if (isa (x, "single") || isa (k, "single") ... || isa (sigma, "single") || isa (mu, "single")); is_class = "single"; else is_class = "double"; endif ## Prepare output p = zeros (size (x), is_class); ## Return NaN for out of range parameter SIGMA. sigma(sigma <= 0) = NaN; ## Calculate z z = (x - mu) ./ sigma; ## Process k == 0 k_0 = (abs(k) < eps); if (uflag) p(k_0) = -expm1 (-exp (-z(k_0))); else p(k_0) = exp (-exp (-z(k_0))); endif ## Process k != 0 k_0 = ! k_0; t = z .* k; if (uflag) p(k_0) = -expm1 (-exp (-(1 ./ k(k_0)) .* log1p (t(k_0)))); else p(k_0) = exp (-exp (-(1 ./ k(k_0)) .* log1p (t(k_0)))); endif ## Return 0 or 1 for 1 + k.*(x-mu)/sigma > 0 k_1 = k_0 & (t<=-1); t(k_1) = 0; if uflag == true p(k_1) = (k(k_1) >= 0); else p(k_1) = (k(k_1) < 0); endif endfunction %!demo %! ## Plot various CDFs from the generalized extreme value distribution %! x = -1:0.001:10; %! p1 = gevcdf (x, 1, 1, 1); %! p2 = gevcdf (x, 0.5, 1, 1); %! p3 = gevcdf (x, 1, 1, 5); %! p4 = gevcdf (x, 1, 2, 5); %! p5 = gevcdf (x, 1, 5, 5); %! p6 = gevcdf (x, 1, 0.5, 5); %! plot (x, p1, "-b", x, p2, "-g", x, p3, "-r", ... %! x, p4, "-c", x, p5, "-m", x, p6, "-k") %! grid on %! xlim ([-1, 10]) %! legend ({"ξ = 1, σ = 1, μ = 1", "ξ = 0.5, σ = 1, μ = 1", ... %! "ξ = 1, σ = 1, μ = 5", "ξ = 1, σ = 2, μ = 5", ... %! "ξ = 1, σ = 5, μ = 5", "ξ = 1, σ = 0.5, μ = 5"}, ... %! "location", "southeast") %! title ("Generalized extreme value CDF") %! xlabel ("values in x") %! ylabel ("probability") ## Test output %!test %! x = 0:0.5:2.5; %! sigma = 1:6; %! k = 1; %! mu = 0; %! p = gevcdf (x, k, sigma, mu); %! expected_p = [0.36788, 0.44933, 0.47237, 0.48323, 0.48954, 0.49367]; %! assert (p, expected_p, 0.001); %!test %! x = -0.5:0.5:2.5; %! sigma = 0.5; %! k = 1; %! mu = 0; %! p = gevcdf (x, k, sigma, mu); %! expected_p = [0, 0.36788, 0.60653, 0.71653, 0.77880, 0.81873, 0.84648]; %! assert (p, expected_p, 0.001); %!test # check for continuity for k near 0 %! x = 1; %! sigma = 0.5; %! k = -0.03:0.01:0.03; %! mu = 0; %! p = gevcdf (x, k, sigma, mu); %! expected_p = [0.88062, 0.87820, 0.87580, 0.87342, 0.87107, 0.86874, 0.86643]; %! assert (p, expected_p, 0.001); ## Test input validation %!error gevcdf () %!error gevcdf (1) %!error gevcdf (1, 2) %!error gevcdf (1, 2, 3) %!error ... %! gevcdf (1, 2, 3, 4, 5, 6) %!error gevcdf (1, 2, 3, 4, "tail") %!error gevcdf (1, 2, 3, 4, 5) %!error ... %! gevcdf (ones (3), ones (2), ones(2), ones(2)) %!error ... %! gevcdf (ones (2), ones (3), ones(2), ones(2)) %!error ... %! gevcdf (ones (2), ones (2), ones(3), ones(2)) %!error ... %! gevcdf (ones (2), ones (2), ones(2), ones(3)) %!error gevcdf (i, 2, 3, 4) %!error gevcdf (1, i, 3, 4) %!error gevcdf (1, 2, i, 4) %!error gevcdf (1, 2, 3, i) statistics-release-1.6.3/inst/dist_fun/gevinv.m000066400000000000000000000137021456127120000215760ustar00rootroot00000000000000## Copyright (C) 2012 Nir Krakauer ## Copyright (C) 2023 Andreas Bertsatos ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{x} =} gevinv (@var{p}, @var{k}, @var{sigma}, @var{mu}) ## ## Inverse of the generalized extreme value (GEV) cumulative distribution ## function (iCDF). ## ## For each element of @var{p}, compute the quantile (the inverse of the CDF) of ## the GEV distribution with shape parameter @var{k}, scale parameter ## @var{sigma}, and location parameter @var{mu}. The size of @var{p} is the ## common size of @var{x}, @var{k}, @var{sigma}, and @var{mu}. A scalar input ## functions as a constant matrix of the same size as the other inputs. ## ## When @qcode{@var{k} < 0}, the GEV is the type III extreme value distribution. ## When @qcode{@var{k} > 0}, the GEV distribution is the type II, or Frechet, ## extreme value distribution. If @var{W} has a Weibull distribution as ## computed by the @code{wblcdf} function, then @qcode{-@var{W}} has a type III ## extreme value distribution and @qcode{1/@var{W}} has a type II extreme value ## distribution. In the limit as @var{k} approaches @qcode{0}, the GEV is the ## mirror image of the type I extreme value distribution as computed by the ## @code{evcdf} function. ## ## The mean of the GEV distribution is not finite when @qcode{@var{k} >= 1}, and ## the variance is not finite when @qcode{@var{k} >= 1/2}. The GEV distribution ## has positive density only for values of @var{x} such that ## @qcode{@var{k} * (@var{x} - @var{mu}) / @var{sigma} > -1}. ## ## Further information about the generalized extreme value distribution can be ## found at ## @url{https://en.wikipedia.org/wiki/Generalized_extreme_value_distribution} ## ## @subheading References ## @enumerate ## @item ## Rolf-Dieter Reiss and Michael Thomas. @cite{Statistical Analysis of Extreme ## Values with Applications to Insurance, Finance, Hydrology and Other Fields}. ## Chapter 1, pages 16-17, Springer, 2007. ## @end enumerate ## ## @seealso{gevcdf, gevpdf, gevrnd, gevfit, gevlike, gevstat} ## @end deftypefn function x = gevinv (p, k, sigma, mu) ## Check for valid number of input arguments if (nargin < 4) error ("gevinv: function called with too few input arguments."); endif ## Check for common size of P, K, SIGMA, and MU [retval, p, k, sigma, mu] = common_size (p, k, sigma, mu); if (retval > 0) error ("gevinv: P, K, SIGMA, and MU must be of common size or scalars."); endif ## Check for P, K, SIGMA, and MU being reals if (iscomplex (p) || iscomplex (k) || iscomplex (sigma) || iscomplex (mu)) error ("gevinv: P, K, SIGMA, and MU must not be complex."); endif x = p; llP = log (-log (p)); kllP = k .* llP; ## Use the Taylor series expansion of the exponential to ## avoid roundoff error or dividing by zero when k is small ii = (abs(kllP) < 1E-4); x(ii) = mu(ii) - sigma(ii) .* llP(ii) .* (1 - kllP(ii) .* (1 - kllP(ii))); x(~ii) = mu(~ii) + (sigma(~ii) ./ k(~ii)) .* (exp(-kllP(~ii)) - 1); endfunction %!demo %! ## Plot various iCDFs from the generalized extreme value distribution %! p = 0.001:0.001:0.999; %! x1 = gevinv (p, 1, 1, 1); %! x2 = gevinv (p, 0.5, 1, 1); %! x3 = gevinv (p, 1, 1, 5); %! x4 = gevinv (p, 1, 2, 5); %! x5 = gevinv (p, 1, 5, 5); %! x6 = gevinv (p, 1, 0.5, 5); %! plot (p, x1, "-b", p, x2, "-g", p, x3, "-r", ... %! p, x4, "-c", p, x5, "-m", p, x6, "-k") %! grid on %! ylim ([-1, 10]) %! legend ({"ξ = 1, σ = 1, μ = 1", "ξ = 0.5, σ = 1, μ = 1", ... %! "ξ = 1, σ = 1, μ = 5", "ξ = 1, σ = 2, μ = 5", ... %! "ξ = 1, σ = 5, μ = 5", "ξ = 1, σ = 0.5, μ = 5"}, ... %! "location", "northwest") %! title ("Generalized extreme value iCDF") %! xlabel ("probability") %! ylabel ("values in x") ## Test output %!test %! p = 0.1:0.1:0.9; %! k = 0; %! sigma = 1; %! mu = 0; %! x = gevinv (p, k, sigma, mu); %! c = gevcdf(x, k, sigma, mu); %! assert (c, p, 0.001); %!test %! p = 0.1:0.1:0.9; %! k = 1; %! sigma = 1; %! mu = 0; %! x = gevinv (p, k, sigma, mu); %! c = gevcdf(x, k, sigma, mu); %! assert (c, p, 0.001); %!test %! p = 0.1:0.1:0.9; %! k = 0.3; %! sigma = 1; %! mu = 0; %! x = gevinv (p, k, sigma, mu); %! c = gevcdf(x, k, sigma, mu); %! assert (c, p, 0.001); ## Test input validation %!error gevinv () %!error gevinv (1) %!error gevinv (1, 2) %!error gevinv (1, 2, 3) %!error ... %! gevinv (ones (3), ones (2), ones(2), ones(2)) %!error ... %! gevinv (ones (2), ones (3), ones(2), ones(2)) %!error ... %! gevinv (ones (2), ones (2), ones(3), ones(2)) %!error ... %! gevinv (ones (2), ones (2), ones(2), ones(3)) %!error gevinv (i, 2, 3, 4) %!error gevinv (1, i, 3, 4) %!error gevinv (1, 2, i, 4) %!error gevinv (1, 2, 3, i) statistics-release-1.6.3/inst/dist_fun/gevpdf.m000066400000000000000000000142541456127120000215560ustar00rootroot00000000000000## Copyright (C) 2012 Nir Krakauer ## Copyright (C) 2023 Andreas Bertsatos ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{y} =} gevpdf (@var{x}, @var{k}, @var{sigma}, @var{mu}) ## ## Generalized extreme value (GEV) probability density function (PDF). ## ## For each element of @var{x}, compute the probability density function (PDF) ## of the GEV distribution with shape parameter @var{k}, scale parameter ## @var{sigma}, and location parameter @var{mu}. The size of @var{y} is the ## common size of @var{x}, @var{k}, @var{sigma}, and @var{mu}. A scalar input ## functions as a constant matrix of the same size as the other inputs. ## ## When @qcode{@var{k} < 0}, the GEV is the type III extreme value distribution. ## When @qcode{@var{k} > 0}, the GEV distribution is the type II, or Frechet, ## extreme value distribution. If @var{W} has a Weibull distribution as ## computed by the @code{wblcdf} function, then @qcode{-@var{W}} has a type III ## extreme value distribution and @qcode{1/@var{W}} has a type II extreme value ## distribution. In the limit as @var{k} approaches @qcode{0}, the GEV is the ## mirror image of the type I extreme value distribution as computed by the ## @code{evcdf} function. ## ## The mean of the GEV distribution is not finite when @qcode{@var{k} >= 1}, and ## the variance is not finite when @qcode{@var{k} >= 1/2}. The GEV distribution ## has positive density only for values of @var{x} such that ## @qcode{@var{k} * (@var{x} - @var{mu}) / @var{sigma} > -1}. ## ## Further information about the generalized extreme value distribution can be ## found at ## @url{https://en.wikipedia.org/wiki/Generalized_extreme_value_distribution} ## ## @subheading References ## @enumerate ## @item ## Rolf-Dieter Reiss and Michael Thomas. @cite{Statistical Analysis of Extreme ## Values with Applications to Insurance, Finance, Hydrology and Other Fields}. ## Chapter 1, pages 16-17, Springer, 2007. ## @end enumerate ## ## @seealso{gevcdf, gevinv, gevrnd, gevfit, gevlike, gevstat} ## @end deftypefn function y = gevpdf (x, k, sigma, mu) ## Check for valid number of input arguments if (nargin < 4) error ("gevpdf: function called with too few input arguments."); endif ## Check for common size of X, K, SIGMA, and MU [retval, x, k, sigma, mu] = common_size (x, k, sigma, mu); if (retval > 0) error ("gevpdf: X, K, SIGMA, and MU must be of common size or scalars."); endif ## Check for X, K, SIGMA, and MU being reals if (iscomplex (x) || iscomplex (k) || iscomplex (sigma) || iscomplex (mu)) error ("gevpdf: X, K, SIGMA, and MU must not be complex."); endif z = 1 + k .* (x - mu) ./ sigma; ## Calculate generalized extreme value PDF y = exp(-(z .^ (-1 ./ k))) .* (z .^ (-1 - 1 ./ k)) ./ sigma; y(z <= 0) = 0; ## Use a different formula if k is very close to zero inds = (abs (k) < (eps^0.7)); if (any (inds)) z = (mu(inds) - x(inds)) ./ sigma(inds); y(inds) = exp (z - exp (z)) ./ sigma(inds); endif endfunction %!demo %! ## Plot various PDFs from the generalized extreme value distribution %! x = -1:0.001:10; %! y1 = gevpdf (x, 1, 1, 1); %! y2 = gevpdf (x, 0.5, 1, 1); %! y3 = gevpdf (x, 1, 1, 5); %! y4 = gevpdf (x, 1, 2, 5); %! y5 = gevpdf (x, 1, 5, 5); %! y6 = gevpdf (x, 1, 0.5, 5); %! plot (x, y1, "-b", x, y2, "-g", x, y3, "-r", ... %! x, y4, "-c", x, y5, "-m", x, y6, "-k") %! grid on %! xlim ([-1, 10]) %! ylim ([0, 1.1]) %! legend ({"ξ = 1, σ = 1, μ = 1", "ξ = 0.5, σ = 1, μ = 1", ... %! "ξ = 1, σ = 1, μ = 5", "ξ = 1, σ = 2, μ = 5", ... %! "ξ = 1, σ = 5, μ = 5", "ξ = 1, σ = 0.5, μ = 5"}, ... %! "location", "northeast") %! title ("Generalized extreme value PDF") %! xlabel ("values in x") %! ylabel ("density") ## Test output %!test %! x = 0:0.5:2.5; %! sigma = 1:6; %! k = 1; %! mu = 0; %! y = gevpdf (x, k, sigma, mu); %! expected_y = [0.367879 0.143785 0.088569 0.063898 0.049953 0.040997]; %! assert (y, expected_y, 0.001); %!test %! x = -0.5:0.5:2.5; %! sigma = 0.5; %! k = 1; %! mu = 0; %! y = gevpdf (x, k, sigma, mu); %! expected_y = [0 0.735759 0.303265 0.159229 0.097350 0.065498 0.047027]; %! assert (y, expected_y, 0.001); %!test # check for continuity for k near 0 %! x = 1; %! sigma = 0.5; %! k = -0.03:0.01:0.03; %! mu = 0; %! y = gevpdf (x, k, sigma, mu); %! expected_y = [0.23820 0.23764 0.23704 0.23641 0.23576 0.23508 0.23438]; %! assert (y, expected_y, 0.001); ## Test input validation %!error gevpdf () %!error gevpdf (1) %!error gevpdf (1, 2) %!error gevpdf (1, 2, 3) %!error ... %! gevpdf (ones (3), ones (2), ones(2), ones(2)) %!error ... %! gevpdf (ones (2), ones (3), ones(2), ones(2)) %!error ... %! gevpdf (ones (2), ones (2), ones(3), ones(2)) %!error ... %! gevpdf (ones (2), ones (2), ones(2), ones(3)) %!error gevpdf (i, 2, 3, 4) %!error gevpdf (1, i, 3, 4) %!error gevpdf (1, 2, i, 4) %!error gevpdf (1, 2, 3, i) statistics-release-1.6.3/inst/dist_fun/gevrnd.m000066400000000000000000000175441456127120000215750ustar00rootroot00000000000000## Copyright (C) 2012 Nir Krakauer ## Copyright (C) 2023 Andreas Bertsatos ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## Octave is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with Octave; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{r} =} gevrnd (@var{k}, @var{sigma}, @var{mu}) ## @deftypefnx {statistics} {@var{r} =} gevrnd (@var{k}, @var{sigma}, @var{mu}, @var{rows}) ## @deftypefnx {statistics} {@var{r} =} gevrnd (@var{k}, @var{sigma}, @var{mu}, @var{rows}, @var{cols}, @dots{}) ## @deftypefnx {statistics} {@var{r} =} gevrnd (@var{k}, @var{sigma}, @var{mu}, [@var{sz}]) ## ## Random arrays from the generalized extreme value (GEV) distribution. ## ## @code{@var{r} = gevrnd (@var{k}, @var{sigma}, @var{mu}} returns an array of ## random numbers chosen from the GEV distribution with shape parameter @var{k}, ## scale parameter @var{sigma}, and location parameter @var{mu}. The size of ## @var{r} is the common size of @var{k}, @var{sigma}, and @var{mu}. A scalar ## input functions as a constant matrix of the same size as the other inputs. ## ## When called with a single size argument, @code{gevrnd} returns a square ## matrix with the dimension specified. When called with more than one scalar ## argument, the first two arguments are taken as the number of rows and columns ## and any further arguments specify additional matrix dimensions. The size may ## also be specified with a row vector of dimensions, @var{sz}. ## ## When @qcode{@var{k} < 0}, the GEV is the type III extreme value distribution. ## When @qcode{@var{k} > 0}, the GEV distribution is the type II, or Frechet, ## extreme value distribution. If @var{W} has a Weibull distribution as ## computed by the @code{wblcdf} function, then @qcode{-@var{W}} has a type III ## extreme value distribution and @qcode{1/@var{W}} has a type II extreme value ## distribution. In the limit as @var{k} approaches @qcode{0}, the GEV is the ## mirror image of the type I extreme value distribution as computed by the ## @code{evcdf} function. ## ## The mean of the GEV distribution is not finite when @qcode{@var{k} >= 1}, and ## the variance is not finite when @qcode{@var{k} >= 1/2}. The GEV distribution ## has positive density only for values of @var{x} such that ## @qcode{@var{k} * (@var{x} - @var{mu}) / @var{sigma} > -1}. ## ## Further information about the generalized extreme value distribution can be ## found at ## @url{https://en.wikipedia.org/wiki/Generalized_extreme_value_distribution} ## ## @subheading References ## @enumerate ## @item ## Rolf-Dieter Reiss and Michael Thomas. @cite{Statistical Analysis of Extreme ## Values with Applications to Insurance, Finance, Hydrology and Other Fields}. ## Chapter 1, pages 16-17, Springer, 2007. ## @end enumerate ## ## @seealso{gevcdf, gevinv, gevpdf, gevfit, gevlike, gevstat} ## @end deftypefn function r = gevrnd (k, sigma, mu, varargin) ## Check for valid number of input arguments if (nargin < 3) error ("gevrnd: function called with too few input arguments."); endif ## Check for common size of K, SIGMA, and MU if (! isscalar (k) || ! isscalar (sigma) || ! isscalar (mu)) [retval, k, sigma, mu] = common_size (k, sigma, mu); if (retval > 0) error ("gevrnd: K, SIGMA, and MU must be of common size or scalars."); endif endif ## Check for K, SIGMA, and MU being reals if (iscomplex (k) || iscomplex (sigma) || iscomplex (mu)) error ("gevrnd: K, SIGMA, and MU must not be complex."); endif ## Parse and check SIZE arguments if (nargin == 3) sz = size (k); elseif (nargin == 4) if (isscalar (varargin{1}) && varargin{1} >= 0 ... && varargin{1} == fix (varargin{1})) sz = [varargin{1}, varargin{1}]; elseif (isrow (varargin{1}) && all (varargin{1} >= 0) ... && all (varargin{1} == fix (varargin{1}))) sz = varargin{1}; elseif error (strcat (["gevrnd: SZ must be a scalar or a row vector"], ... [" of non-negative integers."])); endif elseif (nargin > 4) posint = cellfun (@(x) (! isscalar (x) || x < 0 || x != fix (x)), varargin); if (any (posint)) error ("gevrnd: dimensions must be non-negative integers."); endif sz = [varargin{:}]; endif ## Check that parameters match requested dimensions in size if (!isscalar (k) && ! isequal (size (k), sz)) error ("gevrnd: K, SIGMA, and MU must be scalars or of size SZ."); endif ## Check for class type if (isa (k, "single") || isa (sigma, "single") || isa (mu, "single")) cls = "single"; else cls = "double"; endif ## Generate random sample from Burr type XII distribution r = gevinv (rand(sz), k, sigma, mu); r = cast (r, cls); endfunction ## Test output %!assert(size (gevrnd (1,2,1)), [1, 1]); %!assert(size (gevrnd (ones(2,1), 2, 1)), [2, 1]); %!assert(size (gevrnd (ones(2,2), 2, 1)), [2, 2]); %!assert(size (gevrnd (1, 2*ones(2,1), 1)), [2, 1]); %!assert(size (gevrnd (1, 2*ones(2,2), 1)), [2, 2]); %!assert(size (gevrnd (1, 2, 1, 3)), [3, 3]); %!assert(size (gevrnd (1, 2, 1, [4 1])), [4, 1]); %!assert(size (gevrnd (1, 2, 1, 4, 1)), [4, 1]); ## Test class of input preserved %!assert (class (gevrnd (1,1,1)), "double") %!assert (class (gevrnd (single (1),1,1)), "single") %!assert (class (gevrnd (single ([1 1]),1,1)), "single") %!assert (class (gevrnd (1,single (1),1)), "single") %!assert (class (gevrnd (1,single ([1 1]),1)), "single") %!assert (class (gevrnd (1,1,single (1))), "single") %!assert (class (gevrnd (1,1,single ([1 1]))), "single") ## Test input validation %!error gevrnd () %!error gevrnd (1) %!error gevrnd (1, 2) %!error ... %! gevrnd (ones (3), ones (2), ones (2)) %!error ... %! gevrnd (ones (2), ones (3), ones (2)) %!error ... %! gevrnd (ones (2), ones (2), ones (3)) %!error gevrnd (i, 2, 3) %!error gevrnd (1, i, 3) %!error gevrnd (1, 2, i) %!error ... %! gevrnd (1, 2, 3, -1) %!error ... %! gevrnd (1, 2, 3, 1.2) %!error ... %! gevrnd (1, 2, 3, ones (2)) %!error ... %! gevrnd (1, 2, 3, [2 -1 2]) %!error ... %! gevrnd (1, 2, 3, [2 0 2.5]) %!error ... %! gevrnd (1, 2, 3, 2, -1, 5) %!error ... %! gevrnd (1, 2, 3, 2, 1.5, 5) %!error ... %! gevrnd (2, ones (2), 2, 3) %!error ... %! gevrnd (2, ones (2), 2, [3, 2]) %!error ... %! gevrnd (2, ones (2), 2, 3, 2) statistics-release-1.6.3/inst/dist_fun/gpcdf.m000066400000000000000000000265121456127120000213660ustar00rootroot00000000000000## Copyright (C) 1997-2015 Kurt Hornik ## Copyright (C) 2016 Dag Lyberg ## Copyright (C) 2018 John Donoghue ## Copyright (C) 2022-2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## Octave is free software; you can redistribute it and/or modify it ## under the terms of the GNU General Public License as published by ## the Free Software Foundation; either version 3 of the License, or (at ## your option) any later version. ## ## Octave is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with Octave; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{p} =} gpcdf (@var{x}, @var{k}, @var{sigma}, @var{mu}) ## @deftypefnx {statistics} {@var{p} =} gpcdf (@var{x}, @var{k}, @var{sigma}, @var{mu}, @qcode{"upper"}) ## ## Generalized Pareto cumulative distribution function (CDF). ## ## For each element of @var{x}, compute the cumulative distribution function ## (CDF) of the generalized Pareto distribution with shape parameter @var{k}, ## scale parameter @var{sigma}, and location parameter @var{mu}. The size of ## @var{p} is the common size of @var{x}, @var{k}, @var{sigma}, and @var{mu}. ## A scalar input functions as a constant matrix of the same size as the other ## inputs. ## ## @code{[@dots{}] = gpcdf(@var{x}, @var{k}, @var{sigma}, @var{mu}, "upper")} ## computes the upper tail probability of the generalized Pareto distribution ## with parameters @var{k}, @var{sigma}, and @var{mu}, at the values in @var{x}. ## ## When @qcode{@var{k} = 0} and @qcode{@var{mu} = 0}, the Generalized Pareto CDF ## is equivalent to the exponential distribution. When @qcode{@var{k} > 0} and ## @code{@var{mu} = @var{k} / @var{k}} the Generalized Pareto is equivalent to ## the Pareto distribution. The mean of the Generalized Pareto is not finite ## when @qcode{@var{k} >= 1} and the variance is not finite when ## @qcode{@var{k} >= 1/2}. When @qcode{@var{k} >= 0}, the Generalized Pareto ## has positive density for @qcode{@var{x} > @var{mu}}, or, when ## @qcode{@var{mu} < 0}, for ## @qcode{0 <= (@var{x} - @var{mu}) / @var{sigma} <= -1 / @var{k}}. ## ## Further information about the generalized Pareto distribution can be found at ## @url{https://en.wikipedia.org/wiki/Generalized_Pareto_distribution} ## ## @seealso{gpinv, gppdf, gprnd, gpfit, gplike, gpstat} ## @end deftypefn function p = gpcdf (x, k, sigma, mu, uflag) ## Check for valid number of input arguments if (nargin < 4) error ("gpcdf: function called with too few input arguments."); endif ## Check for valid "upper" flag if (nargin > 4) if (! strcmpi (uflag, "upper")) error ("gpcdf: invalid argument for upper tail."); else uflag = true; endif else uflag = false; endif ## Check for common size of X, K, SIGMA, and MU if (! isscalar (x) || ! isscalar (k) || ! isscalar (sigma) || ! isscalar (mu)) [err, x, k, sigma, mu] = common_size (x, k, sigma, mu); if (err > 0) error ("gpcdf: X, K, SIGMA, and MU must be of common size or scalars."); endif endif ## Check for X, K, SIGMA, and MU being reals if (iscomplex (x) || iscomplex (k) || iscomplex (sigma) || iscomplex (mu)) error ("gpcdf: X, K, SIGMA, and MU must not be complex."); endif ## Check for class type if (isa (x, "single") || isa (k, "single") ... || isa (sigma, "single") || isa (mu, "single")); is_class = "single"; else is_class = "double"; endif ## Prepare output p = zeros (size (x), is_class); ## Return NaNs for out of range values of sigma parameter sigma(sigma <= 0) = NaN; ## Calculate (x-mu)/sigma => 0 and force zero below that z = (x - mu) ./ sigma; z(z < 0) = 0; ## Compute cases for SHAPE == 0 kz = (abs (k) < eps (is_class)); if (uflag) p(kz) = exp (-z(kz)); else p(kz) = -expm1 (-z(kz)); endif ## For SHAPE < 0, calculate 0 <= x/sigma <= -1/k and force zero below that t = z .* k; kt = (t <= -1 & k < -eps (is_class)); t(kt) = 0; ## Compute cases for SHAPE != 0 kz = ! kz; if (uflag) p(kz) = exp ((-1 ./ k(kz)) .* log1p (t(kz))); else p(kz) = -expm1 ((-1 ./ k(kz)) .* log1p (t(kz))); endif if (uflag) p(kt) = 0; else p(kt) = 1; endif ## For SHAPE == NaN force p = NaN p(isnan (k)) = NaN; endfunction %!demo %! ## Plot various CDFs from the generalized Pareto distribution %! x = 0:0.001:5; %! p1 = gpcdf (x, 1, 1, 0); %! p2 = gpcdf (x, 5, 1, 0); %! p3 = gpcdf (x, 20, 1, 0); %! p4 = gpcdf (x, 1, 2, 0); %! p5 = gpcdf (x, 5, 2, 0); %! p6 = gpcdf (x, 20, 2, 0); %! plot (x, p1, "-b", x, p2, "-g", x, p3, "-r", ... %! x, p4, "-c", x, p5, "-m", x, p6, "-k") %! grid on %! xlim ([0, 5]) %! legend ({"ξ = 1, σ = 1, μ = 0", "ξ = 5, σ = 1, μ = 0", ... %! "ξ = 20, σ = 1, μ = 0", "ξ = 1, σ = 2, μ = 0", ... %! "ξ = 5, σ = 2, μ = 0", "ξ = 20, σ = 2, μ = 0"}, ... %! "location", "northwest") %! title ("Generalized Pareto CDF") %! xlabel ("values in x") %! ylabel ("probability") ## Test output %!shared x, y1, y1u, y2, y2u, y3, y3u %! x = [-Inf, -1, 0, 1/2, 1, Inf]; %! y1 = [0, 0, 0, 0.3934693402873666, 0.6321205588285577, 1]; %! y1u = [1, 1, 1, 0.6065306597126334, 0.3678794411714423, 0]; %! y2 = [0, 0, 0, 1/3, 1/2, 1]; %! y2u = [1, 1, 1, 2/3, 1/2, 0]; %! y3 = [0, 0, 0, 1/2, 1, 1]; %! y3u = [1, 1, 1, 1/2, 0, 0]; %!assert (gpcdf (x, zeros (1,6), ones (1,6), zeros (1,6)), y1, eps) %!assert (gpcdf (x, 0, 1, zeros (1,6)), y1, eps) %!assert (gpcdf (x, 0, ones (1,6), 0), y1, eps) %!assert (gpcdf (x, zeros (1,6), 1, 0), y1, eps) %!assert (gpcdf (x, 0, 1, 0), y1, eps) %!assert (gpcdf (x, 0, 1, [0, 0, 0, NaN, 0, 0]), [y1(1:3), NaN, y1(5:6)], eps) %!assert (gpcdf (x, 0, [1, 1, 1, NaN, 1, 1], 0), [y1(1:3), NaN, y1(5:6)], eps) %!assert (gpcdf (x, [0, 0, 0, NaN, 0, 0], 1, 0), [y1(1:3), NaN, y1(5:6)], eps) %!assert (gpcdf ([x(1:3), NaN, x(5:6)], 0, 1, 0), [y1(1:3), NaN, y1(5:6)], eps) %!assert (gpcdf (x, zeros (1,6), ones (1,6), zeros (1,6), "upper"), y1u, eps) %!assert (gpcdf (x, 0, 1, zeros (1,6), "upper"), y1u, eps) %!assert (gpcdf (x, 0, ones (1,6), 0, "upper"), y1u, eps) %!assert (gpcdf (x, zeros (1,6), 1, 0, "upper"), y1u, eps) %!assert (gpcdf (x, 0, 1, 0, "upper"), y1u, eps) %!assert (gpcdf (x, ones (1,6), ones (1,6), zeros (1,6)), y2, eps) %!assert (gpcdf (x, 1, 1, zeros (1,6)), y2, eps) %!assert (gpcdf (x, 1, ones (1,6), 0), y2, eps) %!assert (gpcdf (x, ones (1,6), 1, 0), y2, eps) %!assert (gpcdf (x, 1, 1, 0), y2, eps) %!assert (gpcdf (x, 1, 1, [0, 0, 0, NaN, 0, 0]), [y2(1:3), NaN, y2(5:6)], eps) %!assert (gpcdf (x, 1, [1, 1, 1, NaN, 1, 1], 0), [y2(1:3), NaN, y2(5:6)], eps) %!assert (gpcdf (x, [1, 1, 1, NaN, 1, 1], 1, 0), [y2(1:3), NaN, y2(5:6)], eps) %!assert (gpcdf ([x(1:3), NaN, x(5:6)], 1, 1, 0), [y2(1:3), NaN, y2(5:6)], eps) %!assert (gpcdf (x, ones (1,6), ones (1,6), zeros (1,6), "upper"), y2u, eps) %!assert (gpcdf (x, 1, 1, zeros (1,6), "upper"), y2u, eps) %!assert (gpcdf (x, 1, ones (1,6), 0, "upper"), y2u, eps) %!assert (gpcdf (x, ones (1,6), 1, 0, "upper"), y2u, eps) %!assert (gpcdf (x, 1, 1, 0, "upper"), y2u, eps) %!assert (gpcdf (x, 1, 1, [0, 0, 0, NaN, 0, 0], "upper"), ... %! [y2u(1:3), NaN, y2u(5:6)], eps) %!assert (gpcdf (x, 1, [1, 1, 1, NaN, 1, 1], 0, "upper"), ... %! [y2u(1:3), NaN, y2u(5:6)], eps) %!assert (gpcdf (x, [1, 1, 1, NaN, 1, 1], 1, 0, "upper"), ... %! [y2u(1:3), NaN, y2u(5:6)], eps) %!assert (gpcdf ([x(1:3), NaN, x(5:6)], 1, 1, 0, "upper"), ... %! [y2u(1:3), NaN, y2u(5:6)], eps) %!assert (gpcdf (x, -ones (1,6), ones (1,6), zeros (1,6)), y3, eps) %!assert (gpcdf (x, -1, 1, zeros (1,6)), y3, eps) %!assert (gpcdf (x, -1, ones (1,6), 0), y3, eps) %!assert (gpcdf (x, -ones (1,6), 1, 0), y3, eps) %!assert (gpcdf (x, -1, 1, 0), y3, eps) %!assert (gpcdf (x, -1, 1, [0, 0, 0, NaN, 0, 0]), [y3(1:3), NaN, y3(5:6)], eps) %!assert (gpcdf (x, -1, [1, 1, 1, NaN, 1, 1], 0), [y3(1:3), NaN, y3(5:6)], eps) %!assert (gpcdf (x, [-1, -1, -1, NaN, -1, -1], 1, 0), [y3(1:3), NaN, y3(5:6)], eps) %!assert (gpcdf ([x(1:3), NaN, x(5:6)], -1, 1, 0), [y3(1:3), NaN, y3(5:6)], eps) %!assert (gpcdf (x, -ones (1,6), ones (1,6), zeros (1,6), "upper"), y3u, eps) %!assert (gpcdf (x, -1, 1, zeros (1,6), "upper"), y3u, eps) %!assert (gpcdf (x, -1, ones (1,6), 0, "upper"), y3u, eps) %!assert (gpcdf (x, -ones (1,6), 1, 0, "upper"), y3u, eps) %!assert (gpcdf (x, -1, 1, 0, "upper"), y3u, eps) %!assert (gpcdf (x, -1, 1, [0, 0, 0, NaN, 0, 0], "upper"), ... %! [y3u(1:3), NaN, y3u(5:6)], eps) %!assert (gpcdf (x, -1, [1, 1, 1, NaN, 1, 1], 0, "upper"), ... %! [y3u(1:3), NaN, y3u(5:6)], eps) %!assert (gpcdf (x, [-1, -1, -1, NaN, -1, -1], 1, 0, "upper"), ... %! [y3u(1:3), NaN, y3u(5:6)], eps) %!assert (gpcdf ([x(1:3), NaN, x(5:6)], -1, 1, 0, "upper"), ... %! [y3u(1:3), NaN, y3u(5:6)], eps) ## Test class of input preserved %!assert (gpcdf (single ([x, NaN]), 0, 1, 0), single ([y1, NaN]), eps("single")) %!assert (gpcdf ([x, NaN], 0, 1, single (0)), single ([y1, NaN]), eps("single")) %!assert (gpcdf ([x, NaN], 0, single (1), 0), single ([y1, NaN]), eps("single")) %!assert (gpcdf ([x, NaN], single (0), 1, 0), single ([y1, NaN]), eps("single")) %!assert (gpcdf (single ([x, NaN]), 1, 1, 0), single ([y2, NaN]), eps("single")) %!assert (gpcdf ([x, NaN], 1, 1, single (0)), single ([y2, NaN]), eps("single")) %!assert (gpcdf ([x, NaN], 1, single (1), 0), single ([y2, NaN]), eps("single")) %!assert (gpcdf ([x, NaN], single (1), 1, 0), single ([y2, NaN]), eps("single")) %!assert (gpcdf (single ([x, NaN]), -1, 1, 0), single ([y3, NaN]), eps("single")) %!assert (gpcdf ([x, NaN], -1, 1, single (0)), single ([y3, NaN]), eps("single")) %!assert (gpcdf ([x, NaN], -1, single (1), 0), single ([y3, NaN]), eps("single")) %!assert (gpcdf ([x, NaN], single (-1), 1, 0), single ([y3, NaN]), eps("single")) ## Test input validation %!error gpcdf () %!error gpcdf (1) %!error gpcdf (1, 2) %!error gpcdf (1, 2, 3) %!error gpcdf (1, 2, 3, 4, "tail") %!error gpcdf (1, 2, 3, 4, 5) %!error ... %! gpcdf (ones (3), ones (2), ones(2), ones(2)) %!error ... %! gpcdf (ones (2), ones (3), ones(2), ones(2)) %!error ... %! gpcdf (ones (2), ones (2), ones(3), ones(2)) %!error ... %! gpcdf (ones (2), ones (2), ones(2), ones(3)) %!error gpcdf (i, 2, 3, 4) %!error gpcdf (1, i, 3, 4) %!error gpcdf (1, 2, i, 4) %!error gpcdf (1, 2, 3, i) statistics-release-1.6.3/inst/dist_fun/gpinv.m000066400000000000000000000210071456127120000214200ustar00rootroot00000000000000## Copyright (C) 1997-2015 Kurt Hornik ## Copyright (C) 2016 Dag Lyberg ## Copyright (C) 2018 John Donoghue ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## Octave is free software; you can redistribute it and/or modify it ## under the terms of the GNU General Public License as published by ## the Free Software Foundation; either version 3 of the License, or (at ## your option) any later version. ## ## Octave is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with Octave; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{x} =} gpinv (@var{p}, @var{k}, @var{sigma}, @var{mu}) ## ## Inverse of the generalized Pareto cumulative distribution function (iCDF). ## ## For each element of @var{p}, compute the quantile (the inverse of the CDF) of ## the generalized Pareto distribution with shape parameter @var{k}, scale ## parameter @var{sigma}, and location parameter @var{mu}. The size of @var{x} ## is the common size of @var{p}, @var{k}, @var{sigma}, and @var{mu}. A scalar ## input functions as a constant matrix of the same size as the other inputs. ## ## When @qcode{@var{k} = 0} and @qcode{@var{mu} = 0}, the Generalized Pareto CDF ## is equivalent to the exponential distribution. When @qcode{@var{k} > 0} and ## @code{@var{mu} = @var{k} / @var{k}} the Generalized Pareto is equivalent to ## the Pareto distribution. The mean of the Generalized Pareto is not finite ## when @qcode{@var{k} >= 1} and the variance is not finite when ## @qcode{@var{k} >= 1/2}. When @qcode{@var{k} >= 0}, the Generalized Pareto ## has positive density for @qcode{@var{x} > @var{mu}}, or, when ## @qcode{@var{mu} < 0}, for ## @qcode{0 <= (@var{x} - @var{mu}) / @var{sigma} <= -1 / @var{k}}. ## ## Further information about the generalized Pareto distribution can be found at ## @url{https://en.wikipedia.org/wiki/Generalized_Pareto_distribution} ## ## @seealso{gpcdf, gppdf, gprnd, gpfit, gplike, gpstat} ## @end deftypefn function x = gpinv (p, k, sigma, mu) ## Check for valid number of input arguments if (nargin < 4) error ("gpinv: function called with too few input arguments."); endif ## Check for common size of P, K, SIGMA, and MU [retval, p, k, sigma, mu] = common_size (p, k, sigma, mu); if (retval > 0) error ("gpinv: P, K, SIGMA, and MU must be of common size or scalars."); endif ## Check for P, K, SIGMA, and MU being reals if (iscomplex (p) || iscomplex (k) || iscomplex (sigma) || iscomplex (mu)) error ("gpinv: P, K, SIGMA, and MU must not be complex."); endif ## Check for class type if (isa (p, "single") || isa (mu, "single") ... || isa (sigma, "single") || isa (k, "single")) x = zeros (size (p), "single"); else x = zeros (size (p)); endif ## Return NaNs for out of range values of sigma parameter kx = isnan (p) | ! (0 <= p) | ! (p <= 1) ... | ! (-Inf < mu) | ! (mu < Inf) ... | ! (sigma > 0) | ! (sigma < Inf) ... | ! (-Inf < k) | ! (k < Inf); x(kx) = NaN; kx = (0 <= p) & (p <= 1) & (-Inf < mu) & (mu < Inf) ... & (sigma > 0) & (sigma < Inf) & (-Inf < k) & (k < Inf); if (isscalar (mu) && isscalar (sigma) && isscalar (k)) if (k == 0) x(kx) = -log(1 - p(kx)); x(kx) = sigma * x(kx) + mu; elseif (k > 0) x(kx) = (1 - p(kx)).^(-k) - 1; x(kx) = (sigma / k) * x(kx) + mu; elseif (k < 0) x(kx) = (1 - p(kx)).^(-k) - 1; x(kx) = (sigma / k) * x(kx) + mu; end else j = kx & (k == 0); if (any (j)) x(j) = -log (1 - p(j)); x(j) = sigma(j) .* x(j) + mu(j); endif j = kx & (k > 0); if (any (j)) x(j) = (1 - p(j)).^(-k(j)) - 1; x(j) = (sigma(j) ./ k(j)) .* x(j) + mu(j); endif j = kx & (k < 0); if (any (j)) x(j) = (1 - p(j)).^(-k(j)) - 1; x(j) = (sigma(j) ./ k(j)) .* x(j) + mu(j); endif endif endfunction %!demo %! ## Plot various iCDFs from the generalized Pareto distribution %! p = 0.001:0.001:0.999; %! x1 = gpinv (p, 1, 1, 0); %! x2 = gpinv (p, 5, 1, 0); %! x3 = gpinv (p, 20, 1, 0); %! x4 = gpinv (p, 1, 2, 0); %! x5 = gpinv (p, 5, 2, 0); %! x6 = gpinv (p, 20, 2, 0); %! plot (p, x1, "-b", p, x2, "-g", p, x3, "-r", ... %! p, x4, "-c", p, x5, "-m", p, x6, "-k") %! grid on %! ylim ([0, 5]) %! legend ({"ξ = 1, σ = 1, μ = 0", "ξ = 5, σ = 1, μ = 0", ... %! "ξ = 20, σ = 1, μ = 0", "ξ = 1, σ = 2, μ = 0", ... %! "ξ = 5, σ = 2, μ = 0", "ξ = 20, σ = 2, μ = 0"}, ... %! "location", "southeast") %! title ("Generalized Pareto iCDF") %! xlabel ("probability") %! ylabel ("values in x") ## Test output %!shared p, y1, y2, y3 %! p = [-1, 0, 1/2, 1, 2]; %! y1 = [NaN, 0, 0.6931471805599453, Inf, NaN]; %! y2 = [NaN, 0, 1, Inf, NaN]; %! y3 = [NaN, 0, 1/2, 1, NaN]; %!assert (gpinv (p, zeros (1,5), ones (1,5), zeros (1,5)), y1) %!assert (gpinv (p, 0, 1, zeros (1,5)), y1) %!assert (gpinv (p, 0, ones (1,5), 0), y1) %!assert (gpinv (p, zeros (1,5), 1, 0), y1) %!assert (gpinv (p, 0, 1, 0), y1) %!assert (gpinv (p, 0, 1, [0, 0, NaN, 0, 0]), [y1(1:2), NaN, y1(4:5)]) %!assert (gpinv (p, 0, [1, 1, NaN, 1, 1], 0), [y1(1:2), NaN, y1(4:5)]) %!assert (gpinv (p, [0, 0, NaN, 0, 0], 1, 0), [y1(1:2), NaN, y1(4:5)]) %!assert (gpinv ([p(1:2), NaN, p(4:5)], 0, 1, 0), [y1(1:2), NaN, y1(4:5)]) %!assert (gpinv (p, ones (1,5), ones (1,5), zeros (1,5)), y2) %!assert (gpinv (p, 1, 1, zeros (1,5)), y2) %!assert (gpinv (p, 1, ones (1,5), 0), y2) %!assert (gpinv (p, ones (1,5), 1, 0), y2) %!assert (gpinv (p, 1, 1, 0), y2) %!assert (gpinv (p, 1, 1, [0, 0, NaN, 0, 0]), [y2(1:2), NaN, y2(4:5)]) %!assert (gpinv (p, 1, [1, 1, NaN, 1, 1], 0), [y2(1:2), NaN, y2(4:5)]) %!assert (gpinv (p, [1, 1, NaN, 1, 1], 1, 0), [y2(1:2), NaN, y2(4:5)]) %!assert (gpinv ([p(1:2), NaN, p(4:5)], 1, 1, 0), [y2(1:2), NaN, y2(4:5)]) %!assert (gpinv (p, -ones (1,5), ones (1,5), zeros (1,5)), y3) %!assert (gpinv (p, -1, 1, zeros (1,5)), y3) %!assert (gpinv (p, -1, ones (1,5), 0), y3) %!assert (gpinv (p, -ones (1,5), 1, 0), y3) %!assert (gpinv (p, -1, 1, 0), y3) %!assert (gpinv (p, -1, 1, [0, 0, NaN, 0, 0]), [y3(1:2), NaN, y3(4:5)]) %!assert (gpinv (p, -1, [1, 1, NaN, 1, 1], 0), [y3(1:2), NaN, y3(4:5)]) %!assert (gpinv (p, -[1, 1, NaN, 1, 1], 1, 0), [y3(1:2), NaN, y3(4:5)]) %!assert (gpinv ([p(1:2), NaN, p(4:5)], -1, 1, 0), [y3(1:2), NaN, y3(4:5)]) ## Test class of input preserved %!assert (gpinv (single ([p, NaN]), 0, 1, 0), single ([y1, NaN])) %!assert (gpinv ([p, NaN], 0, 1, single (0)), single ([y1, NaN])) %!assert (gpinv ([p, NaN], 0, single (1), 0), single ([y1, NaN])) %!assert (gpinv ([p, NaN], single (0), 1, 0), single ([y1, NaN])) %!assert (gpinv (single ([p, NaN]), 1, 1, 0), single ([y2, NaN])) %!assert (gpinv ([p, NaN], 1, 1, single (0)), single ([y2, NaN])) %!assert (gpinv ([p, NaN], 1, single (1), 0), single ([y2, NaN])) %!assert (gpinv ([p, NaN], single (1), 1, 0), single ([y2, NaN])) %!assert (gpinv (single ([p, NaN]), -1, 1, 0), single ([y3, NaN])) %!assert (gpinv ([p, NaN], -1, 1, single (0)), single ([y3, NaN])) %!assert (gpinv ([p, NaN], -1, single (1), 0), single ([y3, NaN])) %!assert (gpinv ([p, NaN], single (-1), 1, 0), single ([y3, NaN])) ## Test input validation %!error gpinv () %!error gpinv (1) %!error gpinv (1, 2) %!error gpinv (1, 2, 3) %!error ... %! gpinv (ones (3), ones (2), ones(2), ones(2)) %!error ... %! gpinv (ones (2), ones (3), ones(2), ones(2)) %!error ... %! gpinv (ones (2), ones (2), ones(3), ones(2)) %!error ... %! gpinv (ones (2), ones (2), ones(2), ones(3)) %!error gpinv (i, 2, 3, 4) %!error gpinv (1, i, 3, 4) %!error gpinv (1, 2, i, 4) %!error gpinv (1, 2, 3, i) statistics-release-1.6.3/inst/dist_fun/gppdf.m000066400000000000000000000216241456127120000214020ustar00rootroot00000000000000## Copyright (C) 2018 John Donoghue ## Copyright (C) 2016 Dag Lyberg ## Copyright (C) 1997-2015 Kurt Hornik ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## Octave is free software; you can redistribute it and/or modify it ## under the terms of the GNU General Public License as published by ## the Free Software Foundation; either version 3 of the License, or (at ## your option) any later version. ## ## Octave is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with Octave; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{y} =} gppdf (@var{x}, @var{k}, @var{sigma}, @var{mu}) ## ## Generalized Pareto probability density function (PDF). ## ## For each element of @var{x}, compute the probability density function (PDF) ## of the generalized Pareto distribution with shape parameter @var{k}, scale ## parameter @var{sigma}, and location parameter @var{mu}. The size of @var{y} ## is the common size of @var{p}, @var{k}, @var{sigma}, and @var{mu}. A scalar ## input functions as a constant matrix of the same size as the other inputs. ## ## When @qcode{@var{k} = 0} and @qcode{@var{mu} = 0}, the Generalized Pareto CDF ## is equivalent to the exponential distribution. When @qcode{@var{k} > 0} and ## @code{@var{mu} = @var{k} / @var{k}} the Generalized Pareto is equivalent to ## the Pareto distribution. The mean of the Generalized Pareto is not finite ## when @qcode{@var{k} >= 1} and the variance is not finite when ## @qcode{@var{k} >= 1/2}. When @qcode{@var{k} >= 0}, the Generalized Pareto ## has positive density for @qcode{@var{x} > @var{mu}}, or, when ## @qcode{@var{mu} < 0}, for ## @qcode{0 <= (@var{x} - @var{mu}) / @var{sigma} <= -1 / @var{k}}. ## ## Further information about the generalized Pareto distribution can be found at ## @url{https://en.wikipedia.org/wiki/Generalized_Pareto_distribution} ## ## @seealso{gpcdf, gpinv, gprnd, gpfit, gplike, gpstat} ## @end deftypefn function y = gppdf (x, k, sigma, mu) ## Check for valid number of input arguments if (nargin < 4) error ("gppdf: function called with too few input arguments."); endif ## Check for common size of X, K, SIGMA, and MU if (! isscalar (x) || ! isscalar (k) || ! isscalar (sigma) || ! isscalar (mu)) [err, x, k, sigma, mu] = common_size (x, k, sigma, mu); if (err > 0) error ("gppdf: X, K, SIGMA, and MU must be of common size or scalars."); endif endif ## Check for X, K, SIGMA, and MU being reals if (iscomplex (x) || iscomplex (k) || iscomplex (sigma) || iscomplex (mu)) error ("gppdf: X, K, SIGMA, and MU must not be complex."); endif ## Check for class type if (isa (x, "single") || isa (mu, "single") || isa (sigma, "single") ... || isa (k, "single")) y = zeros (size (x), "single"); else y = zeros (size (x)); endif ## Return NaNs for out of range values of sigma parameter ky = isnan (x) | ! (-Inf < mu) | ! (mu < Inf) | ... ! (sigma > 0) | ! (sigma < Inf) | ... ! (-Inf < k) | ! (k < Inf); y(ky) = NaN; ky = (-Inf < x) & (x < Inf) & (-Inf < mu) & (mu < Inf) & ... (sigma > 0) & (sigma < Inf) & (-Inf < k) & (k < Inf); if (isscalar (mu) && isscalar (sigma) && isscalar (k)) z = (x - mu) / sigma; j = ky & (k == 0) & (z >= 0); if (any (j)) y(j) = exp (-z(j)); endif j = ky & (k > 0) & (z >= 0); if (any (j)) y(j) = (k * z(j) + 1) .^ (-(k + 1) / k) ./ sigma; endif if (k < 0) j = ky & (k < 0) & (0 <= z) & (z <= -1. / k); if (any (j)) y(j) = (k * z(j) + 1) .^ (-(k + 1) / k) ./ sigma; endif endif else z = (x - mu) ./ sigma; j = ky & (k == 0) & (z >= 0); if (any (j)) y(j) = exp( -z(j)); endif j = ky & (k > 0) & (z >= 0); if (any (j)) y(j) = (k(j) .* z(j) + 1) .^ (-(k(j) + 1) ./ k(j)) ... ./ sigma(j); endif if (any (k < 0)) j = ky & (k < 0) & (0 <= z) & (z <= -1 ./ k); if (any (j)) y(j) = (k(j) .* z(j) + 1) .^ (-(k(j) + 1) ./ k(j)) ... ./ sigma(j); endif endif endif endfunction %!demo %! ## Plot various PDFs from the generalized Pareto distribution %! x = 0:0.001:5; %! y1 = gppdf (x, 1, 1, 0); %! y2 = gppdf (x, 5, 1, 0); %! y3 = gppdf (x, 20, 1, 0); %! y4 = gppdf (x, 1, 2, 0); %! y5 = gppdf (x, 5, 2, 0); %! y6 = gppdf (x, 20, 2, 0); %! plot (x, y1, "-b", x, y2, "-g", x, y3, "-r", ... %! x, y4, "-c", x, y5, "-m", x, y6, "-k") %! grid on %! xlim ([0, 5]) %! ylim ([0, 1]) %! legend ({"ξ = 1, σ = 1, μ = 0", "ξ = 5, σ = 1, μ = 0", ... %! "ξ = 20, σ = 1, μ = 0", "ξ = 1, σ = 2, μ = 0", ... %! "ξ = 5, σ = 2, μ = 0", "ξ = 20, σ = 2, μ = 0"}, ... %! "location", "northeast") %! title ("Generalized Pareto PDF") %! xlabel ("values in x") %! ylabel ("density") ## Test output %!shared x, y1, y2, y3 %! x = [-Inf, -1, 0, 1/2, 1, Inf]; %! y1 = [0, 0, 1, 0.6065306597126334, 0.36787944117144233, 0]; %! y2 = [0, 0, 1, 4/9, 1/4, 0]; %! y3 = [0, 0, 1, 1, 1, 0]; %!assert (gppdf (x, zeros (1,6), ones (1,6), zeros (1,6)), y1, eps) %!assert (gppdf (x, 0, 1, zeros (1,6)), y1, eps) %!assert (gppdf (x, 0, ones (1,6), 0), y1, eps) %!assert (gppdf (x, zeros (1,6), 1, 0), y1, eps) %!assert (gppdf (x, 0, 1, 0), y1, eps) %!assert (gppdf (x, 0, 1, [0, 0, 0, NaN, 0, 0]), [y1(1:3), NaN, y1(5:6)]) %!assert (gppdf (x, 0, [1, 1, 1, NaN, 1, 1], 0), [y1(1:3), NaN, y1(5:6)]) %!assert (gppdf (x, [0, 0, 0, NaN, 0, 0], 1, 0), [y1(1:3), NaN, y1(5:6)]) %!assert (gppdf ([x(1:3), NaN, x(5:6)], 0, 1, 0), [y1(1:3), NaN, y1(5:6)]) %!assert (gppdf (x, ones (1,6), ones (1,6), zeros (1,6)), y2, eps) %!assert (gppdf (x, 1, 1, zeros (1,6)), y2, eps) %!assert (gppdf (x, 1, ones (1,6), 0), y2, eps) %!assert (gppdf (x, ones (1,6), 1, 0), y2, eps) %!assert (gppdf (x, 1, 1, 0), y2, eps) %!assert (gppdf (x, 1, 1, [0, 0, 0, NaN, 0, 0]), [y2(1:3), NaN, y2(5:6)]) %!assert (gppdf (x, 1, [1, 1, 1, NaN, 1, 1], 0), [y2(1:3), NaN, y2(5:6)]) %!assert (gppdf (x, [1, 1, 1, NaN, 1, 1], 1, 0), [y2(1:3), NaN, y2(5:6)]) %!assert (gppdf ([x(1:3), NaN, x(5:6)], 1, 1, 0), [y2(1:3), NaN, y2(5:6)]) %!assert (gppdf (x, -ones (1,6), ones (1,6), zeros (1,6)), y3, eps) %!assert (gppdf (x, -1, 1, zeros (1,6)), y3, eps) %!assert (gppdf (x, -1, ones (1,6), 0), y3, eps) %!assert (gppdf (x, -ones (1,6), 1, 0), y3, eps) %!assert (gppdf (x, -1, 1, 0), y3, eps) %!assert (gppdf (x, -1, 1, [0, 0, 0, NaN, 0, 0]), [y3(1:3), NaN, y3(5:6)]) %!assert (gppdf (x, -1, [1, 1, 1, NaN, 1, 1], 0), [y3(1:3), NaN, y3(5:6)]) %!assert (gppdf (x, [-1, -1, -1, NaN, -1, -1], 1, 0), [y3(1:3), NaN, y3(5:6)]) %!assert (gppdf ([x(1:3), NaN, x(5:6)], -1, 1, 0), [y3(1:3), NaN, y3(5:6)]) ## Test class of input preserved %!assert (gppdf (single ([x, NaN]), 0, 1, 0), single ([y1, NaN])) %!assert (gppdf ([x, NaN], 0, 1, single (0)), single ([y1, NaN])) %!assert (gppdf ([x, NaN], 0, single (1), 0), single ([y1, NaN])) %!assert (gppdf ([x, NaN], single (0), 1, 0), single ([y1, NaN])) %!assert (gppdf (single ([x, NaN]), 1, 1, 0), single ([y2, NaN])) %!assert (gppdf ([x, NaN], 1, 1, single (0)), single ([y2, NaN])) %!assert (gppdf ([x, NaN], 1, single (1), 0), single ([y2, NaN])) %!assert (gppdf ([x, NaN], single (1), 1, 0), single ([y2, NaN])) %!assert (gppdf (single ([x, NaN]), -1, 1, 0), single ([y3, NaN])) %!assert (gppdf ([x, NaN], -1, 1, single (0)), single ([y3, NaN])) %!assert (gppdf ([x, NaN], -1, single (1), 0), single ([y3, NaN])) %!assert (gppdf ([x, NaN], single (-1), 1, 0), single ([y3, NaN])) ## Test input validation %!error gpcdf () %!error gpcdf (1) %!error gpcdf (1, 2) %!error gpcdf (1, 2, 3) %!error ... %! gpcdf (ones (3), ones (2), ones(2), ones(2)) %!error ... %! gpcdf (ones (2), ones (3), ones(2), ones(2)) %!error ... %! gpcdf (ones (2), ones (2), ones(3), ones(2)) %!error ... %! gpcdf (ones (2), ones (2), ones(2), ones(3)) %!error gpcdf (i, 2, 3, 4) %!error gpcdf (1, i, 3, 4) %!error gpcdf (1, 2, i, 4) %!error gpcdf (1, 2, 3, i) statistics-release-1.6.3/inst/dist_fun/gprnd.m000066400000000000000000000217741456127120000214220ustar00rootroot00000000000000## Copyright (C) 1995-2015 Kurt Hornik ## Copyright (C) 2016 Dag Lyberg ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## Octave is free software; you can redistribute it and/or modify it ## under the terms of the GNU General Public License as published by ## the Free Software Foundation; either version 3 of the License, or (at ## your option) any later version. ## ## Octave is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with Octave; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{r} =} gprnd (@var{k}, @var{sigma}, @var{mu}) ## @deftypefnx {statistics} {@var{r} =} gprnd (@var{k}, @var{sigma}, @var{mu}, @var{rows}) ## @deftypefnx {statistics} {@var{r} =} gprnd (@var{k}, @var{sigma}, @var{mu}, @var{rows}, @var{cols}, @dots{}) ## @deftypefnx {statistics} {@var{r} =} gprnd (@var{k}, @var{sigma}, @var{mu}, [@var{sz}]) ## ## Random arrays from the generalized Pareto distribution. ## ## @code{@var{r} = gprnd (@var{k}, @var{sigma}, @var{mu})} returns an array of ## random numbers chosen from the generalized Pareto distribution with shape ## parameter @var{k}, scale parameter @var{sigma}, and location parameter ## @var{mu}. The size of @var{r} is the common size of @var{k}, @var{sigma}, ## and @var{mu}. A scalar input functions as a constant matrix of the same size ## as the other inputs. ## ## When called with a single size argument, @code{gprnd} returns a square ## matrix with the dimension specified. When called with more than one scalar ## argument, the first two arguments are taken as the number of rows and columns ## and any further arguments specify additional matrix dimensions. The size may ## also be specified with a row vector of dimensions, @var{sz}. ## ## When @qcode{@var{k} = 0} and @qcode{@var{mu} = 0}, the Generalized Pareto CDF ## is equivalent to the exponential distribution. When @qcode{@var{k} > 0} and ## @code{@var{mu} = @var{k} / @var{k}} the Generalized Pareto is equivalent to ## the Pareto distribution. The mean of the Generalized Pareto is not finite ## when @qcode{@var{k} >= 1} and the variance is not finite when ## @qcode{@var{k} >= 1/2}. When @qcode{@var{k} >= 0}, the Generalized Pareto ## has positive density for @qcode{@var{x} > @var{mu}}, or, when ## @qcode{@var{mu} < 0}, for ## @qcode{0 <= (@var{x} - @var{mu}) / @var{sigma} <= -1 / @var{k}}. ## ## Further information about the generalized Pareto distribution can be found at ## @url{https://en.wikipedia.org/wiki/Generalized_Pareto_distribution} ## ## @seealso{gpcdf, gpinv, gppdf, gpfit, gplike, gpstat} ## @end deftypefn function r = gprnd (k, sigma, mu, varargin) ## Check for valid number of input arguments if (nargin < 3) error ("gprnd: function called with too few input arguments."); endif ## Check for common size of K, SIGMA, and MU if (! isscalar (k) || ! isscalar (sigma) || ! isscalar (mu)) [retval, k, sigma, mu] = common_size (k, sigma, mu); if (retval > 0) error ("gprnd: K, SIGMA, and MU must be of common size or scalars."); endif endif ## Check for K, SIGMA, and MU being reals if (iscomplex (k) || iscomplex (sigma) || iscomplex (mu)) error ("gprnd: K, SIGMA, and MU must not be complex."); endif ## Parse and check SIZE arguments if (nargin == 3) sz = size (k); elseif (nargin == 4) if (isscalar (varargin{1}) && varargin{1} >= 0 ... && varargin{1} == fix (varargin{1})) sz = [varargin{1}, varargin{1}]; elseif (isrow (varargin{1}) && all (varargin{1} >= 0) ... && all (varargin{1} == fix (varargin{1}))) sz = varargin{1}; elseif error (strcat (["gprnd: SZ must be a scalar or a row vector"], ... [" of non-negative integers."])); endif elseif (nargin > 4) posint = cellfun (@(x) (! isscalar (x) || x < 0 || x != fix (x)), varargin); if (any (posint)) error ("gprnd: dimensions must be non-negative integers."); endif sz = [varargin{:}]; endif ## Check that parameters match requested dimensions in size if (!isscalar (k) && ! isequal (size (k), sz)) error ("gprnd: K, SIGMA, and MU must be scalars or of size SZ."); endif ## Check for class type if (isa (k, "single") || isa (sigma, "single") || isa (mu, "single")) cls = "single"; else cls = "double"; endif ## Generate random sample from generalized Pareto distribution r = rand (sz, cls); ## Find valid parameters vr = (isfinite (r)) & (mu > -Inf) & (mu < Inf) ... & (sigma > 0) & (sigma < Inf) ... & (-Inf < k) & (k < Inf); ## Force invalid parameters to NaN r(! vr) = NaN; if (isscalar (k)) if (k == 0) r(vr) = mu - (sigma .* log (1 - r(vr))); else r(vr) = mu + ((sigma .* ((r(vr) .^ -k) - 1)) ./ k); endif else if (any (k == 0)) r(vr) = mu(vr) - (sigma(vr) .* log (1 - r(vr))); endif if (any (k < 0 | k > 0)) r(vr) = mu(vr) + ((sigma(vr) .* ((r(vr) .^ -k(vr)) - 1)) ./ k(vr)); endif endif endfunction ## Test output %!assert (size (gprnd (0, 1, 0)), [1, 1]) %!assert (size (gprnd (0, 1, zeros (2,1))), [2, 1]) %!assert (size (gprnd (0, 1, zeros (2,2))), [2, 2]) %!assert (size (gprnd (0, ones (2,1), 0)), [2, 1]) %!assert (size (gprnd (0, ones (2,2), 0)), [2, 2]) %!assert (size (gprnd (zeros (2,1), 1, 0)), [2, 1]) %!assert (size (gprnd (zeros (2,2), 1, 0)), [2, 2]) %!assert (size (gprnd (0, 1, 0, 3)), [3, 3]) %!assert (size (gprnd (0, 1, 0, [4 1])), [4, 1]) %!assert (size (gprnd (0, 1, 0, 4, 1)), [4, 1]) %!assert (size (gprnd (1,1,0)), [1, 1]) %!assert (size (gprnd (1, 1, zeros (2,1))), [2, 1]) %!assert (size (gprnd (1, 1, zeros (2,2))), [2, 2]) %!assert (size (gprnd (1, ones (2,1), 0)), [2, 1]) %!assert (size (gprnd (1, ones (2,2), 0)), [2, 2]) %!assert (size (gprnd (ones (2,1), 1, 0)), [2, 1]) %!assert (size (gprnd (ones (2,2), 1, 0)), [2, 2]) %!assert (size (gprnd (1, 1, 0, 3)), [3, 3]) %!assert (size (gprnd (1, 1, 0, [4 1])), [4, 1]) %!assert (size (gprnd (1, 1, 0, 4, 1)), [4, 1]) %!assert (size (gprnd (-1, 1, 0)), [1, 1]) %!assert (size (gprnd (-1, 1, zeros (2,1))), [2, 1]) %!assert (size (gprnd (1, -1, zeros (2,2))), [2, 2]) %!assert (size (gprnd (-1, ones (2,1), 0)), [2, 1]) %!assert (size (gprnd (-1, ones (2,2), 0)), [2, 2]) %!assert (size (gprnd (-ones (2,1), 1, 0)), [2, 1]) %!assert (size (gprnd (-ones (2,2), 1, 0)), [2, 2]) %!assert (size (gprnd (-1, 1, 0, 3)), [3, 3]) %!assert (size (gprnd (-1, 1, 0, [4, 1])), [4, 1]) %!assert (size (gprnd (-1, 1, 0, 4, 1)), [4, 1]) ## Test class of input preserved %!assert (class (gprnd (0, 1, 0)), "double") %!assert (class (gprnd (0, 1, single (0))), "single") %!assert (class (gprnd (0, 1, single ([0, 0]))), "single") %!assert (class (gprnd (0, single (1),0)), "single") %!assert (class (gprnd (0, single ([1, 1]),0)), "single") %!assert (class (gprnd (single (0), 1, 0)), "single") %!assert (class (gprnd (single ([0, 0]), 1, 0)), "single") ## Test input validation %!error gprnd () %!error gprnd (1) %!error gprnd (1, 2) %!error ... %! gprnd (ones (3), ones (2), ones (2)) %!error ... %! gprnd (ones (2), ones (3), ones (2)) %!error ... %! gprnd (ones (2), ones (2), ones (3)) %!error gprnd (i, 2, 3) %!error gprnd (1, i, 3) %!error gprnd (1, 2, i) %!error ... %! gprnd (1, 2, 3, -1) %!error ... %! gprnd (1, 2, 3, 1.2) %!error ... %! gprnd (1, 2, 3, ones (2)) %!error ... %! gprnd (1, 2, 3, [2 -1 2]) %!error ... %! gprnd (1, 2, 3, [2 0 2.5]) %!error ... %! gprnd (1, 2, 3, 2, -1, 5) %!error ... %! gprnd (1, 2, 3, 2, 1.5, 5) %!error ... %! gprnd (2, ones (2), 2, 3) %!error ... %! gprnd (2, ones (2), 2, [3, 2]) %!error ... %! gprnd (2, ones (2), 2, 3, 2) statistics-release-1.6.3/inst/dist_fun/gumbelcdf.m000066400000000000000000000222131456127120000222250ustar00rootroot00000000000000## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{p} =} gumbelcdf (@var{x}) ## @deftypefnx {statistics} {@var{p} =} gumbelcdf (@var{x}, @var{mu}) ## @deftypefnx {statistics} {@var{p} =} gumbelcdf (@var{x}, @var{mu}, @var{beta}) ## @deftypefnx {statistics} {@var{p} =} gumbelcdf (@dots{}, @qcode{"upper"}) ## @deftypefnx {statistics} {[@var{p}, @var{plo}, @var{pup}] =} gumbelcdf (@var{x}, @var{mu}, @var{beta}, @var{pcov}) ## @deftypefnx {statistics} {[@var{p}, @var{plo}, @var{pup}] =} gumbelcdf (@var{x}, @var{mu}, @var{beta}, @var{pcov}, @var{alpha}) ## @deftypefnx {statistics} {[@var{p}, @var{plo}, @var{pup}] =} gumbelcdf (@dots{}, @qcode{"upper"}) ## ## Gumbel cumulative distribution function (CDF). ## ## For each element of @var{x}, compute the cumulative distribution function ## (CDF) of the Gumbel distribution (also known as the extreme value or the type ## I generalized extreme value distribution) with location parameter @var{mu} ## and scale parameter @var{beta}. The size of @var{p} is the common size of ## @var{x}, @var{mu} and @var{beta}. A scalar input functions as a constant ## matrix of the same size as the other inputs. ## ## Default values are @var{mu} = 0 and @var{beta} = 1. ## ## When called with three output arguments, i.e. @code{[@var{p}, @var{plo}, ## @var{pup}]}, @code{gumbelcdf} computes the confidence bounds for @var{p} when ## the input parameters @var{mu} and @var{beta} are estimates. In such case, ## @var{pcov}, a @math{2x2} matrix containing the covariance matrix of the ## estimated parameters, is necessary. Optionally, @var{alpha}, which has a ## default value of 0.05, specifies the @qcode{100 * (1 - @var{alpha})} percent ## confidence bounds. @var{plo} and @var{pup} are arrays of the same size as ## @var{p} containing the lower and upper confidence bounds. ## ## @code{[@dots{}] = gumbelcdf (@dots{}, "upper")} computes the upper tail ## probability of the Gumbel distribution with parameters @var{mu} and ## @var{beta}, at the values in @var{x}. ## ## The Gumbel distribution is used to model the distribution of the maximum (or ## the minimum) of a number of samples of various distributions. This version ## is suitable for modeling maxima. For modeling minima, use the alternative ## extreme value CDF, @code{evcdf}. ## ## @code{[@dots{}] = gumbelcdf (@dots{}, "upper")} computes the upper tail ## probability of the extreme value (Gumbel) distribution. ## ## Further information about the Gumbel distribution can be found at ## @url{https://en.wikipedia.org/wiki/Gumbel_distribution} ## ## @seealso{gumbelinv, gumbelpdf, gumbelrnd, gumbelfit, gumbellike, gumbelstat, ## evcdf} ## @end deftypefn function [varargout] = gumbelcdf (x, varargin) ## Check for valid number of input arguments if (nargin < 1 || nargin > 6) error ("gumbelcdf: invalid number of input arguments."); endif ## Check for 'upper' flag if (nargin > 1 && strcmpi (varargin{end}, "upper")) uflag = true; varargin(end) = []; elseif (nargin > 1 && ischar (varargin{end}) && ... ! strcmpi (varargin{end}, "upper")) error ("gumbelcdf: invalid argument for upper tail."); else uflag = false; endif ## Get extra arguments (if they exist) or add defaults if (numel (varargin) > 0) mu = varargin{1}; else mu = 0; endif if (numel (varargin) > 1) beta = varargin{2}; else beta = 1; endif if (numel (varargin) > 2) pcov = varargin{3}; ## Check for valid covariance matrix 2x2 if (! isequal (size (pcov), [2, 2])) error ("gumbelcdf: invalid size of covariance matrix."); endif else ## Check that cov matrix is provided if 3 output arguments are requested if (nargout > 1) error ("gumbelcdf: covariance matrix is required for confidence bounds."); endif pcov = []; endif if (numel (varargin) > 3) alpha = varargin{4}; ## Check for valid alpha value if (! isnumeric (alpha) || numel (alpha) !=1 || alpha <= 0 || alpha >= 1) error ("gumbelcdf: invalid value for alpha."); endif else alpha = 0.05; endif ## Check for common size of X, MU, and BETA if (! isscalar (x) || ! isscalar (mu) || ! isscalar (beta)) [err, x, mu, beta] = common_size (x, mu, beta); if (err > 0) error ("gumbelcdf: X, MU, and BETA must be of common size or scalars."); endif endif ## Check for X, MU, and BETA being reals if (iscomplex (x) || iscomplex (mu) || iscomplex (beta)) error ("gumbelcdf: X, MU, and BETA must not be complex."); endif ## Return NaNs for out of range parameters. beta(beta <= 0) = NaN; ## Compute extreme value cdf z = (x - mu) ./ beta; if (uflag) p = -expm1 (-exp (-z)); else p = exp (-exp (-z)); endif ## Check for class type if (isa (x, "single") || isa (mu, "single") || isa (beta, "single")); is_class = "single"; else is_class = "double"; endif ## Prepare output varargout{1} = cast (p, is_class); if (nargout > 1) plo = NaN (size (z), is_class); pup = NaN (size (z), is_class); endif ## Check beta if (isscalar (beta)) if (beta > 0) sigma_p = true (size (z)); else if (nargout == 3) varargout{2} = plo; varargout{3} = pup; endif return; endif else sigma_p = beta > 0; endif ## Compute confidence bounds (if requested) if (nargout >= 2) zvar = (pcov(1,1) + 2 * pcov(1,2) * z(sigma_p) + ... pcov(2,2) * z(sigma_p) .^ 2) ./ (beta .^ 2); if (any (zvar < 0)) error ("gumbelcdf: bad covariance matrix."); endif normz = -norminv (alpha / 2); halfwidth = normz * sqrt (zvar); zlo = z(sigma_p) - halfwidth; zup = z(sigma_p) + halfwidth; if (uflag) plo(sigma_p) = -expm1 (-exp (-zup)); pup(sigma_p) = -expm1 (-exp (-zlo)); else plo(sigma_p) = exp (-exp (-zlo)); pup(sigma_p) = exp (-exp (-zup)); endif varargout{2} = plo; varargout{3} = pup; endif endfunction %!demo %! ## Plot various CDFs from the Gumbel distribution %! x = -5:0.01:20; %! p1 = gumbelcdf (x, 0.5, 2); %! p2 = gumbelcdf (x, 1.0, 2); %! p3 = gumbelcdf (x, 1.5, 3); %! p4 = gumbelcdf (x, 3.0, 4); %! plot (x, p1, "-b", x, p2, "-g", x, p3, "-r", x, p4, "-c") %! grid on %! legend ({"μ = 0.5, β = 2", "μ = 1.0, β = 2", ... %! "μ = 1.5, β = 3", "μ = 3.0, β = 4"}, "location", "southeast") %! title ("Gumbel CDF") %! xlabel ("values in x") %! ylabel ("probability") ## Test output %!shared x, y %! x = [-Inf, 1, 2, Inf]; %! y = [0, 0.3679, 0.6922, 1]; %!assert (gumbelcdf (x, ones (1,4), ones (1,4)), y, 1e-4) %!assert (gumbelcdf (x, 1, ones (1,4)), y, 1e-4) %!assert (gumbelcdf (x, ones (1,4), 1), y, 1e-4) %!assert (gumbelcdf (x, [0, -Inf, NaN, Inf], 1), [0, 1, NaN, NaN], 1e-4) %!assert (gumbelcdf (x, 1, [Inf, NaN, -1, 0]), [NaN, NaN, NaN, NaN], 1e-4) %!assert (gumbelcdf ([x(1:2), NaN, x(4)], 1, 1), [y(1:2), NaN, y(4)], 1e-4) %!assert (gumbelcdf (x, "upper"), [1, 0.3078, 0.1266, 0], 1e-4) ## Test class of input preserved %!assert (gumbelcdf ([x, NaN], 1, 1), [y, NaN], 1e-4) %!assert (gumbelcdf (single ([x, NaN]), 1, 1), single ([y, NaN]), 1e-4) %!assert (gumbelcdf ([x, NaN], single (1), 1), single ([y, NaN]), 1e-4) %!assert (gumbelcdf ([x, NaN], 1, single (1)), single ([y, NaN]), 1e-4) ## Test input validation %!error gumbelcdf () %!error gumbelcdf (1,2,3,4,5,6,7) %!error gumbelcdf (1, 2, 3, 4, "uper") %!error ... %! gumbelcdf (ones (3), ones (2), ones (2)) %!error gumbelcdf (2, 3, 4, [1, 2]) %!error ... %! [p, plo, pup] = gumbelcdf (1, 2, 3) %!error [p, plo, pup] = ... %! gumbelcdf (1, 2, 3, [1, 0; 0, 1], 0) %!error [p, plo, pup] = ... %! gumbelcdf (1, 2, 3, [1, 0; 0, 1], 1.22) %!error [p, plo, pup] = ... %! gumbelcdf (1, 2, 3, [1, 0; 0, 1], "alpha", "upper") %!error gumbelcdf (i, 2, 2) %!error gumbelcdf (2, i, 2) %!error gumbelcdf (2, 2, i) %!error ... %! [p, plo, pup] = gumbelcdf (1, 2, 3, [1, 0; 0, -inf], 0.04) statistics-release-1.6.3/inst/dist_fun/gumbelinv.m000066400000000000000000000170321456127120000222700ustar00rootroot00000000000000## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{x} =} gumbelinv (@var{p}) ## @deftypefnx {statistics} {@var{x} =} gumbelinv (@var{p}, @var{mu}) ## @deftypefnx {statistics} {@var{x} =} gumbelinv (@var{p}, @var{mu}, @var{beta}) ## @deftypefnx {statistics} {[@var{x}, @var{xlo}, @var{xup}] =} gumbelinv (@var{p}, @var{mu}, @var{beta}, @var{pcov}) ## @deftypefnx {statistics} {[@var{x}, @var{xlo}, @var{xup}] =} gumbelinv (@var{p}, @var{mu}, @var{beta}, @var{pcov}, @var{alpha}) ## ## Inverse of the Gumbel cumulative distribution function (iCDF). ## ## For each element of @var{p}, compute the quantile (the inverse of the CDF) of ## the Gumbel distribution (also known as the extreme value or the type I ## generalized extreme value distribution) with location parameter @var{mu} and ## scale parameter @var{beta}. The size of @var{x} is the common size of ## @var{p}, @var{mu} and @var{beta}. A scalar input functions as a constant ## matrix of the same size as the other inputs. ## ## Default values are @var{mu} = 0 and @var{beta} = 1. ## ## When called with three output arguments, i.e. @qcode{[@var{x}, @var{xlo}, ## @var{xup}]}, @code{gumbelinv} computes the confidence bounds for @var{x} when ## the input parameters @var{mu} and @var{beta} are estimates. In such case, ## @var{pcov}, a @math{2x2} matrix containing the covariance matrix of the ## estimated parameters, is necessary. Optionally, @var{alpha}, which has a ## default value of 0.05, specifies the @qcode{100 * (1 - @var{alpha})} percent ## confidence bounds. @var{xlo} and @var{xup} are arrays of the same size as ## @var{x} containing the lower and upper confidence bounds. ## ## The Gumbel distribution is used to model the distribution of the maximum (or ## the minimum) of a number of samples of various distributions. This version ## is suitable for modeling maxima. For modeling minima, use the alternative ## extreme value iCDF, @code{evinv}. ## ## Further information about the Gumbel distribution can be found at ## @url{https://en.wikipedia.org/wiki/Gumbel_distribution} ## ## @seealso{gumbelcdf, gumbelpdf, gumbelrnd, gumbelfit, gumbellike, gumbelstat, ## evinv} ## @end deftypefn function [x, xlo, xup] = gumbelinv (p, mu, beta, pcov, alpha) ## Check for valid number of input arguments if (nargin < 1 || nargin > 5) error ("gumbelinv: invalid number of input arguments."); endif ## Add defaults (if missing input arguments) if (nargin < 2) mu = 0; endif if (nargin < 3) beta = 1; endif ## Check if PCOV is provided when confidence bounds are requested if (nargout > 2) if (nargin < 4) error ("gumbelinv: covariance matrix is required for confidence bounds."); endif ## Check for valid covariance matrix 2x2 if (! isequal (size (pcov), [2, 2])) error ("gumbelinv: invalid size of covariance matrix."); endif ## Check for valid alpha value if (nargin < 5) alpha = 0.05; elseif (! isnumeric (alpha) || numel (alpha) !=1 || alpha <= 0 || alpha >= 1) error ("gumbelinv: invalid value for alpha."); endif endif ## Check for common size of P, MU, and BETA if (! isscalar (p) || ! isscalar (mu) || ! isscalar (beta)) [err, p, mu, beta] = common_size (p, mu, beta); if (err > 0) error ("gumbelinv: P, MU, and BETA must be of common size or scalars."); endif endif ## Check for P, MU, and BETA being reals if (iscomplex (p) || iscomplex (mu) || iscomplex (beta)) error ("gumbelinv: P, MU, and BETA must not be complex."); endif ## Check for class type if (isa (p, "single") || isa (mu, "single") || isa (beta, "single")); is_class = "single"; else is_class = "double"; endif ## Compute inverse of type 1 extreme value cdf k = (eps <= p & p < 1); if (all (k(:))) q = log (-log (p)); else q = zeros (size (p), is_class); q(k) = log (-log (p(k))); ## Return -Inf for p = 0 and Inf for p = 1 q(p < eps) = Inf; q(p == 1) = -Inf; ## Return NaN for out of range values of P q(p < 0 | 1 < p | isnan (p)) = NaN; endif ## Return NaN for out of range values of BETA beta(beta <= 0) = NaN; x = -(beta .* q + mu); ## Compute confidence bounds if requested. if (nargout >= 2) xvar = pcov(1,1) + 2 * pcov(1,2) * q + pcov(2,2) * q .^ 2; if (any (xvar < 0)) error ("gumbelinv: bad covariance matrix."); endif z = -norminv (alpha / 2); halfwidth = z * sqrt (xvar); xlo = x - halfwidth; xup = x + halfwidth; endif endfunction %!demo %! ## Plot various iCDFs from the Gumbel distribution %! p = 0.001:0.001:0.999; %! x1 = gumbelinv (p, 0.5, 2); %! x2 = gumbelinv (p, 1.0, 2); %! x3 = gumbelinv (p, 1.5, 3); %! x4 = gumbelinv (p, 3.0, 4); %! plot (p, x1, "-b", p, x2, "-g", p, x3, "-r", p, x4, "-c") %! grid on %! ylim ([-5, 20]) %! legend ({"μ = 0.5, β = 2", "μ = 1.0, β = 2", ... %! "μ = 1.5, β = 3", "μ = 3.0, β = 4"}, "location", "northwest") %! title ("Gumbel iCDF") %! xlabel ("probability") %! ylabel ("values in x") ## Test output %!shared p, x %! p = [0, 0.05, 0.5 0.95]; %! x = [-Inf, -1.0972, 0.3665, 2.9702]; %!assert (gumbelinv (p), x, 1e-4) %!assert (gumbelinv (p, zeros (1,4), ones (1,4)), x, 1e-4) %!assert (gumbelinv (p, 0, ones (1,4)), x, 1e-4) %!assert (gumbelinv (p, zeros (1,4), 1), x, 1e-4) %!assert (gumbelinv (p, [0, -Inf, NaN, Inf], 1), [-Inf, Inf, NaN, -Inf], 1e-4) %!assert (gumbelinv (p, 0, [Inf, NaN, -1, 0]), [-Inf, NaN, NaN, NaN], 1e-4) %!assert (gumbelinv ([p(1:2), NaN, p(4)], 0, 1), [x(1:2), NaN, x(4)], 1e-4) ## Test class of input preserved %!assert (gumbelinv ([p, NaN], 0, 1), [x, NaN], 1e-4) %!assert (gumbelinv (single ([p, NaN]), 0, 1), single ([x, NaN]), 1e-4) %!assert (gumbelinv ([p, NaN], single (0), 1), single ([x, NaN]), 1e-4) %!assert (gumbelinv ([p, NaN], 0, single (1)), single ([x, NaN]), 1e-4) ## Test input validation %!error gumbelinv () %!error gumbelinv (1,2,3,4,5,6) %!error ... %! gumbelinv (ones (3), ones (2), ones (2)) %!error ... %! [p, plo, pup] = gumbelinv (2, 3, 4, [1, 2]) %!error ... %! [p, plo, pup] = gumbelinv (1, 2, 3) %!error [p, plo, pup] = ... %! gumbelinv (1, 2, 3, [1, 0; 0, 1], 0) %!error [p, plo, pup] = ... %! gumbelinv (1, 2, 3, [1, 0; 0, 1], 1.22) %!error gumbelinv (i, 2, 2) %!error gumbelinv (2, i, 2) %!error gumbelinv (2, 2, i) %!error ... %! [p, plo, pup] = gumbelinv (1, 2, 3, [-1, 10; -Inf, -Inf], 0.04) statistics-release-1.6.3/inst/dist_fun/gumbelpdf.m000066400000000000000000000104511456127120000222430ustar00rootroot00000000000000## Copyright (C) 2022-2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{y} =} gumbelpdf (@var{x}) ## @deftypefnx {statistics} {@var{y} =} gumbelpdf (@var{x}, @var{mu}) ## @deftypefnx {statistics} {@var{y} =} gumbelpdf (@var{x}, @var{mu}, @var{beta}) ## ## Gumbel probability density function (PDF). ## ## For each element of @var{x}, compute the probability density function (PDF) ## of the Gumbel distribution (also known as the extreme value or the type I ## generalized extreme value distribution) with location parameter @var{mu} and ## scale parameter @var{beta}. The size of @var{y} is the common size of ## @var{x}, @var{mu} and @var{beta}. A scalar input functions as a constant ## matrix of the same size as the other inputs. ## ## Default values are @var{mu} = 0 and @var{beta} = 1. ## ## The Gumbel distribution is used to model the distribution of the maximum (or ## the minimum) of a number of samples of various distributions. This version ## is suitable for modeling maxima. For modeling minima, use the alternative ## extreme value iCDF, @code{evpdf}. ## ## Further information about the Gumbel distribution can be found at ## @url{https://en.wikipedia.org/wiki/Gumbel_distribution} ## ## @seealso{gumbelcdf, gumbelinv, gumbelrnd, gumbelfit, gumbellike, gumbelstat, ## evpdf} ## @end deftypefn function y = gumbelpdf (x, mu, beta) ## Check for valid number of input arguments if (nargin < 1) error ("gumbelpdf: too few input arguments."); endif ## Add defaults (if missing input arguments) if (nargin < 2) mu = 0; endif if (nargin < 3) beta = 1; endif ## Check for common size of X, MU, and BETA if (! isscalar (x) || ! isscalar (mu) || ! isscalar (beta)) [err, x, mu, beta] = common_size (x, mu, beta); if (err > 0) error ("gumbelpdf: X, MU, and BETA must be of common size or scalars."); endif endif ## Check for X, MU, and BETA being reals if (iscomplex (x) || iscomplex (mu) || iscomplex (beta)) error ("gumbelpdf: X, MU, and BETA must not be complex."); endif ## Return NaNs for out of range parameters beta(beta <= 0) = NaN; ## Compute pdf of type 1 extreme value distribution z = -(x - mu) ./ beta; y = exp (z - exp (z)) ./ beta; ## Force 0 for extreme right tail, instead of getting exp (Inf - Inf) = NaN y(z == Inf) = 0; endfunction %!demo %! ## Plot various PDFs from the Extreme value distribution %! x = -5:0.001:20; %! y1 = gumbelpdf (x, 0.5, 2); %! y2 = gumbelpdf (x, 1.0, 2); %! y3 = gumbelpdf (x, 1.5, 3); %! y4 = gumbelpdf (x, 3.0, 4); %! plot (x, y1, "-b", x, y2, "-g", x, y3, "-r", x, y4, "-c") %! grid on %! ylim ([0, 0.2]) %! legend ({"μ = 0.5, β = 2", "μ = 1.0, β = 2", ... %! "μ = 1.5, β = 3", "μ = 3.0, β = 4"}, "location", "northeast") %! title ("Extreme value PDF") %! xlabel ("values in x") %! ylabel ("density") ## Test output %!shared x, y0, y1 %! x = [-5, 0, 1, 2, 3]; %! y0 = [0, 0.3679, 0.2547, 0.1182, 0.0474]; %! y1 = [0, 0.1794, 0.3679, 0.2547, 0.1182]; %!assert (gumbelpdf (x), y0, 1e-4) %!assert (gumbelpdf (x, zeros (1,5), ones (1,5)), y0, 1e-4) %!assert (gumbelpdf (x, ones (1,5), ones (1,5)), y1, 1e-4) ## Test input validation %!error gumbelpdf () %!error ... %! gumbelpdf (ones (3), ones (2), ones (2)) %!error gumbelpdf (i, 2, 2) %!error gumbelpdf (2, i, 2) %!error gumbelpdf (2, 2, i) statistics-release-1.6.3/inst/dist_fun/gumbelrnd.m000066400000000000000000000156171456127120000222660ustar00rootroot00000000000000## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{r} =} gumbelrnd (@var{mu}, @var{beta}) ## @deftypefnx {statistics} {@var{r} =} gumbelrnd (@var{mu}, @var{beta}, @var{rows}) ## @deftypefnx {statistics} {@var{r} =} gumbelrnd (@var{mu}, @var{beta}, @var{rows}, @var{cols}, @dots{}) ## @deftypefnx {statistics} {@var{r} =} gumbelrnd (@var{mu}, @var{beta}, [@var{sz}]) ## ## Random arrays from the Gumbel distribution. ## ## @code{@var{r} = gumbelrnd (@var{mu}, @var{beta})} returns an array of random ## numbers chosen from the Gumbel distribution (also known as the extreme value ## or the type I generalized extreme value distribution) with location ## parameter @var{mu} and scale parameter @var{beta}. The size of @var{r} is ## the common size of @var{mu} and @var{beta}. A scalar input functions as a ## constant matrix of the same size as the other inputs. ## ## When called with a single size argument, @code{gumbelrnd} returns a square ## matrix with the dimension specified. When called with more than one scalar ## argument, the first two arguments are taken as the number of rows and columns ## and any further arguments specify additional matrix dimensions. The size may ## also be specified with a row vector of dimensions, @var{sz}. ## ## The Gumbel distribution is used to model the distribution of the maximum (or ## the minimum) of a number of samples of various distributions. This version ## is suitable for modeling maxima. For modeling minima, use the alternative ## extreme value iCDF, @code{evinv}. ## ## Further information about the Gumbel distribution can be found at ## @url{https://en.wikipedia.org/wiki/Gumbel_distribution} ## ## @seealso{gumbelcdf, gumbelinv, gumbelpdf, gumbelfit, gumbellike, gumbelstat, ## evrnd} ## @end deftypefn function r = gumbelrnd (mu, beta, varargin) ## Check for valid number of input arguments if (nargin < 2) error ("gumbelrnd: function called with too few input arguments."); endif ## Check for common size of MU and BETA if (! isscalar (mu) || ! isscalar (beta)) [retval, mu, beta] = common_size (mu, beta); if (retval > 0) error ("gumbelrnd: MU and BETA must be of common size or scalars."); endif endif ## Check for MU and BETA being reals if (iscomplex (mu) || iscomplex (beta)) error ("gumbelrnd: MU and BETA must not be complex."); endif ## Parse and check SIZE arguments if (nargin == 2) sz = size (mu); elseif (nargin == 3) if (isscalar (varargin{1}) && varargin{1} >= 0 ... && varargin{1} == fix (varargin{1})) sz = [varargin{1}, varargin{1}]; elseif (isrow (varargin{1}) && all (varargin{1} >= 0) ... && all (varargin{1} == fix (varargin{1}))) sz = varargin{1}; elseif error (strcat (["gumbelrnd: SZ must be a scalar or a row vector"], ... [" of non-negative integers."])); endif elseif (nargin > 3) posint = cellfun (@(x) (! isscalar (x) || x < 0 || x != fix (x)), varargin); if (any (posint)) error ("gumbelrnd: dimensions must be non-negative integers."); endif sz = [varargin{:}]; endif ## Check that parameters match requested dimensions in size if (! isscalar (mu) && ! isequal (size (mu), sz)) error ("gumbelrnd: MU and BETA must be scalars or of size SZ."); endif ## Check for class type if (isa (mu, "single") || isa (beta, "single")) cls = "single"; else cls = "double"; endif ## Return NaNs for out of range values of BETA beta(beta < 0) = NaN; ## Generate uniform random values, and apply the extreme value inverse CDF. r = -log (-log (rand (sz, cls))) .* beta + mu; endfunction ## Test output %!assert (size (gumbelrnd (1, 1)), [1 1]) %!assert (size (gumbelrnd (1, ones (2,1))), [2, 1]) %!assert (size (gumbelrnd (1, ones (2,2))), [2, 2]) %!assert (size (gumbelrnd (ones (2,1), 1)), [2, 1]) %!assert (size (gumbelrnd (ones (2,2), 1)), [2, 2]) %!assert (size (gumbelrnd (1, 1, 3)), [3, 3]) %!assert (size (gumbelrnd (1, 1, [4, 1])), [4, 1]) %!assert (size (gumbelrnd (1, 1, 4, 1)), [4, 1]) %!assert (size (gumbelrnd (1, 1, 4, 1, 5)), [4, 1, 5]) %!assert (size (gumbelrnd (1, 1, 0, 1)), [0, 1]) %!assert (size (gumbelrnd (1, 1, 1, 0)), [1, 0]) %!assert (size (gumbelrnd (1, 1, 1, 2, 0, 5)), [1, 2, 0, 5]) ## Test class of input preserved %!assert (class (gumbelrnd (1, 1)), "double") %!assert (class (gumbelrnd (1, single (1))), "single") %!assert (class (gumbelrnd (1, single ([1, 1]))), "single") %!assert (class (gumbelrnd (single (1), 1)), "single") %!assert (class (gumbelrnd (single ([1, 1]), 1)), "single") ## Test input validation %!error gumbelrnd () %!error gumbelrnd (1) %!error ... %! gumbelrnd (ones (3), ones (2)) %!error ... %! gumbelrnd (ones (2), ones (3)) %!error gumbelrnd (i, 2, 3) %!error gumbelrnd (1, i, 3) %!error ... %! gumbelrnd (1, 2, -1) %!error ... %! gumbelrnd (1, 2, 1.2) %!error ... %! gumbelrnd (1, 2, ones (2)) %!error ... %! gumbelrnd (1, 2, [2 -1 2]) %!error ... %! gumbelrnd (1, 2, [2 0 2.5]) %!error ... %! gumbelrnd (1, 2, 2, -1, 5) %!error ... %! gumbelrnd (1, 2, 2, 1.5, 5) %!error ... %! gumbelrnd (2, ones (2), 3) %!error ... %! gumbelrnd (2, ones (2), [3, 2]) %!error ... %! gumbelrnd (2, ones (2), 3, 2) statistics-release-1.6.3/inst/dist_fun/hncdf.m000066400000000000000000000137571456127120000213740ustar00rootroot00000000000000## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## Octave is free software; you can redistribute it and/or modify it ## under the terms of the GNU General Public License as published by ## the Free Software Foundation; either version 3 of the License, or (at ## your option) any later version. ## ## Octave is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with Octave; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{p} =} hncdf (@var{x}, @var{mu}, @var{sigma}) ## @deftypefnx {statistics} {@var{p} =} hncdf (@var{x}, @var{mu}, @var{sigma}, @qcode{"upper"}) ## ## Half-normal cumulative distribution function (CDF). ## ## For each element of @var{x}, compute the cumulative distribution function ## (CDF) of the half-normal distribution with location parameter @var{mu} and ## scale parameter @var{sigma}. The size of @var{p} is the common size of ## @var{x}, @var{mu} and @var{sigma}. A scalar input functions as a constant ## matrix of the same size as the other inputs. ## ## @code{[@dots{}] = hncdf (@var{x}, @var{mu}, @var{sigma}, "upper")} computes ## the upper tail probability of the half-normal distribution with parameters ## @var{mu} and @var{sigma}, at the values in @var{x}. ## ## The half-normal CDF is only defined for @qcode{@var{x} >= @var{mu}}. ## ## Further information about the half-normal distribution can be found at ## @url{https://en.wikipedia.org/wiki/Half-normal_distribution} ## ## @seealso{hninv, hnpdf, hnrnd, hnfit, hnlike} ## @end deftypefn function p = hncdf (x, mu, sigma, uflag) ## Check for valid number of input arguments if (nargin < 3) error ("hncdf: function called with too few input arguments."); endif ## Check for valid "upper" flag if (nargin > 3) if (! strcmpi (uflag, "upper")) error ("hncdf: invalid argument for upper tail."); else uflag = true; endif else uflag = false; endif ## Check for common size of X, MU, and SIGMA if (! isscalar (x) || ! isscalar (mu) || ! isscalar (sigma)) [err, x, mu, sigma] = common_size (x, mu, sigma); if (err > 0) error ("hncdf: X, MU, and SIGMA must be of common size or scalars."); endif endif ## Check for X, MU, and SIGMA being reals if (iscomplex (x) || iscomplex (mu) || iscomplex (sigma)) error ("hncdf: X, MU, and SIGMA must not be complex."); endif ## Check for class type if (isa (x, "single") || isa (mu, "single") || isa (sigma, "single")) is_class = "single"; else is_class = "double"; endif ## Prepare output p = zeros (size (x), is_class); ## Return NaNs for out of range values of SIGMA parameter sigma(sigma <= 0) = NaN; ## Calculate (x-mu)/sigma => 0 and force zero below that z = (x - mu) ./ sigma; z(z < 0) = 0; if (uflag) p = erfc(z./sqrt(2)); else p = erf(z./sqrt(2)); endif endfunction %!demo %! ## Plot various CDFs from the half-normal distribution %! x = 0:0.001:10; %! p1 = hncdf (x, 0, 1); %! p2 = hncdf (x, 0, 2); %! p3 = hncdf (x, 0, 3); %! p4 = hncdf (x, 0, 5); %! plot (x, p1, "-b", x, p2, "-g", x, p3, "-r", x, p4, "-c") %! grid on %! xlim ([0, 10]) %! legend ({"μ = 0, σ = 1", "μ = 0, σ = 2", ... %! "μ = 0, σ = 3", "μ = 0, σ = 5"}, "location", "southeast") %! title ("Half-normal CDF") %! xlabel ("values in x") %! ylabel ("probability") %!demo %! ## Plot half-normal against normal cumulative distribution function %! x = -5:0.001:5; %! p1 = hncdf (x, 0, 1); %! p2 = normcdf (x); %! plot (x, p1, "-b", x, p2, "-g") %! grid on %! xlim ([-5, 5]) %! legend ({"half-normal with μ = 0, σ = 1", ... %! "standart normal (μ = 0, σ = 1)"}, "location", "southeast") %! title ("Half-normal against standard normal CDF") %! xlabel ("values in x") %! ylabel ("probability") ## Test output %!shared x, p1, p1u, y2, y2u, y3, y3u %! x = [-Inf, -1, 0, 1/2, 1, Inf]; %! p1 = [0, 0, 0, 0.3829, 0.6827, 1]; %! p1u = [1, 1, 1, 0.6171, 0.3173, 0]; %!assert (hncdf (x, zeros (1,6), ones (1,6)), p1, 1e-4) %!assert (hncdf (x, 0, 1), p1, 1e-4) %!assert (hncdf (x, 0, ones (1,6)), p1, 1e-4) %!assert (hncdf (x, zeros (1,6), 1), p1, 1e-4) %!assert (hncdf (x, 0, [1, 1, 1, NaN, 1, 1]), [p1(1:3), NaN, p1(5:6)], 1e-4) %!assert (hncdf (x, [0, 0, 0, NaN, 0, 0], 1), [p1(1:3), NaN, p1(5:6)], 1e-4) %!assert (hncdf ([x(1:3), NaN, x(5:6)], 0, 1), [p1(1:3), NaN, p1(5:6)], 1e-4) %!assert (hncdf (x, zeros (1,6), ones (1,6), "upper"), p1u, 1e-4) %!assert (hncdf (x, 0, 1, "upper"), p1u, 1e-4) %!assert (hncdf (x, 0, ones (1,6), "upper"), p1u, 1e-4) %!assert (hncdf (x, zeros (1,6), 1, "upper"), p1u, 1e-4) ## Test class of input preserved %!assert (class (hncdf (single ([x, NaN]), 0, 1)), "single") %!assert (class (hncdf ([x, NaN], 0, single (1))), "single") %!assert (class (hncdf ([x, NaN], single (0), 1)), "single") ## Test input validation %!error hncdf () %!error hncdf (1) %!error hncdf (1, 2) %!error hncdf (1, 2, 3, "tail") %!error hncdf (1, 2, 3, 5) %!error ... %! hncdf (ones (3), ones (2), ones(2)) %!error ... %! hncdf (ones (2), ones (3), ones(2)) %!error ... %! hncdf (ones (2), ones (2), ones(3)) %!error hncdf (i, 2, 3) %!error hncdf (1, i, 3) %!error hncdf (1, 2, i) statistics-release-1.6.3/inst/dist_fun/hninv.m000066400000000000000000000103001456127120000214110ustar00rootroot00000000000000## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{x} =} hninv (@var{p}, @var{mu}, @var{sigma}) ## ## Inverse of the half-normal cumulative distribution function (iCDF). ## ## For each element of @var{p}, compute the quantile (the inverse of the CDF) of ## the half-normal distribution with location parameter @var{mu} and scale ## parameter @var{sigma}. The size of @var{x} is the common size of @var{p}, ## @var{mu}, and @var{sigma}. A scalar input functions as a constant matrix of ## the same size as the other inputs. ## ## Further information about the half-normal distribution can be found at ## @url{https://en.wikipedia.org/wiki/Half-normal_distribution} ## ## @seealso{hncdf, hnpdf, hnrnd, hnfit, hnlike} ## @end deftypefn function x = hninv (p, mu, sigma) ## Check for valid number of input arguments if (nargin < 3) error ("hninv: function called with too few input arguments."); endif ## Check for common size of P, MU, and SIGMA if (! isscalar (p) || ! isscalar (mu) || ! isscalar(sigma)) [retval, p, mu, sigma] = common_size (p, mu, sigma); if (retval > 0) error ("hninv: P, MU, and SIGMA must be of common size or scalars."); endif endif ## Check for X, MU, and SIGMA being reals if (iscomplex (p) || iscomplex (mu) || iscomplex (sigma)) error ("hninv: P, MU, and SIGMA must not be complex."); endif ## Check for class type if (isa (p, "single") || isa (mu, "single") || isa (sigma, "single")); x = NaN (size (p), "single"); else x = NaN (size (p)); endif ## Return NaNs for out of range values of SIGMA parameter sigma(sigma <= 0) = NaN; ## Return NaNs for out of range P values p(p < 0 | 1 < p) = NaN; ## Calculate the quantile of half-normal distribution x = sqrt(2) * sigma .* erfinv (p) + mu; endfunction %!demo %! ## Plot various iCDFs from the half-normal distribution %! p = 0.001:0.001:0.999; %! x1 = hninv (p, 0, 1); %! x2 = hninv (p, 0, 2); %! x3 = hninv (p, 0, 3); %! x4 = hninv (p, 0, 5); %! plot (p, x1, "-b", p, x2, "-g", p, x3, "-r", p, x4, "-c") %! grid on %! ylim ([0, 10]) %! legend ({"μ = 0, σ = 1", "μ = 0, σ = 2", ... %! "μ = 0, σ = 3", "μ = 0, σ = 5"}, "location", "northwest") %! title ("Half-normal iCDF") %! xlabel ("probability") %! ylabel ("x") ## Test output %!shared p, x %! p = [0, 0.3829, 0.6827, 1]; %! x = [0, 1/2, 1, Inf]; %!assert (hninv (p, 0, 1), x, 1e-4); %!assert (hninv (p, 5, 1), x + 5, 1e-4); %!assert (hninv (p, 0, ones (1,4)), x, 1e-4); %!assert (hninv (p, 0, [-1, 0, 1, 1]), [NaN, NaN, x(3:4)], 1e-4) ## Test class of input preserved %!assert (class (hninv (single ([p, NaN]), 0, 1)), "single") %!assert (class (hninv ([p, NaN], single (0), 1)), "single") %!assert (class (hninv ([p, NaN], 0, single (1))), "single") ## Test input validation %!error hninv (1) %!error hninv (1, 2) %!error ... %! hninv (1, ones (2), ones (3)) %!error ... %! hninv (ones (2), 1, ones (3)) %!error ... %! hninv (ones (2), ones (3), 1) %!error hninv (i, 2, 3) %!error hninv (1, i, 3) %!error hninv (1, 2, i) statistics-release-1.6.3/inst/dist_fun/hnpdf.m000066400000000000000000000113321456127120000213740ustar00rootroot00000000000000## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{y} =} hnpdf (@var{x}, @var{mu}, @var{sigma}) ## ## Half-normal probability density function (PDF). ## ## For each element of @var{x}, compute the probability density function (PDF) ## of the half-normal distribution with location parameter @var{mu} and scale ## parameter @var{sigma}. The size of @var{y} is the common size of @var{x}, ## @var{mu}, and @var{sigma}. A scalar input functions as a constant matrix of ## the same size as the other inputs. ## ## The half-normal CDF is only defined for @qcode{@var{x} >= @var{mu}}. ## ## Further information about the half-normal distribution can be found at ## @url{https://en.wikipedia.org/wiki/Half-normal_distribution} ## ## @seealso{hncdf, hninv, hnrnd, hnfit, hnlike} ## @end deftypefn function y = hnpdf (x, mu, sigma) ## Check for valid number of input arguments if (nargin < 3) error ("hnpdf: function called with too few input arguments."); endif ## Check for common size of X, MU, and SIGMA if (! isscalar (x) || ! isscalar (mu) || ! isscalar(sigma)) [retval, x, mu, sigma] = common_size (x, mu, sigma); if (retval > 0) error ("hnpdf: X, MU, and SIGMA must be of common size or scalars."); endif endif ## Check for X, MU, and SIGMA being reals if (iscomplex (x) || iscomplex (mu) || iscomplex (sigma)) error ("hnpdf: X, MU, and SIGMA must not be complex."); endif ## Check for class type if (isa (x, "single") || isa (mu, "single") || isa (sigma, "single")); y = NaN (size (x), "single"); else y = NaN (size (x)); endif ## Return NaNs for out of range values of SIGMA parameter sigma(sigma <= 0) = NaN; ## Compute half-normal PDF z = (x - mu) ./ sigma; y = sqrt (2 / pi) ./ sigma .* exp (-0.5 * z .^ 2); ## Force zero for unsupported X y(z < 0) = 0; endfunction %!demo %! ## Plot various PDFs from the half-normal distribution %! x = 0:0.001:10; %! y1 = hnpdf (x, 0, 1); %! y2 = hnpdf (x, 0, 2); %! y3 = hnpdf (x, 0, 3); %! y4 = hnpdf (x, 0, 5); %! plot (x, y1, "-b", x, y2, "-g", x, y3, "-r", x, y4, "-c") %! grid on %! xlim ([0, 10]) %! ylim ([0, 0.9]) %! legend ({"μ = 0, σ = 1", "μ = 0, σ = 2", ... %! "μ = 0, σ = 3", "μ = 0, σ = 5"}, "location", "northeast") %! title ("Half-normal PDF") %! xlabel ("values in x") %! ylabel ("density") %!demo %! ## Plot half-normal against normal probability density function %! x = -5:0.001:5; %! y1 = hnpdf (x, 0, 1); %! y2 = normpdf (x); %! plot (x, y1, "-b", x, y2, "-g") %! grid on %! xlim ([-5, 5]) %! ylim ([0, 0.9]) %! legend ({"half-normal with μ = 0, σ = 1", ... %! "standart normal (μ = 0, σ = 1)"}, "location", "northeast") %! title ("Half-normal against standard normal PDF") %! xlabel ("values in x") %! ylabel ("density") ## Test output %!shared x, y %! x = [-Inf, -1, 0, 1/2, 1, Inf]; %! y = [0, 0, 0.7979, 0.7041, 0.4839, 0]; %!assert (hnpdf ([x, NaN], 0, 1), [y, NaN], 1e-4) %!assert (hnpdf (x, 0, [-2, -1, 0, 1, 1, 1]), [nan(1,3), y([4:6])], 1e-4) ## Test class of input preserved %!assert (class (hncdf (single ([x, NaN]), 0, 1)), "single") %!assert (class (hncdf ([x, NaN], 0, single (1))), "single") %!assert (class (hncdf ([x, NaN], single (0), 1)), "single") ## Test input validation %!error hnpdf () %!error hnpdf (1) %!error hnpdf (1, 2) %!error ... %! hnpdf (1, ones (2), ones (3)) %!error ... %! hnpdf (ones (2), 1, ones (3)) %!error ... %! hnpdf (ones (2), ones (3), 1) %!error hnpdf (i, 2, 3) %!error hnpdf (1, i, 3) %!error hnpdf (1, 2, i) statistics-release-1.6.3/inst/dist_fun/hnrnd.m000066400000000000000000000146441456127120000214170ustar00rootroot00000000000000## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{r} =} hnrnd (@var{mu}, @var{sigma}) ## @deftypefnx {statistics} {@var{r} =} hnrnd (@var{mu}, @var{sigma}, @var{rows}) ## @deftypefnx {statistics} {@var{r} =} hnrnd (@var{mu}, @var{sigma}, @var{rows}, @var{cols}, @dots{}) ## @deftypefnx {statistics} {@var{r} =} hnrnd (@var{mu}, @var{sigma}, [@var{sz}]) ## ## Random arrays from the half-normal distribution. ## ## @code{@var{r} = hnrnd (@var{mu}, @var{sigma})} returns an array of random ## numbers chosen from the half-normal distribution with location parameter ## @var{mu} and scale parameter @var{sigma}. The size of @var{r} is the common ## size of @var{mu} and @var{sigma}. A scalar input functions as a constant ## matrix of the same size as the other inputs. ## ## When called with a single size argument, @code{hnrnd} returns a square ## matrix with the dimension specified. When called with more than one scalar ## argument, the first two arguments are taken as the number of rows and columns ## and any further arguments specify additional matrix dimensions. The size may ## also be specified with a row vector of dimensions, @var{sz}. ## ## Further information about the half-normal distribution can be found at ## @url{https://en.wikipedia.org/wiki/Half-normal_distribution} ## ## @seealso{hncdf, hninv, hnpdf, hnfit, hnlike} ## @end deftypefn function r = hnrnd (mu, sigma, varargin) ## Check for valid number of input arguments if (nargin < 2) error ("hnrnd: function called with too few input arguments."); endif ## Check for common size of MU, and SIGMA if (! isscalar (mu) || ! isscalar (sigma)) [retval, mu, sigma] = common_size (mu, sigma); if (retval > 0) error ("hnrnd: MU and SIGMA must be of common size or scalars."); endif endif ## Check for X, MU, and SIGMA being reals if (iscomplex (mu) || iscomplex (sigma)) error ("hnrnd: MU and SIGMA must not be complex."); endif ## Parse and check SIZE arguments if (nargin == 2) sz = size (mu); elseif (nargin == 3) if (isscalar (varargin{1}) && varargin{1} >= 0 ... && varargin{1} == fix (varargin{1})) sz = [varargin{1}, varargin{1}]; elseif (isrow (varargin{1}) && all (varargin{1} >= 0) ... && all (varargin{1} == fix (varargin{1}))) sz = varargin{1}; elseif error (strcat (["hnrnd: SZ must be a scalar or a row vector"], ... [" of non-negative integers."])); endif elseif (nargin > 3) posint = cellfun (@(x) (! isscalar (x) || x < 0 || x != fix (x)), varargin); if (any (posint)) error ("hnrnd: dimensions must be non-negative integers."); endif sz = [varargin{:}]; endif ## Check that parameters match requested dimensions in size if (! isscalar (mu) && ! isequal (size (mu), sz)) error ("hnrnd: MU and SIGMA must be scalars or of size SZ."); endif ## Check for class type if (isa (mu, "single") || isa (sigma, "single")) cls = "single"; else cls = "double"; endif ## Generate random sample from half-normal distribution r = abs (randn (sz, cls)) .* sigma + mu; ## Force output to NaN for invalid parameter SIGMA <= 0 k = (sigma <= 0); r(k) = NaN; endfunction ## Test output %!assert (size (hnrnd (1, 1, 1)), [1, 1]) %!assert (size (hnrnd (1, 1, 2)), [2, 2]) %!assert (size (hnrnd (1, 1, [2, 1])), [2, 1]) %!assert (size (hnrnd (1, zeros (2, 2))), [2, 2]) %!assert (size (hnrnd (1, ones (2, 1))), [2, 1]) %!assert (size (hnrnd (1, ones (2, 2))), [2, 2]) %!assert (size (hnrnd (ones (2, 1), 1)), [2, 1]) %!assert (size (hnrnd (ones (2, 2), 1)), [2, 2]) %!assert (size (hnrnd (1, 1, 3)), [3, 3]) %!assert (size (hnrnd (1, 1, [4 1])), [4, 1]) %!assert (size (hnrnd (1, 1, 4, 1)), [4, 1]) %!test %! r = hnrnd (1, [1, 0, -1]); %! assert (r([2:3]), [NaN, NaN]) ## Test class of input preserved %!assert (class (hnrnd (1, 0)), "double") %!assert (class (hnrnd (1, single (0))), "single") %!assert (class (hnrnd (1, single ([0 0]))), "single") %!assert (class (hnrnd (1, single (1))), "single") %!assert (class (hnrnd (1, single ([1 1]))), "single") %!assert (class (hnrnd (single (1), 1)), "single") %!assert (class (hnrnd (single ([1 1]), 1)), "single") ## Test input validation %!error hnrnd () %!error hnrnd (1) %!error ... %! hnrnd (ones (3), ones (2)) %!error ... %! hnrnd (ones (2), ones (3)) %!error hnrnd (i, 2, 3) %!error hnrnd (1, i, 3) %!error ... %! hnrnd (1, 2, -1) %!error ... %! hnrnd (1, 2, 1.2) %!error ... %! hnrnd (1, 2, ones (2)) %!error ... %! hnrnd (1, 2, [2 -1 2]) %!error ... %! hnrnd (1, 2, [2 0 2.5]) %!error ... %! hnrnd (1, 2, 2, -1, 5) %!error ... %! hnrnd (1, 2, 2, 1.5, 5) %!error ... %! hnrnd (2, ones (2), 3) %!error ... %! hnrnd (2, ones (2), [3, 2]) %!error ... %! hnrnd (2, ones (2), 3, 2) statistics-release-1.6.3/inst/dist_fun/hygecdf.m000066400000000000000000000202711456127120000217100ustar00rootroot00000000000000## Copyright (C) 2012 Rik Wehbring ## Copyright (C) 1997-2016 Kurt Hornik ## Copyright (C) 2022 Nicholas R. Jankowski ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{p} =} hygecdf (@var{x}, @var{m}, @var{k}, @var{n}) ## @deftypefnx {statistics} {@var{p} =} hygecdf (@var{x}, @var{m}, @var{k}, @var{n}, @qcode{"upper"}) ## ## Hypergeometric cumulative distribution function (CDF). ## ## For each element of @var{x}, compute the cumulative distribution function ## (CDF) of the hypergeometric distribution with parameters @var{m}, @var{k}, ## and @var{n}. The size of @var{p} is the common size of @var{x}, @var{m}, ## @var{k}, and @var{n}. A scalar input functions as a constant matrix of the ## same size as the other inputs. ## ## This is the cumulative probability of obtaining not more than @var{x} marked ## items when randomly drawing a sample of size @var{n} without replacement from ## a population of total size @var{m} containing @var{k} marked items. The ## parameters @var{m}, @var{k}, and @var{n} must be positive integers with ## @var{k} and @var{n} not greater than @var{m}. ## ## @code{[@dots{}] = hygecdf (@var{x}, @var{m}, @var{k}, @var{n}, "upper")} ## computes the upper tail probability of the hypergeometric distribution with ## parameters @var{m}, @var{k}, and @var{n}, at the values in @var{x}. ## ## Further information about the hypergeometric distribution can be found at ## @url{https://en.wikipedia.org/wiki/Hypergeometric_distribution} ## ## @seealso{hygeinv, hygepdf, hygernd, hygestat} ## @end deftypefn function p = hygecdf (x, m, k, n, uflag) ## Check for valid number of input arguments if (nargin < 4) error ("hygecdf: function called with too few input arguments."); endif ## Check for common size of X, T, k, and N if (! isscalar (x) || ! isscalar (m) || ! isscalar (k) || ! isscalar (n)) [retval, x, m, k, n] = common_size (x, m, k, n); if (retval > 0) error ("hygecdf: X, T, k, and N must be of common size or scalars."); endif endif ## Check for X, T, k, and N being reals if (iscomplex (x) || iscomplex (m) || iscomplex (k) || iscomplex (n)) error ("hygecdf: X, T, k, and N must not be complex."); endif ## Check for class type if (isa (x, "single") || isa (m, "single") || isa (k, "single") || isa (n, "single")) p = zeros (size (x), "single"); else p = zeros (size (x)); endif ## Check for "upper" flag if (nargin > 4 && strcmpi (uflag, "upper")) x = n - floor(x) - 1; k = m - k; elseif (nargin > 4 && ! strcmpi (uflag, "upper")) error ("hygecdf: invalid argument for upper tail."); endif ## Force 1 where required is_1 = (x >= n | x >= k); p(is_1) = 1; ## Force NaNs where required is_nan = (isnan (x) | isnan (m) | isnan (k) | isnan (n) | ... m < 0 | k < 0 | n < 0 | round (m) != m | round (k) != k | ... round (n) != n | n > m | k > m); p(is_nan) = NaN; ## Get values for which P = 0 is_0 = (m - k - n + x + 1 <= 0 | x < 0); ok = ! (is_1 | is_nan | is_0); ## Compute hypergeometric CDF if (any (ok(:))) ## For improved accuracy, compute the upper tail 1-p instead ## of the lower tail pfor x values that are larger than the mean lo = (x <= k .* n ./ m); ok_lo = ok & lo; if (any (ok_lo(:))) p(ok_lo) = localPDF (floor (x(ok_lo)), m(ok_lo), k(ok_lo), n(ok_lo)); endif ok_hi = ok & ! lo; if (any (ok_hi(:))) p(ok_hi) = 1 - localPDF (n(ok_hi) - floor (x(ok_hi)) - 1, ... m(ok_hi), m(ok_hi) - k(ok_hi), n(ok_hi)); endif endif endfunction function p = localPDF (x, m, k, n) HPDF = hygepdf (x, m, k, n); ## Compute hygecdf(x,m,k,n)/hygepdf(x,m,k,n) with a series ## whose terms can be computed recursively, backwards. xmax = max (x(:)); ybig = repmat((0:xmax)', 1, length (x)); xbig = repmat(x(:)', xmax + 1, 1); mbig = repmat(m(:)', xmax + 1, 1); kbig = repmat(k(:)', xmax + 1, 1); nbig = repmat(n(:)', xmax + 1, 1); terms = ((ybig+1) .* (mbig-kbig-nbig+ybig+1)) ./ ((nbig-ybig) .* (kbig-ybig)); terms(ybig >= xbig) = 1; terms = flip (cumprod (flip (terms))); terms(ybig > xbig) = 0; ratio = sum(terms,1); ratio = reshape(ratio,size(x)); p = ratio.*HPDF; ## Correct round-off errors p(p > 1) = 1; endfunction %!demo %! ## Plot various CDFs from the hypergeometric distribution %! x = 0:60; %! p1 = hygecdf (x, 500, 50, 100); %! p2 = hygecdf (x, 500, 60, 200); %! p3 = hygecdf (x, 500, 70, 300); %! plot (x, p1, "*b", x, p2, "*g", x, p3, "*r") %! grid on %! xlim ([0, 60]) %! legend ({"m = 500, k = 50, n = 100", "m = 500, k = 60, n = 200", ... %! "m = 500, k = 70, n = 300"}, "location", "southeast") %! title ("Hypergeometric CDF") %! xlabel ("values in x (number of successes)") %! ylabel ("probability") ## Test output %!shared x, y %! x = [-1 0 1 2 3]; %! y = [0 1/6 5/6 1 1]; %!assert (hygecdf (x, 4*ones (1,5), 2, 2), y, 5*eps) %!assert (hygecdf (x, 4, 2*ones (1,5), 2), y, 5*eps) %!assert (hygecdf (x, 4, 2, 2*ones (1,5)), y, 5*eps) %!assert (hygecdf (x, 4*[1 -1 NaN 1.1 1], 2, 2), [y(1) NaN NaN NaN y(5)], 5*eps) %!assert (hygecdf (x, 4*[1 -1 NaN 1.1 1], 2, 2, "upper"), ... %! [y(5) NaN NaN NaN y(1)], 5*eps) %!assert (hygecdf (x, 4, 2*[1 -1 NaN 1.1 1], 2), [y(1) NaN NaN NaN y(5)], 5*eps) %!assert (hygecdf (x, 4, 2*[1 -1 NaN 1.1 1], 2, "upper"), ... %! [y(5) NaN NaN NaN y(1)], 5*eps) %!assert (hygecdf (x, 4, 5, 2), [NaN NaN NaN NaN NaN]) %!assert (hygecdf (x, 4, 2, 2*[1 -1 NaN 1.1 1]), [y(1) NaN NaN NaN y(5)], 5*eps) %!assert (hygecdf (x, 4, 2, 2*[1 -1 NaN 1.1 1], "upper"), ... %! [y(5) NaN NaN NaN y(1)], 5*eps) %!assert (hygecdf (x, 4, 2, 5), [NaN NaN NaN NaN NaN]) %!assert (hygecdf ([x(1:2) NaN x(4:5)], 4, 2, 2), [y(1:2) NaN y(4:5)], 5*eps) %!test %! p = hygecdf (x, 10, [1 2 3 4 5], 2, "upper"); %! assert (p, [1, 34/90, 2/30, 0, 0], 10*eps); %!test %! p = hygecdf (2*x, 10, [1 2 3 4 5], 2, "upper"); %! assert (p, [1, 34/90, 0, 0, 0], 10*eps); ## Test class of input preserved %!assert (hygecdf ([x, NaN], 4, 2, 2), [y, NaN], 5*eps) %!assert (hygecdf (single ([x, NaN]), 4, 2, 2), single ([y, NaN]), ... %! eps ("single")) %!assert (hygecdf ([x, NaN], single (4), 2, 2), single ([y, NaN]), ... %! eps ("single")) %!assert (hygecdf ([x, NaN], 4, single (2), 2), single ([y, NaN]), ... %! eps ("single")) %!assert (hygecdf ([x, NaN], 4, 2, single (2)), single ([y, NaN]), ... %! eps ("single")) ## Test input validation %!error hygecdf () %!error hygecdf (1) %!error hygecdf (1,2) %!error hygecdf (1,2,3) %!error hygecdf (1,2,3,4,5) %!error hygecdf (1,2,3,4,"uper") %!error ... %! hygecdf (ones (2), ones (3), 1, 1) %!error ... %! hygecdf (1, ones (2), ones (3), 1) %!error ... %! hygecdf (1, 1, ones (2), ones (3)) %!error hygecdf (i, 2, 2, 2) %!error hygecdf (2, i, 2, 2) %!error hygecdf (2, 2, i, 2) %!error hygecdf (2, 2, 2, i) statistics-release-1.6.3/inst/dist_fun/hygeinv.m000066400000000000000000000154001456127120000217460ustar00rootroot00000000000000## Copyright (C) 2012 Rik Wehbring ## Copyright (C) 1997-2016 Kurt Hornik ## Copyright (C) 2022 Nicholas R. Jankowski ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{x} =} hygeinv (@var{p}, @var{m}, @var{k}, @var{n}) ## ## Inverse of the hypergeometric cumulative distribution function (iCDF). ## ## For each element of @var{p}, compute the quantile (the inverse of the CDF) of ## the hypergeometric distribution with parameters @var{m}, @var{k}, and @var{n}. ## The size of @var{x} is the common size of @var{p}, @var{m}, @var{k}, and ## @var{n}. A scalar input functions as a constant matrix of the same size as ## the other inputs. ## ## This is the number of drawn marked items @var{x} given a probability @var{p}, ## when randomly drawing a sample of size @var{n} without replacement from a ## population of total size @var{m} containing @var{k} marked items. The ## parameters @var{m}, @var{k}, and @var{n} must be positive integers with ## @var{k} and @var{n} not greater than @var{m}. ## ## Further information about the hypergeometric distribution can be found at ## @url{https://en.wikipedia.org/wiki/Hypergeometric_distribution} ## ## @seealso{hygeinv, hygepdf, hygernd, hygestat} ## @end deftypefn function x = hygeinv (p, m, k, n) ## Check for valid number of input arguments if (nargin < 4) error ("hygeinv: function called with too few input arguments."); endif ## Check for common size of P, T, M, and N if (! isscalar (p) || ! isscalar (m) || ! isscalar (k) || ! isscalar (n)) [retval, p, m, k, n] = common_size (p, m, k, n); if (retval > 0) error ("hygeinv: P, T, M, and N must be of common size or scalars."); endif endif ## Check for P, T, M, and N being reals if (iscomplex (p) || iscomplex (m) || iscomplex (k) || iscomplex (n)) error ("hygeinv: P, T, M, and N must not be complex."); endif ## Check for class type if (isa (p, "single") || isa (m, "single") || isa (k, "single") || isa (n, "single")) x = NaN (size (p), "single"); else x = NaN (size (p)); endif ok = ((m >= 0) & (k >= 0) & (n > 0) & (k <= m) & (n <= m) & (m == fix (m)) & (k == fix (k)) & (n == fix (n))); if (isscalar (m)) if (ok) x = discrete_inv (p, 0 : n, hygepdf (0 : n, m, k, n)); x(p == 0) = 0; # Hack to return correct value for start of distribution endif else p_0 = (p == 0); x (ok & p_0) = 0; # set any p=0 to 0 if not already set to output NaN p_0 = (p == 1); x (ok & p_0) = n(ok & p_0); ok &= (p>0 & p<1); # remove 0's and p's outside (0,1), leave unfilled as NaN if (any (ok(:))) n = n(ok); v = 0 : max (n(:)); ## Manually perform discrete_inv to enable vectorizing with array input p_tmp = cumsum (hygepdf (v, m(ok), k(ok), n, "vectorexpand"), 2); sz_p = size (p_tmp); end_locs = sub2ind (sz_p, [1 : numel(n)]', n(:) + 1); ## Manual row-wise vectorization of lookup, which returns index of element ## less than or equal to test value, zero if test value less than lowest ## number, and max index if greater than highest number. operated on ## flipped p_tmp, adjusting for different vector lengths in array rows. p_tmp = (p_tmp ./ p_tmp(end_locs))(:, end:-1:1) - p(ok)(:); p_tmp(p_tmp>=0) = NaN; [p_match, p_match_idx] = max (p_tmp, [], 2); p_match_idx(isnan(p_match)) = v(end) + 2; x(ok) = v(v(end) - p_match_idx + 3); endif endif endfunction %!demo %! ## Plot various iCDFs from the hypergeometric distribution %! p = 0.001:0.001:0.999; %! x1 = hygeinv (p, 500, 50, 100); %! x2 = hygeinv (p, 500, 60, 200); %! x3 = hygeinv (p, 500, 70, 300); %! plot (p, x1, "-b", p, x2, "-g", p, x3, "-r") %! grid on %! ylim ([0, 60]) %! legend ({"m = 500, k = 50, n = 100", "m = 500, k = 60, n = 200", ... %! "m = 500, k = 70, n = 300"}, "location", "northwest") %! title ("Hypergeometric iCDF") %! xlabel ("probability") %! ylabel ("values in p (number of successes)") ## Test output %!shared p %! p = [-1 0 0.5 1 2]; %!assert (hygeinv (p, 4*ones (1,5), 2*ones (1,5), 2*ones (1,5)), [NaN 0 1 2 NaN]) %!assert (hygeinv (p, 4*ones (1,5), 2, 2), [NaN 0 1 2 NaN]) %!assert (hygeinv (p, 4, 2*ones (1,5), 2), [NaN 0 1 2 NaN]) %!assert (hygeinv (p, 4, 2, 2*ones (1,5)), [NaN 0 1 2 NaN]) %!assert (hygeinv (p, 4*[1 -1 NaN 1.1 1], 2, 2), [NaN NaN NaN NaN NaN]) %!assert (hygeinv (p, 4, 2*[1 -1 NaN 1.1 1], 2), [NaN NaN NaN NaN NaN]) %!assert (hygeinv (p, 4, 5, 2), [NaN NaN NaN NaN NaN]) %!assert (hygeinv (p, 4, 2, 2*[1 -1 NaN 1.1 1]), [NaN NaN NaN NaN NaN]) %!assert (hygeinv (p, 4, 2, 5), [NaN NaN NaN NaN NaN]) %!assert (hygeinv ([p(1:2) NaN p(4:5)], 4, 2, 2), [NaN 0 NaN 2 NaN]) ## Test class of input preserved %!assert (hygeinv ([p, NaN], 4, 2, 2), [NaN 0 1 2 NaN NaN]) %!assert (hygeinv (single ([p, NaN]), 4, 2, 2), single ([NaN 0 1 2 NaN NaN])) %!assert (hygeinv ([p, NaN], single (4), 2, 2), single ([NaN 0 1 2 NaN NaN])) %!assert (hygeinv ([p, NaN], 4, single (2), 2), single ([NaN 0 1 2 NaN NaN])) %!assert (hygeinv ([p, NaN], 4, 2, single (2)), single ([NaN 0 1 2 NaN NaN])) ## Test input validation %!error hygeinv () %!error hygeinv (1) %!error hygeinv (1,2) %!error hygeinv (1,2,3) %!error ... %! hygeinv (ones (2), ones (3), 1, 1) %!error ... %! hygeinv (1, ones (2), ones (3), 1) %!error ... %! hygeinv (1, 1, ones (2), ones (3)) %!error hygeinv (i, 2, 2, 2) %!error hygeinv (2, i, 2, 2) %!error hygeinv (2, 2, i, 2) %!error hygeinv (2, 2, 2, i) statistics-release-1.6.3/inst/dist_fun/hygepdf.m000066400000000000000000000204651456127120000217320ustar00rootroot00000000000000## Copyright (C) 2012 Rik Wehbring ## Copyright (C) 1996-2016 Kurt Hornik ## Copyright (C) 2022 Nicholas R. Jankowski ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{y} =} hygepdf (@var{x}, @var{m}, @var{k}, @var{n}) ## @deftypefnx {statistics} {@var{y} =} hygepdf (@dots{}, @qcode{"vectorexpand"}) ## ## Hypergeometric probability density function (PDF). ## ## For each element of @var{x}, compute the probability density function (PDF) ## of the hypergeometric distribution with parameters @var{m}, @var{k}, and ## @var{n}. The size of @var{y} is the common size of @var{x}, @var{m}, ## @var{k}, and @var{n}. A scalar input functions as a constant matrix of the ## same size as the other inputs. ## ## This is the probability of obtaining @var{x} marked items when randomly ## drawing a sample of size @var{n} without replacement from a population of ## total size @var{m} containing @var{k} marked items. The parameters @var{m}, ## @var{k}, and @var{n} must be positive integers with @var{k} and @var{n} not ## greater than @var{m}. ## ## If the optional parameter @qcode{vectorexpand} is provided, @var{x} may be an ## array with size different from parameters @var{m}, @var{k}, and @var{n} ## (which must still be of a common size or scalar). Each element of @var{x} ## will be evaluated against each set of parameters @var{m}, @var{k}, and ## @var{n} in columnwise order. The output @var{y} will be an array of size ## @qcode{@var{r} x @var{s}}, where @qcode{@var{r} = numel (@var{m})}, and ## @qcode{@var{s} = numel (@var{x})}. ## ## Further information about the hypergeometric distribution can be found at ## @url{https://en.wikipedia.org/wiki/Hypergeometric_distribution} ## ## @seealso{hygecdf, hygeinv, hygernd, hygestat} ## @end deftypefn function y = hygepdf (x, m, k, n, vect_expand) ## Check for valid number of input arguments if (nargin < 4) error ("hygepdf: function called with too few input arguments."); endif ## Check for X, T, M, and N being reals if (iscomplex (x) || iscomplex (m) || iscomplex (k) || iscomplex (n)) error ("hygepdf: X, T, M, and N must not be complex."); endif ## Check for 5th argument or add default if (nargin < 5) vect_expand = []; endif if strcmpi (vect_expand, "vectorexpand") ## Expansion to improve vectorization of hyge calling functions. ## Project inputs over a 2D array with x(:) as a row vector and m,k,n as ## a column vector. each y(i,j) is hygepdf(x(j), m(i), k(i), n(i)) ## Following expansion, remainder of algorithm processes as normal. if (! isscalar (m) || ! isscalar (k) || ! isscalar (n)) [retval, m, k, n] = common_size (m, k, n); if (retval > 0) error ("hygepdf: T, M, and N must be of common size or scalars."); endif ## Ensure col vectors before expansion m = m(:); k = k(:); n = n(:); endif ## Expand x,m,k,n to arrays of size numel(m) x numel(x) sz = [numel(m), numel(x)]; x = x(:)'; # ensure row vector before expansion x = x(ones (sz(1), 1), :); m = m(:, ones (sz(2), 1)); k = k(:, ones (sz(2), 1)); n = n(:, ones (sz(2), 1)); else ## Check for common size of X, T, M, and N if (! isscalar (m) || ! isscalar (k) || ! isscalar (n)) [retval, x, m, k, n] = common_size (x, m, k, n); if (retval > 0) error ("hygepdf: X, T, M, and N must be of common size or scalars."); endif endif sz = size (x); endif ## Check for class type if (isa (x, "single") || isa (m, "single") || isa (k, "single") || isa (n, "single")) y = zeros (sz, "single"); else y = zeros (sz); endif ## Everything in nel gives NaN nel = (isnan (x) | (m < 0) | (k < 0) | (n <= 0) | (k > m) | (n > m) | (m != fix (m)) | (k != fix (k)) | (n != fix (n))); ## Everything in zel gives 0 unless in nel zel = ((x != fix (x)) | (x < 0) | (x > k) | (n < x) | (n-x > m-k)); y(nel) = NaN; ok = ! nel & ! zel; if (any (ok(:))) if (isscalar (m)) y(ok) = exp (gammaln (k+1) - gammaln (k-x(ok)+1) - gammaln (x(ok)+1) + ... gammaln (m-k+1) - gammaln (m-k-n+x(ok)+1) - ... gammaln (n-x(ok)+1) - gammaln (m+1) + gammaln (m-n+1) + ... gammaln (n+1)); else y(ok) = exp (gammaln (k(ok)+1) - gammaln (k(ok)-x(ok)+1) - ... gammaln (x(ok)+1) + gammaln (m(ok)-k(ok)+1) - ... gammaln (m(ok)-k(ok)-n(ok)+x(ok)+1) - ... gammaln (n(ok)-x(ok)+1) - gammaln (m(ok)+1) + ... gammaln (m(ok)-n(ok)+1) + gammaln (n(ok)+1)); endif endif endfunction %!demo %! ## Plot various PDFs from the hypergeometric distribution %! x = 0:60; %! y1 = hygepdf (x, 500, 50, 100); %! y2 = hygepdf (x, 500, 60, 200); %! y3 = hygepdf (x, 500, 70, 300); %! plot (x, y1, "*b", x, y2, "*g", x, y3, "*r") %! grid on %! xlim ([0, 60]) %! ylim ([0, 0.18]) %! legend ({"m = 500, k = 50, μ = 100", "m = 500, k = 60, μ = 200", ... %! "m = 500, k = 70, μ = 300"}, "location", "northeast") %! title ("Hypergeometric PDF") %! xlabel ("values in x (number of successes)") %! ylabel ("density") ## Test output %!shared x, y %! x = [-1 0 1 2 3]; %! y = [0 1/6 4/6 1/6 0]; %!assert (hygepdf (x, 4*ones (1,5), 2, 2), y, eps) %!assert (hygepdf (x, 4, 2*ones (1,5), 2), y, eps) %!assert (hygepdf (x, 4, 2, 2*ones (1,5)), y, eps) %!assert (hygepdf (x, 4*[1 -1 NaN 1.1 1], 2, 2), [0 NaN NaN NaN 0], eps) %!assert (hygepdf (x, 4, 2*[1 -1 NaN 1.1 1], 2), [0 NaN NaN NaN 0], eps) %!assert (hygepdf (x, 4, 5, 2), [NaN NaN NaN NaN NaN], eps) %!assert (hygepdf (x, 4, 2, 2*[1 -1 NaN 1.1 1]), [0 NaN NaN NaN 0], eps) %!assert (hygepdf (x, 4, 2, 5), [NaN NaN NaN NaN NaN], eps) %!assert (hygepdf ([x, NaN], 4, 2, 2), [y, NaN], eps) ## Test class of input preserved %!assert (hygepdf (single ([x, NaN]), 4, 2, 2), single ([y, NaN]), eps("single")) %!assert (hygepdf ([x, NaN], single (4), 2, 2), single ([y, NaN]), eps("single")) %!assert (hygepdf ([x, NaN], 4, single (2), 2), single ([y, NaN]), eps("single")) %!assert (hygepdf ([x, NaN], 4, 2, single (2)), single ([y, NaN]), eps("single")) ## Test vector expansion %!test %! z = zeros(3,5); %! z([4,5,6,8,9,12]) = [1, 0.5, 1/6, 0.5, 2/3, 1/6]; %! assert (hygepdf (x, 4, [0, 1, 2], 2, "vectorexpand"), z, eps); %! assert (hygepdf (x, 4, [0, 1, 2]', 2, "vectorexpand"), z, eps); %! assert (hygepdf (x', 4, [0, 1, 2], 2, "vectorexpand"), z, eps); %! assert (hygepdf (2, 4, [0 ,1, 2], 2, "vectorexpand"), z(:,4), eps); %! assert (hygepdf (x, 4, 1, 2, "vectorexpand"), z(2,:), eps); %! assert (hygepdf ([NaN, x], 4, [0 1 2]', 2, "vectorexpand"), [NaN(3,1), z], eps); ## Test input validation %!error hygepdf () %!error hygepdf (1) %!error hygepdf (1,2) %!error hygepdf (1,2,3) %!error ... %! hygepdf (1, ones (3), ones (2), ones (2)) %!error ... %! hygepdf (1, ones (2), ones (3), ones (2)) %!error ... %! hygepdf (1, ones (2), ones (2), ones (3)) %!error hygepdf (i, 2, 2, 2) %!error hygepdf (2, i, 2, 2) %!error hygepdf (2, 2, i, 2) %!error hygepdf (2, 2, 2, i) statistics-release-1.6.3/inst/dist_fun/hygernd.m000066400000000000000000000177621456127120000217520ustar00rootroot00000000000000## Copyright (C) 2012 Rik Wehbring ## Copyright (C) 1997-2016 Kurt Hornik ## Copyright (C) 2022 Nicholas R. Jankowski ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{r} =} hygernd (@var{m}, @var{k}, @var{n}) ## @deftypefnx {statistics} {@var{r} =} hygernd (@var{m}, @var{k}, @var{n}, @var{rows}) ## @deftypefnx {statistics} {@var{r} =} hygernd (@var{m}, @var{k}, @var{n}, @var{rows}, @var{cols}, @dots{}) ## @deftypefnx {statistics} {@var{r} =} hygernd (@var{m}, @var{k}, @var{n}, [@var{sz}]) ## ## Random arrays from the hypergeometric distribution. ## ## @code{@var{r} = hygernd ((@var{m}, @var{k}, @var{n}} returns an array of ## random numbers chosen from the hypergeometric distribution with parameters ## @var{m}, @var{k}, and @var{n}. The size of @var{r} is the common size of ## @var{m}, @var{k}, and @var{n}. A scalar input functions as a constant matrix ## of the same size as the other inputs. ## ## The parameters @var{m}, @var{k}, and @var{n} must be positive integers ## with @var{k} and @var{n} not greater than @var{m}. ## ## When called with a single size argument, @code{hygernd} returns a square ## matrix with the dimension specified. When called with more than one scalar ## argument, the first two arguments are taken as the number of rows and columns ## and any further arguments specify additional matrix dimensions. The size may ## also be specified with a row vector of dimensions, @var{sz}. ## ## Further information about the hypergeometric distribution can be found at ## @url{https://en.wikipedia.org/wiki/Hypergeometric_distribution} ## ## @seealso{hygecdf, hygeinv, hygepdf, hygestat} ## @end deftypefn function r = hygernd (m, k, n, varargin) ## Check for valid number of input arguments if (nargin < 3) error ("hygernd: function called with too few input arguments."); endif ## Check for common size of T, M, and N if (! isscalar (m) || ! isscalar (k) || ! isscalar (n)) [retval, m, k, n] = common_size (m, k, n); if (retval > 0) error ("hygernd: T, M, and N must be of common size or scalars."); endif endif ## Check for T, M, and N being reals if (iscomplex (m) || iscomplex (k) || iscomplex (n)) error ("hygernd: T, M, and N must not be complex."); endif ## Parse and check SIZE arguments if (nargin == 3) sz = size (m); elseif (nargin == 4) if (isscalar (varargin{1}) && varargin{1} >= 0 ... && varargin{1} == fix (varargin{1})) sz = [varargin{1}, varargin{1}]; elseif (isrow (varargin{1}) && all (varargin{1} >= 0) ... && all (varargin{1} == fix (varargin{1}))) sz = varargin{1}; elseif error (strcat (["hygernd: SZ must be a scalar or a row vector"], ... [" of non-negative integers."])); endif elseif (nargin > 4) posint = cellfun (@(x) (! isscalar (x) || x < 0 || x != fix (x)), varargin); if (any (posint)) error ("hygernd: dimensions must be non-negative integers."); endif sz = [varargin{:}]; endif ## Check that parameters match requested dimensions in size if (! isscalar (m) && ! isequal (size (m), sz)) error ("hygernd: T, M, and N must be scalars or of size SZ."); endif ## Check for class type if (isa (m, "single") || isa (k, "single") || isa (n, "single")) cls = "single"; else cls = "double"; endif ok = ((m >= 0) & (k >= 0) & (n > 0) & (k <= m) & (n <= m) & (m == fix (m)) & (k == fix (k)) & (n == fix (n))); ## Generate random sample from the hypergeometric distribution if (isscalar (m)) if (ok) v = 0:n; p = hygepdf (v, m, k, n); r = v(lookup (cumsum (p(1:end-1)) / sum (p), rand (sz)) + 1); r = reshape (r, sz); if (strcmp (cls, "single")) r = single (r); endif else r = NaN (sz, cls); endif else r = NaN (sz, cls); n = n(ok); num_n = numel (n); v = 0 : max (n(:)); p = cumsum (hygepdf (v, m(ok), k(ok), n, "vectorexpand"), 2); ## Manual row-wise vectorization of lookup, which returns index of element ## less than or equal to test value, zero if test value is less than lowest ## number, and max index if greater than highest number. end_locs = sub2ind (size (p), [1 : num_n]', n(:) + 1); p = (p ./ p(end_locs)) - rand (num_n, 1); p(p>=0) = NaN; # NaN values ignored by max [p_match, p_match_idx] = max (p, [], 2); p_match_idx(isnan(p_match)) = 0; # rand < min(p) gives NaN, reset to 0 r(ok) = v(p_match_idx + 1); endif endfunction ## Test output %!assert (size (hygernd (4,2,2)), [1, 1]) %!assert (size (hygernd (4*ones (2,1), 2,2)), [2, 1]) %!assert (size (hygernd (4*ones (2,2), 2,2)), [2, 2]) %!assert (size (hygernd (4, 2*ones (2,1), 2)), [2, 1]) %!assert (size (hygernd (4, 2*ones (2,2), 2)), [2, 2]) %!assert (size (hygernd (4, 2, 2*ones (2,1))), [2, 1]) %!assert (size (hygernd (4, 2, 2*ones (2,2))), [2, 2]) %!assert (size (hygernd (4, 2, 2, 3)), [3, 3]) %!assert (size (hygernd (4, 2, 2, [4 1])), [4, 1]) %!assert (size (hygernd (4, 2, 2, 4, 1)), [4, 1]) ## Test class of input preserved %!assert (class (hygernd (4,2,2)), "double") %!assert (class (hygernd (single (4),2,2)), "single") %!assert (class (hygernd (single ([4 4]),2,2)), "single") %!assert (class (hygernd (4,single (2),2)), "single") %!assert (class (hygernd (4,single ([2 2]),2)), "single") %!assert (class (hygernd (4,2,single (2))), "single") %!assert (class (hygernd (4,2,single ([2 2]))), "single") ## Test input validation %!error hygernd () %!error hygernd (1) %!error hygernd (1, 2) %!error ... %! hygernd (ones (3), ones (2), ones (2)) %!error ... %! hygernd (ones (2), ones (3), ones (2)) %!error ... %! hygernd (ones (2), ones (2), ones (3)) %!error hygernd (i, 2, 3) %!error hygernd (1, i, 3) %!error hygernd (1, 2, i) %!error ... %! hygernd (1, 2, 3, -1) %!error ... %! hygernd (1, 2, 3, 1.2) %!error ... %! hygernd (1, 2, 3, ones (2)) %!error ... %! hygernd (1, 2, 3, [2 -1 2]) %!error ... %! hygernd (1, 2, 3, [2 0 2.5]) %!error ... %! hygernd (1, 2, 3, 2, -1, 5) %!error ... %! hygernd (1, 2, 3, 2, 1.5, 5) %!error ... %! hygernd (2, ones (2), 2, 3) %!error ... %! hygernd (2, ones (2), 2, [3, 2]) %!error ... %! hygernd (2, ones (2), 2, 3, 2) statistics-release-1.6.3/inst/dist_fun/invgcdf.m000066400000000000000000000143651456127120000217260ustar00rootroot00000000000000## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## Octave is free software; you can redistribute it and/or modify it ## under the terms of the GNU General Public License as published by ## the Free Software Foundation; either version 3 of the License, or (at ## your option) any later version. ## ## Octave is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with Octave; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{p} =} invgcdf (@var{x}, @var{mu}, @var{lambda}) ## @deftypefnx {statistics} {@var{p} =} invgcdf (@var{x}, @var{mu}, @var{lambda}, @qcode{"upper"}) ## ## Inverse Gaussian cumulative distribution function (CDF). ## ## For each element of @var{x}, compute the cumulative distribution function ## (CDF) of the inverse Gaussian distribution with scale parameter @var{mu} and ## shape parameter @var{lambda}. The size of @var{p} is the common size of ## @var{x}, @var{mu} and @var{lambda}. A scalar input functions as a constant ## matrix of the same size as the other inputs. ## ## @code{@var{p} = invgcdf (@var{x}, @var{mu}, @var{lambda}, "upper")} computes ## the upper tail probability of the inverse Gaussian distribution with ## parameters @var{mu} and @var{lambda}, at the values in @var{x}. ## ## The inverse Gaussian CDF is only defined for @qcode{@var{mu} > 0} and ## @qcode{@var{lambda} > 0}. ## ## Further information about the inverse Gaussian distribution can be found at ## @url{https://en.wikipedia.org/wiki/Inverse_Gaussian_distribution} ## ## @seealso{invginv, invgpdf, invgrnd, invgfit, invglike} ## @end deftypefn function p = invgcdf (x, mu, lambda, uflag) ## Check for valid number of input arguments if (nargin < 3) error ("invgcdf: function called with too few input arguments."); endif ## Check for valid "upper" flag if (nargin > 3) if (! strcmpi (uflag, "upper")) error ("invgcdf: invalid argument for upper tail."); else uflag = true; endif else uflag = false; endif ## Check for common size of X, MU, and LAMBDA if (! isscalar (x) || ! isscalar (mu) || ! isscalar (lambda)) [err, x, mu, lambda] = common_size (x, mu, lambda); if (err > 0) error ("invgcdf: X, MU, and LAMBDA must be of common size or scalars."); endif endif ## Check for X, MU, and LAMBDA being reals if (iscomplex (x) || iscomplex (mu) || iscomplex (lambda)) error ("invgcdf: X, MU, and LAMBDA must not be complex."); endif ## Check for class type if (isa (x, "single") || isa (mu, "single") || isa (lambda, "single")) is_class = "single"; else is_class = "double"; endif ## Prepare output p = zeros (size (x), is_class); ## Return NaNs for out of range values of MU and LAMBDA parameters mu(mu <= 0) = NaN; lambda(lambda <= 0) = NaN; ## Check for valid support of X is_zero = (x <= 0); x(is_zero) = realmin; is_inf = (x == Inf); ## Calculate z1, z2 z1 = sqrt (lambda ./ x) .* (x ./ mu - 1); z2 = -sqrt (lambda ./ x) .* (x ./ mu + 1); ## Compute the CDF if the inverse Gaussian if (uflag) p = 0.5 .* erfc (z1 ./ sqrt (2)) - ... exp (2 .* lambda ./ mu) .* 0.5 .* erfc (-z2 ./ sqrt (2)); p(is_zero) = 1; p(is_inf) = 0; else p = 0.5 .* erfc (-z1 ./ sqrt (2)) + ... exp (2 .* lambda ./ mu) .* 0.5 .* erfc (-z2 ./ sqrt (2)); p(is_zero) = 0; p(is_inf) = 1; endif endfunction %!demo %! ## Plot various CDFs from the inverse Gaussian distribution %! x = 0:0.001:3; %! p1 = invgcdf (x, 1, 0.2); %! p2 = invgcdf (x, 1, 1); %! p3 = invgcdf (x, 1, 3); %! p4 = invgcdf (x, 3, 0.2); %! p5 = invgcdf (x, 3, 1); %! plot (x, p1, "-b", x, p2, "-g", x, p3, "-r", x, p4, "-c", x, p5, "-y") %! grid on %! xlim ([0, 3]) %! legend ({"μ = 1, σ = 0.2", "μ = 1, σ = 1", "μ = 1, σ = 3", ... %! "μ = 3, σ = 0.2", "μ = 3, σ = 1"}, "location", "southeast") %! title ("Inverse Gaussian CDF") %! xlabel ("values in x") %! ylabel ("probability") ## Test output %!shared x, p1, p1u, y2, y2u, y3, y3u %! x = [-Inf, -1, 0, 1/2, 1, Inf]; %! p1 = [0, 0, 0, 0.3650, 0.6681, 1]; %! p1u = [1, 1, 1, 0.6350, 0.3319, 0]; %!assert (invgcdf (x, ones (1,6), ones (1,6)), p1, 1e-4) %!assert (invgcdf (x, 1, 1), p1, 1e-4) %!assert (invgcdf (x, 1, ones (1,6)), p1, 1e-4) %!assert (invgcdf (x, ones (1,6), 1), p1, 1e-4) %!assert (invgcdf (x, 1, [1, 1, 1, NaN, 1, 1]), [p1(1:3), NaN, p1(5:6)], 1e-4) %!assert (invgcdf (x, [1, 1, 1, NaN, 1, 1], 1), [p1(1:3), NaN, p1(5:6)], 1e-4) %!assert (invgcdf ([x(1:3), NaN, x(5:6)], 1, 1), [p1(1:3), NaN, p1(5:6)], 1e-4) %!assert (invgcdf (x, ones (1,6), ones (1,6), "upper"), p1u, 1e-4) %!assert (invgcdf (x, 1, 1, "upper"), p1u, 1e-4) %!assert (invgcdf (x, 1, ones (1,6), "upper"), p1u, 1e-4) %!assert (invgcdf (x, ones (1,6), 1, "upper"), p1u, 1e-4) ## Test class of input preserved %!assert (class (invgcdf (single ([x, NaN]), 1, 1)), "single") %!assert (class (invgcdf ([x, NaN], 1, single (1))), "single") %!assert (class (invgcdf ([x, NaN], single (1), 1)), "single") ## Test input validation %!error invgcdf () %!error invgcdf (1) %!error invgcdf (1, 2) %!error invgcdf (1, 2, 3, "tail") %!error invgcdf (1, 2, 3, 5) %!error ... %! invgcdf (ones (3), ones (2), ones(2)) %!error ... %! invgcdf (ones (2), ones (3), ones(2)) %!error ... %! invgcdf (ones (2), ones (2), ones(3)) %!error invgcdf (i, 2, 3) %!error invgcdf (1, i, 3) %!error invgcdf (1, 2, i) statistics-release-1.6.3/inst/dist_fun/invginv.m000066400000000000000000000151101456127120000217530ustar00rootroot00000000000000## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{x} =} invginv (@var{p}, @var{mu}, @var{lambda}) ## ## Inverse of the inverse Gaussian cumulative distribution function (iCDF). ## ## For each element of @var{p}, compute the quantile (the inverse of the CDF) of ## the inverse Gaussian distribution with scale parameter @var{mu} and shape ## parameter @var{lambda}. The size of @var{x} is the common size of @var{p}, ## @var{mu}, and @var{lambda}. A scalar input functions as a constant matrix of ## the same size as the other inputs. ## ## The inverse Gaussian CDF is only defined for @qcode{@var{mu} > 0} and ## @qcode{@var{lambda} > 0}. ## ## Further information about the inverse Gaussian distribution can be found at ## @url{https://en.wikipedia.org/wiki/Inverse_Gaussian_distribution} ## ## @seealso{invgcdf, invgpdf, invgrnd, invgfit, invglike} ## @end deftypefn function x = invginv (p, mu, lambda) ## Check for valid number of input arguments if (nargin < 3) error ("invginv: function called with too few input arguments."); endif ## Check for common size of P, MU, and LAMBDA if (! isscalar (p) || ! isscalar (mu) || ! isscalar(lambda)) [retval, p, mu, lambda] = common_size (p, mu, lambda); vec = true; if (retval > 0) error ("invginv: P, MU, and LAMBDA must be of common size or scalars."); endif else vec = false; endif ## Check for X, MU, and LAMBDA being reals if (iscomplex (p) || iscomplex (mu) || iscomplex (lambda)) error ("invginv: P, MU, and LAMBDA must not be complex."); endif ## Check for class type if (isa (p, "single") || isa (mu, "single") || isa (lambda, "single")); x = NaN (size (p), "single"); else x = NaN (size (p)); endif ## Return NaNs for out of range values of MU and LAMBDA parameters mu(mu <= 0) = NaN; lambda(lambda <= 0) = NaN; ## Find valid parameters and p-values (handle edges cases below) validmulambda = (mu > 0) & (lambda > 0) & (lambda < Inf); validp_values = (validmulambda & (p > 0) & (p < 1)); valid_all = all (validp_values(:)); valid_any = any (validp_values(:)); ## Handle edges cases here if (! valid_all) x(p == 0 & validmulambda) = 0; x(p == 1 & validmulambda) = Inf; ## Keep valid cases (if any left) if (valid_any) if (vec) p = p(validp_values); mu = mu(validp_values); lambda = lambda(validp_values); endif else return; endif endif ## Apply Newton's Method to find a root of p = invgcdf(x,mu,lambda) ## Choose a starting guess for x0. Use quantiles from a lognormal ## distribution with the same mean (==1) and variance (==lambda0) lambda0 = lambda ./ mu; lognorm = log (1 ./ lambda0 + 1); mulnorm = -0.5 .* lognorm; x0 = exp (mulnorm - sqrt (2 .* lognorm) .* erfcinv (2 * p)); ## Set maximum iterations and tolerance for Newton's Method mit = 500; tol = eps (class (x0)) .^ (3/4); ## Get quantiles F = invgcdf (x0, 1, lambda0); dF = F - p; for it = 1:mit ## Compute the Newton step f = invgpdf (x0, 1, lambda0); h = dF ./ f; x0_1 = max (x0/10, min (10 * x0, x0 - h)); ## Check if tolerance is reached complete = (abs (h) <= tol * x0); if (all (complete(:))) x0 = x0_1; break endif ## Check for increasing error unless tolerance is reached dFold = dF; for j = 1:25 F = invgcdf (x0_1, 1, lambda0); dF = F - p; worse = (abs (dF) > abs (dFold)) & ! complete; if (! any (worse(:))) break endif x0_1(worse) = (x0(worse) + x0_1(worse)) / 2; endfor ## Update for next step x0 = x0_1; endfor ## Issue a warning for exceeding iterations or not converging to tolerance notconv = (abs(dF./F) > tol.^(2/3)); if (it > mit || any (notconv(:))) warning (strcat (["invginv: Newton's Method did not converge"], ... [" or exceeded maximum iterations."])); endif ## Apply the scale factor if (valid_all) x = x0 .* mu; else x(validp_values) = x0 .* mu; endif endfunction %!demo %! ## Plot various iCDFs from the inverse Gaussian distribution %! p = 0.001:0.001:0.999; %! x1 = invginv (p, 1, 0.2); %! x2 = invginv (p, 1, 1); %! x3 = invginv (p, 1, 3); %! x4 = invginv (p, 3, 0.2); %! x5 = invginv (p, 3, 1); %! plot (p, x1, "-b", p, x2, "-g", p, x3, "-r", p, x4, "-c", p, x5, "-y") %! grid on %! ylim ([0, 3]) %! legend ({"μ = 1, σ = 0.2", "μ = 1, σ = 1", "μ = 1, σ = 3", ... %! "μ = 3, σ = 0.2", "μ = 3, σ = 1"}, "location", "northwest") %! title ("Inverse Gaussian iCDF") %! xlabel ("probability") %! ylabel ("x") ## Test output %!shared p, x %! p = [0, 0.3829, 0.6827, 1]; %! x = [0, 0.5207, 1.0376, Inf]; %!assert (invginv (p, 1, 1), x, 1e-4); %!assert (invginv (p, 1, ones (1,4)), x, 1e-4); %!assert (invginv (p, 1, [-1, 0, 1, 1]), [NaN, NaN, x(3:4)], 1e-4) %!assert (invginv (p, [-1, 0, 1, 1], 1), [NaN, NaN, x(3:4)], 1e-4) ## Test class of input preserved %!assert (class (invginv (single ([p, NaN]), 0, 1)), "single") %!assert (class (invginv ([p, NaN], single (0), 1)), "single") %!assert (class (invginv ([p, NaN], 0, single (1))), "single") ## Test input validation %!error invginv (1) %!error invginv (1, 2) %!error ... %! invginv (1, ones (2), ones (3)) %!error ... %! invginv (ones (2), 1, ones (3)) %!error ... %! invginv (ones (2), ones (3), 1) %!error invginv (i, 2, 3) %!error invginv (1, i, 3) %!error invginv (1, 2, i) statistics-release-1.6.3/inst/dist_fun/invgpdf.m000066400000000000000000000113431456127120000217340ustar00rootroot00000000000000## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{y} =} invgpdf (@var{x}, @var{mu}, @var{lambda}) ## ## Inverse Gaussian probability density function (PDF). ## ## For each element of @var{x}, compute the probability density function (PDF) ## of the inverse Gaussian distribution with scale parameter @var{mu} and shape ## parameter @var{lambda}. The size of @var{y} is the common size of @var{x}, ## @var{mu}, and @var{lambda}. A scalar input functions as a constant matrix of ## the same size as the other inputs. ## ## The inverse Gaussian CDF is only defined for @qcode{@var{mu} > 0} and ## @qcode{@var{lambda} > 0}. ## ## Further information about the inverse Gaussian distribution can be found at ## @url{https://en.wikipedia.org/wiki/Inverse_Gaussian_distribution} ## ## @seealso{invgcdf, invginv, invgrnd, invgfit, invglike} ## @end deftypefn function y = invgpdf (x, mu, lambda) ## Check for valid number of input arguments if (nargin < 3) error ("invgpdf: function called with too few input arguments."); endif ## Check for common size of X, MU, and LAMBDA if (! isscalar (x) || ! isscalar (mu) || ! isscalar(lambda)) [retval, x, mu, lambda] = common_size (x, mu, lambda); if (retval > 0) error ("invgpdf: X, MU, and LAMBDA must be of common size or scalars."); endif endif ## Check for X, MU, and LAMBDA being reals if (iscomplex (x) || iscomplex (mu) || iscomplex (lambda)) error ("invgpdf: X, MU, and LAMBDA must not be complex."); endif ## Check for class type if (isa (x, "single") || isa (mu, "single") || isa (lambda, "single")); is_class = "single"; else is_class = "double"; endif ## Return NaNs for out of range values of MU and LAMBDA parameters mu(mu <= 0) = NaN; lambda(lambda <= 0) = NaN; ## Check for valid support of X is_zero = (x <= 0); x(is_zero) = realmin; ## Compute inverse Gaussian PDF y = sqrt (lambda ./ (2 .* pi .* x .^ 3)) .* ... exp (-0.5 .* lambda .* (x ./ mu - 2 + mu ./ x) ./ mu); ## Force zero for unsupported X but valid parameters k0 = is_zero & mu > 0 & lambda > 0; y(k0) = 0; ## Cast to appropriate class y = cast (y, is_class); endfunction %!demo %! ## Plot various PDFs from the inverse Gaussian distribution %! x = 0:0.001:3; %! y1 = invgpdf (x, 1, 0.2); %! y2 = invgpdf (x, 1, 1); %! y3 = invgpdf (x, 1, 3); %! y4 = invgpdf (x, 3, 0.2); %! y5 = invgpdf (x, 3, 1); %! plot (x, y1, "-b", x, y2, "-g", x, y3, "-r", x, y4, "-c", x, y5, "-y") %! grid on %! xlim ([0, 3]) %! ylim ([0, 3]) %! legend ({"μ = 1, σ = 0.2", "μ = 1, σ = 1", "μ = 1, σ = 3", ... %! "μ = 3, σ = 0.2", "μ = 3, σ = 1"}, "location", "northeast") %! title ("Inverse Gaussian PDF") %! xlabel ("values in x") %! ylabel ("density") ## Test output %!shared x, y %! x = [-Inf, -1, 0, 1/2, 1, Inf]; %! y = [0, 0, 0, 0.8788, 0.3989, 0]; %!assert (invgpdf ([x, NaN], 1, 1), [y, NaN], 1e-4) %!assert (invgpdf (x, 1, [-2, -1, 0, 1, 1, 1]), [nan(1,3), y([4:6])], 1e-4) ## Test class of input preserved %!assert (class (hncdf (single ([x, NaN]), 1, 1)), "single") %!assert (class (hncdf ([x, NaN], 1, single (1))), "single") %!assert (class (hncdf ([x, NaN], single (1), 1)), "single") ## Test input validation %!error invgpdf () %!error invgpdf (1) %!error invgpdf (1, 2) %!error ... %! invgpdf (1, ones (2), ones (3)) %!error ... %! invgpdf (ones (2), 1, ones (3)) %!error ... %! invgpdf (ones (2), ones (3), 1) %!error invgpdf (i, 2, 3) %!error invgpdf (1, i, 3) %!error invgpdf (1, 2, i) statistics-release-1.6.3/inst/dist_fun/invgrnd.m000066400000000000000000000161111456127120000217440ustar00rootroot00000000000000## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{r} =} invgrnd (@var{mu}, @var{lambda}) ## @deftypefnx {statistics} {@var{r} =} invgrnd (@var{mu}, @var{lambda}, @var{rows}) ## @deftypefnx {statistics} {@var{r} =} invgrnd (@var{mu}, @var{lambda}, @var{rows}, @var{cols}, @dots{}) ## @deftypefnx {statistics} {@var{r} =} invgrnd (@var{mu}, @var{lambda}, [@var{sz}]) ## ## Random arrays from the inverse Gaussian distribution. ## ## @code{@var{r} = invgrnd (@var{mu}, @var{lambda})} returns an array of random ## numbers chosen from the inverse Gaussian distribution with location parameter ## @var{mu} and scale parameter @var{lambda}. The size of @var{r} is the common ## size of @var{mu} and @var{lambda}. A scalar input functions as a constant ## matrix of the same size as the other inputs. ## ## When called with a single size argument, @code{invgrnd} returns a square ## matrix with the dimension specified. When called with more than one scalar ## argument, the first two arguments are taken as the number of rows and columns ## and any further arguments specify additional matrix dimensions. The size may ## also be specified with a row vector of dimensions, @var{sz}. ## ## The inverse Gaussian CDF is only defined for @qcode{@var{mu} > 0} and ## @qcode{@var{lambda} > 0}. ## ## Further information about the inverse Gaussian distribution can be found at ## @url{https://en.wikipedia.org/wiki/Inverse_Gaussian_distribution} ## ## @seealso{invgcdf, invginv, invgpdf, invgfit, invglike} ## @end deftypefn function r = invgrnd (mu, lambda, varargin) ## Check for valid number of input arguments if (nargin < 2) error ("invgrnd: function called with too few input arguments."); endif ## Check for common size of MU, and LAMBDA if (! isscalar (mu) || ! isscalar (lambda)) [retval, mu, lambda] = common_size (mu, lambda); vec = true; if (retval > 0) error ("invgrnd: MU and LAMBDA must be of common size or scalars."); endif else vec = false; endif ## Check for X, MU, and LAMBDA being reals if (iscomplex (mu) || iscomplex (lambda)) error ("invgrnd: MU and LAMBDA must not be complex."); endif ## Parse and check SIZE arguments if (nargin == 2) sz = size (mu); elseif (nargin == 3) if (isscalar (varargin{1}) && varargin{1} >= 0 ... && varargin{1} == fix (varargin{1})) sz = [varargin{1}, varargin{1}]; elseif (isrow (varargin{1}) && all (varargin{1} >= 0) ... && all (varargin{1} == fix (varargin{1}))) sz = varargin{1}; elseif error (strcat (["invgrnd: SZ must be a scalar or a row vector"], ... [" of non-negative integers."])); endif elseif (nargin > 3) posint = cellfun (@(x) (! isscalar (x) || x < 0 || x != fix (x)), varargin); if (any (posint)) error ("invgrnd: dimensions must be non-negative integers."); endif sz = [varargin{:}]; endif ## Check that parameters match requested dimensions in size if (! isscalar (mu) && ! isequal (size (mu), sz)) error ("invgrnd: MU and LAMBDA must be scalars or of size SZ."); endif ## Check for class type if (isa (mu, "single") || isa (lambda, "single")) cls = "single"; else cls = "double"; endif ## Expand parameters (if needed) if (! vec) mu = repmat (mu, sz); lambda = repmat (lambda, sz); endif ## Generate random sample from inverse Gaussian distribution v = randn (sz, cls); y = v .^ 2; r = mu + (mu .^ 2 .* y) ./ (2 .* lambda) - (mu ./ (2 .* lambda)) .* ... sqrt (4 * mu .* lambda .* y + mu .* mu .* y .* y); inver = (rand (sz) .* (mu + r) > mu); r(inver) = mu(inver) .^2 ./ r(inver); ## Force output to NaN for invalid parameters MU and LAMBDA k = (mu <= 0 | lambda <= 0); r(k) = NaN; endfunction ## Test results %!assert (size (invgrnd (1, 1, 1)), [1, 1]) %!assert (size (invgrnd (1, 1, 2)), [2, 2]) %!assert (size (invgrnd (1, 1, [2, 1])), [2, 1]) %!assert (size (invgrnd (1, zeros (2, 2))), [2, 2]) %!assert (size (invgrnd (1, ones (2, 1))), [2, 1]) %!assert (size (invgrnd (1, ones (2, 2))), [2, 2]) %!assert (size (invgrnd (ones (2, 1), 1)), [2, 1]) %!assert (size (invgrnd (ones (2, 2), 1)), [2, 2]) %!assert (size (invgrnd (1, 1, 3)), [3, 3]) %!assert (size (invgrnd (1, 1, [4 1])), [4, 1]) %!assert (size (invgrnd (1, 1, 4, 1)), [4, 1]) %!test %! r = invgrnd (1, [1, 0, -1]); %! assert (r([2:3]), [NaN, NaN]) ## Test class of input preserved %!assert (class (invgrnd (1, 0)), "double") %!assert (class (invgrnd (1, single (0))), "single") %!assert (class (invgrnd (1, single ([0 0]))), "single") %!assert (class (invgrnd (1, single (1))), "single") %!assert (class (invgrnd (1, single ([1 1]))), "single") %!assert (class (invgrnd (single (1), 1)), "single") %!assert (class (invgrnd (single ([1 1]), 1)), "single") ## Test input validation %!error invgrnd () %!error invgrnd (1) %!error ... %! invgrnd (ones (3), ones (2)) %!error ... %! invgrnd (ones (2), ones (3)) %!error invgrnd (i, 2, 3) %!error invgrnd (1, i, 3) %!error ... %! invgrnd (1, 2, -1) %!error ... %! invgrnd (1, 2, 1.2) %!error ... %! invgrnd (1, 2, ones (2)) %!error ... %! invgrnd (1, 2, [2 -1 2]) %!error ... %! invgrnd (1, 2, [2 0 2.5]) %!error ... %! invgrnd (1, 2, 2, -1, 5) %!error ... %! invgrnd (1, 2, 2, 1.5, 5) %!error ... %! invgrnd (2, ones (2), 3) %!error ... %! invgrnd (2, ones (2), [3, 2]) %!error ... %! invgrnd (2, ones (2), 3, 2) statistics-release-1.6.3/inst/dist_fun/iwishpdf.m000066400000000000000000000054101456127120000221120ustar00rootroot00000000000000## Copyright (C) 2013 Nir Krakauer ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{y} =} iwishpdf (@var{W}, @var{Tau}, @var{df}, @var{log_y}=false) ## ## Compute the probability density function of the inverse Wishart distribution. ## ## Inputs: A @var{p} x @var{p} matrix @var{W} where to find the PDF and the ## @var{p} x @var{p} positive definite scale matrix @var{Tau} and scalar degrees ## of freedom parameter @var{df} characterizing the inverse Wishart distribution. ## (For the density to be finite, need @var{df} > (@var{p} - 1).) ## If the flag @var{log_y} is set, return the log probability density -- this ## helps avoid underflow when the numerical value of the density is very small. ## ## Output: @var{y} is the probability density of Wishart(@var{Sigma}, @var{df}) ## at @var{W}. ## ## @seealso{iwishrnd, wishpdf, wishrnd} ## @end deftypefn function y = iwishpdf (W, Tau, df, log_y=false) if (nargin < 3) print_usage (); endif p = size (Tau, 1); if (df <= (p - 1)) error ("iwishpdf: DF too small, no finite densities exist."); endif ## Calculate the logarithm of G_d(df/2), the multivariate gamma function g = (p * (p - 1) / 4) * log (pi); for i = 1:p g = g + log (gamma ((df + (1 - i)) / 2)); endfor C = chol (W); ## Use formulas for determinant of positive definite matrix for better ## efficiency and numerical accuracy logdet_W = 2*sum(log(diag(C))); logdet_Tau = 2*sum(log(diag(chol(Tau)))); y = -(df * p) / 2 * log (2) + (df / 2) * logdet_Tau - g ... -((df + p + 1) / 2) * logdet_W - trace (Tau * chol2inv (C)) / 2; if (! log_y) y = exp (y); endif endfunction ## Test results cross-checked against diwish function in R MCMCpack library %!assert(iwishpdf(4, 3, 3.1), 0.04226595, 1E-7); %!assert(iwishpdf([2 -0.3;-0.3 4], [1 0.3;0.3 1], 4), 1.60166e-05, 1E-10); %!assert(iwishpdf([6 2 5; 2 10 -5; 5 -5 25], ... %! [9 5 5; 5 10 -8; 5 -8 22], 5.1), 4.946831e-12, 1E-17); ## Test input validation %!error iwishpdf () %!error iwishpdf (1, 2) %!error iwishpdf (1, 2, 0) statistics-release-1.6.3/inst/dist_fun/iwishrnd.m000066400000000000000000000057151456127120000221340ustar00rootroot00000000000000## Copyright (C) 2013 Nir Krakauer ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {[@var{W}, @var{DI}] =} iwishrnd (@var{Tau}, @var{df}, @var{DI}, @var{n}=1) ## ## Return a random matrix sampled from the inverse Wishart distribution with ## given parameters. ## ## Inputs: the @math{p x p} positive definite matrix @var{Tau} and scalar ## degrees of freedom parameter @var{df} (and optionally the transposed Cholesky ## factor @var{DI} of @var{Sigma} = @code{inv(Tau)}). ## ## @var{df} can be non-integer as long as @math{@var{df} > d} ## ## Output: a random @math{p x p} matrix @var{W} from the inverse ## Wishart(@var{Tau}, @var{df}) distribution. (@code{inv(W)} is from the ## Wishart(@code{inv(Tau)}, @var{df}) distribution.) If @var{n} > 1, ## then @var{W} is @var{p} x @var{p} x @var{n} and holds @var{n} such random ## matrices. (Optionally, the transposed Cholesky factor @var{DI} of @var{Sigma} ## is also returned.) ## ## Averaged across many samples, the mean of @var{W} should approach ## @var{Tau} / (@var{df} - @var{p} - 1). ## ## @subheading References ## ## @enumerate ## @item ## Yu-Cheng Ku and Peter Bloomfield (2010), Generating Random Wishart Matrices ## with Fractional Degrees of Freedom in OX, ## http://www.gwu.edu/~forcpgm/YuChengKu-030510final-WishartYu-ChengKu.pdf ## @end enumerate ## ## @seealso{iwishpdf, wishpdf, wishrnd} ## @end deftypefn function [W, DI] = iwishrnd (Tau, df, DI, n = 1) if (nargin < 2) print_usage (); endif if (nargin < 3 || isempty (DI)) try D = chol (inv (Tau)); catch error (strcat (["iwishrnd: Cholesky decomposition failed;"], ... [" TAU probably not positive definite."])); end_try_catch DI = D'; else D = DI'; endif w = wishrnd ([], df, D, n); if (n > 1) p = size (D, 1); W = nan (p, p, n); endif for i = 1:n W(:, :, i) = inv (w(:, :, i)); endfor endfunction %!assert(size (iwishrnd (1,2,1)), [1, 1]); %!assert(size (iwishrnd ([],2,1)), [1, 1]); %!assert(size (iwishrnd ([3 1; 1 3], 2.00001, [], 1)), [2, 2]); %!assert(size (iwishrnd (eye(2), 2, [], 3)), [2, 2, 3]); %% Test input validation %!error iwishrnd () %!error iwishrnd (1) %!error iwishrnd ([-3 1; 1 3],1) %!error iwishrnd ([1; 1],1) statistics-release-1.6.3/inst/dist_fun/jsucdf.m000066400000000000000000000045211456127120000215550ustar00rootroot00000000000000## Copyright (C) 2006 Frederick (Rick) A Niles ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{p} =} jsucdf (@var{x}) ## @deftypefnx {statistics} {@var{p} =} jsucdf (@var{x}, @var{alpha1}) ## @deftypefnx {statistics} {@var{p} =} jsucdf (@var{x}, @var{alpha1}, @var{alpha2}) ## ## Johnson SU cumulative distribution function (CDF). ## ## For each element of @var{x}, return the cumulative distribution functions ## (CDF) at @var{x} of the Johnson SU distribution with shape parameters ## @var{alpha1} and @var{alpha2}. The size of @var{p} is the common size of the ## input arguments @var{x}, @var{alpha1}, and @var{alpha2}. A scalar input ## functions as a constant matrix of the same size as the other ## ## Default values are @var{alpha1} = 1, @var{alpha2} = 1. ## ## @seealso{jsupdf} ## @end deftypefn function p = jsucdf (x, alpha1, alpha2) if (nargin < 1 || nargin > 3) print_usage; endif if (nargin == 1) alpha1 = 1; alpha2 = 1; elseif (nargin == 2) alpha2 = 1; endif if (! isscalar (x) || ! isscalar (alpha1) || ! isscalar(alpha2)) [retval, x, alpha1, alpha2] = common_size (x, alpha1, alpha2); if (retval > 0) error (strcat (["jsucdf: X, ALPHA1, and ALPHA2 must be of common"], ... [" size or scalars."])); endif endif one = ones (size (x)); p = stdnormal_cdf (alpha1 .* one + alpha2 .* log (x + sqrt (x .* x + one))); endfunction %!error jsucdf () %!error jsucdf (1, 2, 3, 4) %!error ... %! jsucdf (1, ones (2), ones (3)) statistics-release-1.6.3/inst/dist_fun/jsupdf.m000066400000000000000000000045611456127120000215760ustar00rootroot00000000000000## Copyright (C) 2006 Frederick (Rick) A Niles ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{y} =} jsupdf (@var{x}) ## @deftypefnx {statistics} {@var{y} =} jsupdf (@var{x}, @var{alpha1}) ## @deftypefnx {statistics} {@var{y} =} jsupdf (@var{x}, @var{alpha1}, @var{alpha2}) ## ## Johnson SU probability density function (PDF). ## ## For each element of @var{x}, compute the probability density function (PDF) ## at @var{x} of the Johnson SU distribution with shape parameters @var{alpha1} ## and @var{alpha2}. The size of @var{p} is the common size of the input ## arguments @var{x}, @var{alpha1}, and @var{alpha2}. A scalar input functions ## as a constant matrix of the same size as the other ## ## Default values are @var{alpha1} = 1, @var{alpha2} = 1. ## ## @seealso{jsucdf} ## @end deftypefn function y = jsupdf (x, alpha1, alpha2) if (nargin < 1 || nargin > 3) print_usage; endif if (nargin == 1) alpha1 = 1; alpha2 = 1; elseif (nargin == 2) alpha2 = 1; endif if (! isscalar (x) || ! isscalar (alpha1) || ! isscalar(alpha2)) [retval, x, alpha1, alpha2] = common_size (x, alpha1, alpha2); if (retval > 0) error (strcat (["jsupdf: X, ALPHA1, and ALPHA2 must be of common"], ... [" size or scalars."])); endif endif one = ones (size (x)); sr = sqrt (x .* x + one); y = (alpha2 ./ sr) .* ... stdnormal_pdf (alpha1 .* one + alpha2 .* log (x + sr)); endfunction %!error jsupdf () %!error jsupdf (1, 2, 3, 4) %!error ... %! jsupdf (1, ones (2), ones (3)) statistics-release-1.6.3/inst/dist_fun/laplacecdf.m000066400000000000000000000131451456127120000223570ustar00rootroot00000000000000## Copyright (C) 2012 Rik Wehbring ## Copyright (C) 1995-2016 Kurt Hornik ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{p} =} laplacecdf (@var{x}, @var{mu}, @var{beta}) ## @deftypefnx {statistics} {@var{p} =} laplacecdf (@var{x}, @var{mu}, @var{beta}, @qcode{"upper"}) ## ## Laplace cumulative distribution function (CDF). ## ## For each element of @var{x}, compute the cumulative distribution function ## (CDF) of the Laplace distribution with location parameter @var{mu} and scale ## parameter (i.e. "diversity") @var{beta}. The size of @var{p} is the common ## size of @var{x}, @var{mu}, and @var{beta}. A scalar input functions as a ## constant matrix of the same size as the other inputs. ## ## Both parameters must be reals and @qcode{@var{beta} > 0}. ## For @qcode{@var{beta} <= 0}, @qcode{NaN} is returned. ## ## @code{@var{p} = laplacecdf (@var{x}, @var{mu}, @var{beta}, "upper")} computes ## the upper tail probability of the Laplace distribution with parameters ## @var{mu} and @var{beta}, at the values in @var{x}. ## ## Further information about the Laplace distribution can be found at ## @url{https://en.wikipedia.org/wiki/Laplace_distribution} ## ## @seealso{laplaceinv, laplacepdf, laplacernd} ## @end deftypefn function p = laplacecdf (x, mu, beta, uflag) ## Check for valid number of input arguments if (nargin < 3) error ("laplacecdf: function called with too few input arguments."); endif ## Check for valid "upper" flag if (nargin > 3) if (! strcmpi (uflag, "upper")) error ("laplacecdf: invalid argument for upper tail."); else uflag = true; endif else uflag = false; endif ## Check for common size of X, MU, and BETA if (! isscalar (x) || ! isscalar (mu) || ! isscalar(beta)) [retval, x, mu, beta] = common_size (x, mu, beta); if (retval > 0) error (strcat (["laplacecdf: X, MU, and BETA must be of"], ... [" common size or scalars."])); endif endif ## Check for X, MU, and BETA being reals if (iscomplex (x) || iscomplex (mu) || iscomplex (beta)) error ("laplacecdf: X, MU, and BETA must not be complex."); endif ## Check for class type if (isa (x, "single") || isa (mu, "single") || isa (beta, "single")); p = NaN (size (x), "single"); else p = NaN (size (x)); endif ## Find normal and edge cases k1 = (x == -Inf) & (beta > 0); k2 = (x == Inf) & (beta > 0); k = ! k1 & ! k2 & (beta > 0); ## Compute Laplace CDF if (uflag) p(k1) = 1; p(k2) = 0; p(k) = (1 + sign (-x(k) + mu(k)) .* ... (1 - exp (- abs (-x(k) + mu(k)) ./ beta(k)))) ./ 2; else p(k1) = 0; p(k2) = 1; p(k) = (1 + sign (x(k) - mu(k)) .* ... (1 - exp (- abs (x(k) - mu(k)) ./ beta(k)))) ./ 2; endif endfunction %!demo %! ## Plot various CDFs from the Laplace distribution %! x = -10:0.01:10; %! p1 = laplacecdf (x, 0, 1); %! p2 = laplacecdf (x, 0, 2); %! p3 = laplacecdf (x, 0, 4); %! p4 = laplacecdf (x, -5, 4); %! plot (x, p1, "-b", x, p2, "-g", x, p3, "-r", x, p4, "-c") %! grid on %! xlim ([-10, 10]) %! legend ({"μ = 0, β = 1", "μ = 0, β = 2", ... %! "μ = 0, β = 4", "μ = -5, β = 4"}, "location", "southeast") %! title ("Laplace CDF") %! xlabel ("values in x") %! ylabel ("probability") ## Test output %!shared x, y %! x = [-Inf, -log(2), 0, log(2), Inf]; %! y = [0, 1/4, 1/2, 3/4, 1]; %!assert (laplacecdf ([x, NaN], 0, 1), [y, NaN]) %!assert (laplacecdf (x, 0, [-2, -1, 0, 1, 2]), [nan(1, 3), 0.75, 1]) ## Test class of input preserved %!assert (laplacecdf (single ([x, NaN]), 0, 1), single ([y, NaN]), eps ("single")) %!assert (laplacecdf ([x, NaN], single (0), 1), single ([y, NaN]), eps ("single")) %!assert (laplacecdf ([x, NaN], 0, single (1)), single ([y, NaN]), eps ("single")) ## Test input validation %!error laplacecdf () %!error laplacecdf (1) %!error ... %! laplacecdf (1, 2) %!error ... %! laplacecdf (1, 2, 3, 4, 5) %!error laplacecdf (1, 2, 3, "tail") %!error laplacecdf (1, 2, 3, 4) %!error ... %! laplacecdf (ones (3), ones (2), ones (2)) %!error ... %! laplacecdf (ones (2), ones (3), ones (2)) %!error ... %! laplacecdf (ones (2), ones (2), ones (3)) %!error laplacecdf (i, 2, 2) %!error laplacecdf (2, i, 2) %!error laplacecdf (2, 2, i) statistics-release-1.6.3/inst/dist_fun/laplaceinv.m000066400000000000000000000112001456127120000224050ustar00rootroot00000000000000## Copyright (C) 2012 Rik Wehbring ## Copyright (C) 1995-2016 Kurt Hornik ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{x} =} laplaceinv (@var{p}, @var{mu}, @var{beta}) ## ## Inverse of the Laplace cumulative distribution function (iCDF). ## ## For each element of @var{p}, compute the quantile (the inverse of the CDF) of ## the Laplace distribution with location parameter @var{mu} and scale parameter ## (i.e. "diversity") @var{beta}. The size of @var{x} is the common size of ## @var{p}, @var{mu}, and @var{beta}. A scalar input functions as a constant ## matrix of the same size as the other inputs. ## ## Both parameters must be reals and @qcode{@var{beta} > 0}. ## For @qcode{@var{beta} <= 0}, @qcode{NaN} is returned. ## ## Further information about the Laplace distribution can be found at ## @url{https://en.wikipedia.org/wiki/Laplace_distribution} ## ## @seealso{laplaceinv, laplacepdf, laplacernd} ## @end deftypefn function x = laplaceinv (p, mu, beta) ## Check for valid number of input arguments if (nargin < 3) error ("laplaceinv: function called with too few input arguments."); endif ## Check for common size of P, MU, and BETA if (! isscalar (p) || ! isscalar (mu) || ! isscalar(beta)) [retval, p, mu, beta] = common_size (p, mu, beta); if (retval > 0) error (strcat (["laplaceinv: P, MU, and BETA must be of"], ... [" common size or scalars."])); endif endif ## Check for X, MU, and BETA being reals if (iscomplex (p) || iscomplex (mu) || iscomplex (beta)) error ("laplaceinv: P, MU, and BETA must not be complex."); endif ## Check for class type if (isa (p, "single") || isa (mu, "single") || isa (beta, "single")); x = NaN (size (p), "single"); else x = NaN (size (p)); endif ## Compute Laplace iCDF k = (p >= 0) & (p <= 1) & (beta > 0); x(k) = mu(k) + beta(k) .* ((p(k) < 1/2) .* log (2 .* p(k)) - ... (p(k) > 1/2) .* log (2 .* (1 - p(k)))); endfunction %!demo %! ## Plot various iCDFs from the Laplace distribution %! p = 0.001:0.001:0.999; %! x1 = cauchyinv (p, 0, 1); %! x2 = cauchyinv (p, 0, 2); %! x3 = cauchyinv (p, 0, 4); %! x4 = cauchyinv (p, -5, 4); %! plot (p, x1, "-b", p, x2, "-g", p, x3, "-r", p, x4, "-c") %! grid on %! ylim ([-10, 10]) %! legend ({"μ = 0, β = 1", "μ = 0, β = 2", ... %! "μ = 0, β = 4", "μ = -5, β = 4"}, "location", "northwest") %! title ("Laplace iCDF") %! xlabel ("probability") %! ylabel ("values in x") ## Test output %!shared p, x %! p = [-1 0 0.5 1 2]; %! x = [NaN, -Inf, 0, Inf, NaN]; %!assert (laplaceinv (p, 0, 1), x) %!assert (laplaceinv (p, 0, [-2, -1, 0, 1, 2]), [nan(1, 3), Inf, NaN]) %!assert (laplaceinv ([p, NaN], 0, 1), [x, NaN]) ## Test class of input preserved %!assert (laplaceinv (single ([p, NaN]), 0, 1), single ([x, NaN])) %!assert (laplaceinv ([p, NaN], single (0), 1), single ([x, NaN])) %!assert (laplaceinv ([p, NaN], 0, single (1)), single ([x, NaN])) ## Test input validation %!error laplaceinv () %!error laplaceinv (1) %!error ... %! laplaceinv (1, 2) %!error laplaceinv (1, 2, 3, 4) %!error ... %! laplaceinv (1, ones (2), ones (3)) %!error ... %! laplaceinv (ones (2), 1, ones (3)) %!error ... %! laplaceinv (ones (2), ones (3), 1) %!error laplaceinv (i, 2, 3) %!error laplaceinv (1, i, 3) %!error laplaceinv (1, 2, i) statistics-release-1.6.3/inst/dist_fun/laplacepdf.m000066400000000000000000000111261456127120000223710ustar00rootroot00000000000000## Copyright (C) 2012 Rik Wehbring ## Copyright (C) 1995-2016 Kurt Hornik ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{y} =} laplacepdf (@var{x}, @var{mu}, @var{beta}) ## ## Laplace probability density function (PDF). ## ## For each element of @var{x}, compute the probability density function (PDF) ## of the Laplace distribution with location parameter @var{mu} and scale ## parameter (i.e. "diversity") @var{beta}. The size of @var{y} is the common ## size of @var{x}, @var{mu}, and @var{beta}. A scalar input functions as a ## constant matrix of the same size as the other inputs. ## ## Both parameters must be reals and @qcode{@var{beta} > 0}. ## For @qcode{@var{beta} <= 0}, @qcode{NaN} is returned. ## ## Further information about the Laplace distribution can be found at ## @url{https://en.wikipedia.org/wiki/Laplace_distribution} ## ## @seealso{laplacecdf, laplacepdf, laplacernd} ## @end deftypefn function y = laplacepdf (x, mu, beta) ## Check for valid number of input arguments if (nargin < 3) error ("laplacepdf: function called with too few input arguments."); endif ## Check for common size of X, MU, and BETA if (! isscalar (x) || ! isscalar (mu) || ! isscalar(beta)) [retval, x, mu, beta] = common_size (x, mu, beta); if (retval > 0) error (strcat (["laplacepdf: X, MU, and BETA must be of"], ... [" common size or scalars."])); endif endif ## Check for X, MU, and BETA being reals if (iscomplex (x) || iscomplex (mu) || iscomplex (beta)) error ("laplacepdf: X, MU, and BETA must not be complex."); endif ## Check for class type if (isa (x, "single") || isa (mu, "single") || isa (beta, "single")); y = NaN (size (x), "single"); else y = NaN (size (x)); endif ## Compute Laplace PDF k1 = ((x == -Inf) & (beta > 0)) | ((x == Inf) & (beta > 0)); y(k1) = 0; k = ! k1 & (beta > 0); y(k) = exp (- abs (x(k) - mu(k)) ./ beta(k)) ./ (2 .* beta(k)); endfunction %!demo %! ## Plot various PDFs from the Laplace distribution %! x = -10:0.01:10; %! y1 = laplacepdf (x, 0, 1); %! y2 = laplacepdf (x, 0, 2); %! y3 = laplacepdf (x, 0, 4); %! y4 = laplacepdf (x, -5, 4); %! plot (x, y1, "-b", x, y2, "-g", x, y3, "-r", x, y4, "-c") %! grid on %! xlim ([-10, 10]) %! ylim ([0, 0.6]) %! legend ({"μ = 0, β = 1", "μ = 0, β = 2", ... %! "μ = 0, β = 4", "μ = -5, β = 4"}, "location", "northeast") %! title ("Laplace PDF") %! xlabel ("values in x") %! ylabel ("density") ## Test results %!shared x, y %! x = [-Inf -log(2) 0 log(2) Inf]; %! y = [0, 1/4, 1/2, 1/4, 0]; %!assert (laplacepdf ([x, NaN], 0, 1), [y, NaN]) %!assert (laplacepdf (x, 0, [-2, -1, 0, 1, 2]), [nan(1, 3), 0.25, 0]) ## Test class of input preserved %!assert (laplacepdf (single ([x, NaN]), 0, 1), single ([y, NaN])) %!assert (laplacepdf ([x, NaN], single (0), 1), single ([y, NaN])) %!assert (laplacepdf ([x, NaN], 0, single (1)), single ([y, NaN])) ## Test input validation %!error laplacepdf () %!error laplacepdf (1) %!error ... %! laplacepdf (1, 2) %!error laplacepdf (1, 2, 3, 4) %!error ... %! laplacepdf (1, ones (2), ones (3)) %!error ... %! laplacepdf (ones (2), 1, ones (3)) %!error ... %! laplacepdf (ones (2), ones (3), 1) %!error laplacepdf (i, 2, 3) %!error laplacepdf (1, i, 3) %!error laplacepdf (1, 2, i) statistics-release-1.6.3/inst/dist_fun/laplacernd.m000066400000000000000000000155171456127120000224130ustar00rootroot00000000000000## Copyright (C) 2012 Rik Wehbring ## Copyright (C) 1995-2016 Kurt Hornik ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{r} =} laplacernd (@var{mu}, @var{beta}) ## @deftypefnx {statistics} {@var{r} =} laplacernd (@var{mu}, @var{beta}, @var{rows}) ## @deftypefnx {statistics} {@var{r} =} laplacernd (@var{mu}, @var{beta}, @var{rows}, @var{cols}, @dots{}) ## @deftypefnx {statistics} {@var{r} =} laplacernd (@var{mu}, @var{beta}, [@var{sz}]) ## ## Random arrays from the Laplace distribution. ## ## @code{@var{r} = laplacernd (@var{mu}, @var{beta})} returns an array of ## random numbers chosen from the Laplace distribution with location parameter ## @var{mu} and scale parameter @var{beta}. The size of @var{r} is the common ## size of @var{mu} and @var{beta}. A scalar input functions as a constant ## matrix of the same size as the other inputs. ## ## Both parameters must be reals and @qcode{@var{beta} > 0}. ## For @qcode{@var{beta} <= 0}, @qcode{NaN} is returned. ## ## When called with a single size argument, @code{laplacernd} returns a square ## matrix with the dimension specified. When called with more than one scalar ## argument, the first two arguments are taken as the number of rows and columns ## and any further arguments specify additional matrix dimensions. The size may ## also be specified with a row vector of dimensions, @var{sz}. ## ## Further information about the Laplace distribution can be found at ## @url{https://en.wikipedia.org/wiki/Laplace_distribution} ## ## @seealso{laplacecdf, laplaceinv, laplacernd} ## @end deftypefn function r = laplacernd (mu, beta, varargin) ## Check for valid number of input arguments if (nargin < 2) error ("laplacernd: function called with too few input arguments."); endif ## Check for common size of MU, and BETA if (! isscalar (mu) || ! isscalar (beta)) [retval, mu, beta] = common_size (mu, beta); if (retval > 0) error ("laplacernd: MU and BETA must be of common size or scalars."); endif endif ## Check for X, MU, and BETA being reals if (iscomplex (mu) || iscomplex (beta)) error ("laplacernd: MU and BETA must not be complex."); endif ## Parse and check SIZE arguments if (nargin == 2) sz = size (mu); elseif (nargin == 3) if (isscalar (varargin{1}) && varargin{1} >= 0 ... && varargin{1} == fix (varargin{1})) sz = [varargin{1}, varargin{1}]; elseif (isrow (varargin{1}) && all (varargin{1} >= 0) ... && all (varargin{1} == fix (varargin{1}))) sz = varargin{1}; elseif error (strcat (["laplacernd: SZ must be a scalar or a row vector"], ... [" of non-negative integers."])); endif elseif (nargin > 3) posint = cellfun (@(x) (! isscalar (x) || x < 0 || x != fix (x)), varargin); if (any (posint)) error ("laplacernd: dimensions must be non-negative integers."); endif sz = [varargin{:}]; endif ## Check that parameters match requested dimensions in size if (! isscalar (mu) && ! isequal (size (mu), sz)) error ("laplacernd: MU and BETA must be scalars or of size SZ."); endif ## Check for class type if (isa (mu, "single") || isa (beta, "single")) is_type = "single"; else is_type = "double"; endif ## Generate random sample from Laplace distribution tmp = rand (sz, is_type); r = ((tmp < 1/2) .* log (2 * tmp) - ... (tmp > 1/2) .* log (2 * (1 - tmp))) .* beta + mu; ## Force output to NaN for invalid parameter BETA <= 0 k = (beta <= 0); r(k) = NaN; endfunction ## Test output %!assert (size (laplacernd (1, 1)), [1 1]) %!assert (size (laplacernd (1, ones (2,1))), [2, 1]) %!assert (size (laplacernd (1, ones (2,2))), [2, 2]) %!assert (size (laplacernd (ones (2,1), 1)), [2, 1]) %!assert (size (laplacernd (ones (2,2), 1)), [2, 2]) %!assert (size (laplacernd (1, 1, 3)), [3, 3]) %!assert (size (laplacernd (1, 1, [4, 1])), [4, 1]) %!assert (size (laplacernd (1, 1, 4, 1)), [4, 1]) %!assert (size (laplacernd (1, 1, 4, 1, 5)), [4, 1, 5]) %!assert (size (laplacernd (1, 1, 0, 1)), [0, 1]) %!assert (size (laplacernd (1, 1, 1, 0)), [1, 0]) %!assert (size (laplacernd (1, 1, 1, 2, 0, 5)), [1, 2, 0, 5]) ## Test class of input preserved %!assert (class (laplacernd (1, 1)), "double") %!assert (class (laplacernd (1, single (1))), "single") %!assert (class (laplacernd (1, single ([1, 1]))), "single") %!assert (class (laplacernd (single (1), 1)), "single") %!assert (class (laplacernd (single ([1, 1]), 1)), "single") ## Test input validation %!error laplacernd () %!error laplacernd (1) %!error ... %! laplacernd (ones (3), ones (2)) %!error ... %! laplacernd (ones (2), ones (3)) %!error laplacernd (i, 2, 3) %!error laplacernd (1, i, 3) %!error ... %! laplacernd (1, 2, -1) %!error ... %! laplacernd (1, 2, 1.2) %!error ... %! laplacernd (1, 2, ones (2)) %!error ... %! laplacernd (1, 2, [2 -1 2]) %!error ... %! laplacernd (1, 2, [2 0 2.5]) %!error ... %! laplacernd (1, 2, 2, -1, 5) %!error ... %! laplacernd (1, 2, 2, 1.5, 5) %!error ... %! laplacernd (2, ones (2), 3) %!error ... %! laplacernd (2, ones (2), [3, 2]) %!error ... %! laplacernd (2, ones (2), 3, 2) statistics-release-1.6.3/inst/dist_fun/logicdf.m000066400000000000000000000123561456127120000217130ustar00rootroot00000000000000## Copyright (C) 2012 Rik Wehbring ## Copyright (C) 1995-2016 Kurt Hornik ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{p} =} logicdf (@var{x}, @var{mu}, @var{s}) ## @deftypefnx {statistics} {@var{p} =} logicdf (@var{x}, @var{mu}, @var{s}, @qcode{"upper"}) ## ## Logistic cumulative distribution function (CDF). ## ## For each element of @var{x}, compute the cumulative distribution function ## (CDF) of the logistic distribution with location parameter @var{mu} and scale ## parameter @var{s}. The size of @var{p} is the common size of @var{x}, ## @var{mu}, and @var{s}. A scalar input functions as a constant matrix of ## the same size as the other inputs. ## ## Both parameters must be reals and @qcode{@var{s} > 0}. ## For @qcode{@var{s} <= 0}, @qcode{NaN} is returned. ## ## @code{@var{p} = logicdf (@var{x}, @var{mu}, @var{s}, "upper")} computes ## the upper tail probability of the logistic distribution with parameters ## @var{mu} and @var{s}, at the values in @var{x}. ## ## Further information about the logistic distribution can be found at ## @url{https://en.wikipedia.org/wiki/Logistic_distribution} ## ## @seealso{logiinv, logipdf, logirnd, logifit, logilike, logistat} ## @end deftypefn function p = logicdf (x, mu, s, uflag) ## Check for valid number of input arguments if (nargin < 3) error ("logicdf: function called with too few input arguments."); endif ## Check for valid "upper" flag if (nargin > 3) if (! strcmpi (uflag, "upper")) error ("logicdf: invalid argument for upper tail."); else uflag = true; endif else uflag = false; endif ## Check for common size of X, MU, and S if (! isscalar (x) || ! isscalar (mu) || ! isscalar(s)) [retval, x, mu, s] = common_size (x, mu, s); if (retval > 0) error (strcat (["logicdf: X, MU, and S must be of"], ... [" common size or scalars."])); endif endif ## Check for X, MU, and S being reals if (iscomplex (x) || iscomplex (mu) || iscomplex (s)) error ("logicdf: X, MU, and S must not be complex."); endif ## Check for class type if (isa (x, "single") || isa (mu, "single") || isa (s, "single")); p = NaN (size (x), "single"); else p = NaN (size (x)); endif ## Find normal and edge cases k1 = (x == -Inf) & (s > 0); k2 = (x == Inf) & (s > 0); k = ! k1 & ! k2 & (s > 0); ## Compute logistic CDF if (uflag) p(k1) = 1; p(k2) = 0; p(k) = 1 ./ (1 + exp ((x(k) - mu(k)) ./ s(k))); else p(k1) = 0; p(k2) = 1; p(k) = 1 ./ (1 + exp (- (x(k) - mu(k)) ./ s(k))); endif endfunction %!demo %! ## Plot various CDFs from the logistic distribution %! x = -5:0.01:20; %! p1 = logicdf (x, 5, 2); %! p2 = logicdf (x, 9, 3); %! p3 = logicdf (x, 9, 4); %! p4 = logicdf (x, 6, 2); %! p5 = logicdf (x, 2, 1); %! plot (x, p1, "-b", x, p2, "-g", x, p3, "-r", x, p4, "-c", x, p5, "-m") %! grid on %! legend ({"μ = 5, s = 2", "μ = 9, s = 3", "μ = 9, s = 4", ... %! "μ = 6, s = 2", "μ = 2, s = 1"}, "location", "southeast") %! title ("Logistic CDF") %! xlabel ("values in x") %! ylabel ("probability") ## Test output %!shared x, y %! x = [-Inf -log(3) 0 log(3) Inf]; %! y = [0, 1/4, 1/2, 3/4, 1]; %!assert (logicdf ([x, NaN], 0, 1), [y, NaN], eps) %!assert (logicdf (x, 0, [-2, -1, 0, 1, 2]), [nan(1, 3), 0.75, 1]) ## Test class of input preserved %!assert (logicdf (single ([x, NaN]), 0, 1), single ([y, NaN]), eps ("single")) %!assert (logicdf ([x, NaN], single (0), 1), single ([y, NaN]), eps ("single")) %!assert (logicdf ([x, NaN], 0, single (1)), single ([y, NaN]), eps ("single")) ## Test input validation %!error logicdf () %!error logicdf (1) %!error ... %! logicdf (1, 2) %!error logicdf (1, 2, 3, "tail") %!error logicdf (1, 2, 3, 4) %!error ... %! logicdf (1, ones (2), ones (3)) %!error ... %! logicdf (ones (2), 1, ones (3)) %!error ... %! logicdf (ones (2), ones (3), 1) %!error logicdf (i, 2, 3) %!error logicdf (1, i, 3) %!error logicdf (1, 2, i) statistics-release-1.6.3/inst/dist_fun/logiinv.m000066400000000000000000000110201456127120000217360ustar00rootroot00000000000000## Copyright (C) 2012 Rik Wehbring ## Copyright (C) 1995-2016 Kurt Hornik ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{x} =} logiinv (@var{p}, @var{mu}, @var{s}) ## ## Inverse of the logistic cumulative distribution function (iCDF). ## ## For each element of @var{p}, compute the quantile (the inverse of the CDF) of ## the logistic distribution with location parameter @var{mu} and scale ## parameter @var{s}. The size of @var{p} is the common size of @var{x}, ## @var{mu}, and @var{s}. A scalar input functions as a constant matrix of ## the same size as the other inputs. ## ## Both parameters must be reals and @qcode{@var{s} > 0}. ## For @qcode{@var{s} <= 0}, @qcode{NaN} is returned. ## ## Further information about the logistic distribution can be found at ## @url{https://en.wikipedia.org/wiki/Logistic_distribution} ## ## @seealso{logicdf, logipdf, logirnd, logifit, logilike, logistat} ## @end deftypefn function x = logiinv (p, mu, s) ## Check for valid number of input arguments if (nargin < 3) error ("logiinv: function called with too few input arguments."); endif ## Check for common size of P, MU, and S if (! isscalar (p) || ! isscalar (mu) || ! isscalar(s)) [retval, p, mu, s] = common_size (p, mu, s); if (retval > 0) error (strcat (["logiinv: P, MU, and S must be of"], ... [" common size or scalars."])); endif endif ## Check for X, MU, and S being reals if (iscomplex (p) || iscomplex (mu) || iscomplex (s)) error ("logiinv: P, MU, and S must not be complex."); endif ## Check for class type if (isa (p, "single") || isa (mu, "single") || isa (s, "single")); x = NaN (size (p), "single"); else x = NaN (size (p)); endif k = (p == 0) & (s > 0); x(k) = -Inf; k = (p == 1) & (s > 0); x(k) = Inf; k = (p > 0) & (p < 1) & (s > 0); x(k) = mu(k) + s(k) .* log (p(k) ./ (1 - p(k))); endfunction %!demo %! ## Plot various iCDFs from the logistic distribution %! p = 0.001:0.001:0.999; %! x1 = logiinv (p, 5, 2); %! x2 = logiinv (p, 9, 3); %! x3 = logiinv (p, 9, 4); %! x4 = logiinv (p, 6, 2); %! x5 = logiinv (p, 2, 1); %! plot (p, x1, "-b", p, x2, "-g", p, x3, "-r", p, x4, "-c", p, x5, "-m") %! grid on %! legend ({"μ = 5, s = 2", "μ = 9, s = 3", "μ = 9, s = 4", ... %! "μ = 6, s = 2", "μ = 2, s = 1"}, "location", "southeast") %! title ("Logistic iCDF") %! xlabel ("probability") %! ylabel ("x") ## Test output %!test %! p = [0.01:0.01:0.99]; %! assert (logiinv (p, 0, 1), log (p ./ (1-p)), 25*eps); %!shared p %! p = [-1 0 0.5 1 2]; %!assert (logiinv (p, 0, 1), [NaN -Inf 0 Inf NaN]) %!assert (logiinv (p, 0, [-1, 0, 1, 2, 3]), [NaN NaN 0 Inf NaN]) ## Test class of input preserved %!assert (logiinv ([p, NaN], 0, 1), [NaN -Inf 0 Inf NaN NaN]) %!assert (logiinv (single ([p, NaN]), 0, 1), single ([NaN -Inf 0 Inf NaN NaN])) %!assert (logiinv ([p, NaN], single (0), 1), single ([NaN -Inf 0 Inf NaN NaN])) %!assert (logiinv ([p, NaN], 0, single (1)), single ([NaN -Inf 0 Inf NaN NaN])) ## Test input validation %!error logiinv () %!error logiinv (1) %!error ... %! logiinv (1, 2) %!error ... %! logiinv (1, ones (2), ones (3)) %!error ... %! logiinv (ones (2), 1, ones (3)) %!error ... %! logiinv (ones (2), ones (3), 1) %!error logiinv (i, 2, 3) %!error logiinv (1, i, 3) %!error logiinv (1, 2, i) statistics-release-1.6.3/inst/dist_fun/logipdf.m000066400000000000000000000106621456127120000217260ustar00rootroot00000000000000## Copyright (C) 1995-2017 Kurt Hornik ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{y} =} logipdf (@var{x}, @var{mu}, @var{s}) ## ## Logistic probability density function (PDF). ## ## For each element of @var{x}, compute the probability density function (PDF) ## of the logistic distribution with location parameter @var{mu} and scale ## parameter @var{s}. The size of @var{p} is the common size of @var{x}, ## @var{mu}, and @var{s}. A scalar input functions as a constant matrix of ## the same size as the other inputs. ## ## Both parameters must be reals and @qcode{@var{s} > 0}. ## For @qcode{@var{s} <= 0}, @qcode{NaN} is returned. ## ## Further information about the logistic distribution can be found at ## @url{https://en.wikipedia.org/wiki/Logistic_distribution} ## ## @seealso{logicdf, logiinv, logirnd, logifit, logilike, logistat} ## @end deftypefn function y = logipdf (x, mu, s) ## Check for valid number of input arguments if (nargin < 3) error ("logipdf: function called with too few input arguments."); endif ## Check for common size of X, MU, and S if (! isscalar (x) || ! isscalar (mu) || ! isscalar(s)) [retval, x, mu, s] = common_size (x, mu, s); if (retval > 0) error (strcat (["logipdf: X, MU, and S must be of"], ... [" common size or scalars."])); endif endif ## Check for X, MU, and S being reals if (iscomplex (x) || iscomplex (mu) || iscomplex (s)) error ("logipdf: X, MU, and S must not be complex."); endif ## Check for class type if (isa (x, "single") || isa (mu, "single") || isa (s, "single")); y = NaN (size (x), "single"); else y = NaN (size (x)); endif ## Compute logistic PDF k1 = ((x == -Inf) & (s > 0)) | ((x == Inf) & (s > 0)); y(k1) = 0; k = ! k1 & (s > 0); y(k) = (1 ./ (4 .* s(k))) .* ... (sech ((x(k) - mu(k)) ./ (2 .* s(k))) .^ 2); endfunction %!demo %! ## Plot various PDFs from the logistic distribution %! x = -5:0.01:20; %! y1 = logipdf (x, 5, 2); %! y2 = logipdf (x, 9, 3); %! y3 = logipdf (x, 9, 4); %! y4 = logipdf (x, 6, 2); %! y5 = logipdf (x, 2, 1); %! plot (x, y1, "-b", x, y2, "-g", x, y3, "-r", x, y4, "-c", x, y5, "-m") %! grid on %! ylim ([0, 0.3]) %! legend ({"μ = 5, s = 2", "μ = 9, s = 3", "μ = 9, s = 4", ... %! "μ = 6, s = 2", "μ = 2, s = 1"}, "location", "northeast") %! title ("Logistic PDF") %! xlabel ("values in x") %! ylabel ("density") ## Test output %!shared x, y %! x = [-Inf -log(4) 0 log(4) Inf]; %! y = [0, 0.16, 1/4, 0.16, 0]; %!assert (logipdf ([x, NaN], 0, 1), [y, NaN], eps) %!assert (logipdf (x, 0, [-2, -1, 0, 1, 2]), [nan(1, 3), y([4:5])], eps) ## Test class of input preserved %!assert (logipdf (single ([x, NaN]), 0, 1), single ([y, NaN]), eps ("single")) %!assert (logipdf ([x, NaN], single (0), 1), single ([y, NaN]), eps ("single")) %!assert (logipdf ([x, NaN], 0, single (1)), single ([y, NaN]), eps ("single")) ## Test input validation %!error logipdf () %!error logipdf (1) %!error ... %! logipdf (1, 2) %!error ... %! logipdf (1, ones (2), ones (3)) %!error ... %! logipdf (ones (2), 1, ones (3)) %!error ... %! logipdf (ones (2), ones (3), 1) %!error logipdf (i, 2, 3) %!error logipdf (1, i, 3) %!error logipdf (1, 2, i) statistics-release-1.6.3/inst/dist_fun/logirnd.m000066400000000000000000000150141456127120000217340ustar00rootroot00000000000000## Copyright (C) 2012 Rik Wehbring ## Copyright (C) 1995-2016 Kurt Hornik ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{r} =} logirnd (@var{mu}, @var{s}) ## @deftypefnx {statistics} {@var{r} =} logirnd (@var{mu}, @var{s}, @var{rows}) ## @deftypefnx {statistics} {@var{r} =} logirnd (@var{mu}, @var{s}, @var{rows}, @var{cols}, @dots{}) ## @deftypefnx {statistics} {@var{r} =} logirnd (@var{mu}, @var{s}, [@var{sz}]) ## ## Random arrays from the logistic distribution. ## ## @code{@var{r} = logirnd (@var{mu}, @var{s})} returns an array of ## random numbers chosen from the logistic distribution with location parameter ## @var{mu} and scale parameter @var{s}. The size of @var{r} is the common size ## of @var{mu} and @var{s}. A scalar input functions as a constant matrix of ## the same size as the other inputs. ## ## Both parameters must be reals and @qcode{@var{s} > 0}. ## For @qcode{@var{s} <= 0}, @qcode{NaN} is returned. ## ## When called with a single size argument, @code{logirnd} returns a square ## matrix with the dimension specified. When called with more than one scalar ## argument, the first two arguments are taken as the number of rows and columns ## and any further arguments specify additional matrix dimensions. The size may ## also be specified with a row vector of dimensions, @var{sz}. ## ## Further information about the logistic distribution can be found at ## @url{https://en.wikipedia.org/wiki/Logistic_distribution} ## ## @seealso{logcdf, logiinv, logipdf, logifit, logilike, logistat} ## @end deftypefn function r = logirnd (mu, s, varargin) ## Check for valid number of input arguments if (nargin < 2) error ("logirnd: function called with too few input arguments."); endif ## Check for common size of MU, and S if (! isscalar (mu) || ! isscalar (s)) [retval, mu, s] = common_size (mu, s); if (retval > 0) error ("logirnd: MU and S must be of common size or scalars."); endif endif ## Check for X, MU, and S being reals if (iscomplex (mu) || iscomplex (s)) error ("logirnd: MU and S must not be complex."); endif ## Parse and check SIZE arguments if (nargin == 2) sz = size (mu); elseif (nargin == 3) if (isscalar (varargin{1}) && varargin{1} >= 0 ... && varargin{1} == fix (varargin{1})) sz = [varargin{1}, varargin{1}]; elseif (isrow (varargin{1}) && all (varargin{1} >= 0) ... && all (varargin{1} == fix (varargin{1}))) sz = varargin{1}; elseif error (strcat (["logirnd: SZ must be a scalar or a row vector"], ... [" of non-negative integers."])); endif elseif (nargin > 3) posint = cellfun (@(x) (! isscalar (x) || x < 0 || x != fix (x)), varargin); if (any (posint)) error ("logirnd: dimensions must be non-negative integers."); endif sz = [varargin{:}]; endif ## Check that parameters match requested dimensions in size if (! isscalar (mu) && ! isequal (size (mu), sz)) error ("logirnd: MU and S must be scalars or of size SZ."); endif ## Check for class type if (isa (mu, "single") || isa (s, "single")) is_type = "single"; else is_type = "double"; endif ## Generate random sample from logistic distribution r = - log (1 ./ rand (sz, is_type) - 1) .* s + mu; ## Force output to NaN for invalid parameter S <= 0 k = (s <= 0); r(k) = NaN; endfunction ## Test output %!assert (size (logirnd (1, 1)), [1 1]) %!assert (size (logirnd (1, ones (2,1))), [2, 1]) %!assert (size (logirnd (1, ones (2,2))), [2, 2]) %!assert (size (logirnd (ones (2,1), 1)), [2, 1]) %!assert (size (logirnd (ones (2,2), 1)), [2, 2]) %!assert (size (logirnd (1, 1, 3)), [3, 3]) %!assert (size (logirnd (1, 1, [4, 1])), [4, 1]) %!assert (size (logirnd (1, 1, 4, 1)), [4, 1]) %!assert (size (logirnd (1, 1, 4, 1, 5)), [4, 1, 5]) %!assert (size (logirnd (1, 1, 0, 1)), [0, 1]) %!assert (size (logirnd (1, 1, 1, 0)), [1, 0]) %!assert (size (logirnd (1, 1, 1, 2, 0, 5)), [1, 2, 0, 5]) ## Test class of input preserved %!assert (class (logirnd (1, 1)), "double") %!assert (class (logirnd (1, single (1))), "single") %!assert (class (logirnd (1, single ([1, 1]))), "single") %!assert (class (logirnd (single (1), 1)), "single") %!assert (class (logirnd (single ([1, 1]), 1)), "single") ## Test input validation %!error logirnd () %!error logirnd (1) %!error ... %! logirnd (ones (3), ones (2)) %!error ... %! logirnd (ones (2), ones (3)) %!error logirnd (i, 2, 3) %!error logirnd (1, i, 3) %!error ... %! logirnd (1, 2, -1) %!error ... %! logirnd (1, 2, 1.2) %!error ... %! logirnd (1, 2, ones (2)) %!error ... %! logirnd (1, 2, [2 -1 2]) %!error ... %! logirnd (1, 2, [2 0 2.5]) %!error ... %! logirnd (1, 2, 2, -1, 5) %!error ... %! logirnd (1, 2, 2, 1.5, 5) %!error ... %! logirnd (2, ones (2), 3) %!error ... %! logirnd (2, ones (2), [3, 2]) %!error ... %! logirnd (2, ones (2), 3, 2) statistics-release-1.6.3/inst/dist_fun/loglcdf.m000066400000000000000000000141141456127120000217100ustar00rootroot00000000000000## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{p} =} loglcdf (@var{x}, @var{a}, @var{b}) ## @deftypefnx {statistics} {@var{p} =} loglcdf (@var{x}, @var{a}, @var{b}, @qcode{"upper"}) ## ## Log-logistic cumulative distribution function (CDF). ## ## For each element of @var{x}, compute the cumulative distribution function ## (CDF) of the log-logistic distribution with scale parameter @var{a} and shape ## parameter @var{b}. The size of @var{p} is the common size of @var{x}, ## @var{a}, and @var{b}. A scalar input functions as a constant matrix of the ## same size as the other inputs. ## ## Both parameters, @var{a} and @var{b}, must be positive reals and @var{x} is ## supported in the range @math{[0,inf)}, otherwise @qcode{NaN} is returned. ## ## @code{@var{p} = loglcdf (@var{x}, @var{a}, @var{b}, "upper")} computes the ## upper tail probability of the log-logistic distribution with parameters ## @var{a} and @var{b}, at the values in @var{x}. ## ## Further information about the log-logistic distribution can be found at ## @url{https://en.wikipedia.org/wiki/Log-logistic_distribution} ## ## MATLAB compatibility: MATLAB uses an alternative parameterization given by ## the pair @math{μ, s}, i.e. @var{mu} and @var{s}, in analogy with the logistic ## distribution. Their relation to the @var{a} and @var{b} parameters is given ## below: ## ## @itemize ## @item @qcode{@var{a} = exp (@var{mu})} ## @item @qcode{@var{b} = 1 / @var{s}} ## @end itemize ## ## @seealso{loglinv, loglpdf, loglrnd, loglfit, logllike} ## @end deftypefn function p = loglcdf (x, a, b, uflag) ## Check for valid number of input arguments if (nargin < 3) error ("loglcdf: function called with too few input arguments."); endif ## Check for valid "upper" flag if (nargin > 3) if (! strcmpi (uflag, "upper")) error ("loglcdf: invalid argument for upper tail."); else uflag = true; endif else uflag = false; endif ## Check for common size of X, A, and B if (! isscalar (x) || ! isscalar (a) || ! isscalar(b)) [retval, x, a, b] = common_size (x, a, b); if (retval > 0) error ("loglcdf: X, A, and B must be of common size or scalars."); endif endif ## Check for X, A, and B being reals if (iscomplex (x) || iscomplex (a) || iscomplex (b)) error ("loglcdf: X, A, and B must not be complex."); endif ## Check for invalid points a(a <= 0) = NaN; b(b <= 0) = NaN; x(x < 0) = NaN; ## Compute log-logistic CDF z = (x ./ a) .^ -b; if (uflag) p = 1 - (1 ./ (1 + z)); else p = 1 ./ (1 + z); endif ## Check for class type if (isa (x, "single") || isa (a, "single") || isa (b, "single")); p = cast (p, "single"); endif endfunction %!demo %! ## Plot various CDFs from the log-logistic distribution %! x = 0:0.001:2; %! p1 = loglcdf (x, 1, 0.5); %! p2 = loglcdf (x, 1, 1); %! p3 = loglcdf (x, 1, 2); %! p4 = loglcdf (x, 1, 4); %! p5 = loglcdf (x, 1, 8); %! plot (x, p1, "-b", x, p2, "-g", x, p3, "-r", x, p4, "-c", x, p5, "-m") %! legend ({"β = 0.5", "β = 1", "β = 2", "β = 4", "β = 8"}, ... %! "location", "northwest") %! grid on %! title ("Log-logistic CDF") %! xlabel ("values in x") %! ylabel ("probability") %! text (0.05, 0.64, "α = 1, values of β as shown in legend") ## Test output %!shared out1, out2 %! out1 = [0, 0.5, 0.66666667, 0.75, 0.8, 0.83333333]; %! out2 = [0, 0.4174, 0.4745, 0.5082, 0.5321, 0.5506]; %!assert (loglcdf ([0:5], 1, 1), out1, 1e-8) %!assert (loglcdf ([0:5], 1, 1, "upper"), 1 - out1, 1e-8) %!assert (loglcdf ([0:5], exp (0), 1), out1, 1e-8) %!assert (loglcdf ([0:5], exp (0), 1, "upper"), 1 - out1, 1e-8) %!assert (loglcdf ([0:5], exp (1), 1 / 3), out2, 1e-4) %!assert (loglcdf ([0:5], exp (1), 1 / 3, "upper"), 1 - out2, 1e-4) ## Test class of input preserved %!assert (class (loglcdf (single (1), 2, 3)), "single") %!assert (class (loglcdf (1, single (2), 3)), "single") %!assert (class (loglcdf (1, 2, single (3))), "single") ## Test input validation %!error loglcdf (1) %!error loglcdf (1, 2) %!error ... %! loglcdf (1, 2, 3, 4) %!error ... %! loglcdf (1, 2, 3, "uper") %!error ... %! loglcdf (1, ones (2), ones (3)) %!error ... %! loglcdf (1, ones (2), ones (3), "upper") %!error ... %! loglcdf (ones (2), 1, ones (3)) %!error ... %! loglcdf (ones (2), 1, ones (3), "upper") %!error ... %! loglcdf (ones (2), ones (3), 1) %!error ... %! loglcdf (ones (2), ones (3), 1, "upper") %!error loglcdf (i, 2, 3) %!error loglcdf (i, 2, 3, "upper") %!error loglcdf (1, i, 3) %!error loglcdf (1, i, 3, "upper") %!error loglcdf (1, 2, i) %!error loglcdf (1, 2, i, "upper") statistics-release-1.6.3/inst/dist_fun/loglinv.m000066400000000000000000000114041456127120000217470ustar00rootroot00000000000000## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{x} =} loglinv (@var{p}, @var{a}, @var{b}) ## ## Inverse of the log-logistic cumulative distribution function (iCDF). ## ## For each element of @var{p}, compute the quantile (the inverse of the CDF) of ## the log-logistic distribution with scale parameter @var{a} and shape ## parameter @var{b}. The size of @var{x} is the common size of @var{p}, ## @var{a}, and @var{b}. A scalar input functions as a constant matrix of the ## same size as the other inputs. ## ## Both parameters, @var{a} and @var{b}, must be positive reals and @var{p} is ## supported in the range @math{[0,1]}, otherwise @qcode{NaN} is returned. ## ## Further information about the log-logistic distribution can be found at ## @url{https://en.wikipedia.org/wiki/Log-logistic_distribution} ## ## MATLAB compatibility: MATLAB uses an alternative parameterization given by ## the pair @math{μ, s}, i.e. @var{mu} and @var{s}, in analogy with the logistic ## distribution. Their relation to the @var{a} and @var{b} parameters is given ## below: ## ## @itemize ## @item @qcode{@var{a} = exp (@var{mu})} ## @item @qcode{@var{b} = 1 / @var{s}} ## @end itemize ## ## @seealso{loglcdf, loglpdf, loglrnd, loglfit, logllike} ## @end deftypefn function x = loglinv (p, a, b) ## Check for valid number of input arguments if (nargin < 3) error ("loglinv: function called with too few input arguments."); endif ## Check for common size of P, A, and B if (! isscalar (p) || ! isscalar (a) || ! isscalar(b)) [retval, p, a, b] = common_size (p, a, b); if (retval > 0) error (strcat (["loglinv: P, A, and B must be of"], ... [" common size or scalars."])); endif endif ## Check for X, A, and B being reals if (iscomplex (p) || iscomplex (a) || iscomplex (b)) error ("loglinv: P, A, and B must not be complex."); endif ## Check for class type if (isa (p, "single") || isa (a, "single") || isa (b, "single")); x = NaN (size (p), "single"); else x = NaN (size (p)); endif ## Check for valid points k = (p >= 0) & (p <= 1) & (a > 0) & (b > 0); ## Compute the log-logistic iCDF x(k) = a(k) .* (p(k) ./ (1 - p(k))) .^ (1 ./ b(k)); endfunction %!demo %! ## Plot various iCDFs from the log-logistic distribution %! p = 0.001:0.001:0.999; %! x1 = loglinv (p, 1, 0.5); %! x2 = loglinv (p, 1, 1); %! x3 = loglinv (p, 1, 2); %! x4 = loglinv (p, 1, 4); %! x5 = loglinv (p, 1, 8); %! plot (p, x1, "-b", p, x2, "-g", p, x3, "-r", p, x4, "-c", p, x5, "-m") %! ylim ([0, 20]) %! grid on %! legend ({"β = 0.5", "β = 1", "β = 2", "β = 4", "β = 8"}, ... %! "location", "northwest") %! title ("Log-logistic iCDF") %! xlabel ("probability") %! ylabel ("x") %! text (0.03, 12.5, "α = 1, values of β as shown in legend") ## Test output %!shared p, out1, out2 %! p = [-1, 0, 0.2, 0.5, 0.8, 0.95, 1, 2]; %! out1 = [NaN, 0, 0.25, 1, 4, 19, Inf, NaN]; %! out2 = [NaN, 0, 0.0424732, 2.718282, 173.970037, 18644.695061, Inf, NaN]; %!assert (loglinv (p, 1, 1), out1, 1e-8) %!assert (loglinv (p, exp (0), 1), out1, 1e-8) %!assert (loglinv (p, exp (1), 1 / 3), out2, 1e-6) ## Test class of input preserved %!assert (class (loglinv (single (1), 2, 3)), "single") %!assert (class (loglinv (1, single (2), 3)), "single") %!assert (class (loglinv (1, 2, single (3))), "single") ## Test input validation %!error loglinv (1) %!error loglinv (1, 2) %!error ... %! loglinv (1, ones (2), ones (3)) %!error ... %! loglinv (ones (2), 1, ones (3)) %!error ... %! loglinv (ones (2), ones (3), 1) %!error loglinv (i, 2, 3) %!error loglinv (1, i, 3) %!error loglinv (1, 2, i) statistics-release-1.6.3/inst/dist_fun/loglpdf.m000066400000000000000000000115131456127120000217250ustar00rootroot00000000000000## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{y} =} loglpdf (@var{x}, @var{a}, @var{b}) ## ## Log-logistic probability density function (PDF). ## ## For each element of @var{x}, compute the probability density function (PDF) ## of the log-logistic distribution with with scale parameter @var{a} and shape ## parameter @var{b}. The size of @var{y} is the common size of @var{x}, ## @var{a}, and @var{b}. A scalar input functions as a constant matrix of the ## same size as the other inputs. ## ## Both parameters, @var{a} and @var{b}, must be positive reals, otherwise ## @qcode{NaN} is returned. @var{x} is supported in the range @math{[0,Inf)}, ## otherwise @qcode{0} is returned. ## ## Further information about the log-logistic distribution can be found at ## @url{https://en.wikipedia.org/wiki/Log-logistic_distribution} ## ## MATLAB compatibility: MATLAB uses an alternative parameterization given by ## the pair @math{μ, s}, i.e. @var{mu} and @var{s}, in analogy with the logistic ## distribution. Their relation to the @var{a} and @var{b} parameters is given ## below: ## ## @itemize ## @item @qcode{@var{a} = exp (@var{mu})} ## @item @qcode{@var{b} = 1 / @var{s}} ## @end itemize ## ## @seealso{loglcdf, loglinv, loglrnd, loglfit, logllike} ## @end deftypefn function y = loglpdf (x, a, b) ## Check for valid number of input arguments if (nargin < 3) error ("loglpdf: function called with too few input arguments."); endif ## Check for common size of X, A, and B if (! isscalar (x) || ! isscalar (a) || ! isscalar(b)) [retval, x, a, b] = common_size (x, a, b); if (retval > 0) error (strcat (["loglpdf: X, A, and B must be of"], ... [" common size or scalars."])); endif endif ## Check for X, A, and B being reals if (iscomplex (x) || iscomplex (a) || iscomplex (b)) error ("loglpdf: X, A, and B must not be complex."); endif ## Check for class type if (isa (x, "single") || isa (a, "single") || isa (b, "single")); y = NaN (size (x), "single"); else y = NaN (size (x)); endif ## Compute log-logistic PDF k1 = ((x == Inf) | (x < 0)) & (a > 0) & (b > 0); y(k1) = 0; k = (! k1) & (a > 0) & (b > 0); y(k) = ((b(k) ./ a(k)) .* (x(k) ./ a(k)) .^ (b(k) -1)) ./ ... ((1 + (x(k) ./ a(k)) .^ b(k)) .^ 2); endfunction %!demo %! ## Plot various PDFs from the log-logistic distribution %! x = 0:0.001:2; %! y1 = loglpdf (x, 1, 0.5); %! y2 = loglpdf (x, 1, 1); %! y3 = loglpdf (x, 1, 2); %! y4 = loglpdf (x, 1, 4); %! y5 = loglpdf (x, 1, 8); %! plot (x, y1, "-b", x, y2, "-g", x, y3, "-r", x, y4, "-c", x, y5, "-m") %! grid on %! ylim ([0,3]) %! legend ({"β = 0.5", "β = 1", "β = 2", "β = 4", "β = 8"}, ... %! "location", "northeast") %! title ("Log-logistic PDF") %! xlabel ("values in x") %! ylabel ("density") %! text (0.5, 2.8, "α = 1, values of β as shown in legend") ## Test output %!shared out1, out2 %! out1 = [0, 1, 0.2500, 0.1111, 0.0625, 0.0400, 0.0278, 0]; %! out2 = [0, Inf, 0.0811, 0.0416, 0.0278, 0.0207, 0.0165, 0]; %!assert (loglpdf ([-1:5,Inf], 1, 1), out1, 1e-4) %!assert (loglpdf ([-1:5,Inf], exp (0), 1), out1, 1e-4) %!assert (loglpdf ([-1:5,Inf], exp (1), 1 / 3), out2, 1e-4) ## Test class of input preserved %!assert (class (loglpdf (single (1), 2, 3)), "single") %!assert (class (loglpdf (1, single (2), 3)), "single") %!assert (class (loglpdf (1, 2, single (3))), "single") ## Test input validation %!error loglpdf (1) %!error loglpdf (1, 2) %!error ... %! loglpdf (1, ones (2), ones (3)) %!error ... %! loglpdf (ones (2), 1, ones (3)) %!error ... %! loglpdf (ones (2), ones (3), 1) %!error loglpdf (i, 2, 3) %!error loglpdf (1, i, 3) %!error loglpdf (1, 2, i) statistics-release-1.6.3/inst/dist_fun/loglrnd.m000066400000000000000000000154201456127120000217400ustar00rootroot00000000000000## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{r} =} loglrnd (@var{a}, @var{b}) ## @deftypefnx {statistics} {@var{r} =} loglrnd (@var{a}, @var{b}, @var{rows}) ## @deftypefnx {statistics} {@var{r} =} loglrnd (@var{a}, @var{b}, @var{rows}, @var{cols}, @dots{}) ## @deftypefnx {statistics} {@var{r} =} loglrnd (@var{a}, @var{b}, [@var{sz}]) ## ## Random arrays from the log-logistic distribution. ## ## @code{@var{r} = loglrnd (@var{a}, @var{b})} returns an array of random ## numbers chosen from the log-logistic distribution with scale parameter ## @var{a} and shape parameter @var{b}. The size of @var{r} is the common size ## of @var{a} and @var{b}. A scalar input functions as a constant matrix of the ## same size as the other inputs. ## ## Both parameters must be positive reals, otherwise @qcode{NaN} is returned. ## ## When called with a single size argument, @code{loglrnd} returns a square ## matrix with the dimension specified. When called with more than one scalar ## argument, the first two arguments are taken as the number of rows and columns ## and any further arguments specify additional matrix dimensions. The size may ## also be specified with a row vector of dimensions, @var{sz}. ## ## Further information about the log-logistic distribution can be found at ## @url{https://en.wikipedia.org/wiki/Log-logistic_distribution} ## ## MATLAB compatibility: MATLAB uses an alternative parameterization given by ## the pair @math{μ, s}, i.e. @var{mu} and @var{s}, in analogy with the logistic ## distribution. Their relation to the @var{a} and @var{b} parameters is given ## below: ## ## @itemize ## @item @qcode{@var{a} = exp (@var{mu})} ## @item @qcode{@var{b} = 1 / @var{s}} ## @end itemize ## ## @seealso{loglcdf, loglinv, loglpdf, loglfit, logllike} ## @end deftypefn function r = loglrnd (a, b, varargin) ## Check for valid number of input arguments if (nargin < 2) error ("loglrnd: function called with too few input arguments."); endif ## Check for common size of A, and B if (! isscalar (a) || ! isscalar (b)) [retval, a, b] = common_size (a, b); if (retval > 0) error ("loglrnd: A and B must be of common size or scalars."); endif endif ## Check for X, A, and B being reals if (iscomplex (a) || iscomplex (b)) error ("loglrnd: A and B must not be complex."); endif ## Parse and check SIZE arguments if (nargin == 2) sz = size (a); elseif (nargin == 3) if (isscalar (varargin{1}) && varargin{1} >= 0 ... && varargin{1} == fix (varargin{1})) sz = [varargin{1}, varargin{1}]; elseif (isrow (varargin{1}) && all (varargin{1} >= 0) ... && all (varargin{1} == fix (varargin{1}))) sz = varargin{1}; elseif error (strcat (["loglrnd: SZ must be a scalar or a row vector"], ... [" of non-negative integers."])); endif elseif (nargin > 3) posint = cellfun (@(x) (! isscalar (x) || x < 0 || x != fix (x)), varargin); if (any (posint)) error ("loglrnd: dimensions must be non-negative integers."); endif sz = [varargin{:}]; endif ## Check that parameters match requested dimensions in size if (! isscalar (a) && ! isequal (size (a), sz)) error ("loglrnd: A and B must be scalars or of size SZ."); endif ## Check for class type if (isa (a, "single") || isa (b, "single")) is_type = "single"; else is_type = "double"; endif ## Generate random sample from log-logistic distribution p = rand (sz, is_type); r = a .* (p ./ (1 - p)) .^ (1 ./ b); ## Force output to NaN for invalid parameters A and B k = (a <= 0 | b <= 0); r(k) = NaN; endfunction ## Test output %!assert (size (loglrnd (1, 1)), [1 1]) %!assert (size (loglrnd (1, ones (2,1))), [2, 1]) %!assert (size (loglrnd (1, ones (2,2))), [2, 2]) %!assert (size (loglrnd (ones (2,1), 1)), [2, 1]) %!assert (size (loglrnd (ones (2,2), 1)), [2, 2]) %!assert (size (loglrnd (1, 1, 3)), [3, 3]) %!assert (size (loglrnd (1, 1, [4, 1])), [4, 1]) %!assert (size (loglrnd (1, 1, 4, 1)), [4, 1]) %!assert (size (loglrnd (1, 1, 4, 1, 5)), [4, 1, 5]) %!assert (size (loglrnd (1, 1, 0, 1)), [0, 1]) %!assert (size (loglrnd (1, 1, 1, 0)), [1, 0]) %!assert (size (loglrnd (1, 1, 1, 2, 0, 5)), [1, 2, 0, 5]) ## Test class of input preserved %!assert (class (loglrnd (1, 1)), "double") %!assert (class (loglrnd (1, single (1))), "single") %!assert (class (loglrnd (1, single ([1, 1]))), "single") %!assert (class (loglrnd (single (1), 1)), "single") %!assert (class (loglrnd (single ([1, 1]), 1)), "single") ## Test input validation %!error loglrnd () %!error loglrnd (1) %!error ... %! loglrnd (ones (3), ones (2)) %!error ... %! loglrnd (ones (2), ones (3)) %!error loglrnd (i, 2, 3) %!error loglrnd (1, i, 3) %!error ... %! loglrnd (1, 2, -1) %!error ... %! loglrnd (1, 2, 1.2) %!error ... %! loglrnd (1, 2, ones (2)) %!error ... %! loglrnd (1, 2, [2 -1 2]) %!error ... %! loglrnd (1, 2, [2 0 2.5]) %!error ... %! loglrnd (1, 2, 2, -1, 5) %!error ... %! loglrnd (1, 2, 2, 1.5, 5) %!error ... %! loglrnd (2, ones (2), 3) %!error ... %! loglrnd (2, ones (2), [3, 2]) %!error ... %! loglrnd (2, ones (2), 3, 2) statistics-release-1.6.3/inst/dist_fun/logncdf.m000066400000000000000000000214061456127120000217140ustar00rootroot00000000000000## Copyright (C) 2012 Rik Wehbring ## Copyright (C) 1995-2016 Kurt Hornik ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{p} =} logncdf (@var{x}) ## @deftypefnx {statistics} {@var{p} =} logncdf (@var{x}, @var{mu}) ## @deftypefnx {statistics} {@var{p} =} logncdf (@var{x}, @var{mu}, @var{sigma}) ## @deftypefnx {statistics} {@var{p} =} logncdf (@dots{}, @qcode{"upper"}) ## @deftypefnx {statistics} {[@var{p}, @var{plo}, @var{pup}] =} logncdf (@var{x}, @var{mu}, @var{sigma}, @var{pcov}) ## @deftypefnx {statistics} {[@var{p}, @var{plo}, @var{pup}] =} logncdf (@var{x}, @var{mu}, @var{sigma}, @var{pcov}, @var{alpha}) ## @deftypefnx {statistics} {[@var{p}, @var{plo}, @var{pup}] =} logncdf (@dots{}, @qcode{"upper"}) ## ## Log-normal cumulative distribution function (CDF). ## ## For each element of @var{x}, compute the cumulative distribution function ## (CDF) of the log-normal distribution with mean parameter @var{mu} and ## standard deviation parameter @var{sigma}, each corresponding to the ## associated normal distribution. The size of @var{p} is the common size of ## @var{x}, @var{mu}, and @var{sigma}. A scalar input functions as a constant ## matrix of the same size as the other inputs. ## ## If a random variable follows this distribution, its logarithm is normally ## distributed with mean @var{mu} and standard deviation @var{sigma}. ## ## Default parameter values are @qcode{@var{mu} = 0} and ## @qcode{@var{sigma} = 1}. Both parameters must be reals and ## @qcode{@var{sigma} > 0}. For @qcode{@var{sigma} <= 0}, @qcode{NaN} is ## returned. ## ## When called with three output arguments, i.e. @qcode{[@var{p}, @var{plo}, ## @var{pup}]}, @code{logncdf} computes the confidence bounds for @var{p} when ## the input parameters @var{mu} and @var{sigma} are estimates. In such case, ## @var{pcov}, a @math{2x2} matrix containing the covariance matrix of the ## estimated parameters, is necessary. Optionally, @var{alpha}, which has a ## default value of 0.05, specifies the @qcode{100 * (1 - @var{alpha})} percent ## confidence bounds. @var{plo} and @var{pup} are arrays of the same size as ## @var{p} containing the lower and upper confidence bounds. ## ## @code{[@dots{}] = logncdf (@dots{}, "upper")} computes the upper tail ## probability of the log-normal distribution with parameters @var{mu} and ## @var{sigma}, at the values in @var{x}. ## ## Further information about the log-normal distribution can be found at ## @url{https://en.wikipedia.org/wiki/Log-normal_distribution} ## ## @seealso{logninv, lognpdf, lognrnd, lognfit, lognlike, lognstat} ## @end deftypefn function [varargout] = logncdf (x, varargin) ## Check for valid number of input arguments if (nargin < 1 || nargin > 6) error ("logncdf: invalid number of input arguments."); endif ## Check for "upper" flag if (nargin > 1 && strcmpi (varargin{end}, "upper")) uflag = true; varargin(end) = []; elseif (nargin > 1 && ischar (varargin{end}) && ... ! strcmpi (varargin{end}, "upper")) error ("logncdf: invalid argument for upper tail."); elseif (nargin > 2 && isempty (varargin{end})) uflag = false; varargin(end) = []; else uflag = false; endif ## Get extra arguments (if they exist) or add defaults if (numel (varargin) > 0) mu = varargin{1}; else mu = 0; endif if (numel (varargin) > 1) sigma = varargin{2}; else sigma = 1; endif if (numel (varargin) > 2) pcov = varargin{3}; ## Check for valid covariance matrix 2x2 if (! isequal (size (pcov), [2, 2])) error ("logncdf: invalid size of covariance matrix."); endif else ## Check that cov matrix is provided if 3 output arguments are requested if (nargout > 1) error ("logncdf: covariance matrix is required for confidence bounds."); endif pcov = []; endif if (numel (varargin) > 3) alpha = varargin{4}; ## Check for valid alpha value if (! isnumeric (alpha) || numel (alpha) !=1 || alpha <= 0 || alpha >= 1) error ("logncdf: invalid value for alpha."); endif else alpha = 0.05; endif ## Check for common size of X, MU, and SIGMA if (! isscalar (x) || ! isscalar (mu) || ! isscalar (sigma)) [err, x, mu, sigma] = common_size (x, mu, sigma); if (err > 0) error ("logncdf: X, MU, and SIGMA must be of common size or scalars."); endif endif ## Check for X, MU, and SIGMA being reals if (iscomplex (x) || iscomplex (mu) || iscomplex (sigma)) error ("logncdf: X, MU, and SIGMA must not be complex."); endif ## Return NaN for out of range parameters. sigma(sigma <= 0) = NaN; ## Negative data would create complex values, which erfc cannot handle. x(x < 0) = 0; ## Compute lognormal cdf z = (log (x) - mu) ./ sigma; if (uflag) z = -z; endif p = 0.5 * erfc (-z ./ sqrt(2)); ## Compute confidence bounds (if requested) if (nargout >= 2) zvar = (pcov(1,1) + 2 * pcov(1,2) * z + pcov(2,2) * z .^ 2) ./ (sigma .^ 2); if (any (zvar(:) < 0)) error ("logncdf: bad covariance matrix."); end normz = -norminv (alpha / 2); halfwidth = normz * sqrt (zvar); zlo = z - halfwidth; zup = z + halfwidth; plo = 0.5 * erfc (-zlo ./ sqrt (2)); pup = 0.5 * erfc (-zup ./ sqrt (2)); endif ## Check for class type if (isa (x, "single") || isa (mu, "single") || isa (sigma, "single")); is_class = "single"; else is_class = "double"; endif ## Prepare output varargout{1} = cast (p, is_class); if (nargout > 1) varargout{2} = cast (plo, is_class); varargout{3} = cast (pup, is_class); endif endfunction %!demo %! ## Plot various CDFs from the log-normal distribution %! x = 0:0.01:3; %! p1 = logncdf (x, 0, 1); %! p2 = logncdf (x, 0, 0.5); %! p3 = logncdf (x, 0, 0.25); %! plot (x, p1, "-b", x, p2, "-g", x, p3, "-r") %! grid on %! legend ({"μ = 0, σ = 1", "μ = 0, σ = 0.5", "μ = 0, σ = 0.25"}, ... %! "location", "southeast") %! title ("Log-normal CDF") %! xlabel ("values in x") %! ylabel ("probability") ## Test output %!shared x, y %! x = [-1, 0, 1, e, Inf]; %! y = [0, 0, 0.5, 1/2+1/2*erf(1/2), 1]; %!assert (logncdf (x, zeros (1,5), sqrt(2)*ones (1,5)), y, eps) %!assert (logncdf (x, zeros (1,5), sqrt(2)*ones (1,5), []), y, eps) %!assert (logncdf (x, 0, sqrt(2)*ones (1,5)), y, eps) %!assert (logncdf (x, zeros (1,5), sqrt(2)), y, eps) %!assert (logncdf (x, [0 1 NaN 0 1], sqrt(2)), [0 0 NaN y(4:5)], eps) %!assert (logncdf (x, 0, sqrt(2)*[0 NaN Inf 1 1]), [NaN NaN y(3:5)], eps) %!assert (logncdf ([x(1:3) NaN x(5)], 0, sqrt(2)), [y(1:3) NaN y(5)], eps) ## Test class of input preserved %!assert (logncdf ([x, NaN], 0, sqrt(2)), [y, NaN], eps) %!assert (logncdf (single ([x, NaN]), 0, sqrt(2)), single ([y, NaN]), eps ("single")) %!assert (logncdf ([x, NaN], single (0), sqrt(2)), single ([y, NaN]), eps ("single")) %!assert (logncdf ([x, NaN], 0, single (sqrt(2))), single ([y, NaN]), eps ("single")) ## Test input validation %!error logncdf () %!error logncdf (1,2,3,4,5,6,7) %!error logncdf (1, 2, 3, 4, "uper") %!error ... %! logncdf (ones (3), ones (2), ones (2)) %!error logncdf (2, 3, 4, [1, 2]) %!error ... %! [p, plo, pup] = logncdf (1, 2, 3) %!error [p, plo, pup] = ... %! logncdf (1, 2, 3, [1, 0; 0, 1], 0) %!error [p, plo, pup] = ... %! logncdf (1, 2, 3, [1, 0; 0, 1], 1.22) %!error [p, plo, pup] = ... %! logncdf (1, 2, 3, [1, 0; 0, 1], "alpha", "upper") %!error logncdf (i, 2, 2) %!error logncdf (2, i, 2) %!error logncdf (2, 2, i) %!error ... %! [p, plo, pup] =logncdf (1, 2, 3, [1, 0; 0, -inf], 0.04) statistics-release-1.6.3/inst/dist_fun/logninv.m000066400000000000000000000114471456127120000217600ustar00rootroot00000000000000## Copyright (C) 2012 Rik Wehbring ## Copyright (C) 1995-2016 Kurt Hornik ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{x} =} logninv (@var{p}) ## @deftypefnx {statistics} {@var{x} =} logninv (@var{p}, @var{mu}) ## @deftypefnx {statistics} {@var{x} =} logninv (@var{p}, @var{mu}, @var{sigma}) ## ## Inverse of the log-normal cumulative distribution function (iCDF). ## ## For each element of @var{p}, compute the quantile (the inverse of the CDF) of ## the log-normal distribution with mean parameter @var{mu} and standard ## deviation parameter @var{sigma}, each corresponding to the associated normal ## distribution. The size of @var{x} is the common size of @var{p}, @var{mu}, ## and @var{sigma}. A scalar input functions as a constant matrix of the same ## size as the other inputs. ## ## If a random variable follows this distribution, its logarithm is normally ## distributed with mean @var{mu} and standard deviation @var{sigma}. ## ## Default parameter values are @qcode{@var{mu} = 0} and ## @qcode{@var{sigma} = 1}. Both parameters must be reals and ## @qcode{@var{sigma} > 0}. For @qcode{@var{sigma} <= 0}, @qcode{NaN} is ## returned. ## ## Further information about the log-normal distribution can be found at ## @url{https://en.wikipedia.org/wiki/Log-normal_distribution} ## ## @seealso{logncdf, lognpdf, lognrnd, lognfit, lognlike, lognstat} ## @end deftypefn function x = logninv (p, mu = 0, sigma = 1) ## Check for valid number of input arguments if (nargin < 1 || nargin > 3) print_usage (); endif ## Check for common size of P, MU, and SIGMA if (! isscalar (p) || ! isscalar (mu) || ! isscalar (sigma)) [retval, p, mu, sigma] = common_size (p, mu, sigma); if (retval > 0) error ("logninv: X, MU, and SIGMA must be of common size or scalars."); endif endif ## Check for X, MU, and SIGMA being reals if (iscomplex (p) || iscomplex (mu) || iscomplex (sigma)) error ("logninv: X, MU, and SIGMA must not be complex."); endif ## Check for class type if (isa (p, "single") || isa (mu, "single") || isa (sigma, "single")) x = NaN (size (p), "single"); else x = NaN (size (p)); endif ## Compute lognormal iCDF k = !(p >= 0) | !(p <= 1) | !(sigma > 0) | !(sigma < Inf); x(k) = NaN; k = (p == 1) & (sigma > 0) & (sigma < Inf); x(k) = Inf; k = (p >= 0) & (p < 1) & (sigma > 0) & (sigma < Inf); if (isscalar (mu) && isscalar (sigma)) x(k) = exp (mu) .* exp (sigma .* (-sqrt (2) * erfcinv (2 * p(k)))); else x(k) = exp (mu(k)) .* exp (sigma(k) .* (-sqrt (2) * erfcinv (2 * p(k)))); endif endfunction %!demo %! ## Plot various iCDFs from the log-normal distribution %! p = 0.001:0.001:0.999; %! x1 = logninv (p, 0, 1); %! x2 = logninv (p, 0, 0.5); %! x3 = logninv (p, 0, 0.25); %! plot (p, x1, "-b", p, x2, "-g", p, x3, "-r") %! grid on %! ylim ([0, 3]) %! legend ({"μ = 0, σ = 1", "μ = 0, σ = 0.5", "μ = 0, σ = 0.25"}, ... %! "location", "northwest") %! title ("Log-normal iCDF") %! xlabel ("probability") %! ylabel ("values in x") ## Test output %!shared p %! p = [-1 0 0.5 1 2]; %!assert (logninv (p, ones (1,5), ones (1,5)), [NaN 0 e Inf NaN]) %!assert (logninv (p, 1, ones (1,5)), [NaN 0 e Inf NaN]) %!assert (logninv (p, ones (1,5), 1), [NaN 0 e Inf NaN]) %!assert (logninv (p, [1 1 NaN 0 1], 1), [NaN 0 NaN Inf NaN]) %!assert (logninv (p, 1, [1 0 NaN Inf 1]), [NaN NaN NaN NaN NaN]) %!assert (logninv ([p(1:2) NaN p(4:5)], 1, 2), [NaN 0 NaN Inf NaN]) ## Test class of input preserved %!assert (logninv ([p, NaN], 1, 1), [NaN 0 e Inf NaN NaN]) %!assert (logninv (single ([p, NaN]), 1, 1), single ([NaN 0 e Inf NaN NaN])) %!assert (logninv ([p, NaN], single (1), 1), single ([NaN 0 e Inf NaN NaN])) %!assert (logninv ([p, NaN], 1, single (1)), single ([NaN 0 e Inf NaN NaN])) ## Test input validation %!error logninv () %!error logninv (1,2,3,4) %!error logninv (ones (3), ones (2), ones (2)) %!error logninv (ones (2), ones (3), ones (2)) %!error logninv (ones (2), ones (2), ones (3)) %!error logninv (i, 2, 2) %!error logninv (2, i, 2) %!error logninv (2, 2, i) statistics-release-1.6.3/inst/dist_fun/lognpdf.m000066400000000000000000000111421456127120000217250ustar00rootroot00000000000000## Copyright (C) 2012 Rik Wehbring ## Copyright (C) 1995-2016 Kurt Hornik ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{y} =} lognpdf (@var{x}) ## @deftypefnx {statistics} {@var{y} =} lognpdf (@var{x}, @var{mu}) ## @deftypefnx {statistics} {@var{y} =} lognpdf (@var{x}, @var{mu}, @var{sigma}) ## ## Lognormal probability density function (PDF). ## ## For each element of @var{x}, compute the probability density function (PDF) ## of the lognormal distribution with mean parameter @var{mu} and standard ## deviation parameter @var{sigma}, each corresponding to the associated normal ## distribution. The size of @var{y} is the common size of @var{p}, @var{mu}, ## and @var{sigma}. A scalar input functions as a constant matrix of the same ## size as the other inputs. ## ## If a random variable follows this distribution, its logarithm is normally ## distributed with mean @var{mu} and standard deviation @var{sigma}. ## ## Default parameter values are @qcode{@var{mu} = 0} and ## @qcode{@var{sigma} = 1}. Both parameters must be reals and ## @qcode{@var{sigma} > 0}. For @qcode{@var{sigma} <= 0}, @qcode{NaN} is ## returned. ## ## Further information about the log-normal distribution can be found at ## @url{https://en.wikipedia.org/wiki/Log-normal_distribution} ## ## @seealso{logncdf, logninv, lognrnd, lognfit, lognlike, lognstat} ## @end deftypefn function y = lognpdf (x, mu = 0, sigma = 1) ## Check for valid number of input arguments if (nargin < 1 || nargin > 3) print_usage (); endif ## Check for common size of P, MU, and SIGMA if (! isscalar (x) || ! isscalar (mu) || ! isscalar (sigma)) [retval, x, mu, sigma] = common_size (x, mu, sigma); if (retval > 0) error ("lognpdf: X, MU, and SIGMA must be of common size or scalars"); endif endif ## Check for X, MU, and SIGMA being reals if (iscomplex (x) || iscomplex (mu) || iscomplex (sigma)) error ("lognpdf: X, MU, and SIGMA must not be complex"); endif ## Check for class type if (isa (x, "single") || isa (mu, "single") || isa (sigma, "single")) y = zeros (size (x), "single"); else y = zeros (size (x)); endif ## Compute lognormal PDF k = isnan (x) | !(sigma > 0) | !(sigma < Inf); y(k) = NaN; k = (x > 0) & (x < Inf) & (sigma > 0) & (sigma < Inf); if (isscalar (mu) && isscalar (sigma)) y(k) = normpdf (log (x(k)), mu, sigma) ./ x(k); else y(k) = normpdf (log (x(k)), mu(k), sigma(k)) ./ x(k); endif endfunction %!demo %! ## Plot various PDFs from the log-normal distribution %! x = 0:0.01:5; %! y1 = lognpdf (x, 0, 1); %! y2 = lognpdf (x, 0, 0.5); %! y3 = lognpdf (x, 0, 0.25); %! plot (x, y1, "-b", x, y2, "-g", x, y3, "-r") %! grid on %! ylim ([0, 2]) %! legend ({"μ = 0, σ = 1", "μ = 0, σ = 0.5", "μ = 0, σ = 0.25"}, ... %! "location", "northeast") %! title ("Log-normal PDF") %! xlabel ("values in x") %! ylabel ("density") ## Test output %!shared x, y %! x = [-1 0 e Inf]; %! y = [0, 0, 1/(e*sqrt(2*pi)) * exp(-1/2), 0]; %!assert (lognpdf (x, zeros (1,4), ones (1,4)), y, eps) %!assert (lognpdf (x, 0, ones (1,4)), y, eps) %!assert (lognpdf (x, zeros (1,4), 1), y, eps) %!assert (lognpdf (x, [0 1 NaN 0], 1), [0 0 NaN y(4)], eps) %!assert (lognpdf (x, 0, [0 NaN Inf 1]), [NaN NaN NaN y(4)], eps) %!assert (lognpdf ([x, NaN], 0, 1), [y, NaN], eps) ## Test class of input preserved %!assert (lognpdf (single ([x, NaN]), 0, 1), single ([y, NaN]), eps ("single")) %!assert (lognpdf ([x, NaN], single (0), 1), single ([y, NaN]), eps ("single")) %!assert (lognpdf ([x, NaN], 0, single (1)), single ([y, NaN]), eps ("single")) ## Test input validation %!error lognpdf () %!error lognpdf (1,2,3,4) %!error lognpdf (ones (3), ones (2), ones (2)) %!error lognpdf (ones (2), ones (3), ones (2)) %!error lognpdf (ones (2), ones (2), ones (3)) %!error lognpdf (i, 2, 2) %!error lognpdf (2, i, 2) %!error lognpdf (2, 2, i) statistics-release-1.6.3/inst/dist_fun/lognrnd.m000066400000000000000000000156141456127120000217470ustar00rootroot00000000000000## Copyright (C) 2012 Rik Wehbring ## Copyright (C) 1995-2016 Kurt Hornik ## Copyright (C) 2023 Andreas Bertsatos ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{r} =} lognrnd (@var{mu}, @var{sigma}) ## @deftypefnx {statistics} {@var{r} =} lognrnd (@var{mu}, @var{sigma}, @var{rows}) ## @deftypefnx {statistics} {@var{r} =} lognrnd (@var{mu}, @var{sigma}, @var{rows}, @var{cols}, @dots{}) ## @deftypefnx {statistics} {@var{r} =} lognrnd (@var{mu}, @var{sigma}, [@var{sz}]) ## ## Random arrays from the lognormal distribution. ## ## @code{@var{r} = lognrnd (@var{mu}, @var{sigma})} returns an array of random ## numbers chosen from the lognormal distribution with mean parameter @var{mu} ## and standard deviation parameter @var{sigma}, each corresponding to the ## associated normal distribution. The size of @var{r} is the common size of ## @var{mu}, and @var{sigma}. A scalar input functions as a constant matrix of ## the same size as the other inputs. Both parameters must be reals and ## @qcode{@var{sigma} > 0}. For @qcode{@var{sigma} <= 0}, @qcode{NaN} is ## returned. ## ## Both parameters must be reals and @qcode{@var{sigma} > 0}. ## For @qcode{@var{sigma} <= 0}, @qcode{NaN} is returned. ## ## When called with a single size argument, @code{lognrnd} returns a square ## matrix with the dimension specified. When called with more than one scalar ## argument, the first two arguments are taken as the number of rows and columns ## and any further arguments specify additional matrix dimensions. The size may ## also be specified with a row vector of dimensions, @var{sz}. ## ## Further information about the log-normal distribution can be found at ## @url{https://en.wikipedia.org/wiki/Log-normal_distribution} ## ## @seealso{logncdf, logninv, lognpdf, lognfit, lognlike, lognstat} ## @end deftypefn function r = lognrnd (mu, sigma, varargin) ## Check for valid number of input arguments if (nargin < 2) error ("lognrnd: function called with too few input arguments."); endif ## Check for common size of P, MU, and SIGMA if (! isscalar (mu) || ! isscalar (sigma)) [retval, mu, sigma] = common_size (mu, sigma); if (retval > 0) error ("lognrnd: MU and SIGMA must be of common size or scalars."); endif endif ## Check for X, MU, and SIGMA being reals if (iscomplex (mu) || iscomplex (sigma)) error ("lognrnd: MU and SIGMA must not be complex."); endif ## Parse and check SIZE arguments if (nargin == 2) sz = size (mu); elseif (nargin == 3) if (isscalar (varargin{1}) && varargin{1} >= 0 ... && varargin{1} == fix (varargin{1})) sz = [varargin{1}, varargin{1}]; elseif (isrow (varargin{1}) && all (varargin{1} >= 0) ... && all (varargin{1} == fix (varargin{1}))) sz = varargin{1}; elseif error (strcat (["lognrnd: SZ must be a scalar or a row vector"], ... [" of non-negative integers."])); endif elseif (nargin > 3) posint = cellfun (@(x) (! isscalar (x) || x < 0 || x != fix (x)), varargin); if (any (posint)) error ("lognrnd: dimensions must be non-negative integers."); endif sz = [varargin{:}]; endif ## Check that parameters match requested dimensions in size if (! isscalar (mu) && ! isequal (size (mu), sz)) error ("lognrnd: MU and SIGMA must be scalars or of size SZ."); endif ## Check for class type if (isa (mu, "single") || isa (sigma, "single")) cls = "single"; else cls = "double"; endif ## Generate random sample from lognormal distribution if (isscalar (mu) && isscalar (sigma)) if ((sigma > 0) && (sigma < Inf)) r = exp (mu + sigma * randn (sz, cls)); else r = NaN (sz, cls); endif else r = exp (mu + sigma .* randn (sz, cls)); k = (sigma < 0) | (sigma == Inf); r(k) = NaN; endif endfunction ## Test output %!assert (size (lognrnd (1, 1)), [1 1]) %!assert (size (lognrnd (1, ones (2,1))), [2, 1]) %!assert (size (lognrnd (1, ones (2,2))), [2, 2]) %!assert (size (lognrnd (ones (2,1), 1)), [2, 1]) %!assert (size (lognrnd (ones (2,2), 1)), [2, 2]) %!assert (size (lognrnd (1, 1, 3)), [3, 3]) %!assert (size (lognrnd (1, 1, [4, 1])), [4, 1]) %!assert (size (lognrnd (1, 1, 4, 1)), [4, 1]) %!assert (size (lognrnd (1, 1, 4, 1, 5)), [4, 1, 5]) %!assert (size (lognrnd (1, 1, 0, 1)), [0, 1]) %!assert (size (lognrnd (1, 1, 1, 0)), [1, 0]) %!assert (size (lognrnd (1, 1, 1, 2, 0, 5)), [1, 2, 0, 5]) ## Test class of input preserved %!assert (class (lognrnd (1, 1)), "double") %!assert (class (lognrnd (1, single (1))), "single") %!assert (class (lognrnd (1, single ([1, 1]))), "single") %!assert (class (lognrnd (single (1), 1)), "single") %!assert (class (lognrnd (single ([1, 1]), 1)), "single") ## Test input validation %!error lognrnd () %!error lognrnd (1) %!error ... %! lognrnd (ones (3), ones (2)) %!error ... %! lognrnd (ones (2), ones (3)) %!error lognrnd (i, 2, 3) %!error lognrnd (1, i, 3) %!error ... %! lognrnd (1, 2, -1) %!error ... %! lognrnd (1, 2, 1.2) %!error ... %! lognrnd (1, 2, ones (2)) %!error ... %! lognrnd (1, 2, [2 -1 2]) %!error ... %! lognrnd (1, 2, [2 0 2.5]) %!error ... %! lognrnd (1, 2, 2, -1, 5) %!error ... %! lognrnd (1, 2, 2, 1.5, 5) %!error ... %! lognrnd (2, ones (2), 3) %!error ... %! lognrnd (2, ones (2), [3, 2]) %!error ... %! lognrnd (2, ones (2), 3, 2) statistics-release-1.6.3/inst/dist_fun/mnpdf.m000066400000000000000000000076661456127120000214200ustar00rootroot00000000000000## Copyright (C) 2012 Arno Onken ## ## This program is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program. If not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{y} =} mnpdf (@var{x}, @var{pk}) ## ## Multinomial probability density function (PDF). ## ## @subheading Arguments ## ## @itemize @bullet ## @item ## @var{x} is vector with a single sample of a multinomial distribution with ## parameter @var{pk} or a matrix of random samples from multinomial ## distributions. In the latter case, each row of @var{x} is a sample from a ## multinomial distribution with the corresponding row of @var{pk} being its ## parameter. ## ## @item ## @var{pk} is a vector with the probabilities of the categories or a matrix ## with each row containing the probabilities of a multinomial sample. ## @end itemize ## ## @subheading Return values ## ## @itemize @bullet ## @item ## @var{y} is a vector of probabilites of the random samples @var{x} from the ## multinomial distribution with corresponding parameter @var{pk}. The parameter ## @var{n} of the multinomial distribution is the sum of the elements of each ## row of @var{x}. The length of @var{y} is the number of columns of @var{x}. ## If a row of @var{pk} does not sum to @code{1}, then the corresponding element ## of @var{y} will be @code{NaN}. ## @end itemize ## ## @subheading Examples ## ## @example ## @group ## x = [1, 4, 2]; ## pk = [0.2, 0.5, 0.3]; ## y = mnpdf (x, pk); ## @end group ## ## @group ## x = [1, 4, 2; 1, 0, 9]; ## pk = [0.2, 0.5, 0.3; 0.1, 0.1, 0.8]; ## y = mnpdf (x, pk); ## @end group ## @end example ## ## @subheading References ## ## @enumerate ## @item ## Wendy L. Martinez and Angel R. Martinez. @cite{Computational Statistics ## Handbook with MATLAB}. Appendix E, pages 547-557, Chapman & Hall/CRC, 2001. ## ## @item ## Merran Evans, Nicholas Hastings and Brian Peacock. @cite{Statistical ## Distributions}. pages 134-136, Wiley, New York, third edition, 2000. ## @end enumerate ## ## @seealso{mnrnd} ## @end deftypefn function y = mnpdf (x, pk) # Check arguments if (nargin != 2) print_usage (); endif if (! ismatrix (x) || any (x(:) < 0 | round (x(:) != x(:)))) error ("mnpdf: X must be a matrix of non-negative integer values."); endif if (! ismatrix (pk) || any (pk(:) < 0)) error ("mnpdf: PK must be a non-empty matrix with rows of probabilities."); endif # Adjust input sizes if (! isvector (x) || ! isvector (pk)) if (isvector (x)) x = x(:)'; endif if (isvector (pk)) pk = pk(:)'; endif if (size (x, 1) == 1 && size (pk, 1) > 1) x = repmat (x, size (pk, 1), 1); elseif (size (x, 1) > 1 && size (pk, 1) == 1) pk = repmat (pk, size (x, 1), 1); endif endif # Continue argument check if (any (size (x) != size (pk))) error ("mnpdf: X and PK must have compatible sizes."); endif # Count total number of elements of each multinomial sample n = sum (x, 2); # Compute probability density function of the multinomial distribution t = x .* log (pk); t(x == 0) = 0; y = exp (gammaln (n+1) - sum (gammaln (x+1), 2) + sum (t, 2)); # Set invalid rows to NaN k = (abs (sum (pk, 2) - 1) > 1e-6); y(k) = NaN; endfunction %!test %! x = [1, 4, 2]; %! pk = [0.2, 0.5, 0.3]; %! y = mnpdf (x, pk); %! assert (y, 0.11812, 0.001); %!test %! x = [1, 4, 2; 1, 0, 9]; %! pk = [0.2, 0.5, 0.3; 0.1, 0.1, 0.8]; %! y = mnpdf (x, pk); %! assert (y, [0.11812; 0.13422], 0.001); statistics-release-1.6.3/inst/dist_fun/mnrnd.m000066400000000000000000000134251456127120000214200ustar00rootroot00000000000000## Copyright (C) 2012 Arno Onken ## ## This program is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program. If not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{r} =} mnrnd (@var{n}, @var{pk}) ## @deftypefnx {statistics} {@var{r} =} mnrnd (@var{n}, @var{pk}, @var{s}) ## ## Random arrays from the multinomial distribution. ## ## @subheading Arguments ## ## @itemize @bullet ## @item ## @var{n} is the first parameter of the multinomial distribution. @var{n} can ## be scalar or a vector containing the number of trials of each multinomial ## sample. The elements of @var{n} must be non-negative integers. ## ## @item ## @var{pk} is the second parameter of the multinomial distribution. @var{pk} ## can be a vector with the probabilities of the categories or a matrix with ## each row containing the probabilities of a multinomial sample. If @var{pk} ## has more than one row and @var{n} is non-scalar, then the number of rows of ## @var{pk} must match the number of elements of @var{n}. ## ## @item ## @var{s} is the number of multinomial samples to be generated. @var{s} must ## be a non-negative integer. If @var{s} is specified, then @var{n} must be ## scalar and @var{pk} must be a vector. ## @end itemize ## ## @subheading Return values ## ## @itemize @bullet ## @item ## @var{r} is a matrix of random samples from the multinomial distribution with ## corresponding parameters @var{n} and @var{pk}. Each row corresponds to one ## multinomial sample. The number of columns, therefore, corresponds to the ## number of columns of @var{pk}. If @var{s} is not specified, then the number ## of rows of @var{r} is the maximum of the number of elements of @var{n} and ## the number of rows of @var{pk}. If a row of @var{pk} does not sum to ## @code{1}, then the corresponding row of @var{r} will contain only @code{NaN} ## values. ## @end itemize ## ## @subheading Examples ## ## @example ## @group ## n = 10; ## pk = [0.2, 0.5, 0.3]; ## r = mnrnd (n, pk); ## @end group ## ## @group ## n = 10 * ones (3, 1); ## pk = [0.2, 0.5, 0.3]; ## r = mnrnd (n, pk); ## @end group ## ## @group ## n = (1:2)'; ## pk = [0.2, 0.5, 0.3; 0.1, 0.1, 0.8]; ## r = mnrnd (n, pk); ## @end group ## @end example ## ## @subheading References ## ## @enumerate ## @item ## Wendy L. Martinez and Angel R. Martinez. @cite{Computational Statistics ## Handbook with MATLAB}. Appendix E, pages 547-557, Chapman & Hall/CRC, 2001. ## ## @item ## Merran Evans, Nicholas Hastings and Brian Peacock. @cite{Statistical ## Distributions}. pages 134-136, Wiley, New York, third edition, 2000. ## @end enumerate ## ## @seealso{mnpdf} ## @end deftypefn function r = mnrnd (n, pk, s) # Check arguments if (nargin == 3) if (! isscalar (n) || n < 0 || round (n) != n) error ("mnrnd: N must be a non-negative integer."); endif if (! isvector (pk) || any (pk < 0 | pk > 1)) error ("mnrnd: PK must be a vector of probabilities."); endif if (! isscalar (s) || s < 0 || round (s) != s) error ("mnrnd: S must be a non-negative integer."); endif elseif (nargin == 2) if (isvector (pk) && size (pk, 1) > 1) pk = pk'; endif if (! isvector (n) || any (n < 0 | round (n) != n) || size (n, 2) > 1) error ("mnrnd: N must be a non-negative integer column vector."); endif if (! ismatrix (pk) || isempty (pk) || any (pk < 0 | pk > 1)) error (strcat (["mnrnd: PK must be a non-empty matrix with"], ... [" rows of probabilities."])); endif if (! isscalar (n) && size (pk, 1) > 1 && length (n) != size (pk, 1)) error ("mnrnd: the length of N must match the number of rows of PK."); endif else print_usage (); endif # Adjust input sizes if (nargin == 3) n = n * ones (s, 1); pk = repmat (pk(:)', s, 1); elseif (nargin == 2) if (isscalar (n) && size (pk, 1) > 1) n = n * ones (size (pk, 1), 1); elseif (size (pk, 1) == 1) pk = repmat (pk, length (n), 1); endif endif sz = size (pk); # Upper bounds of categories ub = cumsum (pk, 2); # Make sure that the greatest upper bound is 1 gub = ub(:, end); ub(:, end) = 1; # Lower bounds of categories lb = [zeros(sz(1), 1) ub(:, 1:(end-1))]; # Draw multinomial samples r = zeros (sz); for i = 1:sz(1) # Draw uniform random numbers r_tmp = repmat (rand (n(i), 1), 1, sz(2)); # Compare the random numbers of r_tmp to the cumulated probabilities of pk # and count the number of samples for each category r(i, :) = sum (r_tmp <= repmat (ub(i, :), n(i), 1) & ... r_tmp > repmat (lb(i, :), n(i), 1), 1); endfor # Set invalid rows to NaN k = (abs (gub - 1) > 1e-6); r(k, :) = NaN; endfunction %!test %! n = 10; %! pk = [0.2, 0.5, 0.3]; %! r = mnrnd (n, pk); %! assert (size (r), size (pk)); %! assert (all (r >= 0)); %! assert (all (round (r) == r)); %! assert (sum (r) == n); %!test %! n = 10 * ones (3, 1); %! pk = [0.2, 0.5, 0.3]; %! r = mnrnd (n, pk); %! assert (size (r), [length(n), length(pk)]); %! assert (all (r >= 0)); %! assert (all (round (r) == r)); %! assert (all (sum (r, 2) == n)); %!test %! n = (1:2)'; %! pk = [0.2, 0.5, 0.3; 0.1, 0.1, 0.8]; %! r = mnrnd (n, pk); %! assert (size (r), size (pk)); %! assert (all (r >= 0)); %! assert (all (round (r) == r)); %! assert (all (sum (r, 2) == n)); statistics-release-1.6.3/inst/dist_fun/mvncdf.m000066400000000000000000000403641456127120000215610ustar00rootroot00000000000000## Copyright (C) 2008 Arno Onken ## Copyright (C) 2022-2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{p} =} mvncdf (@var{x}) ## @deftypefnx {statistics} {@var{p} =} mvncdf (@var{x}, @var{mu}, @var{sigma}) ## @deftypefnx {statistics} {@var{p} =} mvncdf (@var{x_lo}, @var{x_up}, @var{mu}, @var{sigma}) ## @deftypefnx {statistics} {@var{p} =} mvncdf (@dots{}, @var{options}) ## @deftypefnx {statistics} {[@var{p}, @var{err}] =} mvncdf (@dots{}) ## ## Multivariate normal cumulative distribution function (CDF). ## ## @code{@var{p} = mvncdf (@var{x})} returns the cumulative probability of the ## multivariate normal distribution evaluated at each row of @var{x} with zero ## mean and an identity covariance matrix. The rows of matrix @var{x} ## correspond to observations and its columns to variables. The return argument ## @var{p} is a column vector with the same number of rows as in @var{x}. ## ## @code{@var{p} = mvncdf (@var{x}, @var{mu}, @var{sigma})} returns cumulative ## probability of the multivariate normal distribution evaluated at each row of ## @var{x} with mean @var{mu} and a covariance matrix @var{sigma}. @var{mu} can ## be either a scalar (the same of every variable) or a row vector with the same ## number of elements as the number of variables in @var{x}. @var{sigma} ## covariance matrix may be specified a row vector if it only contains variances ## along its diagonal and zero covariances of the diagonal. In such a case, the ## diagonal vector @var{sigma} must have the same number of elements as the ## number of variables (columns) in @var{x}. If you only want to specify sigma, ## you can pass an empty matrix for @var{mu}. ## ## The multivariate normal cumulative probability at @var{x} is defined as the ## probability that a random vector @math{V}, distributed as multivariate ## normal, will fall within the semi-infinite rectangle with upper limits ## defined by @var{x}. ## @itemize ## @item @math{Pr@{V(1)<=X(1), V(2)<=X(2), ... V(D)<=X(D)@}}. ## @end itemize ## ## @code{@var{p} = mvncdf (@var{x_lo}, @var{x_hi}, @var{mu}, @var{sigma})} ## returns the multivariate normal cumulative probability evaluated over the ## rectangle (hyper-rectangle for multivariate data in @var{x}) with lower and ## upper limits defined by @var{x_lo} and @var{x_hi}, respectively. ## ## @code{[@var{p}, @var{err}] = mvncdf (@dots{})} also returns an error estimate ## @var{err} in @var{p}. ## ## @code{@var{p} = mvncdf (@dots{}, @var{options})} specifies the structure, ## which controls specific parameters for the numerical integration used to ## compute @var{p}. The required fieds are: ## ## @multitable @columnfractions 0.2 0.05 0.75 ## @item @qcode{"TolFun"} @tab @tab Maximum absolute error tolerance. Default ## is 1e-8 for D < 4, or 1e-4 for D >= 4. Note that for bivariate normal cdf, ## the Octave implementation has a presicion of more than 1e-10. ## ## @item @qcode{"MaxFunEvals"} @tab @tab Maximum number of integrand ## evaluations. Default is 1e7 for D > 4. ## ## @item @qcode{"Display"} @tab @tab Display options. Choices are @qcode{"off"} ## (default), @qcode{"iter"}, which shows the probability and estimated error at ## each repetition, and @qcode{"final"}, which shows the final probability and ## related error after the integrand has converged successfully. ## @end multitable ## ## @seealso{bvncdf, mvnpdf, mvnrnd} ## @end deftypefn function [p, err] = mvncdf (varargin) ## Check for valid number on input and output arguments narginchk (1,5); ## Check for 'options' structure and parse parameters or add defaults if (isstruct (varargin{end})) if (isfield (varargin{end}, "TolFun")) TolFun = varargin{end}.TolFun; else error ("mvncdf: options structure missing 'TolFun' field."); endif if (isempty (TolFun) && size (varargin{1}, 2) < 4) TolFun = 1e-8; elseif (isempty (TolFun) && size (varargin{1}, 2) < 26) TolFun = 1e-4; endif if (isfield (varargin{end}, "MaxFunEvals")) MaxFunEvals = varargin{end}.MaxFunEvals; else error ("mvncdf: options structure missing 'MaxFunEvals' field."); endif if (isempty (MaxFunEvals)) MaxFunEvals = 1e7; endif if (isfield (varargin{end}, "Display")) Display = varargin{end}.Display; else error ("mvncdf: options structure missing 'Display' field."); endif DispOptions = {"off", "final", "iter"}; if (sum (any (strcmpi (Display, DispOptions))) == 0) error ("mvncdf: 'Display' field in 'options' has invalid value."); endif rem_nargin = nargin - 1; else if (size (varargin{1}, 2) < 4) TolFun = 1e-8; elseif (size (varargin{1}, 2) < 26) TolFun = 1e-4; endif MaxFunEvals = 1e7; Display = "off"; rem_nargin = nargin; endif ## Check for X of X_lo and X_up if (rem_nargin < 4) # MVNCDF(X_UP,MU,SIGMA) x_up_Only = true; x_up = varargin{1}; ## Check for x being a matrix if (! ismatrix (x_up)) error ("mvncdf: X must be a matrix."); endif ## Create x_lo according to data type of x_lo x_lo = - Inf (size (x_up)); if isa (x_up, "single") x_lo = single (x_lo); endif ## Check for mu and sigma arguments if (rem_nargin > 1) mu = varargin{2}; else mu = []; endif if (rem_nargin > 2) sigma = varargin{3}; else sigma = []; endif else # MVNCDF(X_LO,X_UP,MU,SIGMA) x_up_Only = false; x_lo = varargin{1}; x_up = varargin{2}; mu = varargin{3}; sigma = varargin{4}; ## Check for x_lo and x_up being matrices of the same size ## and that they define increasing limits if (! ismatrix (x_lo) || ! ismatrix (x_up)) error ("mvncdf: X_LO and X_UP must be matrices."); endif if (size (x_lo) != size (x_up)) error ("mvncdf: X_LO and X_UP must have the same size."); endif if (any (any (x_lo > x_up))) error ("mvncdf: X_LO and X_UP must define increasing limits."); endif endif ## Check if data is single or double class is_type = "double"; if (isa (x_lo, "single")) is_type = "single"; endif ## Get size of data [n_x, d_x] = size (x_lo); ## Center data according to mu if (isempty (mu)) # already centered XLo0 = x_lo; XUp0 = x_up; elseif (isscalar (mu)) # mu is a scalar XLo0 = x_lo - mu; XUp0 = x_up - mu; elseif (isvector (mu)) # mu is a vector ## Get size of mu vector [n_mu, d_mu] = size (mu); if (d_mu != d_x) error ("mvncdf: wrong size of MU vector."); endif if (n_mu == 1 || n_mu == n_x) XLo0 = x_lo - mu; XUp0 = x_up - mu; else error ("mvncdf: wrong size of MU vector."); endif else error ("mvncdf: MU must be either empty, a scalar, or a vector."); endif ## Check how sigma was parsed if (isempty (sigma)) # already standardized ## If x_lo and x_up are column vectors, transpose them to row vectors if (d_x == 1) XLo0 = XLo0'; XUp0 = XUp0'; [n_x, d_x] = size (XUp0); endif sigmaIsDiag = true; sigma = ones (1, d_x); else ## Check if sigma parsed as diagonal vector if (size (sigma, 1) == 1 && size (sigma, 2) > 1) sigmaIsDiag = true; else sigmaIsDiag = false; endif ## If x_lo and x_up are column vectors, transpose them to row vectors if (d_x == 1) if (isequal (size (sigma), [1, n_x])) XLo0 = XLo0'; XUp0 = XUp0'; [n_x, d_x] = size (XUp0); elseif (! isscalar (mu)) error ("mvncdf: MU must be a scalar if SIGMA is a vector."); endif endif ## Check for sigma being a valid covariance matrix if (! sigmaIsDiag && (size (sigma, 1) != size (sigma, 2))) error ("mvncdf: covariance matrix SIGMA is not square."); elseif (! sigmaIsDiag && (! all (size (sigma) == [d_x, d_x]))) error (strcat (["mvncdf: covariance matrix SIGMA does"], ... [" not match dimensions in data."])); else ## If sigma is a covariance matrix check that it is positive semi-definite if (! sigmaIsDiag) [~, err] = chol (sigma); if (err != 0) error (strcat (["mvncdf: covariance matrix SIGMA must be"], ... [" positive semi-definite."])); endif else if (any (sigma) <= 0) error ("mvncdf: invalid SIGMA diagonal vector."); endif endif endif endif ## Standardize sigma and x data if (sigmaIsDiag) XLo0 = XLo0 ./ sqrt (sigma); XUp0 = XUp0 ./ sqrt (sigma); else s = sqrt (diag (sigma))'; XLo0 = XLo0 ./ s; XUp0 = XUp0 ./ s; Rho = sigma ./ (s * s'); endif ## Compute the cdf from standardized values. if (d_x == 1) p = normcdf (XUp0, 0, 1) - normcdf (XLo0, 0, 1); if (nargout > 1) err = NaN (size (p), is_type); endif elseif (sigmaIsDiag) p = prod (normcdf (XUp0, 0, 1) - normcdf (XLo0, 0, 1), 2); if (nargout > 1) err = NaN (size (p), is_type); endif elseif (d_x < 4) if (x_up_Only) # upper limit only if (d_x == 2) p = bvncdf (x_up, mu, sigma); else p = tvncdf (XUp0, Rho([2 3 6]), TolFun); endif else # lower and upper limits present ## Compute the probability over the rectangle as sums and differences ## of integrals over semi-infinite half-rectangles. For degenerate ## rectangles, force an exact zero by making each piece exactly zero. equalLimits = (XUp0 == XLo0); XUp0(equalLimits) = -Inf; XLo0(equalLimits) = -Inf; ## For bvncdf x_up(equalLimits) = -Inf; x_lo(equalLimits) = -Inf; p = zeros(n_x, 1, is_type); for i = 0:d_x k = nchoosek (1:d_x, i); for j = 1:size (k, 1) X = XUp0; X(:,k(j,:)) = XLo0(:,k(j,:)); if d_x == 2 x = x_up; x(:,k(j,:)) = x_lo(:,k(j,:)); p = p + (-1) ^ i * bvncdf (x, mu, sigma); else p = p + (-1) ^ i * tvncdf (X, Rho([2 3 6]), TolFun / 8); endif endfor endfor endif if (nargout > 1) err = repmat (cast (TolFun, is_type), size (p)); endif elseif (d_x < 26) p = zeros (n_x, 1, is_type); err = zeros (n_x, 1, is_type); for i = 1:n_x [p(i), err(i)] = mvtcdfqmc (XLo0(i,:), XUp0(i,:), Rho, Inf, ... TolFun, MaxFunEvals, Display); endfor else error ("mvncdf: too many dimensions in data (limit = 25 columns)."); endif ## Bound p in range [0, 1] p(p < 0) = 0; p(p > 1) = 1; endfunction ## function for computing a trivariate normal cdf function p = tvncdf (x, rho, tol) ## Get size of data n = size(x,1); ## Check if data is single or double class is_type = "double"; if (isa (x, "single") || isa (rho, "single")) is_type = "single"; endif ## Find a permutation that makes rho_32 == max(rho) [dum,imax] = max(abs(rho)); %#ok if imax == 1 % swap 1 and 3 rho_21 = rho(3); rho_31 = rho(2); rho_32 = rho(1); x = x(:,[3 2 1]); elseif imax == 2 % swap 1 and 2 rho_21 = rho(1); rho_31 = rho(3); rho_32 = rho(2); x = x(:,[2 1 3]); else % imax == 3 rho_21 = rho(1); rho_31 = rho(2); rho_32 = rho(3); end phi = 0.5 * erfc (- x(:,1) / sqrt (2)); p1 = phi .* bvncdf (x(:,2:3), [], rho_32); if abs(rho_21) > 0 loLimit = 0; hiLimit = asin(rho_21); rho_j1 = rho_21; rho_k1 = rho_31; p2 = zeros (size (p1), is_type); for i = 1:n b1 = x(i,1); bj = x(i,2); bk = x(i,3); if isfinite(b1) && isfinite(bj) && ~isnan(bk) p2(i) = quadgk(@tvnIntegrand,loLimit,hiLimit,'AbsTol',tol/3,'RelTol',0); endif endfor else p2 = zeros (size (p1), is_type); endif if abs(rho_31) > 0 loLimit = 0; hiLimit = asin(rho_31); rho_j1 = rho_31; rho_k1 = rho_21; p3 = zeros (size (p1), is_type); for i = 1:n b1 = x(i,1); bj = x(i,3); bk = x(i,2); if isfinite(b1) && isfinite(bj) && ~isnan(bk) p3(i) = quadgk(@tvnIntegrand,loLimit,hiLimit,'AbsTol',tol/3,'RelTol',0); endif endfor else p3 = zeros (size (p1), is_type); endif p = cast (p1 + (p2 + p3) ./ (2 .* pi), is_type); function integrand = tvnIntegrand(theta) # Integrand is exp( -(b1.^2 + bj.^2 - 2*b1*bj*sin(theta))/(2*cos(theta).^2)) sintheta = sin (theta); cossqtheta = cos (theta) .^ 2; expon = ((b1 * sintheta - bj) .^ 2 ./ cossqtheta + b1 .^ 2) / 2; sinphi = sintheta .* rho_k1 ./ rho_j1; numeru = bk .* cossqtheta - b1 .* (sinphi - rho_32 .* sintheta) ... - bj .* (rho_32 - sintheta .* sinphi); denomu = sqrt (cossqtheta .* (cossqtheta - sinphi .* sinphi ... - rho_32 .* (rho_32 - 2 .* sintheta .* sinphi))); phi = 0.5 * erfc (- (numeru ./ denomu) / sqrt (2)); integrand = exp (- expon) .* phi; endfunction endfunction %!demo %! mu = [1, -1]; %! Sigma = [0.9, 0.4; 0.4, 0.3]; %! [X1, X2] = meshgrid (linspace (-1, 3, 25)', linspace (-3, 1, 25)'); %! X = [X1(:), X2(:)]; %! p = mvncdf (X, mu, Sigma); %! Z = reshape (p, 25, 25); %! surf (X1, X2, Z); %! title ("Bivariate Normal Distribution"); %! ylabel "X1" %! xlabel "X2" %!demo %! mu = [0, 0]; %! Sigma = [0.25, 0.3; 0.3, 1]; %! p = mvncdf ([0 0], [1 1], mu, Sigma); %! x1 = -3:.2:3; %! x2 = -3:.2:3; %! [X1, X2] = meshgrid (x1, x2); %! X = [X1(:), X2(:)]; %! p = mvnpdf (X, mu, Sigma); %! p = reshape (p, length (x2), length (x1)); %! contour (x1, x2, p, [0.0001, 0.001, 0.01, 0.05, 0.15, 0.25, 0.35]); %! xlabel ("x"); %! ylabel ("p"); %! title ("Probability over Rectangular Region"); %! line ([0, 0, 1, 1, 0], [1, 0, 0, 1, 1], "Linestyle", "--", "Color", "k"); %!test %! fD = (-2:2)'; %! X = repmat (fD, 1, 4); %! p = mvncdf (X); %! assert (p, [0; 0.0006; 0.0625; 0.5011; 0.9121], ones (5, 1) * 1e-4); %!test %! mu = [1, -1]; %! Sigma = [0.9, 0.4; 0.4, 0.3]; %! [X1,X2] = meshgrid (linspace (-1, 3, 25)', linspace (-3, 1, 25)'); %! X = [X1(:), X2(:)]; %! p = mvncdf (X, mu, Sigma); %! p_out = [0.00011878988774500, 0.00034404112322371, ... %! 0.00087682502191813, 0.00195221905058185, ... %! 0.00378235566873474, 0.00638175749734415, ... %! 0.00943764224329656, 0.01239164888125426, ... %! 0.01472750274376648, 0.01623228313374828]'; %! assert (p([1:10]), p_out, 1e-16); %!test %! mu = [1, -1]; %! Sigma = [0.9, 0.4; 0.4, 0.3]; %! [X1,X2] = meshgrid (linspace (-1, 3, 25)', linspace (-3, 1, 25)'); %! X = [X1(:), X2(:)]; %! p = mvncdf (X, mu, Sigma); %! p_out = [0.8180695783608276, 0.8854485749482751, ... %! 0.9308108777385832, 0.9579855743025508, ... %! 0.9722897881414742, 0.9788150170059926, ... %! 0.9813597788804785, 0.9821977956568989, ... %! 0.9824283794464095, 0.9824809345614861]'; %! assert (p([616:625]), p_out, 2e-16); %!test %! mu = [0, 0]; %! Sigma = [0.25, 0.3; 0.3, 1]; %! [p, err] = mvncdf ([0 0], [1 1], mu, Sigma); %! assert (p, 0.2097424404755626, 1e-16); %! assert (err, 1e-08); %!test %! x = [1 2]; %! mu = [0.5 1.5]; %! sigma = [1.0 0.5; 0.5 1.0]; %! p = mvncdf (x, mu, sigma); %! assert (p, 0.546244443857090, 1e-15); %!test %! x = [1 2]; %! mu = [0.5 1.5]; %! sigma = [1.0 0.5; 0.5 1.0]; %! a = [-inf 0]; %! p = mvncdf (a, x, mu, sigma); %! assert (p, 0.482672935215631, 1e-15); %!error p = mvncdf (randn (25,26), [], eye (26)); %!error p = mvncdf (randn (25,8), [], eye (9)); %!error p = mvncdf (randn (25,4), randn (25,5), [], eye (4)); %!error p = mvncdf (randn (25,4), randn (25,4), [2, 3; 2, 3], eye (4)); %!error p = mvncdf (randn (25,4), randn (25,4), ones (1, 5), eye (4)); %!error p = mvncdf ([-inf 0], [1, 2], [0.5 1.5], [1.0 0.5; 0.5 1.0], option); statistics-release-1.6.3/inst/dist_fun/mvnpdf.m000066400000000000000000000177321456127120000216010ustar00rootroot00000000000000## Copyright (C) 2022-2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{y} =} mvnpdf (@var{x}, @var{mu}, @var{sigma}) ## ## Multivariate normal probability density function (PDF). ## ## @code{@var{y} = mvnpdf (@var{x})} returns the probability density of the ## multivariate normal distribution with zero mean and identity covariance ## matrix, evaluated at each row of @var{x}. Rows of the N-by-D matrix @var{x} ## correspond to observations orpoints, and columns correspond to variables or ## coordinates. @var{y} is an N-by-1 vector. ## ## @code{@var{y} = mvnpdf (@var{x}, @var{mu})} returns the density of the ## multivariate normal distribution with mean MU and identity covariance matrix, ## evaluated at each row of @var{x}. @var{mu} is a 1-by-D vector, or an N-by-D ## matrix, in which case the density is evaluated for each row of @var{x} with ## the corresponding row of @var{mu}. @var{mu} can also be a scalar value, ## which MVNPDF replicates to match the size of @var{x}. ## ## @code{@var{y} = mvnpdf (@var{x}, @var{mu}, @var{sigma})} returns the density ## of the multivariate normal distribution with mean @var{mu} and covariance ## @var{sigma}, evaluated at each row of @var{x}. @var{sigma} is a D-by-D ## matrix, or an D-by-D-by-N array, in which case the density is evaluated for ## each row of @var{x} with the corresponding page of @var{sigma}, i.e., ## @code{mvnpdf} computes @var{y(i)} using @var{x(i,:)} and @var{sigma(:,:,i)}. ## If the covariance matrix is diagonal, containing variances along the diagonal ## and zero covariances off the diagonal, @var{sigma} may also be specified as a ## 1-by-D matrix or a 1-by-D-by-N array, containing just the diagonal. Pass in ## the empty matrix for @var{mu} to use its default value when you want to only ## specify @var{sigma}. ## ## If @var{x} is a 1-by-D vector, @code{mvnpdf} replicates it to match the ## leading dimension of @var{mu} or the trailing dimension of @var{sigma}. ## ## @seealso{mvncdf, mvnrnd} ## @end deftypefn function y = mvnpdf (x, mu, sigma) ## Check for valid number of input arguments if (nargin < 1) error ("mvnpdf: too few input arguments."); endif if (nargin < 3) sigma = []; endif ## Check for valid size of data [row, col] = size (x); if (col < 1) error ("mvnpdf: too few dimensions in X."); elseif (ndims (x) != 2) error ("mvnpdf: wrong dimensions in X."); endif ## Check for second input argument or assume zero mean if (nargin < 2 || isempty (mu)) xc = x; # already centered elseif (numel (mu) == 1) xc = x - mu; # mu is a scalar elseif (ndims (mu) == 2) [rm, cm] = size (mu); # mu is a vector if (cm != col) error ("mvnpdf: columns in X and MU mismatch."); elseif (rm == row) xc = x - mu; elseif (rm == 1 || row == 1) xc = bsxfun (@minus, x, mu); else error ("mvnpdf: rows in X and MU mismatch."); endif else error ("mvnpdf: wrong size of MU."); endif [row, col] = size (xc); ## Check for third input argument or assume identity covariance if (nargin < 2 || isempty (sigma)) ## already standardized if (col == 1 && row > 1) xRinv = xc'; # make row vector col == row; else xRinv = xc; col == row; endif lnSDS = 0; elseif (ndims (sigma) == 2) ## Single covariance matrix [rs, cs] = size (sigma); if (rs == 1 && cs > 1) rs = cs; # sigma passed as a diagonal is_diag = true; else is_diag = false; endif if (col == 1 && row > 1 && rs == row) xc = xc'; # make row vector col = row; endif ## Check sigma for correct size if (rs != cs) error ("mvnpdf: bad covariance matrix."); elseif (rs != col) error ("mvnpdf: covariance matrix mismatch."); else if (is_diag) ## Check sigma for invalid values if (any (sigma <= 0)) error ("mvnpdf: sigma diagonal contains negative or zero values."); endif R = sqrt (sigma); xRinv = bsxfun (@rdivide, xc, R); lnSDS = sum (log (R)); else ## Check for valid covariance matrix [R, err] = cholcov (sigma, 0); if (err != 0) error ("mvnpdf: invalid covariance matrix."); endif xRinv = xc / R; lnSDS = sum (log (diag (R))); endif endif elseif (ndims (sigma) == 3) ## Multiple covariance matrices sd = size (sigma); if (sd(1) == 1 && sd(2) > 1) sd(1) = sd(2); # sigma passed as a diagonal sigma = reshape (sigma, sd(2), sd(3))'; is_diag = true; else is_diag = false; endif if (col == 1 && row > 1 && sd(1) == row) xc = xc'; # make row vector [row, col] = size (xc); endif ## If X and MU are row vectors, match them with covariance if (row == 1) row = sd(3); xc = repmat (xc, row, 1); endif ## Check sigma for correct size if (sd(1) != sd(2)) error ("mvnpdf: bad multiple covariance matrix."); elseif (sd(1) != col || sd(2) != col) error ("mvnpdf: multiple covariance matrix mismatch."); elseif (sd(3) != row) error ("mvnpdf: multiple covariance pages mismatch."); else if (is_diag) ## Check sigma for invalid values if (any (any (sigma <= 0))) error ("mvnpdf: sigma diagonals contain negative or zero values."); endif R = sqrt (sigma); xRinv = xc ./ R; lnSDS = sum (log (R), 2); else ## Create arrays according to class type if (isa (x, "single") || isa (mu, "single") || isa (sigma, "single")) xRinv = zeros (row, col," single"); lnSDS = zeros (row, 1, "single"); else xRinv = zeros (row, col); lnSDS = zeros (row, 1); endif for i = 1:row ## Check for valid covariance matrices [R, err] = cholcov (sigma (:,:,i), 0); if (err != 0) error ("mvnpdf:invalid multiple covariance matrix."); endif xRinv(i,:) = xc(i,:) / R; lnSDS(i) = sum(log(diag(R))); endfor endif endif else error ("mvnpdf: wrong dimensions in covariance matrix."); endif ## Compute the PDF y = exp (-0.5 * sum (xRinv .^ 2, 2) - lnSDS - col * log (2 * pi) / 2); endfunction %!demo %! mu = [1, -1]; %! sigma = [0.9, 0.4; 0.4, 0.3]; %! [X1, X2] = meshgrid (linspace (-1, 3, 25)', linspace (-3, 1, 25)'); %! x = [X1(:), X2(:)]; %! p = mvnpdf (x, mu, sigma); %! surf (X1, X2, reshape (p, 25, 25)); ## Input validation tests %!error y = mvnpdf (); %!error y = mvnpdf ([]); %!error y = mvnpdf (ones (3,3,3)); %!error ... %! y = mvnpdf (ones (10, 2), [4, 2, 3]); %!error ... %! y = mvnpdf (ones (10, 2), [4, 2; 3, 2]); %!error ... %! y = mvnpdf (ones (10, 2), ones (3, 3, 3)); ## Output validation tests %!shared x, mu, sigma %! x = [1, 2, 5, 4, 6]; %! mu = [2, 0, -1, 1, 4]; %! sigma = [2, 2, 2, 2, 2]; %!assert (mvnpdf (x), 1.579343404440977e-20, 1e-30); %!assert (mvnpdf (x, mu), 1.899325144348102e-14, 1e-25); %!assert (mvnpdf (x, mu, sigma), 2.449062307156273e-09, 1e-20); statistics-release-1.6.3/inst/dist_fun/mvnrnd.m000066400000000000000000000163041456127120000216050ustar00rootroot00000000000000## Copyright (C) 2022-2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{r} =} mvnrnd (@var{mu}, @var{sigma}) ## @deftypefnx {statistics} {@var{r} =} mvnrnd (@var{mu}, @var{sigma}, @var{n}) ## @deftypefnx {statistics} {@var{r} =} mvnrnd (@var{mu}, @var{sigma}, @var{n}, @var{T}) ## @deftypefnx {statistics} {[@var{r}, @var{T}] =} mvnrnd (@dots{}) ## ## Random vectors from the multivariate normal distribution. ## ## @code{@var{r} = mvnrnd (@var{mu}, @var{sigma})} returns an N-by-D matrix ## @var{r} of random vectors chosen from the multivariate normal distribution ## with mean vector @var{mu} and covariance matrix @var{sigma}. @var{mu} is an ## N-by-D matrix, and @code{mvnrnd} generates each N of @var{r} using the ## corresponding N of @var{mu}. @var{sigma} is a D-by-D symmetric positive ## semi-definite matrix, or a D-by-D-by-N array. If @var{sigma} is an array, ## @code{mvnrnd} generates each N of @var{r} using the corresponding page of ## @var{sigma}, i.e., @code{mvnrnd} computes @var{r(i,:)} using @var{mu(i,:)} ## and @var{sigma(:,:,i)}. If the covariance matrix is diagonal, containing ## variances along the diagonal and zero covariances off the diagonal, ## @var{sigma} may also be specified as a 1-by-D matrix or a 1-by-D-by-N array, ## containing just the diagonal. If @var{mu} is a 1-by-D vector, @code{mvnrnd} ## replicates it to match the trailing dimension of SIGMA. ## ## @code{@var{r} = mvnrnd (@var{mu}, @var{sigma}, @var{n})} returns a N-by-D ## matrix R of random vectors chosen from the multivariate normal distribution ## with 1-by-D mean vector @var{mu}, and D-by-D covariance matrix @var{sigma}. ## ## @code{@var{r} = mvnrnd (@var{mu}, @var{sigma}, @var{n}, @var{T})} supplies ## the Cholesky factor @var{T} of @var{sigma}, so that @var{sigma(:,:,J)} == ## @var{T(:,:,J)}'*@var{T(:,:,J)} if @var{sigma} is a 3D array or @var{sigma} == ## @var{T}'*@var{T} if @var{sigma} is a matrix. No error checking is done on ## @var{T}. ## ## @code{[@var{r}, @var{T}] = mvnrnd (@dots{})} returns the Cholesky factor ## @var{T}, so it can be re-used to make later calls more efficient, although ## there are greater efficiency gains when SIGMA can be specified as a diagonal ## instead. ## ## @seealso{mvncdf, mvnpdf} ## @end deftypefn function [r, T] = mvnrnd (mu, sigma, N, T) ## Check input arguments if (nargin < 2 || isempty (mu) || isempty (sigma)) error ("mvnrnd: too few input arguments."); elseif (ndims (mu) > 2) error ("mvnrnd: wrong size of MU."); elseif (ndims (sigma) > 3) error ("mvnrnd: wrong size of SIGMA."); endif ## Get data type if (isa (mu, "single") || isa (sigma, "single")) is_class = "single"; else is_class = "double"; endif ## Check whether sigma is passed as a diagonal or a matrix sd = size (sigma); if (sd(1) == 1 && sd(2) > 1) sd(1) = sd(2); is_diag = true; else is_diag = false; endif ## Get size of mean vector [rm, cm] = size (mu); ## Make sure MU is a row vector if (cm == 1 && rm == sd(1)) mu = mu'; [rm, cm] = size (mu); endif ## Check for valid N input argument if (nargin < 3 || isempty (N)) N_empty = true; else N_empty = false; ## If MU is a row vector, rep it out to match N if (rm == 1) rm = N; mu = repmat (mu, rm, 1); elseif (rm != N) error ("mvnrnd: size mismatch of N and MU."); endif endif ## For single covariance matrix if (ndims (sigma) == 2) ## Check sigma for correct size if (sd(1) != sd(2)) error ("mvnpdf: bad covariance matrix."); elseif (! sd(1) == cm) error ("mvnpdf: covariance matrix mismatch."); endif ## Check for Cholesky factor T if (nargin > 3) r = randn (rm, size (T, 1), is_class) * T + mu; elseif (is_diag) ## Check sigma for invalid values if (any (sigma <= 0)) error ("mvnpdf: SIGMA diagonal contains negative or zero values."); endif t = sqrt (sigma); if (nargout > 1) T = diag (t); endif r = bsxfun (@times, randn (rm, cm, is_class), t) + mu; else ## Compute a Cholesky factorization [T, err] = cholcov (sigma); if (err != 0) error ("mvnrnd: covariance matrix is not positive definite."); endif r = randn (rm, size (T, 1), is_class) * T + mu; endif endif ## For multiple covariance matrices if (ndims (sigma) == 3) ## If MU is a row vector, rep it out to match sigma if (rm == 1 && N_empty) rm = sd(3); mu = repmat (mu, rm, 1); endif ## Check sigma for correct size if (sd(1) != sd(2)) error ("mvnpdf: bad multiple covariance matrix."); elseif (sd(1) != cm) error ("mvnpdf: multiple covariance matrix mismatch."); elseif (sd(3) != rm) error ("mvnpdf: multiple covariance pages mismatch."); endif ## Check for Cholesky factor T if (nargin < 4) # T not present if (nargout > 1) T = zeros (sd, is_class); endif if (is_diag) sigma = reshape(sigma,sd(2),sd(3))'; ## Check sigma for invalid values if (any (sigma(:) <= 0)) error ("mvnpdf: SIGMA diagonals contain negative or zero values."); endif R = sqrt(sigma); r = bsxfun (@times, randn (rm, cm, is_class), R) + mu; if (nargout > 1) for i = 1:rm T(:,:,i) = diag (R(i,:)); endfor endif else r = zeros (rm, cm, is_class); for i = 1:rm [R, err] = cholcov (sigma(:,:,i)); if (err != 0) error (strcat (["mvnrnd: multiple covariance matrix"], ... [" is not positive definite."])); endif Rrows = size (R,1); r(i,:) = randn (1, Rrows, is_class) * R + mu(i,:); if (nargout > 1) T(1:Rrows,:,i) = R; endif endfor endif else # T present r = zeros (rm, cm, is_class); for i = 1:rm r(i,:) = randn (1, cm, is_class) * T(:,:,i) + mu(i,:); endfor endif endif endfunction ## Test input validation %!error mvnrnd () %!error mvnrnd ([2, 3, 4]) %!error mvnrnd (ones (2, 2, 2), ones (1, 2, 3, 4)) %!error mvnrnd (ones (1, 3), ones (1, 2, 3, 4)) ## Output validation tests %!assert (size (mvnrnd ([2, 3, 4], [2, 2, 2])), [1, 3]) %!assert (size (mvnrnd ([2, 3, 4], [2, 2, 2], 10)), [10, 3]) statistics-release-1.6.3/inst/dist_fun/mvtcdf.m000066400000000000000000000344111456127120000215630ustar00rootroot00000000000000## Copyright (C) 2008 Arno Onken ## Copyright (C) 2022-2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{p} =} mvtcdf (@var{x}, @var{rho}, @var{df}) ## @deftypefnx {statistics} {@var{p} =} mvncdf (@var{x_lo}, @var{x_up}, @var{rho}, @var{df}) ## @deftypefnx {statistics} {@var{p} =} mvncdf (@dots{}, @var{options}) ## @deftypefnx {statistics} {[@var{p}, @var{err}] =} mvncdf (@dots{}) ## ## Multivariate Student's t cumulative distribution function (CDF). ## ## @code{@var{p} = mvtcdf (@var{x}, @var{rho}, @var{df})} returns the cumulative ## probability of the multivariate student's t distribution with correlation ## parameters @var{rho} and degrees of freedom @var{df}, evaluated at each row ## of @var{x}. The rows of the @math{NxD} matrix @var{x} correspond to sample ## observations and its columns correspond to variables or coordinates. The ## return argument @var{p} is a column vector with the same number of rows as in ## @var{x}. ## ## @var{rho} is a symmetric, positive definite, @math{DxD} correlation matrix. ## @var{dF} is a scalar or a vector with @math{N} elements. ## ## Note: @code{mvtcdf} computes the CDF for the standard multivariate Student's ## t distribution, centered at the origin, with no scale parameters. If ## @var{rho} is a covariance matrix, i.e. @code{diag(@var{rho})} is not all ## ones, @code{mvtcdf} rescales @var{rho} to transform it to a correlation ## matrix. @code{mvtcdf} does not rescale @var{x}, though. ## ## The multivariate Student's t cumulative probability at @var{x} is defined as ## the probability that a random vector T, distributed as multivariate normal, ## will fall within the semi-infinite rectangle with upper limits defined by ## @var{x}. ## @itemize ## @item @math{Pr@{T(1)<=X(1), T(2)<=X(2), ... T(D)<=X(D)@}}. ## @end itemize ## ## @code{@var{p} = mvtcdf (@var{x_lo}, @var{x_hi}, @var{rho}, @var{df})} returns ## the multivariate Student's t cumulative probability evaluated over the ## rectangle (hyper-rectangle for multivariate data in @var{x}) with lower and ## upper limits defined by @var{x_lo} and @var{x_hi}, respectively. ## ## @code{[@var{p}, @var{err}] = mvtcdf (@dots{})} also returns an error estimate ## @var{err} in @var{p}. ## ## @code{@var{p} = mvtcdf (@dots{}, @var{options})} specifies the structure, ## which controls specific parameters for the numerical integration used to ## compute @var{p}. The required fieds are: ## ## @multitable @columnfractions 0.2 0.05 0.75 ## @item @qcode{"TolFun"} @tab @tab Maximum absolute error tolerance. Default ## is 1e-8 for D < 4, or 1e-4 for D >= 4. ## ## @item @qcode{"MaxFunEvals"} @tab @tab Maximum number of integrand evaluations ## when @math{D >= 4}. Default is 1e7. Ignored when @math{D < 4}. ## ## @item @qcode{"Display"} @tab @tab Display options. Choices are @qcode{"off"} ## (default), @qcode{"iter"}, which shows the probability and estimated error at ## each repetition, and @qcode{"final"}, which shows the final probability and ## related error after the integrand has converged successfully. Ignored when ## @math{D < 4}. ## @end multitable ## ## @seealso{bvtcdf, mvtpdf, mvtrnd, mvtcdfqmc} ## @end deftypefn function [p, err] = mvtcdf (varargin) ## Check for valid number on input and output arguments narginchk (3,5); ## Check for 'options' structure and parse parameters or add defaults if (isstruct (varargin{end})) if (isfield (varargin{end}, "TolFun")) TolFun = varargin{end}.TolFun; else error ("mvtcdf: options structure missing 'TolFun' field."); endif if (isempty (TolFun) && size (varargin{1}, 2) < 4) TolFun = 1e-8; elseif (isempty (TolFun) && size (varargin{1}, 2) < 26) TolFun = 1e-4; endif if (isfield (varargin{end}, "MaxFunEvals")) MaxFunEvals = varargin{end}.MaxFunEvals; else error ("mvtcdf: options structure missing 'MaxFunEvals' field."); endif if (isempty (MaxFunEvals)) MaxFunEvals = 1e7; endif if (isfield (varargin{end}, "Display")) Display = varargin{end}.Display; else error ("mvtcdf: options structure missing 'Display' field."); endif DispOptions = {"off", "final", "iter"}; if (sum (any (strcmpi (Display, DispOptions))) == 0) error ("mvtcdf: 'Display' field in 'options' has invalid value."); endif rem_nargin = nargin - 1; else if (size (varargin{1}, 2) < 4) TolFun = 1e-8; elseif (size (varargin{1}, 2) < 26) TolFun = 1e-4; endif MaxFunEvals = 1e7; Display = "off"; rem_nargin = nargin; endif ## Check for X of X_lo and X_up if (rem_nargin < 4) # MVTCDF(X_UP,SIGMA,DF) x_up_Only = true; x_up = varargin{1}; ## Check for x being a matrix if (! ismatrix (x_up)) error ("mvtcdf: X must be a matrix."); endif ## Create x_lo according to data type of x_lo x_lo = - Inf (size (x_up)); if isa (x_up, "single") x_lo = single (x_lo); endif ## Get SIGMA and DF arguments rho = varargin{2}; df = varargin{3}; else # MVNCDF(X_LO,X_UP,SIGMA,DF) x_up_Only = false; x_lo = varargin{1}; x_up = varargin{2}; rho = varargin{3}; df = varargin{4}; ## Check for x_lo and x_up being matrices of the same size ## and that they define increasing limits if (! ismatrix (x_lo) || ! ismatrix (x_up)) error ("mvtcdf: X_LO and X_UP must be matrices."); endif if (any (size (x_lo) != size (x_up))) error ("mvtcdf: X_LO and X_UP must be of the same size."); endif if (any (any (x_lo > x_up))) error ("mvtcdf: X_LO and X_UP must define increasing limits."); endif endif ## Check if data is single or double class is_type = "double"; if (isa (x_up, "single") || isa (x_lo, "single") || ... isa (rho, "single") || isa (df, "single")) is_type = "single"; endif ## Get size of data [n_x, d_x] = size (x_lo); if (d_x < 1) error ("mvtcdf: too few dimensions in data."); endif ## Force univariate column vector into a row vector if ((d_x == 1) && (size (rho, 1) == n_x)) x_lo = x_lo'; x_up = x_up'; [n_x, d_x] = size (x_up); endif ## Check rho sz = size(rho); if (sz(1) != sz(2)) error ("mvtcdf: correlation matrix RHO is not square."); elseif (! isequal (sz, [d_x, d_x])) error (strcat (["mvtcdf: correlation matrix RHO does not"], ... [" match dimensions in data."])); endif ## Standardize rho to correlation if necessary (not the data) s = sqrt (diag (rho)); if (any (s != 1)) rho = rho ./ (s * s'); endif ## Continue checking rho for being a valid correlation matrix [~, err] = cholcov (rho, 0); if (err != 0) error (strcat (["mvtcdf: correlation matrix RHO must be"], ... [" positive semi-definite."])); endif ## Check df if (! isscalar (df) && ! (isvector (df) && length (df) == n_x)) error (strcat (["mvtcdf: DF must be a scalar or a vector with"], ... [" the same samples as in data."])); endif if (any (df <= 0) || ! isreal (df)) error ("mvtcdf: DF must contain only positive real numbers."); endif ## Compute the cdf if (d_x == 1) p = tcdf (x_up, df) - tcdf (x_lo, df); if (nargout > 1) err = NaN (size (p), is_type); endif elseif (d_x < 4) if (x_up_Only) # upper limit only if (d_x == 2) p = bvtcdf (x_up, rho(2), df, TolFun); else p = tvtcdf (x_up, rho([2 3 6]), df, TolFun); endif else # lower and upper limits present ## Compute the probability over the rectangle as sums and differences ## of integrals over semi-infinite half-rectangles. For degenerate ## rectangles, force an exact zero by making each piece exactly zero. equalLimits = (x_lo == x_up); x_lo(equalLimits) = -Inf; x_up(equalLimits) = -Inf; p = zeros (n_x, 1, is_type); for i = 0:d_x k = nchoosek (1:d_x, i); for j = 1:size (k, 1) X = x_up; X(:,k(j,:)) = x_lo(:,k(j,:)); if d_x == 2 p = p + (-1)^i * bvtcdf (X, rho(2), df, TolFun/4); else p = p + (-1)^i * tvtcdf (X, rho([2 3 6]), df, TolFun/8); endif endfor endfor endif if (nargout > 1) err = repmat (cast (TolFun, is_type), size (p)); endif elseif (d_x < 26) p = zeros (n_x, 1, is_type); err = zeros (n_x, 1, is_type); if (isscalar (df)) df = repmat (df, n_x, 1); endif for i = 1:n_x [p(i), err(i)] = mvtcdfqmc (x_lo(i,:), x_up(i,:), rho, df(i), ... TolFun, MaxFunEvals, Display); endfor else error ("mvncdf: too many dimensions in data (limit = 25 columns)."); endif ## Bound p in range [0, 1] p(p < 0) = 0; p(p > 1) = 1; endfunction ## CDF for the trivariate Student's T function p = tvtcdf (x, rho, df, TolFun) n_x = size (x, 1); if (isscalar (df)) df = repmat (df, n_x, 1); endif ## Find a permutation that makes rho_23 == max(rho) [~,imax] = max (abs (rho)); if (imax == 1) # swap 1 and 3 rho_12 = rho(3); rho_13 = rho(2); rho_23 = rho(1); x = x(:,[3 2 1]); elseif (imax == 2) # swap 1 and 2 rho_12 = rho(1); rho_13 = rho(3); rho_23 = rho(2); x = x(:,[2 1 3]); else # x already in correct order rho_12 = rho(1); rho_13 = rho(2); rho_23 = rho(3); endif if (rho_23 >= 0) p1 = bvtcdf ([x(:,1) min(x(:,2:3), [], 2)], 0, df, TolFun / 4); p1(any (isnan (x), 2)) = NaN; else p1 = bvtcdf (x(:,1:2), 0, df, TolFun /4) - ... bvtcdf ([x(:,1) -x(:,3)], 0, df, TolFun / 4); p1(p1 < 0) = 0; endif if (abs (rho_23) < 1) lo = asin (rho_23); hi = (sign (rho_23) + (rho_23 == 0)) .* pi ./ 2; p2 = zeros (size (p1), class (p1)); for i = 1:n_x x1 = x(i,1); x2 = x(i,2); x3 = x(i,3); if (isfinite (x2) && isfinite (x3) && ~! isnan (x1)) v = df(i); p2(i) = quadgk (@tvtIntegr1, lo, hi, "AbsTol", TolFun / 4, "RelTol", 0); endif endfor else p2 = zeros (class (p1)); endif if (abs (rho_12) > 0) lo = 0; hi = asin (rho_12); rj = rho_12; rk = rho_13; p3 = zeros (size (p1), class (p1)); for i = 1:n_x x1 = x(i,1); xj = x(i,2); xk = x(i,3); if (isfinite (x1) && isfinite (xj) && ! isnan (xk)) v = df(i); p3(i) = quadgk (@tvtIntegr2, lo, hi, "AbsTol", TolFun / 4, "RelTol", 0); endif endfor else p3 = zeros (class (p1)); endif if (abs (rho_13) > 0) lo = 0; hi = asin (rho_13); rj = rho_13; rk = rho_12; p4 = zeros (size (p1), class (p1)); for i = 1:n_x x1 = x(i,1); xj = x(i,3); xk = x(i,2); if (isfinite (x1) && isfinite (xj) && ! isnan (xk)) v = df(i); p4(i) = quadgk (@tvtIntegr2, lo, hi, "AbsTol", TolFun / 4, "RelTol", 0); endif endfor else p4 = zeros (class (p1)); endif if (isa (x, "single") || isa (rho, "single") || isa (df, "single")) p = cast (p1 + (-p2 + p3 + p4) ./ (2 .* pi), "single"); else p = cast (p1 + (-p2 + p3 + p4) ./ (2 .* pi), "double"); endif ## Functions to compute the integrands function integrand = tvtIntegr1 (theta) st = sin(theta); c2t = cos(theta) .^ 2; w = sqrt (1 ./ (1 + ((x2 * st - x3) .^ 2 ./ c2t + x2 .^ 2) / v)); integrand = w .^ v .* TCDF (x1 .* w, v); endfunction function integrand = tvtIntegr2 (theta) st = sin (theta); c2t = cos (theta) .^ 2; w = sqrt (1 ./ (1 + ((x1 *st - xj) .^ 2 ./ c2t + x1 .^ 2) / v)); integrand = w .^ v .* TCDF (uk (st, c2t) .* w, v); endfunction function uk = uk (st, c2t) sinphi = st .* rk ./ rj; numeru = xk .* c2t - x1 .* (sinphi - rho_23 .* st) ... - xj .* (rho_23 - st .* sinphi); denomu = sqrt (c2t .* (c2t - sinphi .* sinphi ... - rho_23 .* (rho_23 - 2 .* st .* sinphi))); uk = numeru ./ denomu; endfunction endfunction ## CDF for Student's T function p = TCDF (x, df) p = betainc(df ./ (df + x .^ 2), df / 2, 0.5) / 2; reflect = (x > 0); p(reflect) = 1 - p(reflect); endfunction %!demo %! ## Compute the cdf of a multivariate Student's t distribution with %! ## correlation parameters rho = [1, 0.4; 0.4, 1] and 2 degrees of freedom. %! %! rho = [1, 0.4; 0.4, 1]; %! df = 2; %! [X1, X2] = meshgrid (linspace (-2, 2, 25)', linspace (-2, 2, 25)'); %! X = [X1(:), X2(:)]; %! p = mvtcdf (X, rho, df); %! surf (X1, X2, reshape (p, 25, 25)); %! title ("Bivariate Student's t cummulative distribution function"); ## Test output against MATLAB R2018 %!test %! x = [1, 2]; %! rho = [1, 0.5; 0.5, 1]; %! df = 4; %! a = [-1, 0]; %! assert (mvtcdf(a, x, rho, df), 0.294196905339283, 1e-14); %!test %! x = [1, 2;2, 4;1, 5]; %! rho = [1, 0.5; 0.5, 1]; %! df = 4; %! p =[0.790285178602166; 0.938703291727784; 0.81222737321336]; %! assert (mvtcdf(x, rho, df), p, 1e-14); %!test %! x = [1, 2, 2, 4, 1, 5]; %! rho = eye (6); %! rho(rho == 0) = 0.5; %! df = 4; %! assert (mvtcdf(x, rho, df), 0.6874, 1e-4); %!error mvtcdf (1) %!error mvtcdf (1, 2) %!error ... %! mvtcdf (1, [2, 3; 3, 2], 1) %!error ... %! mvtcdf ([2, 3, 4], ones (2), 1) %!error ... %! mvtcdf ([1, 2, 3], [2, 3], ones (2), 1) %!error ... %! mvtcdf ([2, 3], ones (2), [1, 2, 3]) %!error ... %! mvtcdf ([2, 3], [1, 0.5; 0.5, 1], [1, 2, 3]) statistics-release-1.6.3/inst/dist_fun/mvtcdfqmc.m000066400000000000000000000211251456127120000222620ustar00rootroot00000000000000## Copyright (C) 2022 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{p} =} mvtcdfqmc (@var{A}, @var{B}, @var{Rho}, @var{df}) ## @deftypefnx {statistics} {@var{p} =} mvtcdfqmc (@dots{}, @var{TolFun}) ## @deftypefnx {statistics} {@var{p} =} mvtcdfqmc (@dots{}, @var{TolFun}, @var{MaxFunEvals}) ## @deftypefnx {statistics} {@var{p} =} mvtcdfqmc (@dots{}, @var{TolFun}, @var{MaxFunEvals}, @var{Display}) ## @deftypefnx {statistics} {[@var{p}, @var{err}] =} mvtcdfqmc (@dots{}) ## @deftypefnx {statistics} {[@var{p}, @var{err}, @var{FunEvals}] =} mvtcdfqmc (@dots{}) ## ## Quasi-Monte-Carlo computation of the multivariate Student's T CDF. ## ## The QMC multivariate Student's t distribution is evaluated between the lower ## limit @var{A} and upper limit @var{B} of the hyper-rectangle with a ## correlation matrix @var{Rho} and degrees of freedom @var{df}. ## ## @multitable @columnfractions 0.2 0.8 ## @item "TolFun" @tab --- Maximum absolute error tolerance. Default is 1e-4. ## @item "MaxFunEvals" @tab --- Maximum number of integrand evaluations. ## Default is 1e7 for D > 4. ## @item "Display" @tab --- Display options. Choices are "off" (default), ## "iter", which shows the probability and estimated error at each repetition, ## and "final", which shows the final probability and related error after the ## integrand has converged successfully. ## @end multitable ## ## @code{[@var{p}, @var{err}, @var{FunEvals}] = mvtcdfqmc (@dots{})} returns the ## estimated probability, @var{p}, an estimate of the error, @var{err}, and the ## number of iterations until a successful convergence is met, unless the value ## in @var{MaxFunEvals} was reached. ## ## @seealso{mvtcdf, mvtpdf, mvtrnd} ## @end deftypefn function [p, err, FunEvals] = mvtcdfqmc (A, B, Rho, df, varargin) ## Check for input arguments narginchk (4,7); ## Add defaults TolFun = 1e-4; MaxFunEvals = 1e7; Display = "off"; ## Parse optional arguments (TolFun, MaxFunEvals, Display) if (nargin > 4) TolFun = varargin{1}; if (! isscalar (TolFun) || ! isreal (TolFun)) error ("mvtcdfqmc: TolFun must be a scalar."); endif endif if (nargin > 5) MaxFunEvals = varargin{2}; if (! isscalar (MaxFunEvals) || ! isreal (MaxFunEvals)) error ("mvtcdfqmc: MaxFunEvals must be a scalar."); endif MaxFunEvals = floor (MaxFunEvals); endif if (nargin > 6) Display = varargin{3}; DispOptions = {"off", "final", "iter"}; if (sum (any (strcmpi (Display, DispOptions))) == 0) error ("mvncdf: invalid value for 'Display' argument."); endif endif ## Check if input is single or double class is_type = "double"; if (isa (A, "single") || isa (B, "single") || isa (Rho, "single")) is_type = "single"; endif ## Check for appropriate lower upper limits and NaN values in data if (! all (A < B)) if (any (A > B)) error ("mvtcdfqmc: inconsistent lower upper limits."); elseif (any (isnan (A) | isnan (B))) warning ("mvtcdfqmc: NaNs in data."); p = NaN (is_type); err = NaN (is_type); else warning ("mvtcdfqmc: zero distance between lower upper limits."); p = zeros (is_type); err = zeros (is_type); endif FunEvals = 0; return; endif ## Ignore dimensions with infinite limits InfLim_idx = (A == -Inf) & (B == Inf); if (any (InfLim_idx)) if (all (InfLim_idx)) warning ("mvtcdfqmc: infinite distance between lower upper limits."); p = 1; err = 0; FunEvals = 0; return endif A(InfLim_idx) = []; B(InfLim_idx) = []; Rho(:,InfLim_idx) = []; Rho(InfLim_idx,:) = []; endif ## Get size of covariance matrix m = size (Rho, 1); ## Sort the order of integration according to increasing length of interval [~, ord] = sort (B - A); A = A(ord); B = B(ord); Rho = Rho(ord, ord); ## Check for highly correlated covariance matrix if any(any(abs(tril(Rho,-1)) > .999)) warning("mvtcdfqmc: highly correlated covariance matrix Rho."); endif ## Scale the integration limits and the Cholesky factor of Rho C = chol(Rho); c = diag(C); A = A(:) ./ c; B = B(:) ./ c; C = C ./ repmat(c',m,1); ## Set repetitions fof Monte Carlo MCreps = 25; MCdims = m - isinf(df); ## Set initial output p = zeros (is_type); sigsq = Inf (is_type); FunEvals = 0; err = NaN; ## Initialize vector P = [31, 47, 73, 113, 173, 263, 397, 593, 907, 1361, 2053, 3079, 4621, ... 6947, 10427, 15641, 23473, 35221, 52837, 79259, 118891, 178349, ... 267523, 401287, 601942, 902933, 1354471, 2031713]; for i = 5:length (P); if ((FunEvals + 2*MCreps*P(i)) > MaxFunEvals) break; endif ## Compute the Niederreiter point set generator NRgen = 2 .^ ((1:MCdims) / (MCdims + 1)); ## Compute randomized quasi-Monte Carlo estimate with P points [THat,sigsqTHat] = estimate_mvtqmc (MCreps, P(i), NRgen, C, df, ... A, B, is_type); FunEvals = FunEvals + 2 * MCreps *P (i); ## Recursively update the estimate and the error estimate p = p + (THat - p) ./ (1 + sigsqTHat ./ sigsq); sigsq = sigsqTHat ./ (1 + sigsqTHat ./ sigsq); ## Compute a conservative estimate of error err = 3.5 * sqrt (sigsq); ## Display output for every iteration if (strcmpi (Display, "iter")) printf ("mvtcdfqmc: Probability estimate: %0.4f ",p); printf ("Error estimate: %0.4e Iterations: %d\n", err, FunEvals); endif if (err < TolFun) if (strcmpi (Display, "final")) printf ("mvtcdfqmc: Successfully converged!\n"); printf ("Final probability estimate: %0.4f ",p); printf ("Final error estimate: %0.4e Iterations: %d\n", err, FunEvals); endif return endif endfor warning ("mvtcdfqmc: Error tolerance did NOT converge!"); printf ("Error tolerance: %0.4f Total Iterations: %d\n", TolFun, MaxFunEvals); endfunction ## Randomized Quasi-Monte-Carlo estimate of the integral function [THat, sigsqTHat] = estimate_mvtqmc (MCreps, P, NRgen, C, df, A, ... B, is_type) qq = (1:P)' * NRgen; THat = zeros (MCreps,1,is_type); for rep = 1:MCreps ## Generate A new random lattice of P points. For MVT, this is in the ## m-dimensional unit hypercube, for MVN, in the (m-1)-dimensional unit ## hypercube. w = abs (2 * mod (qq + repmat (rand (size (NRgen), is_type), P, 1), 1) - 1); ## Compute the mean of the integrand over all P of the points, and all P ## of the antithetic points. THat(rep) = (F_qrsvn (A, B, C, df, w) + F_qrsvn (A, B, C, df, 1 - w)) ./ 2; endfor ## Return the MC mean and se^2 sigsqTHat = var(THat) ./ MCreps; THat = mean(THat); endfunction ## Integrand for computation of MVT probabilities function TBar = F_qrsvn (A, B, C, df, w) N = size (w, 1); # number of quasirandom points m = length(A); # number of dimensions if isinf (df) rho = 1; else rho = chi_inv (w(:,m), df) ./ sqrt (df); end rA = norm_cdf (rho .* A(1)); # A is already scaled by diag(C) rB = norm_cdf (rho .* B(1)) - rA; # B is already scaled by diag(C) T = rB; Y = zeros (N, m, "like", T); for i = 2:m z = min (max (rA + rB .* w(:,i-1), eps / 2), 1 - eps / 2); Y(:,i-1) = norm_inv (z); Ysum = Y * C(:,i); rA = norm_cdf (rho .* A(i) - Ysum); # A is already scaled by diag(C) rB = norm_cdf (rho .* B(i) - Ysum) - rA; # B is already scaled by diag(C) T = T .* rB; end TBar = sum (T, 1) ./ length (T); endfunction ## Normal cumulative distribution function function a = norm_cdf (b) a = 0.5 * erfc (- b ./ sqrt (2)); endfunction ## Inverse of normal cumulative distribution function function a = norm_inv (b) a = - sqrt (2) .* erfcinv (2 * b); endfunction ## Inverse of chi cumulative distribution function function a = chi_inv (b,df) a = sqrt (gammaincinv (b, df ./ 2) .* 2); endfunction %!error mvtcdfqmc (1, 2, 3); %!error mvtcdfqmc (1, 2, 3, 4, 5, 6, 7, 8); statistics-release-1.6.3/inst/dist_fun/mvtpdf.m000066400000000000000000000104601456127120000215760ustar00rootroot00000000000000## Copyright (C) 2015 Nir Krakauer ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{y} =} mvtpdf (@var{x}, @var{rho}, @var{df}) ## ## Multivariate Student's t probability density function (PDF). ## ## @subheading Arguments ## ## @itemize @bullet ## @item ## @var{x} are the points at which to find the probability, where each row ## corresponds to an observation. (@math{NxD} matrix) ## ## @item ## @var{rho} is the correlation matrix. (@math{DxD} symmetric positive ## definite matrix) ## ## @item ## @var{df} is the degrees of freedom. (scalar or vector of length @math{N}) ## ## @end itemize ## ## The distribution is assumed to be centered (zero mean). ## ## @subheading Return values ## ## @itemize @bullet ## @item ## @var{y} is the probability density for each row of @var{x}. ## (@math{Nx1} vector) ## @end itemize ## ## @subheading Examples ## ## @example ## @group ## x = [1 2]; ## rho = [1.0 0.5; 0.5 1.0]; ## df = 4; ## y = mvtpdf (x, rho, df) ## @end group ## @end example ## ## @subheading References ## ## @enumerate ## @item ## Michael Roth, On the Multivariate t Distribution, Technical report from ## Automatic Control at Linkoepings universitet, ## @url{http://users.isy.liu.se/en/rt/roth/student.pdf} ## @end enumerate ## ## @seealso{mvtcdf, mvtcdfqmc, mvtrnd} ## @end deftypefn function y = mvtpdf (x, rho, df) if (nargin != 3) print_usage (); endif # Dimensions d = size (rho, 1); n = size (x, 1); # Check parameters if (size (x, 2) != d) error ("mvtpdf: x must have the same number of columns as rho."); endif if (! isscalar (df) && (! isvector (df) || numel (df) != n)) error (strcat (["mvtpdf: DF must be a scalar or a vector with the"], ... [" same number of rows as X."])); endif if (d < 1 || size (rho, 2) != d || ! issymmetric (rho)) error ("mvtpdf: SIGMA must be nonempty and symmetric."); endif try U = chol (rho); catch error ("mvtpdf: rho must be positive definite"); end_try_catch df = df(:); sqrt_det_sigma = prod(diag(U)); #square root of determinant of rho ## Scale factor for PDF c = (gamma((df+d)/2) ./ gamma(df/2)) ./ (sqrt_det_sigma * (df*pi).^(d/2)); #note: sumsq(U' \ x') is equivalent to the quadratic form x*inv(rho)*x' y = c ./ ((1 + sumsq(U' \ x') ./ df') .^ ((df' + d)/2))'; endfunction %!demo %! ## Compute the pdf of a multivariate t distribution with correlation %! ## parameters rho = [1 .4; .4 1] and 2 degrees of freedom. %! %! rho = [1, 0.4; 0.4, 1]; %! df = 2; %! [X1, X2] = meshgrid (linspace (-2, 2, 25)', linspace (-2, 2, 25)'); %! X = [X1(:), X2(:)]; %! y = mvtpdf (X, rho, df); %! surf (X1, X2, reshape (y, 25, 25)); %! title ("Bivariate Student's t probability density function"); ## Test results verified with R mvtnorm package dmvt function ## dmvt(x = c(0,0), rho = diag(2), log = FALSE) %!assert (mvtpdf ([0 0], eye(2), 1), 0.1591549, 1E-7) ## dmvt(x = c(1,0), rho = matrix(c(1, 0.5, 0.5, 1), nrow=2, ncol=2), df = 2, log = FALSE) %!assert (mvtpdf ([1 0], [1 0.5; 0.5 1], 2), 0.06615947, 1E-7) ## dmvt(x = c(1,0.4,0), rho = matrix(c(1, 0.5, 0.3, 0.5, 1, 0.6, 0.3, 0.6, ... ## 1), nrow=3, ncol=3), df = 5, log = FALSE); dmvt(x = c(1.2,0.5,0.5), ... ## rho = matrix(c(1, 0.5, 0.3, 0.5, 1, 0.6, 0.3, 0.6, 1), nrow=3, ncol=3), ... ## df = 6, log = FALSE); dmvt(x = c(1.4,0.6,1), rho = matrix(c(1, 0.5, 0.3,... ## 0.5, 1, 0.6, 0.3, 0.6, 1), nrow=3, ncol=3), df = 7, log = FALSE) %!assert (mvtpdf ([1 0.4 0; 1.2 0.5 0.5; 1.4 0.6 1], ... %! [1 0.5 0.3; 0.5 1 0.6; 0.3 0.6 1], [5 6 7]), ... %! [0.04713313 0.03722421 0.02069011]', 1E-7) statistics-release-1.6.3/inst/dist_fun/mvtrnd.m000066400000000000000000000104331456127120000216100ustar00rootroot00000000000000## Copyright (C) 2012 Arno Onken ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{r} =} mvtrnd (@var{rho}, @var{df}) ## @deftypefnx {statistics} {@var{r} =} mvtrnd (@var{rho}, @var{df}, @var{n}) ## ## Random vectors from the multivariate Student's t distribution. ## ## @subheading Arguments ## ## @itemize @bullet ## @item ## @var{rho} is the matrix of correlation coefficients. If there are any ## non-unit diagonal elements then @var{rho} will be normalized, so that the ## resulting covariance of the obtained samples @var{r} follows: ## @code{cov (r) = df/(df-2) * rho ./ (sqrt (diag (rho) * diag (rho)))}. ## In order to obtain samples distributed according to a standard multivariate ## student's t-distribution, @var{rho} must be equal to the identity matrix. To ## generate multivariate student's t-distribution samples @var{r} with arbitrary ## covariance matrix @var{rho}, the following scaling might be used: ## @code{r = mvtrnd (rho, df, n) * diag (sqrt (diag (rho)))}. ## ## @item ## @var{df} is the degrees of freedom for the multivariate t-distribution. ## @var{df} must be a vector with the same number of elements as samples to be ## generated or be scalar. ## ## @item ## @var{n} is the number of rows of the matrix to be generated. @var{n} must be ## a non-negative integer and corresponds to the number of samples to be ## generated. ## @end itemize ## ## @subheading Return values ## ## @itemize @bullet ## @item ## @var{r} is a matrix of random samples from the multivariate t-distribution ## with @var{n} row samples. ## @end itemize ## ## @subheading Examples ## ## @example ## @group ## rho = [1, 0.5; 0.5, 1]; ## df = 3; ## n = 10; ## r = mvtrnd (rho, df, n); ## @end group ## ## @group ## rho = [1, 0.5; 0.5, 1]; ## df = [2; 3]; ## n = 2; ## r = mvtrnd (rho, df, 2); ## @end group ## @end example ## ## @subheading References ## ## @enumerate ## @item ## Wendy L. Martinez and Angel R. Martinez. @cite{Computational Statistics ## Handbook with MATLAB}. Appendix E, pages 547-557, Chapman & Hall/CRC, 2001. ## ## @item ## Samuel Kotz and Saralees Nadarajah. @cite{Multivariate t Distributions and ## Their Applications}. Cambridge University Press, Cambridge, 2004. ## @end enumerate ## ## @seealso{mvtcdf, mvtcdfqmc, mvtpdf} ## @end deftypefn function r = mvtrnd (rho, df, n) # Check arguments if (nargin < 2) print_usage (); endif [jnk, p] = cholcov (rho); # This is a more robust check for positive definite if (! ismatrix (rho) || any (any (rho != rho')) || (p != 0)) error ("mvtrnd: SIGMA must be a positive definite matrix."); endif if (!isvector (df) || any (df <= 0)) error ("mvtrnd: DF must be a positive scalar or vector."); endif df = df(:); if (nargin > 2) if (! isscalar (n) || n < 0 | round (n) != n) error ("mvtrnd: N must be a non-negative integer.") endif if (isscalar (df)) df = df * ones (n, 1); else if (length (df) != n) error ("mvtrnd: N must match the length of DF.") endif endif else n = length (df); endif # Normalize rho if (any (diag (rho) != 1)) rho = rho ./ sqrt (diag (rho) * diag (rho)'); endif # Dimension d = size (rho, 1); # Draw samples y = mvnrnd (zeros (1, d), rho, n); u = repmat (chi2rnd (df), 1, d); r = y .* sqrt (repmat (df, 1, d) ./ u); endfunction %!test %! rho = [1, 0.5; 0.5, 1]; %! df = 3; %! n = 10; %! r = mvtrnd (rho, df, n); %! assert (size (r), [10, 2]); %!test %! rho = [1, 0.5; 0.5, 1]; %! df = [2; 3]; %! n = 2; %! r = mvtrnd (rho, df, 2); %! assert (size (r), [2, 2]); statistics-release-1.6.3/inst/dist_fun/nakacdf.m000066400000000000000000000137241456127120000216730ustar00rootroot00000000000000## Copyright (C) 2016 Dag Lyberg ## Copyright (C) 1995-2015 Kurt Hornik ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## Octave is free software; you can redistribute it and/or modify it ## under the terms of the GNU General Public License as published by ## the Free Software Foundation; either version 3 of the License, or (at ## your option) any later version. ## ## Octave is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with Octave; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{p} =} nakacdf (@var{x}, @var{mu}, @var{omega}) ## @deftypefnx {statistics} {@var{p} =} nakacdf (@var{x}, @var{mu}, @var{omega}, @qcode{"upper"}) ## ## Nakagami cumulative distribution function (CDF). ## ## For each element of @var{x}, compute the cumulative distribution function ## (CDF) of the Nakagami distribution with shape parameter @var{mu} and spread ## parameter @var{omega}. The size of @var{p} is the common size of @var{x}, ## @var{mu}, and @var{omega}. A scalar input functions as a constant matrix of ## the same size as the other inputs. ## ## Both parameters must be positive reals and @qcode{@var{mu} >= 0.5}. For ## @qcode{@var{mu} < 0.5} or @qcode{@var{omega} <= 0}, @qcode{NaN} is returned. ## ## @code{@var{p} = nakacdf (@var{x}, @var{mu}, @var{omega}, "upper")} computes ## the upper tail probability of the Nakagami distribution with parameters ## @var{mu} and @var{beta}, at the values in @var{x}. ## ## Further information about the Nakagami distribution can be found at ## @url{https://en.wikipedia.org/wiki/Nakagami_distribution} ## ## @seealso{nakainv, nakapdf, nakarnd, nakafit, nakalike} ## @end deftypefn function p = nakacdf (x, mu, omega, uflag) ## Check for valid number of input arguments if (nargin < 3) error ("nakacdf: function called with too few input arguments."); endif ## Check for valid "upper" flag if (nargin > 3) if (! strcmpi (uflag, "upper")) error ("nakacdf: invalid argument for upper tail."); else uflag = true; endif else uflag = false; endif ## Check for common size of X, MU, and OMEGA if (! isscalar (x) || ! isscalar (mu) || ! isscalar (omega)) [retval, x, mu, omega] = common_size (x, mu, omega); if (retval > 0) error ("nakacdf: X, MU, and OMEGA must be of common size or scalars."); endif endif ## Check for X, MU, and OMEGA being reals if (iscomplex (x) || iscomplex (mu) || iscomplex (omega)) error ("nakacdf: X, MU, and OMEGA must not be complex."); endif ## Check for class type if (isa (x, "single") || isa (mu, "single") || isa (omega, "single")) p = zeros (size (x), "single"); else p = zeros (size (x)); endif ## Force invalid parameters and missing data to NaN k1 = isnan (x) | ! (mu >= 0.5) | ! (omega > 0); p(k1) = NaN; ## Find normal and edge cases k2 = (x == Inf) & (mu >= 0.5) & (mu < Inf) & (omega > 0) & (omega < Inf); k = (x > 0) & (x < Inf) & (mu >= 0.5) & (mu < Inf) ... & (omega > 0) & (omega < Inf); ## Compute Nakagami CDF if (uflag) p(k2) = 0; left = mu .* ones (size (x)); right = (mu ./ omega) .* x .^ 2; p(k) = gammainc (right(k), left(k), "upper"); else p(k2) = 1; left = mu .* ones (size (x)); right = (mu ./ omega) .* x .^ 2; p(k) = gammainc (right(k), left(k)); endif endfunction %!demo %! ## Plot various CDFs from the Nakagami distribution %! x = 0:0.01:3; %! p1 = nakacdf (x, 0.5, 1); %! p2 = nakacdf (x, 1, 1); %! p3 = nakacdf (x, 1, 2); %! p4 = nakacdf (x, 1, 3); %! p5 = nakacdf (x, 2, 1); %! p6 = nakacdf (x, 2, 2); %! p7 = nakacdf (x, 5, 1); %! plot (x, p1, "-r", x, p2, "-g", x, p3, "-y", x, p4, "-m", ... %! x, p5, "-k", x, p6, "-b", x, p7, "-c") %! grid on %! xlim ([0, 3]) %! legend ({"μ = 0.5, ω = 1", "μ = 1, ω = 1", "μ = 1, ω = 2", ... %! "μ = 1, ω = 3", "μ = 2, ω = 1", "μ = 2, ω = 2", ... %! "μ = 5, ω = 1"}, "location", "southeast") %! title ("Nakagami CDF") %! xlabel ("values in x") %! ylabel ("probability") ## Test output %!shared x, y %! x = [-1, 0, 1, 2, Inf]; %! y = [0, 0, 0.63212055882855778, 0.98168436111126578, 1]; %!assert (nakacdf (x, ones (1,5), ones (1,5)), y, eps) %!assert (nakacdf (x, 1, 1), y, eps) %!assert (nakacdf (x, [1, 1, NaN, 1, 1], 1), [y(1:2), NaN, y(4:5)]) %!assert (nakacdf (x, 1, [1, 1, NaN, 1, 1]), [y(1:2), NaN, y(4:5)]) %!assert (nakacdf ([x, NaN], 1, 1), [y, NaN], eps) ## Test class of input preserved %!assert (nakacdf (single ([x, NaN]), 1, 1), single ([y, NaN]), eps("single")) %!assert (nakacdf ([x, NaN], single (1), 1), single ([y, NaN]), eps("single")) %!assert (nakacdf ([x, NaN], 1, single (1)), single ([y, NaN]), eps("single")) ## Test input validation %!error nakacdf () %!error nakacdf (1) %!error nakacdf (1, 2) %!error nakacdf (1, 2, 3, "tail") %!error nakacdf (1, 2, 3, 4) %!error ... %! nakacdf (ones (3), ones (2), ones (2)) %!error ... %! nakacdf (ones (2), ones (3), ones (2)) %!error ... %! nakacdf (ones (2), ones (2), ones (3)) %!error nakacdf (i, 2, 2) %!error nakacdf (2, i, 2) %!error nakacdf (2, 2, i) statistics-release-1.6.3/inst/dist_fun/nakainv.m000066400000000000000000000125251456127120000217310ustar00rootroot00000000000000## Copyright (C) 2016 Dag Lyberg ## Copyright (C) 1995-2015 Kurt Hornik ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## Octave is free software; you can redistribute it and/or modify it ## under the terms of the GNU General Public License as published by ## the Free Software Foundation; either version 3 of the License, or (at ## your option) any later version. ## ## Octave is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with Octave; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{x} =} nakacdf (@var{x}, @var{mu}, @var{omega}) ## ## Inverse of the Nakagami cumulative distribution function (iCDF). ## ## For each element of @var{p}, compute the quantile (the inverse of the CDF) of ## the Nakagami distribution with shape parameter @var{mu} and spread parameter ## @var{omega}. The size of @var{x} is the common size of @var{x}, @var{mu}, ## and @var{omega}. A scalar input functions as a constant matrix of the same ## size as the other inputs. ## ## Both parameters must be positive reals and @qcode{@var{mu} >= 0.5}. For ## @qcode{@var{mu} < 0.5} or @qcode{@var{omega} <= 0}, @qcode{NaN} is returned. ## ## Further information about the Nakagami distribution can be found at ## @url{https://en.wikipedia.org/wiki/Nakagami_distribution} ## ## @seealso{nakacdf, nakapdf, nakarnd, nakafit, nakalike} ## @end deftypefn function x = nakainv (p, mu, omega) ## Check for valid number of input arguments if (nargin < 3) error ("nakainv: function called with too few input arguments."); endif ## Check for common size of P, MU, and OMEGA if (! isscalar (p) || ! isscalar (mu) || ! isscalar (omega)) [retval, p, mu, omega] = common_size (p, mu, omega); if (retval > 0) error ("nakainv: P, MU, and OMEGA must be of common size or scalars."); endif endif ## Check for P, MU, and OMEGA being reals if (iscomplex (p) || iscomplex (mu) || iscomplex (omega)) error ("nakainv: P, MU, and OMEGA must not be complex."); endif ## Check for class type if (isa (p, "single") || isa (mu, "single") || isa (omega, "single")) x = zeros (size (p), "single"); else x = zeros (size (p)); endif ## Force invalid parameters and missing data to NaN k = isnan (p) | ! (p >= 0) | ! (p <= 1) | ! (mu >= 0.5) | ! (omega > 0); x(k) = NaN; ## Handle edge cases k = (p == 1) & (mu >= 0.5) & (mu < Inf) & (omega > 0) & (omega < Inf); x(k) = Inf; ## Find normal cases k = (0 < p) & (p < 1) & (0.5 <= mu) & (mu < Inf) ... & (0 < omega) & (omega < Inf); ## Compute Nakagami iCDF if (isscalar (mu) && isscalar(omega)) m_gamma = mu; w_gamma = omega / mu; x(k) = gaminv (p(k), m_gamma, w_gamma); x(k) = sqrt (x(k)); else m_gamma = mu; w_gamma = omega ./ mu; x(k) = gaminv (p(k), m_gamma(k), w_gamma(k)); x(k) = sqrt (x(k)); endif endfunction %!demo %! ## Plot various iCDFs from the Nakagami distribution %! p = 0.001:0.001:0.999; %! x1 = nakainv (p, 0.5, 1); %! x2 = nakainv (p, 1, 1); %! x3 = nakainv (p, 1, 2); %! x4 = nakainv (p, 1, 3); %! x5 = nakainv (p, 2, 1); %! x6 = nakainv (p, 2, 2); %! x7 = nakainv (p, 5, 1); %! plot (p, x1, "-r", p, x2, "-g", p, x3, "-y", p, x4, "-m", ... %! p, x5, "-k", p, x6, "-b", p, x7, "-c") %! grid on %! ylim ([0, 3]) %! legend ({"μ = 0.5, ω = 1", "μ = 1, ω = 1", "μ = 1, ω = 2", ... %! "μ = 1, ω = 3", "μ = 2, ω = 1", "μ = 2, ω = 2", ... %! "μ = 5, ω = 1"}, "location", "northwest") %! title ("Nakagami iCDF") %! xlabel ("probability") %! ylabel ("values in x") ## Test output %!shared p, y %! p = [-Inf, -1, 0, 1/2, 1, 2, Inf]; %! y = [NaN, NaN, 0, 0.83255461115769769, Inf, NaN, NaN]; %!assert (nakainv (p, ones (1,7), ones (1,7)), y, eps) %!assert (nakainv (p, 1, 1), y, eps) %!assert (nakainv (p, [1, 1, 1, NaN, 1, 1, 1], 1), [y(1:3), NaN, y(5:7)], eps) %!assert (nakainv (p, 1, [1, 1, 1, NaN, 1, 1, 1]), [y(1:3), NaN, y(5:7)], eps) %!assert (nakainv ([p, NaN], 1, 1), [y, NaN], eps) ## Test class of input preserved %!assert (nakainv (single ([p, NaN]), 1, 1), single ([y, NaN])) %!assert (nakainv ([p, NaN], single (1), 1), single ([y, NaN])) %!assert (nakainv ([p, NaN], 1, single (1)), single ([y, NaN])) ## Test input validation %!error nakainv () %!error nakainv (1) %!error nakainv (1, 2) %!error ... %! nakainv (ones (3), ones (2), ones(2)) %!error ... %! nakainv (ones (2), ones (3), ones(2)) %!error ... %! nakainv (ones (2), ones (2), ones(3)) %!error nakainv (i, 4, 3) %!error nakainv (1, i, 3) %!error nakainv (1, 4, i) statistics-release-1.6.3/inst/dist_fun/nakapdf.m000066400000000000000000000123551456127120000217070ustar00rootroot00000000000000## Copyright (C) 2016 Dag Lyberg ## Copyright (C) 1995-2015 Kurt Hornik ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## Octave is free software; you can redistribute it and/or modify it ## under the terms of the GNU General Public License as published by ## the Free Software Foundation; either version 3 of the License, or (at ## your option) any later version. ## ## Octave is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with Octave; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{y} =} nakapdf (@var{x}, @var{mu}, @var{omega}) ## ## Nakagami probability density function (PDF). ## ## For each element of @var{x}, compute the probability density function (PDF) ## of the Nakagami distribution with shape parameter @var{mu} and spread ## parameter @var{omega}. The size of @var{y} is the common size of @var{x}, ## @var{mu}, and @var{omega}. A scalar input functions as a constant matrix of ## the same size as the other inputs. ## ## Both parameters must be positive reals and @qcode{@var{mu} >= 0.5}. For ## @qcode{@var{mu} < 0.5} or @qcode{@var{omega} <= 0}, @qcode{NaN} is returned. ## ## Further information about the Nakagami distribution can be found at ## @url{https://en.wikipedia.org/wiki/Nakagami_distribution} ## ## @seealso{nakacdf, nakapdf, nakarnd, nakafit, nakalike} ## @end deftypefn function y = nakapdf (x, mu, omega) ## Check for valid number of input arguments if (nargin < 3) error ("nakapdf: function called with too few input arguments."); endif ## Check for common size of X, MU, and OMEGA if (! isscalar (x) || ! isscalar (mu) || ! isscalar (omega)) [retval, x, mu, omega] = common_size (x, mu, omega); if (retval > 0) error ("nakapdf: X, MU, and OMEGA must be of common size or scalars."); endif endif ## Check for X, MU, and OMEGA being reals if (iscomplex (x) || iscomplex (mu) || iscomplex (omega)) error ("nakapdf: X, MU, and OMEGA must not be complex."); endif ## Check for class type if (isa (x, "single") || isa (mu, "single") || isa (omega, "single")) y = zeros (size (x), "single"); else y = zeros (size (x)); endif ## Compute Nakagami PDF k = isnan (x) | ! (mu >= 0.5) | ! (omega > 0); y(k) = NaN; k = (0 < x) & (x < Inf) & (0.5 <= mu) & (mu < Inf) ... & (0 < omega) & (omega < Inf); if (isscalar (mu) && isscalar(omega)) y(k) = exp (log (2) + mu * log (mu) - log (gamma (mu)) - ... mu * log (omega) + (2 * mu-1) * ... log (x(k)) - (mu / omega) * x(k) .^ 2); else y(k) = exp (log(2) + mu(k) .* log (mu(k)) - log (gamma (mu(k))) - ... mu(k) .* log (omega(k)) + (2 * mu(k) - 1) ... .* log (x(k)) - (mu(k) ./ omega(k)) .* x(k) .^ 2); endif endfunction %!demo %! ## Plot various PDFs from the Nakagami distribution %! x = 0:0.01:3; %! y1 = nakapdf (x, 0.5, 1); %! y2 = nakapdf (x, 1, 1); %! y3 = nakapdf (x, 1, 2); %! y4 = nakapdf (x, 1, 3); %! y5 = nakapdf (x, 2, 1); %! y6 = nakapdf (x, 2, 2); %! y7 = nakapdf (x, 5, 1); %! plot (x, y1, "-r", x, y2, "-g", x, y3, "-y", x, y4, "-m", ... %! x, y5, "-k", x, y6, "-b", x, y7, "-c") %! grid on %! xlim ([0, 3]) %! ylim ([0, 2]) %! legend ({"μ = 0.5, ω = 1", "μ = 1, ω = 1", "μ = 1, ω = 2", ... %! "μ = 1, ω = 3", "μ = 2, ω = 1", "μ = 2, ω = 2", ... %! "μ = 5, ω = 1"}, "location", "northeast") %! title ("Nakagami PDF") %! xlabel ("values in x") %! ylabel ("density") ## Test output %!shared x, y %! x = [-1, 0, 1, 2, Inf]; %! y = [0, 0, 0.73575888234288467, 0.073262555554936715, 0]; %!assert (nakapdf (x, ones (1,5), ones (1,5)), y, eps) %!assert (nakapdf (x, 1, 1), y, eps) %!assert (nakapdf (x, [1, 1, NaN, 1, 1], 1), [y(1:2), NaN, y(4:5)], eps) %!assert (nakapdf (x, 1, [1, 1, NaN, 1, 1]), [y(1:2), NaN, y(4:5)], eps) %!assert (nakapdf ([x, NaN], 1, 1), [y, NaN], eps) ## Test class of input preserved %!assert (nakapdf (single ([x, NaN]), 1, 1), single ([y, NaN])) %!assert (nakapdf ([x, NaN], single (1), 1), single ([y, NaN])) %!assert (nakapdf ([x, NaN], 1, single (1)), single ([y, NaN])) ## Test input validation %!error nakapdf () %!error nakapdf (1) %!error nakapdf (1, 2) %!error ... %! nakapdf (ones (3), ones (2), ones(2)) %!error ... %! nakapdf (ones (2), ones (3), ones(2)) %!error ... %! nakapdf (ones (2), ones (2), ones(3)) %!error nakapdf (i, 4, 3) %!error nakapdf (1, i, 3) %!error nakapdf (1, 4, i) statistics-release-1.6.3/inst/dist_fun/nakarnd.m000066400000000000000000000156571456127120000217310ustar00rootroot00000000000000## Copyright (C) 2016 Dag Lyberg ## Copyright (C) 1995-2015 Kurt Hornik ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## Octave is free software; you can redistribute it and/or modify it ## under the terms of the GNU General Public License as published by ## the Free Software Foundation; either version 3 of the License, or (at ## your option) any later version. ## ## Octave is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with Octave; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{r} =} nakarnd (@var{mu}, @var{omega}) ## @deftypefnx {statistics} {@var{r} =} nakarnd (@var{mu}, @var{omega}, @var{rows}) ## @deftypefnx {statistics} {@var{r} =} nakarnd (@var{mu}, @var{omega}, @var{rows}, @var{cols}, @dots{}) ## @deftypefnx {statistics} {@var{r} =} nakarnd (@var{mu}, @var{omega}, [@var{sz}]) ## ## Random arrays from the Nakagami distribution. ## ## @code{@var{r} = nakarnd (@var{mu}, @var{omega})} returns an array of random ## numbers chosen from the Nakagami distribution with shape parameter @var{mu} ## and spread parameter @var{omega}. The size of @var{r} is the common size of ## @var{mu} and @var{omega}. A scalar input functions as a constant matrix of ## the same size as the other inputs. ## ## Both parameters must be positive reals and @qcode{@var{mu} >= 0.5}. For ## @qcode{@var{mu} < 0.5} or @qcode{@var{omega} <= 0}, @qcode{NaN} is returned. ## ## When called with a single size argument, @code{nakarnd} returns a square ## matrix with the dimension specified. When called with more than one scalar ## argument, the first two arguments are taken as the number of rows and columns ## and any further arguments specify additional matrix dimensions. The size may ## also be specified with a row vector of dimensions, @var{sz}. ## ## Further information about the Nakagami distribution can be found at ## @url{https://en.wikipedia.org/wiki/Nakagami_distribution} ## ## @seealso{nakacdf, nakainv, nakapdf} ## @end deftypefn function r = nakarnd (mu, omega, varargin) ## Check for valid number of input arguments if (nargin < 2) error ("nakarnd: function called with too few input arguments."); endif ## Check for common size of MU and OMEGA if (! isscalar (mu) || ! isscalar (omega)) [retval, mu, omega] = common_size (mu, omega); if (retval > 0) error ("nakarnd: MU and OMEGA must be of common size or scalars."); endif endif ## Check for MU and OMEGA being reals if (iscomplex (mu) || iscomplex (omega)) error ("nakarnd: MU and OMEGA must not be complex."); endif ## Parse and check SIZE arguments if (nargin == 2) sz = size (mu); elseif (nargin == 3) if (isscalar (varargin{1}) && varargin{1} >= 0 ... && varargin{1} == fix (varargin{1})) sz = [varargin{1}, varargin{1}]; elseif (isrow (varargin{1}) && all (varargin{1} >= 0) ... && all (varargin{1} == fix (varargin{1}))) sz = varargin{1}; elseif error (strcat (["nakarnd: SZ must be a scalar or a row vector"], ... [" of non-negative integers."])); endif elseif (nargin > 3) posint = cellfun (@(x) (! isscalar (x) || x < 0 || x != fix (x)), varargin); if (any (posint)) error ("nakarnd: dimensions must be non-negative integers."); endif sz = [varargin{:}]; endif ## Check that parameters match requested dimensions in size if (! isscalar (mu) && ! isequal (size (mu), sz)) error ("nakarnd: MU and OMEGA must be scalars or of size SZ."); endif ## Check for class type if (isa (mu, "single") || isa (omega, "single")) cls = "single"; else cls = "double"; endif ## Generate random sample from Nakagami distribution if (isscalar (mu) && isscalar (omega)) if ((0.5 <= mu) && (mu < Inf) && (0 < omega) && (omega < Inf)) m_gamma = mu; w_gamma = omega / mu; r = gamrnd (m_gamma, w_gamma, sz); r = sqrt (r); else r = NaN (sz, cls); endif else r = NaN (sz, cls); k = (0.5 <= mu) & (mu < Inf) & (0 < omega) & (omega < Inf); m_gamma = mu; w_gamma = omega ./ mu; r(k) = gamrnd (m_gamma(k), w_gamma(k)); r(k) = sqrt (r(k)); endif endfunction ## Test output %!assert (size (nakarnd (1, 1)), [1 1]) %!assert (size (nakarnd (1, ones (2,1))), [2, 1]) %!assert (size (nakarnd (1, ones (2,2))), [2, 2]) %!assert (size (nakarnd (ones (2,1), 1)), [2, 1]) %!assert (size (nakarnd (ones (2,2), 1)), [2, 2]) %!assert (size (nakarnd (1, 1, 3)), [3, 3]) %!assert (size (nakarnd (1, 1, [4, 1])), [4, 1]) %!assert (size (nakarnd (1, 1, 4, 1)), [4, 1]) %!assert (size (nakarnd (1, 1, 4, 1, 5)), [4, 1, 5]) %!assert (size (nakarnd (1, 1, 0, 1)), [0, 1]) %!assert (size (nakarnd (1, 1, 1, 0)), [1, 0]) %!assert (size (nakarnd (1, 1, 1, 2, 0, 5)), [1, 2, 0, 5]) ## Test class of input preserved %!assert (class (nakarnd (1, 1)), "double") %!assert (class (nakarnd (1, single (1))), "single") %!assert (class (nakarnd (1, single ([1, 1]))), "single") %!assert (class (nakarnd (single (1), 1)), "single") %!assert (class (nakarnd (single ([1, 1]), 1)), "single") ## Test input validation %!error nakarnd () %!error nakarnd (1) %!error ... %! nakarnd (ones (3), ones (2)) %!error ... %! nakarnd (ones (2), ones (3)) %!error nakarnd (i, 2, 3) %!error nakarnd (1, i, 3) %!error ... %! nakarnd (1, 2, -1) %!error ... %! nakarnd (1, 2, 1.2) %!error ... %! nakarnd (1, 2, ones (2)) %!error ... %! nakarnd (1, 2, [2 -1 2]) %!error ... %! nakarnd (1, 2, [2 0 2.5]) %!error ... %! nakarnd (1, 2, 2, -1, 5) %!error ... %! nakarnd (1, 2, 2, 1.5, 5) %!error ... %! nakarnd (2, ones (2), 3) %!error ... %! nakarnd (2, ones (2), [3, 2]) %!error ... %! nakarnd (2, ones (2), 3, 2) statistics-release-1.6.3/inst/dist_fun/nbincdf.m000066400000000000000000000202031456127120000216750ustar00rootroot00000000000000## Copyright (C) 2012 Rik Wehbring ## Copyright (C) 1995-2016 Kurt Hornik ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{p} =} nbincdf (@var{x}, @var{r}, @var{ps}) ## @deftypefnx {statistics} {@var{p} =} nbincdf (@var{x}, @var{r}, @var{ps}, @qcode{"upper"}) ## ## Negative binomial cumulative distribution function (CDF). ## ## For each element of @var{x}, compute the cumulative distribution function ## (CDF) of the negative binomial distribution with parameters @var{r} and ## @var{ps}, where @var{r} is the number of successes until the experiment is ## stopped and @var{ps} is the probability of success in each experiment, given ## the number of failures in @var{x}. The size of @var{p} is the common size of ## @var{x}, @var{r}, and @var{ps}. A scalar input functions as a constant ## matrix of the same size as the other inputs. ## ## The algorithm uses the cumulative sums of the binomial masses. ## ## @code{@var{p} = nbincdf (@var{x}, @var{r}, @var{ps}, "upper")} computes the ## upper tail probability of the negative binomial distribution with parameters ## @var{r} and @var{ps}, at the values in @var{x}. ## ## When @var{r} is an integer, the negative binomial distribution is also known ## as the Pascal distribution and it models the number of failures in @var{x} ## before a specified number of successes is reached in a series of independent, ## identical trials. Its parameters are the probability of success in a single ## trial, @var{ps}, and the number of successes, @var{r}. A special case of the ## negative binomial distribution, when @qcode{@var{r} = 1}, is the geometric ## distribution, which models the number of failures before the first success. ## ## @var{r} can also have non-integer positive values, in which form the negative ## binomial distribution, also known as the Polya distribution, has no ## interpretation in terms of repeated trials, but, like the Poisson ## distribution, it is useful in modeling count data. The negative binomial ## distribution is more general than the Poisson distribution because it has a ## variance that is greater than its mean, making it suitable for count data ## that do not meet the assumptions of the Poisson distribution. In the limit, ## as @var{r} increases to infinity, the negative binomial distribution ## approaches the Poisson distribution. ## ## Further information about the negative binomial distribution can be found at ## @url{https://en.wikipedia.org/wiki/Negative_binomial_distribution} ## ## @seealso{nbininv, nbinpdf, nbinrnd, nbinfit, nbinlike, nbinstat} ## @end deftypefn function p = nbincdf (x, r, ps, uflag) ## Check for valid number of input arguments if (nargin < 3) error ("nbincdf: function called with too few input arguments."); endif ## Check for "upper" flag if (nargin == 4 && strcmpi (uflag, "upper")) uflag = true; elseif (nargin == 4 && ! strcmpi (uflag, "upper")) error ("nbincdf: invalid argument for upper tail."); else uflag = false; endif ## Check for R and PS being scalars scalarNPS = (isscalar(r) & isscalar(ps)); ## Check for common size of X, R, and PS if (! isscalar (x) || ! isscalar (r) || ! isscalar (ps)) [retval, x, r, ps] = common_size (x, r, ps); if (retval > 0) error ("nbincdf: X, R, and PS must be of common size or scalars."); endif endif ## Check for X, R, and PS being reals if (iscomplex (x) || iscomplex (r) || iscomplex (ps)) error ("nbincdf: X, R, and PS must not be complex."); endif ## Check for class type if (isa (x, "single") || isa (r, "single") || isa (ps, "single")) p = zeros (size (x), "single"); else p = zeros (size (x)); endif ## Force NaN for out of range or missing parameters and missing data NaN is_nan = (isnan (x) | isnan (r) | (r <= 0) | (r == Inf) | (ps < 0) | (ps > 1)); p(is_nan) = NaN; ## Compute P for X >= 0 xf = floor (x); k = find (xf >= 0 & ! is_nan); ## Return 1 for positive infinite values of X, unless "upper" is given: p = 0 k1 = find (isinf (xf(k))); if (any (k1)) if (uflag) p(k(k1)) = 0; else p(k(k1)) = 1; endif k(k1) = []; endif ## Return 1 when X < 0 and "upper" is given k1 = (x < 0 & ! is_nan); if (any (k1)) if (uflag) p(k1) = 1; endif endif ## Accumulate probabilities up to the maximum value in X if (any (k)) if (uflag) p(k) = betainc (ps(k), r(k), xf(k) + 1, "upper"); else max_val = max (xf(k)); if (scalarNPS) tmp = cumsum (nbinpdf (0:max_val, r(1), ps(1))); p(k) = tmp(xf(k) + 1); else idx = (0:max_val)'; compare = idx(:, ones (size (k))); index = xf(k); index = index(:); index = index(:, ones (size (idx)))'; n_big = r(k); n_big = n_big(:); n_big = n_big(:, ones (size (idx)))'; ps_big = ps(k); ps_big = ps_big(:); ps_big = ps_big(:, ones (size (idx)))'; p0 = nbinpdf (compare, n_big, ps_big); indicator = find (compare > index); p0(indicator) = zeros (size (indicator)); p(k) = sum(p0,1); endif endif endif ## Prevent round-off errors p(p > 1) = 1; endfunction %!demo %! ## Plot various CDFs from the negative binomial distribution %! x = 0:50; %! p1 = nbincdf (x, 2, 0.15); %! p2 = nbincdf (x, 5, 0.2); %! p3 = nbincdf (x, 4, 0.4); %! p4 = nbincdf (x, 10, 0.3); %! plot (x, p1, "*r", x, p2, "*g", x, p3, "*k", x, p4, "*m") %! grid on %! xlim ([0, 40]) %! legend ({"r = 2, ps = 0.15", "r = 5, ps = 0.2", "r = 4, p = 0.4", ... %! "r = 10, ps = 0.3"}, "location", "southeast") %! title ("Negative binomial CDF") %! xlabel ("values in x (number of failures)") %! ylabel ("probability") ## Test output %!shared x, y %! x = [-1 0 1 2 Inf]; %! y = [0 1/2 3/4 7/8 1]; %!assert (nbincdf (x, ones (1,5), 0.5*ones (1,5)), y) %!assert (nbincdf (x, 1, 0.5*ones (1,5)), y) %!assert (nbincdf (x, ones (1,5), 0.5), y) %!assert (nbincdf (x, ones (1,5), 0.5, "upper"), 1 - y, eps) %!assert (nbincdf ([x(1:3) 0 x(5)], [0 1 NaN 1.5 Inf], 0.5), ... %! [NaN 1/2 NaN nbinpdf(0,1.5,0.5) NaN], eps) %!assert (nbincdf (x, 1, 0.5*[-1 NaN 4 1 1]), [NaN NaN NaN y(4:5)]) %!assert (nbincdf ([x(1:2) NaN x(4:5)], 1, 0.5), [y(1:2) NaN y(4:5)]) ## Test class of input preserved %!assert (nbincdf ([x, NaN], 1, 0.5), [y, NaN]) %!assert (nbincdf (single ([x, NaN]), 1, 0.5), single ([y, NaN])) %!assert (nbincdf ([x, NaN], single (1), 0.5), single ([y, NaN])) %!assert (nbincdf ([x, NaN], 1, single (0.5)), single ([y, NaN])) ## Test input validation %!error nbincdf () %!error nbincdf (1) %!error nbincdf (1, 2) %!error nbincdf (1, 2, 3, 4) %!error nbincdf (1, 2, 3, "some") %!error ... %! nbincdf (ones (3), ones (2), ones (2)) %!error ... %! nbincdf (ones (2), ones (3), ones (2)) %!error ... %! nbincdf (ones (2), ones (2), ones (3)) %!error nbincdf (i, 2, 2) %!error nbincdf (2, i, 2) %!error nbincdf (2, 2, i) statistics-release-1.6.3/inst/dist_fun/nbininv.m000066400000000000000000000207331456127120000217450ustar00rootroot00000000000000## Copyright (C) 1995-2012 Kurt Hornik ## Copyright (C) 2012-2016 Rik Wehbring ## Copyright (C) 2016-2017 Lachlan Andrew ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{x} =} nbininv (@var{p}, @var{r}, @var{ps}) ## ## Inverse of the negative binomial cumulative distribution function (iCDF). ## ## For each element of @var{p}, compute the quantile (the inverse of the CDF) of ## the negative binomial distribution with parameters @var{r} and @var{ps}, ## where @var{r} is the number of successes until the experiment is stopped and ## @var{ps} is the probability of success in each experiment, given the ## probability in @var{p}. The size of @var{x} is the common size of @var{p}, ## @var{r}, and @var{ps}. A scalar input functions as a constant matrix of the ## same size as the other inputs. ## ## When @var{r} is an integer, the negative binomial distribution is also known ## as the Pascal distribution and it models the number of failures in @var{x} ## before a specified number of successes is reached in a series of independent, ## identical trials. Its parameters are the probability of success in a single ## trial, @var{ps}, and the number of successes, @var{r}. A special case of the ## negative binomial distribution, when @qcode{@var{r} = 1}, is the geometric ## distribution, which models the number of failures before the first success. ## ## @var{r} can also have non-integer positive values, in which form the negative ## binomial distribution, also known as the Polya distribution, has no ## interpretation in terms of repeated trials, but, like the Poisson ## distribution, it is useful in modeling count data. The negative binomial ## distribution is more general than the Poisson distribution because it has a ## variance that is greater than its mean, making it suitable for count data ## that do not meet the assumptions of the Poisson distribution. In the limit, ## as @var{r} increases to infinity, the negative binomial distribution ## approaches the Poisson distribution. ## ## Further information about the negative binomial distribution can be found at ## @url{https://en.wikipedia.org/wiki/Negative_binomial_distribution} ## ## @seealso{nbininv, nbinpdf, nbinrnd, nbinfit, nbinlike, nbinstat} ## @end deftypefn function x = nbininv (p, r, ps) ## Check for valid number of input arguments if (nargin < 3) error ("nbininv: function called with too few input arguments."); endif ## Check for common size of P, R, and PS if (! isscalar (p) || ! isscalar (r) || ! isscalar (ps)) [retval, p, r, ps] = common_size (p, r, ps); if (retval > 0) error ("nbininv: P, R, and PS must be of common size or scalars."); endif endif ## Check for P, R, and PS being reals if (iscomplex (p) || iscomplex (r) || iscomplex (ps)) error ("nbininv: P, R, and PS must not be complex."); endif ## Check for class type if (isa (p, "single") || isa (r, "single") || isa (ps, "single")) x = zeros (size (p), "single"); else x = zeros (size (p)); endif k = (isnan (p) | (p < 0) | (p > 1) | isnan (r) | (r < 1) | (r == Inf) | isnan (ps) | (ps < 0) | (ps > 1)); x(k) = NaN; k = (p == 1) & (r > 0) & (r < Inf) & (ps >= 0) & (ps <= 1); x(k) = Inf; k = find ((p >= 0) & (p < 1) & (r > 0) & (r < Inf) & (ps > 0) & (ps <= 1)); if (! isempty (k)) p = p(k); m = zeros (size (k)); if (isscalar (r) && isscalar (ps)) [m, unfinished] = scalar_nbininv (p(:), r, ps); m(unfinished) = bin_search_nbininv (p(unfinished), r, ps); else m = bin_search_nbininv (p, r(k), ps(k)); endif x(k) = m; endif endfunction ## Core algorithm to calculate the inverse negative binomial, for r and ps real ## scalars and y a column vector, and for which the output is not NaN or Inf. ## Compute CDF in batches of doubling size until CDF > p, or answer > 500. ## Return the locations of unfinished cases in k. function [m, k] = scalar_nbininv (p, r, ps) k = 1:length (p); m = zeros (size (p)); prev_limit = 0; limit = 10; do cdf = nbincdf (prev_limit:limit, r, ps); rr = bsxfun (@le, p(k), cdf); [v, m(k)] = max (rr, [], 2); # find first instance of p <= cdf m(k) += prev_limit - 1; k = k(v == 0); prev_limit = limit; limit += limit; until (isempty (k) || limit >= 1000) endfunction ## Vectorized binary search. ## Can handle vectors r and ps, and is faster than the scalar case when the ## answer is large. ## Could be optimized to call nbincdf only for a subset of the p at each stage, ## but care must be taken to handle both scalar and vector r,ps. Bookkeeping ## may cost more than the extra computations. function m = bin_search_nbininv (p, r, ps) k = 1:length (p); lower = zeros (size (p)); limit = 1; while (any (k) && limit < 1e100) cdf = nbincdf (limit, r, ps); k = (p > cdf); lower(k) = limit; limit += limit; endwhile upper = max (2*lower, 1); k = find (lower != limit/2); # elements for which above loop finished for i = 1:ceil (log2 (max (lower))) mid = (upper + lower)/2; cdf = nbincdf (floor (mid), r, ps); rr = (p <= cdf); upper(rr) = mid(rr); lower(! rr) = mid(! rr); endfor m = ceil (lower); m(p > nbincdf (m, r, ps)) += 1; # fix off-by-one errors from binary search endfunction %!demo %! ## Plot various iCDFs from the negative binomial distribution %! p = 0.001:0.001:0.999; %! x1 = nbininv (p, 2, 0.15); %! x2 = nbininv (p, 5, 0.2); %! x3 = nbininv (p, 4, 0.4); %! x4 = nbininv (p, 10, 0.3); %! plot (p, x1, "-r", p, x2, "-g", p, x3, "-k", p, x4, "-m") %! grid on %! ylim ([0, 40]) %! legend ({"r = 2, ps = 0.15", "r = 5, ps = 0.2", "r = 4, p = 0.4", ... %! "r = 10, ps = 0.3"}, "location", "northwest") %! title ("Negative binomial iCDF") %! xlabel ("probability") %! ylabel ("values in x (number of failures)") ## Test output %!shared p %! p = [-1 0 3/4 1 2]; %!assert (nbininv (p, ones (1,5), 0.5*ones (1,5)), [NaN 0 1 Inf NaN]) %!assert (nbininv (p, 1, 0.5*ones (1,5)), [NaN 0 1 Inf NaN]) %!assert (nbininv (p, ones (1,5), 0.5), [NaN 0 1 Inf NaN]) %!assert (nbininv (p, [1 0 NaN Inf 1], 0.5), [NaN NaN NaN NaN NaN]) %!assert (nbininv (p, [1 0 1.5 Inf 1], 0.5), [NaN NaN 2 NaN NaN]) %!assert (nbininv (p, 1, 0.5*[1 -Inf NaN Inf 1]), [NaN NaN NaN NaN NaN]) %!assert (nbininv ([p(1:2) NaN p(4:5)], 1, 0.5), [NaN 0 NaN Inf NaN]) ## Test class of input preserved %!assert (nbininv ([p, NaN], 1, 0.5), [NaN 0 1 Inf NaN NaN]) %!assert (nbininv (single ([p, NaN]), 1, 0.5), single ([NaN 0 1 Inf NaN NaN])) %!assert (nbininv ([p, NaN], single (1), 0.5), single ([NaN 0 1 Inf NaN NaN])) %!assert (nbininv ([p, NaN], 1, single (0.5)), single ([NaN 0 1 Inf NaN NaN])) ## Test accuracy, to within +/- 1 since it is a discrete distribution %!shared y, tol %! y = magic (3) + 1; %! tol = 1; %!assert (nbininv (nbincdf (1:10, 3, 0.1), 3, 0.1), 1:10, tol) %!assert (nbininv (nbincdf (1:10, 3./(1:10), 0.1), 3./(1:10), 0.1), 1:10, tol) %!assert (nbininv (nbincdf (y, 3./y, 1./y), 3./y, 1./y), y, tol) ## Test input validation %!error nbininv () %!error nbininv (1) %!error nbininv (1, 2) %!error ... %! nbininv (ones (3), ones (2), ones (2)) %!error ... %! nbininv (ones (2), ones (3), ones (2)) %!error ... %! nbininv (ones (2), ones (2), ones (3)) %!error nbininv (i, 2, 2) %!error nbininv (2, i, 2) %!error nbininv (2, 2, i) statistics-release-1.6.3/inst/dist_fun/nbinpdf.m000066400000000000000000000137731456127120000217300ustar00rootroot00000000000000## Copyright (C) 2012 Rik Wehbring ## Copyright (C) 1995-2016 Kurt Hornik ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{y} =} nbinpdf (@var{x}, @var{r}, @var{ps}) ## ## Negative binomial probability density function (PDF). ## ## For each element of @var{x}, compute the probability density function (PDF) ## at @var{x} of the negative binomial distribution with parameters @var{r} and ## @var{ps}, where @var{r} is the number of successes until the experiment is ## stopped and @var{ps} is the probability of success in each experiment, given ## the number of failures in @var{x}. The size of @var{y} is the common size of ## @var{x}, @var{r}, and @var{ps}. A scalar input functions as a constant ## matrix of the same size as the other inputs. ## ## When @var{r} is an integer, the negative binomial distribution is also known ## as the Pascal distribution and it models the number of failures in @var{x} ## before a specified number of successes is reached in a series of independent, ## identical trials. Its parameters are the probability of success in a single ## trial, @var{ps}, and the number of successes, @var{r}. A special case of the ## negative binomial distribution, when @qcode{@var{r} = 1}, is the geometric ## distribution, which models the number of failures before the first success. ## ## @var{r} can also have non-integer positive values, in which form the negative ## binomial distribution, also known as the Polya distribution, has no ## interpretation in terms of repeated trials, but, like the Poisson ## distribution, it is useful in modeling count data. The negative binomial ## distribution is more general than the Poisson distribution because it has a ## variance that is greater than its mean, making it suitable for count data ## that do not meet the assumptions of the Poisson distribution. In the limit, ## as @var{r} increases to infinity, the negative binomial distribution ## approaches the Poisson distribution. ## ## Further information about the negative binomial distribution can be found at ## @url{https://en.wikipedia.org/wiki/Negative_binomial_distribution} ## ## @seealso{nbininv, nbininv, nbinrnd, nbinfit, nbinlike, nbinstat} ## @end deftypefn function y = nbinpdf (x, r, ps) ## Check for valid number of input arguments if (nargin < 3) error ("nbinpdf: function called with too few input arguments."); endif ## Check for common size of X, R, and PS if (! isscalar (x) || ! isscalar (r) || ! isscalar (ps)) [retval, x, r, ps] = common_size (x, r, ps); if (retval > 0) error ("nbinpdf: X, R, and PS must be of common size or scalars."); endif endif ## Check for X, R, and PS being reals if (iscomplex (x) || iscomplex (r) || iscomplex (ps)) error ("nbinpdf: X, R, and PS must not be complex."); endif ## Check for class type if (isa (x, "single") || isa (r, "single") || isa (ps, "single")) y = NaN (size (x), "single"); else y = NaN (size (x)); endif ok = (x < Inf) & (x == fix (x)) & (r > 0) & (r < Inf) & (ps >= 0) & (ps <= 1); k = (x < 0) & ok; y(k) = 0; k = (x >= 0) & ok; if (isscalar (r) && isscalar (ps)) y(k) = bincoeff (-r, x(k)) .* (ps ^ r) .* ((ps - 1) .^ x(k)); else y(k) = bincoeff (-r(k), x(k)) .* (ps(k) .^ r(k)) .* ((ps(k) - 1) .^ x(k)); endif endfunction %!demo %! ## Plot various PDFs from the negative binomial distribution %! x = 0:40; %! y1 = nbinpdf (x, 2, 0.15); %! y2 = nbinpdf (x, 5, 0.2); %! y3 = nbinpdf (x, 4, 0.4); %! y4 = nbinpdf (x, 10, 0.3); %! plot (x, y1, "*r", x, y2, "*g", x, y3, "*k", x, y4, "*m") %! grid on %! xlim ([0, 40]) %! ylim ([0, 0.12]) %! legend ({"r = 2, ps = 0.15", "r = 5, ps = 0.2", "r = 4, p = 0.4", ... %! "r = 10, ps = 0.3"}, "location", "northeast") %! title ("Negative binomial PDF") %! xlabel ("values in x (number of failures)") %! ylabel ("density") ## Test output %!shared x, y %! x = [-1 0 1 2 Inf]; %! y = [0 1/2 1/4 1/8 NaN]; %!assert (nbinpdf (x, ones (1,5), 0.5*ones (1,5)), y) %!assert (nbinpdf (x, 1, 0.5*ones (1,5)), y) %!assert (nbinpdf (x, ones (1,5), 0.5), y) %!assert (nbinpdf (x, [0 1 NaN 1.5 Inf], 0.5), [NaN 1/2 NaN 1.875*0.5^1.5/4 NaN], eps) %!assert (nbinpdf (x, 1, 0.5*[-1 NaN 4 1 1]), [NaN NaN NaN y(4:5)]) %!assert (nbinpdf ([x, NaN], 1, 0.5), [y, NaN]) ## Test class of input preserved %!assert (nbinpdf (single ([x, NaN]), 1, 0.5), single ([y, NaN])) %!assert (nbinpdf ([x, NaN], single (1), 0.5), single ([y, NaN])) %!assert (nbinpdf ([x, NaN], 1, single (0.5)), single ([y, NaN])) ## Test input validation %!error nbinpdf () %!error nbinpdf (1) %!error nbinpdf (1, 2) %!error ... %! nbinpdf (ones (3), ones (2), ones (2)) %!error ... %! nbinpdf (ones (2), ones (3), ones (2)) %!error ... %! nbinpdf (ones (2), ones (2), ones (3)) %!error nbinpdf (i, 2, 2) %!error nbinpdf (2, i, 2) %!error nbinpdf (2, 2, i) statistics-release-1.6.3/inst/dist_fun/nbinrnd.m000066400000000000000000000202101456127120000217220ustar00rootroot00000000000000## Copyright (C) 2012 Rik Wehbring ## Copyright (C) 1995-2016 Kurt Hornik ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{rnd} =} nbinrnd (@var{r}, @var{ps}) ## @deftypefnx {statistics} {@var{rnd} =} nbinrnd (@var{r}, @var{ps}, @var{rows}) ## @deftypefnx {statistics} {@var{rnd} =} nbinrnd (@var{r}, @var{ps}, @var{rows}, @var{cols}, @dots{}) ## @deftypefnx {statistics} {@var{rnd} =} nbinrnd (@var{r}, @var{ps}, [@var{sz}]) ## ## Random arrays from the negative binomial distribution. ## ## @code{@var{rnd} = nbinrnd (@var{r}, @var{ps})} returns an array of random ## numbers chosen from the Laplace distribution with parameters @var{r} and ## @var{ps}, where @var{r} is the number of successes until the experiment is ## stopped and @var{ps} is the probability of success in each experiment, given ## the number of failures in @var{x}. The size of @var{rnd} is the common size ## of @var{r} and @var{ps}. A scalar input functions as a constant matrix of ## the same size as the other inputs. ## ## When called with a single size argument, return a square matrix with ## the dimension specified. When called with more than one scalar argument the ## first two arguments are taken as the number of rows and columns and any ## further arguments specify additional matrix dimensions. The size may also ## be specified with a vector of dimensions @var{sz}. ## ## When @var{r} is an integer, the negative binomial distribution is also known ## as the Pascal distribution and it models the number of failures in @var{x} ## before a specified number of successes is reached in a series of independent, ## identical trials. Its parameters are the probability of success in a single ## trial, @var{ps}, and the number of successes, @var{r}. A special case of the ## negative binomial distribution, when @qcode{@var{r} = 1}, is the geometric ## distribution, which models the number of failures before the first success. ## ## @var{r} can also have non-integer positive values, in which form the negative ## binomial distribution, also known as the Polya distribution, has no ## interpretation in terms of repeated trials, but, like the Poisson ## distribution, it is useful in modeling count data. The negative binomial ## distribution is more general than the Poisson distribution because it has a ## variance that is greater than its mean, making it suitable for count data ## that do not meet the assumptions of the Poisson distribution. In the limit, ## as @var{r} increases to infinity, the negative binomial distribution ## approaches the Poisson distribution. ## ## Further information about the negative binomial distribution can be found at ## @url{https://en.wikipedia.org/wiki/Negative_binomial_distribution} ## ## @seealso{nbininv, nbininv, nbinpdf, nbinfit, nbinlike, nbinstat} ## @end deftypefn function rnd = nbinrnd (r, ps, varargin) ## Check for valid number of input arguments if (nargin < 2) error ("nbinrnd: function called with too few input arguments."); endif ## Check for common size R and PS if (! isscalar (r) || ! isscalar (ps)) [retval, r, ps] = common_size (r, ps); if (retval > 0) error ("nbinrnd: R and PS must be of common size or scalars."); endif endif ## Check for R and PS being reals if (iscomplex (r) || iscomplex (ps)) error ("nbinrnd: R and PS must not be complex."); endif ## Parse and check SIZE arguments if (nargin == 2) sz = size (r); elseif (nargin == 3) if (isscalar (varargin{1}) && varargin{1} >= 0 ... && varargin{1} == fix (varargin{1})) sz = [varargin{1}, varargin{1}]; elseif (isrow (varargin{1}) && all (varargin{1} >= 0) ... && all (varargin{1} == fix (varargin{1}))) sz = varargin{1}; elseif error (strcat (["nbinrnd: SZ must be a scalar or a row vector"], ... [" of non-negative integers."])); endif elseif (nargin > 3) posint = cellfun (@(x) (! isscalar (x) || x < 0 || x != fix (x)), varargin); if (any (posint)) error ("nbinrnd: dimensions must be non-negative integers."); endif sz = [varargin{:}]; endif ## Check that parameters match requested dimensions in size if (! isscalar (r) && ! isequal (size (r), sz)) error ("nbinrnd: R and PS must be scalars or of size SZ."); endif ## Check for class type if (isa (r, "single") || isa (ps, "single")) cls = "single"; else cls = "double"; endif ## Generate random sample from negative binomial distribution if (isscalar (r) && isscalar (ps)) if ((r > 0) && (r < Inf) && (ps > 0) && (ps <= 1)) rnd = randp ((1 - ps) ./ ps .* randg (r, sz, cls), cls); elseif ((r > 0) && (r < Inf) && (ps == 0)) rnd = zeros (sz, cls); else rnd = NaN (sz, cls); endif else rnd = NaN (sz, cls); k = (r > 0) & (r < Inf) & (ps == 0); rnd(k) = 0; k = (r > 0) & (r < Inf) & (ps > 0) & (ps <= 1); rnd(k) = randp ((1 - ps(k)) ./ ps(k) .* randg (r(k), cls)); endif endfunction ## Test output %!assert (size (nbinrnd (1, 0.5)), [1 1]) %!assert (size (nbinrnd (1, 0.5 * ones (2,1))), [2, 1]) %!assert (size (nbinrnd (1, 0.5 * ones (2,2))), [2, 2]) %!assert (size (nbinrnd (ones (2,1), 0.5)), [2, 1]) %!assert (size (nbinrnd (ones (2,2), 0.5)), [2, 2]) %!assert (size (nbinrnd (1, 0.5, 3)), [3, 3]) %!assert (size (nbinrnd (1, 0.5, [4, 1])), [4, 1]) %!assert (size (nbinrnd (1, 0.5, 4, 1)), [4, 1]) %!assert (size (nbinrnd (1, 0.5, 4, 1, 5)), [4, 1, 5]) %!assert (size (nbinrnd (1, 0.5, 0, 1)), [0, 1]) %!assert (size (nbinrnd (1, 0.5, 1, 0)), [1, 0]) %!assert (size (nbinrnd (1, 0.5, 1, 2, 0, 5)), [1, 2, 0, 5]) ## Test class of input preserved %!assert (class (nbinrnd (1, 0.5)), "double") %!assert (class (nbinrnd (1, single (0.5))), "single") %!assert (class (nbinrnd (1, single ([0.5, 0.5]))), "single") %!assert (class (nbinrnd (single (1), 0.5)), "single") %!assert (class (nbinrnd (single ([1, 1]), 0.5)), "single") ## Test input validation %!error nbinrnd () %!error nbinrnd (1) %!error ... %! nbinrnd (ones (3), ones (2)) %!error ... %! nbinrnd (ones (2), ones (3)) %!error nbinrnd (i, 2, 3) %!error nbinrnd (1, i, 3) %!error ... %! nbinrnd (1, 2, -1) %!error ... %! nbinrnd (1, 2, 1.2) %!error ... %! nbinrnd (1, 2, ones (2)) %!error ... %! nbinrnd (1, 2, [2 -1 2]) %!error ... %! nbinrnd (1, 2, [2 0 2.5]) %!error ... %! nbinrnd (1, 2, 2, -1, 5) %!error ... %! nbinrnd (1, 2, 2, 1.5, 5) %!error ... %! nbinrnd (2, ones (2), 3) %!error ... %! nbinrnd (2, ones (2), [3, 2]) %!error ... %! nbinrnd (2, ones (2), 3, 2) statistics-release-1.6.3/inst/dist_fun/ncfcdf.m000066400000000000000000000226221456127120000215240ustar00rootroot00000000000000## Copyright (C) 2022-2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{p} =} ncfcdf (@var{x}, @var{df1}, @var{df2}, @var{lambda}) ## @deftypefnx {statistics} {@var{p} =} ncfcdf (@var{x}, @var{df1}, @var{df2}, @var{lambda}, @qcode{"upper"}) ## ## Noncentral @math{F}-cumulative distribution function (CDF). ## ## For each element of @var{x}, compute the cumulative distribution function ## (CDF) of the noncentral @math{F}-distribution with @var{df1} and @var{df2} ## degrees of freedom and noncentrality parameter @var{lambda}. The size of ## @var{p} is the common size of @var{x}, @var{df1}, @var{df2}, and ## @var{lambda}. A scalar input functions as a constant matrix of the same size ## as the other inputs. ## ## @code{@var{p} = ncfcdf (@var{x}, @var{df1}, @var{df2}, @var{lambda}, "upper")} ## computes the upper tail probability of the noncentral @math{F}-distribution ## with parameters @var{df1}, @var{df2}, and @var{lambda}, at the values in ## @var{x}. ## ## Further information about the noncentral @math{F}-distribution can be found ## at @url{https://en.wikipedia.org/wiki/Noncentral_F-distribution} ## ## @seealso{ncfinv, ncfpdf, ncfrnd, ncfstat, fcdf} ## @end deftypefn function p = ncfcdf (x, df1, df2, lambda, uflag) ## Check for valid number of input arguments if (nargin < 4) error ("ncfcdf: function called with too few input arguments."); endif ## Check for valid "upper" flag if (nargin > 4) if (! strcmpi (uflag, "upper")) error ("ncfcdf: invalid argument for upper tail."); else uflag = true; endif else uflag = false; endif ## Check for common size of X, DF1, DF2, and LAMBDA [err, x, df1, df2, lambda] = common_size (x, df1, df2, lambda); if (err > 0) error ("ncfcdf: X, DF1, DF2, and LAMBDA must be of common size or scalars."); endif ## Check for X, DF1, DF2, and LAMBDA being reals if (iscomplex (x) || iscomplex (df1) || iscomplex (df2) || iscomplex (lambda)) error ("ncfcdf: X, DF1, DF2, and LAMBDA must not be complex."); endif ## Check for class type if (isa (x, "single") || isa (df1, "single") || ... isa (df2, "single") || isa (lambda, "single")) p = zeros (size (x), "single"); c_eps = eps ("single") .^ (3/4); else p = zeros (size (x)); c_eps = eps .^ (3/4); endif ## Find NaNs in input arguments (if any) and propagate them to p is_nan = isnan (x) | isnan (df1) | isnan (df1) | isnan (lambda); p(is_nan) = NaN; ## For "upper" option, force p = 1 for x <= 0, and p = 0 for x == Inf, ## otherwise, force p = 1 for x == Inf. if (uflag) p(x == Inf & ! is_nan) = 0; p(x <= 0 & ! is_nan) = 1; else p(x == Inf & ! is_nan) = 1; endif ## Find invalid values of parameters and propagate them to p as NaN k = (df1 <= 0 | df2 <= 0 | lambda < 0); p(k) = NaN; ## Compute central distribution (lambda == 0) k0 = (lambda==0); if (any (k0(:))) if (uflag) p(k0) = fcdf (x(k0), df1(k0), df2(k0), "upper"); else p(k0) = fcdf (x(k0), df1(k0), df2(k0)); endif endif ## Check if there are remaining elements and reset variables k1 = ! (k0 | k | x == Inf | x <= 0 | is_nan); if (! any (k1(:))) return; else x = x(k1); df1 = df1(k1); df2 = df2(k1); lambda = lambda(k1); endif ## Prepare variables x = x(:); df1 = df1(:) / 2; df2 = df2(:) / 2; lambda = lambda(:) / 2; ## Value passed to Beta distribution function. tmp = df1 .* x ./ (df2 + df1 .* x); logtmp = log (tmp); nu2const = df2 .* log (1 - tmp) - localgammaln (df2); ## Sum the series. The general idea is that we are going to sum terms ## of the form 'poisspdf(j,lambda) .* betacdf(tmp,j+df1,df2)' j0 = floor (lambda(:)); ## Compute Poisson pdf and beta cdf at the starting point if (uflag) bcdf0 = betainc (tmp, j0 + df1, df2, "upper"); else bcdf0 = betacdf (tmp, j0 + df1, df2); endif ppdf0 = exp (-lambda + j0 .* log (lambda) - localgammaln (j0 + 1)); ## Set up for loop over values less than j0 y = ppdf0 .* bcdf0; ppdf = ppdf0; bcdf = bcdf0; olddy = zeros (size (lambda)); delty = zeros (size (lambda)); j = j0 - 1; ok = j >= 0; while (any (ok)) ## Use recurrence relation to compute new pdf and cdf ppdf(ok) = ppdf(ok) .* (j(ok) + 1) ./ lambda(ok); if (uflag) bcdf(ok) = betainc (tmp(ok), j(ok) + df1(ok), df2(ok), "upper"); else db = exp ((j + df1) .* logtmp + nu2const + ... localgammaln (j + df1 + df2) - localgammaln (j + df1 + 1)); bcdf(ok) = bcdf(ok) + db(ok); endif delty(ok) = ppdf(ok) .* bcdf(ok); y(ok) = y(ok) + delty(ok); ## Convergence test: change must be small and not increasing ok = ok & (delty > y*c_eps | abs (delty) > olddy); j = j - 1; ok = ok & j >= 0; olddy(ok) = abs (delty(ok)); endwhile ## Set up again for loop upward from j0 ppdf = ppdf0; bcdf = bcdf0; olddy = zeros (size (lambda)); j = j0 + 1; ok = true(size(j)); ## Set up for loop to avoid endless loop for jj = 1:5000 ppdf = ppdf .* lambda ./ j; if (uflag) bcdf = betainc (tmp, j + df1, df2, "upper"); else bcdf = bcdf - exp ((j + df1 - 1) .* logtmp + nu2const + ... localgammaln (j + df1 + df2 - 1) - localgammaln (j + df1)); endif delty = ppdf.*bcdf; ## ok = indices not converged y(ok) = y(ok) + delty(ok); ## Convergence test: change must be small and not increasing ok = ok & (delty>y*c_eps | abs(delty)>olddy); ## Break if all indices converged if (! any (ok)) break; endif olddy(ok) = abs (delty(ok)); j = j + 1; endfor if (jj == 5000) warning ("ncfcdf: no convergence."); endif ## Save returning p-value p(k1) = y; endfunction function x = localgammaln (y) x = Inf (size (y), class (y)); x(! (y < 0)) = gammaln (y(! (y < 0))); endfunction %!demo %! ## Plot various CDFs from the noncentral F distribution %! x = 0:0.01:5; %! p1 = ncfcdf (x, 2, 5, 1); %! p2 = ncfcdf (x, 2, 5, 2); %! p3 = ncfcdf (x, 5, 10, 1); %! p4 = ncfcdf (x, 10, 20, 10); %! plot (x, p1, "-r", x, p2, "-g", x, p3, "-k", x, p4, "-m") %! grid on %! xlim ([0, 5]) %! legend ({"df1 = 2, df2 = 5, λ = 1", "df1 = 2, df2 = 5, λ = 2", ... %! "df1 = 5, df2 = 10, λ = 1", "df1 = 10, df2 = 20, λ = 10"}, ... %! "location", "southeast") %! title ("Noncentral F CDF") %! xlabel ("values in x") %! ylabel ("probability") %!demo %! ## Compare the noncentral F CDF with LAMBDA = 10 to the F CDF with the %! ## same number of numerator and denominator degrees of freedom (5, 20) %! %! x = 0.01:0.1:10.01; %! p1 = ncfcdf (x, 5, 20, 10); %! p2 = fcdf (x, 5, 20); %! plot (x, p1, "-", x, p2, "-"); %! grid on %! xlim ([0, 10]) %! legend ({"Noncentral F(5,20,10)", "F(5,20)"}, "location", "southeast") %! title ("Noncentral F vs F CDFs") %! xlabel ("values in x") %! ylabel ("probability") ## Test output %!test %! x = -2:0.1:2; %! p = ncfcdf (x, 10, 1, 3); %! assert (p([1:21]), zeros (1, 21), 1e-76); %! assert (p(22), 0.004530737275319753, 1e-14); %! assert (p(30), 0.255842099135669, 1e-14); %! assert (p(41), 0.4379890998457305, 1e-14); %!test %! p = ncfcdf (12, 10, 3, 2); %! assert (p, 0.9582287900447416, 1e-14); %!test %! p = ncfcdf (2, 3, 2, 1); %! assert (p, 0.5731985522994989, 1e-14); %!test %! p = ncfcdf (2, 3, 2, 1, "upper"); %! assert (p, 0.4268014477004823, 1e-14); %!test %! p = ncfcdf ([3, 6], 3, 2, 5, "upper"); %! assert (p, [0.530248523596927, 0.3350482341323044], 1e-14); ## Test input validation %!error ncfcdf () %!error ncfcdf (1) %!error ncfcdf (1, 2) %!error ncfcdf (1, 2, 3) %!error ncfcdf (1, 2, 3, 4, "tail") %!error ncfcdf (1, 2, 3, 4, 5) %!error ... %! ncfcdf (ones (3), ones (2), ones (2), ones (2)) %!error ... %! ncfcdf (ones (2), ones (3), ones (2), ones (2)) %!error ... %! ncfcdf (ones (2), ones (2), ones (3), ones (2)) %!error ... %! ncfcdf (ones (2), ones (2), ones (2), ones (3)) %!error ncfcdf (i, 2, 2, 2) %!error ncfcdf (2, i, 2, 2) %!error ncfcdf (2, 2, i, 2) %!error ncfcdf (2, 2, 2, i) statistics-release-1.6.3/inst/dist_fun/ncfinv.m000066400000000000000000000171771456127120000215750ustar00rootroot00000000000000## Copyright (C) 2022-2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{x} =} ncfinv (@var{p}, @var{df1}, @var{df2}, @var{lambda}) ## ## Inverse of the noncentral @math{F}-cumulative distribution function (iCDF). ## ## For each element of @var{p}, compute the quantile (the inverse of the CDF) of ## the noncentral @math{F}-distribution with @var{df1} and @var{df2} degrees of ## freedom and noncentrality parameter @var{lambda}. The size of @var{x} is the ## common size of @var{p}, @var{df1}, @var{df2}, and @var{lambda}. A scalar ## input functions as a constant matrix of the same size as the other inputs. ## ## @code{ncfinv} uses Newton's method to converge to the solution. ## ## Further information about the noncentral @math{F}-distribution can be found ## at @url{https://en.wikipedia.org/wiki/Noncentral_F-distribution} ## ## @seealso{ncfcdf, ncfpdf, ncfrnd, ncfstat, finv} ## @end deftypefn function x = ncfinv (p, df1, df2, lambda) ## Check for valid number of input arguments if (nargin < 4) error ("ncfinv: function called with too few input arguments."); endif ## Check for common size of P, DF1, DF2, and LAMBDA [err, p, df1, df2, lambda] = common_size (p, df1, df2, lambda); if (err > 0) error ("ncfinv: P, DF1, DF2, and LAMBDA must be of common size or scalars."); endif ## Check for P, DF1, DF2, and LAMBDA being reals if (iscomplex (p) || iscomplex (df1) || iscomplex (df2) || iscomplex (lambda)) error ("ncfinv: P, DF1, DF2, and LAMBDA must not be complex."); endif ## Check for class type if (isa (p, "single") || isa (df1, "single") || ... isa (df2, "single") || isa (lambda, "single")) x = NaN (size (p), "single"); crit = sqrt (eps ("single")); else x = NaN (size (p), "double"); crit = sqrt (eps ("double")); endif ## For lambda == 0, call finv d0 = lambda == 0; if (any (d0(:))) x(d0) = finv (p(d0), df1(d0), df2(d0)); endif ## For lambda > 0 and valid dfs valid = df1 > 0 & df2 > 0 & lambda > 0; ## Force x = 0 for p == 0 ax = Inf for p ==1 x(p == 0 & valid) = 0; x(p == 1 & valid) = Inf; ## Find remaining valid cases within the range of 0 < p < 1 k = find (p > 0 & p < 1 & valid); ## Return if nothing left if isempty(k) return; endif ## Reset input variables to remaining cases p = p(k); df1 = df1(k); df2 = df2(k); lambda = lambda(k); ## Initialize counter count_limit = 100; count = 0; ## Start at the mean (if it exists) mu0 = df2.*(df1+lambda) ./ (df1.*max(1,df2-2)); next = mu0; prev = 0; F = ncfcdf (mu0, df1, df2, lambda); while(count < count_limit) count += 1; next = (F - p) ./ ncfpdf (mu0, df1, df2, lambda); ## Prevent oscillations if (length (next) == length (prev)) t = sign (next) == -sign (prev); next(t) = sign (next(t)) .* min (abs (next(t)), abs (prev(t))) / 2; endif ## Prepare for next step mu1 = max (mu0 / 5, min (5 * mu0, mu0 - next)); ## Check that next step improves, otherwise abort F1 = ncfcdf (mu1, df1, df2, lambda); while (true) worse = (abs (F1-p) > abs (F - p) * (1 + crit)) & ... (abs (mu0 - mu1) > crit * mu0); if (! any (worse)) break; endif mu1(worse) = 0.5 * (mu1(worse) + mu0(worse)); F1(worse) = ncfcdf (mu1(worse), df1(worse), df2(worse), lambda(worse)); endwhile x(k) = mu1; ## Find elements that are not converged yet next = mu0 - mu1; mask = (abs (next) > crit * abs (mu0)); if (! any (mask)) break; endif ## Save parameters for these elements only F = F1(mask); mu0 = mu1(mask); prev = next(mask); if (! all(mask)) df1 = df1(mask); df2 = df2(mask); lambda = lambda(mask); p = p(mask); k = k(mask); endif endwhile if (count == count_limit) warning ("ncfinv: did not converge."); endif endfunction %!demo %! ## Plot various iCDFs from the noncentral F distribution %! p = 0.001:0.001:0.999; %! x1 = ncfinv (p, 2, 5, 1); %! x2 = ncfinv (p, 2, 5, 2); %! x3 = ncfinv (p, 5, 10, 1); %! x4 = ncfinv (p, 10, 20, 10); %! plot (p, x1, "-r", p, x2, "-g", p, x3, "-k", p, x4, "-m") %! grid on %! ylim ([0, 5]) %! legend ({"df1 = 2, df2 = 5, λ = 1", "df1 = 2, df2 = 5, λ = 2", ... %! "df1 = 5, df2 = 10, λ = 1", "df1 = 10, df2 = 20, λ = 10"}, ... %! "location", "northwest") %! title ("Noncentral F iCDF") %! xlabel ("probability") %! ylabel ("values in x") %!demo %! ## Compare the noncentral F iCDF with LAMBDA = 10 to the F iCDF with the %! ## same number of numerator and denominator degrees of freedom (5, 20) %! %! p = 0.001:0.001:0.999; %! x1 = ncfinv (p, 5, 20, 10); %! x2 = finv (p, 5, 20); %! plot (p, x1, "-", p, x2, "-"); %! grid on %! ylim ([0, 10]) %! legend ({"Noncentral F(5,20,10)", "F(5,20)"}, "location", "northwest") %! title ("Noncentral F vs F quantile functions") %! xlabel ("probability") %! ylabel ("values in x") ## Test output %!test %! x = [0,0.1775,0.3864,0.6395,0.9564,1.3712,1.9471,2.8215,4.3679,8.1865,Inf]; %! assert (ncfinv ([0:0.1:1], 2, 3, 1), x, 1e-4); %!test %! x = [0,0.7492,1.3539,2.0025,2.7658,3.7278,5.0324,6.9826,10.3955,18.7665,Inf]; %! assert (ncfinv ([0:0.1:1], 2, 3, 5), x, 1e-4); %!test %! x = [0,0.2890,0.8632,1.5653,2.4088,3.4594,4.8442,6.8286,10.0983,17.3736,Inf]; %! assert (ncfinv ([0:0.1:1], 1, 4, 3), x, 1e-4); %!test %! x = [0.078410, 0.212716, 0.288618, 0.335752, 0.367963, 0.391460]; %! assert (ncfinv (0.05, [1, 2, 3, 4, 5, 6], 10, 3), x, 1e-6); %!test %! x = [0.2574, 0.2966, 0.3188, 0.3331, 0.3432, 0.3507]; %! assert (ncfinv (0.05, 5, [1, 2, 3, 4, 5, 6], 3), x, 1e-4); %!test %! x = [1.6090, 1.8113, 1.9215, 1.9911, NaN, 2.0742]; %! assert (ncfinv (0.05, 1, [1, 2, 3, 4, -1, 6], 10), x, 1e-4); %!test %! assert (ncfinv (0.996, 3, 5, 8), 58.0912074080671, 2e-13); ## Test input validation %!error ncfinv () %!error ncfinv (1) %!error ncfinv (1, 2) %!error ncfinv (1, 2, 3) %!error ... %! ncfinv (ones (3), ones (2), ones (2), ones (2)) %!error ... %! ncfinv (ones (2), ones (3), ones (2), ones (2)) %!error ... %! ncfinv (ones (2), ones (2), ones (3), ones (2)) %!error ... %! ncfinv (ones (2), ones (2), ones (2), ones (3)) %!error ncfinv (i, 2, 2, 2) %!error ncfinv (2, i, 2, 2) %!error ncfinv (2, 2, i, 2) %!error ncfinv (2, 2, 2, i) statistics-release-1.6.3/inst/dist_fun/ncfpdf.m000066400000000000000000000371101456127120000215370ustar00rootroot00000000000000## Copyright (C) 2022-2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{y} =} ncfpdf (@var{x}, @var{df1}, @var{df2}, @var{lambda}) ## ## Noncentral @math{F}-probability density function (PDF). ## ## For each element of @var{x}, compute the probability density function (PDF) ## of the noncentral @math{F}-distribution with @var{df1} and @var{df2} degrees ## of freedom and noncentrality parameter @var{lambda}. The size of @var{y} is ## the common size of @var{x}, @var{df1}, @var{df2}, and @var{lambda}. A scalar ## input functions as a constant matrix of the same size as the other inputs. ## ## Further information about the noncentral @math{F}-distribution can be found ## at @url{https://en.wikipedia.org/wiki/Noncentral_F-distribution} ## ## @seealso{ncfcdf, ncfinv, ncfrnd, ncfstat, fpdf} ## @end deftypefn function y = ncfpdf (x, df1, df2, lambda) ## Check for valid number of input arguments if (nargin < 4) error ("ncfpdf: function called with too few input arguments."); endif ## Check for common size of X, DF1, DF2, and LAMBDA [err, x, df1, df2, lambda] = common_size (x, df1, df2, lambda); if (err > 0) error ("ncfpdf: X, DF1, DF2, and LAMBDA must be of common size or scalars."); endif ## Check for X, DF1, DF2, and LAMBDA being reals if (iscomplex (x) || iscomplex (df1) || iscomplex (df2) || iscomplex (lambda)) error ("ncfpdf: X, DF1, DF2, and LAMBDA must not be complex."); endif ## Check for class type if (isa (x, "single") || isa (df1, "single") || ... isa (df2, "single") || isa (lambda, "single")) y = zeros (size (x), "single"); else y = zeros (size (x)); endif ## Find NaNs in input arguments (if any) and propagate them to p is_nan = isnan (x) | isnan (df1) | isnan (df2) | isnan (lambda); y(is_nan) = NaN; ## Force invalid parameter cases to NaN k1 = df1 <= 0 | df2 <= 0 | lambda < 0; y(k1) = NaN; ## Handle edge cases where x == 0 k2 = x == 0 & df1 < 2 & ! k1; y(k2) = Inf; k3 = x == 0 & df1 == 2 & ! k1; if (any (k3(:))) y(k3) = exp (-lambda(k3) / 2); endif ## Handle central distribution where lambda == 0 k4 = lambda == 0 & ! k1 & x > 0; if any(k4(:)) y(k4) = fpdf (x(k4), df1(k4), df2(k4)); endif ## Handle normal cases td = find (x > 0 & ! (k1 | k4)); ## Return if finished all normal cases if (isempty (td)) return; endif ## Reset input variables to remaining cases and pre-divide df1, df2 and lambda x = x(td); df1 = df1(td) / 2; df2 = df2(td) / 2; lambda = lambda(td) / 2; ## Use z and scaled x for convenience z = df1 .* x ./ (df1 .* x + df2); z1 = df2 ./ (df1 .* x + df2); xs = lambda .* z; % Find max K at which we start the recursion series K = zeros (size (x)); termK = zeros (size (x)); rsum = zeros (size (x)); ## Handy constant lnsr2pi = 0.9189385332046727; ## Process integer and non-integer df2 separately df2int = df2 == floor (df2); if (any (df2int(:))) # integers smallx = xs <= df1 ./ df2; largex = xs >= df2 .* (df1 + df2 - 1) & ! smallx; K(df2int & largex) = df2(df2int & largex); ## Compute K idx = df2int & ! (smallx | largex); if (any (idx(:))) d = 0.5 * (1 - xs(idx) - df1(idx)); K(idx) = floor (d + sqrt (d .^ 2 + xs(idx) .* (df2(idx) + 1))); endif ## For K == df2 K_df2 = df2int & K == df2; idz1 = K_df2 & z < 0.9; termK(idz1) = (df1(idz1) + df2(idz1) - 1) .* log (z(idz1)); idz2 = K_df2 & ! idz1; termK(idz2) = (df1(idz2) + df2(idz2) - 1) .* log1p (-z1(idz2)); ## For K == 0 Kzero = df2int & (df1 + K) <= 1; termK(Kzero) = StirlingError (df1(Kzero) + df2(Kzero)) - ... StirlingError (df1(Kzero)) - StirlingError (df2(Kzero)) - ... BinoPoisson (df1(Kzero), ... (df1(Kzero) + df2(Kzero)) .* z(Kzero)) - ... BinoPoisson (df2(Kzero), ... (df1(Kzero) + df2(Kzero)) .* z1(Kzero)); ## For all other K K_all = df2int & ! (K_df2 | Kzero); termK(K_all) = StirlingError (df1(K_all) + df2(K_all) - 1) - ... StirlingError (df1(K_all) + K(K_all) -1) - ... StirlingError (df2(K_all) - K(K_all)) - ... BinoPoisson (df1(K_all) + K(K_all) - 1, ... (df1(K_all) + df2(K_all) - 1) .* z(K_all)) - ... BinoPoisson (df2(K_all) - K(K_all), ... (df1(K_all) + df2(K_all) - 1) .* z1(K_all)); ## Poisson density for the leading term x1 = lambda .* z1; smallk = df2int & K <= x1 * realmin; y(td(smallk)) = termK(smallk) - x1(smallk); otherk = df2int & ! smallk; y(td(otherk)) = termK(otherk) - lnsr2pi - 0.5 * log (K(otherk)) - ... StirlingError (K(otherk)) - ... BinoPoisson (K(otherk), x1(otherk)); ## Sum recursively downwards term = ones (size (x)); k = K; ok = df2int & k > 0; while (any (ok(:))) k(ok) = k(ok) - 1; term(ok) = term(ok) .* (k(ok) + 1) .* ... (k(ok) + df1(ok)) ./ (df2(ok) - k(ok)) ./ xs(ok); ok = ok & term >= eps (rsum); rsum(ok) = rsum(ok) + term(ok); endwhile ## Sum recursively upwards term = ones (size (x)); k = K; ok = df2int & k < df2; while any(ok(:)) term(ok) = term(ok) .* xs(ok) .* ... (df2(ok) - k(ok)) ./ (k(ok) + df1(ok)) ./ (k(ok) + 1); ok = ok & term >= eps(rsum); rsum(ok) = rsum(ok) + term(ok); k(ok) = k(ok) + 1; endwhile endif if (any (! df2int(:))) # non-integers ## Compute K largex = ! df2int & xs > df1 ./ (df1 + df2); d = 0.5 * (1 + xs(largex) - df1(largex)); K(largex) = floor (d + sqrt (d .^ 2 + xs(largex) .* ... (df1(largex) + df2(largex) - 1))); ## For K == 0 Kzero = ! df2int & (df1 + K) <= 1; termK(Kzero) = StirlingError (df1(Kzero) + df2(Kzero)) - ... StirlingError (df1(Kzero)) - ... StirlingError (df2(Kzero)) - ... BinoPoisson (df1(Kzero), ... (df1(Kzero) + df2(Kzero)) .* z(Kzero)) - ... BinoPoisson (df2(Kzero), ... (df1(Kzero) + df2(Kzero)) .* z1(Kzero)); ## For K != 0 K_all = ! df2int & ! Kzero; termK(K_all) = StirlingError (df1(K_all) + df2(K_all) + K(K_all) - 1) - ... StirlingError (df1(K_all) + K(K_all) - 1) - ... StirlingError (df2(K_all)) - ... BinoPoisson (df1(K_all) + K(K_all) - 1, ... (df1(K_all) + df2(K_all) + K(K_all) - 1) .* ... z(K_all)) - ... BinoPoisson (df2(K_all), ... (df1(K_all) + df2(K_all) + K(K_all) - 1) .* ... z1(K_all)); ## Poisson density for the leading term smallk = ! df2int & K <= lambda * realmin; y(td(smallk)) = termK(smallk) - lambda(smallk); K_all = ! df2int & ! smallk; y(td(K_all)) = termK(K_all) - lnsr2pi - 0.5 * log (K(K_all)) - ... StirlingError (K(K_all)) - ... BinoPoisson (K(K_all), lambda(K_all)); ## Sum recursively downwards term = ones (size (x)); k = K; ok = ! df2int & k > 0; while (any (ok(:))) k(ok) = k(ok) - 1; term(ok) = term(ok) .* (k(ok) + 1) .* (k(ok) + df1(ok)) ./ ... (k(ok) + df1(ok) + df2(ok)) ./ xs(ok); ok = ok & term >= eps (rsum); rsum(ok) = rsum(ok) + term(ok); endwhile ## Sum recursively upwards term = ones (size (x)); k = K; ok = ! df2int; while (any (ok(:))) term(ok) = term(ok) .* xs(ok) .* (k(ok) + df1(ok) + df2(ok)) ./ ... (k(ok) + df1(ok)) ./ (k(ok) + 1); ok = ok & term >= eps (rsum); rsum(ok) = rsum(ok) + term(ok); k(ok) = k(ok)+1; endwhile endif ## Compute density pi2 = 2 * pi; Kzero = (df1 + K) <= 1; y(td(Kzero)) = exp (y(td(Kzero))) .* (1 + rsum(Kzero)) .* ... sqrt (df1(Kzero) .* df2(Kzero) ./ ... (df1(Kzero) + df2(Kzero)) / pi2) ./ ... x(Kzero); K_df2 = ! Kzero & df2int & K == df2; y(td(K_df2)) = exp (y(td(K_df2))) .* (1 + rsum(K_df2)) .* ... df1(K_df2) .* z1(K_df2); idx = ! Kzero & df2int & ! K_df2; y(td(idx)) = exp (y(td(idx))) .* (1 + rsum(idx)) .* df1(idx) .* z1(idx) .* ... sqrt((df1(idx) + df2(idx) - 1) ./ (df2(idx) - K(idx)) ./ ... (df1(idx) + K(idx) - 1) / pi2); idx = ! df2int & ! Kzero; y(td(idx)) = exp (y(td(idx))) .* (1 + rsum(idx)) .* df1(idx) .* z1(idx) .* ... sqrt ((df1(idx) + df2(idx) + K(idx) - 1) ./ ... df2(idx) ./ (df1(idx) + K(idx) - 1) / pi2); endfunction ## Error of Stirling-De Moivre approximation to n factorial. function lambda = StirlingError (n) is_class = class (n); lambda = zeros (size (n), is_class); nn = n .* n; ## Define S0=1/12 S1=1/360 S2=1/1260 S3=1/1680 S4=1/1188 S0 = 8.333333333333333e-02; S1 = 2.777777777777778e-03; S2 = 7.936507936507937e-04; S3 = 5.952380952380952e-04; S4 = 8.417508417508418e-04; ## Define lambda(n) for n<0:0.5:15 sfe=[ 0; 1.534264097200273e-01;... 8.106146679532726e-02; 5.481412105191765e-02;... 4.134069595540929e-02; 3.316287351993629e-02;... 2.767792568499834e-02; 2.374616365629750e-02;... 2.079067210376509e-02; 1.848845053267319e-02;... 1.664469118982119e-02; 1.513497322191738e-02;... 1.387612882307075e-02; 1.281046524292023e-02;... 1.189670994589177e-02; 1.110455975820868e-02;... 1.041126526197210e-02; 9.799416126158803e-03;... 9.255462182712733e-03; 8.768700134139385e-03;... 8.330563433362871e-03; 7.934114564314021e-03;... 7.573675487951841e-03; 7.244554301320383e-03;... 6.942840107209530e-03; 6.665247032707682e-03;... 6.408994188004207e-03; 6.171712263039458e-03;... 5.951370112758848e-03; 5.746216513010116e-03;... 5.554733551962801e-03]; k = find (n <= 15); if (any (k)) n1 = n(k); n2 = 2 * n1; if (all (n2 == round (n2))) lambda(k) = sfe(n2+1); else lnsr2pi = 0.9189385332046728; lambda(k) = gammaln(n1+1)-(n1+0.5).*log(n1)+n1-lnsr2pi; endif endif k = find (n > 15 & n <= 35); if (any (k)) lambda(k) = (S0 - (S1 - (S2 - (S3 - S4 ./ nn(k)) ./ nn(k)) ./ ... nn(k)) ./ nn(k)) ./ n(k); endif k = find (n > 35 & n <= 80); if (any (k)) lambda(k) = (S0 - (S1 - (S2 - S3 ./ nn(k)) ./ nn(k)) ./ nn(k)) ./ n(k); endif k = find(n > 80 & n <= 500); if (any (k)) lambda(k) = (S0 - (S1 - S2 ./ nn(k)) ./ nn(k)) ./ n(k); endif k = find(n > 500); if (any (k)) lambda(k) = (S0 - S1 ./ nn(k)) ./ n(k); endif endfunction ## Deviance term for binomial and Poisson probability calculation. function BP = BinoPoisson (x, np) if (isa (x,'single') || isa (np,'single')) BP = zeros (size (x), "single"); else BP = zeros (size (x)); endif k = abs (x - np) < 0.1 * (x + np); if any(k(:)) s = (x(k) - np(k)) .* (x(k) - np(k)) ./ (x(k) + np(k)); v = (x(k) - np(k)) ./ (x(k) + np(k)); ej = 2 .* x(k) .* v; is_class = class (s); s1 = zeros (size (s), is_class); ok = true (size (s)); j = 0; while any(ok(:)) ej(ok) = ej(ok) .* v(ok) .* v(ok); j = j + 1; s1(ok) = s(ok) + ej(ok) ./ (2 * j + 1); ok = ok & s1 != s; s(ok) = s1(ok); endwhile BP(k) = s; endif k = ! k; if (any (k(:))) BP(k) = x(k) .* log (x(k) ./ np(k)) + np(k) - x(k); endif endfunction %!demo %! ## Plot various PDFs from the noncentral F distribution %! x = 0:0.01:5; %! y1 = ncfpdf (x, 2, 5, 1); %! y2 = ncfpdf (x, 2, 5, 2); %! y3 = ncfpdf (x, 5, 10, 1); %! y4 = ncfpdf (x, 10, 20, 10); %! plot (x, y1, "-r", x, y2, "-g", x, y3, "-k", x, y4, "-m") %! grid on %! xlim ([0, 5]) %! ylim ([0, 0.8]) %! legend ({"df1 = 2, df2 = 5, λ = 1", "df1 = 2, df2 = 5, λ = 2", ... %! "df1 = 5, df2 = 10, λ = 1", "df1 = 10, df2 = 20, λ = 10"}, ... %! "location", "northeast") %! title ("Noncentral F PDF") %! xlabel ("values in x") %! ylabel ("density") %!demo %! ## Compare the noncentral F PDF with LAMBDA = 10 to the F PDF with the %! ## same number of numerator and denominator degrees of freedom (5, 20) %! %! x = 0.01:0.1:10.01; %! y1 = ncfpdf (x, 5, 20, 10); %! y2 = fpdf (x, 5, 20); %! plot (x, y1, "-", x, y2, "-"); %! grid on %! xlim ([0, 10]) %! ylim ([0, 0.8]) %! legend ({"Noncentral F(5,20,10)", "F(5,20)"}, "location", "northeast") %! title ("Noncentral F vs F PDFs") %! xlabel ("values in x") %! ylabel ("density") ## Test output %!shared x1, df1, df2, lambda %! x1 = [-Inf, 2, NaN, 4, Inf]; %! df1 = [2, 0, -1, 1, 4]; %! df2 = [2, 4, 5, 6, 8]; %! lambda = [1, NaN, 3, -1, 2]; %!assert (ncfpdf (x1, df1, df2, lambda), [0, NaN, NaN, NaN, NaN]); %!assert (ncfpdf (x1, df1, df2, 1), [0, NaN, NaN, ... %! 0.05607937264237208, NaN], 1e-14); %!assert (ncfpdf (x1, df1, df2, 3), [0, NaN, NaN, ... %! 0.080125760971946518, NaN], 1e-14); %!assert (ncfpdf (x1, df1, df2, 2), [0, NaN, NaN, ... %! 0.0715902008258656, NaN], 1e-14); %!assert (ncfpdf (x1, 3, 5, lambda), [0, NaN, NaN, NaN, NaN]); %!assert (ncfpdf (2, df1, df2, lambda), [0.1254046999837947, NaN, NaN, ... %! NaN, 0.2152571783045893], 1e-14); %!assert (ncfpdf (4, df1, df2, lambda), [0.05067089541001374, NaN, NaN, ... %! NaN, 0.05560846335398539], 1e-14); ## Test input validation %!error ncfpdf () %!error ncfpdf (1) %!error ncfpdf (1, 2) %!error ncfpdf (1, 2, 3) %!error ... %! ncfpdf (ones (3), ones (2), ones (2), ones (2)) %!error ... %! ncfpdf (ones (2), ones (3), ones (2), ones (2)) %!error ... %! ncfpdf (ones (2), ones (2), ones (3), ones (2)) %!error ... %! ncfpdf (ones (2), ones (2), ones (2), ones (3)) %!error ncfpdf (i, 2, 2, 2) %!error ncfpdf (2, i, 2, 2) %!error ncfpdf (2, 2, i, 2) %!error ncfpdf (2, 2, 2, i) statistics-release-1.6.3/inst/dist_fun/ncfrnd.m000066400000000000000000000166171456127120000215620ustar00rootroot00000000000000## Copyright (C) 2022-2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{r} =} ncfrnd (@var{df1}, @var{df2}, @var{lambda}) ## @deftypefnx {statistics} {@var{r} =} ncfrnd (@var{df1}, @var{df2}, @var{lambda}, @var{rows}, @var{cols}, @dots{}) ## @deftypefnx {statistics} {@var{r} =} ncfrnd (@var{df1}, @var{df2}, @var{lambda}, [@var{sz}]) ## ## Random arrays from the noncentral @math{F}-distribution. ## ## @code{@var{x} = ncfrnd (@var{p}, @var{df1}, @var{df2}, @var{lambda})} returns ## an array of random numbers chosen from the noncentral @math{F}-distribution with ## @var{df1} and @var{df2} degrees of freedom and noncentrality parameter ## @var{lambda}. The size of @var{r} is the common size of @var{df1}, ## @var{df2}, and @var{lambda}. A scalar input functions as a constant matrix ## of the same size as the other input. ## ## @code{ncfrnd} generates values using the definition of a noncentral @math{F} ## random variable, as the ratio of a noncentral chi-squared distribution and a ## (central) chi-squared distribution. ## ## When called with a single size argument, @code{ncfrnd} returns a square ## matrix with the dimension specified. When called with more than one scalar ## argument, the first two arguments are taken as the number of rows and columns ## and any further arguments specify additional matrix dimensions. The size may ## also be specified with a row vector of dimensions, @var{sz}. ## ## Further information about the noncentral @math{F}-distribution can be found ## at @url{https://en.wikipedia.org/wiki/Noncentral_F-distribution} ## ## @seealso{ncfcdf, ncfinv, ncfpdf, ncfstat, frnd, ncx2rnd, chi2rnd} ## @end deftypefn function r = ncfrnd (df1, df2, lambda, varargin) ## Check for valid number of input arguments if (nargin < 3) error ("ncfrnd: function called with too few input arguments."); endif ## Check for common size of DF1, DF2, and LAMBDA if (! isscalar (df1) || ! isscalar (df2) || ! isscalar (lambda)) [retval, df1, df2, lambda] = common_size (df1, df2, lambda); if (retval > 0) error ("ncfrnd: DF1, DF2, and LAMBDA must be of common size or scalars."); endif endif ## Check for DF1, DF2, and LAMBDA being reals if (iscomplex (df1) || iscomplex (df2) || iscomplex (lambda)) error ("ncfrnd: DF1, DF2, and LAMBDA must not be complex."); endif ## Parse and check SIZE arguments if (nargin == 3) sz = size (df1); elseif (nargin == 4) if (isscalar (varargin{1}) && varargin{1} >= 0 ... && varargin{1} == fix (varargin{1})) sz = [varargin{1}, varargin{1}]; elseif (isrow (varargin{1}) && all (varargin{1} >= 0) ... && all (varargin{1} == fix (varargin{1}))) sz = varargin{1}; elseif error (strcat (["ncfrnd: SZ must be a scalar or a row vector"], ... [" of non-negative integers."])); endif elseif (nargin > 4) posint = cellfun (@(x) (! isscalar (x) || x < 0 || x != fix (x)), varargin); if (any (posint)) error ("ncfrnd: dimensions must be non-negative integers."); endif sz = [varargin{:}]; endif ## Check that parameters match requested dimensions in size if (! isscalar (df1) && ! isequal (size (df1), sz)) error ("ncfrnd: DF1, DF2, and LAMBDA must be scalars or of size SZ."); endif ## Check for class type if (isa (df1, "single") || isa (df2, "single") || isa (lambda, "single")); cls = "single"; else cls = "double"; endif ## Return NaNs for out of range values of DF1, DF2, and LAMBDA df1(df1 <= 0) = NaN; df2(df2 <= 0) = NaN; lambda(lambda <= 0) = NaN; ## Generate random sample from noncentral F distribution r = (ncx2rnd (df1, lambda, sz) ./ df1) ./ ... (2 .* randg (df2 ./ 2, sz) ./ df2); ## Cast to appropriate class r = cast (r, cls); endfunction ## Test output %!assert (size (ncfrnd (1, 1, 1)), [1 1]) %!assert (size (ncfrnd (1, ones (2,1), 1)), [2, 1]) %!assert (size (ncfrnd (1, ones (2,2), 1)), [2, 2]) %!assert (size (ncfrnd (ones (2,1), 1, 1)), [2, 1]) %!assert (size (ncfrnd (ones (2,2), 1, 1)), [2, 2]) %!assert (size (ncfrnd (1, 1, 1, 3)), [3, 3]) %!assert (size (ncfrnd (1, 1, 1, [4, 1])), [4, 1]) %!assert (size (ncfrnd (1, 1, 1, 4, 1)), [4, 1]) %!assert (size (ncfrnd (1, 1, 1, 4, 1, 5)), [4, 1, 5]) %!assert (size (ncfrnd (1, 1, 1, 0, 1)), [0, 1]) %!assert (size (ncfrnd (1, 1, 1, 1, 0)), [1, 0]) %!assert (size (ncfrnd (1, 1, 1, 1, 2, 0, 5)), [1, 2, 0, 5]) ## Test class of input preserved %!assert (class (ncfrnd (1, 1, 1)), "double") %!assert (class (ncfrnd (1, single (1), 1)), "single") %!assert (class (ncfrnd (1, 1, single (1))), "single") %!assert (class (ncfrnd (1, single ([1, 1]), 1)), "single") %!assert (class (ncfrnd (1, 1, single ([1, 1]))), "single") %!assert (class (ncfrnd (single (1), 1, 1)), "single") %!assert (class (ncfrnd (single ([1, 1]), 1, 1)), "single") ## Test input validation %!error ncfrnd () %!error ncfrnd (1) %!error ncfrnd (1, 2) %!error ... %! ncfrnd (ones (3), ones (2), ones (2)) %!error ... %! ncfrnd (ones (2), ones (3), ones (2)) %!error ... %! ncfrnd (ones (2), ones (2), ones (3)) %!error ncfrnd (i, 2, 3) %!error ncfrnd (1, i, 3) %!error ncfrnd (1, 2, i) %!error ... %! ncfrnd (1, 2, 3, -1) %!error ... %! ncfrnd (1, 2, 3, 1.2) %!error ... %! ncfrnd (1, 2, 3, ones (2)) %!error ... %! ncfrnd (1, 2, 3, [2 -1 2]) %!error ... %! ncfrnd (1, 2, 3, [2 0 2.5]) %!error ... %! ncfrnd (1, 2, 3, 2, -1, 5) %!error ... %! ncfrnd (1, 2, 3, 2, 1.5, 5) %!error ... %! ncfrnd (2, ones (2), 2, 3) %!error ... %! ncfrnd (2, ones (2), 2, [3, 2]) %!error ... %! ncfrnd (2, ones (2), 2, 3, 2) statistics-release-1.6.3/inst/dist_fun/nctcdf.m000066400000000000000000000277401456127120000215500ustar00rootroot00000000000000## Copyright (C) 2022-2024 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{p} =} nctcdf (@var{x}, @var{df}, @var{mu}) ## @deftypefnx {statistics} {@var{p} =} nctcdf (@var{x}, @var{df}, @var{mu}, @qcode{"upper"}) ## ## Noncentral @math{t}-cumulative distribution function (CDF). ## ## For each element of @var{x}, compute the cumulative distribution function ## (CDF) of the noncentral @math{t}-distribution with @var{df} degrees of ## freedom and noncentrality parameter @var{mu}. The size of @var{p} is the ## common size of @var{x}, @var{df}, and @var{mu}. A scalar input functions ## as a constant matrix of the same size as the other inputs. ## ## @code{@var{p} = nctcdf (@var{x}, @var{df}, @var{mu}, "upper")} computes ## the upper tail probability of the noncentral @math{t}-distribution with ## parameters @var{df} and @var{mu}, at the values in @var{x}. ## ## Further information about the noncentral @math{t}-distribution can be found ## at @url{https://en.wikipedia.org/wiki/Noncentral_t-distribution} ## ## @seealso{nctinv, nctpdf, nctrnd, nctstat, tcdf} ## @end deftypefn function p = nctcdf (x, df, mu, uflag) ## Check for valid number of input arguments if (nargin < 3) error ("nctcdf: function called with too few input arguments."); endif ## Check for valid "upper" flag if (nargin > 3) if (! strcmpi (uflag, "upper")) error ("nctcdf: invalid argument for upper tail."); else uflag = true; endif else uflag = false; endif ## Check for common size of X, DF, and MU [err, x, df, mu] = common_size (x, df, mu); if (err > 0) error ("nctcdf: X, DF, and MU must be of common size or scalars."); endif ## Check for X, DF, and MU being reals if (iscomplex (x) || iscomplex (df) || iscomplex (mu)) error ("nctcdf: X, DF, and MU must not be complex."); endif ## Check for class type if (isa (x, "single") || isa (df, "single") || isa (mu, "single")) p = zeros (size (x), "single"); c_eps = eps ("single"); else p = zeros (size (x)); c_eps = eps; endif ## Find NaNs in input arguments (if any) and propagate them to p is_nan = isnan (x) | isnan (df) | isnan (mu); p(is_nan) = NaN; ## Find special cases for mu==0 and x<0; and x = Inf. case_Dinf = (df <= 0 | isinf(mu)) & ! is_nan; case_Dzero = mu == 0 & ! case_Dinf & ! is_nan; case_Xzero = x < 0 & ! case_Dzero & ! case_Dinf & ! is_nan; case_Xinf = x == Inf & ! case_Dzero & ! case_Dinf & ! is_nan; case_DFbig = df > 2e6 & ! case_Dzero & ! case_Dinf & ! case_Xinf & ! is_nan; flag_Dinf = any (case_Dinf(:)); flag_Dzero = any (case_Dzero(:)); flag_Xzero = any (case_Xzero(:)); flag_Xinf = any (case_Xinf(:)); flag_DFbig = any (case_DFbig(:)); ## Handle special cases if (flag_Dinf || flag_Dzero || flag_Xzero || flag_Xinf || flag_DFbig) if (flag_Dinf) p(case_Dinf) = NaN; endif if (flag_Dzero) if (uflag) p(case_Dzero) = tcdf (x(case_Dzero), df(case_Dzero), "upper"); else p(case_Dzero) = tcdf (x(case_Dzero), df(case_Dzero)); endif endif if (flag_Xinf) if (uflag) p(case_Xinf) = 0; else p(case_Xinf) = 1; endif endif if (flag_DFbig) s = 1 - 1 ./ (4 * df); d = sqrt (1 + x .^ 2 ./ (2 * df)); if (uflag) p(case_DFbig) = normcdf (x(case_DFbig) .* s(case_DFbig), ... mu(case_DFbig), d(case_DFbig), "upper"); else p(case_DFbig) = normcdf (x(case_DFbig) .* s(case_DFbig), ... mu(case_DFbig), d(case_DFbig)); endif endif fp = ! (case_Dinf | case_Dzero | case_Xzero | case_Xinf | case_DFbig); if (any (fp(:))) if (uflag) p(fp) = nctcdf (x(fp), df(fp), mu(fp), "upper"); else p(fp) = nctcdf (x(fp), df(fp), mu(fp)); endif endif if (flag_Xzero) if (uflag) p(case_Xzero) = nctcdf (-x(case_Xzero), df(case_Xzero), ... -mu(case_Xzero)); else p(case_Xzero) = nctcdf (-x(case_Xzero), df(case_Xzero), ... -mu(case_Xzero), "upper"); endif endif return endif ## Compute value for betainc function. x_square = x .^ 2; denom = df + x_square; P = x_square ./ denom; Q = df ./ denom; ## Initialize infinite sum. d_square = mu .^ 2; ## Compute probability P[TD<0] (first term) if (uflag) x_zero = x == 0 & ! is_nan; if (any (x_zero(:))) fx = normcdf (- mu, 0, 1, "upper"); p(x_zero)= fx(x_zero); endif else p(! is_nan) = normcdf (- mu(! is_nan), 0, 1); endif ## Compute probability P[0 (abs (subtotal(TD)) + c_eps) * c_eps); if (! any (TD)) break; end ## Update for next iteration jj = jj+2; E1(TD) = E1(TD) .* d_square(TD) ./ (jj(TD)); E2(TD) = E2(TD) .* d_square(TD) ./ (jj(TD) + 1); if (uflag) B1(TD) = betainc (P(TD), (jj(TD) + 1) / 2, df(TD) / 2, "upper"); B2(TD) = betainc (P(TD), (jj(TD) + 2) / 2, df(TD) / 2, "upper"); else B1(TD) = B1(TD) - R1(TD); B2(TD) = B2(TD) - R2(TD); R1(TD) = R1(TD) .* P(TD) .* (jj(TD)+df(TD)-1) ./ (jj(TD)+1); R2(TD) = R2(TD) .* P(TD) .* (jj(TD)+df(TD) ) ./ (jj(TD)+2); endif endwhile ## Go back to the peak and start looping downward as far as necessary. E1 = E10; E2 = E20; B1 = B10; B2 = B20; R1 = R10; R2 = R20; jj = j0; TD = (jj > 0); while (any (TD)) JJ = jj(TD); E1(TD) = E1(TD) .* (JJ ) ./ d_square(TD); E2(TD) = E2(TD) .* (JJ+1) ./ d_square(TD); R1(TD) = R1(TD) .* (JJ+1) ./ ((JJ+df(TD)-1) .* P(TD)); R2(TD) = R2(TD) .* (JJ+2) ./ ((JJ+df(TD)) .* P(TD)); if (uflag) B1(TD) = betainc (P(TD), (JJ - 1) / 2, df(TD) / 2, "upper"); B2(TD) = betainc (P(TD), JJ / 2, df(TD) / 2, "upper"); else B1(TD) = B1(TD) + R1(TD); B2(TD) = B2(TD) + R2(TD); end twoterms = E1(TD) .* B1(TD) + E2(TD) .* B2(TD); subtotal(TD) = subtotal(TD) + twoterms; jj = jj - 2; TD(TD) = (abs (twoterms) > (abs (subtotal(TD)) + c_eps) * c_eps) & ... (jj(TD) > 0); endwhile p(x_notzero) = min (1, max (0, p(x_notzero) + subtotal / 2)); endif endfunction %!demo %! ## Plot various CDFs from the noncentral Τ distribution %! x = -5:0.01:5; %! p1 = nctcdf (x, 1, 0); %! p2 = nctcdf (x, 4, 0); %! p3 = nctcdf (x, 1, 2); %! p4 = nctcdf (x, 4, 2); %! plot (x, p1, "-r", x, p2, "-g", x, p3, "-k", x, p4, "-m") %! grid on %! xlim ([-5, 5]) %! legend ({"df = 1, μ = 0", "df = 4, μ = 0", ... %! "df = 1, μ = 2", "df = 4, μ = 2"}, "location", "southeast") %! title ("Noncentral Τ CDF") %! xlabel ("values in x") %! ylabel ("probability") %!demo %! ## Compare the noncentral T CDF with MU = 1 to the T CDF %! ## with the same number of degrees of freedom (10). %! %! x = -5:0.1:5; %! p1 = nctcdf (x, 10, 1); %! p2 = tcdf (x, 10); %! plot (x, p1, "-", x, p2, "-") %! grid on %! xlim ([-5, 5]) %! legend ({"Noncentral T(10,1)", "T(10)"}, "location", "southeast") %! title ("Noncentral T vs T CDFs") %! xlabel ("values in x") %! ylabel ("probability") ## Test output %!test %! x = -2:0.1:2; %! p = nctcdf (x, 10, 1); %! assert (p(1), 0.003302485766631558, 1e-14); %! assert (p(2), 0.004084668193532631, 1e-14); %! assert (p(3), 0.005052800319478737, 1e-14); %! assert (p(41), 0.8076115625303751, 1e-14); %!test %! p = nctcdf (12, 10, 3); %! assert (p, 0.9997719343243797, 1e-14); %!test %! p = nctcdf (2, 3, 2); %! assert (p, 0.4430757822176028, 1e-14); %!test %! p = nctcdf (2, 3, 2, "upper"); %! assert (p, 0.5569242177823971, 1e-14); %!test %! p = nctcdf ([3, 6], 3, 2, "upper"); %! assert (p, [0.3199728259444777, 0.07064855592441913], 1e-14); ## Test input validation %!error nctcdf () %!error nctcdf (1) %!error nctcdf (1, 2) %!error nctcdf (1, 2, 3, "tail") %!error nctcdf (1, 2, 3, 4) %!error ... %! nctcdf (ones (3), ones (2), ones (2)) %!error ... %! nctcdf (ones (2), ones (3), ones (2)) %!error ... %! nctcdf (ones (2), ones (2), ones (3)) %!error nctcdf (i, 2, 2) %!error nctcdf (2, i, 2) %!error nctcdf (2, 2, i) statistics-release-1.6.3/inst/dist_fun/nctinv.m000066400000000000000000000152151456127120000216020ustar00rootroot00000000000000## Copyright (C) 2022-2024 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{x} =} ncx2inv (@var{p}, @var{df}, @var{mu}) ## ## Inverse of the non-central @math{t}-cumulative distribution function (iCDF). ## ## For each element of @var{p}, compute the quantile (the inverse of the CDF) of ## the noncentral @math{t}-distribution with @var{df} degrees of freedom and ## noncentrality parameter @var{mu}. The size of @var{x} is the common size ## of @var{p}, @var{df}, and @var{mu}. A scalar input functions as a ## constant matrix of the same size as the other inputs. ## ## @code{nctinv} uses Newton's method to converge to the solution. ## ## Further information about the noncentral @math{t}-distribution can be found ## at @url{https://en.wikipedia.org/wiki/Noncentral_t-distribution} ## ## @seealso{nctcdf, nctpdf, nctrnd, nctstat, tinv} ## @end deftypefn function x = nctinv (p, df, mu) ## Check for valid number of input arguments if (nargin < 3) error ("nctinv: function called with too few input arguments."); endif ## Check for common size of P, DF, and MU [err, p, df, mu] = common_size (p, df, mu); if (err > 0) error ("nctinv: P, DF, and MU must be of common size or scalars."); endif ## Check for P, DF, and MU being reals if (iscomplex (p) || iscomplex (df) || iscomplex (mu)) error ("nctinv: P, DF, and MU must not be complex."); endif ## Check for class type if (isa (p, "single") || isa (df, "single") || isa (mu, "single")) x = NaN (size (p), "single"); crit = sqrt (eps ("single")); else x = NaN (size (p), "double"); crit = sqrt (eps ("double")); endif ## For mu == 0, call chi2inv m0 = mu == 0; if (any (m0(:))) x(m0) = tinv (p(m0), df(m0)); ## If mu == 0 for all entries, then return if (all (m0(:))) return; endif endif ## For all valid entries valid = df > 0 & ! isnan (mu) & ! isinf (mu); ## Force x = -Inf for p == 0 and x = Inf for p == 1 x(p == 0 & valid) = -Inf; x(p == 1 & valid) = Inf; ## Find valid samples within the range of 0 < p < 1 k = find (p > 0 & p < 1 & valid); p_k = p(k); df_k = df(k); mu_k = mu(k); ## Initialize counter count_limit = 100; count = 0; ## Supply a starting guess for the iteration with norminv x_k = norminv (p_k, mu_k, 1); h_k = ones (size (x_k), class (x_k)); ## Start iteration with a break out loop F = nctcdf (x_k, df_k, mu_k); while (any (abs (h_k) > crit * abs (x_k)) && ... max (abs (h_k)) > crit && count < count_limit) count = count + 1; h_k = (F - p_k) ./ nctpdf (x_k, df_k, mu_k); ## Prevent Infs - NaNs infnan = isinf(h_k) | isnan(h_k); if (any (infnan(:))) h_k(infnan) = x_k(infnan) / 10; endif ## Prepare for next step xnew = max (-5 * abs (x_k), min (5 * abs (x_k), x_k - h_k)); ## Check that next step improves, otherwise abort Fnew = nctcdf (xnew, df_k, mu_k); while (true) worse = (abs (Fnew - p_k) > abs (F - p_k) * (1 + crit)) & ... (abs (x_k - xnew) > crit * abs (x_k)); if (! any (worse)) break; endif xnew(worse) = 0.5 * (xnew(worse) + x_k(worse)); Fnew(worse) = nctcdf (xnew(worse), df_k(worse), mu_k(worse)); endwhile x_k = xnew; F = Fnew; endwhile ## Return the converged value(s). x(k) = x_k; if (count == count_limit) warning ("nctinv: did not converge."); endif endfunction %!demo %! ## Plot various iCDFs from the noncentral T distribution %! p = 0.001:0.001:0.999; %! x1 = nctinv (p, 1, 0); %! x2 = nctinv (p, 4, 0); %! x3 = nctinv (p, 1, 2); %! x4 = nctinv (p, 4, 2); %! plot (p, x1, "-r", p, x2, "-g", p, x3, "-k", p, x4, "-m") %! grid on %! ylim ([-5, 5]) %! legend ({"df = 1, μ = 0", "df = 4, μ = 0", ... %! "df = 1, μ = 2", "df = 4, μ = 2"}, "location", "northwest") %! title ("Noncentral T iCDF") %! xlabel ("probability") %! ylabel ("values in x") %!demo %! ## Compare the noncentral T iCDF with MU = 1 to the T iCDF %! ## with the same number of degrees of freedom (10). %! %! p = 0.001:0.001:0.999; %! x1 = nctinv (p, 10, 1); %! x2 = tinv (p, 10); %! plot (p, x1, "-", p, x2, "-"); %! grid on %! ylim ([-5, 5]) %! legend ({"Noncentral T(10,1)", "T(10)"}, "location", "northwest") %! title ("Noncentral T vs T quantile functions") %! xlabel ("probability") %! ylabel ("values in x") ## Test output %!test %! x = [-Inf,-0.3347,0.1756,0.5209,0.8279,1.1424,1.5021,1.9633,2.6571,4.0845,Inf]; %! assert (nctinv ([0:0.1:1], 2, 1), x, 1e-4); %!test %! x = [-Inf,1.5756,2.0827,2.5343,3.0043,3.5406,4.2050,5.1128,6.5510,9.6442,Inf]; %! assert (nctinv ([0:0.1:1], 2, 3), x, 1e-4); %!test %! x = [-Inf,2.2167,2.9567,3.7276,4.6464,5.8455,7.5619,10.3327,15.7569,31.8159,Inf]; %! assert (nctinv ([0:0.1:1], 1, 4), x, 1e-4); %!test %! x = [1.7791 1.9368 2.0239 2.0801 2.1195 2.1489]; %! assert (nctinv (0.05, [1, 2, 3, 4, 5, 6], 4), x, 1e-4); %!test %! x = [-0.7755, 0.3670, 1.2554, 2.0239, 2.7348, 3.4154]; %! assert (nctinv (0.05, 3, [1, 2, 3, 4, 5, 6]), x, 1e-4); %!test %! x = [-0.7183, 0.3624, 1.2878, 2.1195, -3.5413, 3.6430]; %! assert (nctinv (0.05, 5, [1, 2, 3, 4, -1, 6]), x, 1e-4); %!test %! assert (nctinv (0.996, 5, 8), 30.02610554063658, 2e-11); ## Test input validation %!error nctinv () %!error nctinv (1) %!error nctinv (1, 2) %!error ... %! nctinv (ones (3), ones (2), ones (2)) %!error ... %! nctinv (ones (2), ones (3), ones (2)) %!error ... %! nctinv (ones (2), ones (2), ones (3)) %!error nctinv (i, 2, 2) %!error nctinv (2, i, 2) %!error nctinv (2, 2, i) statistics-release-1.6.3/inst/dist_fun/nctpdf.m000066400000000000000000000142571456127120000215640ustar00rootroot00000000000000## Copyright (C) 2022-2024 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{y} =} nctpdf (@var{x}, @var{df}, @var{mu}) ## ## Noncentral @math{t}-probability density function (PDF). ## ## For each element of @var{x}, compute the probability density function (PDF) ## of the noncentral @math{t}-distribution with @var{df} degrees of freedom and ## noncentrality parameter @var{mu}. The size of @var{y} is the common size ## of @var{x}, @var{df}, and @var{mu}. A scalar input functions as a ## constant matrix of the same size as the other inputs. ## ## Further information about the noncentral @math{t}-distribution can be found ## at @url{https://en.wikipedia.org/wiki/Noncentral_t-distribution} ## ## @seealso{nctcdf, nctinv, nctrnd, nctstat, tpdf} ## @end deftypefn function y = nctpdf (x, df, mu) ## Check for valid number of input arguments if (nargin < 3) error ("nctpdf: function called with too few input arguments."); endif ## Check for common size of X, DF, and MU [err, x, df, mu] = common_size (x, df, mu); if (err > 0) error ("nctpdf: X, DF, and MU must be of common size or scalars."); endif ## Check for X, DF, and MU being reals if (iscomplex (x) || iscomplex (df) || iscomplex (mu)) error ("nctpdf: X, DF, and MU must not be complex."); endif ## Check for class type if (isa (x, "single") || isa (df, "single") || isa (mu, "single")) y = zeros (size (x), "single"); else y = zeros (size (x)); endif ## Find NaNs in input arguments (if any) and propagate them to p is_nan = isnan (x) | isnan (df) | isnan (mu); y(is_nan) = NaN; ## Force invalid parameter cases to NaN invalid = df <= 0 | ! isfinite (mu); y(invalid) = NaN; ## Use normal approximation for df > 1e6 bigDF = df > 1e6 & ! is_nan & ! invalid; if (any (bigDF(:))) s = 1 - 1 ./ (4 * df); d = sqrt (1 + x .^ 2 ./ (2 * df)); y(bigDF) = normpdf (x(bigDF) .* s(bigDF), mu(bigDF), d(bigDF)); endif ## For negative x use left tail cdf x_neg = find ((x < 0) & isfinite (x) & df <= 1e6 & ! is_nan & ! invalid); if (any (x_neg)) y(x_neg) = (df(x_neg) ./ x(x_neg)) .* ... (nctcdf (x(x_neg) .* sqrt ((df(x_neg) + 2) ./ df(x_neg)), ... df(x_neg) + 2, mu(x_neg)) - ... nctcdf (x(x_neg), df(x_neg), mu(x_neg))); endif ## For positive x reflect about zero and use left tail cdf x_pos = find ((x > 0) & isfinite (x) & df <= 1e6 & ! is_nan & ! invalid); if (any (x_pos)) y(x_pos) = (-df(x_pos) ./ x(x_pos)) .* ... (nctcdf (-x(x_pos) .* sqrt ((df(x_pos) + 2) ./ df(x_pos)), ... df(x_pos) + 2, -mu(x_pos)) - ... nctcdf (-x(x_pos), df(x_pos), -mu(x_pos))); endif ## For x == 0 use power series xzero = find ((x == 0) & df <= 1e6 & ! is_nan & ! invalid); if (any (xzero)) y(xzero) = exp (-0.5 * mu(xzero) .^ 2 - 0.5 * log (pi * df(xzero)) + ... gammaln (0.5 * (df(xzero) + 1)) - gammaln (0.5 * df(xzero))); endif endfunction %!demo %! ## Plot various PDFs from the noncentral T distribution %! x = -5:0.01:10; %! y1 = nctpdf (x, 1, 0); %! y2 = nctpdf (x, 4, 0); %! y3 = nctpdf (x, 1, 2); %! y4 = nctpdf (x, 4, 2); %! plot (x, y1, "-r", x, y2, "-g", x, y3, "-k", x, y4, "-m") %! grid on %! xlim ([-5, 10]) %! ylim ([0, 0.4]) %! legend ({"df = 1, μ = 0", "df = 4, μ = 0", ... %! "df = 1, μ = 2", "df = 4, μ = 2"}, "location", "northeast") %! title ("Noncentral T PDF") %! xlabel ("values in x") %! ylabel ("density") %!demo %! ## Compare the noncentral T PDF with MU = 1 to the T PDF %! ## with the same number of degrees of freedom (10). %! %! x = -5:0.1:5; %! y1 = nctpdf (x, 10, 1); %! y2 = tpdf (x, 10); %! plot (x, y1, "-", x, y2, "-"); %! grid on %! xlim ([-5, 5]) %! ylim ([0, 0.4]) %! legend ({"Noncentral χ^2(4,2)", "χ^2(4)"}, "location", "northwest") %! title ("Noncentral T vs T PDFs") %! xlabel ("values in x") %! ylabel ("density") ## Test output %!shared x1, df, mu %! x1 = [-Inf, 2, NaN, 4, Inf]; %! df = [2, 0, -1, 1, 4]; %! mu = [1, NaN, 3, -1, 2]; %!assert (nctpdf (x1, df, mu), [0, NaN, NaN, 0.00401787561306999, 0], 1e-14); %!assert (nctpdf (x1, df, 1), [0, NaN, NaN, 0.0482312135423008, 0], 1e-14); %!assert (nctpdf (x1, df, 3), [0, NaN, NaN, 0.1048493126401585, 0], 1e-14); %!assert (nctpdf (x1, df, 2), [0, NaN, NaN, 0.08137377919890307, 0], 1e-14); %!assert (nctpdf (x1, 3, mu), [0, NaN, NaN, 0.001185305171654381, 0], 1e-14); %!assert (nctpdf (2, df, mu), [0.1791097459405861, NaN, NaN, ... %! 0.0146500727180389, 0.3082302682110299], 1e-14); %!assert (nctpdf (4, df, mu), [0.04467929612254971, NaN, NaN, ... %! 0.00401787561306999, 0.0972086534042828], 1e-14); ## Test input validation %!error nctpdf () %!error nctpdf (1) %!error nctpdf (1, 2) %!error ... %! nctpdf (ones (3), ones (2), ones (2)) %!error ... %! nctpdf (ones (2), ones (3), ones (2)) %!error ... %! nctpdf (ones (2), ones (2), ones (3)) %!error nctpdf (i, 2, 2) %!error nctpdf (2, i, 2) %!error nctpdf (2, 2, i) statistics-release-1.6.3/inst/dist_fun/nctrnd.m000066400000000000000000000151231456127120000215670ustar00rootroot00000000000000## Copyright (C) 2022-2024 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{r} =} nctrnd (@var{df}, @var{mu}) ## @deftypefnx {statistics} {@var{r} =} nctrnd (@var{df}, @var{mu}, @var{rows}, @var{cols}, @dots{}) ## @deftypefnx {statistics} {@var{r} =} nctrnd (@var{df}, @var{mu}, [@var{sz}]) ## ## Random arrays from the noncentral @math{t}-distribution. ## ## @code{@var{x} = nctrnd (@var{p}, @var{df}, @var{mu})} returns an array of ## random numbers chosen from the noncentral @math{t}-distribution with @var{df} ## degrees of freedom and noncentrality parameter @var{mu}. The size of ## @var{r} is the common size of @var{df} and @var{mu}. A scalar input ## functions as a constant matrix of the same size as the other input. ## ## @code{nctrnd} generates values using the definition of a noncentral @math{t} ## random variable, as the ratio of a normal distribution with non-zero mean and ## the sqrt of a chi-squared distribution. ## ## When called with a single size argument, @code{nctrnd} returns a square ## matrix with the dimension specified. When called with more than one scalar ## argument, the first two arguments are taken as the number of rows and columns ## and any further arguments specify additional matrix dimensions. The size may ## also be specified with a row vector of dimensions, @var{sz}. ## ## Further information about the noncentral @math{t}-distribution can be found ## at @url{https://en.wikipedia.org/wiki/Noncentral_t-distribution} ## ## @seealso{nctcdf, nctinv, nctpdf, nctstat, trnd, normrnd, chi2rnd} ## @end deftypefn function r = nctrnd (df, mu, varargin) ## Check for valid number of input arguments if (nargin < 2) error ("nctrnd: function called with too few input arguments."); endif ## Check for common size of DF and MU if (! isscalar (df) || ! isscalar (mu)) [retval, df, mu] = common_size (df, mu); if (retval > 0) error ("nctrnd: DF and MU must be of common size or scalars."); endif endif ## Check for DF and MU being reals if (iscomplex (df) || iscomplex (mu)) error ("nctrnd: DF and MU must not be complex."); endif ## Parse and check SIZE arguments if (nargin == 2) sz = size (df); elseif (nargin == 3) if (isscalar (varargin{1}) && varargin{1} >= 0 ... && varargin{1} == fix (varargin{1})) sz = [varargin{1}, varargin{1}]; elseif (isrow (varargin{1}) && all (varargin{1} >= 0) ... && all (varargin{1} == fix (varargin{1}))) sz = varargin{1}; elseif error (strcat (["nctrnd: SZ must be a scalar or a row vector"], ... [" of non-negative integers."])); endif elseif (nargin > 3) posint = cellfun (@(x) (! isscalar (x) || x < 0 || x != fix (x)), varargin); if (any (posint)) error ("nctrnd: dimensions must be non-negative integers."); endif sz = [varargin{:}]; endif ## Check that parameters match requested dimensions in size if (! isscalar (df) && ! isequal (size (df), sz)) error ("nctrnd: DF and MU must be scalars or of size SZ."); endif ## Check for class type if (isa (df, "single") || isa (mu, "single")); cls = "single"; else cls = "double"; endif ## Return NaNs for out of range values of DF df(df <= 0) = NaN; ## Prevent Inf/Inf==NaN for the standardized chi-square in the denom. df(isinf (df)) = realmax; ## Generate random sample from noncentral F distribution r = (randn (sz) + mu) ./ sqrt (2 .* randg (df ./ 2, sz) ./ df); ## Cast to appropriate class r = cast (r, cls); endfunction ## Test output %!assert (size (nctrnd (1, 1)), [1 1]) %!assert (size (nctrnd (1, ones (2,1))), [2, 1]) %!assert (size (nctrnd (1, ones (2,2))), [2, 2]) %!assert (size (nctrnd (ones (2,1), 1)), [2, 1]) %!assert (size (nctrnd (ones (2,2), 1)), [2, 2]) %!assert (size (nctrnd (1, 1, 3)), [3, 3]) %!assert (size (nctrnd (1, 1, [4, 1])), [4, 1]) %!assert (size (nctrnd (1, 1, 4, 1)), [4, 1]) %!assert (size (nctrnd (1, 1, 4, 1, 5)), [4, 1, 5]) %!assert (size (nctrnd (1, 1, 0, 1)), [0, 1]) %!assert (size (nctrnd (1, 1, 1, 0)), [1, 0]) %!assert (size (nctrnd (1, 1, 1, 2, 0, 5)), [1, 2, 0, 5]) ## Test class of input preserved %!assert (class (nctrnd (1, 1)), "double") %!assert (class (nctrnd (1, single (1))), "single") %!assert (class (nctrnd (1, single ([1, 1]))), "single") %!assert (class (nctrnd (single (1), 1)), "single") %!assert (class (nctrnd (single ([1, 1]), 1)), "single") ## Test input validation %!error nctrnd () %!error nctrnd (1) %!error ... %! nctrnd (ones (3), ones (2)) %!error ... %! nctrnd (ones (2), ones (3)) %!error nctrnd (i, 2) %!error nctrnd (1, i) %!error ... %! nctrnd (1, 2, -1) %!error ... %! nctrnd (1, 2, 1.2) %!error ... %! nctrnd (1, 2, ones (2)) %!error ... %! nctrnd (1, 2, [2 -1 2]) %!error ... %! nctrnd (1, 2, [2 0 2.5]) %!error ... %! nctrnd (1, 2, 2, -1, 5) %!error ... %! nctrnd (1, 2, 2, 1.5, 5) %!error ... %! nctrnd (2, ones (2), 3) %!error ... %! nctrnd (2, ones (2), [3, 2]) %!error ... %! nctrnd (2, ones (2), 3, 2) statistics-release-1.6.3/inst/dist_fun/ncx2cdf.m000066400000000000000000000232561456127120000216340ustar00rootroot00000000000000## Copyright (C) 2022-2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{p} =} ncx2cdf (@var{x}, @var{df}, @var{lambda}) ## @deftypefnx {statistics} {@var{p} =} ncx2cdf (@var{x}, @var{df}, @var{lambda}, @qcode{"upper"}) ## ## Noncentral chi-squared cumulative distribution function (CDF). ## ## For each element of @var{x}, compute the cumulative distribution function ## (CDF) of the noncentral chi-squared distribution with @var{df} degrees of ## freedom and noncentrality parameter @var{lambda}. The size of @var{p} is the ## common size of @var{x}, @var{df}, and @var{lambda}. A scalar input functions ## as a constant matrix of the same size as the other inputs. ## ## @code{@var{p} = ncx2cdf (@var{x}, @var{df}, @var{lambda}, "upper")} computes ## the upper tail probability of the noncentral chi-squared distribution with ## parameters @var{df} and @var{lambda}, at the values in @var{x}. ## ## Further information about the noncentral chi-squared distribution can be ## found at @url{https://en.wikipedia.org/wiki/Noncentral_chi-squared_distribution} ## ## @seealso{ncx2inv, ncx2pdf, ncx2rnd, ncx2stat, chi2cdf} ## @end deftypefn function p = ncx2cdf (x, df, lambda, uflag) ## Check for valid number of input arguments if (nargin < 3) error ("ncx2cdf: function called with too few input arguments."); endif ## Check for valid "upper" flag if (nargin > 3) if (! strcmpi (uflag, "upper")) error ("ncx2cdf: invalid argument for upper tail."); else uflag = true; endif else uflag = false; endif ## Check for common size of X, DF, and LAMBDA [err, x, df, lambda] = common_size (x, df, lambda); if (err > 0) error ("ncx2cdf: X, DF, and LAMBDA must be of common size or scalars."); endif ## Check for X, DF, and LAMBDA being reals if (iscomplex (x) || iscomplex (df) || iscomplex (lambda)) error ("ncx2cdf: X, DF, and LAMBDA must not be complex."); endif ## Check for class type if (isa (x, "single") || isa (df, "single") || isa (lambda, "single")) p = zeros (size (x), "single"); c_eps = eps ("single"); c_min = realmin ("single"); else p = zeros (size (x)); c_eps = eps; c_min = realmin; endif ## Find NaNs in input arguments (if any) and propagate them to p is_nan = isnan (x) | isnan (df) | isnan (lambda); p(is_nan) = NaN; if (uflag) p(x == Inf & ! is_nan) = 0; p(x <= 0 & ! is_nan) = 1; else p(x == Inf & ! is_nan) = 1; endif ## Make P = NaN for negative values of noncentrality parameter and DF p(lambda < 0) = NaN; p(df < 0) = NaN; ## For DF == 0 at x == 0 k = df == 0 & x == 0 & lambda >= 0 & ! is_nan; if (uflag) p(k) = -expm1 (-lambda(k) / 2); else p(k) = exp (-lambda(k) / 2); endif ## Central chi2cdf k = df >= 0 & x > 0 & lambda == 0 & isfinite (x) & ! is_nan; if (uflag) p(k) = chi2cdf (x(k), df(k), "upper"); else p(k) = chi2cdf (x(k), df(k)); endif ## Keep only valid samples td = find (df >= 0 & x > 0 & lambda > 0 & isfinite (x) & ! is_nan); lambda = lambda(td) / 2; df = df(td) / 2; x = x(td) / 2; ## Compute Chernoff bounds e0 = log(c_min); e1 = log(c_eps/4); t = 1 - (df + sqrt (df .^ 2 + 4 * lambda .* x)) ./ (2 * x); q = lambda .* t ./ (1 - t) - df .* log(1 - t) - t .* x; peq0 = x < lambda + df & q < e0; peq1 = x > lambda + df & q < e1; if (uflag) p(td(peq0)) = 1; else p(td(peq1)) = 1; endif td(peq0 | peq1) = []; x(peq0 | peq1) = []; df(peq0 | peq1) = []; lambda(peq0 | peq1) = []; ## Find index K of the maximal term in the summation series. ## K1 and K2 are lower and upper bounds for K, respectively. ## Indexing of terms in the summation series starts at 0. K1 = ceil ((sqrt ((df + x) .^ 2 + 4 * x .* lambda) - (df + x)) / 2); K = zeros (size (x)); k1above1 = find (K1 > 1); K2 = floor (lambda(k1above1) .* gammaincratio (x(k1above1), K1(k1above1))); fixK2 = isnan(K2) | isinf(K2); K2(fixK2) = K1(k1above1(fixK2)); K(k1above1) = K2; ## Find Poisson and Poisson*chi2cdf parts for the maximal terms in the ## summation series. if (uflag) k0 = (K==0 & df==0); K(k0) = 1; endif pois = poisspdf (K, lambda); if (uflag) full = pois .* gammainc (x, df + K, "upper"); else full = pois .* gammainc (x, df + K); endif ## Sum the series. First go downward from K and then go upward. ## The term for K is added afterwards - it is not included in either sum. sumK = zeros (size (x)); ## Downward. poisspdf(k-1,lambda)/poisspdf(k,lambda) = k/lambda poisterm = pois; fullterm = full; keep = K > 0 & fullterm > 0; k = K; while any(keep) poisterm(keep) = poisterm(keep) .* k(keep) ./ lambda(keep); k(keep) = k(keep) - 1; if (uflag) fullterm(keep) = poisterm(keep) .* ... gammainc (x(keep), df(keep) + k(keep), "upper"); else fullterm(keep) = poisterm(keep) .* ... gammainc (x(keep), df(keep) + k(keep)); endif sumK(keep) = sumK(keep) + fullterm(keep); keep = keep & k > 0 & fullterm > eps(sumK); endwhile ## Upward. poisspdf(k+1,lambda)/poisspdf(k,lambda) = lambda/(k+1) poisterm = pois; fullterm = full; keep = fullterm > 0; k = K; while any(keep) k(keep) = k(keep)+1; poisterm(keep) = poisterm(keep) .* lambda(keep) ./ k(keep); if (uflag) fullterm(keep) = poisterm(keep) .* ... gammainc (x(keep), df(keep) + k(keep), "upper"); else fullterm(keep) = poisterm(keep) .* ... gammainc (x(keep), df(keep) + k(keep)); end sumK(keep) = sumK(keep) + fullterm(keep); keep = keep & fullterm > eps(sumK); endwhile ## Get probabilities p(td) = full + sumK; p(p > 1) = 1; endfunction ## Ratio of incomplete gamma function values at S and S-1. function r = gammaincratio (x, s) ## Initialize r = zeros (size (s)); ## Finf small small = s < 2 | s <= x; ## For small S, use the ratio computed directly if (any (small(:))) r(small) = gammainc (x(small), s(small)) ./ ... gammainc (x(small), s(small) - 1); endif ## For large S, estimate numerator and denominator using 'scaledlower' option if (any (! small(:))) idx = find (! small); x = x(idx); s = s(idx); r(idx) = gammainc (x, s, "scaledlower") ./ ... gammainc (x, s - 1, "scaledlower") .* x ./ s; endif endfunction %!demo %! ## Plot various CDFs from the noncentral chi-squared distribution %! x = 0:0.1:10; %! p1 = ncx2cdf (x, 2, 1); %! p2 = ncx2cdf (x, 2, 2); %! p3 = ncx2cdf (x, 2, 3); %! p4 = ncx2cdf (x, 4, 1); %! p5 = ncx2cdf (x, 4, 2); %! p6 = ncx2cdf (x, 4, 3); %! plot (x, p1, "-r", x, p2, "-g", x, p3, "-k", ... %! x, p4, "-m", x, p5, "-c", x, p6, "-y") %! grid on %! xlim ([0, 10]) %! legend ({"df = 2, λ = 1", "df = 2, λ = 2", ... %! "df = 2, λ = 3", "df = 4, λ = 1", ... %! "df = 4, λ = 2", "df = 4, λ = 3"}, "location", "southeast") %! title ("Noncentral chi-squared CDF") %! xlabel ("values in x") %! ylabel ("probability") %!demo %! ## Compare the noncentral chi-squared CDF with LAMBDA = 2 to the %! ## chi-squared CDF with the same number of degrees of freedom (4). %! %! x = 0:0.1:10; %! p1 = ncx2cdf (x, 4, 2); %! p2 = chi2cdf (x, 4); %! plot (x, p1, "-", x, p2, "-") %! grid on %! xlim ([0, 10]) %! legend ({"Noncentral χ^2(4,2)", "χ^2(4)"}, "location", "northwest") %! title ("Noncentral chi-squared vs chi-squared CDFs") %! xlabel ("values in x") %! ylabel ("probability") ## Test output %!test %! x = -2:0.1:2; %! p = ncx2cdf (x, 10, 1); %! assert (p([1:21]), zeros (1, 21), 3e-84); %! assert (p(22), 1.521400636466575e-09, 1e-14); %! assert (p(30), 6.665480510026046e-05, 1e-14); %! assert (p(41), 0.002406447308399836, 1e-14); %!test %! p = ncx2cdf (12, 10, 3); %! assert (p, 0.4845555602398649, 1e-14); %!test %! p = ncx2cdf (2, 3, 2); %! assert (p, 0.2207330870741212, 1e-14); %!test %! p = ncx2cdf (2, 3, 2, "upper"); %! assert (p, 0.7792669129258789, 1e-14); %!test %! p = ncx2cdf ([3, 6], 3, 2, "upper"); %! assert (p, [0.6423318186400054, 0.3152299878943012], 1e-14); ## Test input validation %!error ncx2cdf () %!error ncx2cdf (1) %!error ncx2cdf (1, 2) %!error ncx2cdf (1, 2, 3, "tail") %!error ncx2cdf (1, 2, 3, 4) %!error ... %! ncx2cdf (ones (3), ones (2), ones (2)) %!error ... %! ncx2cdf (ones (2), ones (3), ones (2)) %!error ... %! ncx2cdf (ones (2), ones (2), ones (3)) %!error ncx2cdf (i, 2, 2) %!error ncx2cdf (2, i, 2) %!error ncx2cdf (2, 2, i) statistics-release-1.6.3/inst/dist_fun/ncx2inv.m000066400000000000000000000161671456127120000216770ustar00rootroot00000000000000## Copyright (C) 2022-2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{x} =} ncx2inv (@var{p}, @var{df}, @var{lambda}) ## ## Inverse of the noncentral chi-squared cumulative distribution function (iCDF). ## ## For each element of @var{p}, compute the quantile (the inverse of the CDF) of ## the noncentral chi-squared distribution with @var{df} degrees of freedom and ## noncentrality parameter @var{mu}. The size of @var{x} is the common size of ## @var{p}, @var{df}, and @var{mu}. A scalar input functions as a constant ## matrix of the same size as the other inputs. ## ## @code{ncx2inv} uses Newton's method to converge to the solution. ## ## Further information about the noncentral chi-squared distribution can be ## found at @url{https://en.wikipedia.org/wiki/Noncentral_chi-squared_distribution} ## ## @seealso{ncx2cdf, ncx2pdf, ncx2rnd, ncx2stat, chi2inv} ## @end deftypefn function x = ncx2inv (p, df, lambda) ## Check for valid number of input arguments if (nargin < 3) error ("ncx2inv: function called with too few input arguments."); endif ## Check for common size of P, DF, and LAMBDA [err, p, df, lambda] = common_size (p, df, lambda); if (err > 0) error ("ncx2inv: P, DF, and LAMBDA must be of common size or scalars."); endif ## Check for P, DF, and LAMBDA being reals if (iscomplex (p) || iscomplex (df) || iscomplex (lambda)) error ("ncx2inv: P, DF, and LAMBDA must not be complex."); endif ## Check for class type if (isa (p, "single") || isa (df, "single") || isa (lambda, "single")) x = NaN (size (p), "single"); crit = sqrt (eps ("single")); else x = NaN (size (p), "double"); crit = sqrt (eps ("double")); endif ## For lambda == 0, call chi2inv d0 = lambda == 0; if (any (d0(:))) x(d0) = chi2inv (p(d0), df(d0)); ## If lambda == 0 for all entries, then return if (all (d0(:))) return; endif endif ## CDF with 0 d.d0. has a step at x=0. ## Check if CDF at x=0 exceeds the requested p. df0 = df==0 & lambda > 0; if (any (df0(:))) p0 = zeros (size (p)); p0(df0) = ncx2cdf (0, df(df0), lambda(df0)); df0 = df0 & p0 >= p; x(df0) = 0; endif valid = ! df0 & df > 0 & lambda > 0; ## Force x = 0 for p == 0 and x = Inf for p == 1 x(p == 0 & valid) = 0; x(p == 1 & valid) = Inf; ## Find valid samples within the range of 0 < p < 1 k = find (p > 0 & p < 1 & valid); pk = p(k); ## Initialize counter count_limit = 100; count = 0; ## Supply a starting guess for the iteration. mn = df(k) + lambda(k); variance = 2 * (df(k) + 2 * lambda(k)); temp = log (variance + mn .^ 2); mu = 2 * log (mn) - 0.5 * temp; sigma = -2 * log (mn) + temp; xk = exp (norminv (pk, mu, sigma)); F = ncx2cdf (xk, df(k), lambda(k)); h = ones(size(xk), class (xk)); ## Start iteration with a break out loop while (count < count_limit) count = count + 1; h = (F - pk) ./ ncx2pdf (xk, df(k), lambda(k)); xnew = max (xk / 50, min (5 * xk, xk - h)); newF = ncx2cdf (xnew, df(k), lambda(k)); while (true) worse = (abs (newF - pk) > abs (F - pk) * (1 + crit)) & ... (abs (xk - xnew) > crit * xk); if (! any (worse)) break; endif xnew(worse) = 0.5 * (xnew(worse) + xk(worse)); newF(worse) = ncx2cdf (xnew(worse), df(k(worse)), lambda(k(worse))); endwhile h = xk - xnew; x(k) = xnew; mask = (abs (h) > crit * abs (xk)); if (! any (mask)) break; endif k = k(mask); xk = xnew(mask); F = newF(mask); pk = pk(mask); endwhile if (count == count_limit) warning ("ncx2inv: did not converge."); endif endfunction %!demo %! ## Plot various iCDFs from the noncentral chi-squared distribution %! p = 0.001:0.001:0.999; %! x1 = ncx2inv (p, 2, 1); %! x2 = ncx2inv (p, 2, 2); %! x3 = ncx2inv (p, 2, 3); %! x4 = ncx2inv (p, 4, 1); %! x5 = ncx2inv (p, 4, 2); %! x6 = ncx2inv (p, 4, 3); %! plot (p, x1, "-r", p, x2, "-g", p, x3, "-k", ... %! p, x4, "-m", p, x5, "-c", p, x6, "-y") %! grid on %! ylim ([0, 10]) %! legend ({"df = 2, λ = 1", "df = 2, λ = 2", ... %! "df = 2, λ = 3", "df = 4, λ = 1", ... %! "df = 4, λ = 2", "df = 4, λ = 3"}, "location", "northwest") %! title ("Noncentral chi-squared iCDF") %! xlabel ("probability") %! ylabel ("values in x") %!demo %! ## Compare the noncentral chi-squared CDF with LAMBDA = 2 to the %! ## chi-squared CDF with the same number of degrees of freedom (4). %! %! p = 0.001:0.001:0.999; %! x1 = ncx2inv (p, 4, 2); %! x2 = chi2inv (p, 4); %! plot (p, x1, "-", p, x2, "-"); %! grid on %! ylim ([0, 10]) %! legend ({"Noncentral χ^2(4,2)", "χ^2(4)"}, "location", "northwest") %! title ("Noncentral chi-squared vs chi-squared quantile functions") %! xlabel ("probability") %! ylabel ("values in x") ## Test output %!test %! x = [0,0.3443,0.7226,1.1440,1.6220,2.1770,2.8436,3.6854,4.8447,6.7701,Inf]; %! assert (ncx2inv ([0:0.1:1], 2, 1), x, 1e-4); %!test %! x = [0,0.8295,1.6001,2.3708,3.1785,4.0598,5.0644,6.2765,7.8763,10.4199,Inf]; %! assert (ncx2inv ([0:0.1:1], 2, 3), x, 1e-4); %!test %! x = [0,0.5417,1.3483,2.1796,3.0516,4.0003,5.0777,6.3726,8.0748,10.7686,Inf]; %! assert (ncx2inv ([0:0.1:1], 1, 4), x, 1e-4); %!test %! x = [0.1808, 0.6456, 1.1842, 1.7650, 2.3760, 3.0105]; %! assert (ncx2inv (0.05, [1, 2, 3, 4, 5, 6], 4), x, 1e-4); %!test %! x = [0.4887, 0.6699, 0.9012, 1.1842, 1.5164, 1.8927]; %! assert (ncx2inv (0.05, 3, [1, 2, 3, 4, 5, 6]), x, 1e-4); %!test %! x = [1.3941, 1.6824, 2.0103, 2.3760, NaN, 3.2087]; %! assert (ncx2inv (0.05, 5, [1, 2, 3, 4, -1, 6]), x, 1e-4); %!test %! assert (ncx2inv (0.996, 5, 8), 35.51298862765576, 2e-13); ## Test input validation %!error ncx2inv () %!error ncx2inv (1) %!error ncx2inv (1, 2) %!error ... %! ncx2inv (ones (3), ones (2), ones (2)) %!error ... %! ncx2inv (ones (2), ones (3), ones (2)) %!error ... %! ncx2inv (ones (2), ones (2), ones (3)) %!error ncx2inv (i, 2, 2) %!error ncx2inv (2, i, 2) %!error ncx2inv (2, 2, i) statistics-release-1.6.3/inst/dist_fun/ncx2pdf.m000066400000000000000000000303201456127120000216370ustar00rootroot00000000000000## Copyright (C) 2022-2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{y} =} ncx2pdf (@var{x}, @var{df}, @var{lambda}) ## ## Noncentral chi-squared probability distribution function (PDF). ## ## For each element of @var{x}, compute the probability density function (PDF) ## of the noncentral chi-squared distribution with @var{df} degrees of freedom ## and noncentrality parameter @var{lambda}. The size of @var{y} is the common ## size of @var{x}, @var{df}, and @var{lambda}. A scalar input functions as a ## constant matrix of the same size as the other inputs. ## ## Further information about the noncentral chi-squared distribution can be ## found at @url{https://en.wikipedia.org/wiki/Noncentral_chi-squared_distribution} ## ## @seealso{ncx2cdf, ncx2inv, ncx2rnd, ncx2stat, chi2pdf} ## @end deftypefn function y = ncx2pdf (x, df, lambda) ## Check for valid number of input arguments if (nargin < 3) error ("ncx2pdf: function called with too few input arguments."); endif ## Check for common size of X, DF, and LAMBDA [err, x, df, lambda] = common_size (x, df, lambda); if (err > 0) error ("ncx2pdf: X, DF, and LAMBDA must be of common size or scalars."); endif ## Check for X, DF, and LAMBDA being reals if (iscomplex (x) || iscomplex (df) || iscomplex (lambda)) error ("ncx2pdf: X, DF, and LAMBDA must not be complex."); endif ## Check for class type if (isa (x, "single") || isa (df, "single") || isa (lambda, "single")) y = zeros (size (x), "single"); else y = zeros (size (x)); endif ## Find NaNs in input arguments (if any) and propagate them to p is_nan = isnan (x) | isnan (df) | isnan (lambda); y(is_nan) = NaN; ## Make input arguments column vectors and half DF x = x(:); df = df(:); df = df / 2; lambda = lambda(:); ## Handle special cases k1 = x == 0 & df == 1; y(k1) = 0.5 * exp (-0.5 * lambda(k1)); k2 = x == 0 & df < 1; y(k2) = Inf; y(lambda < 0) = NaN; y(df < 0) = NaN; k3 = lambda == 0 & df > 0; y(k3) = gampdf (x(k3), df(k3), 2); ## Handle normal cases td = find(x>0 & x0 & df>=0); ## Return if finished all normal cases if (isempty (td)) return; endif ## Reset input variables to remaining cases x = x(td); lambda = lambda(td); df = df(td) - 1; x_sqrt = sqrt (x); d_sqrt = sqrt (lambda); ## Upper Limit on density small_DF = df <= -0.5; large_DF = ! small_DF; ul = zeros (size (x)); ul(small_DF) = -0.5 * (lambda(small_DF) + x(small_DF)) + ... 0.5 * x_sqrt(small_DF) .* d_sqrt (small_DF) ./ ... (df(small_DF) + 1) + df(small_DF) .* ... (log (x(small_DF)) - log (2)) - log (2) - ... gammaln (df(small_DF) + 1); ul(large_DF) = -0.5 * (d_sqrt(large_DF) - x_sqrt(large_DF)) .^ 2 + ... df(large_DF) .* (log (x(large_DF)) - log (2)) - log (2) - ... gammaln (df(large_DF) + 1) + (df(large_DF) + 0.5) .* ... log ((df(large_DF) + 0.5) ./ (x_sqrt(large_DF) .* ... d_sqrt(large_DF) + df(large_DF) + 0.5)); ULunderflow = ul < log (realmin); y(td(ULunderflow)) = 0; td(ULunderflow) = []; ## Return if finished all normal cases if (isempty (td)) return; endif x(ULunderflow) = []; lambda(ULunderflow) = []; df(ULunderflow) = []; x_sqrt(ULunderflow) = []; d_sqrt(ULunderflow) = []; ## Try the scaled Bess function scaleB = besseli (df, d_sqrt .* x_sqrt, 1); use_SB = scaleB > 0 & scaleB < Inf; y(td(use_SB)) = exp (-log (2) -0.5 * (x_sqrt(use_SB) - ... d_sqrt(use_SB)) .^ 2 + df(use_SB) .* ... log (x_sqrt(use_SB) ./ d_sqrt(use_SB))) .* scaleB(use_SB); td(use_SB) = []; ## Return if finished all normal cases if (isempty (td)) return; endif x(use_SB) = []; lambda(use_SB) = []; df(use_SB) = []; x_sqrt(use_SB) = []; d_sqrt(use_SB) = []; ## Try the Bess function Bess = besseli (df, d_sqrt .* x_sqrt); useB = Bess > 0 & Bess < Inf; y(td(useB)) = exp (-log (2) - 0.5 * (x(useB) + lambda(useB)) + ... df(useB) .* log (x_sqrt(useB) ./ d_sqrt(useB))) .* Bess(useB); td(useB) = []; ## Return if finished all normal cases if isempty(td) return; endif x(useB) = []; lambda(useB) = []; df(useB) = []; ## If neither Bess function works, use recursion. When non-centrality ## parameter is very large, the initial values of the Poisson numbers used ## in the approximation are very small, smaller than epsilon. This would ## cause premature convergence. To avoid that, we start from the peak of the ## Poisson numbers, and go in both directions. lnsr2pi = 0.9189385332046727; % log(sqrt(2*pi)) dx = lambda .* x / 4; K = max (0, floor (0.5 * (sqrt (df .^ 2 + 4 * dx) - df))); lntK = zeros(size(K)); K0 = K == 0; lntK(K0) = -lnsr2pi -0.5 * (lambda(K0) + log(df(K0))) - ... StirlingError (df(K0)) - BinoPoisson (df(K0), x(K0) / 2); K0 = ! K0; lntK(K0) = -2 * lnsr2pi - 0.5 * (log (K(K0)) + log (df(K0) + K(K0))) - ... StirlingError (K(K0)) - StirlingError (df(K0) + K(K0)) - ... BinoPoisson (K(K0), lambda(K0) / 2) - ... BinoPoisson (df(K0) + K(K0), x(K0) / 2); sumK = ones (size (K)); keep = K>0; term = ones (size (K)); k = K; while (any (keep)) term(keep) = term(keep) .* (df(keep) + k(keep)) .* k(keep) ./ dx(keep); sumK(keep) = sumK(keep) + term(keep); keep = keep & k > 0 & term > eps (sumK); k = k - 1; endwhile keep = true (size (K)); term = ones (size (K)); k = K + 1; while (any (keep)) term(keep) = term(keep) ./ (df(keep) + k(keep)) ./ k(keep) .* dx(keep); sumK(keep) = sumK(keep) + term(keep); keep = keep & term > eps (sumK); k = k + 1; end y(td) = 0.5 * exp (lntK + log (sumK)); endfunction ## Error of Stirling-De Moivre approximation to n factorial. function lambda = StirlingError (n) is_class = class (n); lambda = zeros (size (n), is_class); nn = n .* n; ## Define S0=1/12 S1=1/360 S2=1/1260 S3=1/1680 S4=1/1188 S0 = 8.333333333333333e-02; S1 = 2.777777777777778e-03; S2 = 7.936507936507937e-04; S3 = 5.952380952380952e-04; S4 = 8.417508417508418e-04; ## Define lambda(n) for n<0:0.5:15 sfe=[ 0; 1.534264097200273e-01;... 8.106146679532726e-02; 5.481412105191765e-02;... 4.134069595540929e-02; 3.316287351993629e-02;... 2.767792568499834e-02; 2.374616365629750e-02;... 2.079067210376509e-02; 1.848845053267319e-02;... 1.664469118982119e-02; 1.513497322191738e-02;... 1.387612882307075e-02; 1.281046524292023e-02;... 1.189670994589177e-02; 1.110455975820868e-02;... 1.041126526197210e-02; 9.799416126158803e-03;... 9.255462182712733e-03; 8.768700134139385e-03;... 8.330563433362871e-03; 7.934114564314021e-03;... 7.573675487951841e-03; 7.244554301320383e-03;... 6.942840107209530e-03; 6.665247032707682e-03;... 6.408994188004207e-03; 6.171712263039458e-03;... 5.951370112758848e-03; 5.746216513010116e-03;... 5.554733551962801e-03]; k = find (n <= 15); if (any (k)) n1 = n(k); n2 = 2 * n1; if (all (n2 == round (n2))) lambda(k) = sfe(n2+1); else lnsr2pi = 0.9189385332046728; lambda(k) = gammaln(n1+1)-(n1+0.5).*log(n1)+n1-lnsr2pi; endif endif k = find (n > 15 & n <= 35); if (any (k)) lambda(k) = (S0 - (S1 - (S2 - (S3 - S4 ./ nn(k)) ./ nn(k)) ./ ... nn(k)) ./ nn(k)) ./ n(k); endif k = find (n > 35 & n <= 80); if (any (k)) lambda(k) = (S0 - (S1 - (S2 - S3 ./ nn(k)) ./ nn(k)) ./ nn(k)) ./ n(k); endif k = find(n > 80 & n <= 500); if (any (k)) lambda(k) = (S0 - (S1 - S2 ./ nn(k)) ./ nn(k)) ./ n(k); endif k = find(n > 500); if (any (k)) lambda(k) = (S0 - S1 ./ nn(k)) ./ n(k); endif endfunction ## Deviance term for binomial and Poisson probability calculation. function BP = BinoPoisson (x, np) if (isa (x,'single') || isa (np,'single')) BP = zeros (size (x), "single"); else BP = zeros (size (x)); endif k = abs (x - np) < 0.1 * (x + np); if any(k(:)) s = (x(k) - np(k)) .* (x(k) - np(k)) ./ (x(k) + np(k)); v = (x(k) - np(k)) ./ (x(k) + np(k)); ej = 2 .* x(k) .* v; is_class = class (s); s1 = zeros (size (s), is_class); ok = true (size (s)); j = 0; while any(ok(:)) ej(ok) = ej(ok) .* v(ok) .* v(ok); j = j + 1; s1(ok) = s(ok) + ej(ok) ./ (2 * j + 1); ok = ok & s1 != s; s(ok) = s1(ok); endwhile BP(k) = s; endif k = ! k; if (any (k(:))) BP(k) = x(k) .* log (x(k) ./ np(k)) + np(k) - x(k); endif endfunction %!demo %! ## Plot various PDFs from the noncentral chi-squared distribution %! x = 0:0.1:10; %! y1 = ncx2pdf (x, 2, 1); %! y2 = ncx2pdf (x, 2, 2); %! y3 = ncx2pdf (x, 2, 3); %! y4 = ncx2pdf (x, 4, 1); %! y5 = ncx2pdf (x, 4, 2); %! y6 = ncx2pdf (x, 4, 3); %! plot (x, y1, "-r", x, y2, "-g", x, y3, "-k", ... %! x, y4, "-m", x, y5, "-c", x, y6, "-y") %! grid on %! xlim ([0, 10]) %! ylim ([0, 0.32]) %! legend ({"df = 2, λ = 1", "df = 2, λ = 2", ... %! "df = 2, λ = 3", "df = 4, λ = 1", ... %! "df = 4, λ = 2", "df = 4, λ = 3"}, "location", "northeast") %! title ("Noncentral chi-squared PDF") %! xlabel ("values in x") %! ylabel ("density") %!demo %! ## Compare the noncentral chi-squared PDF with LAMBDA = 2 to the %! ## chi-squared PDF with the same number of degrees of freedom (4). %! %! x = 0:0.1:10; %! y1 = ncx2pdf (x, 4, 2); %! y2 = chi2pdf (x, 4); %! plot (x, y1, "-", x, y2, "-"); %! grid on %! xlim ([0, 10]) %! ylim ([0, 0.32]) %! legend ({"Noncentral T(10,1)", "T(10)"}, "location", "northwest") %! title ("Noncentral chi-squared vs chi-squared PDFs") %! xlabel ("values in x") %! ylabel ("density") ## Test output %!shared x1, df, d1 %! x1 = [-Inf, 2, NaN, 4, Inf]; %! df = [2, 0, -1, 1, 4]; %! d1 = [1, NaN, 3, -1, 2]; %!assert (ncx2pdf (x1, df, d1), [0, NaN, NaN, NaN, 0]); %!assert (ncx2pdf (x1, df, 1), [0, 0.07093996461786045, NaN, ... %! 0.06160064323277038, 0], 1e-14); %!assert (ncx2pdf (x1, df, 3), [0, 0.1208364909271113, NaN, ... %! 0.09631299762429098, 0], 1e-14); %!assert (ncx2pdf (x1, df, 2), [0, 0.1076346446244688, NaN, ... %! 0.08430464047296625, 0], 1e-14); %!assert (ncx2pdf (x1, 2, d1), [0, NaN, NaN, NaN, 0]); %!assert (ncx2pdf (2, df, d1), [0.1747201674611283, NaN, NaN, ... %! NaN, 0.1076346446244688], 1e-14); %!assert (ncx2pdf (4, df, d1), [0.09355987820265799, NaN, NaN, ... %! NaN, 0.1192317192431485], 1e-14); ## Test input validation %!error ncx2pdf () %!error ncx2pdf (1) %!error ncx2pdf (1, 2) %!error ... %! ncx2pdf (ones (3), ones (2), ones (2)) %!error ... %! ncx2pdf (ones (2), ones (3), ones (2)) %!error ... %! ncx2pdf (ones (2), ones (2), ones (3)) %!error ncx2pdf (i, 2, 2) %!error ncx2pdf (2, i, 2) %!error ncx2pdf (2, 2, i) statistics-release-1.6.3/inst/dist_fun/ncx2rnd.m000066400000000000000000000153071456127120000216610ustar00rootroot00000000000000## Copyright (C) 2022-2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{r} =} ncx2rnd (@var{df}, @var{lambda}) ## @deftypefnx {statistics} {@var{r} =} ncx2rnd (@var{df}, @var{lambda}, @var{rows}, @var{cols}, @dots{}) ## @deftypefnx {statistics} {@var{r} =} ncx2rnd (@var{df}, @var{lambda}, [@var{sz}]) ## ## Random arrays from the noncentral chi-squared distribution. ## ## @code{@var{r} = ncx2rnd (@var{df}, @var{lambda})} returns an array of random ## numbers chosen from the noncentral chi-squared distribution with @var{df} ## degrees of freedom and noncentrality parameter @var{lambda}. The size of ## @var{r} is the common size of @var{df} and @var{lambda}. A scalar input ## functions as a constant matrix of the same size as the other input. ## ## When called with a single size argument, @code{ncx2rnd} returns a square ## matrix with the dimension specified. When called with more than one scalar ## argument, the first two arguments are taken as the number of rows and columns ## and any further arguments specify additional matrix dimensions. The size may ## also be specified with a row vector of dimensions, @var{sz}. ## ## Further information about the noncentral chi-squared distribution can be ## found at @url{https://en.wikipedia.org/wiki/Noncentral_chi-squared_distribution} ## ## @seealso{ncx2cdf, ncx2inv, ncx2pdf, ncx2stat} ## @end deftypefn function r = ncx2rnd (df, lambda, varargin) ## Check for valid number of input arguments if (nargin < 2) error ("ncx2rnd: function called with too few input arguments."); endif ## Check for common size of DF and LAMBDA if (! isscalar (df) || ! isscalar (lambda)) [retval, df, lambda] = common_size (df, lambda); if (retval > 0) error ("ncx2rnd: DF and LAMBDA must be of common size or scalars."); endif endif ## Check for DF and LAMBDA being reals if (iscomplex (df) || iscomplex (lambda)) error ("ncx2rnd: DF and LAMBDA must not be complex."); endif ## Parse and check SIZE arguments if (nargin == 2) sz = size (df); elseif (nargin == 3) if (isscalar (varargin{1}) && varargin{1} >= 0 ... && varargin{1} == fix (varargin{1})) sz = [varargin{1}, varargin{1}]; elseif (isrow (varargin{1}) && all (varargin{1} >= 0) ... && all (varargin{1} == fix (varargin{1}))) sz = varargin{1}; elseif error (strcat (["ncx2rnd: SZ must be a scalar or a row vector"], ... [" of non-negative integers."])); endif elseif (nargin > 3) posint = cellfun (@(x) (! isscalar (x) || x < 0 || x != fix (x)), varargin); if (any (posint)) error ("ncx2rnd: dimensions must be non-negative integers."); endif sz = [varargin{:}]; endif ## Check that parameters match requested dimensions in size if (! isscalar (df) && ! isequal (size (df), sz)) error ("ncx2rnd: DF and LAMBDA must be scalars or of size SZ."); endif ## Check for class type if (isa (df, "single") || isa (lambda, "single")); cls = "single"; else cls = "double"; endif ## Return NaNs for out of range values of DF and LAMBDA df(df <= 0) = NaN; lambda(lambda <= 0) = NaN; ## Force DF and LAMBDA into the same size as SZ (if necessary) if (isscalar (df)) df = repmat (df, sz); endif if (isscalar (lambda)) lambda = repmat (lambda, sz); endif ## Generate random sample from noncentral chi-squared distribution r = randp (lambda ./ 2); r(r > 0) = 2 * randg (r(r > 0)); r(df > 0) += 2 * randg (df(df > 0) / 2); ## Cast to appropriate class r = cast (r, cls); endfunction ## Test output %!assert (size (ncx2rnd (1, 1)), [1 1]) %!assert (size (ncx2rnd (1, ones (2,1))), [2, 1]) %!assert (size (ncx2rnd (1, ones (2,2))), [2, 2]) %!assert (size (ncx2rnd (ones (2,1), 1)), [2, 1]) %!assert (size (ncx2rnd (ones (2,2), 1)), [2, 2]) %!assert (size (ncx2rnd (1, 1, 3)), [3, 3]) %!assert (size (ncx2rnd (1, 1, [4, 1])), [4, 1]) %!assert (size (ncx2rnd (1, 1, 4, 1)), [4, 1]) %!assert (size (ncx2rnd (1, 1, 4, 1, 5)), [4, 1, 5]) %!assert (size (ncx2rnd (1, 1, 0, 1)), [0, 1]) %!assert (size (ncx2rnd (1, 1, 1, 0)), [1, 0]) %!assert (size (ncx2rnd (1, 1, 1, 2, 0, 5)), [1, 2, 0, 5]) ## Test class of input preserved %!assert (class (ncx2rnd (1, 1)), "double") %!assert (class (ncx2rnd (1, single (1))), "single") %!assert (class (ncx2rnd (1, single ([1, 1]))), "single") %!assert (class (ncx2rnd (single (1), 1)), "single") %!assert (class (ncx2rnd (single ([1, 1]), 1)), "single") ## Test input validation %!error ncx2rnd () %!error ncx2rnd (1) %!error ... %! ncx2rnd (ones (3), ones (2)) %!error ... %! ncx2rnd (ones (2), ones (3)) %!error ncx2rnd (i, 2) %!error ncx2rnd (1, i) %!error ... %! ncx2rnd (1, 2, -1) %!error ... %! ncx2rnd (1, 2, 1.2) %!error ... %! ncx2rnd (1, 2, ones (2)) %!error ... %! ncx2rnd (1, 2, [2 -1 2]) %!error ... %! ncx2rnd (1, 2, [2 0 2.5]) %!error ... %! ncx2rnd (1, 2, 2, -1, 5) %!error ... %! ncx2rnd (1, 2, 2, 1.5, 5) %!error ... %! ncx2rnd (2, ones (2), 3) %!error ... %! ncx2rnd (2, ones (2), [3, 2]) %!error ... %! ncx2rnd (2, ones (2), 3, 2) statistics-release-1.6.3/inst/dist_fun/normcdf.m000066400000000000000000000225631456127120000217350ustar00rootroot00000000000000## Copyright (C) 1995-2016 Kurt Hornik ## Copyright (C) 2012 Rik Wehbring ## Copyright (C) 2022-2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{p} =} normcdf (@var{x}) ## @deftypefnx {statistics} {@var{p} =} normcdf (@var{x}, @var{mu}) ## @deftypefnx {statistics} {@var{p} =} normcdf (@var{x}, @var{mu}, @var{sigma}) ## @deftypefnx {statistics} {@var{p} =} normcdf (@dots{}, @qcode{"upper"}) ## @deftypefnx {statistics} {[@var{p}, @var{plo}, @var{pup}] =} normcdf (@var{x}, @var{mu}, @var{sigma}, @var{pcov}) ## @deftypefnx {statistics} {[@var{p}, @var{plo}, @var{pup}] =} normcdf (@var{x}, @var{mu}, @var{sigma}, @var{pcov}, @var{alpha}) ## @deftypefnx {statistics} {[@var{p}, @var{plo}, @var{pup}] =} normcdf (@dots{}, @qcode{"upper"}) ## ## Normal cumulative distribution function (CDF). ## ## For each element of @var{x}, compute the cumulative distribution function ## (CDF) of the normal distribution with mean @var{mu} and standard deviation ## @var{sigma}. The size of @var{p} is the common size of @var{x}, @var{mu} and ## @var{sigma}. A scalar input functions as a constant matrix of the same size ## as the other inputs. ## ## Default values are @var{mu} = 0, @var{sigma} = 1. ## ## When called with three output arguments, i.e. @qcode{[@var{p}, @var{plo}, ## @var{pup}]}, @code{normcdf} computes the confidence bounds for @var{p} when ## the input parameters @var{mu} and @var{sigma} are estimates. In such case, ## @var{pcov}, a @math{2x2} matrix containing the covariance matrix of the ## estimated parameters, is necessary. Optionally, @var{alpha}, which has a ## default value of 0.05, specifies the @qcode{100 * (1 - @var{alpha})} percent ## confidence bounds. @var{plo} and @var{pup} are arrays of the same size as ## @var{p} containing the lower and upper confidence bounds. ## ## @code{[@dots{}] = normcdf (@dots{}, "upper")} computes the upper tail ## probability of the normal distribution with parameters @var{mu} and ## @var{sigma}, at the values in @var{x}. This can be used to compute a ## right-tailed p-value. To compute a two-tailed p-value, use ## @code{2 * normcdf (-abs (@var{x}), @var{mu}, @var{sigma})}. ## ## Further information about the normal distribution can be found at ## @url{https://en.wikipedia.org/wiki/Normal_distribution} ## ## @seealso{norminv, normpdf, normrnd, normfit, normlike, normstat} ## @end deftypefn function [varargout] = normcdf (x, varargin) ## Check for valid number of input arguments if (nargin < 1 || nargin > 6) error ("normcdf: invalid number of input arguments."); endif ## Check for "upper" flag if (nargin > 1 && strcmpi (varargin{end}, "upper")) uflag = true; varargin(end) = []; elseif (nargin > 1 && ischar (varargin{end}) && ... ! strcmpi (varargin{end}, "upper")) error ("normcdf: invalid argument for upper tail."); else uflag = false; endif ## Get extra arguments (if they exist) or add defaults if (numel (varargin) > 0) mu = varargin{1}; else mu = 0; endif if (numel (varargin) > 1) sigma = varargin{2}; else sigma = 1; endif if (numel (varargin) > 2) pcov = varargin{3}; ## Check for valid covariance matrix 2x2 if (! isequal (size (pcov), [2, 2])) error ("normcdf: invalid size of covariance matrix."); endif else ## Check that cov matrix is provided if 3 output arguments are requested if (nargout > 1) error ("normcdf: covariance matrix is required for confidence bounds."); endif pcov = []; endif if (numel (varargin) > 3) alpha = varargin{4}; ## Check for valid alpha value if (! isnumeric (alpha) || numel (alpha) !=1 || alpha <= 0 || alpha >= 1) error ("normcdf: invalid value for alpha."); end else alpha = 0.05; endif ## Check for common size of X, MU, and SIGMA if (! isscalar (mu) || ! isscalar (sigma)) [err, x, mu, sigma] = common_size (x, mu, sigma); if (err > 0) error ("normcdf: X, MU, and SIGMA must be of common size or scalars."); endif endif ## Check for X, MU, and SIGMA being reals if (iscomplex (x) || iscomplex (mu) || iscomplex (sigma)) error ("normcdf: X, MU, and SIGMA must not be complex."); endif ## Compute normal CDF z = (x - mu) ./ sigma; if (uflag) z = -z; endif ## Check for class type if (isa (x, "single") || isa (mu, "single") || isa (sigma, "single")); is_class = "single"; else is_class = "double"; endif ## Prepare output p = NaN (size (z), is_class); if (nargout > 1) plo = NaN (size (z), is_class); pup = NaN (size (z), is_class); endif ## Check SIGMA if (isscalar (sigma)) if (sigma > 0) sigma_p = true (size (z)); sigma_z = false (size (z)); elseif (sigma == 0) sigma_z = true (size (z)); sigma_p = false (size (z)); else if (nargout <= 1) varargout{1} = p; elseif (nargout == 3) varargout{1} = p; varargout{2} = plo; varargout{3} = pup; endif return; endif else sigma_p = sigma > 0; sigma_z = sigma == 0; endif ## Set edge cases when SIGMA = 0 if (uflag) p(sigma_z & x < mu) = 1; p(sigma_z & x >= mu) = 0; if (nargout > 1) plo(sigma_z & x < mu) = 1; plo(sigma_z & x >= mu) = 0; pup(sigma_z & x < mu) = 1; pup(sigma_z & x >= mu) = 0; endif else p(sigma_z & x < mu) = 0; p(sigma_z & x >= mu) = 1; if (nargout >= 2) plo(sigma_z & x < mu) = 0; plo(sigma_z & x >= mu) = 1; pup(sigma_z & x < mu) = 0; pup(sigma_z & x >= mu) = 1; endif endif ## Compute cases when SIGMA > 0 p(sigma_p) = 0.5 * erfc (-z(sigma_p) ./ sqrt (2)); varargout{1} = p; ## Compute confidence bounds (if requested) if (nargout >= 2) zvar = (pcov(1,1) + 2 * pcov(1,2) * z(sigma_p) + ... pcov(2,2) * z(sigma_p) .^ 2) ./ (sigma .^ 2); if (any (zvar < 0)) error ("normcdf: bad covariance matrix."); endif normz = -norminv (alpha / 2); halfwidth = normz * sqrt (zvar); zlo = z(sigma_p) - halfwidth; zup = z(sigma_p) + halfwidth; plo(sigma_p) = 0.5 * erfc (-zlo ./ sqrt (2)); pup(sigma_p) = 0.5 * erfc (-zup ./ sqrt (2)); varargout{2} = plo; varargout{3} = pup; endif endfunction %!demo %! ## Plot various CDFs from the normal distribution %! x = -5:0.01:5; %! p1 = normcdf (x, 0, 0.5); %! p2 = normcdf (x, 0, 1); %! p3 = normcdf (x, 0, 2); %! p4 = normcdf (x, -2, 0.8); %! plot (x, p1, "-b", x, p2, "-g", x, p3, "-r", x, p4, "-c") %! grid on %! xlim ([-5, 5]) %! legend ({"μ = 0, σ = 0.5", "μ = 0, σ = 1", ... %! "μ = 0, σ = 2", "μ = -2, σ = 0.8"}, "location", "southeast") %! title ("Normal CDF") %! xlabel ("values in x") %! ylabel ("probability") ## Test output %!shared x, y %! x = [-Inf 1 2 Inf]; %! y = [0, 0.5, 1/2*(1+erf(1/sqrt(2))), 1]; %!assert (normcdf (x, ones (1,4), ones (1,4)), y) %!assert (normcdf (x, 1, ones (1,4)), y) %!assert (normcdf (x, ones (1,4), 1), y) %!assert (normcdf (x, [0, -Inf, NaN, Inf], 1), [0, 1, NaN, NaN]) %!assert (normcdf (x, 1, [Inf, NaN, -1, 0]), [NaN, NaN, NaN, 1]) %!assert (normcdf ([x(1:2), NaN, x(4)], 1, 1), [y(1:2), NaN, y(4)]) %!assert (normcdf (x, "upper"), [1, 0.1587, 0.0228, 0], 1e-4) ## Test class of input preserved %!assert (normcdf ([x, NaN], 1, 1), [y, NaN]) %!assert (normcdf (single ([x, NaN]), 1, 1), single ([y, NaN]), eps ("single")) %!assert (normcdf ([x, NaN], single (1), 1), single ([y, NaN]), eps ("single")) %!assert (normcdf ([x, NaN], 1, single (1)), single ([y, NaN]), eps ("single")) ## Test input validation %!error normcdf () %!error normcdf (1,2,3,4,5,6,7) %!error normcdf (1, 2, 3, 4, "uper") %!error ... %! normcdf (ones (3), ones (2), ones (2)) %!error normcdf (2, 3, 4, [1, 2]) %!error ... %! [p, plo, pup] = normcdf (1, 2, 3) %!error [p, plo, pup] = ... %! normcdf (1, 2, 3, [1, 0; 0, 1], 0) %!error [p, plo, pup] = ... %! normcdf (1, 2, 3, [1, 0; 0, 1], 1.22) %!error [p, plo, pup] = ... %! normcdf (1, 2, 3, [1, 0; 0, 1], "alpha", "upper") %!error normcdf (i, 2, 2) %!error normcdf (2, i, 2) %!error normcdf (2, 2, i) %!error ... %! [p, plo, pup] =normcdf (1, 2, 3, [1, 0; 0, -inf], 0.04) statistics-release-1.6.3/inst/dist_fun/norminv.m000066400000000000000000000123151456127120000217670ustar00rootroot00000000000000## Copyright (C) 2012 Rik Wehbring ## Copyright (C) 1995-2016 Kurt Hornik ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{x} =} norminv (@var{p}) ## @deftypefnx {statistics} {@var{x} =} norminv (@var{p}, @var{mu}) ## @deftypefnx {statistics} {@var{x} =} norminv (@var{p}, @var{mu}, @var{sigma}) ## ## Inverse of the normal cumulative distribution function (iCDF). ## ## For each element of @var{p}, compute the quantile (the inverse of the CDF) of ## the normal distribution with mean @var{mu} and standard deviation @var{sigma}. ## The size of @var{p} is the common size of @var{p}, @var{mu} and @var{sigma}. ## A scalar input functions as a constant matrix of the same size as the other ## inputs. ## ## Default values are @var{mu} = 0, @var{sigma} = 1. ## ## The default values correspond to the standard normal distribution and ## computing its quantile function is also possible with the @code{probit} ## function, which is faster but it does not perform any input validation. ## ## Further information about the normal distribution can be found at ## @url{https://en.wikipedia.org/wiki/Normal_distribution} ## ## @seealso{norminv, normpdf, normrnd, normfit, normlike, normstat, probit} ## @end deftypefn function x = norminv (p, mu, sigma) ## Check for valid number of input arguments if (nargin < 1) error ("norminv: function called with too few input arguments."); endif ## Add defaults (if missing input arguments) if (nargin < 2) mu = 0; endif if (nargin < 3) sigma = 1; endif ## Check for common size of P, MU, and SIGMA if (! isscalar (p) || ! isscalar (mu) || ! isscalar (sigma)) [retval, p, mu, sigma] = common_size (p, mu, sigma); if (retval > 0) error ("norminv: P, MU, and SIGMA must be of common size or scalars."); endif endif ## Check for P, MU, and SIGMA being reals if (iscomplex (p) || iscomplex (mu) || iscomplex (sigma)) error ("norminv: P, MU, and SIGMA must not be complex."); endif ## Check for class type if (isa (p, "single") || isa (mu, "single") || isa (sigma, "single")) x = NaN (size (p), "single"); else x = NaN (size (p)); endif ## Compute normal iCDF if (isscalar (mu) && isscalar (sigma)) if (isfinite (mu) && (sigma > 0) && (sigma < Inf)) x = mu + sigma * (-sqrt (2) * erfcinv (2 * p)); endif else k = isfinite (mu) & (sigma > 0) & (sigma < Inf); x(k) = mu(k) + sigma(k) .* (-sqrt (2) * erfcinv (2 * p(k))); endif endfunction %!demo %! ## Plot various iCDFs from the normal distribution %! p = 0.001:0.001:0.999; %! x1 = norminv (p, 0, 0.5); %! x2 = norminv (p, 0, 1); %! x3 = norminv (p, 0, 2); %! x4 = norminv (p, -2, 0.8); %! plot (p, x1, "-b", p, x2, "-g", p, x3, "-r", p, x4, "-c") %! grid on %! ylim ([-5, 5]) %! legend ({"μ = 0, σ = 0.5", "μ = 0, σ = 1", ... %! "μ = 0, σ = 2", "μ = -2, σ = 0.8"}, "location", "northwest") %! title ("Normal iCDF") %! xlabel ("probability") %! ylabel ("values in x") ## Test output %!shared p %! p = [-1 0 0.5 1 2]; %!assert (norminv (p, ones (1,5), ones (1,5)), [NaN -Inf 1 Inf NaN]) %!assert (norminv (p, 1, ones (1,5)), [NaN -Inf 1 Inf NaN]) %!assert (norminv (p, ones (1,5), 1), [NaN -Inf 1 Inf NaN]) %!assert (norminv (p, [1 -Inf NaN Inf 1], 1), [NaN NaN NaN NaN NaN]) %!assert (norminv (p, 1, [1 0 NaN Inf 1]), [NaN NaN NaN NaN NaN]) %!assert (norminv ([p(1:2) NaN p(4:5)], 1, 1), [NaN -Inf NaN Inf NaN]) %!assert (norminv (p), probit (p)) %!assert (norminv (0.31254), probit (0.31254)) ## Test class of input preserved %!assert (norminv ([p, NaN], 1, 1), [NaN -Inf 1 Inf NaN NaN]) %!assert (norminv (single ([p, NaN]), 1, 1), single ([NaN -Inf 1 Inf NaN NaN])) %!assert (norminv ([p, NaN], single (1), 1), single ([NaN -Inf 1 Inf NaN NaN])) %!assert (norminv ([p, NaN], 1, single (1)), single ([NaN -Inf 1 Inf NaN NaN])) ## Test input validation %!error norminv () %!error ... %! norminv (ones (3), ones (2), ones (2)) %!error ... %! norminv (ones (2), ones (3), ones (2)) %!error ... %! norminv (ones (2), ones (2), ones (3)) %!error norminv (i, 2, 2) %!error norminv (2, i, 2) %!error norminv (2, 2, i) statistics-release-1.6.3/inst/dist_fun/normpdf.m000066400000000000000000000120371456127120000217450ustar00rootroot00000000000000## Copyright (C) 2012 Rik Wehbring ## Copyright (C) 1995-2016 Kurt Hornik ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{y} =} normpdf (@var{x}) ## @deftypefnx {statistics} {@var{y} =} normpdf (@var{x}, @var{mu}) ## @deftypefnx {statistics} {@var{y} =} normpdf (@var{x}, @var{mu}, @var{sigma}) ## ## Normal probability density function (PDF). ## ## For each element of @var{x}, compute the probability density function (PDF) ## of the normal distribution with mean @var{mu} and standard deviation ## @var{sigma}. The size of @var{y} is the common size of @var{p}, @var{mu} and ## @var{sigma}. A scalar input functions as a constant matrix of the same size ## as the other inputs. ## ## Default values are @var{mu} = 0, @var{sigma} = 1. ## ## Further information about the normal distribution can be found at ## @url{https://en.wikipedia.org/wiki/Normal_distribution} ## ## @seealso{norminv, norminv, normrnd, normfit, normlike, normstat} ## @end deftypefn function y = normpdf (x, mu, sigma) ## Check for valid number of input arguments if (nargin < 1 || nargin > 3) error ("normpdf: function called with too few input arguments."); endif ## Add defaults (if missing input arguments) if (nargin < 2) mu = 0; endif if (nargin < 3) sigma = 1; endif ## Check for common size of X, MU, and SIGMA if (! isscalar (x) || ! isscalar (mu) || ! isscalar (sigma)) [retval, x, mu, sigma] = common_size (x, mu, sigma); if (retval > 0) error ("normpdf: X, MU, and SIGMA must be of common size or scalars."); endif endif ## Check for X, MU, and SIGMA being reals if (iscomplex (x) || iscomplex (mu) || iscomplex (sigma)) error ("normpdf: X, MU, and SIGMA must not be complex."); endif ## Check for class type if (isa (x, "single") || isa (mu, "single") || isa (sigma, "single")) y = zeros (size (x), "single"); else y = zeros (size (x)); endif ## Compute normal PDF if (isscalar (mu) && isscalar (sigma)) if (isfinite (mu) && (sigma > 0) && (sigma < Inf)) y = stdnormal_pdf ((x - mu) / sigma) / sigma; else y = NaN (size (x), class (y)); endif else k = isinf (mu) | !(sigma > 0) | !(sigma < Inf); y(k) = NaN; k = ! isinf (mu) & (sigma > 0) & (sigma < Inf); y(k) = stdnormal_pdf ((x(k) - mu(k)) ./ sigma(k)) ./ sigma(k); endif endfunction function y = stdnormal_pdf (x) y = (2 * pi)^(- 1/2) * exp (- x .^ 2 / 2); endfunction %!demo %! ## Plot various PDFs from the normal distribution %! x = -5:0.01:5; %! y1 = normpdf (x, 0, 0.5); %! y2 = normpdf (x, 0, 1); %! y3 = normpdf (x, 0, 2); %! y4 = normpdf (x, -2, 0.8); %! plot (x, y1, "-b", x, y2, "-g", x, y3, "-r", x, y4, "-c") %! grid on %! xlim ([-5, 5]) %! ylim ([0, 0.9]) %! legend ({"μ = 0, σ = 0.5", "μ = 0, σ = 1", ... %! "μ = 0, σ = 2", "μ = -2, σ = 0.8"}, "location", "northeast") %! title ("Normal PDF") %! xlabel ("values in x") %! ylabel ("density") ## Test output %!shared x, y %! x = [-Inf, 1, 2, Inf]; %! y = 1 / sqrt (2 * pi) * exp (-(x - 1) .^ 2 / 2); %!assert (normpdf (x, ones (1,4), ones (1,4)), y, eps) %!assert (normpdf (x, 1, ones (1,4)), y, eps) %!assert (normpdf (x, ones (1,4), 1), y, eps) %!assert (normpdf (x, [0 -Inf NaN Inf], 1), [y(1) NaN NaN NaN], eps) %!assert (normpdf (x, 1, [Inf NaN -1 0]), [NaN NaN NaN NaN], eps) %!assert (normpdf ([x, NaN], 1, 1), [y, NaN], eps) ## Test class of input preserved %!assert (normpdf (single ([x, NaN]), 1, 1), single ([y, NaN]), eps ("single")) %!assert (normpdf ([x, NaN], single (1), 1), single ([y, NaN]), eps ("single")) %!assert (normpdf ([x, NaN], 1, single (1)), single ([y, NaN]), eps ("single")) ## Test input validation %!error normpdf () %!error ... %! normpdf (ones (3), ones (2), ones (2)) %!error ... %! normpdf (ones (2), ones (3), ones (2)) %!error ... %! normpdf (ones (2), ones (2), ones (3)) %!error normpdf (i, 2, 2) %!error normpdf (2, i, 2) %!error normpdf (2, 2, i) statistics-release-1.6.3/inst/dist_fun/normrnd.m000066400000000000000000000153471456127120000217660ustar00rootroot00000000000000## Copyright (C) 2012 Rik Wehbring ## Copyright (C) 1995-2016 Kurt Hornik ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{r} =} normrnd (@var{mu}, @var{sigma}) ## @deftypefnx {statistics} {@var{r} =} normrnd (@var{mu}, @var{sigma}, @var{rows}) ## @deftypefnx {statistics} {@var{r} =} normrnd (@var{mu}, @var{sigma}, @var{rows}, @var{cols}, @dots{}) ## @deftypefnx {statistics} {@var{r} =} normrnd (@var{mu}, @var{sigma}, [@var{sz}]) ## ## Random arrays from the normal distribution. ## ## @code{@var{r} = normrnd (@var{mu}, @var{sigma})} returns an array of random ## numbers chosen from the normal distribution with mean @var{mu} and standard ## deviation @var{sigma}. The size of @var{r} is the common size of @var{mu} ## and @var{sigma}. A scalar input functions as a constant matrix of the same ## size as the other inputs. Both parameters must be finite real numbers and ## @var{sigma} > 0, otherwise NaN is returned. ## ## When called with a single size argument, @code{normrnd} returns a square ## matrix with the dimension specified. When called with more than one scalar ## argument, the first two arguments are taken as the number of rows and columns ## and any further arguments specify additional matrix dimensions. The size may ## also be specified with a row vector of dimensions, @var{sz}. ## ## Further information about the normal distribution can be found at ## @url{https://en.wikipedia.org/wiki/Normal_distribution} ## ## @seealso{norminv, norminv, normpdf, normfit, normlike, normstat} ## @end deftypefn function r = normrnd (mu, sigma, varargin) ## Check for valid number of input arguments if (nargin < 2) error ("normrnd: function called with too few input arguments."); endif ## Check for common size of MU and SIGMA if (! isscalar (mu) || ! isscalar (sigma)) [retval, mu, sigma] = common_size (mu, sigma); if (retval > 0) error ("normrnd: MU and SIGMA must be of common size or scalars."); endif endif ## Check for MU and SIGMA being reals if (iscomplex (mu) || iscomplex (sigma)) error ("normrnd: MU and SIGMA must not be complex."); endif ## Parse and check SIZE arguments if (nargin == 2) sz = size (mu); elseif (nargin == 3) if (isscalar (varargin{1}) && varargin{1} >= 0 ... && varargin{1} == fix (varargin{1})) sz = [varargin{1}, varargin{1}]; elseif (isrow (varargin{1}) && all (varargin{1} >= 0) ... && all (varargin{1} == fix (varargin{1}))) sz = varargin{1}; elseif error (strcat (["normrnd: SZ must be a scalar or a row vector"], ... [" of non-negative integers."])); endif elseif (nargin > 3) posint = cellfun (@(x) (! isscalar (x) || x < 0 || x != fix (x)), varargin); if (any (posint)) error ("normrnd: dimensions must be non-negative integers."); endif sz = [varargin{:}]; endif ## Check that parameters match requested dimensions in size if (! isscalar (mu) && ! isequal (size (mu), sz)) error ("normrnd: MU and SIGMA must be scalars or of size SZ."); endif ## Check for class type if (isa (mu, "single") || isa (sigma, "single")) cls = "single"; else cls = "double"; endif ## Generate random sample from normal distribution if (isscalar (mu) && isscalar (sigma)) if (isfinite (mu) && (sigma >= 0) && (sigma < Inf)) r = mu + sigma * randn (sz, cls); else r = NaN (sz, cls); endif else r = mu + sigma .* randn (sz, cls); k = ! isfinite (mu) | !(sigma >= 0) | !(sigma < Inf); r(k) = NaN; endif endfunction ## Test output %!assert (size (normrnd (1, 1)), [1 1]) %!assert (size (normrnd (1, ones (2,1))), [2, 1]) %!assert (size (normrnd (1, ones (2,2))), [2, 2]) %!assert (size (normrnd (ones (2,1), 1)), [2, 1]) %!assert (size (normrnd (ones (2,2), 1)), [2, 2]) %!assert (size (normrnd (1, 1, 3)), [3, 3]) %!assert (size (normrnd (1, 1, [4, 1])), [4, 1]) %!assert (size (normrnd (1, 1, 4, 1)), [4, 1]) %!assert (size (normrnd (1, 1, 4, 1, 5)), [4, 1, 5]) %!assert (size (normrnd (1, 1, 0, 1)), [0, 1]) %!assert (size (normrnd (1, 1, 1, 0)), [1, 0]) %!assert (size (normrnd (1, 1, 1, 2, 0, 5)), [1, 2, 0, 5]) ## Test class of input preserved %!assert (class (normrnd (1, 1)), "double") %!assert (class (normrnd (1, single (1))), "single") %!assert (class (normrnd (1, single ([1, 1]))), "single") %!assert (class (normrnd (single (1), 1)), "single") %!assert (class (normrnd (single ([1, 1]), 1)), "single") ## Test input validation %!error normrnd () %!error normrnd (1) %!error ... %! normrnd (ones (3), ones (2)) %!error ... %! normrnd (ones (2), ones (3)) %!error normrnd (i, 2, 3) %!error normrnd (1, i, 3) %!error ... %! normrnd (1, 2, -1) %!error ... %! normrnd (1, 2, 1.2) %!error ... %! normrnd (1, 2, ones (2)) %!error ... %! normrnd (1, 2, [2 -1 2]) %!error ... %! normrnd (1, 2, [2 0 2.5]) %!error ... %! normrnd (1, 2, 2, -1, 5) %!error ... %! normrnd (1, 2, 2, 1.5, 5) %!error ... %! normrnd (2, ones (2), 3) %!error ... %! normrnd (2, ones (2), [3, 2]) %!error ... %! normrnd (2, ones (2), 3, 2) statistics-release-1.6.3/inst/dist_fun/poisscdf.m000066400000000000000000000122671456127120000221170ustar00rootroot00000000000000## Copyright (C) 2012 Rik Wehbring ## Copyright (C) 1995-2016 Kurt Hornik ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{p} =} poisscdf (@var{x}, @var{lambda}) ## @deftypefnx {statistics} {@var{p} =} poisscdf (@var{x}, @var{lambda}, @qcode{"upper"}) ## ## Poisson cumulative distribution function (CDF). ## ## For each element of @var{x}, compute the cumulative distribution function ## (CDF) of the Poisson distribution with rate parameter @var{lambda}. The ## size of @var{p} is the common size of @var{x} and @var{lambda}. A scalar ## input functions as a constant matrix of the same size as the other inputs. ## ## @code{@var{p} = poisscdf (@var{x}, @var{lambda}, "upper")} computes the ## upper tail probability of the Poisson distribution with parameter ## @var{lambda}, at the values in @var{x}. ## ## Further information about the Poisson distribution can be found at ## @url{https://en.wikipedia.org/wiki/Poisson_distribution} ## ## @seealso{poissinv, poisspdf, poissrnd, poissfit, poisslike, poisstat} ## @end deftypefn function p = poisscdf (x, lambda, uflag) ## Check for valid number of input arguments if (nargin < 2) error ("poisscdf: function called with too few input arguments."); endif ## Check for "upper" flag if (nargin == 3 && strcmpi (uflag, "upper")) uflag = true; elseif (nargin == 3 && ! strcmpi (uflag, "upper")) error ("poisscdf: invalid argument for upper tail."); else uflag = false; endif ## Check for common size of X and LAMBDA if (! isscalar (x) || ! isscalar (lambda)) [retval, x, lambda] = common_size (x, lambda); if (retval > 0) error ("poisscdf: X and LAMBDA must be of common size or scalars."); endif endif ## Check for X and LAMBDA being reals if (iscomplex (x) || iscomplex (lambda)) error ("poisscdf: X and LAMBDA must not be complex."); endif ## Check for class type if (isa (x, "single") || isa (lambda, "single")) p = zeros (size (x), "single"); else p = zeros (size (x)); endif ## Force NaN for out of range parameters or missing data NaN is_nan = isnan (x) | isnan (lambda) | (lambda < 0) ... | (isinf (x) & isinf (lambda)); p(is_nan) = NaN; ## Compute P for X >= 0 x = floor (x); k = x >= 0 & ! is_nan & isfinite (lambda); ## Return 1 for positive infinite values of X, unless "upper" is given: p = 0 k1 = isinf (x) & lambda > 0 & isfinite (lambda); if (any (k1)) if (uflag) p(k1) = 0; else p(k1) = 1; endif endif ## Return 1 when X < 0 and "upper" is given k1 = x < 0 & lambda > 0 & isfinite (lambda); if (any (k1)) if (uflag) p(k1) = 1; endif endif ## Compute Poisson CDF for remaining cases x = x(k); lambda = lambda(k); if (uflag) p(k) = gammainc (lambda, x + 1); else p(k) = gammainc (lambda, x + 1, "upper"); endif endfunction %!demo %! ## Plot various CDFs from the Poisson distribution %! x = 0:20; %! p1 = poisscdf (x, 1); %! p2 = poisscdf (x, 4); %! p3 = poisscdf (x, 10); %! plot (x, p1, "*b", x, p2, "*g", x, p3, "*r") %! grid on %! ylim ([0, 1]) %! legend ({"λ = 1", "λ = 4", "λ = 10"}, "location", "southeast") %! title ("Poisson CDF") %! xlabel ("values in x (number of occurences)") %! ylabel ("probability") ## Test output %!shared x, y %! x = [-1 0 1 2 Inf]; %! y = [0, gammainc(1, (x(2:4) +1), "upper"), 1]; %!assert (poisscdf (x, ones (1,5)), y) %!assert (poisscdf (x, 1), y) %!assert (poisscdf (x, [1 0 NaN 1 1]), [y(1) 1 NaN y(4:5)]) %!assert (poisscdf ([x(1:2) NaN Inf x(5)], 1), [y(1:2) NaN 1 y(5)]) ## Test class of input preserved %!assert (poisscdf ([x, NaN], 1), [y, NaN]) %!assert (poisscdf (single ([x, NaN]), 1), single ([y, NaN]), eps ("single")) %!assert (poisscdf ([x, NaN], single (1)), single ([y, NaN]), eps ("single")) ## Test input validation %!error poisscdf () %!error poisscdf (1) %!error poisscdf (1, 2, 3) %!error poisscdf (1, 2, "tail") %!error ... %! poisscdf (ones (3), ones (2)) %!error ... %! poisscdf (ones (2), ones (3)) %!error poisscdf (i, 2) %!error poisscdf (2, i) statistics-release-1.6.3/inst/dist_fun/poissinv.m000066400000000000000000000166301456127120000221550ustar00rootroot00000000000000## Copyright (C) 2012 Rik Wehbring ## Copyright (C) 2014 Mike Giles ## Copyright (C) 2016 Lachlan Andrew ## Copyright (C) 1995-2017 Kurt Hornik ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{x} =} poissinv (@var{p}, @var{lambda}) ## ## Inverse of the Poisson cumulative distribution function (iCDF). ## ## For each element of @var{p}, compute the quantile (the inverse of the CDF) of ## the Poisson distribution with rate parameter @var{lambda}. The size of ## @var{x} is the common size of @var{p} and @var{lambda}. A scalar input ## functions as a constant matrix of the same size as the other inputs. ## ## Further information about the Poisson distribution can be found at ## @url{https://en.wikipedia.org/wiki/Poisson_distribution} ## ## @seealso{poisscdf, poisspdf, poissrnd, poissfit, poisslike, poisstat} ## @end deftypefn function x = poissinv (p, lambda) ## Check for valid number of input arguments if (nargin < 2) error ("poissinv: function called with too few input arguments."); endif ## Check for common size of P and LAMBDA if (! isscalar (p) || ! isscalar (lambda)) [retval, p, lambda] = common_size (p, lambda); if (retval > 0) error ("poissinv: P and LAMBDA must be of common size or scalars."); endif endif ## Check for P and LAMBDA being reals if (iscomplex (p) || iscomplex (lambda)) error ("poissinv: P and LAMBDA must not be complex."); endif ## Check for class type if (isa (p, "single") || isa (lambda, "single")) x = zeros (size (p), "single"); else x = zeros (size (p)); endif ## Force NaN for out of range parameters or p-values k = (p < 0) | (p > 1) | isnan (p) | ! (lambda > 0); x(k) = NaN; k = (p == 1) & (lambda > 0); x(k) = Inf; k = (p > 0) & (p < 1) & (lambda > 0); if (any (k(:))) limit = 20; # After 'limit' iterations, use approx if (isscalar (lambda)) cdf = [(cumsum (poisspdf (0:limit-1,lambda))), 2]; y = p(:); # force to column r = bsxfun (@le, y(k), cdf); [~, x(k)] = max (r, [], 2); # find first instance of p <= cdf x(k) -= 1; else kk = find (k); cdf = exp (-lambda(kk)); for i = 1:limit m = find (cdf < p(kk)); if (isempty (m)) break; else x(kk(m)) += 1; cdf(m) += poisspdf (i, lambda(kk(m))); endif endfor endif ## Use Mike Giles's magic when x isn't < limit k &= (x == limit); if (any (k(:))) if (isscalar (lambda)) lam = repmat (lambda, size (p)); else lam = lambda; endif x(k) = analytic_approx (p(k), lam(k)); endif endif endfunction ## The following is based on Mike Giles's CUDA implementation, ## [http://people.maths.ox.ac.uk/gilesm/codes/poissinv/poissinv_cuda.h] ## which is copyright by the University of Oxford ## and is provided under the terms of the GNU GPLv3 license: ## http://www.gnu.org/licenses/gpl.html function x = analytic_approx (p, lambda) s = norminv (p, 0, 1) ./ sqrt (lambda); k = (s > -0.6833501) & (s < 1.777993); ## use polynomial approximations in central region if (any (k)) lam = lambda(k); if (isscalar (s)) sk = s; else sk = s(k); endif ## polynomial approximation to f^{-1}(s) - 1 rm = 2.82298751e-07; rm = -2.58136133e-06 + rm.*sk; rm = 1.02118025e-05 + rm.*sk; rm = -2.37996199e-05 + rm.*sk; rm = 4.05347462e-05 + rm.*sk; rm = -6.63730967e-05 + rm.*sk; rm = 0.000124762566 + rm.*sk; rm = -0.000256970731 + rm.*sk; rm = 0.000558953132 + rm.*sk; rm = -0.00133129194 + rm.*sk; rm = 0.00370367937 + rm.*sk; rm = -0.0138888706 + rm.*sk; rm = 0.166666667 + rm.*sk; rm = sk + sk.*(rm.*sk); ## polynomial approximation to correction c0(r) t = 1.86386867e-05; t = -0.000207319499 + t.*rm; t = 0.0009689451 + t.*rm; t = -0.00247340054 + t.*rm; t = 0.00379952985 + t.*rm; t = -0.00386717047 + t.*rm; t = 0.00346960934 + t.*rm; t = -0.00414125511 + t.*rm; t = 0.00586752093 + t.*rm; t = -0.00838583787 + t.*rm; t = 0.0132793933 + t.*rm; t = -0.027775536 + t.*rm; t = 0.333333333 + t.*rm; ## O(1/lam) correction y = -0.00014585224; y = 0.00146121529 + y.*rm; y = -0.00610328845 + y.*rm; y = 0.0138117964 + y.*rm; y = -0.0186988746 + y.*rm; y = 0.0168155118 + y.*rm; y = -0.013394797 + y.*rm; y = 0.0135698573 + y.*rm; y = -0.0155377333 + y.*rm; y = 0.0174065334 + y.*rm; y = -0.0198011178 + y.*rm; y ./= lam; x(k) = floor (lam + (y+t)+lam.*rm); endif k = ! k & (s > -sqrt (2)); if (any (k)) ## Newton iteration r = 1 + s(k); r2 = r + 1; while (any (abs (r - r2) > 1e-5)) t = log (r); r2 = r; s2 = sqrt (2 * ((1-r) + r.*t)); s2(r<1) *= -1; r = r2 - (s2 - s(k)) .* s2 ./ t; if (r < 0.1 * r2) r = 0.1 * r2; endif endwhile t = log (r); y = lambda(k) .* r + log (sqrt (2*r.*((1-r) + r.*t)) ./ abs (r-1)) ./ t; x(k) = floor (y - 0.0218 ./ (y + 0.065 * lambda(k))); endif endfunction %!demo %! ## Plot various iCDFs from the Poisson distribution %! p = 0.001:0.001:0.999; %! x1 = poissinv (p, 13); %! x2 = poissinv (p, 4); %! x3 = poissinv (p, 10); %! plot (p, x1, "-b", p, x2, "-g", p, x3, "-r") %! grid on %! ylim ([0, 20]) %! legend ({"λ = 1", "λ = 4", "λ = 10"}, "location", "northwest") %! title ("Poisson iCDF") %! xlabel ("probability") %! ylabel ("values in x (number of occurences)") ## Test output %!shared p %! p = [-1 0 0.5 1 2]; %!assert (poissinv (p, ones (1,5)), [NaN 0 1 Inf NaN]) %!assert (poissinv (p, 1), [NaN 0 1 Inf NaN]) %!assert (poissinv (p, [1 0 NaN 1 1]), [NaN NaN NaN Inf NaN]) %!assert (poissinv ([p(1:2) NaN p(4:5)], 1), [NaN 0 NaN Inf NaN]) ## Test class of input preserved %!assert (poissinv ([p, NaN], 1), [NaN 0 1 Inf NaN NaN]) %!assert (poissinv (single ([p, NaN]), 1), single ([NaN 0 1 Inf NaN NaN])) %!assert (poissinv ([p, NaN], single (1)), single ([NaN 0 1 Inf NaN NaN])) ## Test input validation %!error poissinv () %!error poissinv (1) %!error ... %! poissinv (ones (3), ones (2)) %!error ... %! poissinv (ones (2), ones (3)) %!error poissinv (i, 2) %!error poissinv (2, i) statistics-release-1.6.3/inst/dist_fun/poisspdf.m000066400000000000000000000077301456127120000221330ustar00rootroot00000000000000## Copyright (C) 2012 Rik Wehbring ## Copyright (C) 1995-2016 Kurt Hornik ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{y} =} poisspdf (@var{x}, @var{lambda}) ## ## Poisson probability density function (PDF). ## ## For each element of @var{x}, compute the probability density function (PDF) ## of the Poisson distribution with rate parameter @var{lambda}. The size of ## @var{y} is the common size of @var{x} and @var{lambda}. A scalar input ## functions as a constant matrix of the same size as the other inputs. ## ## Further information about the Poisson distribution can be found at ## @url{https://en.wikipedia.org/wiki/Poisson_distribution} ## ## @seealso{poisscdf, poissinv, poissrnd, poissfit, poisslike, poisstat} ## @end deftypefn function y = poisspdf (x, lambda) ## Check for valid number of input arguments if (nargin < 2) error ("poisspdf: function called with too few input arguments."); endif ## Check for common size of X and LAMBDA if (! isscalar (x) || ! isscalar (lambda)) [retval, x, lambda] = common_size (x, lambda); if (retval > 0) error ("poisspdf: X and LAMBDA must be of common size or scalars."); endif endif ## Check for X and LAMBDA being reals if (iscomplex (x) || iscomplex (lambda)) error ("poisspdf: X and LAMBDA must not be complex."); endif ## Check for class type if (isa (x, "single") || isa (lambda, "single")) y = zeros (size (x), "single"); else y = zeros (size (x)); endif ## Force NaN for out of range parameters or missing data NaN k = isnan (x) | ! (lambda > 0); y(k) = NaN; k = (x >= 0) & (x < Inf) & (x == fix (x)) & (lambda > 0); if (isscalar (lambda)) y(k) = exp (x(k) * log (lambda) - lambda - gammaln (x(k) + 1)); else y(k) = exp (x(k) .* log (lambda(k)) - lambda(k) - gammaln (x(k) + 1)); endif endfunction %!demo %! ## Plot various PDFs from the Poisson distribution %! x = 0:20; %! y1 = poisspdf (x, 1); %! y2 = poisspdf (x, 4); %! y3 = poisspdf (x, 10); %! plot (x, y1, "*b", x, y2, "*g", x, y3, "*r") %! grid on %! ylim ([0, 0.4]) %! legend ({"λ = 1", "λ = 4", "λ = 10"}, "location", "northeast") %! title ("Poisson PDF") %! xlabel ("values in x (number of occurences)") %! ylabel ("density") ## Test output %!shared x, y %! x = [-1 0 1 2 Inf]; %! y = [0, exp(-1)*[1 1 0.5], 0]; %!assert (poisspdf (x, ones (1,5)), y, eps) %!assert (poisspdf (x, 1), y, eps) %!assert (poisspdf (x, [1 0 NaN 1 1]), [y(1) NaN NaN y(4:5)], eps) %!assert (poisspdf ([x, NaN], 1), [y, NaN], eps) ## Test class of input preserved %!assert (poisspdf (single ([x, NaN]), 1), single ([y, NaN]), eps ("single")) %!assert (poisspdf ([x, NaN], single (1)), single ([y, NaN]), eps ("single")) ## Test input validation %!error poisspdf () %!error poisspdf (1) %!error ... %! poisspdf (ones (3), ones (2)) %!error ... %! poisspdf (ones (2), ones (3)) %!error poisspdf (i, 2) %!error poisspdf (2, i) statistics-release-1.6.3/inst/dist_fun/poissrnd.m000066400000000000000000000137111456127120000221410ustar00rootroot00000000000000## Copyright (C) 2012 Rik Wehbring ## Copyright (C) 1995-2016 Kurt Hornik ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{r} =} poissrnd (@var{lambda}) ## @deftypefnx {statistics} {@var{r} =} poissrnd (@var{lambda}, @var{rows}) ## @deftypefnx {statistics} {@var{r} =} poissrnd (@var{lambda}, @var{rows}, @var{cols}, @dots{}) ## @deftypefnx {statistics} {@var{r} =} poissrnd (@var{lambda}, [@var{sz}]) ## ## Random arrays from the Poisson distribution. ## ## @code{@var{r} = normrnd (@var{lambda})} returns an array of random numbers ## chosen from the Poisson distribution with rate parameter @var{lambda}. The ## size of @var{r} is the common size of @var{lambda}. A scalar input functions ## as a constant matrix of the same size as the other inputs. @var{lambda} must ## be a finite real number and greater or equal to 0, otherwise @qcode{NaN} is ## returned. ## ## When called with a single size argument, @code{poissrnd} returns a square ## matrix with the dimension specified. When called with more than one scalar ## argument, the first two arguments are taken as the number of rows and columns ## and any further arguments specify additional matrix dimensions. The size may ## also be specified with a row vector of dimensions, @var{sz}. ## ## Further information about the Poisson distribution can be found at ## @url{https://en.wikipedia.org/wiki/Poisson_distribution} ## ## @seealso{poisscdf, poissinv, poisspdf, poissfit, poisslike, poisstat} ## @end deftypefn function r = poissrnd (lambda, varargin) ## Check for valid number of input arguments if (nargin < 1) error ("poissrnd: function called with too few input arguments."); endif ## Check for LAMBDA being real if (iscomplex (lambda)) error ("poissrnd: LAMBDA must not be complex."); endif ## Parse and check SIZE arguments if (nargin == 1) sz = size (lambda); elseif (nargin == 2) if (isscalar (varargin{1}) && varargin{1} >= 0 ... && varargin{1} == fix (varargin{1})) sz = [varargin{1}, varargin{1}]; elseif (isrow (varargin{1}) && all (varargin{1} >= 0) ... && all (varargin{1} == fix (varargin{1}))) sz = varargin{1}; elseif error (strcat (["poissrnd: SZ must be a scalar or a row vector"], ... [" of non-negative integers."])); endif elseif (nargin > 2) posint = cellfun (@(x) (! isscalar (x) || x < 0 || x != fix (x)), varargin); if (any (posint)) error ("poissrnd: dimensions must be non-negative integers."); endif sz = [varargin{:}]; endif ## Check that parameters match requested dimensions in size if (! isscalar (lambda) && ! isequal (size (lambda), sz)) error ("poissrnd: LAMBDA must be scalar or of size SZ."); endif ## Check for class type if (isa (lambda, "single")) cls = "single"; else cls = "double"; endif ## Generate random sample from Poisson distribution if (isscalar (lambda)) if (lambda >= 0 && lambda < Inf) r = randp (lambda, sz, cls); else r = NaN (sz, cls); endif else r = NaN (sz, cls); k = (lambda >= 0) & (lambda < Inf); r(k) = randp (lambda(k), cls); endif endfunction ## Test output %!assert (size (poissrnd (2)), [1, 1]) %!assert (size (poissrnd (ones (2,1))), [2, 1]) %!assert (size (poissrnd (ones (2,2))), [2, 2]) %!assert (size (poissrnd (1, 3)), [3, 3]) %!assert (size (poissrnd (1, [4 1])), [4, 1]) %!assert (size (poissrnd (1, 4, 1)), [4, 1]) %!assert (size (poissrnd (1, 4, 1)), [4, 1]) %!assert (size (poissrnd (1, 4, 1, 5)), [4, 1, 5]) %!assert (size (poissrnd (1, 0, 1)), [0, 1]) %!assert (size (poissrnd (1, 1, 0)), [1, 0]) %!assert (size (poissrnd (1, 1, 2, 0, 5)), [1, 2, 0, 5]) %!assert (poissrnd (0, 1, 1), 0) %!assert (poissrnd ([0, 0, 0], [1, 3]), [0 0 0]) ## Test class of input preserved %!assert (class (poissrnd (2)), "double") %!assert (class (poissrnd (single (2))), "single") %!assert (class (poissrnd (single ([2 2]))), "single") ## Test input validation %!error poissrnd () %!error poissrnd (i) %!error ... %! poissrnd (1, -1) %!error ... %! poissrnd (1, 1.2) %!error ... %! poissrnd (1, ones (2)) %!error ... %! poissrnd (1, [2 -1 2]) %!error ... %! poissrnd (1, [2 0 2.5]) %!error ... %! poissrnd (ones (2), ones (2)) %!error ... %! poissrnd (1, 2, -1, 5) %!error ... %! poissrnd (1, 2, 1.5, 5) %!error poissrnd (ones (2,2), 3) %!error poissrnd (ones (2,2), [3, 2]) %!error poissrnd (ones (2,2), 2, 3) statistics-release-1.6.3/inst/dist_fun/raylcdf.m000066400000000000000000000114041456127120000217210ustar00rootroot00000000000000## Copyright (C) 2006, 2007 Arno Onken ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{p} =} raylcdf (@var{x}, @var{sigma}) ## @deftypefnx {statistics} {@var{p} =} raylcdf (@var{x}, @var{sigma}, @qcode{"upper"}) ## ## Rayleigh cumulative distribution function (CDF). ## ## For each element of @var{x}, compute the cumulative distribution function ## (CDF) of the Rayleigh distribution with scale parameter @var{sigma}. The ## size of @var{p} is the common size of @var{x} and @var{sigma}. A scalar ## input functions as a constant matrix of the same size as the other inputs. ## ## @code{@var{p} = raylcdf (@var{x}, @var{sigma}, "upper")} computes the upper ## tail probability of the Rayleigh distribution with parameter @var{sigma}, at ## the values in @var{x}. ## ## Further information about the Rayleigh distribution can be found at ## @url{https://en.wikipedia.org/wiki/Rayleigh_distribution} ## ## @seealso{raylinv, raylpdf, raylrnd, raylfit, rayllike, raylstat} ## @end deftypefn function p = raylcdf (x, sigma, uflag) ## Check for valid number of input arguments if (nargin < 2) error ("raylcdf: function called with too few input arguments."); endif ## Check for "upper" flag if (nargin == 3 && strcmpi (uflag, "upper")) uflag = true; elseif (nargin == 3 && ! strcmpi (uflag, "upper")) error ("raylcdf: invalid argument for upper tail."); else uflag = false; endif ## Check for common size of X and SIGMA if (! isscalar (x) || ! isscalar (sigma)) [retval, x, sigma] = common_size (x, sigma); if (retval > 0) error ("raylcdf: X and SIGMA must be of common size or scalars."); endif endif ## Check for X and SIGMA being reals if (iscomplex (x) || iscomplex (sigma)) error ("raylcdf: X and SIGMA must not be complex."); endif ## Check for class type if (isa (x, "single") || isa (sigma, "single")); p = zeros (size (x), "single"); else p = zeros (size (x)); endif ## Force 1 for upper flag and X <= 0 k0 = sigma > 0 & x <= 0; if (uflag && any (k0(:))) p(k0) = 1; end ## Calculate Rayleigh CDF for valid parameter and data range k = sigma > 0 & x > 0; if (any (k(:))) if (uflag) p(k) = exp (-x(k) .^ 2 ./ (2 * sigma(k) .^ 2)); else p(k) = - expm1 (-x(k) .^ 2 ./ (2 * sigma(k) .^ 2)); endif endif ## Continue argument check p(! (k0 | k)) = NaN; endfunction %!demo %! ## Plot various CDFs from the Rayleigh distribution %! x = 0:0.01:10; %! p1 = raylcdf (x, 0.5); %! p2 = raylcdf (x, 1); %! p3 = raylcdf (x, 2); %! p4 = raylcdf (x, 3); %! p5 = raylcdf (x, 4); %! plot (x, p1, "-b", x, p2, "g", x, p3, "-r", x, p4, "-m", x, p5, "-k") %! grid on %! ylim ([0, 1]) %! legend ({"σ = 0.5", "σ = 1", "σ = 2", ... %! "σ = 3", "σ = 4"}, "location", "southeast") %! title ("Rayleigh CDF") %! xlabel ("values in x") %! ylabel ("probability") ## Test output %!test %! x = 0:0.5:2.5; %! sigma = 1:6; %! p = raylcdf (x, sigma); %! expected_p = [0.0000, 0.0308, 0.0540, 0.0679, 0.0769, 0.0831]; %! assert (p, expected_p, 0.001); %!test %! x = 0:0.5:2.5; %! p = raylcdf (x, 0.5); %! expected_p = [0.0000, 0.3935, 0.8647, 0.9889, 0.9997, 1.0000]; %! assert (p, expected_p, 0.001); %!shared x, p %! x = [-1, 0, 1, 2, Inf]; %! p = [0, 0, 0.39346934028737, 0.86466471676338, 1]; %!assert (raylcdf (x, 1), p, 1e-14) %!assert (raylcdf (x, 1, "upper"), 1 - p, 1e-14) ## Test input validation %!error raylcdf () %!error raylcdf (1) %!error raylcdf (1, 2, "uper") %!error raylcdf (1, 2, 3) %!error ... %! raylcdf (ones (3), ones (2)) %!error ... %! raylcdf (ones (2), ones (3)) %!error raylcdf (i, 2) %!error raylcdf (2, i) statistics-release-1.6.3/inst/dist_fun/raylinv.m000066400000000000000000000072231456127120000217650ustar00rootroot00000000000000## Copyright (C) 2006, 2007 Arno Onken ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{x} =} raylinv (@var{p}, @var{sigma}) ## ## Inverse of the Rayleigh cumulative distribution function (iCDF). ## ## For each element of @var{p}, compute the quantile (the inverse of the CDF) of ## the Rayleigh distribution with scale parameter @var{sigma}. The size of ## @var{x} is the common size of @var{p} and @var{sigma}. A scalar input ## functions as a constant matrix of the same size as the other inputs. ## ## Further information about the Rayleigh distribution can be found at ## @url{https://en.wikipedia.org/wiki/Rayleigh_distribution} ## ## @seealso{raylcdf, raylpdf, raylrnd, raylfit, rayllike, raylstat} ## @end deftypefn function x = raylinv (p, sigma) ## Check for valid number of input arguments if (nargin < 2) error ("raylinv: function called with too few input arguments."); endif ## Check for common size of P and SIGMA if (! isscalar (p) || ! isscalar (sigma)) [retval, p, sigma] = common_size (p, sigma); if (retval > 0) error ("raylinv: P and SIGMA must be of common size or scalars."); endif endif ## Check for X and SIGMA being reals if (iscomplex (p) || iscomplex (sigma)) error ("raylinv: P and SIGMA must not be complex."); endif ## Calculate Rayleigh iCDF x = sqrt (-2 .* log (1 - p) .* sigma .^ 2); ## Check for valid parameter and support k = find (p == 1); if (any (k)) x(k) = Inf; endif k = find (! (p >= 0) | ! (p <= 1) | ! (sigma > 0)); if (any (k)) x(k) = NaN; endif endfunction %!demo %! ## Plot various iCDFs from the Rayleigh distribution %! p = 0.001:0.001:0.999; %! x1 = raylinv (p, 0.5); %! x2 = raylinv (p, 1); %! x3 = raylinv (p, 2); %! x4 = raylinv (p, 3); %! x5 = raylinv (p, 4); %! plot (p, x1, "-b", p, x2, "g", p, x3, "-r", p, x4, "-m", p, x5, "-k") %! grid on %! ylim ([0, 10]) %! legend ({"σ = 0,5", "σ = 1", "σ = 2", ... %! "σ = 3", "σ = 4"}, "location", "northwest") %! title ("Rayleigh iCDF") %! xlabel ("probability") %! ylabel ("values in x") ## Test output %!test %! p = 0:0.1:0.5; %! sigma = 1:6; %! x = raylinv (p, sigma); %! expected_x = [0.0000, 0.9181, 2.0041, 3.3784, 5.0538, 7.0645]; %! assert (x, expected_x, 0.001); %!test %! p = 0:0.1:0.5; %! x = raylinv (p, 0.5); %! expected_x = [0.0000, 0.2295, 0.3340, 0.4223, 0.5054, 0.5887]; %! assert (x, expected_x, 0.001); ## Test input validation %!error raylinv () %!error raylinv (1) %!error ... %! raylinv (ones (3), ones (2)) %!error ... %! raylinv (ones (2), ones (3)) %!error raylinv (i, 2) %!error raylinv (2, i) statistics-release-1.6.3/inst/dist_fun/raylpdf.m000066400000000000000000000070741456127120000217460ustar00rootroot00000000000000## Copyright (C) 2006, 2007 Arno Onken ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{y} =} raylpdf (@var{x}, @var{sigma}) ## ## Rayleigh probability density function (PDF). ## ## For each element of @var{x}, compute the probability density function (PDF) ## of the Rayleigh distribution with scale parameter @var{sigma}. The size of ## @var{p} is the common size of @var{x} and @var{sigma}. A scalar input ## functions as a constant matrix of the same size as the other inputs. ## ## Further information about the Rayleigh distribution can be found at ## @url{https://en.wikipedia.org/wiki/Rayleigh_distribution} ## ## @seealso{raylcdf, raylinv, raylrnd, raylfit, rayllike, raylstat} ## @end deftypefn function y = raylpdf (x, sigma) ## Check for valid number of input arguments if (nargin < 2) error ("raylpdf: function called with too few input arguments."); endif ## Check for common size of X and SIGMA if (! isscalar (x) || ! isscalar (sigma)) [retval, x, sigma] = common_size (x, sigma); if (retval > 0) error ("raylpdf: X and SIGMA must be of common size or scalars."); endif endif ## Check for X and SIGMA being reals if (iscomplex (x) || iscomplex (sigma)) error ("raylpdf: X and SIGMA must not be complex."); endif ## Calculate Rayleigh PDF y = x .* exp ((-x .^ 2) ./ (2 .* sigma .^ 2)) ./ (sigma .^ 2); ## Continue argument check k = find (! (x >= 0) | ! (x < Inf) | ! (sigma > 0)); if (any (k)) y(k) = NaN; endif endfunction %!demo %! ## Plot various PDFs from the Rayleigh distribution %! x = 0:0.01:10; %! y1 = raylpdf (x, 0.5); %! y2 = raylpdf (x, 1); %! y3 = raylpdf (x, 2); %! y4 = raylpdf (x, 3); %! y5 = raylpdf (x, 4); %! plot (x, y1, "-b", x, y2, "g", x, y3, "-r", x, y4, "-m", x, y5, "-k") %! grid on %! ylim ([0, 1.25]) %! legend ({"σ = 0,5", "σ = 1", "σ = 2", ... %! "σ = 3", "σ = 4"}, "location", "northeast") %! title ("Rayleigh PDF") %! xlabel ("values in x") %! ylabel ("density") ## Test output %!test %! x = 0:0.5:2.5; %! sigma = 1:6; %! y = raylpdf (x, sigma); %! expected_y = [0.0000, 0.1212, 0.1051, 0.0874, 0.0738, 0.0637]; %! assert (y, expected_y, 0.001); %!test %! x = 0:0.5:2.5; %! y = raylpdf (x, 0.5); %! expected_y = [0.0000, 1.2131, 0.5413, 0.0667, 0.0027, 0.0000]; %! assert (y, expected_y, 0.001); ## Test input validation %!error raylpdf () %!error raylpdf (1) %!error ... %! raylpdf (ones (3), ones (2)) %!error ... %! raylpdf (ones (2), ones (3)) %!error raylpdf (i, 2) %!error raylpdf (2, i) statistics-release-1.6.3/inst/dist_fun/raylrnd.m000066400000000000000000000133321456127120000217520ustar00rootroot00000000000000## Copyright (C) 2006, 2007 Arno Onken ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{r} =} raylrnd (@var{sigma}) ## @deftypefnx {statistics} {@var{r} =} raylrnd (@var{sigma}, @var{rows}) ## @deftypefnx {statistics} {@var{r} =} raylrnd (@var{sigma}, @var{rows}, @var{cols}, @dots{}) ## @deftypefnx {statistics} {@var{r} =} raylrnd (@var{sigma}, [@var{sz}]) ## ## Random arrays from the Rayleigh distribution. ## ## @code{@var{r} = raylrnd (@var{sigma})} returns an array of random numbers ## chosen from the Rayleigh distribution with scale parameter @var{sigma}. The ## size of @var{r} is the size of @var{sigma}. A scalar input functions as a ## constant matrix of the same size as the other inputs. @var{sigma} must be a ## finite real number greater than 0, otherwise @qcode{NaN} is returned. ## ## When called with a single size argument, @code{raylrnd} returns a square ## matrix with the dimension specified. When called with more than one scalar ## argument, the first two arguments are taken as the number of rows and columns ## and any further arguments specify additional matrix dimensions. The size may ## also be specified with a row vector of dimensions, @var{sz}. ## ## Further information about the Rayleigh distribution can be found at ## @url{https://en.wikipedia.org/wiki/Rayleigh_distribution} ## ## @seealso{raylcdf, raylinv, raylpdf, raylfit, rayllike, raylstat} ## @end deftypefn function r = raylrnd (sigma, varargin) ## Check for valid number of input arguments if (nargin < 1) error ("raylrnd: function called with too few input arguments."); endif ## Check for SIGMA being real if (iscomplex (sigma)) error ("raylrnd: SIGMA must not be complex."); endif ## Parse and check SIZE arguments if (nargin == 1) sz = size (sigma); elseif (nargin == 2) if (isscalar (varargin{1}) && varargin{1} >= 0 ... && varargin{1} == fix (varargin{1})) sz = [varargin{1}, varargin{1}]; elseif (isrow (varargin{1}) && all (varargin{1} >= 0) ... && all (varargin{1} == fix (varargin{1}))) sz = varargin{1}; elseif error (strcat (["raylrnd: SZ must be a scalar or a row vector"], ... [" of non-negative integers."])); endif elseif (nargin > 2) posint = cellfun (@(x) (! isscalar (x) || x < 0 || x != fix (x)), varargin); if (any (posint)) error ("raylrnd: dimensions must be non-negative integers."); endif sz = [varargin{:}]; endif ## Check that parameters match requested dimensions in size if (! isscalar (sigma) && ! isequal (size (sigma), sz)) error ("raylrnd: SIGMA must be scalar or of size SZ."); endif ## Generate random sample from Rayleigh distribution r = sqrt (-2 .* log (1 - rand (sz)) .* sigma .^ 2); ## Check for valid parameter k = find (! (sigma > 0)); if (any (k)) r(k) = NaN; endif ## Cast into appropriate class if (isa (sigma, "single")) r = cast (r, "single"); endif endfunction ## Test output %!assert (size (raylrnd (2)), [1, 1]) %!assert (size (raylrnd (ones (2,1))), [2, 1]) %!assert (size (raylrnd (ones (2,2))), [2, 2]) %!assert (size (raylrnd (1, 3)), [3, 3]) %!assert (size (raylrnd (1, [4 1])), [4, 1]) %!assert (size (raylrnd (1, 4, 1)), [4, 1]) %!assert (size (raylrnd (1, 4, 1)), [4, 1]) %!assert (size (raylrnd (1, 4, 1, 5)), [4, 1, 5]) %!assert (size (raylrnd (1, 0, 1)), [0, 1]) %!assert (size (raylrnd (1, 1, 0)), [1, 0]) %!assert (size (raylrnd (1, 1, 2, 0, 5)), [1, 2, 0, 5]) %!assert (raylrnd (0, 1, 1), NaN) %!assert (raylrnd ([0, 0, 0], [1, 3]), [NaN, NaN, NaN]) ## Test class of input preserved %!assert (class (raylrnd (2)), "double") %!assert (class (raylrnd (single (2))), "single") %!assert (class (raylrnd (single ([2 2]))), "single") ## Test input validation %!error raylrnd () %!error raylrnd (i) %!error ... %! raylrnd (1, -1) %!error ... %! raylrnd (1, 1.2) %!error ... %! raylrnd (1, ones (2)) %!error ... %! raylrnd (1, [2 -1 2]) %!error ... %! raylrnd (1, [2 0 2.5]) %!error ... %! raylrnd (ones (2), ones (2)) %!error ... %! raylrnd (1, 2, -1, 5) %!error ... %! raylrnd (1, 2, 1.5, 5) %!error raylrnd (ones (2,2), 3) %!error raylrnd (ones (2,2), [3, 2]) %!error raylrnd (ones (2,2), 2, 3) statistics-release-1.6.3/inst/dist_fun/ricecdf.m000066400000000000000000000161051456127120000216770ustar00rootroot00000000000000## Copyright (C) 2024 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{p} =} ricecdf (@var{x}, @var{nu}, @var{sigma}) ## @deftypefnx {statistics} {@var{p} =} ricecdf (@var{x}, @var{nu}, @var{sigma}, @qcode{"upper"}) ## ## Rician cumulative distribution function (CDF). ## ## For each element of @var{x}, compute the cumulative distribution function ## (CDF) of the Rician distribution with non-centrality (distance) parameter ## @var{nu} and scale parameter @var{sigma}. The size of @var{p} is the common ## size of @var{x}, @var{nu}, and @var{sigma}. A scalar input functions as a ## constant matrix of the same size as the other inputs. ## ## @code{@var{p} = ricecdf (@var{x}, @var{nu}, @var{sigma}, "upper")} computes ## the upper tail probability of the Rician distribution with parameters ## @var{nu} and @var{sigma}, at the values in @var{x}. ## ## Further information about the Rician distribution can be found at ## @url{https://en.wikipedia.org/wiki/Rice_distribution} ## ## @seealso{riceinv, ricepdf, ricernd, ricefit, ricelike, ricestat} ## @end deftypefn function p = ricecdf (x, nu, sigma, uflag) ## Check for valid number of input arguments if (nargin < 3) error ("ricecdf: function called with too few input arguments."); endif ## Check for "upper" flag if (nargin == 4 && strcmpi (uflag, "upper")) uflag = true; elseif (nargin == 4 && ! strcmpi (uflag, "upper")) error ("ricecdf: invalid argument for upper tail."); else uflag = false; endif ## Check for common size of X and SIGMA if (! isscalar (x) || ! isscalar (nu) || ! isscalar (sigma)) [retval, x, nu, sigma] = common_size (x, nu, sigma); if (retval > 0) error ("ricecdf: X, NU, and SIGMA must be of common size or scalars."); endif endif ## Check for X and SIGMA being reals if (iscomplex (x) || iscomplex (nu) || iscomplex (sigma)) error ("ricecdf: X, NU, and SIGMA must not be complex."); endif ## Check for class type if (isa (x, "single") || isa (nu, "single") || isa (sigma, "single")); p = zeros (size (x), "single"); else p = zeros (size (x)); endif ## Force 1 for upper flag and X <= 0 k0 = nu >= 0 & sigma > 0 & x < 0; if (uflag && any (k0(:))) p(k0) = 1; end ## Calculate Rayleigh CDF for valid parameter and data range k = nu >= 0 & sigma > 0 & x >= 0; if (any (k(:))) if (uflag) p(k) = marcumQ1 (nu(k) ./ sigma(k), x(k) ./ sigma(k)); else p(k) = 1 - marcumQ1 (nu(k) ./ sigma(k), x(k) ./ sigma(k)); endif endif ## Continue argument check p(! (k0 | k)) = NaN; endfunction ## Marcum's "Q" function of order 1 function Q = marcumQ1 (a, b) ## Prepare output matrix if (isa (a, "single") || isa (b, "single")) Q = NaN (size (b), "single"); else Q = NaN (size (b)); endif ## Force marginal cases Q(a != Inf & b == 0) = 1; Q(a != Inf & b == Inf) = 0; Q(a == Inf & b != Inf) = 1; z = isnan (Q) & a == 0 & b != Inf; if (any(z)) Q(z) = exp ((-b(z) .^ 2) ./ 2); end ## Compute the remaining cases z = isnan (Q) & ! isnan (a) & ! isnan (b); if (any(z(:))) aa = (a(z) .^ 2) ./ 2; bb = (b(z) .^ 2) ./ 2; eA = exp (-aa); eB = bb .* exp (-bb); h = eA; d = eB .* h; s = d; j = (d > s.*eps(class(d))); k = 1; while (any (j)) eA = aa .* eA ./ k; h = h + eA; eB = bb .* eB ./ (k + 1); d = eB .* h; s(j) = s (j) + d(j); j = (d > s .* eps (class (d))); k = k + 1; endwhile Q(z) = 1 - s; endif endfunction %!demo %! ## Plot various CDFs from the Rician distribution %! x = 0:0.01:10; %! p1 = ricecdf (x, 0, 1); %! p2 = ricecdf (x, 0.5, 1); %! p3 = ricecdf (x, 1, 1); %! p4 = ricecdf (x, 2, 1); %! p5 = ricecdf (x, 4, 1); %! plot (x, p1, "-b", x, p2, "g", x, p3, "-r", x, p4, "-m", x, p5, "-k") %! grid on %! ylim ([0, 1]) %! xlim ([0, 8]) %! legend ({"ν = 0, σ = 1", "ν = 0.5, σ = 1", "ν = 1, σ = 1", ... %! "ν = 2, σ = 1", "ν = 4, σ = 1"}, "location", "southeast") %! title ("Rician CDF") %! xlabel ("values in x") %! ylabel ("probability") %!demo %! ## Plot various CDFs from the Rician distribution %! x = 0:0.01:10; %! p1 = ricecdf (x, 0, 0.5); %! p2 = ricecdf (x, 0, 2); %! p3 = ricecdf (x, 0, 3); %! p4 = ricecdf (x, 2, 2); %! p5 = ricecdf (x, 4, 2); %! plot (x, p1, "-b", x, p2, "g", x, p3, "-r", x, p4, "-m", x, p5, "-k") %! grid on %! ylim ([0, 1]) %! xlim ([0, 8]) %! legend ({"ν = 0, σ = 0.5", "ν = 0, σ = 2", "ν = 0, σ = 3", ... %! "ν = 2, σ = 2", "ν = 4, σ = 2"}, "location", "southeast") %! title ("Rician CDF") %! xlabel ("values in x") %! ylabel ("probability") ## Test output %!test %! x = 0:0.5:2.5; %! nu = 1:6; %! p = ricecdf (x, nu, 1); %! expected_p = [0.0000, 0.0179, 0.0108, 0.0034, 0.0008, 0.0001]; %! assert (p, expected_p, 0.001); %!test %! x = 0:0.5:2.5; %! sigma = 1:6; %! p = ricecdf (x, 1, sigma); %! expected_p = [0.0000, 0.0272, 0.0512, 0.0659, 0.0754, 0.0820]; %! assert (p, expected_p, 0.001); %!test %! x = 0:0.5:2.5; %! p = ricecdf (x, 0, 1); %! expected_p = [0.0000, 0.1175, 0.3935, 0.6753, 0.8647, 0.9561]; %! assert (p, expected_p, 0.001); %!test %! x = 0:0.5:2.5; %! p = ricecdf (x, 1, 1); %! expected_p = [0.0000, 0.0735, 0.2671, 0.5120, 0.7310, 0.8791]; %! assert (p, expected_p, 0.001); %!shared x, p %! x = [-1, 0, 1, 2, Inf]; %! p = [0, 0, 0.26712019620318, 0.73098793996409, 1]; %!assert (ricecdf (x, 1, 1), p, 1e-14) %!assert (ricecdf (x, 1, 1, "upper"), 1 - p, 1e-14) ## Test input validation %!error ricecdf () %!error ricecdf (1) %!error ricecdf (1, 2) %!error ricecdf (1, 2, 3, "uper") %!error ricecdf (1, 2, 3, 4) %!error ... %! ricecdf (ones (3), ones (2), ones (2)) %!error ... %! ricecdf (ones (2), ones (3), ones (2)) %!error ... %! ricecdf (ones (2), ones (2), ones (3)) %!error ricecdf (i, 2, 3) %!error ricecdf (2, i, 3) %!error ricecdf (2, 2, i) statistics-release-1.6.3/inst/dist_fun/riceinv.m000066400000000000000000000113531456127120000217370ustar00rootroot00000000000000## Copyright (C) 2024 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{x} =} riceinv (@var{p}, @var{nu}, @var{sigma}) ## ## Inverse of the Rician distribution (iCDF). ## ## For each element of @var{p}, compute the quantile (the inverse of the CDF) ## of the Rician distribution with with with non-centrality (distance) parameter ## @var{nu} and scale parameter @var{sigma}. The size of @var{x} is the common ## size of @var{x}, @var{nu}, and @var{sigma}. A scalar input functions as a ## constant matrix of the same size as the other inputs. ## ## Further information about the Rician distribution can be found at ## @url{https://en.wikipedia.org/wiki/Rice_distribution} ## ## @seealso{ricecdf, ricepdf, ricernd, ricefit, ricelike, ricestat} ## @end deftypefn function x = riceinv (p, nu, sigma) ## Check for valid number of input arguments if (nargin < 3) error ("riceinv: function called with too few input arguments."); endif ## Check for common size of P, NU, and B if (! isscalar (p) || ! isscalar (nu) || ! isscalar (sigma)) [retval, p, nu, sigma] = common_size (p, nu, sigma); if (retval > 0) error ("riceinv: P, NU, and B must be of common size or scalars."); endif endif ## Check for P, NU, and B being reals if (iscomplex (p) || iscomplex (nu) || iscomplex (sigma)) error ("riceinv: P, NU, and B must not be complex."); endif ## Check for class type if (isa (p, "single") || isa (nu, "single") || isa (sigma, "single")) x = zeros (size (p), "single"); else x = zeros (size (p)); endif k = nu < 0 | sigma <= 0 | p < 0 | p > 1 | ... isnan (p) | isnan (nu) | isnan (sigma); x(k) = NaN; k = ! k; x(k) = sigma(k) .* sqrt (ncx2inv (p(k), 2, (nu(k) ./ sigma(k)) .^ 2)); endfunction %!demo %! ## Plot various iCDFs from the Beta distribution %! p = 0.001:0.001:0.999; %! x1 = riceinv (p, 0, 1); %! x2 = riceinv (p, 0.5, 1); %! x3 = riceinv (p, 1, 1); %! x4 = riceinv (p, 2, 1); %! x5 = riceinv (p, 4, 1); %! plot (p, x1, "-b", p, x2, "-g", p, x3, "-r", p, x4, "-m", p, x5, "-k") %! grid on %! legend ({"ν = 0, σ = 1", "ν = 0.5, σ = 1", "ν = 1, σ = 1", ... %! "ν = 2, σ = 1", "ν = 4, σ = 1"}, "location", "northwest") %! title ("Beta iCDF") %! xlabel ("probability") %! ylabel ("values in x") ## Test output %!shared p %! p = [-1 0 0.75 1 2]; %!assert (riceinv (p, ones (1,5), 2*ones (1,5)), [NaN 0 3.5354 Inf NaN], 1e-4) %!assert (riceinv (p, 1, 2*ones (1,5)), [NaN 0 3.5354 Inf NaN], 1e-4) %!assert (riceinv (p, ones (1,5), 2), [NaN 0 3.5354 Inf NaN], 1e-4) %!assert (riceinv (p, [1 0 NaN 1 1], 2), [NaN 0 NaN Inf NaN]) %!assert (riceinv (p, 1, 2*[1 0 NaN 1 1]), [NaN NaN NaN Inf NaN]) %!assert (riceinv ([p(1:2) NaN p(4:5)], 1, 2), [NaN 0 NaN Inf NaN]) ## Test class of input preserved %!assert (riceinv ([p, NaN], 1, 2), [NaN 0 3.5354 Inf NaN NaN], 1e-4) %!assert (riceinv (single ([p, NaN]), 1, 2), ... %! single ([NaN 0 3.5354 Inf NaN NaN]), 1e-4) %!assert (riceinv ([p, NaN], single (1), 2), ... %! single ([NaN 0 3.5354 Inf NaN NaN]), 1e-4) %!assert (riceinv ([p, NaN], 1, single (2)), ... %! single ([NaN 0 3.5354 Inf NaN NaN]), 1e-4) ## Test input validation %!error riceinv () %!error riceinv (1) %!error riceinv (1,2) %!error riceinv (1,2,3,4) %!error ... %! riceinv (ones (3), ones (2), ones (2)) %!error ... %! riceinv (ones (2), ones (3), ones (2)) %!error ... %! riceinv (ones (2), ones (2), ones (3)) %!error riceinv (i, 2, 2) %!error riceinv (2, i, 2) %!error riceinv (2, 2, i) statistics-release-1.6.3/inst/dist_fun/ricepdf.m000066400000000000000000000115221456127120000217120ustar00rootroot00000000000000## Copyright (C) 2024 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{y} =} ricepdf (@var{x}, @var{nu}, @var{sigma}) ## ## Rician probability density function (PDF). ## ## For each element of @var{x}, compute the probability density function (PDF) ## of the Rician distribution with with non-centrality (distance) parameter ## @var{nu} and scale parameter @var{sigma}. The size of @var{y} is the common ## size of @var{x}, @var{nu}, and @var{sigma}. A scalar input functions as a ## constant matrix of the same size as the other inputs. ## ## Further information about the Rician distribution can be found at ## @url{https://en.wikipedia.org/wiki/Rice_distribution} ## ## @seealso{ricecdf, riceinv, ricernd, ricefit, ricelike, ricestat} ## @end deftypefn function y = ricepdf (x, nu, sigma) ## Check for valid number of input arguments if (nargin < 3) error ("ricepdf: function called with too few input arguments."); endif ## Check for common size of X, NU, and SIGMA if (! isscalar (x) || ! isscalar (nu) || ! isscalar (sigma)) [retval, x, nu, sigma] = common_size (x, nu, sigma); if (retval > 0) error ("ricepdf: X, NU, and SIGMA must be of common size or scalars."); endif endif ## Check for X, NU, and SIGMA being reals if (iscomplex (x) || iscomplex (nu) || iscomplex (sigma)) error ("ricepdf: X, NU, and SIGMA must not be complex."); endif ## Check for class type if (isa (x, "single") || isa (nu, "single") || isa (sigma, "single")); y = zeros (size (x), "single"); else y = zeros (size (x)); endif k = nu < 0 | sigma <= 0 | x < 0 | isnan (x) | isnan (nu) | isnan (sigma); y(k) = NaN; k = ! k; ## Do the math x_k = x(k); n_k = nu(k); s_sq = sigma(k) .^ 2; x_s2 = x_k ./ s_sq; xnsq = (x_k .^ 2 + n_k .^ 2) ./ (2 .* s_sq); epxt = xnsq - x_s2 .* n_k; term = exp (-epxt); y(k) = x_s2 .* term .* besseli (0, x_s2 .* n_k, 1); ## Fix arithmetic overflow due to exponent y(epxt > (log(realmax(class(y))))) = 0; endfunction %!demo %! ## Plot various PDFs from the Rician distribution %! x = 0:0.01:8; %! y1 = ricepdf (x, 0, 1); %! y2 = ricepdf (x, 0.5, 1); %! y3 = ricepdf (x, 1, 1); %! y4 = ricepdf (x, 2, 1); %! y5 = ricepdf (x, 4, 1); %! plot (x, y1, "-b", x, y2, "-g", x, y3, "-r", x, y4, "-m", x, y5, "-k") %! grid on %! ylim ([0, 0.65]) %! xlim ([0, 8]) %! legend ({"ν = 0, σ = 1", "ν = 0.5, σ = 1", "ν = 1, σ = 1", ... %! "ν = 2, σ = 1", "ν = 4, σ = 1"}, "location", "northeast") %! title ("Rician PDF") %! xlabel ("values in x") %! ylabel ("density") ## Test output %!shared x, y %! x = [-1 0 0.5 1 2]; %! y = [NaN 0 0.1073 0.1978 0.2846]; %!assert (ricepdf (x, ones (1, 5), 2 * ones (1, 5)), y, 1e-4) %!assert (ricepdf (x, 1, 2 * ones (1, 5)), y, 1e-4) %!assert (ricepdf (x, ones (1, 5), 2), y, 1e-4) %!assert (ricepdf (x, [0 NaN 1 1 1], 2), [NaN NaN y(3:5)], 1e-4) %!assert (ricepdf (x, 1, 2 * [0 NaN 1 1 1]), [NaN NaN y(3:5)], 1e-4) %!assert (ricepdf ([x, NaN], 1, 2), [y, NaN], 1e-4) ## Test class of input preserved %!assert (ricepdf (single ([x, NaN]), 1, 2), single ([y, NaN]), 1e-4) %!assert (ricepdf ([x, NaN], single (1), 2), single ([y, NaN]), 1e-4) %!assert (ricepdf ([x, NaN], 1, single (2)), single ([y, NaN]), 1e-4) ## Test input validation %!error ricepdf () %!error ricepdf (1) %!error ricepdf (1,2) %!error ricepdf (1,2,3,4) %!error ... %! ricepdf (ones (3), ones (2), ones (2)) %!error ... %! ricepdf (ones (2), ones (3), ones (2)) %!error ... %! ricepdf (ones (2), ones (2), ones (3)) %!error ricepdf (i, 2, 2) %!error ricepdf (2, i, 2) %!error ricepdf (2, 2, i) statistics-release-1.6.3/inst/dist_fun/ricernd.m000066400000000000000000000163641456127120000217350ustar00rootroot00000000000000## Copyright (C) 2024 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{r} =} ricernd (@var{nu}, @var{sigma}) ## @deftypefnx {statistics} {@var{r} =} ricernd (@var{nu}, @var{sigma}, @var{rows}) ## @deftypefnx {statistics} {@var{r} =} ricernd (@var{nu}, @var{sigma}, @var{rows}, @var{cols}, @dots{}) ## @deftypefnx {statistics} {@var{r} =} ricernd (@var{nu}, @var{sigma}, [@var{sz}]) ## ## Random arrays from the Rician distribution. ## ## @code{@var{r} = ricernd (@var{nu}, @var{sigma})} returns an array of random ## numbers chosen from the Rician distribution with shape parameters @var{nu} ## and @var{sigma}. The size of @var{r} is the common size of @var{nu} and ## @var{sigma}. A scalar input functions as a constant matrix of the same size ## as the other inputs. ## ## When called with a single size argument, @code{ricernd} returns a square ## matrix with the dimension specified. When called with more than one scalar ## argument, the first two arguments are taken as the number of rows and columns ## and any further arguments specify additional matrix dimensions. The size may ## also be specified with a row vector of dimensions, @var{sz}. ## ## Further information about the Rician distribution can be found at ## @url{https://en.wikipedia.org/wiki/Rice_distribution} ## ## @seealso{ricecdf, riceinv, ricepdf, ricefit, ricelike, ricestat} ## @end deftypefn function r = ricernd (nu, sigma, varargin) ## Check for valid number of input arguments if (nargin < 2) error ("ricernd: function called with too few input arguments."); endif ## Check for common size of NU and SIGMA if (! isscalar (nu) || ! isscalar (sigma)) [retval, nu, sigma] = common_size (nu, sigma); if (retval > 0) error ("ricernd: NU and SIGMA must be of common size or scalars."); endif endif ## Check for NU and SIGMA being reals if (iscomplex (nu) || iscomplex (sigma)) error ("ricernd: NU and SIGMA must not be complex."); endif ## Parse and check SIZE arguments if (nargin == 2) sz = size (nu); elseif (nargin == 3) if (isscalar (varargin{1}) && varargin{1} >= 0 ... && varargin{1} == fix (varargin{1})) sz = [varargin{1}, varargin{1}]; elseif (isrow (varargin{1}) && all (varargin{1} >= 0) ... && all (varargin{1} == fix (varargin{1}))) sz = varargin{1}; elseif error (strcat (["ricernd: SZ must be nu scalar or nu row vector"], ... [" of non-negative integers."])); endif elseif (nargin > 3) posint = cellfun (@(x) (! isscalar (x) || x < 0 || x != fix (x)), varargin); if (any (posint)) error ("ricernd: dimensions must be non-negative integers."); endif sz = [varargin{:}]; endif ## Check that parameters match requested dimensions in size if (! isscalar (nu) && ! isequal (size (nu), sz)) error ("ricernd: NU and SIGMA must be scalars or of size SZ."); endif ## Check for class type if (isa (nu, "single") || isa (sigma, "single")) cls = "single"; else cls = "double"; endif ## Return NaNs for out of range values of NU and SIGMA nu(nu < 0) = NaN; sigma(sigma <= 0) = NaN; ## Force NU and SIGMA into the same size as SZ (if necessary) if (isscalar (nu)) nu = repmat (nu, sz); endif if (isscalar (sigma)) sigma = repmat (sigma, sz); endif ## Generate random sample from the Rician distribution r = sigma .* sqrt (ncx2rnd (2, (nu ./ sigma) .^ 2, sz)); ## Cast to appropriate class r = cast (r, cls); endfunction ## Test output %!assert (size (ricernd (2, 1/2)), [1 1]) %!assert (size (ricernd (2 * ones (2, 1), 1/2)), [2, 1]) %!assert (size (ricernd (2 * ones (2, 2), 1/2)), [2, 2]) %!assert (size (ricernd (2, 1/2 * ones (2, 1))), [2, 1]) %!assert (size (ricernd (1, 1/2 * ones (2, 2))), [2, 2]) %!assert (size (ricernd (ones (2, 1), 1)), [2, 1]) %!assert (size (ricernd (ones (2, 2), 1)), [2, 2]) %!assert (size (ricernd (2, 1/2, 3)), [3, 3]) %!assert (size (ricernd (1, 1, [4, 1])), [4, 1]) %!assert (size (ricernd (1, 1, 4, 1)), [4, 1]) %!assert (size (ricernd (1, 1, 4, 1, 5)), [4, 1, 5]) %!assert (size (ricernd (1, 1, 0, 1)), [0, 1]) %!assert (size (ricernd (1, 1, 1, 0)), [1, 0]) %!assert (size (ricernd (1, 1, 1, 2, 0, 5)), [1, 2, 0, 5]) ## Test class of input preserved %!assert (class (ricernd (1, 1)), "double") %!assert (class (ricernd (1, single (0))), "single") %!assert (class (ricernd (1, single ([0, 0]))), "single") %!assert (class (ricernd (1, single (1), 2)), "single") %!assert (class (ricernd (1, single ([1, 1]), 1, 2)), "single") %!assert (class (ricernd (single (1), 1, 2)), "single") %!assert (class (ricernd (single ([1, 1]), 1, 1, 2)), "single") ## Test input validation %!error ricernd () %!error ricernd (1) %!error ... %! ricernd (ones (3), ones (2)) %!error ... %! ricernd (ones (2), ones (3)) %!error ricernd (i, 2) %!error ricernd (1, i) %!error ... %! ricernd (1, 1/2, -1) %!error ... %! ricernd (1, 1/2, 1.2) %!error ... %! ricernd (1, 1/2, ones (2)) %!error ... %! ricernd (1, 1/2, [2 -1 2]) %!error ... %! ricernd (1, 1/2, [2 0 2.5]) %!error ... %! ricernd (1, 1/2, 2, -1, 5) %!error ... %! ricernd (1, 1/2, 2, 1.5, 5) %!error ... %! ricernd (2, 1/2 * ones (2), 3) %!error ... %! ricernd (2, 1/2 * ones (2), [3, 2]) %!error ... %! ricernd (2, 1/2 * ones (2), 3, 2) %!error ... %! ricernd (2 * ones (2), 1/2, 3) %!error ... %! ricernd (2 * ones (2), 1/2, [3, 2]) %!error ... %! ricernd (2 * ones (2), 1/2, 3, 2) statistics-release-1.6.3/inst/dist_fun/tcdf.m000066400000000000000000000206431456127120000212220ustar00rootroot00000000000000## Copyright (C) 1995-2016 Kurt Hornik ## Copyright (C) 2012 Rik Wehbring ## Copyright (C) 2013-2017 Julien Bect ## Copyright (C) 2022-2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{p} =} tcdf (@var{x}, @var{df}) ## @deftypefnx {statistics} {@var{p} =} tcdf (@var{x}, @var{df}, @qcode{"upper"}) ## ## Student's T cumulative distribution function (CDF). ## ## For each element of @var{x}, compute the cumulative distribution function ## (CDF) of the Student's T distribution with @var{df} degrees of freedom. The ## size of @var{p} is the common size of @var{x} and @var{df}. A scalar input ## functions as a constant matrix of the same size as the other input. ## ## @code{@var{p} = tcdf (@var{x}, @var{df}, "upper")} computes the upper tail ## probability of the Student's T distribution with @var{df} degrees of freedom, ## at the values in @var{x}. ## ## Further information about the Student's T distribution can be found at ## @url{https://en.wikipedia.org/wiki/Student%27s_t-distribution} ## ## @seealso{tinv, tpdf, trnd, tstat} ## @end deftypefn function p = tcdf (x, df, uflag) ## Check for valid number of input arguments if (nargin < 2) error ("tcdf: function called with too few input arguments."); endif ## Check for "upper" flag if (nargin > 2 && strcmpi (uflag, "upper")) x = -x; elseif (nargin > 2 && ! strcmpi (uflag, "upper")) error ("tcdf: invalid argument for upper tail."); endif ## Check for common size of X and DF if (! isscalar (x) || ! isscalar (df)) [err, x, df] = common_size (x, df); if (err > 0) error ("tcdf: X and DF must be of common size or scalars."); endif endif ## Check for X and DF being reals if (iscomplex (x) || iscomplex (df)) error ("tcdf: X and DF must not be complex."); endif ## Check for class type if (isa (x, "single") || isa (df, "single")) p = zeros (size (x), "single"); else p = zeros (size (x)); endif ## Check for NaNs or DF <= 0 is_nan = isnan (x) | ! (df > 0); p(is_nan) = NaN; ## Check for Inf where DF > 0 (for -Inf is already 0) k = (x == Inf) & (df > 0); p(k) = 1; ## Find finite values in X where 0 < DF < Inf k = isfinite (x) & (df > 0) & (df < Inf); ## Process more efficiently small positive integer DF up to 1e4 ks = k & (fix (df) == df) & (df <= 1e4); if any (ks(:)) if (isscalar (df)) p(ks) = tcdf_integer_df (x(ks), df); return else vu = unique (df(ks)); for i = 1:numel (vu) ki = k & (df == vu(i)); p(ki) = tcdf_integer_df (x(ki), vu(i)); endfor endif endif ## Proccess remaining values for DF (non-integers and > 1e4) except DF == Inf k &= ! ks; ## Distinguish between small and big abs(x) xx = x .^ 2; x_big_abs = (xx > df); ## Deal with the case "abs(x) big" kk = k & x_big_abs; if (isscalar (df)) p(kk) = betainc (df ./ (df + xx(kk)), df/2, 1/2) / 2; else p(kk) = betainc (df(kk) ./ (df(kk) + xx(kk)), df(kk)/2, 1/2) / 2; endif ## Deal with the case "abs(x) small" kk = k & ! x_big_abs; if (isscalar (df)) p(kk) = 0.5 * (1 - betainc (xx(kk) ./ (df + xx(kk)), 1/2, df/2)); else p(kk) = 0.5 * (1 - betainc (xx(kk) ./ (df(kk) + xx(kk)), 1/2, df(kk)/2)); endif ## For x > 0, F(x) = 1 - F(-|x|). k &= (x > 0); if (any (k(:))) p(k) = 1 - p(k); endif ## Special case for Cauchy distribution ## Use acot(-x) instead of the usual (atan x)/pi + 0.5 to avoid roundoff error xpos = (x > 0); c = (df == 1); p(c) = xpos(c) + acot (-x(c)) / pi; ## Special case for DF == Inf k = isfinite (x) & (df == Inf); p(k) = normcdf (x(k)); ## Make the result exact for the median p(x == 0 & ! is_nan) = 0.5; endfunction ## Compute the t distribution CDF efficiently (without calling betainc) ## for small positive integer DF up to 1e4 function p = tcdf_integer_df (x, df) if (df == 1) p = 0.5 + atan(x)/pi; elseif (df == 2) p = 0.5 + x ./ (2 * sqrt(2 + x .^ 2)); else xs = x ./ sqrt(df); xxf = 1 ./ (1 + xs .^ 2); u = s = 1; if mod (df, 2) ## odd DF m = (df - 1) / 2; for i = 2:m u .*= (1 - 1/(2*i - 1)) .* xxf; s += u; endfor p = 0.5 + (xs .* xxf .* s + atan(xs)) / pi; else ## even DF m = df / 2; for i = 1:(m - 1) u .*= (1 - 1/(2*i)) .* xxf; s += u; endfor p = 0.5 + (xs .* sqrt(xxf) .* s) / 2; endif endif endfunction %!demo %! ## Plot various CDFs from the Student's T distribution %! x = -5:0.01:5; %! p1 = tcdf (x, 1); %! p2 = tcdf (x, 2); %! p3 = tcdf (x, 5); %! p4 = tcdf (x, Inf); %! plot (x, p1, "-b", x, p2, "-g", x, p3, "-r", x, p4, "-m") %! grid on %! xlim ([-5, 5]) %! ylim ([0, 1]) %! legend ({"df = 1", "df = 2", ... %! "df = 5", 'df = \infty'}, "location", "southeast") %! title ("Student's T CDF") %! xlabel ("values in x") %! ylabel ("probability") ## Test output %!shared x,y %! x = [-Inf 0 1 Inf]; %! y = [0 1/2 3/4 1]; %!assert (tcdf (x, ones (1,4)), y, eps) %!assert (tcdf (x, 1), y, eps) %!assert (tcdf (x, [0 1 NaN 1]), [NaN 1/2 NaN 1], eps) %!assert (tcdf ([x(1:2) NaN x(4)], 1), [y(1:2) NaN y(4)], eps) %!assert (tcdf (2, 3, "upper"), 0.0697, 1e-4) %!assert (tcdf (205, 5, "upper"), 2.6206e-11, 1e-14) ## Test class of input preserved %!assert (tcdf ([x, NaN], 1), [y, NaN], eps) %!assert (tcdf (single ([x, NaN]), 1), single ([y, NaN]), eps ("single")) %!assert (tcdf ([x, NaN], single (1)), single ([y, NaN]), eps ("single")) ## Test input validation %!error tcdf () %!error tcdf (1) %!error tcdf (1, 2, "uper") %!error tcdf (1, 2, 3) %!error ... %! tcdf (ones (3), ones (2)) %!error ... %! tcdf (ones (3), ones (2)) %!error ... %! tcdf (ones (3), ones (2), "upper") %!error tcdf (i, 2) %!error tcdf (2, i) ## Check some reference values %!shared tol_rel %! tol_rel = 10 * eps; ## check accuracy for small positive values %!assert (tcdf (10^(-10), 2.5), 0.50000000003618087, -tol_rel) %!assert (tcdf (10^(-11), 2.5), 0.50000000000361809, -tol_rel) %!assert (tcdf (10^(-12), 2.5), 0.50000000000036181, -tol_rel) %!assert (tcdf (10^(-13), 2.5), 0.50000000000003618, -tol_rel) %!assert (tcdf (10^(-14), 2.5), 0.50000000000000362, -tol_rel) %!assert (tcdf (10^(-15), 2.5), 0.50000000000000036, -tol_rel) %!assert (tcdf (10^(-16), 2.5), 0.50000000000000004, -tol_rel) ## check accuracy for large negative values %!assert (tcdf (-10^1, 2.5), 2.2207478836537124e-03, -tol_rel) %!assert (tcdf (-10^2, 2.5), 7.1916492116661878e-06, -tol_rel) %!assert (tcdf (-10^3, 2.5), 2.2747463948307452e-08, -tol_rel) %!assert (tcdf (-10^4, 2.5), 7.1933970159922115e-11, -tol_rel) %!assert (tcdf (-10^5, 2.5), 2.2747519231756221e-13, -tol_rel) ## # Reference values obtained using Python 2.7.4 and mpmath 0.17 ## ## from mpmath import * ## ## mp.dps = 100 ## ## def F(x_in, nu_in): ## x = mpf(x_in); ## nu = mpf(nu_in); ## t = nu / (nu + x*x) ## a = nu / 2 ## b = mpf(0.5) ## F = betainc(a, b, 0, t, regularized=True) / 2 ## if (x > 0): ## F = 1 - F ## return F ## ## nu = 2.5 ## ## for i in range(1, 6): ## x = - power(mpf(10), mpf(i)) ## print "%%!assert (tcdf (-10^%d, 2.5), %s, -eps)" \ ## % (i, nstr(F(x, nu), 17)) ## ## for i in range(10, 17): ## x = power(mpf(10), -mpf(i)) ## print "%%!assert (tcdf (10^(-%d), 2.5), %s, -eps)" \ ## % (i, nstr(F(x, nu), 17)) statistics-release-1.6.3/inst/dist_fun/tinv.m000066400000000000000000000112661456127120000212630ustar00rootroot00000000000000## Copyright (C) 2012 Rik Wehbring ## Copyright (C) 1995-2016 Kurt Hornik ## Copyright (C) 2022-2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{x} =} tinv (@var{p}, @var{df}) ## ## Inverse of the Student's T cumulative distribution function (iCDF). ## ## For each element of @var{p}, compute the quantile (the inverse of the CDF) of ## the Student's T distribution with @var{df} degrees of freedom. The size of ## @var{x} is the common size of @var{x} and @var{df}. A scalar input functions ## as a constant matrix of the same size as the other input. ## ## This function is analogous to looking in a table for the t-value of a ## single-tailed distribution. For very large @var{df} (>10000), the inverse of ## the standard normal distribution is used. ## ## Further information about the Student's T distribution can be found at ## @url{https://en.wikipedia.org/wiki/Student%27s_t-distribution} ## ## @seealso{tcdf, tpdf, trnd, tstat} ## @end deftypefn function x = tinv (p, df) ## Check for valid number of input arguments if (nargin < 2) error ("tinv: function called with too few input arguments."); endif ## Check for common size of P and DF if (! isscalar (p) || ! isscalar (df)) [retval, p, df] = common_size (p, df); if (retval > 0) error ("tinv: P and DF must be of common size or scalars."); endif endif ## Check for P and DF being reals if (iscomplex (p) || iscomplex (df)) error ("tinv: P and DF must not be complex."); endif ## Check for class type if (isa (p, "single") || isa (df, "single")) x = NaN (size (p), "single"); else x = NaN (size (p)); endif k = (p == 0) & (df > 0); x(k) = -Inf; k = (p == 1) & (df > 0); x(k) = Inf; if (isscalar (df)) k = (p > 0) & (p < 1); if ((df > 0) && (df < 10000)) x(k) = (sign (p(k) - 1/2) .* sqrt (df * (1 ./ betainv (2*min (p(k), 1 - p(k)), df/2, 1/2) - 1))); elseif (df >= 10000) ## For large df, use the quantiles of the standard normal x(k) = -sqrt (2) * erfcinv (2 * p(k)); endif else k = (p > 0) & (p < 1) & (df > 0) & (df < 10000); x(k) = (sign (p(k) - 1/2) .* sqrt (df(k) .* (1 ./ betainv (2*min (p(k), 1 - p(k)), df(k)/2, 1/2) - 1))); ## For large df, use the quantiles of the standard normal k = (p > 0) & (p < 1) & (df >= 10000); x(k) = -sqrt (2) * erfcinv (2 * p(k)); endif endfunction %!demo %! ## Plot various iCDFs from the Student's T distribution %! p = 0.001:0.001:0.999; %! x1 = tinv (p, 1); %! x2 = tinv (p, 2); %! x3 = tinv (p, 5); %! x4 = tinv (p, Inf); %! plot (p, x1, "-b", p, x2, "-g", p, x3, "-r", p, x4, "-m") %! grid on %! xlim ([0, 1]) %! ylim ([-5, 5]) %! legend ({"df = 1", "df = 2", ... %! "df = 5", 'df = \infty'}, "location", "northwest") %! title ("Student's T iCDF") %! xlabel ("probability") %! ylabel ("values in x") ## Test output %!shared p %! p = [-1 0 0.5 1 2]; %!assert (tinv (p, ones (1,5)), [NaN -Inf 0 Inf NaN]) %!assert (tinv (p, 1), [NaN -Inf 0 Inf NaN], eps) %!assert (tinv (p, [1 0 NaN 1 1]), [NaN NaN NaN Inf NaN], eps) %!assert (tinv ([p(1:2) NaN p(4:5)], 1), [NaN -Inf NaN Inf NaN]) ## Test class of input preserved %!assert (tinv ([p, NaN], 1), [NaN -Inf 0 Inf NaN NaN], eps) %!assert (tinv (single ([p, NaN]), 1), single ([NaN -Inf 0 Inf NaN NaN]), eps ("single")) %!assert (tinv ([p, NaN], single (1)), single ([NaN -Inf 0 Inf NaN NaN]), eps ("single")) ## Test input validation %!error tinv () %!error tinv (1) %!error ... %! tinv (ones (3), ones (2)) %!error ... %! tinv (ones (2), ones (3)) %!error tinv (i, 2) %!error tinv (2, i) statistics-release-1.6.3/inst/dist_fun/tlscdf.m000066400000000000000000000143331456127120000215600ustar00rootroot00000000000000## Copyright (C) 2024 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{p} =} tlscdf (@var{x}, @var{mu}, @var{sigma}, @var{df}) ## @deftypefnx {statistics} {@var{p} =} tlscdf (@var{x}, @var{mu}, @var{sigma}, @var{df}, @qcode{"upper"}) ## ## Location-scale Student's T cumulative distribution function (CDF). ## ## For each element of @var{x}, compute the cumulative distribution function ## (CDF) of the location-scale Student's T distribution with location parameter ## @var{mu}, scale parameter @var{sigma}, and @var{df} degrees of freedom. The ## size of @var{p} is the common size of @var{x}, @var{mu}, @var{sigma}, and ## @var{df}. A scalar input functions as a constant matrix of the same size as ## the other inputs. ## ## @code{@var{p} = tlscdf (@var{x}, @var{mu}, @var{sigma}, @var{df}, "upper")} ## computes the upper tail probability of the location-scale Student's T ## distribution with parameters @var{mu}, @var{sigma}, and @var{df}, at the ## values in @var{x}. ## ## Further information about the location-scale Student's T distribution can be ## found at @url{https://en.wikipedia.org/wiki/Student%27s_t-distribution#Location-scale_t_distribution} ## ## @seealso{tlsinv, tlspdf, tlsrnd, tlsfit, tlslike, tlsstat} ## @end deftypefn function p = tlscdf (x, mu, sigma, df, uflag) ## Check for valid number of input arguments if (nargin < 4) error ("tlscdf: function called with too few input arguments."); endif ## Check for "upper" flag upper = false; if (nargin > 4 && strcmpi (uflag, "upper")) upper = true; elseif (nargin > 4 && ! strcmpi (uflag, "upper")) error ("tlscdf: invalid argument for upper tail."); endif ## Check for common size of X, MU, SIGMA, and DF if (! isscalar (x) || ! isscalar (mu) || ! isscalar (sigma) || ! isscalar (df)) [err, x, mu, sigma, df] = common_size (x, mu, sigma, df); if (err > 0) error ("tlscdf: X, MU, SIGMA, and DF must be of common size or scalars."); endif endif ## Check for X, MU, SIGMA, and DF being reals if (iscomplex (x) || iscomplex (mu) || iscomplex (sigma) || iscomplex (df)) error ("tlscdf: X, MU, SIGMA, and DF must not be complex."); endif ## Check for class type if (isa (x, "single") || isa (mu, "single") || isa (sigma, "single") || isa (df, "single")) cls = "single"; else cls = "double"; endif ## Force invalid SIGMA parameter to NaN sigma(sigma <= 0) = NaN; ## Call tcdf to do the work if (upper) p = tcdf ((x - mu) ./ sigma, df, "upper"); else p = tcdf ((x - mu) ./ sigma, df); endif ## Force class type p = cast (p, cls); endfunction %!demo %! ## Plot various CDFs from the location-scale Student's T distribution %! x = -8:0.01:8; %! p1 = tlscdf (x, 0, 1, 1); %! p2 = tlscdf (x, 0, 2, 2); %! p3 = tlscdf (x, 3, 2, 5); %! p4 = tlscdf (x, -1, 3, Inf); %! plot (x, p1, "-b", x, p2, "-g", x, p3, "-r", x, p4, "-m") %! grid on %! xlim ([-8, 8]) %! ylim ([0, 1]) %! legend ({"mu = 0, sigma = 1, df = 1", "mu = 0, sigma = 2, df = 2", ... %! "mu = 3, sigma = 2, df = 5", 'mu = -1, sigma = 3, df = \infty'}, ... %! "location", "northwest") %! title ("Location-scale Student's T CDF") %! xlabel ("values in x") %! ylabel ("probability") ## Test output %!shared x,y %! x = [-Inf 0 1 Inf]; %! y = [0 1/2 3/4 1]; %!assert (tlscdf (x, 0, 1, ones (1,4)), y, eps) %!assert (tlscdf (x, 0, 1, 1), y, eps) %!assert (tlscdf (x, 0, 1, [0 1 NaN 1]), [NaN 1/2 NaN 1], eps) %!assert (tlscdf ([x(1:2) NaN x(4)], 0, 1, 1), [y(1:2) NaN y(4)], eps) %!assert (tlscdf (2, 0, 1, 3, "upper"), 0.0697, 1e-4) %!assert (tlscdf (205, 0, 1, 5, "upper"), 2.6206e-11, 1e-14) ## Test class of input preserved %!assert (tlscdf ([x, NaN], 0, 1, 1), [y, NaN], eps) %!assert (tlscdf (single ([x, NaN]), 0, 1, 1), single ([y, NaN]), eps ("single")) %!assert (tlscdf ([x, NaN], single (0), 1, 1), single ([y, NaN]), eps ("single")) %!assert (tlscdf ([x, NaN], 0, single (1), 1), single ([y, NaN]), eps ("single")) %!assert (tlscdf ([x, NaN], 0, 1, single (1)), single ([y, NaN]), eps ("single")) ## Test input validation %!error tlscdf () %!error tlscdf (1) %!error tlscdf (1, 2) %!error tlscdf (1, 2, 3) %!error tlscdf (1, 2, 3, 4, "uper") %!error tlscdf (1, 2, 3, 4, 5) %!error ... %! tlscdf (ones (3), ones (2), 1, 1) %!error ... %! tlscdf (ones (3), 1, ones (2), 1) %!error ... %! tlscdf (ones (3), 1, 1, ones (2)) %!error ... %! tlscdf (ones (3), ones (2), 1, 1, "upper") %!error ... %! tlscdf (ones (3), 1, ones (2), 1, "upper") %!error ... %! tlscdf (ones (3), 1, 1, ones (2), "upper") %!error tlscdf (i, 2, 1, 1) %!error tlscdf (2, i, 1, 1) %!error tlscdf (2, 1, i, 1) %!error tlscdf (2, 1, 1, i) statistics-release-1.6.3/inst/dist_fun/tlsinv.m000066400000000000000000000116531456127120000216220ustar00rootroot00000000000000## Copyright (C) 2024 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{x} =} tlsinv (@var{p}, @var{mu}, @var{sigma}, @var{df}) ## ## Inverse of the location-scale Student's T cumulative distribution function (iCDF). ## ## For each element of @var{p}, compute the quantile (the inverse of the CDF) of ## the location-scale Student's T distribution with location parameter @var{mu}, ## scale parameter @var{sigma}, and @var{df} degrees of freedom. The size of ## @var{x} is the common size of @var{p}, @var{mu}, @var{sigma}, and @var{df}. ## A scalar input functions as a constant matrix of the same size as the other ## inputs. ## ## Further information about the location-scale Student's T distribution can be ## found at @url{https://en.wikipedia.org/wiki/Student%27s_t-distribution#Location-scale_t_distribution} ## ## @seealso{tlscdf, tlspdf, tlsrnd, tlsfit, tlslike, tlsstat} ## @end deftypefn function x = tlsinv (p, mu, sigma, df) ## Check for valid number of input arguments if (nargin < 4) error ("tlsinv: function called with too few input arguments."); endif ## Check for common size of P, MU, SIGMA, and DF if (! isscalar (p) || ! isscalar (mu) || ! isscalar (sigma) || ! isscalar (df)) [retval, p, mu, sigma, df] = common_size (p, mu, sigma, df); if (retval > 0) error ("tlsinv: P, MU, SIGMA, and DF must be of common size or scalars."); endif endif ## Check for P, MU, SIGMA, and DF being reals if (iscomplex (p) || iscomplex (mu) || iscomplex (sigma) || iscomplex (df)) error ("tlsinv: P, MU, SIGMA, and DF must not be complex."); endif ## Check for class type if (isa (p, "single") || isa (mu, "single") || isa (sigma, "single") || isa (df, "single")) cls = "single"; else cls = "double"; endif ## Force invalid SIGMA parameter to NaN sigma(sigma <= 0) = NaN; ## Call tinv to do the work x = tinv (p, df) .* sigma + mu; ## Force class type x = cast (x, cls); endfunction %!demo %! ## Plot various iCDFs from the location-scale Student's T distribution %! p = 0.001:0.001:0.999; %! x1 = tlsinv (p, 0, 1, 1); %! x2 = tlsinv (p, 0, 2, 2); %! x3 = tlsinv (p, 3, 2, 5); %! x4 = tlsinv (p, -1, 3, Inf); %! plot (p, x1, "-b", p, x2, "-g", p, x3, "-r", p, x4, "-m") %! grid on %! xlim ([0, 1]) %! ylim ([-8, 8]) %! legend ({"mu = 0, sigma = 1, df = 1", "mu = 0, sigma = 2, df = 2", ... %! "mu = 3, sigma = 2, df = 5", 'mu = -1, sigma = 3, df = \infty'}, ... %! "location", "southeast") %! title ("Location-scale Student's T iCDF") %! xlabel ("probability") %! ylabel ("values in x") ## Test output %!shared p %! p = [-1 0 0.5 1 2]; %!assert (tlsinv (p, 0, 1, ones (1,5)), [NaN -Inf 0 Inf NaN]) %!assert (tlsinv (p, 0, 1, 1), [NaN -Inf 0 Inf NaN], eps) %!assert (tlsinv (p, 0, 1, [1 0 NaN 1 1]), [NaN NaN NaN Inf NaN], eps) %!assert (tlsinv ([p(1:2) NaN p(4:5)], 0, 1, 1), [NaN -Inf NaN Inf NaN]) ## Test class of input preserved %!assert (class (tlsinv ([p, NaN], 0, 1, 1)), "double") %!assert (class (tlsinv (single ([p, NaN]), 0, 1, 1)), "single") %!assert (class (tlsinv ([p, NaN], single (0), 1, 1)), "single") %!assert (class (tlsinv ([p, NaN], 0, single (1), 1)), "single") %!assert (class (tlsinv ([p, NaN], 0, 1, single (1))), "single") ## Test input validation %!error tlsinv () %!error tlsinv (1) %!error tlsinv (1, 2) %!error tlsinv (1, 2, 3) %!error ... %! tlsinv (ones (3), ones (2), 1, 1) %!error ... %! tlsinv (ones (2), 1, ones (3), 1) %!error ... %! tlsinv (ones (2), 1, 1, ones (3)) %!error tlsinv (i, 2, 3, 4) %!error tlsinv (2, i, 3, 4) %!error tlsinv (2, 2, i, 4) %!error tlsinv (2, 2, 3, i) statistics-release-1.6.3/inst/dist_fun/tlspdf.m000066400000000000000000000120471456127120000215750ustar00rootroot00000000000000## Copyright (C) 2024 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{p} =} tlspdf (@var{x}, @var{mu}, @var{sigma}, @var{df}) ## ## Location-scale Student's T probability density function (PDF). ## ## For each element of @var{x}, compute the probability density function (PDF) ## of the location-scale Student's T distribution with location parameter ## @var{mu}, scale parameter @var{sigma}, and @var{df} degrees of freedom. The ## size of @var{y} is the common size of @var{x}, @var{mu}, @var{sigma}, and ## @var{df}. A scalar input functions as a constant matrix of the same size as ## the other inputs. ## ## Further information about the location-scale Student's T distribution can be ## found at @url{https://en.wikipedia.org/wiki/Student%27s_t-distribution#Location-scale_t_distribution} ## ## @seealso{tlscdf, tlspdf, tlsrnd, tlsfit, tlslike, tlsstat} ## @end deftypefn function y = tlspdf (x, mu, sigma, df) ## Check for valid number of input arguments if (nargin < 4) error ("tlspdf: function called with too few input arguments."); endif ## Check for common size of X, MU, SIGMA, and DF if (! isscalar (x) || ! isscalar (mu) || ! isscalar (sigma) || ! isscalar (df)) [err, x, mu, sigma, df] = common_size (x, mu, sigma, df); if (err > 0) error ("tlspdf: X, MU, SIGMA, and DF must be of common size or scalars."); endif endif ## Check for X, MU, SIGMA, and DF being reals if (iscomplex (x) || iscomplex (mu) || iscomplex (sigma) || iscomplex (df)) error ("tlspdf: X, MU, SIGMA, and DF must not be complex."); endif ## Check for class type if (isa (x, "single") || isa (mu, "single") || isa (sigma, "single") || isa (df, "single")) cls = "single"; else cls = "double"; endif ## Force invalid SIGMA parameter to NaN sigma(sigma <= 0) = NaN; ## Call tpdf to do the work y = tpdf ((x - mu) ./ sigma, df) ./ sigma; ## Force class type y = cast (y, cls); endfunction %!demo %! ## Plot various PDFs from the Student's T distribution %! x = -8:0.01:8; %! y1 = tlspdf (x, 0, 1, 1); %! y2 = tlspdf (x, 0, 2, 2); %! y3 = tlspdf (x, 3, 2, 5); %! y4 = tlspdf (x, -1, 3, Inf); %! plot (x, y1, "-b", x, y2, "-g", x, y3, "-r", x, y4, "-m") %! grid on %! xlim ([-8, 8]) %! ylim ([0, 0.41]) %! legend ({"mu = 0, sigma = 1, df = 1", "mu = 0, sigma = 2, df = 2", ... %! "mu = 3, sigma = 2, df = 5", 'mu = -1, sigma = 3, df = \infty'}, ... %! "location", "northwest") %! title ("Location-scale Student's T PDF") %! xlabel ("values in x") %! ylabel ("density") ## Test output %!test %! x = rand (10,1); %! y = 1./(pi * (1 + x.^2)); %! assert (tlspdf (x, 0, 1, 1), y, 5*eps); %! assert (tlspdf (x+5, 5, 1, 1), y, 5*eps); %! assert (tlspdf (x.*2, 0, 2, 1), y./2, 5*eps); %!shared x, y %! x = [-Inf 0 0.5 1 Inf]; %! y = 1./(pi * (1 + x.^2)); %!assert (tlspdf (x, 0, 1, ones (1,5)), y, eps) %!assert (tlspdf (x, 0, 1, 1), y, eps) %!assert (tlspdf (x, 0, 1, [0 NaN 1 1 1]), [NaN NaN y(3:5)], eps) %!assert (tlspdf (x, 0, 1, Inf), normpdf (x)) ## Test class of input preserved %!assert (class (tlspdf ([x, NaN], 1, 1, 1)), "double") %!assert (class (tlspdf (single ([x, NaN]), 1, 1, 1)), "single") %!assert (class (tlspdf ([x, NaN], single (1), 1, 1)), "single") %!assert (class (tlspdf ([x, NaN], 1, single (1), 1)), "single") %!assert (class (tlspdf ([x, NaN], 1, 1, single (1))), "single") ## Test input validation %!error tlspdf () %!error tlspdf (1) %!error tlspdf (1, 2) %!error tlspdf (1, 2, 3) %!error ... %! tlspdf (ones (3), ones (2), 1, 1) %!error ... %! tlspdf (ones (2), 1, ones (3), 1) %!error ... %! tlspdf (ones (2), 1, 1, ones (3)) %!error tlspdf (i, 2, 1, 1) %!error tlspdf (2, i, 1, 1) %!error tlspdf (2, 1, i, 1) %!error tlspdf (2, 1, 1, i) statistics-release-1.6.3/inst/dist_fun/tlsrnd.m000066400000000000000000000171551456127120000216140ustar00rootroot00000000000000## Copyright (C) 2012 Rik Wehbring ## Copyright (C) 1995-2016 Kurt Hornik ## Copyright (C) 2022-2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{r} =} tlsrnd (@var{mu}, @var{sigma}, @var{df}) ## @deftypefnx {statistics} {@var{r} =} tlsrnd (@var{mu}, @var{sigma}, @var{df}, @var{rows}) ## @deftypefnx {statistics} {@var{r} =} tlsrnd (@var{mu}, @var{sigma}, @var{df}, @var{rows}, @var{cols}, @dots{}) ## @deftypefnx {statistics} {@var{r} =} tlsrnd (@var{mu}, @var{sigma}, @var{df}, [@var{sz}]) ## ## Random arrays from the location-scale Student's T distribution. ## ## Return a matrix of random samples from the location-scale Student's T ## distribution with location parameter @var{mu}, scale parameter @var{sigma}, ## and @var{df} degrees of freedom. ## ## @code{@var{r} = tlsrnd (@var{df})} returns an array of random numbers chosen ## from the location-scale Student's T distribution with location parameter ## @var{mu}, scale parameter @var{sigma}, and @var{df} degrees of freedom. The ## size of @var{r} is the common size of @var{mu}, @var{sigma}, and @var{df}. A ## scalar input functions as a constant matrix of the same size as the other ## inputs. ## ## When called with a single size argument, @code{tlsrnd} returns a square ## matrix with the dimension specified. When called with more than one scalar ## argument, the first two arguments are taken as the number of rows and columns ## and any further arguments specify additional matrix dimensions. The size may ## also be specified with a row vector of dimensions, @var{sz}. ## ## Further information about the location-scale Student's T distribution can be ## found at @url{https://en.wikipedia.org/wiki/Student%27s_t-distribution#Location-scale_t_distribution} ## ## @seealso{tlscdf, tlsinv, tlspdf, tlsfit, tlslike, tlsstat} ## @end deftypefn function r = tlsrnd (mu, sigma, df, varargin) ## Check for valid number of input arguments if (nargin < 3) error ("tlsrnd: function called with too few input arguments."); endif ## Check for common size of MU, SIGMA, and DF if (! isscalar (mu) || ! isscalar (sigma) || ! isscalar (df)) [retval, mu, sigma, df] = common_size (mu, sigma, df); if (retval > 0) error ("tlsrnd: MU, SIGMA, and DF must be of common size or scalars."); endif endif ## Check for DF being real if (iscomplex (mu) || iscomplex (sigma) || iscomplex (df)) error ("tlsrnd: MU, SIGMA, and DF must not be complex."); endif ## Parse and check SIZE arguments if (nargin == 3) sz = size (df); elseif (nargin == 4) if (isscalar (varargin{1}) && varargin{1} >= 0 ... && varargin{1} == fix (varargin{1})) sz = [varargin{1}, varargin{1}]; elseif (isrow (varargin{1}) && all (varargin{1} >= 0) ... && all (varargin{1} == fix (varargin{1}))) sz = varargin{1}; elseif error (strcat (["tlsrnd: SZ must be a scalar or a row vector"], ... [" of non-negative integers."])); endif elseif (nargin > 4) posint = cellfun (@(x) (! isscalar (x) || x < 0 || x != fix (x)), varargin); if (any (posint)) error ("tlsrnd: dimensions must be non-negative integers."); endif sz = [varargin{:}]; endif ## Check that parameters match requested dimensions in size if (! isscalar (df) && ! isequal (size (df), sz)) error ("tlsrnd: MU, SIGMA, and DF must be scalar or of size SZ."); endif ## Check for class type if (isa (mu, "single") || isa (sigma, "single") || isa (df, "single")) cls = "single"; else cls = "double"; endif ## Call trnd to do the work r = mu + sigma .* trnd (df, sz); ## Force class type r = cast (r, cls); endfunction ## Test output %!assert (size (tlsrnd (1, 2, 3)), [1, 1]) %!assert (size (tlsrnd (ones (2,1), 2, 3)), [2, 1]) %!assert (size (tlsrnd (ones (2,2), 2, 3)), [2, 2]) %!assert (size (tlsrnd (1, 2, 3, 3)), [3, 3]) %!assert (size (tlsrnd (1, 2, 3, [4 1])), [4, 1]) %!assert (size (tlsrnd (1, 2, 3, 4, 1)), [4, 1]) %!assert (size (tlsrnd (1, 2, 3, 4, 1)), [4, 1]) %!assert (size (tlsrnd (1, 2, 3, 4, 1, 5)), [4, 1, 5]) %!assert (size (tlsrnd (1, 2, 3, 0, 1)), [0, 1]) %!assert (size (tlsrnd (1, 2, 3, 1, 0)), [1, 0]) %!assert (size (tlsrnd (1, 2, 3, 1, 2, 0, 5)), [1, 2, 0, 5]) %!assert (tlsrnd (1, 2, 0, 1, 1), NaN) %!assert (tlsrnd (1, 2, [0, 0, 0], [1, 3]), [NaN, NaN, NaN]) ## Test class of input preserved %!assert (class (tlsrnd (1, 2, 3)), "double") %!assert (class (tlsrnd (single (1), 2, 3)), "single") %!assert (class (tlsrnd (single ([1, 1]), 2, 3)), "single") %!assert (class (tlsrnd (1, single (2), 3)), "single") %!assert (class (tlsrnd (1, single ([2, 2]), 3)), "single") %!assert (class (tlsrnd (1, 2, single (3))), "single") %!assert (class (tlsrnd (1, 2, single ([3, 3]))), "single") ## Test input validation %!error tlsrnd () %!error tlsrnd (1) %!error tlsrnd (1, 2) %!error ... %! tlsrnd (ones (3), ones (2), 1) %!error ... %! tlsrnd (ones (2), 1, ones (3)) %!error ... %! tlsrnd (1, ones (2), ones (3)) %!error tlsrnd (i, 2, 3) %!error tlsrnd (1, i, 3) %!error tlsrnd (1, 2, i) %!error ... %! tlsrnd (1, 2, 3, -1) %!error ... %! tlsrnd (1, 2, 3, 1.2) %!error ... %! tlsrnd (1, 2, 3, ones (2)) %!error ... %! tlsrnd (1, 2, 3, [2 -1 2]) %!error ... %! tlsrnd (1, 2, 3, [2 0 2.5]) %!error ... %! tlsrnd (ones (2), 2, 3, ones (2)) %!error ... %! tlsrnd (1, 2, 3, 2, -1, 5) %!error ... %! tlsrnd (1, 2, 3, 2, 1.5, 5) %!error ... %! tlsrnd (ones (2,2), 2, 3, 3) %!error ... %! tlsrnd (1, ones (2,2), 3, 3) %!error ... %! tlsrnd (1, 2, ones (2,2), 3) %!error ... %! tlsrnd (1, 2, ones (2,2), [3, 3]) %!error ... %! tlsrnd (1, 2, ones (2,2), 2, 3) statistics-release-1.6.3/inst/dist_fun/tpdf.m000066400000000000000000000077171456127120000212460ustar00rootroot00000000000000## Copyright (C) 2012 Rik Wehbring ## Copyright (C) 1995-2016 Kurt Hornik ## Copyright (C) 2022-2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{p} =} tpdf (@var{x}, @var{df}) ## ## Student's T probability density function (PDF). ## ## For each element of @var{x}, compute the probability density function (PDF) ## of the Student's T distribution with @var{df} degrees of freedom. The size ## of @var{y} is the common size of @var{x} and @var{df}. A scalar input ## functions as a constant matrix of the same size as the other input. ## ## Further information about the Student's T distribution can be found at ## @url{https://en.wikipedia.org/wiki/Student%27s_t-distribution} ## ## @seealso{tcdf, tpdf, trnd, tstat} ## @end deftypefn function y = tpdf (x, df) ## Check for valid number of input arguments if (nargin < 2) error ("tpdf: function called with too few input arguments."); endif ## Check for common size of X and DF if (! isscalar (x) || ! isscalar (df)) [retval, x, df] = common_size (x, df); if (retval > 0) error ("tpdf: X and DF must be of common size or scalars."); endif endif ## Check for X and DF being reals if (iscomplex (x) || iscomplex (df)) error ("tpdf: X and DF must not be complex."); endif ## Check for class type if (isa (x, "single") || isa (df, "single")) y = zeros (size (x), "single"); else y = zeros (size (x)); endif k = isnan (x) | ! (df > 0); y(k) = NaN; k = isfinite (x) & (df > 0) & (df < Inf); kinf = isfinite (x) & isinf (df); if (any (k)) y(k) = exp (- (df(k) + 1) .* log (1 + x(k) .^ 2 ./ df(k)) / 2) ./ ... (sqrt (df(k)) .* beta (df(k)/2, 1/2)); endif if (any (kinf)) y(kinf) = normpdf (x(kinf)); endif endfunction %!demo %! ## Plot various PDFs from the Student's T distribution %! x = -5:0.01:5; %! y1 = tpdf (x, 1); %! y2 = tpdf (x, 2); %! y3 = tpdf (x, 5); %! y4 = tpdf (x, Inf); %! plot (x, y1, "-b", x, y2, "-g", x, y3, "-r", x, y4, "-m") %! grid on %! xlim ([-5, 5]) %! ylim ([0, 0.41]) %! legend ({"df = 1", "df = 2", ... %! "df = 5", 'df = \infty'}, "location", "northeast") %! title ("Student's T PDF") %! xlabel ("values in x") %! ylabel ("density") ## Test output %!test %! x = rand (10,1); %! y = 1./(pi * (1 + x.^2)); %! assert (tpdf (x, 1), y, 5*eps); %!shared x, y %! x = [-Inf 0 0.5 1 Inf]; %! y = 1./(pi * (1 + x.^2)); %!assert (tpdf (x, ones (1,5)), y, eps) %!assert (tpdf (x, 1), y, eps) %!assert (tpdf (x, [0 NaN 1 1 1]), [NaN NaN y(3:5)], eps) %!assert (tpdf (x, Inf), normpdf (x)) ## Test class of input preserved %!assert (tpdf ([x, NaN], 1), [y, NaN], eps) %!assert (tpdf (single ([x, NaN]), 1), single ([y, NaN]), eps ("single")) %!assert (tpdf ([x, NaN], single (1)), single ([y, NaN]), eps ("single")) ## Test input validation %!error tpdf () %!error tpdf (1) %!error ... %! tpdf (ones (3), ones (2)) %!error ... %! tpdf (ones (2), ones (3)) %!error tpdf (i, 2) %!error tpdf (2, i) statistics-release-1.6.3/inst/dist_fun/tricdf.m000066400000000000000000000151441456127120000215550ustar00rootroot00000000000000## Copyright (C) 1997-2015 Kurt Hornik ## Copyright (C) 2016 Dag Lyberg ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## Octave is free software; you can redistribute it and/or modify it ## under the terms of the GNU General Public License as published by ## the Free Software Foundation; either version 3 of the License, or (at ## your option) any later version. ## ## Octave is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with Octave; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{p} =} tricdf (@var{x}, @var{a}, @var{b}, @var{c}) ## @deftypefnx {statistics} {@var{p} =} tricdf (@var{x}, @var{a}, @var{b}, @var{c}, @qcode{"upper"}) ## ## Triangular cumulative distribution function (CDF). ## ## For each element of @var{x}, compute the cumulative distribution function ## (CDF) of the triangular distribution with parameters @var{a}, @var{b}, and ## @var{c} on the interval @qcode{[@var{a}, @var{b}]}. The size of @var{p} is ## the common size of the input arguments. A scalar input functions as a ## constant matrix of the same size as the other inputs. ## ## @code{@var{p} = tricdf (@var{x}, @var{a}, @var{b}, @var{c}, "upper")} ## computes the upper tail probability of the triangular distribution with ## parameters @var{a}, @var{b}, and @var{c}, at the values in @var{x}. ## ## Further information about the triangular distribution can be found at ## @url{https://en.wikipedia.org/wiki/Triangular_distribution} ## ## @seealso{triinv, tripdf, trirnd} ## @end deftypefn function p = tricdf (x, a, b, c, uflag) ## Check for valid number of input arguments if (nargin < 4) error ("tricdf: function called with too few input arguments."); endif ## Check for valid "upper" flag if (nargin > 4) if (! strcmpi (uflag, "upper")) error ("tricdf: invalid argument for upper tail."); else uflag = true; endif else uflag = false; endif ## Check for common size of A, B, and C if (! isscalar (x) || ! isscalar (a) || ! isscalar (b) || ! isscalar (c)) [retval, x, a, b, c] = common_size (x, a, b, c); if (retval > 0) error ("tricdf: X, A, B, and C must be of common size or scalars."); endif endif ## Check for X, BETA, and GAMMA being reals if (iscomplex (x) || iscomplex (a) || iscomplex (b) || iscomplex (c)) error ("tricdf: X, A, B, and C must not be complex."); endif ## Check for class type if (isa (x, "single") || isa (a, "single") || isa (b, "single") ... || isa (c, "single")) p = zeros (size (x), "single"); else p = zeros (size (x)); endif ## Force NaNs for out of range parameters. k = isnan (x) | ! (a < b) | ! (c >= a) | ! (c <= b); p(k) = NaN; ## Find valid values in parameters and data k = (a < b) & (a <= c) & (c <= b); k1 = (x <= a) & k; k2 = (x > a) & (x <= c) & k; k3 = (x > c) & (x < b) & k; k4 = (x >= b) & k; ## Compute triangular CDF if (uflag) p(k1) = 1; p(k2) = 1 - ((x(k2) - a(k2)) .^ 2) ./ ((b(k2) - a(k2)) .* (c(k2) - a(k2))); p(k3) = ((b(k3) - x(k3)) .^ 2) ./ ((b(k3) - a(k3)) .* (b(k3) - c(k3))); else p(k2) = ((x(k2) - a(k2)) .^ 2) ./ ((b(k2) - a(k2)) .* (c(k2) - a(k2))); p(k3) = 1 - ((b(k3) - x(k3)) .^ 2) ./ ((b(k3) - a(k3)) .* (b(k3) - c(k3))); p(k4) = 1; endif endfunction %!demo %! ## Plot various CDFs from the triangular distribution %! x = 0.001:0.001:10; %! p1 = tricdf (x, 3, 6, 4); %! p2 = tricdf (x, 1, 5, 2); %! p3 = tricdf (x, 2, 9, 3); %! p4 = tricdf (x, 2, 9, 5); %! plot (x, p1, "-b", x, p2, "-g", x, p3, "-r", x, p4, "-c") %! grid on %! xlim ([0, 10]) %! legend ({"a = 3, b = 6, c = 4", "a = 1, b = 5, c = 2", ... %! "a = 2, b = 9, c = 3", "a = 2, b = 9, c = 5"}, ... %! "location", "southeast") %! title ("Triangular CDF") %! xlabel ("values in x") %! ylabel ("probability") ## Test output %!shared x, y %! x = [-1, 0, 0.1, 0.5, 0.9, 1, 2] + 1; %! y = [0, 0, 0.02, 0.5, 0.98, 1 1]; %!assert (tricdf (x, ones (1,7), 2*ones (1,7), 1.5*ones (1,7)), y, eps) %!assert (tricdf (x, 1*ones (1,7), 2, 1.5), y, eps) %!assert (tricdf (x, 1*ones (1,7), 2, 1.5, "upper"), 1 - y, eps) %!assert (tricdf (x, 1, 2*ones (1,7), 1.5), y, eps) %!assert (tricdf (x, 1, 2, 1.5*ones (1,7)), y, eps) %!assert (tricdf (x, 1, 2, 1.5), y, eps) %!assert (tricdf (x, [1, 1, NaN, 1, 1, 1, 1], 2, 1.5), ... %! [y(1:2), NaN, y(4:7)], eps) %!assert (tricdf (x, 1, 2*[1, 1, NaN, 1, 1, 1, 1], 1.5), ... %! [y(1:2), NaN, y(4:7)], eps) %!assert (tricdf (x, 1, 2, 1.5*[1, 1, NaN, 1, 1, 1, 1]), ... %! [y(1:2), NaN, y(4:7)], eps) %!assert (tricdf ([x, NaN], 1, 2, 1.5), [y, NaN], eps) ## Test class of input preserved %!assert (tricdf (single ([x, NaN]), 1, 2, 1.5), ... %! single ([y, NaN]), eps("single")) %!assert (tricdf ([x, NaN], single (1), 2, 1.5), ... %! single ([y, NaN]), eps("single")) %!assert (tricdf ([x, NaN], 1, single (2), 1.5), ... %! single ([y, NaN]), eps("single")) %!assert (tricdf ([x, NaN], 1, 2, single (1.5)), ... %! single ([y, NaN]), eps("single")) ## Test input validation %!error tricdf () %!error tricdf (1) %!error tricdf (1, 2) %!error tricdf (1, 2, 3) %!error ... %! tricdf (1, 2, 3, 4, 5, 6) %!error tricdf (1, 2, 3, 4, "tail") %!error tricdf (1, 2, 3, 4, 5) %!error ... %! tricdf (ones (3), ones (2), ones(2), ones(2)) %!error ... %! tricdf (ones (2), ones (3), ones(2), ones(2)) %!error ... %! tricdf (ones (2), ones (2), ones(3), ones(2)) %!error ... %! tricdf (ones (2), ones (2), ones(2), ones(3)) %!error tricdf (i, 2, 3, 4) %!error tricdf (1, i, 3, 4) %!error tricdf (1, 2, i, 4) %!error tricdf (1, 2, 3, i) statistics-release-1.6.3/inst/dist_fun/triinv.m000066400000000000000000000130401456127120000216060ustar00rootroot00000000000000## Copyright (B) 1995-2015 Kurt Hornik ## Copyright (B) 2016 Dag Lyberg ## Copyright (B) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## Octave is free software; you can redistribute it and/or modify it ## under the terms of the GNU General Public License as published by ## the Free Software Foundation; either version 3 of the License, or (at ## your option) any later version. ## ## Octave is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with Octave; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{x} =} triinv (@var{p}, @var{a}, @var{b}, @var{c}) ## ## Inverse of the triangular cumulative distribution function (iCDF). ## ## For each element of @var{p}, compute the quantile (the inverse of the CDF) of ## the triangular distribution with parameters @var{a}, @var{b}, and @var{c} on ## the interval @qcode{[@var{a}, @var{b}]}. The size of @var{x} is the common ## size of the input arguments. A scalar input functions as a constant matrix ## of the same size as the other inputs. ## ## Further information about the triangular distribution can be found at ## @url{https://en.wikipedia.org/wiki/Triangular_distribution} ## ## @seealso{tricdf, tripdf, trirnd} ## @end deftypefn function x = triinv (p, a, b, c) ## Check for valid number of input arguments if (nargin < 4) error ("triinv: function called with too few input arguments."); endif ## Check for common size of P, A, B, and C if (! isscalar (p) || ! isscalar (a) || ! isscalar (b) || ! isscalar (c)) [retval, p, a, b, c] = common_size (p, a, b, c); if (retval > 0) error ("triinv: P, A, B, and C must be of common size or scalars."); endif endif ## Check for P, A, B, and C being reals if (iscomplex (p) || iscomplex (a) || iscomplex (b) || iscomplex (c)) error ("triinv: P, A, B, and C must not be complex."); endif ## Check for class type if (isa (p, "single") || isa (a, "single") || isa (b, "single") ... || isa (c, "single")) x = NaN (size (p), "single"); else x = NaN (size (p)); endif ## Force zeros for within range parameters. k = (p >= 0) & (p <= 1) & (a < b) & (a <= c) & (c <= b); x(k) = 0; ## Compute triangular iCDF h = 2 ./ (b-a); w = c-a; area1 = h .* w / 2; j = k & (p <= area1); x(j) += (2 * p(j) .* (w(j) ./ h(j))).^0.5 + a(j); w = b-c; j = k & (area1 < p) & (p < 1); x(j) += b(j) - (2 * (1-p(j)) .* (w(j) ./ h(j))).^0.5; j = k & (p == 1); x(j) = b(j); endfunction %!demo %! ## Plot various iCDFs from the triangular distribution %! p = 0.001:0.001:0.999; %! x1 = triinv (p, 3, 6, 4); %! x2 = triinv (p, 1, 5, 2); %! x3 = triinv (p, 2, 9, 3); %! x4 = triinv (p, 2, 9, 5); %! plot (p, x1, "-b", p, x2, "-g", p, x3, "-r", p, x4, "-c") %! grid on %! ylim ([0, 10]) %! legend ({"a = 3, b = 6, c = 4", "a = 1, b = 5, c = 2", ... %! "a = 2, b = 9, c = 3", "a = 2, b = 9, c = 5"}, ... %! "location", "northwest") %! title ("Triangular CDF") %! xlabel ("probability") %! ylabel ("values in x") ## Test output %!shared p, y %! p = [-1, 0, 0.02, 0.5, 0.98, 1, 2]; %! y = [NaN, 0, 0.1, 0.5, 0.9, 1, NaN] + 1; %!assert (triinv (p, ones (1,7), 2*ones (1,7), 1.5*ones (1,7)), y, eps) %!assert (triinv (p, 1*ones (1,7), 2, 1.5), y, eps) %!assert (triinv (p, 1, 2*ones (1,7), 1.5), y, eps) %!assert (triinv (p, 1, 2, 1.5*ones (1,7)), y, eps) %!assert (triinv (p, 1, 2, 1.5), y, eps) %!assert (triinv (p, [1, 1, NaN, 1, 1, 1, 1], 2, 1.5), [y(1:2), NaN, y(4:7)], eps) %!assert (triinv (p, 1, 2*[1, 1, NaN, 1, 1, 1, 1], 1.5), [y(1:2), NaN, y(4:7)], eps) %!assert (triinv (p, 1, 2, 1.5*[1, 1, NaN, 1, 1, 1, 1]), [y(1:2), NaN, y(4:7)], eps) %!assert (triinv ([p, NaN], 1, 2, 1.5), [y, NaN], eps) ## Test class of input preserved %!assert (triinv (single ([p, NaN]), 1, 2, 1.5), single ([y, NaN]), eps('single')) %!assert (triinv ([p, NaN], single (1), 2, 1.5), single ([y, NaN]), eps('single')) %!assert (triinv ([p, NaN], 1, single (2), 1.5), single ([y, NaN]), eps('single')) %!assert (triinv ([p, NaN], 1, 2, single (1.5)), single ([y, NaN]), eps('single')) ## Test input validation %!error triinv () %!error triinv (1) %!error triinv (1, 2) %!error triinv (1, 2, 3) %!error ... %! triinv (1, 2, 3, 4, 5) %!error ... %! triinv (ones (3), ones (2), ones(2), ones(2)) %!error ... %! triinv (ones (2), ones (3), ones(2), ones(2)) %!error ... %! triinv (ones (2), ones (2), ones(3), ones(2)) %!error ... %! triinv (ones (2), ones (2), ones(2), ones(3)) %!error triinv (i, 2, 3, 4) %!error triinv (1, i, 3, 4) %!error triinv (1, 2, i, 4) %!error triinv (1, 2, 3, i) statistics-release-1.6.3/inst/dist_fun/tripdf.m000066400000000000000000000130021456127120000215610ustar00rootroot00000000000000## Copyright (C) 1997-2015 Kurt Hornik ## Copyright (C) 2016 Dag Lyberg ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## Octave is free software; you can redistribute it and/or modify it ## under the terms of the GNU General Public License as published by ## the Free Software Foundation; either version 3 of the License, or (at ## your option) any later version. ## ## Octave is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with Octave; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{y} =} tripdf (@var{x}, @var{a}, @var{b}, @var{c}) ## ## Triangular probability density function (PDF). ## ## For each element of @var{x}, compute the probability density function (PDF) ## of the triangular distribution with parameters @var{a}, @var{b}, and @var{c} ## on the interval @qcode{[@var{a}, @var{b}]}. The size of @var{y} is the ## common size of the input arguments. A scalar input functions as a constant ## matrix of the same size as the other inputs. ## ## Further information about the triangular distribution can be found at ## @url{https://en.wikipedia.org/wiki/Triangular_distribution} ## ## @seealso{tricdf, triinv, trirnd} ## @end deftypefn function y = tripdf (x, a, b, c) ## Check for valid number of input arguments if (nargin < 4) error ("tripdf: function called with too few input arguments."); endif ## Check for common size of X, A, B, and C if (! isscalar (x) || ! isscalar (a) || ! isscalar (b) || ! isscalar (c)) [retval, x, a, b, c] = common_size (x, a, b, c); if (retval > 0) error ("tripdf: X, A, B, and C must be of common size or scalars."); endif endif ## Check for X, A, B, and C being reals if (iscomplex (x) || iscomplex (a) || iscomplex (b) || iscomplex (c)) error ("tripdf: X, A, B, and C must not be complex."); endif ## Check for class type if (isa (x, "single") || isa (a, "single") || isa (b, "single") ... || isa (c, "single")) y = zeros (size (x), "single"); else y = zeros (size (x)); endif ## Force NaNs for out of range parameters. k = isnan (x) | !(a < b) | !(c >= a) | !(c <= b) ; y(k) = NaN; k = (x >= a) & (x <= b) & (a < b) & (a <= c) & (c <= b); h = 2 ./ (b-a); j = k & (a <= x) & (x < c); y(j) = h(j) .* (x(j)-a(j)) ./ (c(j)-a(j)); j = k & (x == c); y(j) = h(j); j = k & (c < x) & (x <= b); y(j) = h(j) .* (b(j)-x(j)) ./ (b(j)-c(j)); endfunction %!demo %! ## Plot various CDFs from the triangular distribution %! x = 0.001:0.001:10; %! y1 = tripdf (x, 3, 6, 4); %! y2 = tripdf (x, 1, 5, 2); %! y3 = tripdf (x, 2, 9, 3); %! y4 = tripdf (x, 2, 9, 5); %! plot (x, y1, "-b", x, y2, "-g", x, y3, "-r", x, y4, "-c") %! grid on %! xlim ([0, 10]) %! legend ({"a = 3, b = 6, c = 4", "a = 1, b = 5, c = 2", ... %! "a = 2, b = 9, c = 3", "a = 2, b = 9, c = 5"}, ... %! "location", "northeast") %! title ("Triangular CDF") %! xlabel ("values in x") %! ylabel ("probability") ## Test output %!shared x, y, deps %! x = [-1, 0, 0.1, 0.5, 0.9, 1, 2] + 1; %! y = [0, 0, 0.4, 2, 0.4, 0, 0]; %! deps = 2*eps; %!assert (tripdf (x, ones (1,7), 2*ones (1,7), 1.5*ones (1,7)), y, deps) %!assert (tripdf (x, 1*ones (1,7), 2, 1.5), y, deps) %!assert (tripdf (x, 1, 2*ones (1,7), 1.5), y, deps) %!assert (tripdf (x, 1, 2, 1.5*ones (1,7)), y, deps) %!assert (tripdf (x, 1, 2, 1.5), y, deps) %!assert (tripdf (x, [1, 1, NaN, 1, 1, 1, 1], 2, 1.5), [y(1:2), NaN, y(4:7)], deps) %!assert (tripdf (x, 1, 2*[1, 1, NaN, 1, 1, 1, 1], 1.5), [y(1:2), NaN, y(4:7)], deps) %!assert (tripdf (x, 1, 2, 1.5*[1, 1, NaN, 1, 1, 1, 1]), [y(1:2), NaN, y(4:7)], deps) %!assert (tripdf ([x, NaN], 1, 2, 1.5), [y, NaN], deps) ## Test class of input preserved %!assert (tripdf (single ([x, NaN]), 1, 2, 1.5), single ([y, NaN]), eps("single")) %!assert (tripdf ([x, NaN], single (1), 2, 1.5), single ([y, NaN]), eps("single")) %!assert (tripdf ([x, NaN], 1, single (2), 1.5), single ([y, NaN]), eps("single")) %!assert (tripdf ([x, NaN], 1, 2, single (1.5)), single ([y, NaN]), eps("single")) ## Test input validation %!error tripdf () %!error tripdf (1) %!error tripdf (1, 2) %!error tripdf (1, 2, 3) %!error ... %! tripdf (1, 2, 3, 4, 5) %!error ... %! tripdf (ones (3), ones (2), ones(2), ones(2)) %!error ... %! tripdf (ones (2), ones (3), ones(2), ones(2)) %!error ... %! tripdf (ones (2), ones (2), ones(3), ones(2)) %!error ... %! tripdf (ones (2), ones (2), ones(2), ones(3)) %!error tripdf (i, 2, 3, 4) %!error tripdf (1, i, 3, 4) %!error tripdf (1, 2, i, 4) %!error tripdf (1, 2, 3, i) statistics-release-1.6.3/inst/dist_fun/trirnd.m000066400000000000000000000170651456127120000216100ustar00rootroot00000000000000## Copyright (C) 1997-2015 Kurt Hornik ## Copyright (C) 2016 Dag Lyberg ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## Octave is free software; you can redistribute it and/or modify it ## under the terms of the GNU General Public License as published by ## the Free Software Foundation; either version 3 of the License, or (at ## your option) any later version. ## ## Octave is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with Octave; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{r} =} trirnd (@var{a}, @var{b}, @var{c}) ## @deftypefnx {statistics} {@var{r} =} trirnd (@var{a}, @var{b}, @var{c}, @var{rows}) ## @deftypefnx {statistics} {@var{r} =} trirnd (@var{a}, @var{b}, @var{c}, @var{rows}, @var{cols}, @dots{}) ## @deftypefnx {statistics} {@var{r} =} trirnd (@var{a}, @var{b}, @var{c}, [@var{sz}]) ## ## Random arrays from the triangular distribution. ## ## @code{@var{r} = trirnd (@var{sigma})} returns an array of random numbers ## chosen from the triangular distribution with parameters @var{a}, @var{b}, and ## @var{c} on the interval [@var{a}, @var{b}]. The size of @var{r} is the ## common size of @var{a}, @var{b}, and @var{c}. A scalar input functions as a ## constant matrix of the same size as the other inputs. ## ## When called with a single size argument, @code{trirnd} returns a square ## matrix with the dimension specified. When called with more than one scalar ## argument, the first two arguments are taken as the number of rows and columns ## and any further arguments specify additional matrix dimensions. The size may ## also be specified with a row vector of dimensions, @var{sz}. ## ## Further information about the triangular distribution can be found at ## @url{https://en.wikipedia.org/wiki/Triangular_distribution} ## ## @seealso{tricdf, triinv, tripdf} ## @end deftypefn function rnd = trirnd (a, b, c, varargin) ## Check for valid number of input arguments if (nargin < 3) error ("trirnd: function called with too few input arguments."); endif ## Check for common size of A, B, and C if (! isscalar (a) || ! isscalar (b) || ! isscalar (c)) [retval, a, b, c] = common_size (a, b, c); scalarABC = false; if (retval > 0) error ("trirnd: A, B, and C must be of common size or scalars."); endif else scalarABC = true; endif ## Check for A, B, and C being reals if (iscomplex (a) || iscomplex (b) || iscomplex (c)) error ("trirnd: A, B, and C must not be complex."); endif ## Parse and check SIZE arguments if (nargin == 3) sz = size (a); elseif (nargin == 4) if (isscalar (varargin{1}) && varargin{1} >= 0 ... && varargin{1} == fix (varargin{1})) sz = [varargin{1}, varargin{1}]; elseif (isrow (varargin{1}) && all (varargin{1} >= 0) ... && all (varargin{1} == fix (varargin{1}))) sz = varargin{1}; elseif error (strcat (["trirnd: SZ must be a scalar or a row vector"], ... [" of non-negative integers."])); endif elseif (nargin > 4) posint = cellfun (@(x) (! isscalar (x) || x < 0 || x != fix (x)), varargin); if (any (posint)) error ("trirnd: dimensions must be non-negative integers."); endif sz = [varargin{:}]; endif ## Check that parameters match requested dimensions in size if (! isscalar (a) && ! isequal (size (a), sz)) error ("trirnd: A, B, and C must be scalar or of size SZ."); endif ## Check for class type if (isa (a, "single") || isa (b, "single") || isa (c, "single")) cls = "single"; else cls = "double"; endif ## Generate random sample from triangular distribution if (scalarABC) if ((-Inf < a) && (a < b) && (a <= c) && (c <= b) && (b < Inf)) w = b-a; left_width = c-a; right_width = b-c; h = 2 / w; left_area = h * left_width / 2; rnd = rand (sz, cls); idx = rnd < left_area; rnd(idx) = a + (rnd(idx) * w * left_width).^0.5; rnd(~idx) = b - ((1-rnd(~idx)) * w * right_width).^0.5; else rnd = NaN (sz, cls); endif else w = b-a; left_width = c-a; right_width = b-c; h = 2 ./ w; left_area = h .* left_width / 2; rnd = rand (sz, cls); k = rnd < left_area; rnd(k) = a(k) + (rnd(k) .* w(k) .* left_width(k)).^0.5; rnd(~k) = b(~k) - ((1-rnd(~k)) .* w(~k) .* right_width(~k)).^0.5; k = ! (-Inf < a) | ! (a < b) | ! (a <= c) | ! (c <= b) | ! (b < Inf); rnd(k) = NaN; endif endfunction ## Test results %!assert (size (trirnd (1,2,1.5)), [1, 1]) %!assert (size (trirnd (1*ones (2,1), 2,1.5)), [2, 1]) %!assert (size (trirnd (1*ones (2,2), 2,1.5)), [2, 2]) %!assert (size (trirnd (1, 2*ones (2,1), 1.5)), [2, 1]) %!assert (size (trirnd (1, 2*ones (2,2), 1.5)), [2, 2]) %!assert (size (trirnd (1, 2, 1.5*ones (2,1))), [2, 1]) %!assert (size (trirnd (1, 2, 1.5*ones (2,2))), [2, 2]) %!assert (size (trirnd (1, 2, 1.5, 3)), [3, 3]) %!assert (size (trirnd (1, 2, 1.5, [4 1])), [4, 1]) %!assert (size (trirnd (1, 2, 1.5, 4, 1)), [4, 1]) ## Test class of input preserved %!assert (class (trirnd (1,2,1.5)), "double") %!assert (class (trirnd (single (1),2,1.5)), "single") %!assert (class (trirnd (single ([1 1]),2,1.5)), "single") %!assert (class (trirnd (1,single (2),1.5)), "single") %!assert (class (trirnd (1,single ([2 2]),1.5)), "single") %!assert (class (trirnd (1,2,single (1.5))), "single") %!assert (class (trirnd (1,2,single ([1.5 1.5]))), "single") ## Test input validation %!error trirnd () %!error trirnd (1) %!error trirnd (1, 2) %!error ... %! trirnd (ones (3), 5 * ones (2), ones (2)) %!error ... %! trirnd (ones (2), 5 * ones (3), ones (2)) %!error ... %! trirnd (ones (2), 5 * ones (2), ones (3)) %!error trirnd (i, 5, 3) %!error trirnd (1, 5+i, 3) %!error trirnd (1, 5, i) %!error ... %! trirnd (1, 5, 3, -1) %!error ... %! trirnd (1, 5, 3, 1.2) %!error ... %! trirnd (1, 5, 3, ones (2)) %!error ... %! trirnd (1, 5, 3, [2 -1 2]) %!error ... %! trirnd (1, 5, 3, [2 0 2.5]) %!error ... %! trirnd (1, 5, 3, 2, -1, 5) %!error ... %! trirnd (1, 5, 3, 2, 1.5, 5) %!error ... %! trirnd (2, 5 * ones (2), 2, 3) %!error ... %! trirnd (2, 5 * ones (2), 2, [3, 2]) %!error ... %! trirnd (2, 5 * ones (2), 2, 3, 2) statistics-release-1.6.3/inst/dist_fun/trnd.m000066400000000000000000000136011456127120000212450ustar00rootroot00000000000000## Copyright (C) 2012 Rik Wehbring ## Copyright (C) 1995-2016 Kurt Hornik ## Copyright (C) 2022-2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{r} =} trnd (@var{df}) ## @deftypefnx {statistics} {@var{r} =} trnd (@var{df}, @var{rows}) ## @deftypefnx {statistics} {@var{r} =} trnd (@var{df}, @var{rows}, @var{cols}, @dots{}) ## @deftypefnx {statistics} {@var{r} =} trnd (@var{df}, [@var{sz}]) ## ## Random arrays from the Student's T distribution. ## ## Return a matrix of random samples from the Students's T distribution with ## @var{df} degrees of freedom. ## ## @code{@var{r} = trnd (@var{df})} returns an array of random numbers chosen ## from the Student's T distribution with @var{df} degrees of freedom. The size ## of @var{r} is the size of @var{df}. A scalar input functions as a constant ## matrix of the same size as the other inputs. @var{df} must be a finite real ## number greater than 0, otherwise NaN is returned. ## ## When called with a single size argument, @code{trnd} returns a square matrix ## with the dimension specified. When called with more than one scalar ## argument, the first two arguments are taken as the number of rows and columns ## and any further arguments specify additional matrix dimensions. The size may ## also be specified with a row vector of dimensions, @var{sz}. ## ## Further information about the Student's T distribution can be found at ## @url{https://en.wikipedia.org/wiki/Student%27s_t-distribution} ## ## @seealso{tcdf, tpdf, tpdf, tstat} ## @end deftypefn function r = trnd (df, varargin) ## Check for valid number of input arguments if (nargin < 1) error ("trnd: function called with too few input arguments."); endif ## Check for DF being real if (iscomplex (df)) error ("trnd: DF must not be complex."); endif ## Parse and check SIZE arguments if (nargin == 1) sz = size (df); elseif (nargin == 2) if (isscalar (varargin{1}) && varargin{1} >= 0 ... && varargin{1} == fix (varargin{1})) sz = [varargin{1}, varargin{1}]; elseif (isrow (varargin{1}) && all (varargin{1} >= 0) ... && all (varargin{1} == fix (varargin{1}))) sz = varargin{1}; elseif error (strcat (["trnd: SZ must be a scalar or a row vector"], ... [" of non-negative integers."])); endif elseif (nargin > 2) posint = cellfun (@(x) (! isscalar (x) || x < 0 || x != fix (x)), varargin); if (any (posint)) error ("trnd: dimensions must be non-negative integers."); endif sz = [varargin{:}]; endif ## Check that parameters match requested dimensions in size if (! isscalar (df) && ! isequal (size (df), sz)) error ("trnd: DF must be scalar or of size SZ."); endif ## Check for class type if (isa (df, "single")) cls = "single"; else cls = "double"; endif if (isscalar (df)) if ((df > 0) && (df < Inf)) r = randn (sz, cls) ./ sqrt (2*randg (df/2, sz, cls) / df); elseif (isinf (df)) r = randn (sz, cls); else r = NaN (sz, cls); endif else r = NaN (sz, cls); k = (df > 0) & (df < Inf); kinf = isinf (df); r(k) = randn (sum (k(:)), 1, cls) ./ ... sqrt (2*randg (df(k)/2, cls) ./ df(k))(:); r(kinf) = randn (sum (kinf(:)), 1, cls); endif endfunction ## Test output %!assert (size (trnd (2)), [1, 1]) %!assert (size (trnd (ones (2,1))), [2, 1]) %!assert (size (trnd (ones (2,2))), [2, 2]) %!assert (size (trnd (1, 3)), [3, 3]) %!assert (size (trnd (1, [4 1])), [4, 1]) %!assert (size (trnd (1, 4, 1)), [4, 1]) %!assert (size (trnd (1, 4, 1)), [4, 1]) %!assert (size (trnd (1, 4, 1, 5)), [4, 1, 5]) %!assert (size (trnd (1, 0, 1)), [0, 1]) %!assert (size (trnd (1, 1, 0)), [1, 0]) %!assert (size (trnd (1, 1, 2, 0, 5)), [1, 2, 0, 5]) %!assert (trnd (0, 1, 1), NaN) %!assert (trnd ([0, 0, 0], [1, 3]), [NaN, NaN, NaN]) ## Test class of input preserved %!assert (class (trnd (2)), "double") %!assert (class (trnd (single (2))), "single") %!assert (class (trnd (single ([2 2]))), "single") ## Test input validation %!error trnd () %!error trnd (i) %!error ... %! trnd (1, -1) %!error ... %! trnd (1, 1.2) %!error ... %! trnd (1, ones (2)) %!error ... %! trnd (1, [2 -1 2]) %!error ... %! trnd (1, [2 0 2.5]) %!error ... %! trnd (ones (2), ones (2)) %!error ... %! trnd (1, 2, -1, 5) %!error ... %! trnd (1, 2, 1.5, 5) %!error trnd (ones (2,2), 3) %!error trnd (ones (2,2), [3, 2]) %!error trnd (ones (2,2), 2, 3) statistics-release-1.6.3/inst/dist_fun/unidcdf.m000066400000000000000000000122021456127120000217060ustar00rootroot00000000000000## Copyright (C) 2012 Rik Wehbring ## Copyright (C) 2007-2016 David Bateman ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{p} =} unidcdf (@var{x}, @var{N}) ## @deftypefnx {statistics} {@var{p} =} unidcdf (@var{x}, @var{N}, @qcode{"upper"}) ## ## Discrete uniform cumulative distribution function (CDF). ## ## For each element of @var{x}, compute the cumulative distribution function ## (CDF) of a discrete uniform distribution with parameter @var{N}, which ## corresponds to the maximum observable value. @code{unidcdf} assumes the ## integer values in the range @math{[1,N]} with equal probability. The size of ## @var{p} is the common size of @var{x} and @var{N}. A scalar input functions ## as a constant matrix of the same size as the other inputs. ## ## The maximum observable values in @var{N} must be positive integers, otherwise ## @qcode{NaN} is returned. ## ## @code{[@dots{}] = unidcdf (@var{x}, @var{N}, "upper")} computes the upper ## tail probability of the discrete uniform distribution with maximum observable ## value @var{N}, at the values in @var{x}. ## ## Warning: The underlying implementation uses the double class and will only ## be accurate for @var{N} < @code{flintmax} (@w{@math{2^{53}}} on ## IEEE 754 compatible systems). ## ## Further information about the discrete uniform distribution can be found at ## @url{https://en.wikipedia.org/wiki/Discrete_uniform_distribution} ## ## @seealso{unidinv, unidpdf, unidrnd, unidfit, unidstat} ## @end deftypefn function p = unidcdf (x, N, uflag) ## Check for valid number of input arguments if (nargin < 2) error ("unidcdf: function called with too few input arguments."); endif ## Check for "upper" flag if (nargin > 2 && strcmpi (uflag, "upper")) uflag = true; elseif (nargin > 2 && ! strcmpi (uflag, "upper")) error ("unidcdf: invalid argument for upper tail."); else uflag = false; endif ## Check for common size of X and N if (! isscalar (x) || ! isscalar (N)) [retval, x, N] = common_size (x, N); if (retval > 0) error ("unidcdf: X and N must be of common size or scalars."); endif endif ## Check for X and N being reals if (iscomplex (x) || iscomplex (N)) error ("unidcdf: X and N must not be complex."); endif ## Check for class type if (isa (x, "single") || isa (N, "single")) p = zeros (size (x), "single"); else p = zeros (size (x)); endif ## Return 1 for X >= N p(x >= N) = 1; ## Floor X xf = floor (x); ## Compute uniform discrete CDF k = find (xf >= 1 & xf <= N); if any(k) p(k) = xf(k) ./ N(k); endif ## Check for NaNs or floored N <= 0 is_nan = isnan (x) | ! (N > 0 & N == fix (N)); if (any (is_nan(:))) p(is_nan) = NaN; endif p(N < 1 | round(N) != N) = NaN; if (uflag) # Compute upper tail p = 1 - unidcdf (x, N); endif endfunction %!demo %! ## Plot various CDFs from the discrete uniform distribution %! x = 0:10; %! p1 = unidcdf (x, 5); %! p2 = unidcdf (x, 9); %! plot (x, p1, "*b", x, p2, "*g") %! grid on %! xlim ([0, 10]) %! ylim ([0, 1]) %! legend ({"N = 5", "N = 9"}, "location", "southeast") %! title ("Discrete uniform CDF") %! xlabel ("values in x") %! ylabel ("probability") ## Test output %!shared x, y %! x = [0 1 2.5 10 11]; %! y = [0, 0.1 0.2 1.0 1.0]; %!assert (unidcdf (x, 10*ones (1,5)), y) %!assert (unidcdf (x, 10*ones (1,5), "upper"), 1 - y) %!assert (unidcdf (x, 10), y) %!assert (unidcdf (x, 10, "upper"), 1 - y) %!assert (unidcdf (x, 10*[0 1 NaN 1 1]), [NaN 0.1 NaN y(4:5)]) %!assert (unidcdf ([x(1:2) NaN Inf x(5)], 10), [y(1:2) NaN 1 y(5)]) ## Test class of input preserved %!assert (unidcdf ([x, NaN], 10), [y, NaN]) %!assert (unidcdf (single ([x, NaN]), 10), single ([y, NaN])) %!assert (unidcdf ([x, NaN], single (10)), single ([y, NaN])) ## Test input validation %!error unidcdf () %!error unidcdf (1) %!error unidcdf (1, 2, 3) %!error unidcdf (1, 2, "tail") %!error ... %! unidcdf (ones (3), ones (2)) %!error ... %! unidcdf (ones (2), ones (3)) %!error unidcdf (i, 2) %!error unidcdf (2, i) statistics-release-1.6.3/inst/dist_fun/unidinv.m000066400000000000000000000102621456127120000217520ustar00rootroot00000000000000## Copyright (C) 2012 Rik Wehbring ## Copyright (C) 2007-2016 David Bateman ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{x} =} unidinv (@var{p}, @var{N}) ## ## Inverse of the discrete uniform cumulative distribution function (iCDF). ## ## For each element of @var{p}, compute the quantile (the inverse of the CDF) of ## the discrete uniform distribution with parameter @var{N}, which corresponds ## to the maximum observable value. @code{unidinv} assumes the integer values ## in the range @math{[1,N]} with equal probability. The size of @var{x} is the ## common size of @var{p} and @var{N}. A scalar input functions as a constant ## matrix of the same size as the other inputs. ## ## The maximum observable values in @var{N} must be positive integers, otherwise ## @qcode{NaN} is returned. ## ## Warning: The underlying implementation uses the double class and will only ## be accurate for @var{N} < @code{flintmax} (@w{@math{2^{53}}} on ## IEEE 754 compatible systems). ## ## Further information about the discrete uniform distribution can be found at ## @url{https://en.wikipedia.org/wiki/Discrete_uniform_distribution} ## ## @seealso{unidcdf, unidpdf, unidrnd, unidfit, unidstat} ## @end deftypefn function x = unidinv (p, N) ## Check for valid number of input arguments if (nargin < 2) error ("unidinv: function called with too few input arguments."); endif ## Check for common size of P and N if (! isscalar (p) || ! isscalar (N)) [retval, p, N] = common_size (p, N); if (retval > 0) error ("unidinv: P and N must be of common size or scalars."); endif endif ## Check for P and N being reals if (iscomplex (p) || iscomplex (N)) error ("unidinv: P and N must not be complex."); endif ## Check for class type if (isa (p, "single") || isa (N, "single")) x = NaN (size (p), "single"); else x = NaN (size (p)); endif ## For Matlab compatibility, unidinv(0) = NaN k = (p > 0) & (p <= 1) & (N > 0 & N == fix (N)); x(k) = floor (p(k) .* N(k)); endfunction %!demo %! ## Plot various iCDFs from the discrete uniform distribution %! p = 0.001:0.001:0.999; %! x1 = unidinv (p, 5); %! x2 = unidinv (p, 9); %! plot (p, x1, "-b", p, x2, "-g") %! grid on %! xlim ([0, 1]) %! ylim ([0, 10]) %! legend ({"N = 5", "N = 9"}, "location", "northwest") %! title ("Discrete uniform iCDF") %! xlabel ("probability") %! ylabel ("values in x") ## Test output %!shared p %! p = [-1 0 0.5 1 2]; %!assert (unidinv (p, 10*ones (1,5)), [NaN NaN 5 10 NaN], eps) %!assert (unidinv (p, 10), [NaN NaN 5 10 NaN], eps) %!assert (unidinv (p, 10*[0 1 NaN 1 1]), [NaN NaN NaN 10 NaN], eps) %!assert (unidinv ([p(1:2) NaN p(4:5)], 10), [NaN NaN NaN 10 NaN], eps) ## Test class of input preserved %!assert (unidinv ([p, NaN], 10), [NaN NaN 5 10 NaN NaN], eps) %!assert (unidinv (single ([p, NaN]), 10), single ([NaN NaN 5 10 NaN NaN]), eps) %!assert (unidinv ([p, NaN], single (10)), single ([NaN NaN 5 10 NaN NaN]), eps) ## Test input validation %!error unidinv () %!error unidinv (1) %!error ... %! unidinv (ones (3), ones (2)) %!error ... %! unidinv (ones (2), ones (3)) %!error unidinv (i, 2) %!error unidinv (2, i) statistics-release-1.6.3/inst/dist_fun/unidpdf.m000066400000000000000000000077771456127120000217500ustar00rootroot00000000000000## Copyright (C) 2012 Rik Wehbring ## Copyright (C) 2007-2016 David Bateman ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{y} =} unidpdf (@var{x}, @var{N}) ## ## Discrete uniform probability density function (PDF). ## ## For each element of @var{x}, compute the probability density function (PDF) ## of the discrete uniform distribution with parameter @var{N}, which ## corresponds to the maximum observable value. @code{unidpdf} assumes the ## integer values in the range @math{[1,N]} with equal probability. The size of ## @var{x} is the common size of @var{p} and @var{N}. A scalar input functions ## as a constant matrix of the same size as the other inputs. ## ## The maximum observable values in @var{N} must be positive integers, otherwise ## @qcode{NaN} is returned. ## ## Warning: The underlying implementation uses the double class and will only ## be accurate for @var{N} < @code{flintmax} (@w{@math{2^{53}}} on ## IEEE 754 compatible systems). ## ## Further information about the discrete uniform distribution can be found at ## @url{https://en.wikipedia.org/wiki/Discrete_uniform_distribution} ## ## @seealso{unidcdf, unidinv, unidrnd, unidfit, unidstat} ## @end deftypefn function y = unidpdf (x, N) ## Check for valid number of input arguments if (nargin < 2) error ("unidpdf: function called with too few input arguments."); endif ## Check for common size of X and N if (! isscalar (x) || ! isscalar (N)) [retval, x, N] = common_size (x, N); if (retval > 0) error ("unidpdf: X and N must be of common size or scalars."); endif endif ## Check for X and N being reals if (iscomplex (x) || iscomplex (N)) error ("unidpdf: X and N must not be complex."); endif ## Check for class type if (isa (x, "single") || isa (N, "single")) y = zeros (size (x), "single"); else y = zeros (size (x)); endif k = isnan (x) | ! (N > 0 & N == fix (N)); y(k) = NaN; k = ! k & (x >= 1) & (x <= N) & (x == fix (x)); y(k) = 1 ./ N(k); endfunction %!demo %! ## Plot various PDFs from the discrete uniform distribution %! x = 0:10; %! y1 = unidpdf (x, 5); %! y2 = unidpdf (x, 9); %! plot (x, y1, "*b", x, y2, "*g") %! grid on %! xlim ([0, 10]) %! ylim ([0, 0.25]) %! legend ({"N = 5", "N = 9"}, "location", "northeast") %! title ("Descrete uniform PDF") %! xlabel ("values in x") %! ylabel ("density") ## Test output %!shared x, y %! x = [-1 0 1 2 10 11]; %! y = [0 0 0.1 0.1 0.1 0]; %!assert (unidpdf (x, 10*ones (1,6)), y) %!assert (unidpdf (x, 10), y) %!assert (unidpdf (x, 10*[0 NaN 1 1 1 1]), [NaN NaN y(3:6)]) %!assert (unidpdf ([x, NaN], 10), [y, NaN]) ## Test class of input preserved %!assert (unidpdf (single ([x, NaN]), 10), single ([y, NaN])) %!assert (unidpdf ([x, NaN], single (10)), single ([y, NaN])) ## Test input validation %!error unidpdf () %!error unidpdf (1) %!error ... %! unidpdf (ones (3), ones (2)) %!error ... %! unidpdf (ones (2), ones (3)) %!error unidpdf (i, 2) %!error unidpdf (2, i) statistics-release-1.6.3/inst/dist_fun/unidrnd.m000066400000000000000000000141071456127120000217430ustar00rootroot00000000000000## Copyright (C) 2012 Rik Wehbring ## Copyright (C) 2005-2016 John W. Eaton ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{r} =} unidrnd (@var{N}) ## @deftypefnx {statistics} {@var{r} =} unidrnd (@var{N}, @var{rows}) ## @deftypefnx {statistics} {@var{r} =} unidrnd (@var{N}, @var{rows}, @var{cols}, @dots{}) ## @deftypefnx {statistics} {@var{r} =} unidrnd (@var{N}, [@var{sz}]) ## ## Random arrays from the discrete uniform distribution. ## ## @code{@var{r} = unidrnd (@var{N})} returns an array of random numbers chosen ## from the discrete uniform distribution with parameter @var{N}, which ## corresponds to the maximum observable value. @code{unidrnd} assumes the ## integer values in the range @math{[1,N]} with equal probability. The size of ## @var{r} is the size of @var{N}. A scalar input functions as a constant ## matrix of the same size as the other inputs. ## ## The maximum observable values in @var{N} must be positive integers, otherwise ## @qcode{NaN} is returned. ## ## When called with a single size argument, @code{unidrnd} returns a square ## matrix with the dimension specified. When called with more than one scalar ## argument, the first two arguments are taken as the number of rows and columns ## and any further arguments specify additional matrix dimensions. The size may ## also be specified with a row vector of dimensions, @var{sz}. ## ## Warning: The underlying implementation uses the double class and will only ## be accurate for @var{N} < @code{flintmax} (@w{@math{2^{53}}} on ## IEEE 754 compatible systems). ## ## Further information about the discrete uniform distribution can be found at ## @url{https://en.wikipedia.org/wiki/Discrete_uniform_distribution} ## ## @seealso{unidcdf, unidinv, unidrnd, unidfit, unidstat} ## @end deftypefn function r = unidrnd (N, varargin) ## Check for valid number of input arguments if (nargin < 1) error ("unidrnd: function called with too few input arguments."); endif ## Check for N being real if (iscomplex (N)) error ("unidrnd: N must not be complex."); endif ## Parse and check SIZE arguments if (nargin == 1) sz = size (N); elseif (nargin == 2) if (isscalar (varargin{1}) && varargin{1} >= 0 ... && varargin{1} == fix (varargin{1})) sz = [varargin{1}, varargin{1}]; elseif (isrow (varargin{1}) && all (varargin{1} >= 0) ... && all (varargin{1} == fix (varargin{1}))) sz = varargin{1}; elseif error (strcat (["unidrnd: SZ must be a scalar or a row vector"], ... [" of non-negative integers."])); endif elseif (nargin > 2) posint = cellfun (@(x) (! isscalar (x) || x < 0 || x != fix (x)), varargin); if (any (posint)) error ("unidrnd: dimensions must be non-negative integers."); endif sz = [varargin{:}]; endif ## Check that parameters match requested dimensions in size if (! isscalar (N) && ! isequal (size (N), sz)) error ("unidrnd: N must be scalar or of size SZ."); endif ## Check for class type if (isa (N, "single")) cls = "single"; else cls = "double"; endif if (isscalar (N)) if (N > 0 && N == fix (N)) r = ceil (rand (sz, cls) * N); else r = NaN (sz, cls); endif else r = ceil (rand (sz, cls) .* N); k = ! (N > 0 & N == fix (N)); r(k) = NaN; endif endfunction ## Test output %!assert (size (unidrnd (2)), [1, 1]) %!assert (size (unidrnd (ones (2,1))), [2, 1]) %!assert (size (unidrnd (ones (2,2))), [2, 2]) %!assert (size (unidrnd (1, 3)), [3, 3]) %!assert (size (unidrnd (1, [4 1])), [4, 1]) %!assert (size (unidrnd (1, 4, 1)), [4, 1]) %!assert (size (unidrnd (1, 4, 1)), [4, 1]) %!assert (size (unidrnd (1, 4, 1, 5)), [4, 1, 5]) %!assert (size (unidrnd (1, 0, 1)), [0, 1]) %!assert (size (unidrnd (1, 1, 0)), [1, 0]) %!assert (size (unidrnd (1, 1, 2, 0, 5)), [1, 2, 0, 5]) %!assert (unidrnd (0, 1, 1), NaN) %!assert (unidrnd ([0, 0, 0], [1, 3]), [NaN, NaN, NaN]) ## Test class of input preserved %!assert (class (unidrnd (2)), "double") %!assert (class (unidrnd (single (2))), "single") %!assert (class (unidrnd (single ([2 2]))), "single") ## Test input validation %!error unidrnd () %!error unidrnd (i) %!error ... %! unidrnd (1, -1) %!error ... %! unidrnd (1, 1.2) %!error ... %! unidrnd (1, ones (2)) %!error ... %! unidrnd (1, [2 -1 2]) %!error ... %! unidrnd (1, [2 0 2.5]) %!error ... %! unidrnd (ones (2), ones (2)) %!error ... %! unidrnd (1, 2, -1, 5) %!error ... %! unidrnd (1, 2, 1.5, 5) %!error unidrnd (ones (2,2), 3) %!error unidrnd (ones (2,2), [3, 2]) %!error unidrnd (ones (2,2), 2, 3) statistics-release-1.6.3/inst/dist_fun/unifcdf.m000066400000000000000000000134131456127120000217150ustar00rootroot00000000000000## Copyright (C) 2012 Rik Wehbring ## Copyright (C) 1995-2016 Kurt Hornik ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{p} =} unifcdf (@var{x}, @var{a}, @var{b}) ## @deftypefnx {statistics} {@var{p} =} unifcdf (@var{x}, @var{a}, @var{b}, @qcode{"upper"}) ## ## Continuous uniform cumulative distribution function (CDF). ## ## For each element of @var{x}, compute the cumulative distribution function ## (CDF) of the continuous uniform distribution with parameters @var{a} and ## @var{b}, which define the lower and upper bounds of the interval ## @qcode{[@var{a}, @var{b}]}. The size of @var{p} is the common size of ## @var{x}, @var{a}, and @var{b}. A scalar input functions as a constant matrix ## of the same size as the other inputs. ## ## @code{[@dots{}] = unifcdf (@var{x}, @var{a}, @var{b}, "upper")} computes the ## upper tail probability of the continuous uniform distribution with parameters ## @var{a}, and @var{b}, at the values in @var{x}. ## ## Further information about the continuous uniform distribution can be found at ## @url{https://en.wikipedia.org/wiki/Continuous_uniform_distribution} ## ## @seealso{unifinv, unifpdf, unifrnd, unifit, unifstat} ## @end deftypefn function p = unifcdf (x, a, b, uflag) ## Check for valid number of input arguments if (nargin < 3) error ("unifcdf: function called with too few input arguments."); endif ## Check for "upper" flag if (nargin > 3 && strcmpi (uflag, "upper")) uflag = true; elseif (nargin > 3 && ! strcmpi (uflag, "upper")) error ("unifcdf: invalid argument for upper tail."); else uflag = false; endif ## Check for common size of X, A, and B if (! isscalar (x) || ! isscalar (a) || ! isscalar (b)) [retval, x, a, b] = common_size (x, a, b); if (retval > 0) error ("unifcdf: X, A, and B must be of common size or scalars."); endif endif ## Check for X, A, and B being reals if (iscomplex (x) || iscomplex (a) || iscomplex (b)) error ("unifcdf: X, A, and B must not be complex."); endif ## Check for class type if (isa (x, "single") || isa (a, "single") || isa (b, "single")) p = zeros (size (x), "single"); else p = zeros (size (x)); endif ## Calculate continuous uniform CDF for valid parameter and data range k = find(x > a & x < b & a < b); if (uflag) p(x <= a & a < b) = 1; p(x >= b & a < b) = 0; if any(k) p(k) = (b(k)- x(k)) ./ (b(k) - a(k)); endif else p(x <= a & a < b) = 0; p(x >= b & a < b) = 1; if any(k) p(k) = (x(k) - a(k)) ./ (b(k) - a(k)); endif endif ## Continue argument check p(a >= b) = NaN; p(isnan(x) | isnan(a) | isnan(b)) = NaN; endfunction %!demo %! ## Plot various CDFs from the continuous uniform distribution %! x = 0:0.1:10; %! p1 = unifcdf (x, 2, 5); %! p2 = unifcdf (x, 3, 9); %! plot (x, p1, "-b", x, p2, "-g") %! grid on %! xlim ([0, 10]) %! ylim ([0, 1]) %! legend ({"a = 2, b = 5", "a = 3, b = 9"}, "location", "southeast") %! title ("Continuous uniform CDF") %! xlabel ("values in x") %! ylabel ("probability") ## Test output %!shared x, y %! x = [-1 0 0.5 1 2] + 1; %! y = [0 0 0.5 1 1]; %!assert (unifcdf (x, ones (1,5), 2*ones (1,5)), y) %!assert (unifcdf (x, ones (1,5), 2*ones (1,5), "upper"), 1 - y) %!assert (unifcdf (x, 1, 2*ones (1,5)), y) %!assert (unifcdf (x, 1, 2*ones (1,5), "upper"), 1 - y) %!assert (unifcdf (x, ones (1,5), 2), y) %!assert (unifcdf (x, ones (1,5), 2, "upper"), 1 - y) %!assert (unifcdf (x, [2 1 NaN 1 1], 2), [NaN 0 NaN 1 1]) %!assert (unifcdf (x, [2 1 NaN 1 1], 2, "upper"), 1 - [NaN 0 NaN 1 1]) %!assert (unifcdf (x, 1, 2*[0 1 NaN 1 1]), [NaN 0 NaN 1 1]) %!assert (unifcdf (x, 1, 2*[0 1 NaN 1 1], "upper"), 1 - [NaN 0 NaN 1 1]) %!assert (unifcdf ([x(1:2) NaN x(4:5)], 1, 2), [y(1:2) NaN y(4:5)]) %!assert (unifcdf ([x(1:2) NaN x(4:5)], 1, 2, "upper"), 1 - [y(1:2) NaN y(4:5)]) ## Test class of input preserved %!assert (unifcdf ([x, NaN], 1, 2), [y, NaN]) %!assert (unifcdf (single ([x, NaN]), 1, 2), single ([y, NaN])) %!assert (unifcdf ([x, NaN], single (1), 2), single ([y, NaN])) %!assert (unifcdf ([x, NaN], 1, single (2)), single ([y, NaN])) ## Test input validation %!error unifcdf () %!error unifcdf (1) %!error unifcdf (1, 2) %!error unifcdf (1, 2, 3, 4) %!error unifcdf (1, 2, 3, "tail") %!error ... %! unifcdf (ones (3), ones (2), ones (2)) %!error ... %! unifcdf (ones (2), ones (3), ones (2)) %!error ... %! unifcdf (ones (2), ones (2), ones (3)) %!error unifcdf (i, 2, 2) %!error unifcdf (2, i, 2) %!error unifcdf (2, 2, i) statistics-release-1.6.3/inst/dist_fun/unifinv.m000066400000000000000000000107021456127120000217530ustar00rootroot00000000000000## Copyright (C) 2012 Rik Wehbring ## Copyright (C) 1995-2016 Kurt Hornik ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{x} =} unifinv (@var{p}, @var{a}, @var{b}) ## ## Inverse of the continuous uniform cumulative distribution function (iCDF). ## ## For each element of @var{p}, compute the quantile (the inverse of the CDF) of ## the continuous uniform distribution with parameters @var{a} and @var{b}, ## which define the lower and upper bounds of the interval ## @qcode{[@var{a}, @var{b}]}. The size of @var{x} is the common size of ## @var{p}, @var{a}, and @var{b}. A scalar input functions as a constant matrix ## of the same size as the other inputs. ## ## Further information about the continuous uniform distribution can be found at ## @url{https://en.wikipedia.org/wiki/Continuous_uniform_distribution} ## ## @seealso{unifcdf, unifpdf, unifrnd, unifit, unifstat} ## @end deftypefn function x = unifinv (p, a, b) ## Check for valid number of input arguments if (nargin < 3) error ("unifinv: function called with too few input arguments."); endif ## Check for common size of P, A, and B if (! isscalar (p) || ! isscalar (a) || ! isscalar (b)) [retval, p, a, b] = common_size (p, a, b); if (retval > 0) error ("unifinv: P, A, and B must be of common size or scalars."); endif endif ## Check for P, A, and B being reals if (iscomplex (p) || iscomplex (a) || iscomplex (b)) error ("unifinv: P, A, and B must not be complex."); endif ## Check for class type if (isa (p, "single") || isa (a, "single") || isa (b, "single")) x = NaN (size (p), "single"); else x = NaN (size (p)); endif ## Calculate continuous uniform iCDF for valid parameter and data range k = (p >= 0) & (p <= 1) & (a < b); x(k) = a(k) + p(k) .* (b(k) - a(k)); endfunction %!demo %! ## Plot various iCDFs from the continuous uniform distribution %! p = 0.001:0.001:0.999; %! x1 = unifinv (p, 2, 5); %! x2 = unifinv (p, 3, 9); %! plot (p, x1, "-b", p, x2, "-g") %! grid on %! xlim ([0, 1]) %! ylim ([0, 10]) %! legend ({"a = 2, b = 5", "a = 3, b = 9"}, "location", "northwest") %! title ("Continuous uniform iCDF") %! xlabel ("probability") %! ylabel ("values in x") ## Test output %!shared p %! p = [-1 0 0.5 1 2]; %!assert (unifinv (p, ones (1,5), 2*ones (1,5)), [NaN 1 1.5 2 NaN]) %!assert (unifinv (p, 0, 1), [NaN 1 1.5 2 NaN] - 1) %!assert (unifinv (p, 1, 2*ones (1,5)), [NaN 1 1.5 2 NaN]) %!assert (unifinv (p, ones (1,5), 2), [NaN 1 1.5 2 NaN]) %!assert (unifinv (p, [1 2 NaN 1 1], 2), [NaN NaN NaN 2 NaN]) %!assert (unifinv (p, 1, 2*[1 0 NaN 1 1]), [NaN NaN NaN 2 NaN]) %!assert (unifinv ([p(1:2) NaN p(4:5)], 1, 2), [NaN 1 NaN 2 NaN]) ## Test class of input preserved %!assert (unifinv ([p, NaN], 1, 2), [NaN 1 1.5 2 NaN NaN]) %!assert (unifinv (single ([p, NaN]), 1, 2), single ([NaN 1 1.5 2 NaN NaN])) %!assert (unifinv ([p, NaN], single (1), 2), single ([NaN 1 1.5 2 NaN NaN])) %!assert (unifinv ([p, NaN], 1, single (2)), single ([NaN 1 1.5 2 NaN NaN])) ## Test input validation %!error unifinv () %!error unifinv (1, 2) %!error ... %! unifinv (ones (3), ones (2), ones (2)) %!error ... %! unifinv (ones (2), ones (3), ones (2)) %!error ... %! unifinv (ones (2), ones (2), ones (3)) %!error unifinv (i, 2, 2) %!error unifinv (2, i, 2) %!error unifinv (2, 2, i) statistics-release-1.6.3/inst/dist_fun/unifpdf.m000066400000000000000000000106031456127120000217300ustar00rootroot00000000000000## Copyright (C) 2012 Rik Wehbring ## Copyright (C) 1995-2016 Kurt Hornik ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{y} =} unifpdf (@var{x}, @var{a}, @var{b}) ## ## Continuous uniform probability density function (PDF). ## ## For each element of @var{x}, compute the probability density function (PDF) ## of the continuous uniform distribution with parameters @var{a} and @var{b}, ## which define the lower and upper bounds of the interval ## @qcode{[@var{a}, @var{b}]}. The size of @var{y} is the common size of ## @var{x}, @var{a}, and @var{b}. A scalar input functions as a constant matrix ## of the same size as the other inputs. ## ## Further information about the continuous uniform distribution can be found at ## @url{https://en.wikipedia.org/wiki/Continuous_uniform_distribution} ## ## @seealso{unifcdf, unifinv, unifrnd, unifit, unifstat} ## @end deftypefn function y = unifpdf (x, a, b) ## Check for valid number of input arguments if (nargin < 3) error ("unifpdf: function called with too few input arguments."); endif ## Check for common size of X, A, and B if (! isscalar (x) || ! isscalar (a) || ! isscalar (b)) [retval, x, a, b] = common_size (x, a, b); if (retval > 0) error ("unifpdf: X, A, and B must be of common size or scalars."); endif endif ## Check for X, A, and B being reals if (iscomplex (x) || iscomplex (a) || iscomplex (b)) error ("unifpdf: X, A, and B must not be complex."); endif ## Check for class type if (isa (x, "single") || isa (a, "single") || isa (b, "single")) y = zeros (size (x), "single"); else y = zeros (size (x)); endif ## Calculate continuous uniform PDF for valid parameter and data range k = isnan (x) | ! (a < b); y(k) = NaN; k = (x >= a) & (x <= b) & (a < b); y(k) = 1 ./ (b(k) - a(k)); endfunction %!demo %! ## Plot various PDFs from the continuous uniform distribution %! x = 0:0.001:10; %! y1 = unifpdf (x, 2, 5); %! y2 = unifpdf (x, 3, 9); %! plot (x, y1, "-b", x, y2, "-g") %! grid on %! xlim ([0, 10]) %! ylim ([0, 0.4]) %! legend ({"a = 2, b = 5", "a = 3, b = 9"}, "location", "northeast") %! title ("Continuous uniform PDF") %! xlabel ("values in x") %! ylabel ("density") ## Test output %!shared x, y %! x = [-1 0 0.5 1 2] + 1; %! y = [0 1 1 1 0]; %!assert (unifpdf (x, ones (1,5), 2*ones (1,5)), y) %!assert (unifpdf (x, 1, 2*ones (1,5)), y) %!assert (unifpdf (x, ones (1,5), 2), y) %!assert (unifpdf (x, [2 NaN 1 1 1], 2), [NaN NaN y(3:5)]) %!assert (unifpdf (x, 1, 2*[0 NaN 1 1 1]), [NaN NaN y(3:5)]) %!assert (unifpdf ([x, NaN], 1, 2), [y, NaN]) %!assert (unifpdf (x, 0, 1), [1 1 0 0 0]) ## Test class of input preserved %!assert (unifpdf (single ([x, NaN]), 1, 2), single ([y, NaN])) %!assert (unifpdf (single ([x, NaN]), single (1), 2), single ([y, NaN])) %!assert (unifpdf ([x, NaN], 1, single (2)), single ([y, NaN])) ## Test input validation %!error unifpdf () %!error unifpdf (1) %!error unifpdf (1, 2) %!error ... %! unifpdf (ones (3), ones (2), ones (2)) %!error ... %! unifpdf (ones (2), ones (3), ones (2)) %!error ... %! unifpdf (ones (2), ones (2), ones (3)) %!error unifpdf (i, 2, 2) %!error unifpdf (2, i, 2) %!error unifpdf (2, 2, i) statistics-release-1.6.3/inst/dist_fun/unifrnd.m000066400000000000000000000151131456127120000217430ustar00rootroot00000000000000## Copyright (C) 2012 Rik Wehbring ## Copyright (C) 1995-2016 Kurt Hornik ## Copyright (C) 2019 Anthony Morast ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{r} =} unifrnd (@var{a}, @var{b}) ## @deftypefnx {statistics} {@var{r} =} unifrnd (@var{a}, @var{b}, @var{rows}) ## @deftypefnx {statistics} {@var{r} =} unifrnd (@var{a}, @var{b}, @var{rows}, @var{cols}, @dots{}) ## @deftypefnx {statistics} {@var{r} =} unifrnd (@var{a}, @var{b}, [@var{sz}]) ## ## Random arrays from the continuous uniform distribution. ## ## @code{@var{r} = unifrnd (@var{a}, @var{b})} returns an array of random ## numbers chosen from the continuous uniform distribution with parameters ## @var{a} and @var{b}, which define the lower and upper bounds of the interval ## @qcode{[@var{a}, @var{b}]}. The size of @var{r} is the common size of ## @var{a} and @var{b}. A scalar input functions as a constant matrix of the ## same size as the other inputs. ## ## When called with a single size argument, @code{unifrnd} returns a square ## matrix with the dimension specified. When called with more than one scalar ## argument, the first two arguments are taken as the number of rows and columns ## and any further arguments specify additional matrix dimensions. The size may ## also be specified with a row vector of dimensions, @var{sz}. ## ## Further information about the continuous uniform distribution can be found at ## @url{https://en.wikipedia.org/wiki/Continuous_uniform_distribution} ## ## @seealso{unifcdf, unifinv, unifpdf, unifit, unifstat} ## @end deftypefn function r = unifrnd (a, b, varargin) ## Check for valid number of input arguments if (nargin < 2) error ("unifrnd: function called with too few input arguments."); endif ## Check for common size of A and B if (! isscalar (a) || ! isscalar (b)) [retval, a, b] = common_size (a, b); if (retval > 0) error ("unifrnd: A and B must be of common size or scalars."); endif endif ## Check for A and B being reals if (iscomplex (a) || iscomplex (b)) error ("unifrnd: A and B must not be complex."); endif ## Parse and check SIZE arguments if (nargin == 2) sz = size (a); elseif (nargin == 3) if (isscalar (varargin{1}) && varargin{1} >= 0 ... && varargin{1} == fix (varargin{1})) sz = [varargin{1}, varargin{1}]; elseif (isrow (varargin{1}) && all (varargin{1} >= 0) ... && all (varargin{1} == fix (varargin{1}))) sz = varargin{1}; elseif error (strcat (["unifrnd: SZ must be a scalar or a row vector"], ... [" of non-negative integers."])); endif elseif (nargin > 3) posint = cellfun (@(x) (! isscalar (x) || x < 0 || x != fix (x)), varargin); if (any (posint)) error ("unifrnd: dimensions must be non-negative integers."); endif sz = [varargin{:}]; endif ## Check that parameters match requested dimensions in size if (! isscalar (a) && ! isequal (size (a), sz)) error ("unifrnd: A and B must be scalars or of size SZ."); endif ## Check for class type if (isa (a, "single") || isa (b, "single")) cls = "single"; else cls = "double"; endif if (isscalar (a) && isscalar (b)) if ((-Inf < a) && (a <= b) && (b < Inf)) r = a + (b - a) * rand (sz, cls); else r = NaN (sz, cls); endif else r = a + (b - a) .* rand (sz, cls); k = !(-Inf < a) | !(a <= b) | !(b < Inf); r(k) = NaN; endif endfunction ## Test output %!assert (size (unifrnd (1, 1)), [1 1]) %!assert (size (unifrnd (1, ones (2,1))), [2, 1]) %!assert (size (unifrnd (1, ones (2,2))), [2, 2]) %!assert (size (unifrnd (ones (2,1), 1)), [2, 1]) %!assert (size (unifrnd (ones (2,2), 1)), [2, 2]) %!assert (size (unifrnd (1, 1, 3)), [3, 3]) %!assert (size (unifrnd (1, 1, [4, 1])), [4, 1]) %!assert (size (unifrnd (1, 1, 4, 1)), [4, 1]) %!assert (size (unifrnd (1, 1, 4, 1, 5)), [4, 1, 5]) %!assert (size (unifrnd (1, 1, 0, 1)), [0, 1]) %!assert (size (unifrnd (1, 1, 1, 0)), [1, 0]) %!assert (size (unifrnd (1, 1, 1, 2, 0, 5)), [1, 2, 0, 5]) ## Test class of input preserved %!assert (class (unifrnd (1, 1)), "double") %!assert (class (unifrnd (1, single (1))), "single") %!assert (class (unifrnd (1, single ([1, 1]))), "single") %!assert (class (unifrnd (single (1), 1)), "single") %!assert (class (unifrnd (single ([1, 1]), 1)), "single") ## Test input validation %!error unifrnd () %!error unifrnd (1) %!error ... %! unifrnd (ones (3), ones (2)) %!error ... %! unifrnd (ones (2), ones (3)) %!error unifrnd (i, 2, 3) %!error unifrnd (1, i, 3) %!error ... %! unifrnd (1, 2, -1) %!error ... %! unifrnd (1, 2, 1.2) %!error ... %! unifrnd (1, 2, ones (2)) %!error ... %! unifrnd (1, 2, [2 -1 2]) %!error ... %! unifrnd (1, 2, [2 0 2.5]) %!error ... %! unifrnd (1, 2, 2, -1, 5) %!error ... %! unifrnd (1, 2, 2, 1.5, 5) %!error ... %! unifrnd (2, ones (2), 3) %!error ... %! unifrnd (2, ones (2), [3, 2]) %!error ... %! unifrnd (2, ones (2), 3, 2) statistics-release-1.6.3/inst/dist_fun/vmcdf.m000066400000000000000000000130351456127120000213760ustar00rootroot00000000000000## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{p} =} vmcdf (@var{x}, @var{mu}, @var{k}) ## @deftypefnx {statistics} {@var{p} =} vmcdf (@var{x}, @var{mu}, @var{k}, @qcode{"upper"}) ## ## Von Mises probability density function (PDF). ## ## For each element of @var{x}, compute the cumulative distribution function ## (CDF) of the von Mises distribution with location parameter @var{mu} and ## concentration parameter @var{k} on the interval @math{[-pi,pi]}. The size of ## @var{p} is the common size of @var{x}, @var{mu}, and @var{k}. A scalar input ## functions as a constant matrix of the same same size as the other inputs. ## ## @code{@var{p} = vmcdf (@var{x}, @var{mu}, @var{k}, "upper")} computes the ## upper tail probability of the von Mises distribution with parameters @var{mu} ## and @var{k}, at the values in @var{x}. ## ## Note: the CDF of the von Mises distribution is not analytic. Hence, it is ## calculated by integrating its probability density which is expressed as a ## series of Bessel functions. Balancing between performance and accuracy, the ## integration uses a step of @qcode{1e-5} on the interval @math{[-pi,pi]}, ## which results to an accuracy of about 10 significant digits. ## ## Further information about the von Mises distribution can be found at ## @url{https://en.wikipedia.org/wiki/Von_Mises_distribution} ## ## @seealso{vminv, vmpdf, vmrnd} ## @end deftypefn function p = vmcdf (x, mu, k, uflag) ## Check for valid number of input arguments if (nargin < 3) error ("vmcdf: function called with too few input arguments."); endif ## Check for valid "upper" flag if (nargin > 3) if (! strcmpi (uflag, "upper")) error ("vmcdf: invalid argument for upper tail."); else uflag = true; endif else uflag = false; endif ## Check for common size of X, MU, and K if (! isscalar (x) || ! isscalar (mu) || ! isscalar (k)) [retval, x, mu, k] = common_size (x, mu, k); if (retval > 0) error ("vmcdf: X, MU, and K must be of common size or scalars."); endif endif ## Check for X, MU, and K being reals if (iscomplex (x) || iscomplex (mu) || iscomplex (k)) error ("vmcdf: X, MU, and K must not be complex."); endif ## Check for class type if (isa (x, "single") || isa (mu, "single") || isa (k, "single")) p = zeros (size (x), "single"); else p = zeros (size (x)); endif ## Evaluate Von Mises CDF by integrating from -PI to PI interval = linspace (-pi, pi, 1e5)'; # accurate to >10 significant digits f = exp (k .* cos (interval)) ./ (2 .* pi .* besseli (0, k)); c = cumtrapz (interval, f); p = diag (interp1 (interval, c, x - mu, "spline"))'; ## Force Nan for negative K p(k < 0) = NaN; ## Apply upper flag (if required) if (uflag) p = 1 - p; endif endfunction %!demo %! ## Plot various CDFs from the von Mises distribution %! x1 = [-pi:0.1:pi]; %! p1 = vmcdf (x1, 0, 0.5); %! p2 = vmcdf (x1, 0, 1); %! p3 = vmcdf (x1, 0, 2); %! p4 = vmcdf (x1, 0, 4); %! plot (x1, p1, "-r", x1, p2, "-g", x1, p3, "-b", x1, p4, "-c") %! grid on %! xlim ([-pi, pi]) %! legend ({"μ = 0, k = 0.5", "μ = 0, k = 1", ... %! "μ = 0, k = 2", "μ = 0, k = 4"}, "location", "northwest") %! title ("Von Mises CDF") %! xlabel ("values in x") %! ylabel ("probability") ## Test output %!shared x, p0, p1 %! x = [-pi:pi/2:pi]; %! p0 = [0, 0.10975, 0.5, 0.89025, 1]; %! p1 = [0, 0.03752, 0.5, 0.99622, 1]; %!assert (vmcdf (x, 0, 1), p0, 1e-5) %!assert (vmcdf (x, 0, 1, "upper"), 1 - p0, 1e-5) %!assert (vmcdf (x, zeros (1,5), ones (1,5)), p0, 1e-5) %!assert (vmcdf (x, zeros (1,5), ones (1,5), "upper"), 1 - p0, 1e-5) %!assert (vmcdf (x, 0, [1 2 3 4 5]), p1, 1e-5) %!assert (vmcdf (x, 0, [1 2 3 4 5], "upper"), 1 - p1, 1e-5) ## Test class of input preserved %!assert (isa (vmcdf (single (pi), 0, 1), "single"), true) %!assert (isa (vmcdf (pi, single (0), 1), "single"), true) %!assert (isa (vmcdf (pi, 0, single (1)), "single"), true) ## Test input validation %!error vmcdf () %!error vmcdf (1) %!error vmcdf (1, 2) %!error vmcdf (1, 2, 3, "tail") %!error vmcdf (1, 2, 3, 4) %!error ... %! vmcdf (ones (3), ones (2), ones (2)) %!error ... %! vmcdf (ones (2), ones (3), ones (2)) %!error ... %! vmcdf (ones (2), ones (2), ones (3)) %!error vmcdf (i, 2, 2) %!error vmcdf (2, i, 2) %!error vmcdf (2, 2, i) statistics-release-1.6.3/inst/dist_fun/vminv.m000066400000000000000000000131571456127120000214430ustar00rootroot00000000000000## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{x} =} vminv (@var{p}, @var{mu}, @var{k}) ## ## Inverse of the von Mises cumulative distribution function (iCDF). ## ## For each element of @var{p}, compute the quantile (the inverse of the CDF) of ## the von Mises distribution with location parameter @var{mu} and concentration ## parameter @var{k} on the interval @math{[-pi,pi]}. The size of @var{x} is ## the common size of @var{p}, @var{mu}, and @var{k}. A scalar input functions ## as a constant matrix of the same size as the other inputs. ## ## Note: the quantile of the von Mises distribution is not analytic. Hence, it ## is approximated by a custom searching algorithm using its CDF until it ## converges up to a tolerance of @qcode{1e-5} or 100 iterations. As a result, ## balancing between performance and accuracy, the accuracy is about ## @qcode{5e-5} for @qcode{@var{k} = 1} and it drops to @qcode{5e-5} as @var{k} ## increases. ## ## Further information about the von Mises distribution can be found at ## @url{https://en.wikipedia.org/wiki/Von_Mises_distribution} ## ## @seealso{vmcdf, vmpdf, vmrnd} ## @end deftypefn function x = vminv (p, mu, k) ## Check for valid number of input arguments if (nargin < 3) error ("vminv: function called with too few input arguments."); endif ## Check for common size of P, MU, and K [err, p, mu, k] = common_size (p, mu, k); if (err > 0) error ("vminv: P, MU, and K must be of common size or scalars."); endif ## Check for P, MU, and K being reals if (iscomplex (p) || iscomplex (mu) || iscomplex (k)) error ("vminv: P, MU, and K must not be complex."); endif ## Check for class type if (isa (p, "single") || isa (mu, "single") || isa (k, "single")) x = NaN (size (p), "single"); else x = NaN (size (p), "double"); endif ## Process edge cases p=0, p=0.5, p=1 p_0 = p < eps (class (x)) & k > 0 & isfinite (mu); x(p_0) = -pi + mu(p_0); p_5 = abs (p - 0.5) < eps (class (x)) & k > 0 & isfinite (mu); x(p_5) = mu(p_5); p_1 = 1 - p < eps (class (x)) & k > 0 & isfinite (mu); x(p_1) = pi + mu(p_1); ## Get remaining valid cases valc = p > 0 & p < 1 & ! p_5 & k > 0 & isfinite (mu); if (! any (valc)) return endif p = p(valc); mu = mu(valc); k = k(valc); ## Complement cases of p<0.5 to 1-p and keep track to invert them at the end comp = p < 0.5; p(comp) = 1 - p(comp); ## Initialize counter and threshold crit = 1e-5; count_limit = 100; count = 0; ## Supply a starting guess for the iteration by linear interpolation with k=0 x0 = 2 * pi .* p - pi; xz = zeros (size (p)); xa = xz; ## Compute p0 and compare to target p p0 = vmcdf (x0, 0, k); ## Solution is always 0 < x < x0. Search for x until p == p0 within threshold while (any (abs (p - p0) > crit) && count < count_limit) count = count + 1; xnew = xz + (abs (x0) - abs (xz)) .* 0.5; p0 = vmcdf (xnew, 0, k); ## Prepare for next step xdec = (p0 - p) > crit; xinc = (p - p0) > crit; if (any (xdec)) x0(xdec) = xnew(xdec); endif if (any (xinc)) xz(xinc) = xnew(xinc); endif endwhile ## Return the converged value(s). xnew(comp) = -xnew(comp); x(valc) = xnew + mu; if (count == count_limit) warning ("vminv: did not converge."); endif endfunction %!demo %! ## Plot various iCDFs from the von Mises distribution %! p1 = [0,0.005,0.01:0.01:0.1,0.15,0.2:0.1:0.8,0.85,0.9:0.01:0.99,0.995,1]; %! x1 = vminv (p1, 0, 0.5); %! x2 = vminv (p1, 0, 1); %! x3 = vminv (p1, 0, 2); %! x4 = vminv (p1, 0, 4); %! plot (p1, x1, "-r", p1, x2, "-g", p1, x3, "-b", p1, x4, "-c") %! grid on %! ylim ([-pi, pi]) %! legend ({"μ = 0, k = 0.5", "μ = 0, k = 1", ... %! "μ = 0, k = 2", "μ = 0, k = 4"}, "location", "northwest") %! title ("Von Mises iCDF") %! xlabel ("probability") %! ylabel ("values in x") ## Test output %!shared x, p0, p1 %! x = [-pi:pi/2:pi]; %! p0 = [0, 0.10975, 0.5, 0.89025, 1]; %! p1 = [0, 0.03752, 0.5, 0.99622, 1]; %!assert (vminv (p0, 0, 1), x, 5e-5) %!assert (vminv (p0, zeros (1,5), ones (1,5)), x, 5e-5) %!assert (vminv (p1, 0, [1 2 3 4 5]), x, [5e-5, 5e-4, 5e-5, 5e-4, 5e-5]) ## Test input validation %!error vminv () %!error vminv (1) %!error vminv (1, 2) %!error ... %! vminv (ones (3), ones (2), ones (2)) %!error ... %! vminv (ones (2), ones (3), ones (2)) %!error ... %! vminv (ones (2), ones (2), ones (3)) %!error vminv (i, 2, 2) %!error vminv (2, i, 2) %!error vminv (2, 2, i) statistics-release-1.6.3/inst/dist_fun/vmpdf.m000066400000000000000000000102621456127120000214120ustar00rootroot00000000000000## Copyright (C) 2009 Soren Hauberg ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{y} =} vmpdf (@var{x}, @var{mu}, @var{k}) ## ## Von Mises probability density function (PDF). ## ## For each element of @var{x}, compute the probability density function (PDF) ## of the von Mises distribution with location parameter @var{mu} and ## concentration parameter @var{k} on the interval [-pi, pi]. The size of ## @var{y} is the common size of @var{x}, @var{mu}, and @var{k}. A scalar input ## functions as a constant matrix of the same size as the other inputs. ## ## Further information about the von Mises distribution can be found at ## @url{https://en.wikipedia.org/wiki/Von_Mises_distribution} ## ## @seealso{vmcdf, vminv, vmrnd} ## @end deftypefn function y = vmpdf (x, mu, k) ## Check for valid number of input arguments if (nargin < 3) error ("vmpdf: function called with too few input arguments."); endif ## Check for common size of X, MU, and K if (! isscalar (x) || ! isscalar (mu) || ! isscalar (k)) [retval, x, mu, k] = common_size (x, mu, k); if (retval > 0) error ("vmpdf: X, MU, and K must be of common size or scalars."); endif endif ## Check for X, MU, and K being reals if (iscomplex (x) || iscomplex (mu) || iscomplex (k)) error ("vmpdf: X, MU, and K must not be complex."); endif ## Evaluate Von Mises PDF Z = 2 .* pi .* besseli (0, k); y = exp (k .* cos (x - mu)) ./ Z; ## Force Nan for negative K y(k < 0) = NaN; ## Check for class type if (isa (x, "single") || isa (mu, "single") || isa (k, "single")) y = cast (y, "single"); else y = cast (y, "double"); endif endfunction %!demo %! ## Plot various PDFs from the von Mises distribution %! x1 = [-pi:0.1:pi]; %! y1 = vmpdf (x1, 0, 0.5); %! y2 = vmpdf (x1, 0, 1); %! y3 = vmpdf (x1, 0, 2); %! y4 = vmpdf (x1, 0, 4); %! plot (x1, y1, "-r", x1, y2, "-g", x1, y3, "-b", x1, y4, "-c") %! grid on %! xlim ([-pi, pi]) %! ylim ([0, 0.8]) %! legend ({"μ = 0, k = 0.5", "μ = 0, k = 1", ... %! "μ = 0, k = 2", "μ = 0, k = 4"}, "location", "northwest") %! title ("Von Mises PDF") %! xlabel ("values in x") %! ylabel ("density") ## Test output %!shared x, y0, y1 %! x = [-pi:pi/2:pi]; %! y0 = [0.046245, 0.125708, 0.341710, 0.125708, 0.046245]; %! y1 = [0.046245, 0.069817, 0.654958, 0.014082, 0.000039]; %!assert (vmpdf (x, 0, 1), y0, 1e-5) %!assert (vmpdf (x, zeros (1,5), ones (1,5)), y0, 1e-6) %!assert (vmpdf (x, 0, [1 2 3 4 5]), y1, 1e-6) ## Test class of input preserved %!assert (isa (vmpdf (single (pi), 0, 1), "single"), true) %!assert (isa (vmpdf (pi, single (0), 1), "single"), true) %!assert (isa (vmpdf (pi, 0, single (1)), "single"), true) ## Test input validation %!error vmpdf () %!error vmpdf (1) %!error vmpdf (1, 2) %!error ... %! vmpdf (ones (3), ones (2), ones (2)) %!error ... %! vmpdf (ones (2), ones (3), ones (2)) %!error ... %! vmpdf (ones (2), ones (2), ones (3)) %!error vmpdf (i, 2, 2) %!error vmpdf (2, i, 2) %!error vmpdf (2, 2, i) statistics-release-1.6.3/inst/dist_fun/vmrnd.m000066400000000000000000000164461456127120000214360ustar00rootroot00000000000000## Copyright (C) 2009 Soren Hauberg ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{r} =} vmrnd (@var{mu}, @var{k}) ## @deftypefnx {statistics} {@var{r} =} vmrnd (@var{mu}, @var{k}, @var{rows}) ## @deftypefnx {statistics} {@var{r} =} vmrnd (@var{mu}, @var{k}, @var{rows}, @var{cols}, @dots{}) ## @deftypefnx {statistics} {@var{r} =} vmrnd (@var{mu}, @var{k}, [@var{sz}]) ## ## Random arrays from the von Mises distribution. ## ## @code{@var{r} = vmrnd (@var{mu}, @var{k})} returns an array of random angles ## chosen from a von Mises distribution with location parameter @var{mu} and ## concentration parameter @var{k} on the interval [-pi, pi]. The size of ## @var{r} is the common size of @var{mu} and @var{k}. A scalar input functions ## as a constant matrix of the same size as the other inputs. Both parameters ## must be finite real numbers and @var{k} > 0, otherwise NaN is returned. ## ## When called with a single size argument, @code{vmrnd} returns a square ## matrix with the dimension specified. When called with more than one scalar ## argument, the first two arguments are taken as the number of rows and columns ## and any further arguments specify additional matrix dimensions. The size may ## also be specified with a row vector of dimensions, @var{sz}. ## ## Further information about the von Mises distribution can be found at ## @url{https://en.wikipedia.org/wiki/Von_Mises_distribution} ## ## @seealso{vmcdf, vminv, vmpdf} ## @end deftypefn function r = vmrnd (mu, k, varargin) ## Check for valid number of input arguments if (nargin < 2) error ("vmrnd: function called with too few input arguments."); endif ## Check for common size of MU and Κ if (! isscalar (mu) || ! isscalar (k)) [retval, mu, k] = common_size (mu, k); if (retval > 0) error ("vmrnd: MU and K must be of common size or scalars."); endif endif ## Check for MU and Κ being reals if (iscomplex (mu) || iscomplex (k)) error ("vmrnd: MU and K must not be complex."); endif ## Parse and check SIZE arguments if (nargin == 2) sz = size (mu); elseif (nargin == 3) if (isscalar (varargin{1}) && varargin{1} >= 0 ... && varargin{1} == fix (varargin{1})) sz = [varargin{1}, varargin{1}]; elseif (isrow (varargin{1}) && all (varargin{1} >= 0) ... && all (varargin{1} == fix (varargin{1}))) sz = varargin{1}; elseif error (strcat (["vmrnd: SZ must be a scalar or a row vector"], ... [" of non-negative integers."])); endif elseif (nargin > 3) posint = cellfun (@(x) (! isscalar (x) || x < 0 || x != fix (x)), varargin); if (any (posint)) error ("vmrnd: dimensions must be non-negative integers."); endif sz = [varargin{:}]; endif ## Check that parameters match requested dimensions in size if (! isscalar (mu) && ! isequal (size (k), sz)) error ("vmrnd: MU and K must be scalars or of size SZ."); endif ## Check for class type if (isa (mu, "single") || isa (k, "single")) cls = "single"; else cls = "double"; endif ## Handle zero size dimensions if (any (sz == 0)) r = nan (sz, cls); return endif ## Simulate! if (all (k < 1e-6)) ## k is small: sample uniformly on circle r = mu + (2 * pi * rand (sz) - pi); else a = 1 + sqrt (1 + 4 .* k .^ 2); b = (a - sqrt (2 .* a)) ./ (2 .* k); r_tmp = (1 + b .^ 2) ./ (2 .* b); N = prod (sz); if (isscalar (k)) r_tmp = repmat (r_tmp, 1, N); k_tmp = repmat (k, 1, N); mu_rs = repmat (mu, 1, N); else r_tmp = reshape (r_tmp, 1, N); k_tmp = reshape (k, 1, N); mu_rs = reshape (mu, 1, N); endif notdone = true (N, 1); while (any (notdone)) u (:, notdone) = (rand (3, N))(:,notdone); z (notdone) = (cos (pi .* u (1, :)))(notdone); f (notdone) = ((1 + r_tmp(notdone) .* z(notdone)) ./ (r_tmp(notdone) + z(notdone))); c (notdone) = (k_tmp(notdone) .* (r_tmp(notdone) - f(notdone))); notdone = (u (2, :) >= c .* (2 - c)) & (log (c) - log (u (2, :)) + 1 - c < 0); #N = sum (notdone); endwhile r = mu_rs + sign (u (3, :) - 0.5) .* acos (f); r = reshape (r, sz); endif ## Cast to appropriate class r = cast (r, cls); endfunction ## Test output %!assert (size (vmrnd (1, 1)), [1 1]) %!assert (size (vmrnd (1, ones (2,1))), [2, 1]) %!assert (size (vmrnd (1, ones (2,2))), [2, 2]) %!assert (size (vmrnd (ones (2,1), 1)), [2, 1]) %!assert (size (vmrnd (ones (2,2), 1)), [2, 2]) %!assert (size (vmrnd (1, 1, 3)), [3, 3]) %!assert (size (vmrnd (1, 1, [4, 1])), [4, 1]) %!assert (size (vmrnd (1, 1, 4, 1)), [4, 1]) %!assert (size (vmrnd (1, 1, 4, 1, 5)), [4, 1, 5]) %!assert (size (vmrnd (1, 1, 0, 1)), [0, 1]) %!assert (size (vmrnd (1, 1, 1, 0)), [1, 0]) %!assert (size (vmrnd (1, 1, 1, 2, 0, 5)), [1, 2, 0, 5]) ## Test class of input preserved %!assert (class (vmrnd (1, 1)), "double") %!assert (class (vmrnd (1, single (1))), "single") %!assert (class (vmrnd (1, single ([1, 1]))), "single") %!assert (class (vmrnd (single (1), 1)), "single") %!assert (class (vmrnd (single ([1, 1]), 1)), "single") ## Test input validation %!error vmrnd () %!error vmrnd (1) %!error ... %! vmrnd (ones (3), ones (2)) %!error ... %! vmrnd (ones (2), ones (3)) %!error vmrnd (i, 2, 3) %!error vmrnd (1, i, 3) %!error ... %! vmrnd (1, 2, -1) %!error ... %! vmrnd (1, 2, 1.2) %!error ... %! vmrnd (1, 2, ones (2)) %!error ... %! vmrnd (1, 2, [2 -1 2]) %!error ... %! vmrnd (1, 2, [2 0 2.5]) %!error ... %! vmrnd (1, 2, 2, -1, 5) %!error ... %! vmrnd (1, 2, 2, 1.5, 5) %!error ... %! vmrnd (2, ones (2), 3) %!error ... %! vmrnd (2, ones (2), [3, 2]) %!error ... %! vmrnd (2, ones (2), 3, 2) statistics-release-1.6.3/inst/dist_fun/wblcdf.m000066400000000000000000000212021456127120000215330ustar00rootroot00000000000000## Copyright (C) 2012 Rik Wehbring ## Copyright (C) 1995-2016 Kurt Hornik ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received lambda copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{p} =} wblcdf (@var{x}) ## @deftypefnx {statistics} {@var{p} =} wblcdf (@var{x}, @var{lambda}) ## @deftypefnx {statistics} {@var{p} =} wblcdf (@var{x}, @var{lambda}, @var{k}) ## @deftypefnx {statistics} {@var{p} =} wblcdf (@dots{}, @qcode{"upper"}) ## @deftypefnx {statistics} {[@var{p}, @var{plo}, @var{pup}] =} wblcdf (@var{x}, @var{lambda}, @var{k}, @var{pcov}) ## @deftypefnx {statistics} {[@var{p}, @var{plo}, @var{pup}] =} wblcdf (@var{x}, @var{lambda}, @var{k}, @var{pcov}, @var{alpha}) ## @deftypefnx {statistics} {[@var{p}, @var{plo}, @var{pup}] =} wblcdf (@dots{}, @qcode{"upper"}) ## ## Weibull cumulative distribution function (CDF). ## ## For each element of @var{x}, compute the cumulative distribution function ## (CDF) of the Weibull distribution with scale parameter @var{lambda} and shape ## parameter @var{k}. The size of @var{p} is the common size of @var{x}, ## @var{lambda} and @var{k}. A scalar input functions as a constant matrix of ## the same size as the other inputs. ## ## Default values are @var{lambda} = 1, @var{k} = 1. ## ## When called with three output arguments, @code{[@var{p}, @var{plo}, ## @var{pup}]} it computes the confidence bounds for @var{p} when the input ## parameters @var{lambda} and @var{k} are estimates. In such case, @var{pcov}, ## a 2-by-2 matrix containing the covariance matrix of the estimated parameters, ## is necessary. Optionally, @var{alpha} has a default value of 0.05, and ## specifies 100 * (1 - @var{alpha})% confidence bounds. @var{plo} and @var{pup} ## are arrays of the same size as @var{p} containing the lower and upper ## confidence bounds. ## ## @code{[@dots{}] = wblcdf (@dots{}, "upper")} computes the upper tail ## probability of the lognormal distribution. ## ## Further information about the Weibull distribution can be found at ## @url{https://en.wikipedia.org/wiki/Weibull_distribution} ## ## @seealso{wblinv, wblpdf, wblrnd, wblstat, wblplot} ## @end deftypefn function [varargout] = wblcdf (x, varargin) ## Check for valid number of input arguments if (nargin < 1 || nargin > 6) error ("wblcdf: invalid number of input arguments."); endif ## Check for "upper" flag if (nargin > 1 && strcmpi (varargin{end}, "upper")) uflag = true; varargin(end) = []; elseif (nargin > 1 && ischar (varargin{end}) && ... ! strcmpi (varargin{end}, "upper")) error ("wblcdf: invalid argument for upper tail."); elseif (nargin > 1 && isempty (varargin{end})) uflag = false; varargin(end) = []; else uflag = false; endif ## Get extra arguments (if they exist) or add defaults if (numel (varargin) > 0) lambda = varargin{1}; else lambda = 1; endif if (numel (varargin) > 1) k = varargin{2}; else k = 1; endif if (numel (varargin) > 2) pcov = varargin{3}; ## Check for valid covariance matrix 2x2 if (! isequal (size (pcov), [2, 2])) error ("wblcdf: invalid size of covariance matrix."); endif else ## Check that cov matrix is provided if 3 output arguments are requested if (nargout > 1) error ("wblcdf: covariance matrix is required for confidence bounds."); endif pcov = []; endif if (numel (varargin) > 3) alpha = varargin{4}; ## Check for valid alpha value if (! isnumeric (alpha) || numel (alpha) !=1 || alpha <= 0 || alpha >= 1) error ("wblcdf: invalid value for alpha."); endif else alpha = 0.05; endif ## Check for common size of X, LAMBDA, and K if (! isscalar (x) || ! isscalar (lambda) || ! isscalar (k)) [err, x, lambda, k] = common_size (x, lambda, k); if (err > 0) error ("wblcdf: X, LAMBDA, and K must be of common size or scalars."); endif endif ## Check for X, LAMBDA, and K being reals if (iscomplex (x) || iscomplex (lambda) || iscomplex (k)) error ("wblcdf: X, LAMBDA, and K must not be complex."); endif ## Check for class type if (isa (x, "single") || isa (lambda, "single") || isa (k, "single")); is_class = "single"; else is_class = "double"; endif ## Return NaN for out of range parameters. lambda(lambda <= 0) = NaN; k(k <= 0) = NaN; ## Force 0 for negative data x(x < 0) = 0; ## Compute z z = (x ./ lambda) .^ k; if (uflag) p = exp(-z); else p = -expm1(-z); endif ## Compute confidence bounds (if requested) if (nargout >= 2) ## Work on log scale log_z = log (z); d_lambda = 1 ./ lambda; d_k = -1 ./ (k .^ 2); log_zvar = (pcov(1,1) .* d_lambda .^ 2 + ... 2 * pcov(1,2) .* d_lambda .* d_k .* log_z + ... pcov(2,2) .* (d_k .* log_z) .^ 2) .* (k .^ 2); if (any(log_zvar < 0)) error ("wblcdf: bad covariance matrix."); endif normz = -norminv (alpha / 2); halfwidth = normz * sqrt (log_zvar); zlo = log_z - halfwidth; zup = log_z + halfwidth; ## Convert back from log scale if uflag == true plo = exp (-exp (zup)); pup = exp (-exp (zlo)); else plo = -expm1 (-exp (zlo)); pup = -expm1 (-exp (zup)); endif endif ## Prepare output varargout{1} = cast (p, is_class); if (nargout > 1) varargout{2} = cast (plo, is_class); varargout{3} = cast (pup, is_class); endif endfunction %!demo %! ## Plot various CDFs from the Weibull distribution %! x = 0:0.001:2.5; %! p1 = wblcdf (x, 1, 0.5); %! p2 = wblcdf (x, 1, 1); %! p3 = wblcdf (x, 1, 1.5); %! p4 = wblcdf (x, 1, 5); %! plot (x, p1, "-b", x, p2, "-r", x, p3, "-m", x, p4, "-g") %! grid on %! legend ({"λ = 1, k = 0.5", "λ = 1, k = 1", ... %! "λ = 1, k = 1.5", "λ = 1, k = 5"}, "location", "southeast") %! title ("Weibull CDF") %! xlabel ("values in x") %! ylabel ("probability") ## Test output %!shared x, y %! x = [-1 0 0.5 1 Inf]; %! y = [0, 1-exp(-x(2:4)), 1]; %!assert (wblcdf (x, ones (1,5), ones (1,5)), y) %!assert (wblcdf (x, ones (1,5), ones (1,5), "upper"), 1 - y) %!assert (wblcdf (x, "upper"), 1 - y) %!assert (wblcdf (x, 1, ones (1,5)), y) %!assert (wblcdf (x, ones (1,5), 1), y) %!assert (wblcdf (x, [0 1 NaN Inf 1], 1), [NaN 0 NaN 0 1]) %!assert (wblcdf (x, [0 1 NaN Inf 1], 1, "upper"), 1 - [NaN 0 NaN 0 1]) %!assert (wblcdf (x, 1, [0 1 NaN Inf 1]), [NaN 0 NaN y(4:5)]) %!assert (wblcdf (x, 1, [0 1 NaN Inf 1], "upper"), 1 - [NaN 0 NaN y(4:5)]) %!assert (wblcdf ([x(1:2) NaN x(4:5)], 1, 1), [y(1:2) NaN y(4:5)]) %!assert (wblcdf ([x(1:2) NaN x(4:5)], 1, 1, "upper"), 1 - [y(1:2) NaN y(4:5)]) ## Test class of input preserved %!assert (wblcdf ([x, NaN], 1, 1), [y, NaN]) %!assert (wblcdf (single ([x, NaN]), 1, 1), single ([y, NaN])) %!assert (wblcdf ([x, NaN], single (1), 1), single ([y, NaN])) %!assert (wblcdf ([x, NaN], 1, single (1)), single ([y, NaN])) ## Test input validation %!error wblcdf () %!error wblcdf (1,2,3,4,5,6,7) %!error wblcdf (1, 2, 3, 4, "uper") %!error ... %! wblcdf (ones (3), ones (2), ones (2)) %!error wblcdf (2, 3, 4, [1, 2]) %!error ... %! [p, plo, pup] = wblcdf (1, 2, 3) %!error [p, plo, pup] = ... %! wblcdf (1, 2, 3, [1, 0; 0, 1], 0) %!error [p, plo, pup] = ... %! wblcdf (1, 2, 3, [1, 0; 0, 1], 1.22) %!error [p, plo, pup] = ... %! wblcdf (1, 2, 3, [1, 0; 0, 1], "alpha", "upper") %!error wblcdf (i, 2, 2) %!error wblcdf (2, i, 2) %!error wblcdf (2, 2, i) %!error ... %! [p, plo, pup] =wblcdf (1, 2, 3, [1, 0; 0, -inf], 0.04) statistics-release-1.6.3/inst/dist_fun/wblinv.m000066400000000000000000000121151456127120000215760ustar00rootroot00000000000000## Copyright (C) 2012 Rik Wehbring ## Copyright (C) 1995-2016 Kurt Hornik ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{x} =} wblinv (@var{p}) ## @deftypefnx {statistics} {@var{x} =} wblinv (@var{p}, @var{lambda}) ## @deftypefnx {statistics} {@var{x} =} wblinv (@var{p}, @var{lambda}, @var{k}) ## ## Inverse of the Weibull cumulative distribution function (iCDF). ## ## For each element of @var{p}, compute the quantile (the inverse of the CDF) ## of the Weibull distribution with scale parameter @var{lambda} and shape ## parameter @var{k}. The size of @var{x} is the common size of @var{p}, ## @var{lambda}, and @var{k}. A scalar input functions as a constant matrix of ## the same size as the other inputs. ## ## Default values are @var{lambda} = 1, @var{k} = 1. ## ## Further information about the Weibull distribution can be found at ## @url{https://en.wikipedia.org/wiki/Weibull_distribution} ## ## @seealso{wblcdf, wblpdf, wblrnd, wblstat, wblplot} ## @end deftypefn function x = wblinv (p, varargin) ## Check for valid number of input arguments if (nargin < 1 || nargin > 3) error ("wblinv: invalid number of input arguments."); endif ## Get extra arguments (if they exist) or add defaults if (numel (varargin) > 0) lambda = varargin{1}; else lambda = 1; endif if (numel (varargin) > 1) k = varargin{2}; else k = 1; endif ## Check for common size of P, LAMBDA, and K if (! isscalar (p) || ! isscalar (lambda) || ! isscalar (k)) [retval, p, lambda, k] = common_size (p, lambda, k); if (retval > 0) error ("wblinv: P, LAMBDA, and K must be of common size or scalars."); endif endif ## Check for P, LAMBDA, and K being reals if (iscomplex (p) || iscomplex (lambda) || iscomplex (k)) error ("wblinv: P, LAMBDA, and K must not be complex."); endif ## Check for class type if (isa (p, "single") || isa (lambda, "single") || isa (k, "single")) x = NaN (size (p), "single"); else x = NaN (size (p)); endif ok = (lambda > 0) & (lambda < Inf) & (k > 0) & (k < Inf); pk = (p == 0) & ok; x(pk) = 0; pk = (p == 1) & ok; x(pk) = Inf; pk = (p > 0) & (p < 1) & ok; if (isscalar (lambda) && isscalar (k)) x(pk) = lambda * (- log (1 - p(pk))) .^ (1 / k); else x(pk) = lambda(pk) .* (- log (1 - p(pk))) .^ (1 ./ k(pk)); endif endfunction %!demo %! ## Plot various iCDFs from the Weibull distribution %! p = 0.001:0.001:0.999; %! x1 = wblinv (p, 1, 0.5); %! x2 = wblinv (p, 1, 1); %! x3 = wblinv (p, 1, 1.5); %! x4 = wblinv (p, 1, 5); %! plot (p, x1, "-b", p, x2, "-r", p, x3, "-m", p, x4, "-g") %! ylim ([0, 2.5]) %! grid on %! legend ({"λ = 1, k = 0.5", "λ = 1, k = 1", ... %! "λ = 1, k = 1.5", "λ = 1, k = 5"}, "location", "northwest") %! title ("Weibull iCDF") %! xlabel ("probability") %! ylabel ("x") ## Test output %!shared p %! p = [-1 0 0.63212055882855778 1 2]; %!assert (wblinv (p, ones (1,5), ones (1,5)), [NaN 0 1 Inf NaN], eps) %!assert (wblinv (p, 1, ones (1,5)), [NaN 0 1 Inf NaN], eps) %!assert (wblinv (p, ones (1,5), 1), [NaN 0 1 Inf NaN], eps) %!assert (wblinv (p, [1 -1 NaN Inf 1], 1), [NaN NaN NaN NaN NaN]) %!assert (wblinv (p, 1, [1 -1 NaN Inf 1]), [NaN NaN NaN NaN NaN]) %!assert (wblinv ([p(1:2) NaN p(4:5)], 1, 1), [NaN 0 NaN Inf NaN]) ## Test class of input preserved %!assert (wblinv ([p, NaN], 1, 1), [NaN 0 1 Inf NaN NaN], eps) %!assert (wblinv (single ([p, NaN]), 1, 1), single ([NaN 0 1 Inf NaN NaN]), eps ("single")) %!assert (wblinv ([p, NaN], single (1), 1), single ([NaN 0 1 Inf NaN NaN]), eps ("single")) %!assert (wblinv ([p, NaN], 1, single (1)), single ([NaN 0 1 Inf NaN NaN]), eps ("single")) ## Test input validation %!error wblinv () %!error wblinv (1,2,3,4) %!error ... %! wblinv (ones (3), ones (2), ones (2)) %!error ... %! wblinv (ones (2), ones (3), ones (2)) %!error ... %! wblinv (ones (2), ones (2), ones (3)) %!error wblinv (i, 2, 2) %!error wblinv (2, i, 2) %!error wblinv (2, 2, i) statistics-release-1.6.3/inst/dist_fun/wblpdf.m000066400000000000000000000110441456127120000215530ustar00rootroot00000000000000## Copyright (C) 2012 Rik Wehbring ## Copyright (C) 1995-2016 Kurt Hornik ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{y} =} wblinv (@var{x}) ## @deftypefnx {statistics} {@var{y} =} wblinv (@var{x}, @var{lambda}) ## @deftypefnx {statistics} {@var{y} =} wblinv (@var{x}, @var{lambda}, @var{k}) ## ## Weibull probability density function (PDF). ## ## For each element of @var{x}, compute the probability density function (PDF) ## of the Weibull distribution with scale parameter @var{lambda} and shpe ## parameter @var{k}. The size of @var{y} is the common size of @var{x}, ## @var{lambda}, and @var{k}. A scalar input functions as a constant matrix of ## the same size as the other inputs. ## ## Default values are @var{lambda} = 1, @var{k} = 1. ## ## Further information about the Weibull distribution can be found at ## @url{https://en.wikipedia.org/wiki/Weibull_distribution} ## ## @seealso{wblcdf, wblinv, wblrnd, wblfit, wbllike, wblstat, wblplot} ## @end deftypefn function y = wblpdf (x, varargin) ## Check for valid number of input arguments if (nargin < 1 || nargin > 3) error ("wblpdf: invalid number of input arguments."); endif ## Get extra arguments (if they exist) or add defaults if (numel (varargin) > 0) lambda = varargin{1}; else lambda = 1; endif if (numel (varargin) > 1) k = varargin{2}; else k = 1; endif ## Check for common size of X, LAMBDA, and K if (! isscalar (lambda) || ! isscalar (k)) [retval, x, lambda, k] = common_size (x, lambda, k); if (retval > 0) error ("wblpdf: X, LAMBDA, and K must be of common size or scalars."); endif endif ## Check for X, LAMBDA, and K being reals if (iscomplex (x) || iscomplex (lambda) || iscomplex (k)) error ("wblpdf: X, LAMBDA, and K must not be complex."); endif ## Check for class type if (isa (x, "single") || isa (lambda, "single") || isa (k, "single")) y = NaN (size (x), "single"); else y = NaN (size (x)); endif ok = ((lambda > 0) & (lambda < Inf) & (k > 0) & (k < Inf)); xk = (x < 0) & ok; y(xk) = 0; xk = (x >= 0) & (x < Inf) & ok; if (isscalar (lambda) && isscalar (k)) y(xk) = (k * (lambda .^ -k) ... .* (x(xk) .^ (k - 1)) ... .* exp (- (x(xk) / lambda) .^ k)); else y(xk) = (k(xk) .* (lambda(xk) .^ -k(xk)) ... .* (x(xk) .^ (k(xk) - 1)) ... .* exp (- (x(xk) ./ lambda(xk)) .^ k(xk))); endif endfunction %!demo %! ## Plot various PDFs from the Weibul distribution %! x = 0:0.001:2.5; %! y1 = wblpdf (x, 1, 0.5); %! y2 = wblpdf (x, 1, 1); %! y3 = wblpdf (x, 1, 1.5); %! y4 = wblpdf (x, 1, 5); %! plot (x, y1, "-b", x, y2, "-r", x, y3, "-m", x, y4, "-g") %! grid on %! ylim ([0, 2.5]) %! legend ({"λ = 5, k = 0.5", "λ = 9, k = 1", ... %! "λ = 6, k = 1.5", "λ = 2, k = 5"}, "location", "northeast") %! title ("Weibul PDF") %! xlabel ("values in x") %! ylabel ("density") ## Test output %!shared x,y %! x = [-1 0 0.5 1 Inf]; %! y = [0, exp(-x(2:4)), NaN]; %!assert (wblpdf (x, ones (1,5), ones (1,5)), y) %!assert (wblpdf (x, 1, ones (1,5)), y) %!assert (wblpdf (x, ones (1,5), 1), y) %!assert (wblpdf (x, [0 NaN Inf 1 1], 1), [NaN NaN NaN y(4:5)]) %!assert (wblpdf (x, 1, [0 NaN Inf 1 1]), [NaN NaN NaN y(4:5)]) %!assert (wblpdf ([x, NaN], 1, 1), [y, NaN]) ## Test class of input preserved %!assert (wblpdf (single ([x, NaN]), 1, 1), single ([y, NaN])) %!assert (wblpdf ([x, NaN], single (1), 1), single ([y, NaN])) %!assert (wblpdf ([x, NaN], 1, single (1)), single ([y, NaN])) ## Test input validation %!error wblpdf () %!error wblpdf (1,2,3,4) %!error wblpdf (ones (3), ones (2), ones (2)) %!error wblpdf (ones (2), ones (3), ones (2)) %!error wblpdf (ones (2), ones (2), ones (3)) %!error wblpdf (i, 2, 2) %!error wblpdf (2, i, 2) %!error wblpdf (2, 2, i) statistics-release-1.6.3/inst/dist_fun/wblrnd.m000066400000000000000000000152621456127120000215730ustar00rootroot00000000000000## Copyright (C) 2012 Rik Wehbring ## Copyright (C) 1995-2016 Kurt Hornik ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{r} =} wblrnd (@var{lambda}, @var{k}) ## @deftypefnx {statistics} {@var{r} =} wblrnd (@var{lambda}, @var{k}, @var{rows}) ## @deftypefnx {statistics} {@var{r} =} wblrnd (@var{lambda}, @var{k}, @var{rows}, @var{cols}, @dots{}) ## @deftypefnx {statistics} {@var{r} =} wblrnd (@var{lambda}, @var{k}, [@var{sz}]) ## ## Random arrays from the Weibull distribution. ## ## @code{@var{r} = wblrnd (@var{lambda}, @var{k})} returns an array of random ## numbers chosen from the Weibull distribution with scale parameter ## @var{lambda} and shape parameter @var{k}. The size of @var{r} is the common ## size of @var{lambda} and @var{k}. A scalar input functions as a constant ## matrix of the same size as the other inputs. Both parameters must be ## positive reals. ## ## When called with a single size argument, @code{wblrnd} returns a square ## matrix with the dimension specified. When called with more than one scalar ## argument, the first two arguments are taken as the number of rows and columns ## and any further arguments specify additional matrix dimensions. The size may ## also be specified with a row vector of dimensions, @var{sz}. ## ## Further information about the Weibull distribution can be found at ## @url{https://en.wikipedia.org/wiki/Weibull_distribution} ## ## @seealso{wblcdf, wblinv, wblpdf, wblfit, wbllike, wblstat, wblplot} ## @end deftypefn function r = wblrnd (lambda, k, varargin) ## Check for valid number of input arguments if (nargin < 2) error ("wblrnd: function called with too few input arguments."); endif ## Check for common size of LAMBDA and K if (! isscalar (lambda) || ! isscalar (k)) [retval, lambda, k] = common_size (lambda, k); if (retval > 0) error ("wblrnd: LAMBDA and K must be of common size or scalars."); endif endif ## Check for LAMBDA and K being reals if (iscomplex (lambda) || iscomplex (k)) error ("wblrnd: LAMBDA and K must not be complex."); endif ## Parse and check SIZE arguments if (nargin == 2) sz = size (lambda); elseif (nargin == 3) if (isscalar (varargin{1}) && varargin{1} >= 0 ... && varargin{1} == fix (varargin{1})) sz = [varargin{1}, varargin{1}]; elseif (isrow (varargin{1}) && all (varargin{1} >= 0) ... && all (varargin{1} == fix (varargin{1}))) sz = varargin{1}; elseif error (strcat (["wblrnd: SZ must be a scalar or a row vector"], ... [" of non-negative integers."])); endif elseif (nargin > 3) posint = cellfun (@(x) (! isscalar (x) || x < 0 || x != fix (x)), varargin); if (any (posint)) error ("wblrnd: dimensions must be non-negative integers."); endif sz = [varargin{:}]; endif ## Check that parameters match requested dimensions in size if (! isscalar (lambda) && ! isequal (size (lambda), sz)) error ("wblrnd: LAMBDA and K must be scalar or of size SZ."); endif ## Check for class type if (isa (lambda, "single") || isa (k, "single")) cls = "single"; else cls = "double"; endif ## Generate random sample from Weibull distribution if (isscalar (lambda) && isscalar (k)) if ((lambda > 0) && (lambda < Inf) && (k > 0) && (k < Inf)) r = lambda * rande (sz, cls) .^ (1/k); else r = NaN (sz, cls); endif else r = lambda .* rande (sz, cls) .^ (1./k); is_nan = (lambda <= 0) | (lambda == Inf) | (k <= 0) | (k == Inf); r(is_nan) = NaN; endif endfunction ## Test output %!assert (size (wblrnd (1, 1)), [1 1]) %!assert (size (wblrnd (1, ones (2,1))), [2, 1]) %!assert (size (wblrnd (1, ones (2,2))), [2, 2]) %!assert (size (wblrnd (ones (2,1), 1)), [2, 1]) %!assert (size (wblrnd (ones (2,2), 1)), [2, 2]) %!assert (size (wblrnd (1, 1, 3)), [3, 3]) %!assert (size (wblrnd (1, 1, [4, 1])), [4, 1]) %!assert (size (wblrnd (1, 1, 4, 1)), [4, 1]) %!assert (size (wblrnd (1, 1, 4, 1, 5)), [4, 1, 5]) %!assert (size (wblrnd (1, 1, 0, 1)), [0, 1]) %!assert (size (wblrnd (1, 1, 1, 0)), [1, 0]) %!assert (size (wblrnd (1, 1, 1, 2, 0, 5)), [1, 2, 0, 5]) ## Test class of input preserved %!assert (class (wblrnd (1, 1)), "double") %!assert (class (wblrnd (1, single (1))), "single") %!assert (class (wblrnd (1, single ([1, 1]))), "single") %!assert (class (wblrnd (single (1), 1)), "single") %!assert (class (wblrnd (single ([1, 1]), 1)), "single") ## Test input validation %!error wblrnd () %!error wblrnd (1) %!error ... %! wblrnd (ones (3), ones (2)) %!error ... %! wblrnd (ones (2), ones (3)) %!error wblrnd (i, 2, 3) %!error wblrnd (1, i, 3) %!error ... %! wblrnd (1, 2, -1) %!error ... %! wblrnd (1, 2, 1.2) %!error ... %! wblrnd (1, 2, ones (2)) %!error ... %! wblrnd (1, 2, [2 -1 2]) %!error ... %! wblrnd (1, 2, [2 0 2.5]) %!error ... %! wblrnd (1, 2, 2, -1, 5) %!error ... %! wblrnd (1, 2, 2, 1.5, 5) %!error ... %! wblrnd (2, ones (2), 3) %!error ... %! wblrnd (2, ones (2), [3, 2]) %!error ... %! wblrnd (2, ones (2), 3, 2) statistics-release-1.6.3/inst/dist_fun/wienrnd.m000066400000000000000000000041211456127120000217410ustar00rootroot00000000000000## Copyright (C) 1995-2017 Friedrich Leisch ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{r} =} wienrnd (@var{t}, @var{d}, @var{n}) ## ## Return a simulated realization of the @var{d}-dimensional Wiener Process ## on the interval [0, @var{t}]. ## ## If @var{d} is omitted, @var{d} = 1 is used. The first column of the ## return matrix contains time, the remaining columns contain the Wiener ## process. ## ## The optional parameter @var{n} defines the number of summands used for ## simulating the process over an interval of length 1. If @var{n} is ## omitted, @var{n} = 1000 is used. ## @end deftypefn function r = wienrnd (t, d, n) if (nargin == 1) d = 1; n = 1000; elseif (nargin == 2) n = 1000; elseif (nargin > 3) print_usage (); endif if (! isscalar (t) || ! isscalar (d) || ! isscalar (n)) error ("wienrnd: T, D, and N must all be scalars."); endif if (! (fix (t) == t) || ! (fix (d) == d) || ! (fix (n) == n) || t <= 0 || d <= 0 || n <= 0) error ("wienrnd: T, D, and N must all be positive integers."); endif r = randn (n * t, d); r = cumsum (r) / sqrt (n); r = [((1: n*t)' / n), r]; endfunction %!error wienrnd (0) %!error wienrnd (1, 3, -50) %!error wienrnd (5, 0) %!error wienrnd (0.4, 3, 5) %!error wienrnd ([1 4], 3, 5) statistics-release-1.6.3/inst/dist_fun/wishpdf.m000066400000000000000000000053431456127120000217460ustar00rootroot00000000000000## Copyright (C) 2013 Nir Krakauer ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{y} =} wishpdf (@var{W}, @var{Sigma}, @var{df}, @var{log_y}=false) ## ## Compute the probability density function of the Wishart distribution ## ## Inputs: A @var{p} x @var{p} matrix @var{W} where to find the PDF. The @var{p} ## x @var{p} positive definite matrix @var{Sigma} and scalar degrees of freedom ## parameter @var{df} characterizing the Wishart distribution. (For the density ## to be finite, need @var{df} > (@var{p} - 1).) ## ## If the flag @var{log_y} is set, return the log probability density -- this ## helps avoid underflow when the numerical value of the density is very small ## ## Output: @var{y} is the probability density of Wishart(@var{Sigma}, @var{df}) ## at @var{W}. ## ## @seealso{wishrnd, iwishpdf, iwishrnd} ## @end deftypefn function y = wishpdf (W, Sigma, df, log_y=false) if (nargin < 3) print_usage (); endif p = size(Sigma, 1); if (df <= (p - 1)) error ("wishpdf: DF too small, no finite densities exist."); endif ## calculate the logarithm of G_d(df/2), the multivariate gamma function g = (p * (p-1) / 4) * log(pi); for i = 1:p g = g + log(gamma((df + (1 - i))/2)); endfor C = chol(Sigma); ## use formulas for determinant of positive definite matrix for better ## efficiency and numerical accuracy logdet_W = 2*sum(log(diag(chol(W)))); logdet_Sigma = 2*sum(log(diag(C))); y = -(df*p)/2 * log(2) - (df/2)*logdet_Sigma - g + ... ((df - p - 1)/2)*logdet_W - trace(chol2inv(C)*W)/2; if ~log_y y = exp(y); endif endfunction ##test results cross-checked against dwish function in R MCMCpack library %!assert(wishpdf(4, 3, 3.1), 0.07702496, 1E-7); %!assert(wishpdf([2 -0.3;-0.3 4], [1 0.3;0.3 1], 4), 0.004529741, 1E-7); %!assert(wishpdf([6 2 5; 2 10 -5; 5 -5 25], [9 5 5; 5 10 -8; 5 -8 22], 5.1), 4.474865e-10, 1E-15); %% Test input validation %!error wishpdf () %!error wishpdf (1, 2) %!error wishpdf (1, 2, 0) %!error wishpdf (1, 2) statistics-release-1.6.3/inst/dist_fun/wishrnd.m000066400000000000000000000066401456127120000217610ustar00rootroot00000000000000## Copyright (C) 2013-2019 Nir Krakauer ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {[@var{W}, @var{D}] =} wishrnd (@var{Sigma}, @var{df}, @var{D}, @var{n}=1) ## ## Return a random matrix sampled from the Wishart distribution with given ## parameters ## ## Inputs: the @math{p x p} positive definite matrix @var{Sigma} (or the ## lower-triangular Cholesky factor @var{D} of @var{Sigma}) and scalar degrees ## of freedom parameter @var{df}. ## ## @var{df} can be non-integer as long as @math{@var{df} > p} ## ## Output: a random @math{p x p} matrix @var{W} from the ## Wishart(@var{Sigma}, @var{df}) distribution. If @var{n} > 1, then @var{W} is ## @var{p} x @var{p} x @var{n} and holds @var{n} such random matrices. ## (Optionally, the lower-triangular Cholesky factor @var{D} of @var{Sigma} is ## also returned.) ## ## Averaged across many samples, the mean of @var{W} should approach ## @var{df}*@var{Sigma}, and the variance of each element @var{W}_ij should ## approach @var{df}*(@var{Sigma}_ij^2 + @var{Sigma}_ii*@var{Sigma}_jj) ## ## @subheading References ## ## @enumerate ## @item ## Yu-Cheng Ku and Peter Bloomfield (2010), Generating Random Wishart Matrices ## with Fractional Degrees of Freedom in OX, ## http://www.gwu.edu/~forcpgm/YuChengKu-030510final-WishartYu-ChengKu.pdf ## @end enumerate ## ## @seealso{wishpdf, iwishpdf, iwishrnd} ## @end deftypefn function [W, D] = wishrnd (Sigma, df, D, n=1) if (nargin < 2) print_usage (); endif if nargin < 3 || isempty(D) try D = chol(Sigma, 'lower'); catch error (strcat (["iwishrnd: Cholesky decomposition failed;"], ... [" SIGMA probably not positive definite."])); end_try_catch endif p = size(D, 1); if df < p df = floor(df); #distribution not defined for small noninteger df df_isint = 1; else #check for integer degrees of freedom df_isint = (df == floor(df)); endif if ~df_isint [ii, jj] = ind2sub([p, p], 1:(p*p)); endif if n > 1 W = nan(p, p, n); endif for i = 1:n if df_isint Z = D * randn(p, df); else Z = diag(sqrt(chi2rnd(df - (0:(p-1))))); #fill diagonal ##note: chi2rnd(x) is equivalent to 2*randg(x/2), but the latter seems to ## offer no performance advantage Z(ii > jj) = randn(p*(p-1)/2, 1); #fill lower triangle Z = D * Z; endif W(:, :, i) = Z*Z'; endfor endfunction %!assert(size (wishrnd (1,2)), [1, 1]); %!assert(size (wishrnd (1,2,[])), [1, 1]); %!assert(size (wishrnd (1,2,1)), [1, 1]); %!assert(size (wishrnd ([],2,1)), [1, 1]); %!assert(size (wishrnd ([3 1; 1 3], 2.00001, [], 1)), [2, 2]); %!assert(size (wishrnd (eye(2), 2, [], 3)), [2, 2, 3]); %% Test input validation %!error wishrnd () %!error wishrnd (1) %!error wishrnd ([1; 1], 2) statistics-release-1.6.3/inst/dist_stat/000077500000000000000000000000001456127120000203025ustar00rootroot00000000000000statistics-release-1.6.3/inst/dist_stat/betastat.m000066400000000000000000000101011456127120000222600ustar00rootroot00000000000000## Copyright (C) 2006, 2007 Arno Onken ## Copyright (C) 2024 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {[@var{m}, @var{v}] =} betastat (@var{a}, @var{b}) ## ## Compute statistics of the Beta distribution. ## ## @code{[@var{m}, @var{v}] = betastat (@var{a}, @var{b})} returns the mean ## and variance of the Beta distribution with shape parameters @var{a} and ## @var{b}. ## ## The size of @var{m} (mean) and @var{v} (variance) is the common size of the ## input arguments. A scalar input functions as a constant matrix of the ## same size as the other inputs. ## ## Further information about the Beta distribution can be found at ## @url{https://en.wikipedia.org/wiki/Beta_distribution} ## ## @seealso{betacdf, betainv, betapdf, betarnd, betafit, betalike} ## @end deftypefn function [m, v] = betastat (a, b) ## Check for valid number of input arguments if (nargin < 2) error ("betastat: function called with too few input arguments."); endif ## Check for A and B being numeric if (! (isnumeric (a) && isnumeric (b))) error ("betastat: A and B must be numeric."); endif ## Check for A and B being real if (iscomplex (a) || iscomplex (b)) error ("betastat: A and B must not be complex."); endif ## Check for common size of A and B if (! isscalar (a) || ! isscalar (b)) [retval, a, b] = common_size (a, b); if (retval > 0) error ("betastat: A and B must be of common size or scalars."); endif endif ## Catch invalid parameters k = find (! (a > 0 & b > 0)); ## Calculate moments a_b = a + b; m = a ./ (a_b); m(k) = NaN; if (nargout > 1) v = (a .* b) ./ ((a_b .^ 2) .* (a_b + 1)); v(k) = NaN; endif endfunction ## Input validation tests %!error betastat () %!error betastat (1) %!error betastat ({}, 2) %!error betastat (1, "") %!error betastat (i, 2) %!error betastat (1, i) %!error ... %! betastat (ones (3), ones (2)) %!error ... %! betastat (ones (2), ones (3)) ## Output validation tests %!test %! a = -2:6; %! b = 0.4:0.2:2; %! [m, v] = betastat (a, b); %! expected_m = [NaN NaN NaN 1/2 2/3.2 3/4.4 4/5.6 5/6.8 6/8]; %! expected_v = [NaN NaN NaN 0.0833, 0.0558, 0.0402, 0.0309, 0.0250, 0.0208]; %! assert (m, expected_m, eps*100); %! assert (v, expected_v, 0.001); %!test %! a = -2:1:6; %! [m, v] = betastat (a, 1.5); %! expected_m = [NaN NaN NaN 1/2.5 2/3.5 3/4.5 4/5.5 5/6.5 6/7.5]; %! expected_v = [NaN NaN NaN 0.0686, 0.0544, 0.0404, 0.0305, 0.0237, 0.0188]; %! assert (m, expected_m); %! assert (v, expected_v, 0.001); %!test %! a = [14 Inf 10 NaN 10]; %! b = [12 9 NaN Inf 12]; %! [m, v] = betastat (a, b); %! expected_m = [14/26 NaN NaN NaN 10/22]; %! expected_v = [168/18252 NaN NaN NaN 120/11132]; %! assert (m, expected_m); %! assert (v, expected_v); %!assert (nthargout (1:2, @betastat, 5, []), {[], []}) %!assert (nthargout (1:2, @betastat, [], 5), {[], []}) %!assert (size (betastat (rand (10, 5, 4), rand (10, 5, 4))), [10 5 4]) %!assert (size (betastat (rand (10, 5, 4), 7)), [10 5 4]) statistics-release-1.6.3/inst/dist_stat/binostat.m000066400000000000000000000103021456127120000222770ustar00rootroot00000000000000## Copyright (C) 2006, 2007 Arno Onken ## Copyright (C) 2015 Carnë Draug ## Copyright (C) 2024 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {[@var{m}, @var{v}] =} binostat (@var{n}, @var{ps}) ## ## Compute statistics of the binomial distribution. ## ## @code{[@var{m}, @var{v}] = binostat (@var{n}, @var{ps})} returns the mean and ## variance of the binomial distribution with parameters @var{n} and @var{ps}, ## where @var{n} is the number of trials and @var{ps} is the probability of ## success. ## ## The size of @var{m} (mean) and @var{v} (variance) is the common size of the ## input arguments. A scalar input functions as a constant matrix of the ## same size as the other inputs. ## ## Further information about the binomial distribution can be found at ## @url{https://en.wikipedia.org/wiki/Binomial_distribution} ## ## @seealso{binocdf, binoinv, binopdf, binornd, binofit, binolike, binotest} ## @end deftypefn function [m, v] = binostat (n, ps) ## Check for valid number of input arguments if (nargin < 2) error ("binostat: function called with too few input arguments."); endif ## Check for N and PS being numeric if (! (isnumeric (n) && isnumeric (ps))) error ("binostat: N and PS must be numeric."); endif ## Check for N and PS being real if (iscomplex (n) || iscomplex (ps)) error ("binostat: N and PS must not be complex."); endif ## Check for common size of N and PS if (! isscalar (n) || ! isscalar (ps)) [retval, n, ps] = common_size (n, ps); if (retval > 0) error ("binostat: N and PS must be of common size or scalars."); endif endif ## Catch invalid parameters k = find (! (n > 0 & fix (n) == n & ps >= 0 & ps <= 1)); ## Calculate moments m = n .* ps; m(k) = NaN; if (nargout > 1) v = m .* (1 - ps); v(k) = NaN; endif endfunction ## Input validation tests %!error binostat () %!error binostat (1) %!error binostat ({}, 2) %!error binostat (1, "") %!error binostat (i, 2) %!error binostat (1, i) %!error ... %! binostat (ones (3), ones (2)) %!error ... %! binostat (ones (2), ones (3)) ## Output validation tests %!test %! n = 1:6; %! ps = 0:0.2:1; %! [m, v] = binostat (n, ps); %! expected_m = [0.00, 0.40, 1.20, 2.40, 4.00, 6.00]; %! expected_v = [0.00, 0.32, 0.72, 0.96, 0.80, 0.00]; %! assert (m, expected_m, 0.001); %! assert (v, expected_v, 0.001); %!test %! n = 1:6; %! [m, v] = binostat (n, 0.5); %! expected_m = [0.50, 1.00, 1.50, 2.00, 2.50, 3.00]; %! expected_v = [0.25, 0.50, 0.75, 1.00, 1.25, 1.50]; %! assert (m, expected_m, 0.001); %! assert (v, expected_v, 0.001); %!test %! n = [-Inf -3 5 0.5 3 NaN 100, Inf]; %! [m, v] = binostat (n, 0.5); %! assert (isnan (m), [true true false true false true false false]) %! assert (isnan (v), [true true false true false true false false]) %! assert (m(end), Inf); %! assert (v(end), Inf); %!assert (nthargout (1:2, @binostat, 5, []), {[], []}) %!assert (nthargout (1:2, @binostat, [], 5), {[], []}) %!assert (size (binostat (randi (100, 10, 5, 4), rand (10, 5, 4))), [10 5 4]) %!assert (size (binostat (randi (100, 10, 5, 4), 7)), [10 5 4]) statistics-release-1.6.3/inst/dist_stat/chi2stat.m000066400000000000000000000046041456127120000222050ustar00rootroot00000000000000## Copyright (C) 2006, 2007 Arno Onken ## Copyright (C) 2024 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {[@var{m}, @var{v}] =} chi2stat (@var{df}) ## ## Compute statistics of the chi-squared distribution. ## ## @code{[@var{m}, @var{v}] = chi2stat (@var{df})} returns the mean and ## variance of the chi-squared distribution with @var{df} degrees of freedom. ## ## The size of @var{m} (mean) and @var{v} (variance) is the same size of the ## input argument. ## ## Further information about the chi-squared distribution can be found at ## @url{https://en.wikipedia.org/wiki/Chi-squared_distribution} ## ## @seealso{chi2cdf, chi2inv, chi2pdf, chi2rnd} ## @end deftypefn function [m, v] = chi2stat (df) ## Check for valid number of input arguments if (nargin < 1) error ("chi2stat: function called with too few input arguments."); endif ## Check for DF being numeric if (! isnumeric (df)) error ("chi2stat: DF must be numeric."); endif ## Check for DF being real if (iscomplex (df)) error ("chi2stat: DF must not be complex."); endif ## Calculate moments m = df; v = 2 .* df; ## Continue argument check k = find (! (df > 0) | ! (df < Inf)); if (any (k)) m(k) = NaN; v(k) = NaN; endif endfunction ## Input validation tests %!error chi2stat () %!error chi2stat ({}) %!error chi2stat ("") %!error chi2stat (i) ## Output validation tests %!test %! df = 1:6; %! [m, v] = chi2stat (df); %! assert (m, df); %! assert (v, [2, 4, 6, 8, 10, 12], 0.001); statistics-release-1.6.3/inst/dist_stat/evstat.m000066400000000000000000000073641456127120000220000ustar00rootroot00000000000000## Copyright (C) 2022-2024 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {[@var{m}, @var{v}] =} evstat (@var{mu}, @var{sigma}) ## ## Compute statistics of the extreme value distribution. ## ## @code{[@var{m}, @var{v}] = evstat (@var{mu}, @var{sigma})} returns the mean ## and variance of the extreme value distribution (also known as the Gumbel ## or the type I generalized extreme value distribution) with location parameter ## @var{mu} and scale parameter @var{sigma}. ## ## The size of @var{m} (mean) and @var{v} (variance) is the common size of the ## input arguments. A scalar input functions as a constant matrix of the ## same size as the other inputs. ## ## The type 1 extreme value distribution is also known as the Gumbel ## distribution. This version is suitable for modeling minima. The mirror image ## of this distribution can be used to model maxima by negating @var{x}. If ## @var{y} has a Weibull distribution, then @code{@var{x} = log (@var{y})} has ## the type 1 extreme value distribution. ## ## Further information about the Gumbel distribution can be found at ## @url{https://en.wikipedia.org/wiki/Gumbel_distribution} ## ## @seealso{evcdf, evinv, evpdf, evrnd, evfit, evlike} ## @end deftypefn function [m, v] = evstat (mu, sigma) ## Check for valid number of input arguments if (nargin < 2) error ("evstat: function called with too few input arguments."); endif ## Check for MU and SIGMA being numeric if (! (isnumeric (mu) && isnumeric (sigma))) error ("evstat: MU and SIGMA must be numeric."); endif ## Check for MU and SIGMA being real if (iscomplex (mu) || iscomplex (sigma)) error ("evstat: MU and SIGMA must not be complex."); endif ## Check for common size of MU and SIGMA if (! isscalar (mu) || ! isscalar (sigma)) [retval, mu, sigma] = common_size (mu, sigma); if (retval > 0) error ("evstat: MU and SIGMA must be of common size or scalars."); endif endif ## Return NaNs for out of range values of SIGMA sigma(sigma <= 0) = NaN; ## Calculate mean and variance m = mu + psi(1) .* sigma; v = (pi .* sigma) .^ 2 ./ 6; endfunction ## Input validation tests %!error evstat () %!error evstat (1) %!error evstat ({}, 2) %!error evstat (1, "") %!error evstat (i, 2) %!error evstat (1, i) %!error ... %! evstat (ones (3), ones (2)) %!error ... %! evstat (ones (2), ones (3)) ## Output validation tests %!shared x, y0, y1 %! x = [-5, 0, 1, 2, 3]; %! y0 = [NaN, NaN, 0.4228, 0.8456, 1.2684]; %! y1 = [-5.5772, -3.4633, -3.0405, -2.6177, -2.1949]; %!assert (evstat (x, x), y0, 1e-4) %!assert (evstat (x, x+6), y1, 1e-4) %!assert (evstat (x, x-6), NaN (1,5)) statistics-release-1.6.3/inst/dist_stat/expstat.m000066400000000000000000000053371456127120000221600ustar00rootroot00000000000000## Copyright (C) 2006, 2007 Arno Onken ## Copyright (C) 2024 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {[@var{m}, @var{v}] =} expstat (@var{mu}) ## ## Compute statistics of the exponential distribution. ## ## @code{[@var{m}, @var{v}] = expstat (@var{mu})} returns the mean and ## variance of the exponential distribution with mean parameter @var{mu}. ## ## The size of @var{m} (mean) and @var{v} (variance) is the same size of the ## input argument. ## ## A common alternative parameterization of the exponential distribution is to ## use the parameter @math{λ} defined as the mean number of events in an ## interval as opposed to the parameter @math{μ}, which is the mean wait time ## for an event to occur. @math{λ} and @math{μ} are reciprocals, ## i.e. @math{μ = 1 / λ}. ## ## Further information about the exponential distribution can be found at ## @url{https://en.wikipedia.org/wiki/Exponential_distribution} ## ## @seealso{expcdf, expinv, exppdf, exprnd, expfit, explike} ## @end deftypefn function [m, v] = expstat (mu) ## Check for valid number of input arguments if (nargin < 1) error ("expstat: function called with too few input arguments."); endif ## Check for MU being numeric if (! isnumeric (mu)) error ("expstat: MU must be numeric."); endif ## Check for MU being real if (iscomplex (mu)) error ("expstat: MU must not be complex."); endif ## Calculate moments m = mu; v = m .^ 2; ## Continue argument check k = find (! (mu > 0) | ! (mu < Inf)); if (any (k)) m(k) = NaN; v(k) = NaN; endif endfunction ## Input validation tests %!error expstat () %!error expstat ({}) %!error expstat ("") %!error expstat (i) ## Output validation tests %!test %! mu = 1:6; %! [m, v] = expstat (mu); %! assert (m, [1, 2, 3, 4, 5, 6], 0.001); %! assert (v, [1, 4, 9, 16, 25, 36], 0.001); statistics-release-1.6.3/inst/dist_stat/fstat.m000066400000000000000000000073231456127120000216060ustar00rootroot00000000000000## Copyright (C) 2006, 2007 Arno Onken ## Copyright (C) 2024 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {[@var{m}, @var{v}] =} fstat (@var{df1}, @var{df2}) ## ## Compute statistics of the @math{F}-distribution. ## ## @code{[@var{m}, @var{v}] = fstat (@var{df1}, @var{df2})} returns the mean and ## variance of the @math{F}-distribution with @var{df1} and @var{df2} degrees ## of freedom. ## ## The size of @var{m} (mean) and @var{v} (variance) is the common size of the ## input arguments. A scalar input functions as a constant matrix of the ## same size as the other inputs. ## ## Further information about the @math{F}-distribution can be found at ## @url{https://en.wikipedia.org/wiki/F-distribution} ## ## @seealso{fcdf, finv, fpdf, frnd} ## @end deftypefn function [m, v] = fstat (df1, df2) ## Check for valid number of input arguments if (nargin < 2) error ("fstat: function called with too few input arguments."); endif ## Check for DF1 and DF2 being numeric if (! (isnumeric (df1) && isnumeric (df2))) error ("fstat: DF1 and DF2 must be numeric."); endif ## Check for DF1 and DF2 being real if (iscomplex (df1) || iscomplex (df2)) error ("fstat: DF1 and DF2 must not be complex."); endif ## Check for common size of DF1 and DF2 if (! isscalar (df1) || ! isscalar (df2)) [retval, df1, df2] = common_size (df1, df2); if (retval > 0) error ("fstat: DF1 and DF2 must be of common size or scalars."); endif endif ## Calculate moments m = df2 ./ (df2 - 2); v = (2 .* (df2 .^ 2) .* (df1 + df2 - 2)) ./ ... (df1 .* ((df2 - 2) .^ 2) .* (df2 - 4)); ## Continue argument check k = find (! (df1 > 0) | ! (df1 < Inf) | ! (df2 > 2) | ! (df2 < Inf)); if (any (k)) m(k) = NaN; v(k) = NaN; endif k = find (! (df2 > 4)); if (any (k)) v(k) = NaN; endif endfunction ## Input validation tests %!error fstat () %!error fstat (1) %!error fstat ({}, 2) %!error fstat (1, "") %!error fstat (i, 2) %!error fstat (1, i) %!error ... %! fstat (ones (3), ones (2)) %!error ... %! fstat (ones (2), ones (3)) ## Output validation tests %!test %! df1 = 1:6; %! df2 = 5:10; %! [m, v] = fstat (df1, df2); %! expected_mn = [1.6667, 1.5000, 1.4000, 1.3333, 1.2857, 1.2500]; %! expected_v = [22.2222, 6.7500, 3.4844, 2.2222, 1.5869, 1.2153]; %! assert (m, expected_mn, 0.001); %! assert (v, expected_v, 0.001); %!test %! df1 = 1:6; %! [m, v] = fstat (df1, 5); %! expected_mn = [1.6667, 1.6667, 1.6667, 1.6667, 1.6667, 1.6667]; %! expected_v = [22.2222, 13.8889, 11.1111, 9.7222, 8.8889, 8.3333]; %! assert (m, expected_mn, 0.001); %! assert (v, expected_v, 0.001); statistics-release-1.6.3/inst/dist_stat/gamstat.m000066400000000000000000000076711456127120000221330ustar00rootroot00000000000000## Copyright (C) 2006, 2007 Arno Onken ## Copyright (C) 2024 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received k copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {[@var{m}, @var{v}] =} gamstat (@var{k}, @var{theta}) ## ## Compute statistics of the Gamma distribution. ## ## @code{[@var{m}, @var{v}] = gamstat (@var{k}, @var{theta})} returns the mean ## and variance of the Gamma distribution with with shape parameter @var{k} and ## scale parameter @var{theta}. ## ## The size of @var{m} (mean) and @var{v} (variance) is the common size of the ## input arguments. A scalar input functions as a constant matrix of the ## same size as the other inputs. ## ## There are two equivalent parameterizations in common use: ## @enumerate ## @item With a shape parameter @math{k} and a scale parameter @math{θ}, which ## is used by @code{gamrnd}. ## @item With a shape parameter @math{α = k} and an inverse scale parameter ## @math{β = 1 / θ}, called a rate parameter. ## @end enumerate ## ## Further information about the Gamma distribution can be found at ## @url{https://en.wikipedia.org/wiki/Gamma_distribution} ## ## @seealso{gamcdf, gaminv, gampdf, gamrnd, gamfit, gamlike} ## @end deftypefn function [m, v] = gamstat (k, theta) ## Check for valid number of input arguments if (nargin < 2) error ("gamstat: function called with too few input arguments."); endif ## Check for K and THETA being numeric if (! (isnumeric (k) && isnumeric (theta))) error ("gamstat: K and THETA must be numeric."); endif ## Check for K and THETA being real if (iscomplex (k) || iscomplex (theta)) error ("gamstat: K and THETA must not be complex."); endif ## Check for common size of K and THETA if (! isscalar (k) || ! isscalar (theta)) [retval, k, theta] = common_size (k, theta); if (retval > 0) error ("gamstat: K and THETA must be of common size or scalars."); endif endif ## Calculate moments m = k .* theta; v = k .* (theta .^ 2); ## Continue argument check k = find (! (k > 0) | ! (k < Inf) | ! (theta > 0) | ! (theta < Inf)); if (any (k)) m(k) = NaN; v(k) = NaN; endif endfunction ## Input validation tests %!error gamstat () %!error gamstat (1) %!error gamstat ({}, 2) %!error gamstat (1, "") %!error gamstat (i, 2) %!error gamstat (1, i) %!error ... %! gamstat (ones (3), ones (2)) %!error ... %! gamstat (ones (2), ones (3)) ## Output validation tests %!test %! k = 1:6; %! theta = 1:0.2:2; %! [m, v] = gamstat (k, theta); %! expected_m = [1.00, 2.40, 4.20, 6.40, 9.00, 12.00]; %! expected_v = [1.00, 2.88, 5.88, 10.24, 16.20, 24.00]; %! assert (m, expected_m, 0.001); %! assert (v, expected_v, 0.001); %!test %! k = 1:6; %! [m, v] = gamstat (k, 1.5); %! expected_m = [1.50, 3.00, 4.50, 6.00, 7.50, 9.00]; %! expected_v = [2.25, 4.50, 6.75, 9.00, 11.25, 13.50]; %! assert (m, expected_m, 0.001); %! assert (v, expected_v, 0.001); statistics-release-1.6.3/inst/dist_stat/geostat.m000066400000000000000000000046731456127120000221400ustar00rootroot00000000000000## Copyright (C) 2006, 2007 Arno Onken ## Copyright (C) 2024 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {[@var{m}, @var{v}] =} geostat (@var{ps}) ## ## Compute statistics of the geometric distribution. ## ## @code{[@var{m}, @var{v}] = geostat (@var{ps})} returns the mean and ## variance of the geometric distribution with probability of success parameter ## @var{ps}. ## ## The size of @var{m} (mean) and @var{v} (variance) is the same size of the ## input argument. ## ## Further information about the geometric distribution can be found at ## @url{https://en.wikipedia.org/wiki/Geometric_distribution} ## ## @seealso{geocdf, geoinv, geopdf, geornd, geofit} ## @end deftypefn function [m, v] = geostat (ps) ## Check for valid number of input arguments if (nargin < 1) error ("geostat: function called with too few input arguments."); endif ## Check for PS being numeric if (! isnumeric (ps)) error ("geostat: PS must be numeric."); endif ## Check for PS being real if (iscomplex (ps)) error ("geostat: PS must not be complex."); endif ## Calculate moments q = 1 - ps; m = q ./ ps; v = q ./ (ps .^ 2); ## Continue argument check k = find (! (ps >= 0) | ! (ps <= 1)); if (any (k)) m(k) = NaN; v(k) = NaN; endif endfunction ## Input validation tests %!error geostat () %!error geostat ({}) %!error geostat ("") %!error geostat (i) ## Output validation tests %!test %! ps = 1 ./ (1:6); %! [m, v] = geostat (ps); %! assert (m, [0, 1, 2, 3, 4, 5], 0.001); %! assert (v, [0, 2, 6, 12, 20, 30], 0.001); statistics-release-1.6.3/inst/dist_stat/gevstat.m000066400000000000000000000117311456127120000221400ustar00rootroot00000000000000## Copyright (C) 2012 Nir Krakauer ## Copyright (C) 2022-2024 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {[@var{m}, @var{v}] =} gevstat (@var{k}, @var{sigma}, @var{mu}) ## ## Compute statistics of the generalized extreme value distribution. ## ## @code{[@var{m}, @var{v}] = gevstat (@var{k}, @var{sigma}, @var{mu})} returns ## the mean and variance of the generalized extreme value distribution with ## shape parameter @var{k}, scale parameter @var{sigma}, and location parameter ## @var{mu}. ## ## The size of @var{m} (mean) and @var{v} (variance) is the common size of the ## input arguments. A scalar input functions as a constant matrix of the ## same size as the other inputs. ## ## The mean of the GEV distribution is not finite when @qcode{@var{k} >= 1}, and ## the variance is not finite when @qcode{@var{k} >= 1/2}. The GEV distribution ## has positive density only for values of @var{x} such that ## @qcode{@var{k} * (@var{x} - @var{mu}) / @var{sigma} > -1}. ## ## Further information about the generalized extreme value distribution can be ## found at ## @url{https://en.wikipedia.org/wiki/Generalized_extreme_value_distribution} ## ## @subheading References ## @enumerate ## @item ## Rolf-Dieter Reiss and Michael Thomas. @cite{Statistical Analysis of Extreme ## Values with Applications to Insurance, Finance, Hydrology and Other Fields}. ## Chapter 1, pages 16-17, Springer, 2007. ## @end enumerate ## ## @seealso{gevcdf, gevinv, gevpdf, gevrnd, gevfit, gevlike} ## @end deftypefn function [m, v] = gevstat (k, sigma, mu) ## Check for valid number of input arguments if (nargin < 3) error ("gevstat: function called with too few input arguments."); endif ## Check for K, SIGMA, and MU being numeric if (! (isnumeric (k) && isnumeric (sigma) && isnumeric (mu))) error ("gevstat: K, SIGMA, and MU must be numeric."); endif ## Check for K, SIGMA, and MU being real if (iscomplex (k) || iscomplex (sigma) || iscomplex (mu)) error ("gevstat: K, SIGMA, and MU must not be complex."); endif ## Check for common size of K, SIGMA, and MU if (! isscalar (k) || ! isscalar (sigma) || ! isscalar (mu)) [retval, k, sigma, mu] = common_size (k, sigma, mu); if (retval > 0) error ("gevstat: K, SIGMA, and MU must be of common size or scalars."); endif endif ## Euler-Mascheroni constant eg = 0.57721566490153286; m = v = k; ## Find the mean m(k >= 1) = Inf; m(k == 0) = mu(k == 0) + eg*sigma(k == 0); m(k < 1 & k != 0) = mu(k < 1 & k != 0) + sigma(k < 1 & k != 0) .* ... (gamma(1-k(k < 1 & k != 0)) - 1) ./ k(k < 1 & k != 0); ## Find the variance v(k >= 0.5) = Inf; v(k == 0) = (pi^2 / 6) * sigma(k == 0) .^ 2; v(k < 0.5 & k != 0) = (gamma(1-2*k(k < 0.5 & k != 0)) - ... gamma(1-k(k < 0.5 & k != 0)).^2) .* ... (sigma(k < 0.5 & k != 0) ./ k(k < 0.5 & k != 0)) .^ 2; endfunction ## Input validation tests %!error gevstat () %!error gevstat (1) %!error gevstat (1, 2) %!error gevstat ({}, 2, 3) %!error gevstat (1, "", 3) %!error gevstat (1, 2, "") %!error gevstat (i, 2, 3) %!error gevstat (1, i, 3) %!error gevstat (1, 2, i) %!error ... %! gevstat (ones (3), ones (2), 3) %!error ... %! gevstat (ones (2), 2, ones (3)) %!error ... %! gevstat (1, ones (2), ones (3)) ## Output validation tests %!test %! k = [-1, -0.5, 0, 0.2, 0.4, 0.5, 1]; %! sigma = 2; %! mu = 1; %! [m, v] = gevstat (k, sigma, mu); %! expected_m = [1, 1.4551, 2.1544, 2.6423, 3.4460, 4.0898, Inf]; %! expected_v = [4, 3.4336, 6.5797, 13.3761, 59.3288, Inf, Inf]; %! assert (m, expected_m, -0.001); %! assert (v, expected_v, -0.001); statistics-release-1.6.3/inst/dist_stat/gpstat.m000066400000000000000000000130021456127120000217560ustar00rootroot00000000000000## Copyright (C) 2022-2024 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {[@var{m}, @var{v}] =} gpstat (@var{k}, @var{sigma}, @var{mu}) ## ## Compute statistics of the generalized Pareto distribution. ## ## @code{[@var{m}, @var{v}] = gpstat (@var{k}, @var{sigma}, @var{mu})} ## returns the mean and variance of the generalized Pareto distribution with ## shape parameter @var{k}, scale parameter @var{sigma}, and location parameter ## @var{mu}. ## ## The size of @var{m} (mean) and @var{v} (variance) is the common size of the ## input arguments. A scalar input functions as a constant matrix of the ## same size as the other inputs. ## ## When @var{k} = 0 and @var{mu} = 0, the generalized Pareto ## distribution is equivalent to the exponential distribution. When ## @code{@var{k} > 0} and @code{@var{mu} = @var{sigma} / @var{k}}, ## the generalized Pareto distribution is equivalent to the Pareto distribution. ## The mean of the generalized Pareto distribution is not finite when ## @code{@var{k} >= 1}, and the variance is not finite when ## @code{@var{k} >= 1/2}. When @code{@var{k} >= 0}, the generalized ## Pareto distribution has positive density for @code{@var{x} > @var{mu}}, ## or, when @code{@var{k} < 0}, for ## @code{0 <= (@var{x} - @var{mu}) / @var{sigma} <= -1 / @var{k}}. ## ## Further information about the generalized Pareto distribution can be found at ## @url{https://en.wikipedia.org/wiki/Generalized_Pareto_distribution} ## ## @seealso{gpcdf, gpinv, gppdf, gprnd, gpfit, gplike} ## @end deftypefn function [m, v] = gpstat (k, sigma, mu) ## Check for valid number of input arguments if (nargin < 3) error ("gpstat: function called with too few input arguments."); endif ## Check for K, SIGMA, and MU being numeric if (! (isnumeric (k) && isnumeric (sigma) && isnumeric (mu))) error ("gpstat: K, SIGMA, and MU must be numeric."); endif ## Check for K, SIGMA, and MU being real if (iscomplex (k) || iscomplex (sigma) || iscomplex (mu)) error ("gpstat: K, SIGMA, and MU must not be complex."); endif ## Check for common size of K, SIGMA, and MU if (! isscalar (k) || ! isscalar (sigma) || ! isscalar (mu)) [retval, k, sigma, mu] = common_size (k, sigma, mu); if (retval > 0) error ("gpstat: K, SIGMA, and MU must be of common size or scalars."); endif endif ## Return NaNs for out of range SCALE parameters. sigma(sigma <= 0) = NaN; ## Check for appropriate class if (isa (k, "single") || isa (sigma, "single") || isa (mu, "single")); is_class = "single"; else is_class = "double"; endif ## Prepare output m = NaN (size (k), is_class); v = NaN (size (k), is_class); ## Compute cases for SHAPE == 0 knot0 = (abs (k) < eps (is_class)); m(knot0) = 1; v(knot0) = 1; ## Compute cases for SHAPE != 0 knot0 = ! knot0; ## SHAPE < 1 kless = knot0 & (k < 1); m(kless) = 1 ./ (1 - k(kless)); ## SHAPE > 1 m(k >= 1) = Inf; ## SHAPE < 1/2 ## Find the k~=0 cases and fill in the variance. kless = knot0 & (k < 1/2); v(kless) = 1 ./ ((1-k(kless)).^2 .* (1-2.*k(kless))); ## SHAPE > 1/2 v(k >= 1/2) = Inf; ## Compute mean and variance m = mu + sigma .* m; v = sigma .^ 2 .* v; endfunction ## Input validation tests %!error gpstat () %!error gpstat (1) %!error gpstat (1, 2) %!error gpstat ({}, 2, 3) %!error gpstat (1, "", 3) %!error gpstat (1, 2, "") %!error gpstat (i, 2, 3) %!error gpstat (1, i, 3) %!error gpstat (1, 2, i) %!error ... %! gpstat (ones (3), ones (2), 3) %!error ... %! gpstat (ones (2), 2, ones (3)) %!error ... %! gpstat (1, ones (2), ones (3)) ## Output validation tests %!shared x, y %! x = [-Inf, -1, 0, 1/2, 1, Inf]; %! y = [0, 0.5, 1, 2, Inf, Inf]; %!assert (gpstat (x, ones (1,6), zeros (1,6)), y, eps) ## Test class of input preserved %!assert (gpstat (single (x), 1, 0), single (y), eps("single")) %!assert (gpstat (x, single (1), 0), single (y), eps("single")) %!assert (gpstat (x, 1, single (0)), single (y), eps("single")) %!assert (gpstat (single ([x, NaN]), 1, 0), single ([y, NaN]), eps("single")) %!assert (gpstat ([x, NaN], single (1), 0), single ([y, NaN]), eps("single")) %!assert (gpstat ([x, NaN], 1, single (0)), single ([y, NaN]), eps("single")) statistics-release-1.6.3/inst/dist_stat/hygestat.m000066400000000000000000000113261456127120000223130ustar00rootroot00000000000000## Copyright (C) 2006, 2007 Arno Onken ## Copyright (C) 2024 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {[@var{mn}, @var{v}] =} hygestat (@var{m}, @var{k}, @var{n}) ## ## Compute statistics of the hypergeometric distribution. ## ## @code{[@var{mn}, @var{v}] = hygestat (@var{m}, @var{k}, @var{n})} returns the ## mean and variance of the hypergeometric distribution parameters @var{m}, ## @var{k}, and @var{n}. ## ## @itemize ## @item ## @var{m} is the total size of the population of the hypergeometric ## distribution. The elements of @var{m} must be positive natural numbers. ## ## @item ## @var{k} is the number of marked items of the hypergeometric distribution. ## The elements of @var{k} must be natural numbers. ## ## @item ## @var{n} is the size of the drawn sample of the hypergeometric ## distribution. The elements of @var{n} must be positive natural numbers. ## @end itemize ## ## The size of @var{mn} (mean) and @var{v} (variance) is the common size of the ## input arguments. A scalar input functions as a constant matrix of the ## same size as the other inputs. ## ## Further information about the hypergeometric distribution can be found at ## @url{https://en.wikipedia.org/wiki/Hypergeometric_distribution} ## ## @seealso{hygecdf, hygeinv, hygepdf, hygernd} ## @end deftypefn function [mn, v] = hygestat (m, k, n) ## Check for valid number of input arguments if (nargin < 3) error ("hygestat: function called with too few input arguments."); endif ## Check for M, K, and N being numeric if (! (isnumeric (m) && isnumeric (k) && isnumeric (n))) error ("hygestat: M, K, and N must be numeric."); endif ## Check for M, K, and N being real if (iscomplex (m) || iscomplex (k) || iscomplex (n)) error ("hygestat: M, K, and N must not be complex."); endif ## Check for common size of M, K, and N if (! isscalar (m) || ! isscalar (k) || ! isscalar (n)) [retval, m, k, n] = common_size (m, k, n); if (retval > 0) error ("hygestat: M, K, and N must be of common size or scalars."); endif endif ## Calculate moments mn = (n .* k) ./ m; v = (n .* (k ./ m) .* (1 - k ./ m) .* (m - n)) ./ (m - 1); ## Continue argument check is_nan = find (! (m >= 0) | ! (k >= 0) | ! (n > 0) | ! (m == round (m)) | ... ! (k == round (k)) | ! (n == round (n)) | ! (k <= m) | ... ! (n <= m)); if (any (is_nan)) mn(is_nan) = NaN; v(is_nan) = NaN; endif endfunction ## Input validation tests %!error hygestat () %!error hygestat (1) %!error hygestat (1, 2) %!error hygestat ({}, 2, 3) %!error hygestat (1, "", 3) %!error hygestat (1, 2, "") %!error hygestat (i, 2, 3) %!error hygestat (1, i, 3) %!error hygestat (1, 2, i) %!error ... %! hygestat (ones (3), ones (2), 3) %!error ... %! hygestat (ones (2), 2, ones (3)) %!error ... %! hygestat (1, ones (2), ones (3)) ## Output validation tests %!test %! m = 4:9; %! k = 0:5; %! n = 1:6; %! [mn, v] = hygestat (m, k, n); %! expected_mn = [0.0000, 0.4000, 1.0000, 1.7143, 2.5000, 3.3333]; %! expected_v = [0.0000, 0.2400, 0.4000, 0.4898, 0.5357, 0.5556]; %! assert (mn, expected_mn, 0.001); %! assert (v, expected_v, 0.001); %!test %! m = 4:9; %! k = 0:5; %! [mn, v] = hygestat (m, k, 2); %! expected_mn = [0.0000, 0.4000, 0.6667, 0.8571, 1.0000, 1.1111]; %! expected_v = [0.0000, 0.2400, 0.3556, 0.4082, 0.4286, 0.4321]; %! assert (mn, expected_mn, 0.001); %! assert (v, expected_v, 0.001); statistics-release-1.6.3/inst/dist_stat/lognstat.m000066400000000000000000000075351456127120000223250ustar00rootroot00000000000000## Copyright (C) 2006, 2007 Arno Onken ## Copyright (C) 2024 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {[@var{m}, @var{v}] =} lognstat (@var{mu}, @var{sigma}) ## ## Compute statistics of the log-normal distribution. ## ## @code{[@var{m}, @var{v}] = lognstat (@var{mu}, @var{sigma})} returns the mean ## and variance of the log-normal distribution with mean parameter @var{mu} and ## standard deviation parameter @var{sigma}, each corresponding to the ## associated normal distribution. ## ## The size of @var{m} (mean) and @var{v} (variance) is the common size of the ## input arguments. A scalar input functions as a constant matrix of the ## same size as the other inputs. ## ## Further information about the log-normal distribution can be found at ## @url{https://en.wikipedia.org/wiki/Log-normal_distribution} ## ## @seealso{logncdf, logninv, lognpdf, lognrnd, lognfit, lognlike} ## @end deftypefn function [m, v] = lognstat (mu, sigma) ## Check for valid number of input arguments if (nargin < 2) error ("lognstat: function called with too few input arguments."); endif ## Check for MU and SIGMA being numeric if (! (isnumeric (mu) && isnumeric (sigma))) error ("lognstat: MU and SIGMA must be numeric."); endif ## Check for MU and SIGMA being real if (iscomplex (mu) || iscomplex (sigma)) error ("lognstat: MU and SIGMA must not be complex."); endif ## Check for common size of MU and SIGMA if (! isscalar (mu) || ! isscalar (sigma)) [retval, mu, sigma] = common_size (mu, sigma); if (retval > 0) error ("lognstat: MU and SIGMA must be of common size or scalars."); endif endif ## Calculate moments m = exp (mu + (sigma .^ 2) ./ 2); v = (exp (sigma .^ 2) - 1) .* exp (2 .* mu + sigma .^ 2); ## Continue argument check k = find (! (sigma >= 0) | ! (sigma < Inf)); if (any (k)) m(k) = NaN; v(k) = NaN; endif endfunction ## Input validation tests %!error lognstat () %!error lognstat (1) %!error lognstat ({}, 2) %!error lognstat (1, "") %!error lognstat (i, 2) %!error lognstat (1, i) %!error ... %! lognstat (ones (3), ones (2)) %!error ... %! lognstat (ones (2), ones (3)) ## Output validation tests %!test %! mu = 0:0.2:1; %! sigma = 0.2:0.2:1.2; %! [m, v] = lognstat (mu, sigma); %! expected_m = [1.0202, 1.3231, 1.7860, 2.5093, 3.6693, 5.5845]; %! expected_v = [0.0425, 0.3038, 1.3823, 5.6447, 23.1345, 100.4437]; %! assert (m, expected_m, 0.001); %! assert (v, expected_v, 0.001); %!test %! sigma = 0.2:0.2:1.2; %! [m, v] = lognstat (0, sigma); %! expected_m = [1.0202, 1.0833, 1.1972, 1.3771, 1.6487, 2.0544]; %! expected_v = [0.0425, 0.2036, 0.6211, 1.7002, 4.6708, 13.5936]; %! assert (m, expected_m, 0.001); %! assert (v, expected_v, 0.001); statistics-release-1.6.3/inst/dist_stat/nbinstat.m000066400000000000000000000073441456127120000223120ustar00rootroot00000000000000## Copyright (C) 2006, 2007 Arno Onken ## Copyright (C) 2024 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {[@var{m}, @var{v}] =} nbinstat (@var{r}, @var{ps}) ## ## Compute statistics of the negative binomial distribution. ## ## @code{[@var{m}, @var{v}] = nbinstat (@var{r}, @var{ps})} returns the mean ## and variance of the negative binomial distribution with parameters @var{r} ## and @var{ps}, where @var{r} is the number of successes until the experiment ## is stopped and @var{ps} is the probability of success in each experiment, ## given the number of failures in @var{x}. ## ## The size of @var{m} (mean) and @var{v} (variance) is the common size of the ## input arguments. A scalar input functions as a constant matrix of the ## same size as the other inputs. ## ## Further information about the negative binomial distribution can be found at ## @url{https://en.wikipedia.org/wiki/Negative_binomial_distribution} ## ## @seealso{nbincdf, nbininv, nbininv, nbinrnd, nbinfit, nbinlike} ## @end deftypefn function [m, v] = nbinstat (r, ps) ## Check for valid number of input arguments if (nargin < 2) error ("nbinstat: function called with too few input arguments."); endif ## Check for R and PS being numeric if (! (isnumeric (r) && isnumeric (ps))) error ("nbinstat: R and PS must be numeric."); endif ## Check for R and PS being real if (iscomplex (r) || iscomplex (ps)) error ("nbinstat: R and PS must not be complex."); endif ## Check for common size of R and PS if (! isscalar (r) || ! isscalar (ps)) [retval, r, ps] = common_size (r, ps); if (retval > 0) error ("nbinstat: R and PS must be of common size or scalars."); endif endif ## Calculate moments q = 1 - ps; m = r .* q ./ ps; v = r .* q ./ (ps .^ 2); ## Continue argument check k = find (! (r > 0) | ! (r < Inf) | ! (ps > 0) | ! (ps < 1)); if (any (k)) m(k) = NaN; v(k) = NaN; endif endfunction ## Input validation tests %!error nbinstat () %!error nbinstat (1) %!error nbinstat ({}, 2) %!error nbinstat (1, "") %!error nbinstat (i, 2) %!error nbinstat (1, i) %!error ... %! nbinstat (ones (3), ones (2)) %!error ... %! nbinstat (ones (2), ones (3)) ## Output validation tests %!test %! r = 1:4; %! ps = 0.2:0.2:0.8; %! [m, v] = nbinstat (r, ps); %! expected_m = [ 4.0000, 3.0000, 2.0000, 1.0000]; %! expected_v = [20.0000, 7.5000, 3.3333, 1.2500]; %! assert (m, expected_m, 0.001); %! assert (v, expected_v, 0.001); %!test %! r = 1:4; %! [m, v] = nbinstat (r, 0.5); %! expected_m = [1, 2, 3, 4]; %! expected_v = [2, 4, 6, 8]; %! assert (m, expected_m, 0.001); %! assert (v, expected_v, 0.001); statistics-release-1.6.3/inst/dist_stat/ncfstat.m000066400000000000000000000117351456127120000221310ustar00rootroot00000000000000## Copyright (C) 2022-2024 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {[@var{m}, @var{v}] =} ncfstat (@var{df1}, @var{df1}, @var{lambda}) ## ## Compute statistics for the noncentral @math{F}-distribution. ## ## @code{[@var{m}, @var{v}] = ncfstat (@var{df1}, @var{df1}, @var{lambda})} ## returns the mean and variance of the noncentral @math{F}-distribution with ## @var{df1} and @var{df2} degrees of freedom and noncentrality parameter ## @var{lambda}. ## ## The size of @var{m} (mean) and @var{v} (variance) is the common size of the ## input arguments. A scalar input functions as a constant matrix of the same ## size as the other inputs. ## ## Further information about the noncentral @math{F}-distribution can be found ## at @url{https://en.wikipedia.org/wiki/Noncentral_F-distribution} ## ## @seealso{ncfcdf, ncfinv, ncfpdf, ncfrnd, fstat} ## @end deftypefn function [m, v] = ncfstat (df1, df2, lambda) ## Check for valid number of input arguments if (nargin < 3) error ("ncfstat: function called with too few input arguments."); endif ## Check for DF1, DF2, and LAMBDA being numeric if (! (isnumeric (df1) && isnumeric (df2) && isnumeric (lambda))) error ("ncfstat: DF1, DF2, and LAMBDA must be numeric."); endif ## Check for DF1, DF2, and LAMBDA being reals if (iscomplex (df1) || iscomplex (df2) || iscomplex (lambda)) error ("ncfstat: DF1, DF2, and LAMBDA must not be complex."); endif ## Check for common size of DF1, DF2, and LAMBDA if (! isscalar (df1) || ! isscalar (df2) || ! isscalar (lambda)) [retval, df1, df2, lambda] = common_size (df1, df2, lambda); if (retval > 0) error ("ncfstat: DF1, DF2, and LAMBDA must be of common size or scalars."); endif endif ## Initialize mean and variance if (isa (df1, "single") || isa (df2, "single") || isa (lambda, "single")) m = zeros (size (df1), "single"); v = m; else m = zeros (size (df1)); v = m; endif ## Return NaNs for invalid df2 parameters m(df2 <= 2) = NaN; v(df2 <= 4) = NaN; ## Compute mean and variance for valid parameter values. k = (df2 > 2); if (any (k(:))) m(k) = df2(k) .* (df1(k) + lambda(k)) ./ (df1(k) .* (df2(k) - 2)); endif k = (df2 > 4); if (any (k(:))) df1_idx = df1(k) + lambda(k); df2_idx = df2(k) - 2; df1_df2 = (df2(k) ./ df1(k)) .^ 2; v(k) = 2 * df1_df2 .* (df1_idx .^ 2 + (df1_idx + lambda(k)) .* ... df2_idx) ./ ((df2(k) - 4) .* df2_idx .^ 2); endif endfunction ## Input validation tests %!error ncfstat () %!error ncfstat (1) %!error ncfstat (1, 2) %!error ncfstat ({}, 2, 3) %!error ncfstat (1, "", 3) %!error ncfstat (1, 2, "") %!error ncfstat (i, 2, 3) %!error ncfstat (1, i, 3) %!error ncfstat (1, 2, i) %!error ... %! ncfstat (ones (3), ones (2), 3) %!error ... %! ncfstat (ones (2), 2, ones (3)) %!error ... %! ncfstat (1, ones (2), ones (3)) ## Output validation tests %!shared df1, df2, lambda %! df1 = [2, 0, -1, 1, 4, 5]; %! df2 = [2, 4, -1, 5, 6, 7]; %! lambda = [1, NaN, 3, 0, 2, -1]; %!assert (ncfstat (df1, df2, lambda), [NaN, NaN, NaN, 1.6667, 2.25, 1.12], 1e-4); %!assert (ncfstat (df1(4:6), df2(4:6), 1), [3.3333, 1.8750, 1.6800], 1e-4); %!assert (ncfstat (df1(4:6), df2(4:6), 2), [5.0000, 2.2500, 1.9600], 1e-4); %!assert (ncfstat (df1(4:6), df2(4:6), 3), [6.6667, 2.6250, 2.2400], 1e-4); %!assert (ncfstat (2, [df2(1), df2(4:6)], 5), [NaN,5.8333,5.2500,4.9000], 1e-4); %!assert (ncfstat (0, [df2(1), df2(4:6)], 5), [NaN, Inf, Inf, Inf]); %!assert (ncfstat (1, [df2(1), df2(4:6)], 5), [NaN, 10, 9, 8.4], 1e-14); %!assert (ncfstat (4, [df2(1), df2(4:6)], 5), [NaN, 3.75, 3.375, 3.15], 1e-14); statistics-release-1.6.3/inst/dist_stat/nctstat.m000066400000000000000000000101731456127120000221420ustar00rootroot00000000000000## Copyright (C) 2022-2024 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {[@var{m}, @var{v}] =} nctstat (@var{df}, @var{mu}) ## ## Compute statistics for the noncentral @math{t}-distribution. ## ## @code{[@var{m}, @var{v}] = nctstat (@var{df}, @var{mu})} returns the mean ## and variance of the noncentral @math{t}-distribution with @var{df} degrees ## of freedom and noncentrality parameter @var{mu}. ## ## The size of @var{m} (mean) and @var{v} (variance) is the common size of the ## input arguments. A scalar input functions as a constant matrix of the same ## size as the other inputs. ## ## Further information about the noncentral @math{t}-distribution can be found ## at @url{https://en.wikipedia.org/wiki/Noncentral_t-distribution} ## ## @seealso{nctcdf, nctinv, nctpdf, nctrnd, tstat} ## @end deftypefn function [m, v] = nctstat (df, mu) ## Check for valid number of input arguments if (nargin < 2) error ("nctstat: function called with too few input arguments."); endif ## Check for DF and MU being numeric if (! (isnumeric (df) && isnumeric (mu))) error ("nctstat: DF and MU must be numeric."); endif ## Check for DF and MU being real if (iscomplex (df) || iscomplex (mu)) error ("nctstat: DF and MU must not be complex."); endif ## Check for common size of DF and MU if (! isscalar (df) || ! isscalar (mu)) [retval, df, mu] = common_size (df, mu); if (retval > 0) error ("nctstat: DF and MU must be of common size or scalars."); endif endif ## Initialize mean and variance if (isa (df, "single") || isa (mu, "single")) m = NaN (size (df), "single"); v = m; else m = NaN (size (df)); v = m; endif ## Compute mean and variance for valid parameter values. mk = df > 1; if (any (mk(:))) m(mk) = mu(mk) .* sqrt ((df(mk) / 2)) .* ... gamma ((df(mk) - 1) / 2) ./ gamma (df(mk) / 2); endif vk = df > 2; if (any (vk(:))) v(vk) = (df(vk) ./ (df(vk) - 2)) .* ... (1 + mu(vk) .^2) - 0.5 * (df(vk) .* mu(vk) .^ 2) .* ... exp (2 * (gammaln ((df(vk) - 1) / 2) - gammaln (df(vk) / 2))); endif endfunction ## Input validation tests %!error nctstat () %!error nctstat (1) %!error nctstat ({}, 2) %!error nctstat (1, "") %!error nctstat (i, 2) %!error nctstat (1, i) %!error ... %! nctstat (ones (3), ones (2)) %!error ... %! nctstat (ones (2), ones (3)) ## Output validation tests %!shared df, mu %! df = [2, 0, -1, 1, 4]; %! mu = [1, NaN, 3, -1, 2]; %!assert (nctstat (df, mu), [1.7725, NaN, NaN, NaN, 2.5066], 1e-4); %!assert (nctstat ([df(1:2), df(4:5)], 1), [1.7725, NaN, NaN, 1.2533], 1e-4); %!assert (nctstat ([df(1:2), df(4:5)], 3), [5.3174, NaN, NaN, 3.7599], 1e-4); %!assert (nctstat ([df(1:2), df(4:5)], 2), [3.5449, NaN, NaN, 2.5066], 1e-4); %!assert (nctstat (2, [mu(1), mu(3:5)]), [1.7725,5.3174,-1.7725,3.5449], 1e-4); %!assert (nctstat (0, [mu(1), mu(3:5)]), [NaN, NaN, NaN, NaN]); %!assert (nctstat (1, [mu(1), mu(3:5)]), [NaN, NaN, NaN, NaN]); %!assert (nctstat (4, [mu(1), mu(3:5)]), [1.2533,3.7599,-1.2533,2.5066], 1e-4); statistics-release-1.6.3/inst/dist_stat/ncx2stat.m000066400000000000000000000076161456127120000222400ustar00rootroot00000000000000## Copyright (C) 2022-2024 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {[@var{m}, @var{v}] =} ncx2stat (@var{df}, @var{lambda}) ## ## Compute statistics for the noncentral chi-squared distribution. ## ## @code{[@var{m}, @var{v}] = ncx2stat (@var{df}, @var{lambda})} returns the ## mean and variance of the noncentral chi-squared distribution with @var{df} ## degrees of freedom and noncentrality parameter @var{lambda}. ## ## The size of @var{m} (mean) and @var{v} (variance) is the common size of the ## input arguments. A scalar input functions as a constant matrix of the ## same size as the other inputs. ## ## Further information about the noncentral chi-squared distribution can be ## found at @url{https://en.wikipedia.org/wiki/Noncentral_chi-squared_distribution} ## ## @seealso{ncx2cdf, ncx2inv, ncx2pdf, ncx2rnd} ## @end deftypefn function [m, v] = ncx2stat (df, lambda) ## Check for valid number of input arguments if (nargin < 2) error ("ncx2stat: function called with too few input arguments."); endif ## Check for DF and LAMBDA being numeric if (! (isnumeric (df) && isnumeric (lambda))) error ("ncx2stat: DF and LAMBDA must be numeric."); endif ## Check for DF and LAMBDA being real if (iscomplex (df) || iscomplex (lambda)) error ("ncx2stat: DF and LAMBDA must not be complex."); endif ## Check for common size of DF and LAMBDA if (! isscalar (df) || ! isscalar (lambda)) [retval, df, lambda] = common_size (df, lambda); if (retval > 0) error ("ncx2stat: DF and LAMBDA must be of common size or scalars."); endif endif ## Initialize mean and variance if (isa (df, "single") || isa (lambda, "single")) m = NaN (size (df), "single"); v = m; else m = NaN (size (df)); v = m; endif ## Compute mean and variance for valid parameter values. k = (df > 0 & lambda >= 0); if (any (k(:))) m(k) = lambda(k) + df(k); v(k) = 2 * (df(k) + 2 * (lambda(k))); endif endfunction ## Input validation tests %!error ncx2stat () %!error ncx2stat (1) %!error ncx2stat ({}, 2) %!error ncx2stat (1, "") %!error ncx2stat (i, 2) %!error ncx2stat (1, i) %!error ... %! ncx2stat (ones (3), ones (2)) %!error ... %! ncx2stat (ones (2), ones (3)) ## Output validation tests %!shared df, d1 %! df = [2, 0, -1, 1, 4]; %! d1 = [1, NaN, 3, -1, 2]; %!assert (ncx2stat (df, d1), [3, NaN, NaN, NaN, 6]); %!assert (ncx2stat ([df(1:2), df(4:5)], 1), [3, NaN, 2, 5]); %!assert (ncx2stat ([df(1:2), df(4:5)], 3), [5, NaN, 4, 7]); %!assert (ncx2stat ([df(1:2), df(4:5)], 2), [4, NaN, 3, 6]); %!assert (ncx2stat (2, [d1(1), d1(3:5)]), [3, 5, NaN, 4]); %!assert (ncx2stat (0, [d1(1), d1(3:5)]), [NaN, NaN, NaN, NaN]); %!assert (ncx2stat (1, [d1(1), d1(3:5)]), [2, 4, NaN, 3]); %!assert (ncx2stat (4, [d1(1), d1(3:5)]), [5, 7, NaN, 6]); statistics-release-1.6.3/inst/dist_stat/normstat.m000066400000000000000000000071361456127120000223360ustar00rootroot00000000000000## Copyright (C) 2006, 2007 Arno Onken ## Copyright (C) 2024 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {[@var{m}, @var{v}] =} normstat (@var{mu}, @var{sigma}) ## ## Compute statistics of the normal distribution. ## ## @code{[@var{m}, @var{v}] = normstat (@var{mu}, @var{sigma})} returns the mean ## and variance of the normal distribution with non-centrality (distance) ## parameter @var{mu} and scale parameter @var{sigma}. ## ## The size of @var{m} (mean) and @var{v} (variance) is the common size of the ## input arguments. A scalar input functions as a constant matrix of the ## same size as the other inputs. ## ## Further information about the normal distribution can be found at ## @url{https://en.wikipedia.org/wiki/Normal_distribution} ## ## @seealso{norminv, norminv, normpdf, normrnd, normfit, normlike} ## @end deftypefn function [m, v] = normstat (mu, sigma) ## Check for valid number of input arguments if (nargin < 2) error ("normstat: function called with too few input arguments."); endif ## Check for MU and SIGMA being numeric if (! (isnumeric (mu) && isnumeric (sigma))) error ("normstat: MU and SIGMA must be numeric."); endif ## Check for MU and SIGMA being real if (iscomplex (mu) || iscomplex (sigma)) error ("normstat: MU and SIGMA must not be complex."); endif ## Check for common size of MU and SIGMA if (! isscalar (mu) || ! isscalar (sigma)) [retval, mu, sigma] = common_size (mu, sigma); if (retval > 0) error ("normstat: MU and SIGMA must be of common size or scalars."); endif endif ## Calculate moments m = mu; v = sigma .* sigma; ## Continue argument check k = find (! (sigma > 0) | ! (sigma < Inf)); if (any (k)) m(k) = NaN; v(k) = NaN; endif endfunction ## Input validation tests %!error normstat () %!error normstat (1) %!error normstat ({}, 2) %!error normstat (1, "") %!error normstat (i, 2) %!error normstat (1, i) %!error ... %! normstat (ones (3), ones (2)) %!error ... %! normstat (ones (2), ones (3)) ## Output validation tests %!test %! mu = 1:6; %! sigma = 0.2:0.2:1.2; %! [m, v] = normstat (mu, sigma); %! expected_v = [0.0400, 0.1600, 0.3600, 0.6400, 1.0000, 1.4400]; %! assert (m, mu); %! assert (v, expected_v, 0.001); %!test %! sigma = 0.2:0.2:1.2; %! [m, v] = normstat (0, sigma); %! expected_mn = [0, 0, 0, 0, 0, 0]; %! expected_v = [0.0400, 0.1600, 0.3600, 0.6400, 1.0000, 1.4400]; %! assert (m, expected_mn, 0.001); %! assert (v, expected_v, 0.001); statistics-release-1.6.3/inst/dist_stat/poisstat.m000066400000000000000000000046711456127120000223360ustar00rootroot00000000000000## Copyright (C) 2006, 2007 Arno Onken ## Copyright (C) 2024 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {[@var{m}, @var{v}] =} poisstat (@var{lambda}) ## ## Compute statistics of the Poisson distribution. ## ## @code{[@var{m}, @var{v}] = poisstat (@var{lambda})} returns the mean and ## variance of the Poisson distribution with rate parameter @var{lambda}. ## ## The size of @var{m} (mean) and @var{v} (variance) is the same size of the ## input argument. ## ## Further information about the Poisson distribution can be found at ## @url{https://en.wikipedia.org/wiki/Poisson_distribution} ## ## @seealso{poisscdf, poissinv, poisspdf, poissrnd, poissfit, poisslike} ## @end deftypefn function [m, v] = poisstat (lambda) ## Check for valid number of input arguments if (nargin < 1) error ("poisstat: function called with too few input arguments."); endif ## Check for SIGMA being numeric if (! isnumeric (lambda)) error ("poisstat: SIGMA must be numeric."); endif ## Check for SIGMA being real if (iscomplex (lambda)) error ("poisstat: SIGMA must not be complex."); endif ## Set moments m = lambda; v = lambda; ## Continue argument check k = find (! (lambda > 0) | ! (lambda < Inf)); if (any (k)) m(k) = NaN; v(k) = NaN; endif endfunction ## Input validation tests %!error poisstat () %!error poisstat ({}) %!error poisstat ("") %!error poisstat (i) ## Output validation tests %!test %! lambda = 1 ./ (1:6); %! [m, v] = poisstat (lambda); %! assert (m, lambda); %! assert (v, lambda); statistics-release-1.6.3/inst/dist_stat/raylstat.m000066400000000000000000000051351456127120000223270ustar00rootroot00000000000000## Copyright (C) 2006, 2007 Arno Onken ## Copyright (C) 2024 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {[@var{m}, @var{v}] =} raylstat (@var{sigma}) ## ## Compute statistics of the Rayleigh distribution. ## ## @code{[@var{m}, @var{v}] = raylstat (@var{sigma})} returns the mean and ## variance of the Rayleigh distribution with scale parameter @var{sigma}. ## ## The size of @var{m} (mean) and @var{v} (variance) is the same size of the ## input argument. ## ## Further information about the Rayleigh distribution can be found at ## @url{https://en.wikipedia.org/wiki/Rayleigh_distribution} ## ## @seealso{raylcdf, raylinv, raylpdf, raylrnd, raylfit, rayllike} ## @end deftypefn function [m, v] = raylstat (sigma) ## Check for valid number of input arguments if (nargin < 1) error ("raylstat: function called with too few input arguments."); endif ## Check for SIGMA being numeric if (! isnumeric (sigma)) error ("raylstat: SIGMA must be numeric."); endif ## Check for SIGMA being real if (iscomplex (sigma)) error ("raylstat: SIGMA must not be complex."); endif ## Calculate moments m = sigma .* sqrt (pi ./ 2); v = (2 - pi ./ 2) .* sigma .^ 2; ## Continue argument check k = find (! (sigma > 0)); if (any (k)) m(k) = NaN; v(k) = NaN; endif endfunction ## Input validation tests %!error raylstat () %!error raylstat ({}) %!error raylstat ("") %!error raylstat (i) ## Output validation tests %!test %! sigma = 1:6; %! [m, v] = raylstat (sigma); %! expected_m = [1.2533, 2.5066, 3.7599, 5.0133, 6.2666, 7.5199]; %! expected_v = [0.4292, 1.7168, 3.8628, 6.8673, 10.7301, 15.4513]; %! assert (m, expected_m, 0.001); %! assert (v, expected_v, 0.001); statistics-release-1.6.3/inst/dist_stat/ricestat.m000066400000000000000000000104321456127120000222760ustar00rootroot00000000000000## Copyright (C) 2024 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {[@var{m}, @var{v}] =} ricestat (@var{nu}, @var{sigma}) ## ## Compute statistics of the Rician distribution. ## ## @code{[@var{m}, @var{v}] = ricestat (@var{nu}, @var{sigma})} returns the mean ## and variance of the Rician distribution with non-centrality (distance) ## parameter @var{nu} and scale parameter @var{sigma}. ## ## The size of @var{m} (mean) and @var{v} (variance) is the common size of the ## input arguments. A scalar input functions as a constant matrix of the ## same size as the other inputs. ## ## Further information about the Rician distribution can be found at ## @url{https://en.wikipedia.org/wiki/Rice_distribution} ## ## @seealso{ricecdf, riceinv, ricepdf, ricernd, ricefit, ricelike} ## @end deftypefn function [m, v] = ricestat (nu, sigma) ## Check for valid number of input arguments if (nargin < 2) error ("ricestat: function called with too few input arguments."); endif ## Check for NU and SIGMA being numeric if (! (isnumeric (nu) && isnumeric (sigma))) error ("ricestat: NU and SIGMA must be numeric."); endif ## Check for NU and SIGMA being real if (iscomplex (nu) || iscomplex (sigma)) error ("ricestat: NU and SIGMA must not be complex."); endif ## Check for common size of NU and SIGMA if (! isscalar (nu) || ! isscalar (sigma)) [retval, nu, sigma] = common_size (nu, sigma); if (retval > 0) error ("ricestat: NU and SIGMA must be of common size or scalars."); endif endif ## Initialize mean and variance if (isa (nu, "single") || isa (sigma, "single")) m = NaN (size (nu), "single"); v = m; else m = NaN (size (nu)); v = m; endif ## Compute mean and variance for valid parameter values. k = (nu >= 0 & sigma > 0); if (any (k(:))) thetasq = (nu(k) .^ 2) ./ (sigma(k) .^ 2); L = Laguerre_half (-0.5 .* thetasq); m(k) = sigma(k) .* sqrt (pi / 2) .* L; v(k) = 2 * (sigma(k) .^ 2) + nu(k) .^ 2 - (0.5 .* pi .* sigma(k) .^ 2) .* L; endif endfunction function L = Laguerre_half(x) L = exp (x ./ 2) .* ((1 - x) .* besseli (0, -x./2) - x .* besseli (1, -x./2)); endfunction ## Input validation tests %!error ricestat () %!error ricestat (1) %!error ricestat ({}, 2) %!error ricestat (1, "") %!error ricestat (i, 2) %!error ricestat (1, i) %!error ... %! ricestat (ones (3), ones (2)) %!error ... %! ricestat (ones (2), ones (3)) ## Output validation tests %!shared nu, sigma %! nu = [2, 0, -1, 1, 4]; %! sigma = [1, NaN, 3, -1, 2]; %!assert (ricestat (nu, sigma), [2.2724, NaN, NaN, NaN, 4.5448], 1e-4); %!assert (ricestat ([nu(1:2), nu(4:5)], 1), [2.2724, 1.2533, 1.5486, 4.1272], 1e-4); %!assert (ricestat ([nu(1:2), nu(4:5)], 3), [4.1665, 3.7599, 3.8637, 5.2695], 1e-4); %!assert (ricestat ([nu(1:2), nu(4:5)], 2), [3.0971, 2.5066, 2.6609, 4.5448], 1e-4); %!assert (ricestat (2, [sigma(1), sigma(3:5)]), [2.2724, 4.1665, NaN, 3.0971], 1e-4); %!assert (ricestat (0, [sigma(1), sigma(3:5)]), [1.2533, 3.7599, NaN, 2.5066], 1e-4); %!assert (ricestat (1, [sigma(1), sigma(3:5)]), [1.5486, 3.8637, NaN, 2.6609], 1e-4); %!assert (ricestat (4, [sigma(1), sigma(3:5)]), [4.1272, 5.2695, NaN, 4.5448], 1e-4); statistics-release-1.6.3/inst/dist_stat/tlsstat.m000066400000000000000000000117511456127120000221630ustar00rootroot00000000000000## Copyright (C) 2024 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {[@var{m}, @var{v}] =} tlsstat (@var{mu}, @var{sigma}, @var{df}) ## ## Compute statistics of the location-scale Student's T distribution. ## ## @code{[@var{m}, @var{v}] = tlsstat (@var{mu}, @var{sigma}, @var{df})} returns ## the mean and variance of the location-scale Student's T distribution with ## location parameter @var{mu}, scale parameter @var{sigma}, and @var{df} ## degrees of freedom. ## ## The size of @var{m} (mean) and @var{v} (variance) is the common size of the ## input arguments. A scalar input functions as a constant matrix of the ## same size as the other inputs. ## ## Further information about the location-scale Student's T distribution can be ## found at @url{https://en.wikipedia.org/wiki/Student%27s_t-distribution#Location-scale_t_distribution} ## ## @seealso{tlscdf, tlsinv, tlspdf, tlsrnd, tlsfit, tlslike} ## @end deftypefn function [m, v] = tlsstat (mu, sigma, df) ## Check for valid number of input arguments if (nargin < 3) error ("tlsstat: function called with too few input arguments."); endif ## Check for MU, SIGMA, and DF being numeric if (! (isnumeric (mu) && isnumeric (sigma) && isnumeric (df))) error ("tlsstat: MU, SIGMA, and DF must be numeric."); endif ## Check for MU, SIGMA, and DF being real if (iscomplex (mu) || iscomplex (sigma) || iscomplex (df)) error ("tlsstat: MU, SIGMA, and DF must not be complex."); endif ## Check for common size of MU, SIGMA, and DF if (! isscalar (mu) || ! isscalar (sigma) || ! isscalar (df)) [retval, mu, sigma, df] = common_size (mu, sigma, df); if (retval > 0) error ("tlsstat: MU, SIGMA, and DF must be of common size or scalars."); endif endif ## Calculate moments m = zeros (size (df)) + mu; v = sigma .* (df ./ (df - 2)); ## Continue argument check k = find (! (df > 1) | ! (df < Inf)); if (any (k)) m(k) = NaN; v(k) = NaN; endif k = find (! (df > 2) & (df < Inf)); if (any (k)) v(k) = NaN; endif endfunction ## Input validation tests %!error tlsstat () %!error tlsstat (1) %!error tlsstat (1, 2) %!error tlsstat ({}, 2, 3) %!error tlsstat (1, "", 3) %!error tlsstat (1, 2, ["d"]) %!error tlsstat (i, 2, 3) %!error tlsstat (1, i, 3) %!error tlsstat (1, 2, i) %!error ... %! tlsstat (ones (3), ones (2), 1) %!error ... %! tlsstat (ones (2), 1, ones (3)) %!error ... %! tlsstat (1, ones (2), ones (3)) ## Output validation tests %!test %! [m, v] = tlsstat (0, 1, 0); %! assert (m, NaN); %! assert (v, NaN); %!test %! [m, v] = tlsstat (0, 1, 1); %! assert (m, NaN); %! assert (v, NaN); %!test %! [m, v] = tlsstat (2, 1, 1); %! assert (m, NaN); %! assert (v, NaN); %!test %! [m, v] = tlsstat (-2, 1, 1); %! assert (m, NaN); %! assert (v, NaN); %!test %! [m, v] = tlsstat (0, 1, 2); %! assert (m, 0); %! assert (v, NaN); %!test %! [m, v] = tlsstat (2, 1, 2); %! assert (m, 2); %! assert (v, NaN); %!test %! [m, v] = tlsstat (-2, 1, 2); %! assert (m, -2); %! assert (v, NaN); %!test %! [m, v] = tlsstat (0, 2, 2); %! assert (m, 0); %! assert (v, NaN); %!test %! [m, v] = tlsstat (2, 2, 2); %! assert (m, 2); %! assert (v, NaN); %!test %! [m, v] = tlsstat (-2, 2, 2); %! assert (m, -2); %! assert (v, NaN); %!test %! [m, v] = tlsstat (0, 1, 3); %! assert (m, 0); %! assert (v, 3); %!test %! [m, v] = tlsstat (0, 2, 3); %! assert (m, 0); %! assert (v, 6); %!test %! [m, v] = tlsstat (2, 1, 3); %! assert (m, 2); %! assert (v, 3); %!test %! [m, v] = tlsstat (2, 2, 3); %! assert (m, 2); %! assert (v, 6); %!test %! [m, v] = tlsstat (-2, 1, 3); %! assert (m, -2); %! assert (v, 3); %!test %! [m, v] = tlsstat (-2, 2, 3); %! assert (m, -2); %! assert (v, 6); statistics-release-1.6.3/inst/dist_stat/tstat.m000066400000000000000000000050241456127120000216200ustar00rootroot00000000000000## Copyright (C) 2006, 2007 Arno Onken ## Copyright (C) 2024 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {[@var{m}, @var{v}] =} tstat (@var{df}) ## ## Compute statistics of the Student's T distribution. ## ## @code{[@var{m}, @var{v}] = tstat (@var{df})} returns the mean and variance of ## the Student's T distribution with @var{df} degrees of freedom. ## ## The size of @var{m} (mean) and @var{v} (variance) is the same size of the ## input argument. ## ## Further information about the Student's T distribution can be found at ## @url{https://en.wikipedia.org/wiki/Student%27s_t-distribution} ## ## @seealso{tcdf, tinv, tpdf, trnd} ## @end deftypefn function [m, v] = tstat (df) ## Check for valid number of input arguments if (nargin < 1) error ("tstat: function called with too few input arguments."); endif ## Check for DF being numeric if (! isnumeric (df)) error ("tstat: DF must be numeric."); endif ## Check for DF being real if (iscomplex (df)) error ("tstat: DF must not be complex."); endif ## Calculate moments m = zeros (size (df)); v = df ./ (df - 2); ## Continue argument check k = find (! (df > 1) | ! (df < Inf)); if (any (k)) m(k) = NaN; v(k) = NaN; endif k = find (! (df > 2) & (df < Inf)); if (any (k)) v(k) = Inf; endif endfunction ## Input validation tests %!error tstat () %!error tstat ({}) %!error tstat ("") %!error tstat (i) ## Output validation tests %!test %! df = 3:8; %! [m, v] = tstat (df); %! expected_m = [0, 0, 0, 0, 0, 0]; %! expected_v = [3.0000, 2.0000, 1.6667, 1.5000, 1.4000, 1.3333]; %! assert (m, expected_m); %! assert (v, expected_v, 0.001); statistics-release-1.6.3/inst/dist_stat/unidstat.m000066400000000000000000000052631456127120000223210ustar00rootroot00000000000000## Copyright (C) 2006, 2007 Arno Onken ## Copyright (C) 2024 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {[@var{m}, @var{v}] =} unidstat (@var{df}) ## ## Compute statistics of the discrete uniform cumulative distribution. ## ## @code{[@var{m}, @var{v}] = unidstat (@var{df})} returns the mean and variance ## of the discrete uniform cumulative distribution with parameter @var{N}, which ## corresponds to the maximum observable value and must be a positive natural ## number. ## ## The size of @var{m} (mean) and @var{v} (variance) is the same size of the ## input argument. ## ## Further information about the discrete uniform distribution can be found at ## @url{https://en.wikipedia.org/wiki/Discrete_uniform_distribution} ## ## @seealso{unidcdf, unidinv, unidpdf, unidrnd, unidfit} ## @end deftypefn function [m, v] = unidstat (N) ## Check for valid number of input arguments if (nargin < 1) error ("unidstat: function called with too few input arguments."); endif ## Check for N being numeric if (! isnumeric (N)) error ("unidstat: N must be numeric."); endif ## Check for N being real if (iscomplex (N)) error ("unidstat: N must not be complex."); endif ## Calculate moments m = (N + 1) ./ 2; v = ((N .^ 2) - 1) ./ 12; ## Continue argument check k = find (! (N > 0) | ! (N < Inf) | ! (N == round (N))); if (any (k)) m(k) = NaN; v(k) = NaN; endif endfunction ## Input validation tests %!error unidstat () %!error unidstat ({}) %!error unidstat ("") %!error unidstat (i) ## Output validation tests %!test %! N = 1:6; %! [m, v] = unidstat (N); %! expected_m = [1.0000, 1.5000, 2.0000, 2.5000, 3.0000, 3.5000]; %! expected_v = [0.0000, 0.2500, 0.6667, 1.2500, 2.0000, 2.9167]; %! assert (m, expected_m, 0.001); %! assert (v, expected_v, 0.001); statistics-release-1.6.3/inst/dist_stat/unifstat.m000066400000000000000000000072641456127120000223260ustar00rootroot00000000000000## Copyright (C) 2006, 2007 Arno Onken ## Copyright (C) 2024 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {[@var{m}, @var{v}] =} unifstat (@var{df}) ## ## Compute statistics of the continuous uniform cumulative distribution. ## ## @code{[@var{m}, @var{v}] = unifstat (@var{df})} returns the mean and variance ## of the continuous uniform cumulative distribution with parameters @var{a} and ## @var{b}, which define the lower and upper bounds of the interval ## @qcode{[@var{a}, @var{b}]}. ## ## The size of @var{m} (mean) and @var{v} (variance) is the common size of the ## input arguments. A scalar input functions as a constant matrix of the ## same size as the other inputs. ## ## Further information about the continuous uniform distribution can be found at ## @url{https://en.wikipedia.org/wiki/Continuous_uniform_distribution} ## ## @seealso{unifcdf, unifinv, unifpdf, unifrnd, unifit} ## @end deftypefn function [m, v] = unifstat (a, b) ## Check for valid number of input arguments if (nargin < 2) error ("unifstat: function called with too few input arguments."); endif ## Check for A and B being numeric if (! (isnumeric (a) && isnumeric (b))) error ("unifstat: A and B must be numeric."); endif ## Check for A and B being real if (iscomplex (a) || iscomplex (b)) error ("unifstat: A and B must not be complex."); endif ## Check for common size of A and B if (! isscalar (a) || ! isscalar (b)) [retval, a, b] = common_size (a, b); if (retval > 0) error ("unifstat: A and B must be of common size or scalars."); endif endif ## Calculate moments m = (a + b) ./ 2; v = ((b - a) .^ 2) ./ 12; ## Continue argument check k = find (! (-Inf < a) | ! (a < b) | ! (b < Inf)); if (any (k)) m(k) = NaN; v(k) = NaN; endif endfunction ## Input validation tests %!error unifstat () %!error unifstat (1) %!error unifstat ({}, 2) %!error unifstat (1, "") %!error unifstat (i, 2) %!error unifstat (1, i) %!error ... %! unifstat (ones (3), ones (2)) %!error ... %! unifstat (ones (2), ones (3)) ## Output validation tests %!test %! a = 1:6; %! b = 2:2:12; %! [m, v] = unifstat (a, b); %! expected_m = [1.5000, 3.0000, 4.5000, 6.0000, 7.5000, 9.0000]; %! expected_v = [0.0833, 0.3333, 0.7500, 1.3333, 2.0833, 3.0000]; %! assert (m, expected_m, 0.001); %! assert (v, expected_v, 0.001); %!test %! a = 1:6; %! [m, v] = unifstat (a, 10); %! expected_m = [5.5000, 6.0000, 6.5000, 7.0000, 7.5000, 8.0000]; %! expected_v = [6.7500, 5.3333, 4.0833, 3.0000, 2.0833, 1.3333]; %! assert (m, expected_m, 0.001); %! assert (v, expected_v, 0.001); statistics-release-1.6.3/inst/dist_stat/wblstat.m000066400000000000000000000073671456127120000221550ustar00rootroot00000000000000## Copyright (C) 2006, 2007 Arno Onken ## Copyright (C) 2024 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {[@var{m}, @var{v}] =} wblstat (@var{lambda}, @var{k}) ## ## Compute statistics of the Weibull distribution. ## ## @code{[@var{m}, @var{v}] = wblstat (@var{lambda}, @var{k})} returns the mean ## and variance of the Weibull distribution with scale parameter @var{lambda} ## and shape parameter @var{k}. ## ## The size of @var{m} (mean) and @var{v} (variance) is the common size of the ## input arguments. A scalar input functions as a constant matrix of the ## same size as the other inputs. ## ## Further information about the Weibull distribution can be found at ## @url{https://en.wikipedia.org/wiki/Weibull_distribution} ## ## @seealso{wblcdf, wblinv, wblpdf, wblrnd, wblfit, wbllike, wblplot} ## @end deftypefn function [m, v] = wblstat (lambda, k) ## Check for valid number of input arguments if (nargin < 2) error ("wblstat: function called with too few input arguments."); endif ## Check for LAMBDA and K being numeric if (! (isnumeric (lambda) && isnumeric (k))) error ("wblstat: LAMBDA and K must be numeric."); endif ## Check for LAMBDA and K being real if (iscomplex (lambda) || iscomplex (k)) error ("wblstat: LAMBDA and K must not be complex."); endif ## Check for common size of LAMBDA and K if (! isscalar (lambda) || ! isscalar (k)) [retval, lambda, k] = common_size (lambda, k); if (retval > 0) error ("wblstat: LAMBDA and K must be of common size or scalars."); endif endif ## Calculate moments m = lambda .* gamma (1 + 1 ./ k); v = (lambda .^ 2) .* gamma (1 + 2 ./ k) - m .^ 2; ## Continue argument check is_nan = find (! (lambda > 0) | ! (lambda < Inf) | ! (k > 0) | ! (k < Inf)); if (any (is_nan)) m(is_nan) = NaN; v(is_nan) = NaN; endif endfunction ## Input validation tests %!error wblstat () %!error wblstat (1) %!error wblstat ({}, 2) %!error wblstat (1, "") %!error wblstat (i, 2) %!error wblstat (1, i) %!error ... %! wblstat (ones (3), ones (2)) %!error ... %! wblstat (ones (2), ones (3)) ## Output validation tests %!test %! lambda = 3:8; %! k = 1:6; %! [m, v] = wblstat (lambda, k); %! expected_m = [3.0000, 3.5449, 4.4649, 5.4384, 6.4272, 7.4218]; %! expected_v = [9.0000, 3.4336, 2.6333, 2.3278, 2.1673, 2.0682]; %! assert (m, expected_m, 0.001); %! assert (v, expected_v, 0.001); %!test %! k = 1:6; %! [m, v] = wblstat (6, k); %! expected_m = [ 6.0000, 5.3174, 5.3579, 5.4384, 5.5090, 5.5663]; %! expected_v = [36.0000, 7.7257, 3.7920, 2.3278, 1.5923, 1.1634]; %! assert (m, expected_m, 0.001); %! assert (v, expected_v, 0.001); statistics-release-1.6.3/inst/ecdf.m000066400000000000000000000257531456127120000173770ustar00rootroot00000000000000## Copyright (C) 2022 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {[@var{f}, @var{x}] =} ecdf (@var{y}) ## @deftypefnx {statistics} {[@var{f}, @var{x}, @var{flo}, @var{fup}] =} ecdf (@var{y}) ## @deftypefnx {statistics} {} ecdf (@dots{}) ## @deftypefnx {statistics} {} ecdf (@var{ax}, @dots{}) ## @deftypefnx {statistics} {[@dots{}] =} ecdf (@var{y}, @var{name}, @var{value}, @dots{}) ## @deftypefnx {statistics} {[@dots{}] =} ecdf (@var{ax}, @var{y}, @var{name}, @var{value}, @dots{}) ## ## Empirical (Kaplan-Meier) cumulative distribution function. ## ## @code{[@var{f}, @var{x}] = ecdf (@var{y})} calculates the Kaplan-Meier ## estimate of the cumulative distribution function (cdf), also known as the ## empirical cdf. @var{y} is a vector of data values. @var{f} is a vector of ## values of the empirical cdf evaluated at @var{x}. ## ## @code{[@var{f}, @var{x}, @var{flo}, @var{fup}] = ecdf (@var{y})} also returns ## lower and upper confidence bounds for the cdf. These bounds are calculated ## using Greenwood's formula, and are not simultaneous confidence bounds. ## ## @code{ecdf (@dots{})} without output arguments produces a plot of the ## empirical cdf. ## ## @code{ecdf (@var{ax}, @dots{})} plots into existing axes @var{ax}. ## ## @code{[@dots{}] = ecdf (@var{y}, @var{name}, @var{value}, @dots{})} specifies ## additional parameter name/value pairs chosen from the following: ## ## @multitable @columnfractions 0.20 0.8 ## @headitem @var{name} @tab @var{value} ## @item "censoring" @tab A boolean vector of the same size as Y that is 1 for ## observations that are right-censored and 0 for observations that are observed ## exactly. Default is all observations observed exactly. ## ## @item "frequency" @tab A vector of the same size as Y containing non-negative ## integer counts. The jth element of this vector gives the number of times the ## jth element of Y was observed. Default is 1 observation per Y element. ## ## @item "alpha" @tab A value @var{alpha} between 0 and 1 specifying the ## significance level. Default is 0.05 for 5% significance. ## ## @item "function" @tab The type of function returned as the F output argument, ## chosen from "cdf" (the default), "survivor", or "cumulative hazard". ## ## @item "bounds" @tab Either "on" to include bounds or "off" (the default) to ## omit them. Used only for plotting. ## @end multitable ## ## Type @code{demo ecdf} to see examples of usage. ## ## @seealso{cdfplot, ecdfhist} ## @end deftypefn function [Fout, x, Flo, Fup] = ecdf (y, varargin) ## Check for valid input arguments narginchk (1, Inf); ## Parse input arguments if (nargin == 1) ax = []; if (! isvector (y) || ! isreal (y)) error ("ecdf: Y must be a vector of real numbers."); endif ##x = varargin{1}; else ## ax = y; ## Check that ax is a valid axis handle try isstruct (get (y)); ax = y; y = varargin{1}; varargin{1} = []; catch ##error ("ecdf: invalid handle %f.", ax); ax = []; end_try_catch #y = varargin{1}; #varargin{1} = []; endif ## Make y a column vector x = y(:); ## Add defaults cens = zeros (size (x)); freq = ones (size (x)); alpha = 0.05; fname = "cdf"; bound = "off"; ## Check for remaining varargins and parse extra parameters if (length (varargin) > 0 && mod (numel (varargin), 2) == 0) [~, prop] = parseparams (varargin); while (!isempty (prop)) switch (lower (prop{1})) case "censoring" cens = prop{2}; ## Check for valid input if (! isequal (size (x), size (cens))) error ("ecdf: censoring data mismatch Y vector."); endif ## Make double in case censoring data is logical if (islogical (cens)) cens = double (cens); endif case "frequency" freq = prop{2}; ## Check for valid input if (! isequal (size (x), size (freq))) error ("ecdf: frequency data mismatch Y vector."); endif ## Make double in case frequency data is logical if (islogical (freq)) freq = double (freq); endif case "alpha" alpha = prop{2}; ## Check for valid alpha value if (numel (alpha) != 1 || ! isnumeric (alpha) || alpha <= 0 || alpha >= 1) error ("ecdf: alpha must be a numeric scalar in the range (0,1)."); endif case "function" fname = prop{2}; ## Check for valid function name option if (sum (strcmpi (fname, {"cdf", "survivor", "cumulative hazard"})) < 1) error ("ecdf: wrong function name."); endif case "bounds" bound = prop{2}; ## Check for valid bounds option if (! (strcmpi (bound, "off") || strcmpi (bound, "on"))) error ("ecdf: wrong bounds."); endif otherwise error ("ecdf: unknown option %s", prop{1}); endswitch prop = prop(3:end); endwhile elseif nargin > 2 error ("ecdf: optional parameters must be in name/value pairs."); endif ## Remove NaNs from data rn = ! isnan (x) & ! isnan (cens) & freq > 0; x = x(rn); if (length (x) == 0) error ("ecdf: not enought data."); endif cens = cens(rn); freq = freq(rn); ## Sort data in ascending order [x, sr] = sort (x); cens = cens(sr); freq = freq(sr); ## Keep class for data (single | double) if (isa (x, "single")) freq = single (freq); endif ## Calculate cumulative frequencies tcfreq = cumsum (freq); ocfreq = cumsum (freq .* ! cens); x_diff = (diff (x) == 0); if (any (x_diff)) x(x_diff) = []; tcfreq(x_diff) = []; ocfreq(x_diff) = []; endif max_count = tcfreq(end); ## Get Deaths and Number at Risk for each unique X Death = [ocfreq(1); diff(ocfreq)]; NRisk = max_count - [0; tcfreq(1:end-1)]; ## Remove no Death observations x = x(Death > 0); NRisk = NRisk(Death > 0); Death = Death(Death > 0); ## Estimate function switch (fname) case "cdf" S = cumprod (1 - Death ./ NRisk); Fun_x = 1 - S; Fzero = 0; fdisp = "F(x)"; case "survivor" S = cumprod (1 - Death ./ NRisk); Fun_x = S; Fzero = 1; fdisp = "S(x)"; case "cumulative hazard" Fun_x = cumsum (Death ./ NRisk); Fzero = 0; fdisp = "H(x)"; endswitch ## Check for remaining non-censored data and add a starting value if (! isempty (Death)) x = [min(y); x]; F = [Fzero; Fun_x]; else warning("ecdf: No Death in data"); F = Fun_x; endif ## Calculate lower and upper confidence bounds if requested if (nargout > 2 || (nargout == 0 && strcmpi (bound, "on"))) switch (fname) case {"cdf", "survivor"} se = NaN (size (Death)); if (! isempty (Death)) if (NRisk(end) == Death(end)) t = 1:length (NRisk) - 1; else t = 1:length (NRisk); endif se(t) = S(t) .* sqrt (cumsum (Death(t) ./ ... (NRisk(t) .* (NRisk(t) - Death(t))))); endif case "cumulative hazard" se = sqrt (cumsum (Death ./ (NRisk .* NRisk))); endswitch ## Calculate confidence limits if (! isempty (se)) z_a = - norminv (alpha / 2); h_w = z_a * se; Flo = max (0, Fun_x - h_w); Flo(isnan (h_w)) = NaN; switch (fname) case {"cdf", "survivor"} Fup = min (1, Fun_x + h_w); Fup(isnan (h_w)) = NaN; case "cumulative hazard" Fup = Fun_x + h_w; endswitch Flo = [NaN; Flo]; Fup = [NaN; Fup]; else Flo = []; Fup = []; endif else Flo = []; Fup = []; endif ## Plot stairs if no output is requested if (nargout == 0) if (isempty (ax)) ax = newplot(); end h = stairs(ax, x , [F, Flo, Fup]); xlabel (ax, "x"); ylabel (ax, fdisp); title ("ecdf"); else Fout = F; endif endfunction %!demo %! y = exprnd (10, 50, 1); ## random failure times are exponential(10) %! d = exprnd (20, 50, 1); ## drop-out times are exponential(20) %! t = min (y, d); ## we observe the minimum of these times %! censored = (y > d); ## we also observe whether the subject failed %! %! ## Calculate and plot the empirical cdf and confidence bounds %! [f, x, flo, fup] = ecdf (t, "censoring", censored); %! stairs (x, f); %! hold on; %! stairs (x, flo, "r:"); stairs (x, fup, "r:"); %! %! ## Superimpose a plot of the known true cdf %! xx = 0:.1:max (t); yy = 1 - exp (-xx / 10); plot (xx, yy, "g-"); %! hold off; %!demo %! R = wblrnd (100, 2, 100, 1); %! ecdf (R, "Function", "survivor", "Alpha", 0.01, "Bounds", "on"); %! hold on %! x = 1:1:250; %! wblsurv = 1 - cdf ("weibull", x, 100, 2); %! plot (x, wblsurv, "g-", "LineWidth", 2) %! legend ("Empirical survivor function", "Lower confidence bound", ... %! "Upper confidence bound", "Weibull survivor function", ... %! "Location", "northeast"); %! hold off ## Test input %!error ecdf (); %!error ecdf (randi (15,2)); %!error ecdf ([3,2,4,3+2i,5]); %!error kstest ([2,3,4,5,6],"tail"); %!error kstest ([2,3,4,5,6],"tail", "whatever"); %!error kstest ([2,3,4,5,6],"function", ""); %!error kstest ([2,3,4,5,6],"badoption", 0.51); %!error kstest ([2,3,4,5,6],"tail", 0); %!error kstest ([2,3,4,5,6],"alpha", 0); %!error kstest ([2,3,4,5,6],"alpha", NaN); %!error kstest ([NaN,NaN,NaN,NaN,NaN],"tail", "unequal"); %!error kstest ([2,3,4,5,6],"alpha", 0.05, "CDF", [2,3,4;1,3,4;1,2,1]); ## Test output against MATLAB results %!test %! hf = figure ("visible", "off"); %! unwind_protect %! x = [2, 3, 4, 3, 5, 4, 6, 5, 8, 3, 7, 8, 9, 0]; %! [F, x, Flo, Fup] = ecdf (x); %! F_out = [0; 0.0714; 0.1429; 0.3571; 0.5; 0.6429; 0.7143; 0.7857; 0.9286; 1]; %! assert (F, F_out, ones (10,1) * 1e-4); %! x_out = [0 0 2 3 4 5 6 7 8 9]'; %! assert (x, x_out); %! Flo_out = [NaN, 0, 0, 0.1061, 0.2381, 0.3919, 0.4776, 0.5708, 0.7937, NaN]'; %! assert (Flo, Flo_out, ones (10,1) * 1e-4); %! Fup_out = [NaN, 0.2063, 0.3262, 0.6081, 0.7619, 0.8939, 0.9509, 1, 1, NaN]'; %! assert (Fup, Fup_out, ones (10,1) * 1e-4); %! unwind_protect_cleanup %! close (hf); %! end_unwind_protect %!test %! hf = figure ("visible", "off"); %! unwind_protect %! x = [2, 3, 4, 3, 5, 4, 6, 5, 8, 3, 7, 8, 9, 0]; %! ecdf (x); %! unwind_protect_cleanup %! close (hf); %! end_unwind_protect statistics-release-1.6.3/inst/einstein.m000066400000000000000000000225361456127120000203100ustar00rootroot00000000000000## Copyright (C) 2023 Arun Giridhar ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {} einstein () ## @deftypefnx {statistics} {@var{tiles} =} einstein (@var{a}, @var{b}) ## @deftypefnx {statistics} {[@var{tiles}, @var{rhat}] =} einstein (@var{a}, @var{b}) ## @deftypefnx {statistics} {[@var{tiles}, @var{rhat}, @var{that}] =} einstein (@var{a}, @var{b}) ## @deftypefnx {statistics} {[@var{tiles}, @var{rhat}, @var{that}, @var{shat}] =} einstein (@var{a}, @var{b}) ## @deftypefnx {statistics} {[@var{tiles}, @var{rhat}, @var{that}, @var{shat}, @var{phat}] =} einstein (@var{a}, @var{b}) ## @deftypefnx {statistics} {[@var{tiles}, @var{rhat}, @var{that}, @var{shat}, @var{phat}, @var{fhat}] =} einstein (@var{a}, @var{b}) ## ## Plots the tiling of the basic clusters of einstein tiles. ## ## Scalars @var{a} and @var{b} define the shape of the einstein tile. ## See Smith et al (2023) for details: @url{https://arxiv.org/abs/2303.10798} ## ## @itemize ## @item @var{tiles} is a structure containing the coordinates of the einstein ## tiles that are tiled on the plot. Each field contains the tile coordinates ## of the corresponding clusters. ## @itemize ## @item @var{tiles}@qcode{.rhat} contains the reflected einstein tiles ## @item @var{tiles}@qcode{.that} contains the three-hat shells ## @item @var{tiles}@qcode{.shat} contains the single-hat clusters ## @item @var{tiles}@qcode{.phat} contains the paired-hat clusters ## @item @var{tiles}@qcode{.fhat} contains the fylfot clusters ## @end itemize ## ## @item @var{rhat} contains the coordinates of the first reflected tile ## @item @var{that} contains the coordinates of the first three-hat shell ## @item @var{shat} contains the coordinates of the first single-hat cluster ## @item @var{phat} contains the coordinates of the first paired-hat cluster ## @item @var{fhat} contains the coordinates of the first fylfot cluster ## @end itemize ## ## @end deftypefn function [varargout] = einstein (a, b, varargin) ## Check for valid number of input arguments if (nargin < 2) print_usage; endif ## Check A and B for valid type and range if (! (isscalar (a) && isscalar (b) && isnumeric (a) && isnumeric (b)... && isreal (a) && isreal (b))) error ("einstein: A and B must real scalars."); end if (a <= 0 || a >= 1 || b <= 0 || b >= 1) error ("einstein: A and B must be within the open interval (0,1)."); endif ## Get initial hat points single_hat = getpoly (a, b) * rotz (-150)([1,2],[1,2]); ## Make a reflected-hat reflecthat = [-1, 1] .* single_hat; ## Make a three-hat shell cluster three_hat1 = rotatehat (single_hat, -60); three_hat1 = three_hat1 + (reflecthat(1,:) - three_hat1(9,:)); three_hat2 = three_hat1 + (reflecthat(6,:) - three_hat1(11,:)); three_hat3 = rotatehat (three_hat1, 120); three_hat3 = three_hat3 + (reflecthat(5,:) - three_hat1(9,:)); three_hat = [three_hat1, three_hat2, three_hat3]; ## Translate another four-tile cluster translate = three_hat(5,[3,4]) - three_hat(12,[1,2]); all.rhat = translatehat (reflecthat, translate); all.that = translatehat (three_hat, translate); all.rhat = [reflecthat, all.rhat]; all.that = [three_hat, all.that]; ## Rotate and translate another four-tile cluster tmp_3hat = rotatehat (three_hat, 120); tmp_rhat = rotatehat (reflecthat, 120); translate = three_hat(12,[5,6]) - tmp_3hat(5,[3,4]); tmp_3hat = translatehat (tmp_3hat, translate); tmp_rhat = translatehat (tmp_rhat, translate); all.that = [all.that, tmp_3hat]; all.rhat = [all.rhat, tmp_rhat]; ## Plot four-tile clusters patch (all.that(:,[1:2:end]), all.that(:,[2:2:end]), "LineWidth", 2, ... "FaceColor", "c", "EdgeColor","k"); patch (all.rhat(:,[1:2:end]), all.rhat(:,[2:2:end]), "LineWidth", 2, ... "FaceColor", "b", "EdgeColor","k"); title (sprintf ("a = %4.2f b = %4.2f", a, b), "FontSize",30) ## Make a single-hat cluster translate = three_hat(10,[3,4]) - single_hat(2,:); singlehat = translatehat (single_hat, translate); all.shat = singlehat; ## Plot single-tile cluster patch (all.shat(:,1), all.shat(:,2), "LineWidth", 2, ... "FaceColor", [0.95, 0.95, 0.95], "EdgeColor","k"); axis ("equal") ## Make a paired-hat cluster paired_hat1 = all.shat + (all.rhat(9,[5,6]) - all.shat(5,:)); paired_hat2 = three_hat(:,[3,4]) + (three_hat(1,[5,6]) - three_hat(3,[3,4])); paired_hat = [paired_hat1, paired_hat2]; ## Rotate and translate another two paired-hat clusters tmp_phat = rotatehat (paired_hat, -120); translate = three_hat(4,[3,4]) - tmp_phat(4,[3,4]); tmp_phat = translatehat (tmp_phat, translate); all.phat = [paired_hat, tmp_phat]; tmp_phat = rotatehat (paired_hat, -60); translate = all.that(10,[11,12]) - tmp_phat(11,[3,4]); tmp_phat = translatehat (tmp_phat, translate); all.phat = [all.phat, tmp_phat]; ## Plot paired-tiles clusters patch (all.phat(:,[1:2:end]), all.phat(:,[2:2:end]), "LineWidth", 2, ... "FaceColor", [0.9, 0.9, 0.9], "EdgeColor","k"); ## Make a fylfot cluster fylfot_hat = paired_hat; tmp_fhat = rotatehat (paired_hat, -120); translate = fylfot_hat(1,[3,4]) - tmp_fhat(3,[3,4]); tmp_fhat = translatehat (tmp_fhat, translate); fylfot_hat = [fylfot_hat, tmp_fhat]; tmp_fhat = rotatehat (paired_hat, 120); translate = fylfot_hat(3,[3,4]) - tmp_fhat(1,[3,4]); tmp_fhat = translatehat (tmp_fhat, translate); fylfot_hat = [fylfot_hat, tmp_fhat]; translate = all.that(13,[13,14]) - fylfot_hat(13,[7,8]); fylfot_hat = translatehat (fylfot_hat, translate); ## Translate another two fylfot clusters translate = all.that(4,[9,10]) - fylfot_hat(13,[3,4]); tmp_fhat = translatehat (fylfot_hat, translate); all.fhat = [fylfot_hat, tmp_fhat]; translate = all.that(13,[1,2]) - fylfot_hat(4,[3,4]); tmp_fhat = translatehat (fylfot_hat, translate); all.fhat = [all.fhat, tmp_fhat]; ## Plot fylfot clusters patch (all.fhat(:,[1:2:end]), all.fhat(:,[2:2:end]), "LineWidth", 2, ... "FaceColor", "r", "EdgeColor","k"); if (nargout > 0) varargout{1} = all; endif if (nargout > 1) varargout{2} = reflecthat; endif if (nargout > 2) varargout{3} = three_hat; endif if (nargout > 3) varargout{4} = singlehat; endif if (nargout > 4) varargout{5} = paired_hat; endif if (nargout > 5) varargout{6} = fylfot_hat; endif endfunction ## Rotates a cluster of hats. function newhat = rotatehat (hat, degrees) rotM = rotz (degrees)([1,2],[1,2]); newhat = zeros (size (hat)); for i=1:2:columns (hat) newhat(:,[i,i+1]) = hat(:,[i,i+1]) * rotM; endfor endfunction ## Translates a cluster of hats. function newhat = translatehat (hat, dist) nhats = columns (hat) / 2; newhat = hat + repmat (dist, 1, nhats); endfunction ## Returns a unit vector given a direction angle in degrees function ret = u (t) persistent angles = (0:30:330); persistent tbl = [cosd(angles); sind(angles)]'; ret = tbl(angles == mod (t, 360), :); endfunction ## Returns the hat polygon function single_hat = getpoly (a, b) single_hat = zeros (14, 2); pos = 0; single_hat(++pos, :) = [0 0]; t = 270; single_hat(pos+1, :) = single_hat(pos++, :) + a * u(t); t += 60; single_hat(pos+1, :) = single_hat(pos++, :) + a * u(t); t -= 90; single_hat(pos+1, :) = single_hat(pos++, :) + b * u(t); t += 60; single_hat(pos+1, :) = single_hat(pos++, :) + b * u(t); t += 90; single_hat(pos+1, :) = single_hat(pos++, :) + a * u(t); t -= 60; single_hat(pos+1, :) = single_hat(pos++, :) + a * u(t); t += 90; single_hat(pos+1, :) = single_hat(pos++, :) + b * u(t); t -= 60; single_hat(pos+1, :) = single_hat(pos++, :) + b * u(t); t += 90; single_hat(pos+1, :) = single_hat(pos++, :) + a * u(t); t += 60; single_hat(pos+1, :) = single_hat(pos++, :) + a * u(t) * 2; t += 60; single_hat(pos+1, :) = single_hat(pos++, :) + a * u(t); t -= 90; single_hat(pos+1, :) = single_hat(pos++, :) + b * u(t); t += 60; single_hat(pos+1, :) = single_hat(pos++, :) + b * u(t); ## t(14, :) == t(1, :) to within numerical roundoff endfunction %!demo %! einstein (0.4, 0.6) %!demo %! einstein (0.2, 0.5) %!demo %! einstein (0.6, 0.1) ## Test plotting %!test %! hf = figure ("visible", "off"); %! unwind_protect %! tiles = einstein (0.4, 0.6); %! assert (isstruct (tiles), true); %! unwind_protect_cleanup %! close (hf); %! end_unwind_protect ## Test input validation %!error einstein %!error einstein (0.5) %!error einstein (0, 0.9) %!error einstein (0.4, 1) %!error einstein (-0.4, 1) statistics-release-1.6.3/inst/evalclusters.m000066400000000000000000000361451456127120000212070ustar00rootroot00000000000000## Copyright (C) 2021 Stefano Guidoni ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{eva} =} evalclusters (@var{x}, @var{clust}, @var{criterion}) ## @deftypefnx {statistics} {@var{eva} =} evalclusters (@dots{}, @qcode{Name}, @qcode{Value}) ## ## Create a clustering evaluation object to find the optimal number of clusters. ## ## @code{evalclusters} creates a clustering evaluation object to evaluate the ## optimal number of clusters for data @var{x}, using criterion @var{criterion}. ## The input data @var{x} is a matrix with @code{n} observations of @code{p} ## variables. ## The evaluation criterion @var{criterion} is one of the following: ## @table @code ## @item @qcode{CalinskiHarabasz} ## to create a @code{CalinskiHarabaszEvaluation} object. ## ## @item @qcode{DaviesBouldin} ## to create a @code{DaviesBouldinEvaluation} object. ## ## @item @qcode{gap} ## to create a @code{GapEvaluation} object. ## ## @item @qcode{silhouette} ## to create a @code{SilhouetteEvaluation} object. ## ## @end table ## The clustering algorithm @var{clust} is one of the following: ## @table @code ## @item @qcode{kmeans} ## to cluster the data using @code{kmeans} with @code{EmptyAction} set to ## @code{singleton} and @code{Replicates} set to 5. ## ## @item @qcode{linkage} ## to cluster the data using @code{clusterdata} with @code{linkage} set to ## @code{Ward}. ## ## @item @qcode{gmdistribution} ## to cluster the data using @code{fitgmdist} with @code{SharedCov} set to ## @code{true} and @code{Replicates} set to 5. ## ## @end table ## If the @var{criterion} is @code{CalinskiHarabasz}, @code{DaviesBouldin}, or ## @code{silhouette}, @var{clust} can also be a function handle to a function ## of the form @code{c = clust(x, k)}, where @var{x} is the input data, ## @var{k} the number of clusters to evaluate and @var{c} the clustering result. ## The clustering result can be either an array of size @code{n} with @code{k} ## different integer values, or a matrix of size @code{n} by @code{k} with a ## likelihood value assigned to each one of the @code{n} observations for each ## one of the @var{k} clusters. In the latter case, each observation is assigned ## to the cluster with the higher value. ## If the @var{criterion} is @code{CalinskiHarabasz}, @code{DaviesBouldin}, or ## @code{silhouette}, @var{clust} can also be a matrix of size @code{n} by ## @code{k}, where @code{k} is the number of proposed clustering solutions, so ## that each column of @var{clust} is a clustering solution. ## ## In addition to the obligatory @var{x}, @var{clust} and @var{criterion} inputs ## there is a number of optional arguments, specified as pairs of @code{Name} ## and @code{Value} options. The known @code{Name} arguments are: ## @table @code ## @item @qcode{KList} ## a vector of positive integer numbers, that is the cluster sizes to evaluate. ## This option is necessary, unless @var{clust} is a matrix of proposed ## clustering solutions. ## ## @item @qcode{Distance} ## a distance metric as accepted by the chosen @var{clust}. It can be the ## name of the distance metric as a string or a function handle. When ## @var{criterion} is @code{silhouette}, it can be a vector as created by ## function @code{pdist}. Valid distance metric strings are: @code{sqEuclidean} ## (default), @code{Euclidean}, @code{cityblock}, @code{cosine}, ## @code{correlation}, @code{Hamming}, @code{Jaccard}. ## Only used by @code{silhouette} and @code{gap} evaluation. ## ## @item @qcode{ClusterPriors} ## the prior probabilities of each cluster, which can be either @code{empirical} ## (default), or @code{equal}. When @code{empirical} the silhouette value is ## the average of the silhouette values of all points; when @code{equal} the ## silhouette value is the average of the average silhouette value of each ## cluster. Only used by @code{silhouette} evaluation. ## ## @item @qcode{B} ## the number of reference datasets generated from the reference distribution. ## Only used by @code{gap} evaluation. ## ## @item @qcode{ReferenceDistribution} ## the reference distribution used to create the reference data. It can be ## @code{PCA} (default) for a distribution based on the principal components of ## @var{X}, or @code{uniform} for a uniform distribution based on the range of ## the observed data. @code{PCA} is currently not implemented. ## Only used by @code{gap} evaluation. ## ## @item @qcode{SearchMethod} ## the method for selecting the optimal value with a @code{gap} evaluation. It ## can be either @code{globalMaxSE} (default) for selecting the smallest number ## of clusters which is inside the standard error of the maximum gap value, or ## @code{firstMaxSE} for selecting the first number of clusters which is inside ## the standard error of the following cluster number. ## Only used by @code{gap} evaluation. ## ## @end table ## ## Output @var{eva} is a clustering evaluation object. ## ## @end deftypefn ## ## @seealso{CalinskiHarabaszEvaluation, DaviesBouldinEvaluation, GapEvaluation, ## SilhouetteEvaluation} function cc = evalclusters (x, clust, criterion, varargin) ## input check if (nargin < 3) print_usage (); endif ## parsing input data if ((! ismatrix (x)) || (! isnumeric (x))) error ("evalclusters: 'x' must be a numeric matrix"); endif ## useful values for input check n = rows (x); p = columns (x); ## parsing the clustering algorithm if (ischar (clust)) clust = lower (clust); if (! any (strcmpi (clust, {"kmeans", "linkage", "gmdistribution"}))) error ("evalclusters: unknown clustering algorithm '%s'", clust); endif elseif (! isscalar (clust)) if ((! isnumeric (clust)) || (length (size (clust)) != 2) || ... (rows (clust) != n)) error ("evalclusters: invalid matrix of clustering solutions"); endif elseif (! isa (clust, "function_handle")) error ("evalclusters: invalid argument for 'clust'"); endif ## parsing the criterion parameter ## we check the rest later, as the check depends on the chosen criterion if (! ischar (criterion)) error ("evalclusters: invalid criterion, it must be a string"); else criterion = lower (criterion); if (! any (strcmpi (criterion, {"calinskiharabasz", "daviesbouldin", ... "silhouette", "gap"}))) error ("evalclusters: unknown criterion '%s'", criterion); endif endif ## some default value klist = []; distance = "sqeuclidean"; clusterpriors = "empirical"; b = 100; referencedistribution = "pca"; searchmethod = "globalmaxse"; ## parse the name/value pairs pair_index = 1; while (pair_index < (nargin - 3)) ## type check if (! ischar (varargin{pair_index})) error ("evalclusters: invalid property, string expected"); endif ## now parse the parameter switch (lower (varargin{pair_index})) case "klist" ## klist must be an array of positive interger numbers; ## there is a special case when it can be empty, but that is not the ## suggested way to use it (it is better to omit it instead) if (isempty (varargin{pair_index + 1})) if (ischar (clust) || isa (clust, "function_handle")) error (["evalclusters: 'KList' can be empty only when 'clust' "... "is a matrix"]); endif elseif ((! isnumeric (varargin{pair_index + 1})) || ... (! isvector (varargin{pair_index + 1})) || ... any (find (varargin{pair_index + 1} < 1)) || ... any (floor (varargin{pair_index + 1}) != varargin{pair_index + 1})) error ("evalclusters: 'KList' must be an array of positive integers"); endif klist = varargin{pair_index + 1}; case "distance" ## used by silhouette and gap if (! (strcmpi (criterion, "silhouette") || strcmpi (criterion, "gap"))) error (["evalclusters: distance metric cannot be used with '%s'"... " criterion"], criterion); endif if (ischar (varargin{pair_index + 1})) if (! any (strcmpi (varargin{pair_index + 1}, ... {"sqeuclidean", "euclidean", "cityblock", "cosine", ... "correlation", "hamming", "jaccard"}))) error ("evalclusters: unknown distance criterion '%s'", ... varargin{pair_index + 1}); endif elseif (! isa (varargin{pair_index + 1}, "function_handle") || ! ((isvector (varargin{pair_index + 1}) && ... isnumeric (varargin{pair_index + 1})))) error ("evalclusters: invalid distance metric"); endif distance = varargin{pair_index + 1}; case "clusterpriors" ## used by silhouette evaluation if (! strcmpi (criterion, "silhouette")) error (["evalclusters: cluster prior probabilities cannot be used "... "with '%s' criterion"], criterion); endif if (any (strcmpi (varargin{pair_index + 1}, {"empirical", "equal"}))) clusterpriors = lower (varargin{pair_index + 1}); else error ("evalclusters: invalid cluster prior probabilities value"); endif case "b" ## used by gap evaluation if (! isnumeric (varargin{pair_index + 1}) || ... ! isscalar (varargin{pair_index + 1}) || ... varargin{pair_index + 1} != floor (varargin{pair_index + 1}) || ... varargin{pair_index + 1} < 1) error ("evalclusters: b must a be positive integer number"); endif b = varargin{pair_index + 1}; case "referencedistribution" ## used by gap evaluation if (! ischar (varargin{pair_index + 1}) || any (strcmpi ... (varargin{pair_index + 1}, {"pca", "uniform"}))) error (["evalclusters: the reference distribution must be either" ... "'PCA' or 'uniform'"]); endif referencedistribution = lower (varargin{pair_index + 1}); case "searchmethod" ## used by gap evaluation if (! ischar (varargin{pair_index + 1}) || any (strcmpi ... (varargin{pair_index + 1}, {"globalmaxse", "uniform"}))) error (["evalclusters: the search method must be either" ... "'globalMaxSE' or 'firstmaxse'"]); endif searchmethod = lower (varargin{pair_index + 1}); otherwise error ("evalclusters: unknown property %s", varargin{pair_index}); endswitch pair_index += 2; endwhile ## check if there are parameters without a value or a name left if (nargin - 2 - pair_index) if (ischar (varargin{pair_index})) error ("evalclusters: invalid parameter '%s'", varargin{pair_index}); else error ("evalclusters: invalid parameter '%d'", varargin{pair_index}); endif endif ## another check on klist if (isempty (klist) && (ischar (clust) || isa (clust, "function_handle"))) error (["evalclusters: 'KList' can be empty only when 'clust' ", ... "is a matrix"]); endif ## main switch (lower (criterion)) case "calinskiharabasz" ## further compatibility checks between the chosen parameters are ## delegated to the class constructor if (isempty (klist)) klist = 1 : columns (clust); endif cc = CalinskiHarabaszEvaluation (x, clust, klist); case "daviesbouldin" ## further compatibility checks between the chosen parameters are ## delegated to the class constructor if (isempty (klist)) klist = 1 : columns (clust); endif cc = DaviesBouldinEvaluation (x, clust, klist); case "silhouette" ## further compatibility checks between the chosen parameters are ## delegated to the class constructor if (isempty (klist)) klist = 1 : columns (clust); endif cc = SilhouetteEvaluation (x, clust, klist, distance, clusterpriors); case "gap" ## gap cannot be used with a pre-computed solution, i.e. a matrix for ## 'clust', and klist must be specified if (isnumeric (clust)) error (["evalclusters: 'clust' must be a clustering algorithm when "... "using the gap criterion"]); endif if (isempty (klist)) error (["evalclusters: 'klist' cannot be empty when using the gap " ... "criterion"]); endif cc = GapEvaluation (x, clust, klist, b, distance, ... referencedistribution, searchmethod); otherwise error ("evalclusters: invalid criterion '%s'", criterion); endswitch endfunction ## input tests %!error evalclusters () %!error evalclusters ([1 1;0 1]) %!error evalclusters ([1 1;0 1], "kmeans") %!error <'x' must be a numeric*> evalclusters ("abc", "kmeans", "gap") %!error evalclusters ([1 1;0 1], "xxx", "gap") %!error evalclusters ([1 1;0 1], [1 2], "gap") %!error evalclusters ([1 1;0 1], 1.2, "gap") %!error evalclusters ([1 1;0 1], [1; 2], 123) %!error evalclusters ([1 1;0 1], [1; 2], "xxx") %!error <'KList' can be empty*> evalclusters ([1 1;0 1], "kmeans", "gap") %!error evalclusters ([1 1;0 1], [1; 2], "gap", 1) %!error evalclusters ([1 1;0 1], [1; 2], "gap", 1, 1) %!error evalclusters ([1 1;0 1], [1; 2], "gap", "xxx", 1) %!error <'KList'*> evalclusters ([1 1;0 1], [1; 2], "gap", "KList", [-1 0]) %!error <'KList'*> evalclusters ([1 1;0 1], [1; 2], "gap", "KList", [1 .5]) %!error <'KList'*> evalclusters ([1 1;0 1], [1; 2], "gap", "KList", [1 1; 1 1]) %!error evalclusters ([1 1;0 1], [1; 2], "gap", ... %! "distance", "a") %!error evalclusters ([1 1;0 1], [1; 2], "daviesbouldin", ... %! "distance", "a") %!error evalclusters ([1 1;0 1], [1; 2], "gap", ... %! "clusterpriors", "equal") %!error evalclusters ([1 1;0 1], [1; 2], ... %! "silhouette", "clusterpriors", "xxx") %!error <'clust' must be a clustering*> evalclusters ([1 1;0 1], [1; 2], "gap") %!test %! load fisheriris; %! eva = evalclusters(meas, "kmeans", "calinskiharabasz", "KList", [1:6]); %! assert(isa(eva, "CalinskiHarabaszEvaluation")); %! assert(eva.NumObservations, 150); %! assert(eva.OptimalK, 3); %! assert(eva.InspectedK, [1 2 3 4 5 6]); ## demonstration #%!demo #%! load fisheriris; #%! eva = evalclusters(meas, "kmeans", "calinskiharabasz", "KList", [1:6]) statistics-release-1.6.3/inst/ff2n.m000066400000000000000000000037051456127120000173220ustar00rootroot00000000000000## Copyright (C) 2022 Andreas Bertsatos ## based on public domain work by Paul Kienzle ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{dFF2} =} ff2n (@var{n}) ## ## Two-level full factorial design. ## ## @code{@var{dFF2} = ff2n (@var{n})} gives factor settings dFF2 for a two-level ## full factorial design with n factors. @var{dFF2} is m-by-n, where m is the ## number of treatments in the full-factorial design. Each row of @var{dFF2} ## corresponds to a single treatment. Each column contains the settings for a ## single factor, with values of 0 and 1 for the two levels. ## ## @seealso {fullfact} ## @end deftypefn function A = ff2n (n) if (nargin != 1) error ("ff2n: wrong number of input arguments."); endif if (floor (n) != n || numel (n) != 1 || n < 1 ... || ! isfinite (n) || ! isreal (n)) error ("ff2n: @var{N} must be a positive integer scalar."); endif A = fullfact (2 * ones (1, n)) - 1; endfunction %!error ff2n (); %!error ff2n (2, 5); %!error ff2n (2.5); %!error ff2n (0); %!error ff2n (-3); %!error ff2n (3+2i); %!error ff2n (Inf); %!error ff2n (NaN); %!test %! A = ff2n (3); %! assert (A, fullfact (3)); %!test %! A = ff2n (8); %! assert (A, fullfact (8)); statistics-release-1.6.3/inst/fillmissing.m000066400000000000000000003305671456127120000210200ustar00rootroot00000000000000## Copyright (C) 1995-2023 The Octave Project Developers ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{B} =} fillmissing (@var{A}, 'constant', @var{v}) ## @deftypefnx {statistics} {@var{B} =} fillmissing (@var{A}, @var{method}) ## @deftypefnx {statistics} {@var{B} =} fillmissing (@var{A}, @var{move_method}, @var{window_size}) ## @deftypefnx {statistics} {@var{B} =} fillmissing (@var{A}, @var{fill_function}, @var{window_size}) ## @deftypefnx {statistics} {@var{B} =} fillmissing (@dots{}, @var{dim}) ## @deftypefnx {statistics} {@var{B} =} fillmissing (@dots{}, @var{PropertyName}, @var{PropertyValue}) ## @deftypefnx {statistics} {[@var{B}, @var{idx}] =} fillmissing (@dots{}) ## ## Replace missing entries of array @var{A} either with values in @var{v} or ## as determined by other specified methods. 'missing' values are determined ## by the data type of @var{A} as identified by the function @ref{ismissing}, ## curently defined as: ## ## @itemize ## @item ## @qcode{NaN}: @code{single}, @code{double} ## ## @item ## @qcode{" "} (white space): @code{char} ## ## @item ## @qcode{@{""@}} (white space in cell): string cells. ## @end itemize ## ## @var{A} can be a numeric scalar or array, a character vector or array, or ## a cell array of character vectors (a.k.a. string cells). ## ## @var{v} can be a scalar or an array containing values for replacing the ## missing values in @var{A} with a compatible data type for isertion into ## @var{A}. The shape of @var{v} must be a scalar or an array with number ## of elements in @var{v} equal to the number of elements orthoganal to the ## operating dimension. E.g., if @code{size(@var{A})} = [3 5 4], operating ## along @code{dim} = 2 requires @var{v} to contain either 1 or 3x4=12 ## elements. ## ## If requested, the optional output @var{idx} will contain a logical array ## the same shape as @var{A} indicating with 1's which locations in @var{A} ## were filled. ## ## Alternate Input Arguments and Values: ## @itemize ## @item @var{method} - replace missing values with: ## @table @code ## ## @item next ## @itemx previous ## @itemx nearest ## next, previous, or nearest non-missing value (nearest defaults to next ## when equidistant as determined by @code{SamplePoints}.) ## ## @item linear ## linear interpolation of neigboring, non-missing values ## ## @item spline ## piecewise cubic spline interpolation of neigboring, non-missing values ## ## @item pchip ## 'shape preserving' piecewise cubic spline interposaliton of neighboring, ## non-missing values ## @end table ## ## @item @var{move_method} - moving window calculated replacement values: ## @table @code ## ## @item movmean ## @itemx movmedian ## moving average or median using a window determined by @var{window_size}. ## @var{window_size} must be either a positive scalar value or a two element ## positive vector of sizes @w{@code{[@var{nb}, @var{na}]}} measured in the ## same units as @code{SamplePoints}. For scalar values, the window is ## centered on the missing element and includes all data points within a ## distance of half of @var{window_size} on either side of the window center ## point. Note that for compatability, when using a scalar value, the backward ## window limit is inclusive and the forward limit is exclusive. If a ## two-element @var{window_size} vector is specified, the window includes all ## points within a distance of @var{nb} backward and @var{na} forward from the ## current element at the window center (both limits inclusive). ## @end table ## ## @item @var{fill_function} - custom method specified as a function handle. ## The supplied fill function must accept three inputs in the following order ## for each missing gap in the data: ## @table @var ## @item A_values - ## elements of @var{A} within the window on either side of the gap as ## determined by @var{window_size}. (Note these elements can include missing ## values from other nearby gaps.) ## @item A_locs - ## locations of the reference data, @var{A_values}, in terms of the default ## or specified @code{SamplePoints}. ## @item gap_locs - ## location of the gap data points that need to be filled in terms of the ## default or specified @code{SamplePoints}. ## @end table ## ## The supplied function must return a scalar or vector with the same number of ## elements in @var{gap_locs}. The required @var{window_size} parameter ## follows similar rules as for the moving average and median methods ## described above, with the two exceptions that (1) each gap is processed as a ## single element, rather than gap elements being processed individually, and ## (2) the window extended on either side of the gap has inclusive endpoints ## regardless of how @var{window_size} is specified. ## ## @item @var{dim} - specify a dimension for vector operation (default = ## first non-singeton dimension) ## ## @item @var{PropertyName}-@var{PropertyValue} pairs ## @table @code ## @item SamplePoints ## @var{PropertyValue} is a vector of sample point values representing the ## sorted and unique x-axis values of the data in @var{A}. If unspecified, ## the default is assumed to be the vector @var{[1 : size (A, dim)]}. The ## values in @code{SamplePoints} will affect methods and properties that rely ## on the effective distance between data points in @var{A}, such as ## interpolants and moving window functions where the @var{window_size} ## specified for moving window functions is measured relative to the ## @code{SamplePoints}. ## ## @item EndValues ## Apply a separate handling method for missing values at the front or back of ## the array. @var{PropertyValue} can be: ## @itemize ## @item A constant scalar or array with the same shape requirments as @var{v}. ## @item @code{none} - Do not fill end gap values. ## @item @code{extrap} - Use the same procedure as @var{method} to fill the ## end gap values. ## @item Any valid @var{method} listed above except for @code{movmean}, ## @code{movmedian}, and @code{fill_function}. Those methods can only be ## applied to end gap values with @code{extrap}. ## @end itemize ## ## @item MissingLocations ## @var{PropertyValue} must be a logical array the same size as @var{A} ## indicating locations of known missing data with a value of @code{true}. ## (cannot be combined with MaxGap) ## ## @item MaxGap ## @var{PropertyValue} is a numeric scalar indicating the maximum gap length ## to fill, and assumes the same distance scale as the sample points. Gap ## length is calculated by the difference in locations of the sample points ## on either side of the gap, and gaps larger than MaxGap are ignored by ## @var{fillmissing}. (cannot be combined with MissingLocations) ## @end table ## @end itemize ## ## Compatibility Notes: ## @itemize ## @item ## Numerical and logical inputs for @var{A} and @var{v} may be specified ## in any combination. The output will be the same class as @var{A}, with the ## @var{v} converted to that data type for filling. Only @code{single} and ## @code{double} have defined 'missing' values, so except for when the ## @code{missinglocations} option specifies the missing value identification of ## logical and other numeric data types, the output will always be ## @code{@var{B} = @var{A}} with @code{@var{idx} = false(size(@var{A}))}. ## @item ## All interpolation methods can be individually applied to @code{EndValues}. ## @item ## @sc{Matlab}'s @var{fill_function} method currently has several ## inconsistencies with the other methods (tested against version 2022a), and ## Octave's implementation has chosen the following consistent behavior over ## compatibility: (1) a column full of missing data is considered part of ## @code{EndValues}, (2) such columns are then excluded from ## @var{fill_function} processing because the moving window is always empty. ## (3) operation in dimensions higher than 2 perform identically to operations ## in dims 1 and 2, most notable on vectors. ## @item ## Method "makima" is not yet implemented in @code{interp1}, which is ## used by @code{fillmissing}. Attempting to call this method will produce ## an error until the method is implemented in @code{interp1}. ## @end itemize ## ## @seealso{ismissing, rmmissing, standardizeMissing} ## @end deftypefn function [A, idx_out] = fillmissing (A, varargin) if (nargin < 2)|| (nargin > 12) print_usage (); endif method = varargin{1}; if (ischar (method)) method = lower (method); elseif (! is_function_handle (method)) error ("fillmissing: second input must be a string or function handle."); endif sz_A = size (A); ndims_A = numel (sz_A); dim = []; missing_locs = []; endgap_method = []; endgap_locs = []; endgap_val = []; fill_vals = []; idx_flag = (nargout > 1); maxgap = []; missinglocations = false; reshape_flag = false; samplepoints = []; standard_samplepoints = true; v = []; if (idx_flag) idx_out = false (sz_A); endif ## process input arguments if (is_function_handle (method)) ## verify function handle and window if ((nargin < 3) || ! isnumeric (varargin{2}) || ... ! any( numel (varargin{2})==[1 2])) error (["fillmissing: fill function handle must be followed by ", ... "a numeric scalar or two-element vector window size."]); elseif (nargin (method) < 3) error ("fillmissing: fill function must accept at least three inputs."); endif move_fcn = method; method = "movfcn"; window_size = varargin{2}; next_varg = 3; else switch (method) case {"previous", "next", "nearest"} next_varg = 2; case {"linear", "spline", "pchip", "makima"} next_varg = 2; if (! (isnumeric (A) || islogical (A))) error (["fillmissing: interpolation methods only valid for ", ... "numeric input types."]); endif case "constant" if ((nargin < 3)) error (["fillmissing: 'constant' method must be followed by a ", ... "numeric scalar or array."]); endif v = varargin{2}; if ! (isscalar (v) || isempty (v)) v = v(:); endif if ((! ischar(v)) && isempty (v)) error ("fillmissing: a numeric fill value cannot be emtpy."); endif ## type check v against A if (iscellstr (A) && ischar (v) && ! iscellstr (v)) v = {v}; endif if ((! isempty (v)) && ... ((isnumeric (A) && ! (isnumeric (v) || islogical (v))) || ... (ischar (A) && ! ischar (v)) || ... (iscellstr (A) && ! (iscellstr (v))))) error ("fillmissing: fill value must be the same data type as 'A'."); endif ## v can't be size checked until after processing rest of inputs next_varg = 3; case {"movmean", "movmedian"} if (! (isnumeric (A) || islogical (A))) error (["fillmissing: 'movmean' and 'movmedian' methods only ", ... "valid for numeric input types."]); endif if ((nargin < 3) || ! isnumeric (varargin{2}) || ... ! any( numel (varargin{2})==[1 2])) error (["fillmissing: moving window method must be followed by ", ... "a numeric scalar or two-element vector."]); endif window_size = varargin{2}; next_varg = 3; otherwise error ("fillmissing: unknown fill method '%s'.", method); endswitch endif ## process any more parameters if (next_varg < nargin) #set dim. if specified, it is the only numeric option that can appear next if isnumeric (varargin{next_varg}) dim = varargin{next_varg}; if (! (isscalar (dim) && (dim > 0))) error ("fillmissing: DIM must be a positive scalar."); endif next_varg++; else ## default dim is first nonsingleton dimension of A if isscalar (A) dim = 1; else dim = find (sz_A > 1, 1, "first"); endif endif sz_A_dim = size (A, dim); ## process any remaining inputs, must be name-value pairs while (next_varg < nargin) propname = varargin{next_varg}; if (next_varg + 1 == nargin) ## must be at least one more input with 1st containing value error ("fillmissing: properties must be given as name-value pairs."); else propval = varargin{next_varg + 1}; next_varg = next_varg + 2; if (! ischar (propname)) error ("fillmissing: invalid parameter name specified."); else propname = lower (propname); endif ## input validation for names and values switch (propname) case "samplepoints" ## val must be sorted, unique, numeric vector the same size ## as size(A,dim) if (! (isnumeric (propval) && isvector (propval) && (numel (propval) == sz_A_dim) && issorted (propval) && (numel (propval) == numel (unique (propval))))) error (["fillmissing: SamplePoints must be a sorted ", ... "non-repeating, numeric vector with %d elements."], ... sz_A_dim); endif samplepoints = propval(:); standard_samplepoints = all (diff (samplepoints, 1, 1) == 1); case "endvalues" ## for numeric A, val must be numeric scalar, a numeric ## array with numel equal to the elements orthogonal to ## the dim or certain string methads. For non-numeric A, ## "constant" method is not valid. if ischar (propval) switch (lower (propval)) case {"extrap", "previous", "next", "nearest", "none", ... "linear", "spline", "pchip", "makima"} endgap_method = propval; otherwise error ("fillmissing: invalid EndValues method '%s'.", propval); endswitch elseif (isnumeric (propval)) if (! (isnumeric (A) || islogical (A))) error (["fillmissing: EndValues method 'constant' only ", ... "valid for numeric arrays."]); endif endgap_method = 'constant'; endgap_val = propval; else error (["fillmissing: EndValues must be numeric or a ", ... "valid method name."]); endif case "missinglocations" if !(isnumeric (A) || islogical (A) || isinteger (A) || ... ischar (A) || iscellstr (A)) error (["fillmissing: MissingLocations option is not ", ... "compatible with data type '%s'."], class (A)); endif if (! isempty (maxgap)) error (["fillmissing: MissingLocations and MaxGap options ", ... "cannot be used simultaneously."]); endif ## val must be logical array same size as A if (! (islogical (propval) && isequal (sz_A, size (propval)))) error (["fillmissing: MissingLocations must be a logical ", ... "array the same size as A."]); endif missinglocations = true; missing_locs = propval; case "maxgap" ## val must be positive numeric scalar if (! (isnumeric (propval) && isscalar (propval) && (propval > 0))) error ("fillmissing: MaxGap must be a positive numeric scalar."); endif if (! isempty (missing_locs)) error (["fillmissing: MissingLocations and MaxGap options ", ... "cannot be used simultaneously."]); endif maxgap = propval; case {"replacevalues", "datavariables"} error ("fillmissing: the '%s' option has not been implemented.", ... propname); otherwise error ("invalid parameter name '%s'.", propname); endswitch endif endwhile else ## no inputs after method ## set default dim if isscalar (A) dim = 1; else dim = find (sz_A > 1, 1, "first"); endif sz_A_dim = size (A, dim); endif ## reduce calls to size and avoid overruns checking sz_A for high dims if dim > ndims_A sz_A = [sz_A, ones(1, dim - ndims_A)]; ndims_A = numel (sz_A); endif ## set defaults for any unspecified parameters if (isempty (samplepoints)) samplepoints = [1 : sz_A_dim]'; endif if isempty (missing_locs) missing_locs = ismissing (A); endif ## endvalues treated separately from interior missing_locs if (isempty (endgap_method) || strcmp (endgap_method, "extrap")) endgap_method = method; if strcmp(endgap_method, "constant") endgap_val = v; endif endif ## missingvalues option not compatible with some methods and inputs: if (isinteger (A) || islogical (A)) if (any (ismember (method, ... {"linear", "spline", "pchip", "makima", "movmean", "movmedian"}))) error (["fillmissing: MissingLocations cannot be used ", ... "with method '%s' and inputs of type '%s'."], method, class (A)); elseif (any (ismember (endgap_method, ... {"linear", "spline", "pchip", "makima"}))) error (["fillmissing: MissingLocations cannot be used with EndValues", ... " method '%s' and inputs of type '%s'."], method, class (A)); endif endif ## verify size of v and endgap_val for 'constant' methods, resize for A orthogonal_size = [sz_A(1:dim-1), 1, sz_A(dim+1:end)]; # orthog. to dim size numel_orthogonal = prod (orthogonal_size); # numel perpen. to dim if (strcmp (method, "constant") && (! isscalar (v))) if (numel (v) != numel_orthogonal) error (["fillmissing: fill value 'V' must be a scalar or a %d ", ... " element array."], numel_orthogonal); else v = reshape (v, orthogonal_size); endif endif if (strcmp (endgap_method, "constant") && (! isscalar (endgap_val))) if (numel (endgap_val) != numel_orthogonal) error (["fillmissing: EndValues must be a scalar or a %d element ", ... "array."], numel_orthogonal); else endgap_val = reshape (endgap_val, orthogonal_size); endif endif ## simplify processing by temporarily permuting A so operation always on dim1 ## revert permutation at the end dim_idx_perm = [1 : ndims_A]; dim_idx_flip(1 : max(dim, ndims_A)) = {':'}; dim_idx_flip(1) = [sz_A_dim:-1:1]; if (dim != 1) dim_idx_perm([1, dim]) = [dim, 1]; A = permute (A, dim_idx_perm); sz_A([1, dim]) = sz_A([dim, 1]); missing_locs = permute (missing_locs, dim_idx_perm); reshape_flag = true; orthogonal_size = [1, sz_A(2:end)]; if (idx_flag) idx_out = false (sz_A); endif if (! isempty (v) && ! isscalar (v)) v = permute (v, dim_idx_perm); endif if (! isempty (endgap_val) && ! isscalar (endgap_val)) endgap_val = permute (endgap_val, dim_idx_perm); endif endif ## precalculate fill data for several methods zero_padding = zeros (orthogonal_size); samplepoints_expand = samplepoints(:, ones (1, prod (sz_A(2:end)))); ##find endgap locations if (sz_A_dim < 3) ## all missing are endgaps endgap_locs = missing_locs; else ## use cumsums starting from first and last part in dim to find missing ## values in and adjacent to end locations. endgap_locs = cumprod (missing_locs,1) | ... (cumprod (missing_locs(dim_idx_flip{:}),1))(dim_idx_flip{:}); endif ## remove endgap_locs from missing_locs to avoid double processing missing_locs(endgap_locs) = false; ## remove elements from missing and end location arrays if maxgap is specified if (! isempty (maxgap)) ## missing_locs: if samplepoints value diff on either side of missing ## elements is > maxgap, remove those values. ## for endgaps, use diff of inside and missing end samplepoint values ## and remove from endgaps ## First check gapsize of any interior missings in missing_locs if (any (missing_locs(:))) ## locations in front of gaps loc_before = [diff(missing_locs,1,1); zero_padding] == 1; ## locations in back of gaps loc_after = diff ([zero_padding; missing_locs],1,1) == -1; ## value of samplepoints at front and back locs sampvals_before = samplepoints_expand(loc_before); sampvals_after = samplepoints_expand(loc_after); ## evaluate which gaps are too big to fill gaps_to_remove = (sampvals_after - sampvals_before) > maxgap; ## convert those gaps into an array element list idxs_to_remove = arrayfun ('colon', ... ((find (loc_before))(gaps_to_remove ) + 1), ... ((find (loc_after))(gaps_to_remove ) - 1), ... "UniformOutput", false); ## remove those elements from missing_locs missing_locs([idxs_to_remove{:}]) = false; endif ##then do any endgaps if (any (endgap_locs(:))) ## if any are all missing, remove for any value of maxgap endgap_locs &= ! prod (endgap_locs, 1); if ((sz_A_dim < 3) && (abs (samplepoints(2) - samplepoints(1)) > maxgap)) ## shortcut - all missings are ends and exceed maxgap. endgap_locs(:) = false; else ## check gap size of front endgaps ##find loc element after gap nextvals = sum (cumprod (endgap_locs,1)) + 1; ## compare diff between values at those points and at base with maxgap ends_to_remove = abs (samplepoints(nextvals) - samplepoints(1)) ... > maxgap; ## remove any with gap>maxgap endgap_locs((cumprod (endgap_locs,1)) & ends_to_remove) = false; ## flip, repeat for back endgaps, then unflip and remove. nextvals = sum (cumprod (endgap_locs(dim_idx_flip{:}),1)) + 1; ends_to_remove = abs (samplepoints(end:-1:1)(nextvals) ... - samplepoints(end)) > maxgap; endgap_locs((cumprod (... endgap_locs(dim_idx_flip{:}), 1)(dim_idx_flip{:})) & ... ends_to_remove) = false; endif endif endif if (any (strcmp (endgap_method, {"movmean", "movmedian", "movfcn"}))) ## These methods only called for endgaps with "extrap", so endgaps ## are processed together in the missing_locs section. missing_locs = missing_locs | endgap_locs; endgap_locs(:) = false; endif ## Actaully fill the missing data ## process central missing values (all gaps bound by two valid datapoints) ## for each method, calcualte fill_vals, which will be used in assignment ## A(missing_locs) = fill_vals, and if idx_flag, populate idx_out if (any (missing_locs(:))) switch (method) case "constant" if (isscalar (v)) fill_vals = v; else fill_vals = (missing_locs .* v)(missing_locs); endif if (idx_flag) ## if any v are the missing type, those get removed from idx_out ## unless using 'missinglocations' if (! missinglocations) && any (miss_v = ismissing (v)) idx_out(missing_locs) = true; idx_out(missing_locs & miss_v) = false; else idx_out(missing_locs) = true; endif endif case {"previous", "next", "nearest", "linear"} ## find element locations bounding each gap loc_before = [diff(missing_locs, 1, 1); zero_padding] == 1; loc_after = diff ([zero_padding; missing_locs], 1, 1) == -1; gapsizes = find (loc_after) - find (loc_before) - 1; gap_count_idx = [1 : numel(gapsizes); gapsizes']; switch (method) case "previous" fill_vals = repelems (A(loc_before), gap_count_idx)'; case "next" fill_vals = repelems (A(loc_after), gap_count_idx)'; case {"nearest", "linear"} ## determine which missings go with values before or after ## gap based on samplevalue distance. (equal dist goes to after) ## find sample values before and after gaps sampvals_before = samplepoints_expand(loc_before); sampvals_after = samplepoints_expand(loc_after); ## build cell with linear indices of elements in each gap gap_locations = arrayfun ('colon', (find (loc_before)) + 1, ... (find (loc_after)) - 1, "UniformOutput", false); ## get sample values at those elements [sampvals_in_gaps, ~] = ind2sub (sz_A, [gap_locations{:}]); sampvals_in_gaps = samplepoints(sampvals_in_gaps); ## expand first and last vectors for each gap point Avals_before = repelems (A(loc_before), gap_count_idx)'; Avals_after = repelems (A(loc_after), gap_count_idx)'; switch (method) case "nearest" ## calculate gap mid point for each gap element sampvals_midgap = repelems ( ... (sampvals_before + sampvals_after)/2, gap_count_idx)'; ## generate fill vectors sorting elements into nearest before ## or after prev_fill = (sampvals_in_gaps < sampvals_midgap); next_fill = (sampvals_in_gaps >= sampvals_midgap); fill_vals = A(missing_locs); fill_vals(prev_fill) = Avals_before(prev_fill); fill_vals(next_fill) = Avals_after(next_fill); case "linear" ## expand samplepoint values for interpolation x-values sampvals_before = repelems (sampvals_before, gap_count_idx)'; sampvals_after = repelems (sampvals_after, gap_count_idx)'; ## linearly interpolate fill_vals = ((Avals_after - Avals_before) ... ./ (sampvals_after - sampvals_before)) ... .* (sampvals_in_gaps - sampvals_before) ... + Avals_before; endswitch endswitch if (idx_flag) ## mid gaps will always be filled by above methods. idx_out(missing_locs) = true; endif case {"spline", "pchip", "makima"} ## pass more complex interpolations to interp1 ## TODO: vectorized 'linear' is ~10-100x faster than using interp1. ## look to speed these up as well. ## identify columns needing interpolation to reduce empty operations cols_to_use = any (missing_locs, 1); ## missinglocations may send columns with NaN and less than 2 ## real values resulting in interp1 error. Trim those columns, ## prepopulate fill_vals with NaN, mark as filled. if (missinglocations) fill_vals = NaN (sum (missing_locs(:, cols_to_use)(:)), 1); cols_enough_points = (sum ( ... !isnan(A) & (! missing_locs), 1) > 1) & cols_to_use; interp_cols = (cols_enough_points & cols_to_use); interp_vals = (missing_locs & cols_enough_points)(missing_locs & ... cols_to_use); fill_vals(interp_vals) = other_interpolants (A(:, interp_cols), missing_locs(:, interp_cols), endgap_locs(:, interp_cols), ... method, samplepoints); else fill_vals = other_interpolants (A(:, cols_to_use), missing_locs(:, cols_to_use), endgap_locs(:, cols_to_use), ... method, samplepoints); endif if (idx_flag) idx_out(missing_locs) = true; endif case {"movmean","movmedian"} ## check window size versus smallest sample gaps. if window smaller, ## nothing to do, break out early. if ((isscalar (window_size) && ... (window_size/2 >= min (diff (samplepoints)))) || ... (isvector (window_size) && (sum (window_size) >= min (diff (samplepoints))))) switch (method) case "movmean" if sz_A_dim > 1 allmissing = (missing_locs | endgap_locs)(:,:); ## create temporary flattened array for processing, A_sum = A(:,:); A_sum (allmissing) = 0; if (standard_samplepoints && ... all (round (window_size) == window_size)) ## window size based on vector elements ## faster codepath for uniform, unit-spacing samplepoints ## and integer valued window sizes. if (isscalar (window_size)) window_width = window_size; if (mod (window_size, 2)) ## odd window size ## equal number of values on either side of gap window_size = (window_width - 1) .* [0.5, 0.5]; else ## even window size ## one extra element on previous side of gap window_size(1) = window_width/2; window_size(2) = window_size(1) - 1; endif else window_width = window_size(1) + window_size(2) + 1; endif ## use columnwise convolution of windowing vector and A for ## vectorized summation. conv_vector = ones (window_width, 1); A_sum = convn (A_sum, conv_vector, ... "full")(1 + window_size(2):end - window_size(1), :); ## get count of values contributing to convolution to account ## for missing elements and to calculate mean. A_sum_count = convn (! allmissing, conv_vector, ... "full")(1 + window_size(2):end - window_size(1), :); else ## window size based on sample point distance. Works for non ## integer, non uniform values. ## use A_sum (flattened to 2D), project slice windows in dim3 ## automatic broadcasting to get window summations & counts samplepoints_shift = ... samplepoints(:, ones(1, sz_A_dim)) - samplepoints'; if (isscalar (window_size)) ## [nb, na) window_size = window_size * [-0.5, 0.5]; samplepoints_slice_windows = permute (... samplepoints_shift >= window_size(1) & ... samplepoints_shift < window_size(2), [1,3,2]); else ## [nb, na] window_size(1) = -window_size(1); samplepoints_slice_windows = permute (... samplepoints_shift >= window_size(1) & ... samplepoints_shift <= window_size(2), [1,3,2]); endif if (missinglocations) ## NaNs left in A_sum will cause all sums to produce NaN ## FIXME: when sum can handle nanflag, the 'else' path ## should be able to be made to handle the vectorized ## summation even with 'missinglocations' A_nan = isnan (A_sum); A_temp = A_sum .* samplepoints_slice_windows; A_temp(!samplepoints_slice_windows & A_nan) = 0; A_sum = permute (sum (A_temp, 1), [3,2,1]); else A_sum = permute (... sum (A_sum .* samplepoints_slice_windows, 1), [3,2,1]); endif A_sum_count = permute (... sum (! allmissing & samplepoints_slice_windows, 1), ... [3,2,1]); endif ## build fill values fill_vals = A(missing_locs); # prefill to include missing vals fillable_gaps = missing_locs(:,:) & A_sum_count; fill_vals(fillable_gaps(missing_locs(:,:))) = ... A_sum(fillable_gaps) ./ A_sum_count(fillable_gaps); endif case "movmedian" if sz_A_dim > 1 cols_to_use = any (missing_locs(:,:), 1); samplepoints_shift = ... samplepoints(:, ones(1, sz_A_dim)) - samplepoints'; if (isscalar (window_size)) window_size = window_size * [-0.5, 0.5]; ## [nb, na) samplepoints_slice_windows = permute (... samplepoints_shift >= window_size(1) & ... samplepoints_shift < window_size(2), [1,3,2]); else window_size(1) = -window_size(1); ## [nb, na] samplepoints_slice_windows = permute (... samplepoints_shift >= window_size(1) & ... samplepoints_shift <= window_size(2), [1,3,2]); endif ## use moving window slices to project A and use ## custom function for vectorized full array median computation A_med = A(:, cols_to_use); nan_slice_windows = double (samplepoints_slice_windows); nan_slice_windows(! samplepoints_slice_windows) = NaN; A_med_slices = A_med .* nan_slice_windows; A_med = permute (columnwise_median (A_med_slices), [3 2 1]); fillable_gaps = missing_locs(:, cols_to_use); fill_vals = A_med(fillable_gaps); endif endswitch if (idx_flag) ## Matlab compatiblity - NaNs filled back in by movmean and ## movmedian should _not_ show as filled. idx_out(fillable_gaps) = true; still_nan = missing_locs; still_nan(missing_locs) = isnan (fill_vals); idx_out(still_nan) = false; endif endif case "movfcn" ## for each gap construct: ## xval - data values in window on either side of gap, including ## other missing values ## xloc - sample point values for those xval ## gap_loc - sample point values for gap elements ## if window has xval fully empty skip processing gap ## missing_locs might include endgap_locs ## need to build gap locations accounting for both types ## missing values can include more than just numeric inputs ## windows containing no data points (e.g., endgaps when window ## is one sided [3 0] or [0 2], will be dropped from processing, ## not being passed to the mov_fcn. if (isscalar (window_size)) window_size = window_size * [-0.5, 0.5]; else window_size(1) = -window_size(1); endif ## midgap bounds loc_before = [diff(missing_locs, 1, 1); zero_padding] == 1; loc_after = diff ([zero_padding; missing_locs], 1, 1) == -1; ## front/back endgap locations and bounds front_gap_locs = logical (cumprod (missing_locs, 1)); front_next_locs = diff ([zero_padding; front_gap_locs], 1, 1) == -1; back_gap_locs = logical ( ... cumprod (missing_locs(dim_idx_flip{:}), 1)(dim_idx_flip{:})); back_prev_locs = [diff(back_gap_locs, 1, 1); zero_padding] == 1; ## remove gap double counting back_gap_locs &= ! front_gap_locs; loc_before &= ! back_prev_locs; loc_after &= ! front_next_locs; ## build gap location array using gap starts and lengths. ## simplest to use front / mid / back ordering, track later with sort. gap_start_locs = ... [find(front_gap_locs & [true; false(sz_A_dim-1,1)])(:); ... find(circshift (loc_before, 1, 1))(:); find(circshift (back_prev_locs, 1, 1))(:)]; gapsizes = [(sum (front_gap_locs, 1))(any (front_gap_locs, 1))(:);... find(loc_after) - find(loc_before) - 1;... (sum (back_gap_locs, 1))(any (back_gap_locs, 1))(:)]; ## separate arrayfun/cellfun faster than single fun with ## composite anonymous function gap_locations = arrayfun ('colon', gap_start_locs, ... gap_start_locs + gapsizes - 1, "UniformOutput", false); gap_locations = cellfun('transpose', ... gap_locations, "UniformOutput", false); ## sorting index to bridge front-mid-back and linear index ordering [~, gap_full_sort_idx] = sort (vertcat (gap_locations{:})); ## remove front or back gaps from gapsizes & gap_locations ## if front/back window size = 0, or if full column is missing ## index to track empty/removed elements removed_element_idx = true (numel (gap_full_sort_idx), 1); removed_front_elements = 0; removed_back_elements = 0; ## simple front/back gap trimming for either window size = 0 if (! window_size(2)) ## if no back facing window, ignore front gaps removed_front_gap_count = sum (front_gap_locs(1,:)); removed_front_elements = sum (gapsizes(1 : removed_front_gap_count)); removed_element_idx(1 : removed_front_elements) = false; gap_locations(1 : removed_front_gap_count ) = []; gapsizes(1 : removed_front_gap_count ) = []; elseif (any (missing_col_gaps = (gapsizes == sz_A_dim))) missing_col_elements = ... repelems (missing_col_gaps, [1:numel(gapsizes); gapsizes'])'; removed_element_idx(missing_col_elements) = false; gap_locations(missing_col_gaps) = []; gapsizes(missing_col_gaps) = []; endif if (! window_size(1)) ## if no front facing window, ignore back gaps. removed_back_gap_count = sum (back_gap_locs(end,:)); removed_back_elements = sum (... gapsizes(end - removed_back_gap_count + 1 : end)); removed_element_idx(end - removed_back_elements + 1 : end) = false; gap_locations(end - removed_back_gap_count + 1 : end) = []; gapsizes(end - removed_back_gap_count + 1 : end) = []; endif if (! isempty (gapsizes)) gap_sample_values = cellfun_subsref (gap_locations, false, ... {samplepoints_expand}); ## build [row,column] locations array for windows around each gap window_points_r_c = cell (numel (gapsizes), 2); window_points_r_c(:,1) = cellfun (@(x) ... ([1:sz_A_dim]')((samplepoints= ... max (x(1) + window_size(1), samplepoints(1))) | ... (samplepoints>x(end) & samplepoints <= ... min (x(end) + window_size(2), samplepoints(end)))), ... gap_sample_values, "UniformOutput", false); window_points_r_c(:,2) = cellfun ( ... @(x,y) (fix ((x(1)-1)/sz_A_dim)+1)(ones (size(y))), ... gap_locations, window_points_r_c(:,1),"UniformOutput",false); ## if any window is emtpy, do not pass that gap to the move_fcn empty_gaps = cellfun ('isempty', window_points_r_c(:,1)); if (any (empty_gaps)) removed_element_idx(... repelems (empty_gaps, [1:numel(gapsizes); gapsizes'])')... = false; window_points_r_c(empty_gaps,:) = []; gap_sample_values(empty_gaps) = []; gapsizes(empty_gaps) = []; gap_locations(empty_gaps) = []; endif if (! isempty (gapsizes)) ## Aval = A values at window locations ## Aloc = sample values at window locations A_window_indexes = cellfun ('sub2ind', {sz_A}, ... window_points_r_c(:,1), window_points_r_c(:,2), ... "UniformOutput", false); Aval = cellfun_subsref (A_window_indexes, false, {A}); Aloc = cellfun_subsref (window_points_r_c(:,1), false, ... {samplepoints}); ##build fill values fill_vals_C = cellfun (move_fcn, Aval, Aloc, ... gap_sample_values(:,1), "UniformOutput", false); ## check for output of move_fcn having different size than gaps if (! all (cellfun ('numel', fill_vals_C) == gapsizes)) error (["fillmissing: fill function return values must be ", ... "the same size as the gaps."]); endif [~, gap_trim_sort_idx] = sort (vertcat (gap_locations{:})); fill_vals_trim = cell2mat (fill_vals_C); if (! isempty (fill_vals_trim)) fill_vals = A(missing_locs); # prefill to include missing vals fill_vals(removed_element_idx(gap_full_sort_idx)) = ... fill_vals_trim(gap_trim_sort_idx); if (idx_flag) ## for movfcn with A of type having missing: ## any outputs still containing class's 'missing' values ## are counted as not filled in idx_out, even if the value ## was put there by the movfcn. This is true even if ## missinglocations is used. If missinglocations changed ## a value with no apparent change, it still shows up ## as filled. ## if A has no missing value (int or logical), then without ## missinglocations, idx_out is always empty. With ## missinglocations, compatible behavior is undefined as ## Matlab 2022a has an apparent bug producing a error message ## saying missinglocations with int/logical needs a method that ## incldues function handle. Expect behavior should match other ## methods, where any processed missing value should be marked ## as filled no matter the fill value. if ((isnumeric(A) && !isinteger(A)) || ischar (A) || iscellstr (A)) idx_out(missing_locs) = ! ismissing (fill_vals); elseif (missinglocations) ## any missing_loc processed and not skipped must become true idx_out(missing_locs) = removed_element_idx(gap_full_sort_idx); endif endif endif endif endif endswitch if (! isempty (fill_vals)) A(missing_locs) = fill_vals; fill_vals = []; endif endif ## process endgaps if (any (endgap_locs(:))) switch (endgap_method) case "none" endgap_locs(:) = false; case "constant" if (isscalar (endgap_val)) fill_vals = endgap_val; else fill_vals = (endgap_locs .* endgap_val)(endgap_locs); endif if (idx_flag) ## if any v are the missing type, those get removed from idx_out ## unless using 'missinglocations' idx_out(endgap_locs) = true; if (! missinglocations) && any (miss_ev = ismissing (endgap_val)) idx_out(endgap_locs & miss_ev) = false; endif endif case {"previous", "next", "nearest", "linear", ... "spline", "pchip", "makima"} ## all of these methods require sz_A_dim >= 2. shortcut path otherwise if (sz_A_dim < 2) endgap_locs(:) = false; else switch (endgap_method) case "previous" ## remove any gaps at front of array, includes all-missing cols endgap_locs (logical (cumprod (endgap_locs,1))) = false; if (any (endgap_locs(:))) ##find locations of the 'prev' value to use for filling subsval_loc = [diff(endgap_locs, 1, 1); zero_padding] == 1; ##calculate the number of spots each 'prev' needs to fill gapsizes = (sum (endgap_locs, 1))(any (endgap_locs, 1))(:); ## construct substitution value vector fill_vals = repelems (A(subsval_loc), ... [1:numel(gapsizes); gapsizes'])'; endif case "next" ## remove any gaps at back of array from endgap_locs ## includes any all-missing columns endgap_locs(logical (cumprod ( ... endgap_locs(dim_idx_flip{:}), 1)(dim_idx_flip{:}))) ... = false; if (any (endgap_locs(:))) ##find locations of the 'next' value to use for filling subsval_loc = diff ([zero_padding; endgap_locs],1,1) == -1; ##calculate the number of spots each 'next' needs to fill gapsizes = (sum (endgap_locs, 1))(any (endgap_locs, 1))(:); ## construct substitution value vector fill_vals = repelems (A(subsval_loc), ... [1:numel(gapsizes); gapsizes'])'; endif case "nearest" ## remove any all-missing columns endgap_locs &= (! prod (endgap_locs, 1)); if (any (endgap_locs(:))) ## find front end info front_gap_locs = logical (cumprod (endgap_locs, 1)); front_next_loc = diff ( ... [zero_padding; front_gap_locs], 1, 1) == -1; front_gapsizes = (sum (front_gap_locs, 1))(any ... (front_gap_locs,1)); ## find back end info back_gap_locs = logical ( ... cumprod (endgap_locs(dim_idx_flip{:}), 1)(dim_idx_flip{:})); back_prev_loc = [diff(back_gap_locs, 1, 1); zero_padding] == 1; back_gapsizes = (sum (back_gap_locs, 1))(any (back_gap_locs,1)); ## combine into fill variables [~, fb_sort_idx] = sort ... ([find(front_gap_locs); find(back_gap_locs)]); fillval_loc = [find(front_next_loc); find(back_prev_loc)]; gapsizes = [front_gapsizes; back_gapsizes]; ## construct substitution value vector with sort order to mix ## fronts and backs in column order fill_vals = (repelems (A(fillval_loc), ... [1:numel(gapsizes); gapsizes'])')(fb_sort_idx); endif case "linear" ## endgaps not guaranteed to have enough points to interpolate cols_to_use = (sum (! (missing_locs | endgap_locs), 1) > 1) ... & any (endgap_locs, 1); interp_locs = ! (missing_locs | endgap_locs) & cols_to_use; endgap_locs &= cols_to_use; if (any (endgap_locs(:))) ## process front endgaps front_gap_locs = logical (cumprod (endgap_locs, 1)); fill_vals_front = []; if (any (front_gap_locs(:))) front_gapsizes = (sum (front_gap_locs, 1))(any ... (front_gap_locs,1)); ## collect first data point after gap & expand to gapsize front_interppoint_1 = repelems ( find (... diff ([zero_padding; front_gap_locs], 1, 1) == -1), ... [1:numel(front_gapsizes); front_gapsizes'])'; ## collect second data point after gap & expand to gapsize front_interppoint_2 = repelems ( find ( ... diff ([zero_padding; ((cumsum (interp_locs, 1) .* ... any (front_gap_locs, 1)) == 2)], 1, 1) == 1), ... [1:numel(front_gapsizes); front_gapsizes'])'; front_interp_Avals = A([front_interppoint_1, ... front_interppoint_2]); front_interp_sampvals = samplepoints_expand( ... [front_interppoint_1, front_interppoint_2]); front_gap_loc_sampvals = samplepoints_expand(front_gap_locs); ## hack for vector automatic orientation forcing col vector if (isvector (front_interp_Avals)) front_interp_Avals = (front_interp_Avals(:)).'; endif if (isvector (front_interp_sampvals)) front_interp_sampvals = (front_interp_sampvals(:)).'; endif ## perform interpolation for every gap point interp_slopes_front = diff (front_interp_Avals, 1, 2) ... ./ diff (front_interp_sampvals, 1, 2); fill_vals_front = interp_slopes_front .* ... (front_gap_loc_sampvals - ... front_interp_sampvals(:,1)) + ... front_interp_Avals(:,1); endif ## process back endgaps back_gap_locs = logical ( ... cumprod (endgap_locs(dim_idx_flip{:}), 1)(dim_idx_flip{:})); fill_vals_back = []; if (any (back_gap_locs(:))) back_gapsizes = (sum ( ... back_gap_locs, 1))(any (back_gap_locs,1)); ## collect last data point before gap & expand to gapsize back_interppoint_2 = repelems ( ... find ([diff(back_gap_locs, 1, 1); zero_padding] == 1), ... [1:numel(back_gapsizes); back_gapsizes'])'; ## collect 2nd to last data point before gap & expand to gap back_interppoint_1 = repelems ( ... find ((diff ([zero_padding; ... ((cumsum (interp_locs(dim_idx_flip{:}), 1) .* ... any (back_gap_locs, 1)) == 2)], ... 1, 1) == 1)(dim_idx_flip{:})), ... [1:numel(back_gapsizes); back_gapsizes'])'; ## build linear interpolant vectors back_interp_Avals = A([back_interppoint_1, ... back_interppoint_2]); back_interp_sampvals = samplepoints_expand( ... [back_interppoint_1, back_interppoint_2]); back_gap_loc_sampvals = samplepoints_expand(back_gap_locs); ## hack for vector automatic orientation forcing col vector if (isvector (back_interp_Avals)) back_interp_Avals = (back_interp_Avals(:)).'; endif if (isvector (back_interp_sampvals)) back_interp_sampvals = (back_interp_sampvals(:)).'; endif ## perform interpolation for every gap point interp_slopes_back = diff (back_interp_Avals, 1, 2) ... ./ diff (back_interp_sampvals, 1, 2); fill_vals_back = interp_slopes_back .* ... (back_gap_loc_sampvals - ... back_interp_sampvals(:,1)) + ... back_interp_Avals(:,1); endif [~, fb_sort_idx] = sort ... ([find(front_gap_locs); find(back_gap_locs)]); fill_vals = [fill_vals_front; fill_vals_back](fb_sort_idx); endif case {"spline", "pchip", "makima"} ## endgap_locs not guaranteed to have 2 points. ## need to ignore columns with < 2 values, or with nothing to do cols_to_use = (sum (! (endgap_locs | missing_locs), 1) > 1) ... & any (endgap_locs, 1); ## trim out unused cols from endgap_locs endgap_locs &= cols_to_use; if (missinglocations) ## missinglocations may send columns with NaN and less than 2 ## real values resulting in interp1 error. Trim those columns, ## prepopulate fill_vals with NaN, mark as filled. fill_vals = NaN (sum (endgap_locs(:, cols_to_use)(:)), 1); cols_enough_points = (sum ( ... !isnan(A) & (! endgap_locs), 1) > 1) & cols_to_use; interp_cols = (cols_enough_points & cols_to_use); interp_vals = (endgap_locs & ... cols_enough_points)(endgap_locs & cols_to_use); fill_vals(interp_vals) = other_interpolants ( ... A(:, interp_cols),endgap_locs(:, interp_cols), ... missing_locs(:, interp_cols), endgap_method, ... samplepoints); else fill_vals = other_interpolants ( A(:, cols_to_use), endgap_locs(:, cols_to_use), ... missing_locs(:, cols_to_use), endgap_method, samplepoints); endif endswitch endif if (idx_flag) idx_out(endgap_locs) = true; endif endswitch ## some methods remove fill locations, only process if not empty if (any (endgap_locs(:))) ## replace missings with appropriate fill values A(endgap_locs) = fill_vals; endif endif if (reshape_flag) ## revert permutation A = permute (A, dim_idx_perm); if (idx_flag) idx_out = permute (idx_out, dim_idx_perm); endif endif endfunction function varargout = cellfun_subsref (x, TF, varargin) ## utility fcn for cellfun (@(x) A(x), x, "UniformOutput", true/false) ## ~50% faster than anonymous function call ## pass A, x, and truefalse. ## if nargaut > 1, repeat for C2 with B(x), C3 with C(x), etc. x_C = num2cell (struct ("type", "()", "subs", num2cell (x))); for (idx = 1 : numel (varargin)) varargout{idx} = cellfun ('subsref', varargin{idx}, x_C, ... "UniformOutput", TF); endfor endfunction function fill_vals = other_interpolants (data_array, primary_locs, secondary_locs, method, samplepoints) ## use interp1 to perform more complex interpolations. will only be performed ## on numerical data. ## primary_locs is missing_locs or endgap_locs, whichever the fill_vals are ## being returned for. secondary_locs is the other. ## ## TODO: splitting out from columnwise cellfun to interp1 would increase ## speed a lot, but cannot count on same number of elements being processed ## in each column. ## ## Will error on any columns without at least two non-NaN interpolation ## values. ## ## Matlab incompatibility - using interp1, if 'missinglocations' sends through ## data_array values with NaN, interp1 will ignore those and interpolate with ## other data points. Matlab will instead return NaN. ## find logical data and empty location indices for each column to be ## used in columnwise call to interp1 (cast as columnwise cell arrays) interp_data_locs = num2cell ((! (primary_locs | secondary_locs)), 1); interp_locs_tofill = num2cell (primary_locs, 1); ## build cell arrays with sample and interp values to pass to interp1 [A_interpvalues, interp_samplevals] = cellfun_subsref (interp_data_locs, ... false, num2cell (data_array, 1), {samplepoints}); interp_empty_samplevals = cellfun_subsref (interp_locs_tofill, false, ... {samplepoints}); ## generate fill_vals using interp1 for missing locations. fill_vals = vertcat (cellfun ('interp1', interp_samplevals, ... A_interpvalues, interp_empty_samplevals, {method}, ... {'extrap'}, 'UniformOutput', false){:}); endfunction function med = columnwise_median (x) ## takes a column of values, ignores any NaN values, retuns the median ## of what's left. returns NaN if no values. ## uses only built-in fns to avoid 3x 'median' slowdown szx = size (x); if (isempty (x)) med = NaN ([1, szx(2:end)]); elseif (isvector (x)) x = x(! isnan (x)); x = sort (x); n = numel (x); if (mod (n, 2)) ## odd med = x((n+1)/2); elseif (n > 0) ## even med = (x(n/2) + x((n/2)+1))/2; else ## only called for types with missing = NaN med = NaN; endif else x = sort (x, 1); # NaNs sent to bottom n = sum (! isnan (x), 1); m_idx_odd = logical (mod (n, 2)); # 0 even or zero, 1 odd m_idx_even = !m_idx_odd & (n != 0); k = floor ((n + 1) ./ 2); med = NaN ([1, szx(2:end)]); if (! ismatrix (x)) szx = [szx(1), prod(szx(2:end))]; endif if any (m_idx_odd(:)) x_idx_odd = sub2ind (szx, k(m_idx_odd)(:), find(m_idx_odd)(:));; med(m_idx_odd) = x(x_idx_odd); endif if any (m_idx_even(:)) k_even = k(m_idx_even)(:); x_idx_even = sub2ind (szx, [k_even, k_even+1], ... (find (m_idx_even))(:,[1 1]));; med(m_idx_even) = sum (x(x_idx_even), 2) / 2; endif endif endfunction %!assert (fillmissing ([1 2 3], "constant", 99), [1 2 3]) %!assert (fillmissing ([1 2 NaN], "constant", 99), [1 2 99]) %!assert (fillmissing ([NaN 2 NaN], "constant", 99), [99 2 99]) %!assert (fillmissing ([1 2 3]', "constant", 99), [1 2 3]') %!assert (fillmissing ([1 2 NaN]', "constant", 99), [1 2 99]') %!assert (fillmissing ([1 2 3; 4 5 6], "constant", 99), [1 2 3; 4 5 6]) %!assert (fillmissing ([1 2 NaN; 4 NaN 6], "constant", 99), [1 2 99; 4 99 6]) %!assert (fillmissing ([NaN 2 NaN; 4 NaN 6], "constant", [97, 98, 99]), [97 2 99; 4 98 6]) %!test %! x = cat (3, [1, 2, NaN; 4, NaN, 6], [NaN, 2, 3; 4, 5, NaN]); %! y = cat (3, [1, 2, 99; 4, 99, 6], [99, 2, 3; 4, 5, 99]); %! assert (fillmissing (x, "constant", 99), y); %! y = cat (3, [1, 2, 96; 4, 95, 6], [97, 2, 3; 4, 5, 99]); %! assert (fillmissing (x, "constant", [94:99]), y); %! assert (fillmissing (x, "constant", [94:99]'), y); %! assert (fillmissing (x, "constant", permute ([94:99], [1 3 2])), y); %! assert (fillmissing (x, "constant", [94, 96, 98; 95, 97, 99]), y); %! assert (fillmissing (x, "constant", [94:99], 1), y); %! y = cat (3, [1, 2, 96; 4, 97, 6], [98, 2, 3; 4, 5, 99]); %! assert (fillmissing (x, "constant", [96:99], 2), y); %! y = cat (3, [1, 2, 98; 4, 97, 6], [94, 2, 3; 4, 5, 99]); %! assert (fillmissing (x, "constant", [94:99], 3), y); %! y = cat (3, [1, 2, 92; 4, 91, 6], [94, 2, 3; 4, 5, 99]); %! assert (fillmissing (x, "constant", [88:99], 99), y); %!test %! x = reshape([1:24],4,3,2); %! x([1, 6, 7, 9, 12, 14, 16, 19, 22, 23]) = NaN; %! y = x; %! y([1,6,7,9 12, 14, 16, 19, 22, 23]) = [94 95 95 96 96 97 97 98 99 99]; %! assert (fillmissing (x, "constant", [94:99], 1), y); %! y([1,6,7,9 12, 14, 16, 19, 22, 23]) = [92 93 94 92 95 97 99 98 97 98]; %! assert (fillmissing (x, "constant", [92:99], 2), y); %! y([1,6,7,9 12, 14, 16, 19, 22, 23]) = [88 93 94 96 99 89 91 94 97 98]; %! assert (fillmissing (x, "constant", [88:99], 3), y); %! y([1,6,7,9 12, 14, 16, 19, 22, 23]) = [76 81 82 84 87 89 91 94 97 98]; %! assert (fillmissing (x, "constant", [76:99], 99), y); ## tests with different endvalues behavior %!assert (fillmissing ([1 2 3], "constant", 99, "endvalues", 88), [1 2 3]) %!assert (fillmissing ([1 NaN 3], "constant", 99, "endvalues", 88), [1 99 3]) %!assert (fillmissing ([1 2 NaN], "constant", 99, "endvalues", 88), [1 2 88]) %!assert (fillmissing ([NaN 2 3], "constant", 99, "endvalues", 88), [88 2 3]) %!assert (fillmissing ([NaN NaN 3], "constant", 99, "endvalues", 88), [88 88 3]) %!assert (fillmissing ([1 NaN NaN], "constant", 99, "endvalues", 88), [1 88 88]) %!assert (fillmissing ([NaN 2 NaN], "constant", 99, "endvalues", 88), [88 2 88]) %!assert (fillmissing ([NaN 2 NaN]', "constant", 99, "endvalues", 88), [88 2 88]') %!assert (fillmissing ([1 NaN 3 NaN 5], "constant", 99, "endvalues", 88), [1 99 3 99 5]) %!assert (fillmissing ([1 NaN NaN NaN 5], "constant", 99, "endvalues", 88), [1 99 99 99 5]) %!assert (fillmissing ([NaN NaN NaN NaN 5], "constant", 99, "endvalues", 88), [88 88 88 88 5]) %!assert (fillmissing ([1 NaN 3 4 NaN], "constant", 99, "endvalues", 88), [1 99 3 4 88]) %!assert (fillmissing ([1 NaN 3 4 NaN], "constant", 99, 1, "endvalues", 88), [1 88 3 4 88]) %!assert (fillmissing ([1 NaN 3 4 NaN], "constant", 99, 1, "endvalues", "extrap"), [1 99 3 4 99]) %!test %! x = reshape ([1:24], 3, 4, 2); %! y = x; %! x([1,2,5,6,8,10,13,16,18,19,20,21,22]) = NaN; %! y([1,2,5,6,10,13,16,18,19,20,21,22])= 88; y([8])=99; %! assert (fillmissing (x, "constant", 99, "endvalues", 88), y); %! assert (fillmissing (x, "constant", 99, 1, "endvalues", 88), y); %! y = x; y([1,2,5,8,10,13,16,19,22])= 88; y([6,18,20,21])=99; %! assert (fillmissing (x, "constant", 99, 2, "endvalues", 88), y); %! y(y==99) = 88; %! assert (fillmissing (x, "constant", 99, 3, "endvalues", 88), y); %! assert (fillmissing (x, "constant", 99, 4, "endvalues", 88), y); %! assert (fillmissing (x, "constant", 99, 99, "endvalues", 88), y); %! y([8]) = 94; %! assert (fillmissing (x, "constant", [92:99], 1, "endvalues", 88), y); %! y([6,8,18,20,21]) = [96,88,99,98,99]; %! assert (fillmissing (x, "constant", [94:99], 2, "endvalues", 88), y); %! y = x; y(isnan(y)) = 88; %! assert (fillmissing (x, "constant", [88:99], 3, "endvalues", 88), y); %! y = x; y(isnan(y)) = [82,82,83,83,94,85,86,87,87,88,88,88,89]; %! assert (fillmissing (x, "constant", [92:99], 1, "endvalues", [82:89]), y); %! y = x; y(isnan(y)) = [84,85,85,96,85,84,87,87,99,87,98,99,87]; %! assert (fillmissing (x, "constant", [94:99], 2, "endvalues", [84:89]), y); %! y = x; y(isnan(y)) = [68,69,72,73,75,77,68,71,73,74,75,76,77]; %! assert (fillmissing (x, "constant", [88:99], 3, "endvalues", [68:79]), y); %! assert (fillmissing (x, "constant", [88:93;94:99]', 3, "endvalues", [68:73;74:79]'), y) %!test %! x = reshape([1:24],4,3,2); %! x([1, 6, 7, 9, 12, 14, 16, 19, 22, 23]) = NaN; %! y = x; %! y([1,6,7,9 12, 14, 16, 19, 22, 23]) = [94 95 95 96 96 97 97 98 99 99]; %! assert (fillmissing (x, "constant", [94:99], 1), y); %! y([1,6,7,9 12, 14, 16, 19, 22, 23]) = [92 93 94 92 95 97 99 98 97 98]; %! assert (fillmissing (x, "constant", [92:99], 2), y); %! y([1,6,7,9 12, 14, 16, 19, 22, 23]) = [88 93 94 96 99 89 91 94 97 98]; %! assert (fillmissing (x, "constant", [88:99], 3), y); %! y([1,6,7,9 12, 14, 16, 19, 22, 23]) = [76 81 82 84 87 89 91 94 97 98]; %! assert (fillmissing (x, "constant", [76:99], 99), y); ## next/previous tests %!assert (fillmissing ([1 2 3], "previous"), [1 2 3]) %!assert (fillmissing ([1 2 3], "next"), [1 2 3]) %!assert (fillmissing ([1 2 3]', "previous"), [1 2 3]') %!assert (fillmissing ([1 2 3]', "next"), [1 2 3]') %!assert (fillmissing ([1 2 NaN], "previous"), [1 2 2]) %!assert (fillmissing ([1 2 NaN], "next"), [1 2 NaN]) %!assert (fillmissing ([NaN 2 NaN], "previous"), [NaN 2 2]) %!assert (fillmissing ([NaN 2 NaN], "next"), [2 2 NaN]) %!assert (fillmissing ([1 NaN 3], "previous"), [1 1 3]) %!assert (fillmissing ([1 NaN 3], "next"), [1 3 3]) %!assert (fillmissing ([1 2 NaN; 4 NaN 6], "previous", 1), [1 2 NaN; 4 2 6]) %!assert (fillmissing ([1 2 NaN; 4 NaN 6], "previous", 2), [1 2 2; 4 4 6]) %!assert (fillmissing ([1 2 NaN; 4 NaN 6], "previous", 3), [1 2 NaN; 4 NaN 6]) %!assert (fillmissing ([1 2 NaN; 4 NaN 6], "next", 1), [1 2 6; 4 NaN 6]) %!assert (fillmissing ([1 2 NaN; 4 NaN 6], "next", 2), [1 2 NaN; 4 6 6]) %!assert (fillmissing ([1 2 NaN; 4 NaN 6], "next", 3), [1 2 NaN; 4 NaN 6]) %!test %! x = reshape([1:24],4,3,2); %! x([1, 6, 7, 9, 12, 14, 16, 19, 22, 23]) = NaN; %! y = x; %! y([1, 6, 7, 9, 14, 19, 22, 23]) = [2 8 8 10 15 20 24 24]; %! assert (fillmissing (x, "next", 1), y); %! y = x; %! y([1, 6, 7, 14, 16]) = [5, 10, 11, 18, 20]; %! assert (fillmissing (x, "next", 2), y); %! y = x; %! y([1, 6, 9, 12]) = [13 18 21 24]; %! assert (fillmissing (x, "next", 3), y); %! assert (fillmissing (x, "next", 99), x); %! y = x; %! y([6, 7, 12, 14, 16, 19, 22, 23]) = [5 5 11 13 15 18 21 21]; %! assert (fillmissing (x, "previous", 1), y); %! y = x; %! y([6, 7, 9, 12, 19, 22, 23]) = [2 3 5 8 15 18 15]; %! assert (fillmissing (x, "previous", 2), y); %! y = x; %! y([14, 16, 22, 23]) = [2 4 10 11]; %! assert (fillmissing (x, "previous", 3), y); %! assert (fillmissing (x, "previous", 99), x); ## next/previous tests with different endvalue behavior %!assert (fillmissing ([1 2 3], "constant", 0, "endvalues", "previous"), [1 2 3]) %!assert (fillmissing ([1 2 3], "constant", 0, "endvalues", "next"), [1 2 3]) %!assert (fillmissing ([1 NaN 3], "constant", 0, "endvalues", "previous"), [1 0 3]) %!assert (fillmissing ([1 NaN 3], "constant", 0, "endvalues", "next"), [1 0 3]) %!assert (fillmissing ([1 2 NaN], "constant", 0, "endvalues", "previous"), [1 2 2]) %!assert (fillmissing ([1 2 NaN], "constant", 0, "endvalues", "next"), [1 2 NaN]) %!assert (fillmissing ([1 NaN NaN], "constant", 0, "endvalues", "previous"), [1 1 1]) %!assert (fillmissing ([1 NaN NaN], "constant", 0, "endvalues", "next"), [1 NaN NaN]) %!assert (fillmissing ([NaN 2 3], "constant", 0, "endvalues", "previous"), [NaN 2 3]) %!assert (fillmissing ([NaN 2 3], "constant", 0, "endvalues", "next"), [2 2 3]) %!assert (fillmissing ([NaN NaN 3], "constant", 0, "endvalues", "previous"), [NaN NaN 3]) %!assert (fillmissing ([NaN NaN 3], "constant", 0, "endvalues", "next"), [3 3 3]) %!assert (fillmissing ([NaN NaN NaN], "constant", 0, "endvalues", "previous"), [NaN NaN NaN]) %!assert (fillmissing ([NaN NaN NaN], "constant", 0, "endvalues", "next"), [NaN NaN NaN]) %!assert (fillmissing ([NaN 2 NaN 4 NaN], "constant", 0, "endvalues", "previous"), [NaN 2 0 4 4]) %!assert (fillmissing ([NaN 2 NaN 4 NaN], "constant", 0, "endvalues", "next"), [2 2 0 4 NaN]) %!assert (fillmissing ([NaN 2 NaN 4 NaN], "constant", 0, 1, "endvalues", "previous"), [NaN 2 NaN 4 NaN]) %!assert (fillmissing ([NaN 2 NaN 4 NaN], "constant", 0, 1, "endvalues", "next"), [NaN 2 NaN 4 NaN]) %!assert (fillmissing ([NaN 2 NaN 4 NaN], "constant", 0, 2, "endvalues", "previous"), [NaN 2 0 4 4]) %!assert (fillmissing ([NaN 2 NaN 4 NaN], "constant", 0, 2, "endvalues", "next"), [2 2 0 4 NaN]) %!assert (fillmissing ([NaN 2 NaN 4 NaN], "constant", 0, 3, "endvalues", "previous"), [NaN 2 NaN 4 NaN]) %!assert (fillmissing ([NaN 2 NaN 4 NaN], "constant", 0, 3, "endvalues", "next"), [NaN 2 NaN 4 NaN]) %!test %! x = reshape ([1:24], 3, 4, 2); %! x([1,2,5,6,8,10,13,16,18,19,20,21,22]) = NaN; %! y = x; %! y([5,6,8,18])=[4,4,0,17]; %! assert (fillmissing (x, "constant", 0, "endvalues", "previous"), y); %! assert (fillmissing (x, "constant", 0, 1, "endvalues", "previous"), y); %! y = x; %! y([6,10,18,20,21])=[0,7,0,0,0]; %! assert (fillmissing (x, "constant", 0, 2, "endvalues", "previous"), y); %! y = x; %! y([16,19,21])=[4,7,9]; %! assert (fillmissing (x, "constant", 0, 3, "endvalues", "previous"), y); %! assert (fillmissing (x, "constant", 0, 4, "endvalues", "previous"), x); %! assert (fillmissing (x, "constant", 0, 99, "endvalues", "previous"), x); %! y = x; %! y([1,2,8,10,13,16,22])=[3,3,0,11,14,17,23]; %! assert (fillmissing (x, "constant", 0, "endvalues", "next"), y); %! assert (fillmissing (x, "constant", 0, 1, "endvalues", "next"), y); %! y = x; %! y([1,2,5,6,8,18,20,21])=[4,11,11,0,11,0,0,0]; %! assert (fillmissing (x, "constant", 0, 2, "endvalues", "next"), y); %! y = x; %! y([2,5])=[14,17]; %! assert (fillmissing (x, "constant", 0, 3, "endvalues", "next"), y); %! assert (fillmissing (x, "constant", 0, 4, "endvalues", "next"), x); %! assert (fillmissing (x, "constant", 0, 99, "endvalues", "next"), x); ##tests for nearest %!assert (fillmissing ([1 2 3], "nearest"), [1 2 3]) %!assert (fillmissing ([1 2 3]', "nearest"), [1 2 3]') %!assert (fillmissing ([1 2 NaN], "nearest"), [1 2 2]) %!assert (fillmissing ([NaN 2 NaN], "nearest"), [2 2 2]) %!assert (fillmissing ([1 NaN 3], "nearest"), [1 3 3]) %!assert (fillmissing ([1 2 NaN; 4 NaN 6], "nearest", 1), [1 2 6; 4 2 6]) %!assert (fillmissing ([1 2 NaN; 4 NaN 6], "nearest", 2), [1 2 2; 4 6 6]) %!assert (fillmissing ([1 2 NaN; 4 NaN 6], "nearest", 3), [1 2 NaN; 4 NaN 6]) %!assert (fillmissing ([1 NaN 3 NaN 5], "nearest"), [1 3 3 5 5]) %!assert (fillmissing ([1 NaN 3 NaN 5], "nearest", "samplepoints", [0 1 2 3 4]), [1 3 3 5 5]) %!assert (fillmissing ([1 NaN 3 NaN 5], "nearest", "samplepoints", [0.5 1 2 3 5]), [1 1 3 3 5]) %!test %! x = reshape([1:24],4,3,2); %! x([1, 6, 7, 9, 12, 14, 16, 19, 22, 23]) = NaN; %! y = x; %! y([1, 6, 7, 9, 12, 14, 16, 19, 22, 23]) = [2 5 8 10 11 15 15 20 21 24]; %! assert (fillmissing (x, "nearest", 1), y); %! y = x; %! y([1, 6, 7, 9, 12, 14, 16, 19, 22, 23]) = [5 10 11 5 8 18 20 15 18 15]; %! assert (fillmissing (x, "nearest", 2), y); %! y = x; %! y([1, 6, 9, 12, 14, 16, 22, 23]) = [13 18 21 24 2 4 10 11]; %! assert (fillmissing (x, "nearest", 3), y); %! assert (fillmissing (x, "nearest", 99), x); ##tests for nearest with diff endvalue behavior %!assert (fillmissing ([1 2 3], "constant", 0, "endvalues", "nearest"), [1 2 3]) %!assert (fillmissing ([1 NaN 3], "constant", 0, "endvalues", "nearest"), [1 0 3]) %!assert (fillmissing ([1 2 NaN], "constant", 0, "endvalues", "nearest"), [1 2 2]) %!assert (fillmissing ([1 NaN NaN], "constant", 0, "endvalues", "nearest"), [1 1 1]) %!assert (fillmissing ([NaN 2 3], "constant", 0, "endvalues", "nearest"), [2 2 3]) %!assert (fillmissing ([NaN NaN 3], "constant", 0, "endvalues", "nearest"), [3 3 3]) %!assert (fillmissing ([NaN NaN NaN], "constant", 0, "endvalues", "nearest"), [NaN NaN NaN]) %!assert (fillmissing ([NaN 2 NaN 4 NaN], "constant", 0, "endvalues", "nearest"), [2 2 0 4 4]) %!assert (fillmissing ([NaN 2 NaN 4 NaN], "constant", 0, 1, "endvalues", "nearest"), [NaN 2 NaN 4 NaN]) %!assert (fillmissing ([NaN 2 NaN 4 NaN], "constant", 0, 2, "endvalues", "nearest"), [2 2 0 4 4]) %!assert (fillmissing ([NaN 2 NaN 4 NaN], "constant", 0, 3, "endvalues", "nearest"), [NaN 2 NaN 4 NaN]) %!test %! x = reshape ([1:24], 3, 4, 2); %! x([1,2,5,6,8,10,13,16,18,19,20,21,22]) = NaN; %! y = x; %! y([1,2,5,6,8,10,13,16,18,22])=[3 3 4 4 0 11 14 17 17 23]; %! assert (fillmissing (x, "constant", 0, "endvalues", "nearest"), y); %! assert (fillmissing (x, "constant", 0, 1, "endvalues", "nearest"), y); %! y = x; %! y([1,2,5,6,8,10,18,20,21])=[4 11 11 0 11 7 0 0 0]; %! assert (fillmissing (x, "constant", 0, 2, "endvalues", "nearest"), y); %! y = x; %! y([2,5,16,19,21])=[14 17 4 7 9]; %! assert (fillmissing (x, "constant", 0, 3, "endvalues", "nearest"), y); %! assert (fillmissing (x, "constant", 0, 99, "endvalues", "nearest"), x); ##tests for linear %!assert (fillmissing ([1 2 3], "linear"), [1 2 3]) %!assert (fillmissing ([1 2 3]', "linear"), [1 2 3]') %!assert (fillmissing ([1 2 NaN], "linear"), [1 2 3]) %!assert (fillmissing ([NaN 2 NaN], "linear"), [NaN 2 NaN]) %!assert (fillmissing ([1 NaN 3], "linear"), [1 2 3]) %!assert (fillmissing ([1 2 NaN; 4 NaN 6], "linear", 1), [1 2 NaN; 4 NaN 6]) %!assert (fillmissing ([1 2 NaN; 4 NaN 6], "linear", 2), [1 2 3; 4 5 6]) %!assert (fillmissing ([1 2 NaN; 4 NaN 6], "linear", 3), [1 2 NaN; 4 NaN 6]) %!assert (fillmissing ([1 NaN 3 NaN 5], "linear"), [1 2 3 4 5]) %!assert (fillmissing ([1 NaN 3 NaN 5], "linear", "samplepoints", [0 1 2 3 4]), [1 2 3 4 5]) %!assert (fillmissing ([1 NaN 3 NaN 5], "linear", "samplepoints", [0 1.5 2 5 14]), [1 2.5 3 3.5 5], eps) %!test %! x = reshape([1:24],4,3,2); %! x([1, 6, 7, 9, 12, 14, 16, 19, 22, 23]) = NaN; %! assert (fillmissing (x, "linear", 1), reshape([1:24],4,3,2)); %! y = reshape([1:24],4,3,2); %! y([1 9 14 19 22 23]) = NaN; %! assert (fillmissing (x, "linear", 2), y); %! y = reshape([1:24],4,3,2); %! y([1, 6, 7, 9, 12, 14, 16, 19, 22, 23]) = NaN; %! assert (fillmissing (x, "linear", 3), y); %! assert (fillmissing (x, "linear", 99), x); ##tests for linear with diff endvalue behavior %!assert (fillmissing ([1 2 3], "linear", "endvalues", 0), [1 2 3]) %!assert (fillmissing ([1 NaN 3], "linear", "endvalues", 0), [1 2 3]) %!assert (fillmissing ([1 2 NaN], "linear", "endvalues", 0), [1 2 0]) %!assert (fillmissing ([1 NaN NaN], "linear", "endvalues", 0), [1 0 0]) %!assert (fillmissing ([NaN 2 3], "linear", "endvalues", 0), [0 2 3]) %!assert (fillmissing ([NaN NaN 3], "linear", "endvalues", 0), [0 0 3]) %!assert (fillmissing ([NaN NaN NaN], "linear", "endvalues", 0), [0 0 0]) %!assert (fillmissing ([NaN 2 NaN 4 NaN], "linear", "endvalues", 0), [0 2 3 4 0]) %!assert (fillmissing ([NaN 2 NaN 4 NaN], "linear", 1, "endvalues", 0), [0 2 0 4 0]) %!assert (fillmissing ([NaN 2 NaN 4 NaN], "linear", 2, "endvalues", 0), [0 2 3 4 0]) %!assert (fillmissing ([NaN 2 NaN 4 NaN], "linear", 3, "endvalues", 0), [0 2 0 4 0]) %!test %! x = reshape ([1:24], 3, 4, 2); %! x([1,2,5,6,8,10,13,16,18,19,20,21,22]) = NaN; %! y = x; %! y([1,2,5,6,10,13,16,18,19,20,21,22])=0; y(8)=8; %! assert (fillmissing (x, "linear", "endvalues", 0), y); %! assert (fillmissing (x, "linear", 1, "endvalues", 0), y); %! y = x; %! y([1,2,5,8,10,13,16,19,22])=0; y([6,18,20,21])=[6,18,20,21]; %! assert (fillmissing (x, "linear", 2, "endvalues", 0), y); %! y = x; %! y(isnan(y))=0; %! assert (fillmissing (x, "linear", 3, "endvalues", 0), y); %! assert (fillmissing (x, "linear", 99, "endvalues", 0), y); ##tests with linear only on endvalues %!assert (fillmissing ([1 2 3], "constant", 99, "endvalues", "linear"), [1 2 3]) %!assert (fillmissing ([1 NaN 3], "constant", 99, "endvalues", "linear"), [1 99 3]) %!assert (fillmissing ([1 NaN 3 NaN], "constant", 99, "endvalues", "linear"), [1 99 3 4]) %!assert (fillmissing ([NaN 2 NaN 4 NaN], "constant", 99, "endvalues", "linear"), [1 2 99 4 5]) %!assert (fillmissing ([NaN 2 NaN NaN], "constant", 99, "endvalues", "linear"), [NaN 2 NaN NaN]) %!assert (fillmissing ([NaN 2 NaN 4 NaN], "constant", 99, "endvalues", "linear", "samplepoints", [1 2 3 4 5]), [1 2 99 4 5]) %!assert (fillmissing ([NaN 2 NaN 4 NaN], "constant", 99, "endvalues", "linear", "samplepoints", [0 2 3 4 10]), [0 2 99 4 10]) ##test other interpolants %! x = reshape ([1:24], 3, 4, 2); %! x([1,2,5,6,8,10,13,16,18,19,20,21,22]) = NaN; %! y = x; %! y([1,6,10,18,20,21]) = [2.5, 5, 8.5, 17.25, 21, 21.75]; %! assert (fillmissing (x, "linear", 2, "samplepoints", [2 4 8 10]), y, eps); %! y([1,6,10,18,20,21]) = [2.5, 4.5, 8.5, 17.25, 21.5, 21.75]; %! assert (fillmissing (x, "spline", 2, "samplepoints", [2 4 8 10]), y, eps); %! y([1,6,10,18,20,21]) = [2.5, 4.559386973180077, 8.5, 17.25, 21.440613026819925, 21.75]; %! assert (fillmissing (x, "pchip", 2, "samplepoints", [2 4 8 10]), y, 10*eps); ## known fail: makima method not yet implemented in interp1 %!test <60965> %! x = reshape ([1:24], 3, 4, 2); %! x([1,2,5,6,8,10,13,16,18,19,20,21,22]) = NaN; %! y = x; %! y([1,6,10,18,20,21]) = [2.5, 4.609523809523809, 8.5, 17.25, 21.390476190476186, 21.75]; %! assert (fillmissing (x, "makima", 2, "samplepoints", [2 4 8 10]), y, 10*eps); ##test other interpolants code path on endvalues %!assert (fillmissing ([1 2 3], "constant", 99, "endvalues", "spline"), [1 2 3]) %!assert (fillmissing ([1 NaN 3], "constant", 99, "endvalues", "spline"), [1 99 3]) %!assert (fillmissing ([1 NaN 3 NaN], "constant", 99, "endvalues", "spline"), [1 99 3 4]) %!assert (fillmissing ([NaN 2 NaN 4 NaN], "constant", 99, "endvalues", "spline"), [1 2 99 4 5]) %!assert (fillmissing ([NaN 2 NaN NaN], "constant", 99, "endvalues", "spline"), [NaN 2 NaN NaN]) %!assert (fillmissing ([NaN 2 NaN 4 NaN], "constant", 99, "endvalues", "spline", "samplepoints", [1 2 3 4 5]), [1 2 99 4 5]) %!assert (fillmissing ([NaN 2 NaN 4 NaN], "constant", 99, "endvalues", "spline", "samplepoints", [0 2 3 4 10]), [0 2 99 4 10]) ## test movmean %!assert (fillmissing ([1 2 3], "movmean", 1), [1 2 3]) %!assert (fillmissing ([1 2 NaN], "movmean", 1), [1 2 NaN]) %!assert (fillmissing ([1 2 3], "movmean", 2), [1 2 3]) %!assert (fillmissing ([1 2 3], "movmean", [1 0]), [1 2 3]) %!assert (fillmissing ([1 2 3]', "movmean", 2), [1 2 3]') %!assert (fillmissing ([1 2 NaN], "movmean", 2), [1 2 2]) %!assert (fillmissing ([1 2 NaN], "movmean", [1 0]), [1 2 2]) %!assert (fillmissing ([1 2 NaN], "movmean", [1 0]'), [1 2 2]) %!assert (fillmissing ([NaN 2 NaN], "movmean", 2), [NaN 2 2]) %!assert (fillmissing ([NaN 2 NaN], "movmean", [1 0]), [NaN 2 2]) %!assert (fillmissing ([NaN 2 NaN], "movmean", [0 1]), [2 2 NaN]) %!assert (fillmissing ([NaN 2 NaN], "movmean", [0 1.1]), [2 2 NaN]) %!assert (fillmissing ([1 NaN 3 NaN 5], "movmean", [3 0]), [1 1 3 2 5]) %!assert (fillmissing ([1 2 NaN; 4 NaN 6], "movmean", 3, 1), [1 2 6; 4 2 6]) %!assert (fillmissing ([1 2 NaN; 4 NaN 6], "movmean", 3, 2), [1 2 2; 4 5 6]) %!assert (fillmissing ([1 2 NaN; 4 NaN 6], "movmean", 3, 3), [1 2 NaN; 4 NaN 6]) %!assert (fillmissing ([1 NaN 3 NaN 5], "movmean", 99), [1 3 3 3 5]) %!assert (fillmissing ([1 NaN 3 NaN 5], "movmean", 99, 1), [1 NaN 3 NaN 5]) %!assert (fillmissing ([1 NaN 3 NaN 5]', "movmean", 99, 1), [1 3 3 3 5]') %!assert (fillmissing ([1 NaN 3 NaN 5], "movmean", 99, 2), [1 3 3 3 5]) %!assert (fillmissing ([1 NaN 3 NaN 5]', "movmean", 99, 2), [1 NaN 3 NaN 5]') %!assert (fillmissing ([1 NaN NaN NaN 5], "movmean", 3, "samplepoints", [1 2 3 4 5]), [1 1 NaN 5 5]) %!assert (fillmissing ([1 NaN NaN NaN 5], "movmean", [1 1], "samplepoints", [1 2 3 4 5]), [1 1 NaN 5 5]) %!assert (fillmissing ([1 NaN NaN NaN 5], "movmean", [1.5 1.5], "samplepoints", [1 2 3 4 5]), [1 1 NaN 5 5]) %!assert (fillmissing ([1 NaN NaN NaN 5], "movmean", 4, "samplepoints", [1 2 3 4 5]), [1 1 1 5 5]) %!assert (fillmissing ([1 NaN NaN NaN 5], "movmean", [2 2], "samplepoints", [1 2 3 4 5]), [1 1 3 5 5]) %!assert (fillmissing ([1 NaN NaN NaN 5], "movmean", 4.0001, "samplepoints", [1 2 3 4 5]), [1 1 3 5 5]) %!assert (fillmissing ([1 NaN NaN NaN 5], "movmean", 3, "samplepoints", [1.5 2 3 4 5]), [1 1 1 5 5]) %!assert (fillmissing ([1 NaN NaN NaN 5], "movmean", 3, "samplepoints", [1 2 3 4 4.5]), [1 1 NaN 5 5]) %!assert (fillmissing ([1 NaN NaN NaN 5], "movmean", 3, "samplepoints", [1.5 2 3 4 4.5]), [1 1 1 5 5]) %!assert (fillmissing ([1 NaN NaN NaN 5], "movmean", [1.5 1.5], "samplepoints", [1.5 2 3 4 5]), [1 1 1 5 5]) %!assert (fillmissing ([1 NaN NaN NaN 5], "movmean", [1.5 1.5], "samplepoints", [1 2 3 4 4.5]), [1 1 5 5 5]) %!assert (fillmissing ([1 NaN NaN NaN 5], "movmean", [1.5 1.5], "samplepoints", [1.5 2 3 4 4.5]), [1 1 3 5 5]) %!test %! x = reshape ([1:24], 3, 4, 2); %! x([1,2,5,6,8,10,13,16,18,19,20,21,22]) = NaN; %! y = x; %! y([2,5,8,10,13,16,18,22]) = [3,4,8,11,14,17,17,23]; %! assert (fillmissing (x, "movmean", 3), y); %! assert (fillmissing (x, "movmean", [1 1]), y); %! assert (fillmissing (x, "movmean", 3, "endvalues", "extrap"), y); %! assert (fillmissing (x, "movmean", 3, "samplepoints", [1 2 3]), y); %! y = x; %! y([1,6,8,10,18,20,21]) = [4,6,11,7,15,20,24]; %! assert (fillmissing (x, "movmean", 3, 2), y); %! assert (fillmissing (x, "movmean", [1 1], 2), y); %! assert (fillmissing (x, "movmean", 3, 2, "endvalues", "extrap"), y); %! assert (fillmissing (x, "movmean", 3, 2, "samplepoints", [1 2 3 4]), y); %! y([1,18]) = NaN; y(6) = 9; %! assert (fillmissing (x, "movmean", 3, 2, "samplepoints", [0 2 3 4]), y); %! y = x; %! y([1,2,5,6,10,13,16,18,19,20,21,22]) = 99; y(8) = 8; %! assert (fillmissing (x, "movmean", 3, "endvalues", 99), y); %! y = x; %! y([1,2,5,8,10,13,16,19,22]) = 99; y([6,18,20,21]) = [6,15,20,24]; %! assert (fillmissing (x, "movmean", 3, 2, "endvalues", 99), y); ## test movmedian %!assert (fillmissing ([1 2 3], "movmedian", 1), [1 2 3]) %!assert (fillmissing ([1 2 NaN], "movmedian", 1), [1 2 NaN]) %!assert (fillmissing ([1 2 3], "movmedian", 2), [1 2 3]) %!assert (fillmissing ([1 2 3], "movmedian", [1 0]), [1 2 3]) %!assert (fillmissing ([1 2 3]', "movmedian", 2), [1 2 3]') %!assert (fillmissing ([1 2 NaN], "movmedian", 2), [1 2 2]) %!assert (fillmissing ([1 2 NaN], "movmedian", [1 0]), [1 2 2]) %!assert (fillmissing ([1 2 NaN], "movmedian", [1 0]'), [1 2 2]) %!assert (fillmissing ([NaN 2 NaN], "movmedian", 2), [NaN 2 2]) %!assert (fillmissing ([NaN 2 NaN], "movmedian", [1 0]), [NaN 2 2]) %!assert (fillmissing ([NaN 2 NaN], "movmedian", [0 1]), [2 2 NaN]) %!assert (fillmissing ([NaN 2 NaN], "movmedian", [0 1.1]), [2 2 NaN]) %!assert (fillmissing ([1 NaN 3 NaN 5], "movmedian", [3 0]), [1 1 3 2 5]) %!assert (fillmissing ([1 2 NaN; 4 NaN 6], "movmedian", 3, 1), [1 2 6; 4 2 6]) %!assert (fillmissing ([1 2 NaN; 4 NaN 6], "movmedian", 3, 2), [1 2 2; 4 5 6]) %!assert (fillmissing ([1 2 NaN; 4 NaN 6], "movmedian", 3, 3), [1 2 NaN; 4 NaN 6]) %!assert (fillmissing ([1 NaN 3 NaN 5], "movmedian", 99), [1 3 3 3 5]) %!assert (fillmissing ([1 NaN 3 NaN 5], "movmedian", 99, 1), [1 NaN 3 NaN 5]) %!assert (fillmissing ([1 NaN 3 NaN 5]', "movmedian", 99, 1), [1 3 3 3 5]') %!assert (fillmissing ([1 NaN 3 NaN 5], "movmedian", 99, 2), [1 3 3 3 5]) %!assert (fillmissing ([1 NaN 3 NaN 5]', "movmedian", 99, 2), [1 NaN 3 NaN 5]') %!assert (fillmissing ([1 NaN NaN NaN 5], "movmedian", 3, "samplepoints", [1 2 3 4 5]), [1 1 NaN 5 5]) %!assert (fillmissing ([1 NaN NaN NaN 5], "movmedian", [1 1], "samplepoints", [1 2 3 4 5]), [1 1 NaN 5 5]) %!assert (fillmissing ([1 NaN NaN NaN 5], "movmedian", [1.5 1.5], "samplepoints", [1 2 3 4 5]), [1 1 NaN 5 5]) %!assert (fillmissing ([1 NaN NaN NaN 5], "movmedian", 4, "samplepoints", [1 2 3 4 5]), [1 1 1 5 5]) %!assert (fillmissing ([1 NaN NaN NaN 5], "movmedian", [2 2], "samplepoints", [1 2 3 4 5]), [1 1 3 5 5]) %!assert (fillmissing ([1 NaN NaN NaN 5], "movmedian", 4.0001, "samplepoints", [1 2 3 4 5]), [1 1 3 5 5]) %!assert (fillmissing ([1 NaN NaN NaN 5], "movmedian", 3, "samplepoints", [1.5 2 3 4 5]), [1 1 1 5 5]) %!assert (fillmissing ([1 NaN NaN NaN 5], "movmedian", 3, "samplepoints", [1 2 3 4 4.5]), [1 1 NaN 5 5]) %!assert (fillmissing ([1 NaN NaN NaN 5], "movmedian", 3, "samplepoints", [1.5 2 3 4 4.5]), [1 1 1 5 5]) %!assert (fillmissing ([1 NaN NaN NaN 5], "movmedian", [1.5 1.5], "samplepoints", [1.5 2 3 4 5]), [1 1 1 5 5]) %!assert (fillmissing ([1 NaN NaN NaN 5], "movmedian", [1.5 1.5], "samplepoints", [1 2 3 4 4.5]), [1 1 5 5 5]) %!assert (fillmissing ([1 NaN NaN NaN 5], "movmedian", [1.5 1.5], "samplepoints", [1.5 2 3 4 4.5]), [1 1 3 5 5]) %!test %! x = reshape ([1:24], 3, 4, 2); %! x([1,2,5,6,8,10,13,16,18,19,20,21,22]) = NaN; %! y = x; %! y([2 5 8 10 13 16 18 22]) = [3 4 8 11 14 17 17 23]; %! assert (fillmissing (x, "movmedian", 3), y); %! assert (fillmissing (x, "movmedian", [1 1]), y); %! assert (fillmissing (x, "movmedian", 3, "endvalues", "extrap"), y); %! assert (fillmissing (x, "movmedian", 3, "samplepoints", [1 2 3]), y); %! y = x; %! y([1 6 8 10 18 20 21]) = [4 6 11 7 15 20 24]; %! assert (fillmissing (x, "movmedian", 3, 2), y); %! assert (fillmissing (x, "movmedian", [1 1], 2), y); %! assert (fillmissing (x, "movmedian", 3, 2, "endvalues", "extrap"), y); %! assert (fillmissing (x, "movmedian", 3, 2, "samplepoints", [1 2 3 4]), y); %! y([1,18]) = NaN; y(6) = 9; %! assert (fillmissing (x, "movmedian", 3, 2, "samplepoints", [0 2 3 4]), y); %! y = x; %! y([1,2,5,6,10,13,16,18,19,20,21,22]) = 99; y(8) = 8; %! assert (fillmissing (x, "movmedian", 3, "endvalues", 99), y); %! y = x; %! y([1,2,5,8,10,13,16,19,22]) = 99; y([6,18,20,21]) = [6,15,20,24]; %! assert (fillmissing (x, "movmedian", 3, 2, "endvalues", 99), y); ## test movfcn %!assert (fillmissing ([1 2 3], @(x,y,z) x+y+z, 2), [1 2 3]) %!assert (fillmissing ([1 2 NaN], @(x,y,z) x+y+z, 1), [1 2 NaN]) %!assert (fillmissing ([1 2 3], @(x,y,z) x+y+z, 2), [1 2 3]) %!assert (fillmissing ([1 2 3], @(x,y,z) x+y+z, [1 0]), [1 2 3]) %!assert (fillmissing ([1 2 3]', @(x,y,z) x+y+z, 2), [1 2 3]') %!assert (fillmissing ([1 2 NaN], @(x,y,z) x+y+z, 2), [1 2 7]) %!assert (fillmissing ([1 2 NaN], @(x,y,z) x+y+z, [1 0]), [1 2 7]) %!assert (fillmissing ([1 2 NaN], @(x,y,z) x+y+z, [1 0]'), [1 2 7]) %!assert (fillmissing ([NaN 2 NaN], @(x,y,z) x+y+z, 2), [5 2 7]) %!assert (fillmissing ([NaN 2 NaN], @(x,y,z) x+y+z, [1 0]), [NaN 2 7]) %!assert (fillmissing ([NaN 2 NaN], @(x,y,z) x+y+z, [0 1]), [5 2 NaN]) %!assert (fillmissing ([NaN 2 NaN], @(x,y,z) x+y+z, [0 1.1]), [5 2 NaN]) %!assert (fillmissing ([1 2 NaN NaN 3 4], @(x,y,z) x+y+z, 2),[1 2 7 12 3 4]) %!assert (fillmissing ([1 2 NaN NaN 3 4], @(x,y,z) x+y+z, 0.5),[1 2 NaN NaN 3 4]) %!function A = testfcn (x,y,z) %! if isempty (y) %! A = z; %! elseif (numel (y) == 1) %! A = repelem (x(1), numel(z)); %! else %! A = interp1 (y, x, z, "linear","extrap"); %! endif %!endfunction %!assert (fillmissing ([1 NaN 3 NaN 5], @testfcn, [3 0]), [1 1 3 NaN 5]) %!assert (fillmissing ([1 2 NaN; 4 NaN 6], @testfcn, 3, 1), [1 2 6; 4 2 6]) %!assert (fillmissing ([1 2 NaN; 4 NaN 6], @testfcn, 3, 2), [1 2 2; 4 5 6]) %!assert (fillmissing ([1 2 NaN; 4 NaN 6], @testfcn, 3, 3), [1 2 NaN; 4 NaN 6]) %!assert (fillmissing ([1 NaN 3 NaN 5], @testfcn, 99), [1 2 3 4 5]) %!assert (fillmissing ([1 NaN 3 NaN 5], @testfcn, 99, 1), [1 NaN 3 NaN 5]) ##known not-compatible. matlab bug ML2022a: [1 1 3 1 5] %!assert (fillmissing ([1 NaN 3 NaN 5]', @testfcn, 99, 1), [1 2 3 4 5]') %!assert (fillmissing ([1 NaN 3 NaN 5], @testfcn, 99, 2), [1 2 3 4 5]) %!assert (fillmissing ([1 NaN 3 NaN 5]', @testfcn, 99, 2), [1 NaN 3 NaN 5]') ##known not-compatible. matlab bug ML2022a: [1 1 3 1 5]' %!assert (fillmissing ([1 NaN 3 NaN 5], @testfcn, 99, 3), [1 NaN 3 NaN 5]) %!assert (fillmissing ([1 NaN 3 NaN 5]', @testfcn, 99, 3), [1 NaN 3 NaN 5]') %!assert (fillmissing ([1 NaN NaN NaN 5], @testfcn, 3, "samplepoints", [1 2 3 4 5]), [1 2 3 4 5]) %!assert (fillmissing ([1 NaN NaN NaN 5], @testfcn, [1 1], "samplepoints", [1 2 3 4 5]), [1 2 3 4 5]) %!assert (fillmissing ([1 NaN NaN NaN 5], @testfcn, [1.5 1.5], "samplepoints", [1 2 3 4 5]), [1 2 3 4 5]) %!assert (fillmissing ([1 NaN NaN NaN 5], @testfcn, 4, "samplepoints", [1 2 3 4 5]), [1 2 3 4 5]) %!assert (fillmissing ([1 NaN NaN NaN 5], @testfcn, [2 2], "samplepoints", [1 2 3 4 5]), [1 2 3 4 5]) %!assert (fillmissing ([1 NaN NaN NaN 5], @testfcn, 3, "samplepoints", [1 2 2.5 3 3.5]), [1 2.6 3.4 4.2 5], 10*eps) %!assert (fillmissing ([NaN NaN 3 NaN 5], @testfcn, 99, 1), [NaN NaN 3 NaN 5]) ##known not-compatible. matlab bug ML2022a: [1 1 3 1 5] ## known noncompatible. for move_fcn method, ML2021b (1) ignores windowsize ## for full missing column and processes it anyway, (2) doesn't consider it ## part of endvalues unlike all other methods, (3) ignores samplepoint values ## when calcuating move_fcn results. should review against future versions. %!test %!function A = testfcn (x,y,z) %! if isempty (y) %! A = z; %! elseif (numel (y) == 1) %! A = repelem (x(1), numel(z)); %! else %! A = interp1 (y, x, z, "linear","extrap"); %! endif %!endfunction %! x = reshape ([1:24], 3, 4, 2); %! x([1,2,5,6,8,10,13,16,18,19,20,21,22]) = NaN; %! y = x; %! y([1,2,5,6,8,10,13,16,18,22]) = [3,3,4,4,8,11,14,17,17,23]; %! assert (fillmissing (x, @testfcn, 3), y); %! assert (fillmissing (x, @testfcn, [1 1]), y); %! assert (fillmissing (x, @testfcn, 3, "endvalues", "extrap"), y); %! assert (fillmissing (x, @testfcn, 3, "samplepoints", [1 2 3]), y); %! y= x; %! y(isnan(x)) = 99; y(8) = 8; %! assert (fillmissing (x, @testfcn, 3, "endvalues", 99), y) %! y = x; %! y([1,2,5,6,8,10,18,20,21]) = [4,11,11,6,11,7,18,20,21]; %! assert (fillmissing (x, @testfcn, 3, 2), y); %! assert (fillmissing (x, @testfcn, [1 1], 2), y); %! assert (fillmissing (x, @testfcn, 3, 2, "endvalues", "extrap"), y); %! assert (fillmissing (x, @testfcn, 3, 2, "samplepoints", [1 2 3 4]), y); %! y(1) = NaN; y([6,18,21]) = [9,24,24]; %! assert (fillmissing (x, @testfcn, 3, 2, "samplepoints", [0 2 3 4]), y); %! y = x; %! y([1,2,5,6,10,13,16,18,19,20,21,22]) = 99; y(8) = [8]; %! assert (fillmissing (x, @testfcn, 3, "endvalues", 99), y); %! y([6,18,20,21]) = [6,18,20,21]; y(8)=99; %! assert (fillmissing (x, @testfcn, 3, 2, "endvalues", 99), y); %! y([6,18,20,21]) = 99; %! assert (fillmissing (x, @testfcn, 3, 3, "endvalues", 99), y); ##test maxgap for mid and end points %!assert (fillmissing ([1 2 3], "constant", 0, "maxgap", 1), [1 2 3]) %!assert (fillmissing ([1 2 3], "constant", 0, "maxgap", 99), [1 2 3]) %!assert (fillmissing ([1 NaN 3], "constant", 0, "maxgap", 1), [1 NaN 3]) %!assert (fillmissing ([1 NaN 3], "constant", 0, "maxgap", 1.999), [1 NaN 3]) %!assert (fillmissing ([1 NaN 3], "constant", 0, "maxgap", 2), [1 0 3]) %!assert (fillmissing ([1 NaN NaN 4], "constant", 0, "maxgap", 2), [1 NaN NaN 4]) %!assert (fillmissing ([1 NaN NaN 4], "constant", 0, "maxgap", 3), [1 0 0 4]) %!assert (fillmissing ([1 NaN 3 NaN 5], "constant", 0, "maxgap", 2), [1 0 3 0 5]) %!assert (fillmissing ([NaN 2 NaN], "constant", 0, "maxgap", 0.999), [NaN 2 NaN]) %!assert (fillmissing ([NaN 2 NaN], "constant", 0, "maxgap", 1), [0 2 0]) %!assert (fillmissing ([NaN 2 NaN NaN], "constant", 0, "maxgap", 1), [0 2 NaN NaN]) %!assert (fillmissing ([NaN 2 NaN NaN], "constant", 0, "maxgap", 2), [0 2 0 0]) %!assert (fillmissing ([NaN NaN NaN], "constant", 0, "maxgap", 1), [NaN NaN NaN]) %!assert (fillmissing ([NaN NaN NaN], "constant", 0, "maxgap", 3), [NaN NaN NaN]) %!assert (fillmissing ([NaN NaN NaN], "constant", 0, "maxgap", 999), [NaN NaN NaN]) %!assert (fillmissing ([1 NaN 3 NaN 5], "constant", 0, "maxgap", 2, "samplepoints", [0 1 2 3 5]), [1 0 3 NaN 5]) %!assert (fillmissing ([1 NaN 3 NaN 5]', "constant", 0, "maxgap", 2, "samplepoints", [0 1 2 3 5]), [1 0 3 NaN 5]') %!assert (fillmissing ([1 NaN 3 NaN 5], "constant", 0, "maxgap", 2, "samplepoints", [0 2 3 4 5]), [1 NaN 3 0 5]) %!assert (fillmissing ([1 NaN 3 NaN 5; 1 NaN 3 NaN 5], "constant", 0, 2, "maxgap", 2, "samplepoints", [0 2 3 4 5]), [1 NaN 3 0 5; 1 NaN 3 0 5]) %!test %! x = cat (3, [1, 2, NaN; 4, NaN, NaN], [NaN, 2, 3; 4, 5, NaN]); %! assert (fillmissing (x, "constant", 0, "maxgap", 0.1), x); %! y = x; %! y([4,7,12]) = 0; %! assert (fillmissing (x, "constant", 0, "maxgap", 1), y); %! assert (fillmissing (x, "constant", 0, 1, "maxgap", 1), y); %! y = x; %! y([5,7,12]) = 0; %! assert (fillmissing (x, "constant", 0, 2, "maxgap", 1), y); %! y = x; %! y([4,5,7]) = 0; %! assert (fillmissing (x, "constant", 0, 3, "maxgap", 1), y); ## 2nd output ## verify consistent with dim %!test %! x = cat (3, [1, 2, NaN; 4, NaN, NaN], [NaN, 2, 3; 4, 5, NaN]); %! [~, idx] = fillmissing (x, "constant", 0, "maxgap", 1); %! assert (idx, logical (cat (3, [0 0 0; 0 1 0], [1 0 0; 0 0 1]))); %! [~, idx] = fillmissing (x, "constant", 0, 1, "maxgap", 1); %! assert (idx, logical (cat (3, [0 0 0; 0 1 0], [1 0 0; 0 0 1]))); %! [~, idx] = fillmissing (x, "constant", 0, 2, "maxgap", 1); %! assert (idx, logical (cat (3, [0 0 1; 0 0 0], [1 0 0; 0 0 1]))); %! [~, idx] = fillmissing (x, "constant", 0, 3, "maxgap", 1); %! assert (idx, logical (cat (3, [0 0 1; 0 1 0], [1 0 0; 0 0 0]))); ## verify idx matches when methods leave gaps unfilled, or when fill looks ## the same %!test %! x = [NaN, 2, 3]; %! [~,idx] = fillmissing (x, "previous"); %! assert (idx, logical ([0 0 0])); %! [~,idx] = fillmissing (x, "movmean", 1); %! assert (idx, logical ([0 0 0])); %! x = [1:3;4:6;7:9]; %! x([2,4,7,9]) = NaN; %! [~,idx] = fillmissing (x, "linear"); %! assert (idx, logical ([0 1 0;1 0 0;0 0 0])); %! [~,idx] = fillmissing (x, "movmean", 2); %! assert (idx, logical ([0 0 0;1 0 0;0 0 1])); %! [A, idx] = fillmissing ([1 2 3 NaN NaN], 'movmean',2); %! assert (A, [1 2 3 3 NaN]); %! assert (idx, logical([0 0 0 1 0])); %! [A, idx] = fillmissing ([1 2 3 NaN NaN], 'movmean',3); %! assert (A, [1 2 3 3 NaN]); %! assert (idx, logical([0 0 0 1 0])); %! [A, idx] = fillmissing ([1 2 NaN NaN NaN], 'movmedian', 2); %! assert (A, [1 2 2 NaN NaN]); %! assert (idx, logical([0 0 1 0 0])); %! [A, idx] = fillmissing ([1 2 3 NaN NaN], 'movmedian', 3); %! assert (A, [1 2 3 3 NaN]); %! assert (idx, logical([0 0 0 1 0])); %! [A, idx] = fillmissing ([1 NaN 1 NaN 1], @(x,y,z) z, 3); %! assert (A, [1 2 1 4 1]); %! assert (idx, logical([0 1 0 1 0])); %! [A, idx] = fillmissing ([1 NaN 1 NaN 1], @(x,y,z) NaN (size (z)), 3); %! assert (A, [1 NaN 1 NaN 1]); %! assert (idx, logical([0 0 0 0 0])); #test missinglocations %!assert (fillmissing ([1 2 3], "constant", 99, "missinglocations", logical([0 0 0])), [1 2 3]) %!assert (fillmissing ([1 2 3], "constant", 99, "missinglocations", logical([1 1 1])), [99 99 99]) %!assert (fillmissing ([1 NaN 2 3 NaN], "constant", 99, "missinglocations", logical([1 0 1 0 1])), [99 NaN 99 3 99]) %!assert (fillmissing ([1 NaN 3 NaN 5], "constant", NaN, "missinglocations", logical([0 1 1 1 0])), [1 NaN NaN NaN 5]) %!assert (fillmissing (["foo ";" bar"], "constant", 'X', "missinglocations", logical([0 0 0 0; 0 0 0 0])), ["foo ";" bar"]) %!assert (fillmissing (["foo ";" bar"], "constant", 'X', "missinglocations", logical([1 0 1 0; 0 1 1 0])), ["XoX ";" XXr"]) %!assert (fillmissing ({"foo","", "bar"}, "constant", 'X', "missinglocations", logical([0 0 0])), {"foo","", "bar"}) %!assert (fillmissing ({"foo","", "bar"}, "constant", 'X', "missinglocations", logical([1 1 0])), {"X","X","bar"}) %!test %! [~,idx] = fillmissing ([1 NaN 3 NaN 5], "constant", NaN); %! assert (idx, logical([0 0 0 0 0])); %! [~,idx] = fillmissing ([1 NaN 3 NaN 5], "constant", NaN, "missinglocations", logical([0 1 1 1 0])); %! assert (idx, logical([0 1 1 1 0])); %! [A, idx] = fillmissing ([1 2 NaN 1 NaN], 'movmean', 3.1, 'missinglocations', logical([0 0 1 1 0])); %! assert (A, [1 2 2 NaN NaN]); %! assert (idx, logical([0 0 1 0 0])); %! [A, idx] = fillmissing ([1 2 NaN NaN NaN], 'movmean', 2, 'missinglocations', logical([0 0 1 1 0])); %! assert (A, [1 2 2 NaN NaN]); %! assert (idx, logical([0 0 1 0 0])); %! [A, idx] = fillmissing ([1 2 NaN 1 NaN], 'movmean', 3, 'missinglocations', logical([0 0 1 1 0])); %! assert (A, [1 2 2 NaN NaN]); %! assert (idx, logical([0 0 1 0 0])); %! [A, idx] = fillmissing ([1 2 NaN NaN NaN], 'movmean', 3, 'missinglocations', logical([0 0 1 1 0])); %! assert (A, [1 2 2 NaN NaN]); %! assert (idx, logical([0 0 1 0 0])); %! [A, idx] = fillmissing ([1 2 NaN NaN NaN], 'movmedian', 2, 'missinglocations', logical([0 0 1 1 0])); %! assert (A, [1 2 2 NaN NaN]); %! assert (idx, logical([0 0 1 0 0])); %! [A, idx] = fillmissing ([1 2 NaN NaN NaN], 'movmedian', 3, 'missinglocations', logical([0 0 1 1 0])); %! assert (A, [1 2 2 NaN NaN]); %! assert (idx, logical([0 0 1 0 0])); %! [A, idx] = fillmissing ([1 2 NaN NaN NaN], 'movmedian', 3.1, 'missinglocations', logical([0 0 1 1 0])); %! assert (A, [1 2 2 NaN NaN]); %! assert (idx, logical([0 0 1 0 0])); %! [A, idx] = fillmissing ([1 NaN 1 NaN 1], @(x,y,z) ones (size (z)), 3, "missinglocations", logical([0 1 0 1 1])); %! assert (A, [1 1 1 1 1]); %! assert (idx, logical([0 1 0 1 1])); %! [A, idx] = fillmissing ([1 NaN 1 NaN 1], @(x,y,z) NaN (size (z)), 3, "missinglocations", logical([0 1 0 1 1])); %! assert (A, [1 NaN 1 NaN NaN]); %! assert (idx, logical([0 0 0 0 0])); ##Test char and cellstr %!assert (fillmissing (' foo bar ', "constant", 'X'), 'XfooXbarX') %!assert (fillmissing ([' foo';'bar '], "constant", 'X'), ['Xfoo';'barX']) %!assert (fillmissing ([' foo';'bar '], "next"), ['bfoo';'bar ']) %!assert (fillmissing ([' foo';'bar '], "next", 1), ['bfoo';'bar ']) %!assert (fillmissing ([' foo';'bar '], "previous"), [' foo';'baro']) %!assert (fillmissing ([' foo';'bar '], "previous", 1), [' foo';'baro']) %!assert (fillmissing ([' foo';'bar '], "nearest"), ['bfoo';'baro']) %!assert (fillmissing ([' foo';'bar '], "nearest", 1), ['bfoo';'baro']) %!assert (fillmissing ([' foo';'bar '], "next", 2), ['ffoo';'bar ']) %!assert (fillmissing ([' foo';'bar '], "previous", 2), [' foo';'barr']) %!assert (fillmissing ([' foo';'bar '], "nearest", 2), ['ffoo';'barr']) %!assert (fillmissing ([' foo';'bar '], "next", 3), [' foo';'bar ']) %!assert (fillmissing ([' foo';'bar '], "previous", 3), [' foo';'bar ']) %!assert (fillmissing ([' foo';'bar '], "nearest", 3), [' foo';'bar ']) %!assert (fillmissing ({'foo','bar'}, "constant", 'a'), {'foo','bar'}) %!assert (fillmissing ({'foo','bar'}, "constant", {'a'}), {'foo','bar'}) %!assert (fillmissing ({'foo', '', 'bar'}, "constant", 'a'), {'foo', 'a', 'bar'}) %!assert (fillmissing ({'foo', '', 'bar'}, "constant", {'a'}), {'foo', 'a', 'bar'}) %!assert (fillmissing ({'foo', '', 'bar'}, "previous"), {'foo', 'foo', 'bar'}) %!assert (fillmissing ({'foo', '', 'bar'}, "next"), {'foo', 'bar', 'bar'}) %!assert (fillmissing ({'foo', '', 'bar'}, "nearest"), {'foo', 'bar', 'bar'}) %!assert (fillmissing ({'foo', '', 'bar'}, "previous", 2), {'foo', 'foo', 'bar'}) %!assert (fillmissing ({'foo', '', 'bar'}, "next", 2), {'foo', 'bar', 'bar'}) %!assert (fillmissing ({'foo', '', 'bar'}, "nearest", 2), {'foo', 'bar', 'bar'}) %!assert (fillmissing ({'foo', '', 'bar'}, "previous", 1), {'foo', '', 'bar'}) %!assert (fillmissing ({'foo', '', 'bar'}, "previous", 1), {'foo', '', 'bar'}) %!assert (fillmissing ({'foo', '', 'bar'}, "next", 1), {'foo', '', 'bar'}) %!assert (fillmissing ({'foo', '', 'bar'}, "nearest", 1), {'foo', '', 'bar'}) %!assert (fillmissing ("abc ", @(x,y,z) x+y+z, 2), "abcj") %!assert (fillmissing ({'foo', '', 'bar'}, @(x,y,z) x(1), 3), {'foo','foo','bar'}) %!test %! [A, idx] = fillmissing (" a b c", "constant", " "); %! assert (A, " a b c"); %! assert (idx, logical([0 0 0 0 0 0])); %! [A, idx] = fillmissing ({"foo", "", "bar", ""}, "constant", ""); %! assert (A, {"foo", "", "bar", ""}); %! assert (idx, logical([0 0 0 0])); %! [A, idx] = fillmissing ({"foo", "", "bar", ""}, "constant", {""}); %! assert (A, {"foo", "", "bar", ""}); %! assert (idx, logical([0 0 0 0])); %! [A,idx] = fillmissing (' f o o ', @(x,y,z) repelem ("a", numel (z)), 3); %! assert (A, "afaoaoa"); %! assert (idx, logical([1 0 1 0 1 0 1])); %! [A,idx] = fillmissing (' f o o ', @(x,y,z) repelem (" ", numel (z)), 3); %! assert (A, " f o o "); %! assert (idx, logical([0 0 0 0 0 0 0])); %! [A,idx] = fillmissing ({'','foo',''}, @(x,y,z) repelem ({'a'}, numel (z)), 3); %! assert (A, {'a','foo','a'}); %! assert (idx, logical([1 0 1])); %! [A,idx] = fillmissing ({'','foo',''}, @(x,y,z) repelem ({''}, numel (z)), 3); %! assert (A, {'','foo',''}); %! assert (idx, logical([0 0 0])); ##types without a defined 'missing' (currently logical, int) that can be filled %!assert (fillmissing (logical ([1 0 1 0 1]), "constant", true), logical ([1 0 1 0 1])) %!assert (fillmissing (logical ([1 0 1 0 1]), "constant", false, 'missinglocations', logical([1 0 1 0 1])), logical ([0 0 0 0 0])) %!assert (fillmissing (logical ([1 0 1 0 1]), "previous", 'missinglocations', logical([1 0 1 0 1])), logical ([1 0 0 0 0])) %!assert (fillmissing (logical ([1 0 1 0 1]), "next", 'missinglocations', logical([1 0 1 0 1])), logical ([0 0 0 0 1])) %!assert (fillmissing (logical ([1 0 1 0 1]), "nearest", 'missinglocations', logical([1 0 1 0 1])), logical ([0 0 0 0 0])) %!assert (fillmissing (logical ([1 0 1 0 1]), @(x,y,z) false(size(z)), 3), logical ([1 0 1 0 1])) %!assert (fillmissing (logical ([1 0 1 0 1]), @(x,y,z) false(size(z)), 3, 'missinglocations', logical([1 0 1 0 1])), logical ([0 0 0 0 0])) %!assert (fillmissing (logical ([1 0 1 0 1]), @(x,y,z) false(size(z)), [2 0], 'missinglocations', logical([1 0 1 0 1])), logical ([1 0 0 0 0])) %!test %! x = logical ([1 0 1 0 1]); %! [~,idx] = fillmissing (x, "constant", true); %! assert (idx, logical([0 0 0 0 0])); %! [~,idx] = fillmissing (x, "constant", false, 'missinglocations', logical([1 0 1 0 1])); %! assert (idx, logical([1 0 1 0 1])); %! [~,idx] = fillmissing (x, "constant", true, 'missinglocations', logical([1 0 1 0 1])); %! assert (idx, logical([1 0 1 0 1])); %! [~,idx] = fillmissing (x, "previous", 'missinglocations', logical([1 0 1 0 1])); %! assert (idx, logical([0 0 1 0 1])); %! [~,idx] = fillmissing (x, "next", 'missinglocations', logical([1 0 1 0 1])); %! assert (idx, logical([1 0 1 0 0])); %! [~,idx] = fillmissing (x, "nearest", 'missinglocations', logical([1 0 1 0 1])); %! assert (idx, logical([1 0 1 0 1])); %! [~,idx] = fillmissing (x, @(x,y,z) false(size(z)), 3); %! assert (idx, logical ([0 0 0 0 0])) %! [~,idx] = fillmissing (x, @(x,y,z) false(size(z)), 3, 'missinglocations', logical([1 0 1 0 1])); %! assert (idx, logical ([1 0 1 0 1])) %! [~,idx] = fillmissing (x, @(x,y,z) false(size(z)), [2 0], 'missinglocations', logical([1 0 1 0 1])); %! assert (idx, logical ([0 0 1 0 1])) %!assert (fillmissing (int32 ([1 2 3 4 5]), "constant", 0), int32 ([1 2 3 4 5])) %!assert (fillmissing (int32 ([1 2 3 4 5]), "constant", 0,'missinglocations', logical([1 0 1 0 1])), int32 ([0 2 0 4 0])) %!assert (fillmissing (int32 ([1 2 3 4 5]), "previous", 'missinglocations', logical([1 0 1 0 1])), int32 ([1 2 2 4 4])) %!assert (fillmissing (int32 ([1 2 3 4 5]), "next", 'missinglocations', logical([1 0 1 0 1])), int32 ([2 2 4 4 5])) %!assert (fillmissing (int32 ([1 2 3 4 5]), "nearest", 'missinglocations', logical([1 0 1 0 1])), int32 ([2 2 4 4 4])) %!assert (fillmissing (int32 ([1 2 3 4 5]), @(x,y,z) z+10, 3), int32 ([1 2 3 4 5])) %!assert (fillmissing (int32 ([1 2 3 4 5]), @(x,y,z) z+10, 3, 'missinglocations', logical([1 0 1 0 1])), int32 ([11 2 13 4 15])) %!assert (fillmissing (int32 ([1 2 3 4 5]), @(x,y,z) z+10, [2 0], 'missinglocations', logical([1 0 1 0 1])), int32 ([1 2 13 4 15])) %!test %! x = int32 ([1 2 3 4 5]); %! [~,idx] = fillmissing (x, "constant", 0); %! assert (idx, logical([0 0 0 0 0])); %! [~,idx] = fillmissing (x, "constant", 0, 'missinglocations', logical([1 0 1 0 1])); %! assert (idx, logical([1 0 1 0 1])); %! [~,idx] = fillmissing (x, "constant", 3, 'missinglocations', logical([0 0 1 0 0])); %! assert (idx, logical([0 0 1 0 0])); %! [~,idx] = fillmissing (x, "previous", 'missinglocations', logical([1 0 1 0 1])); %! assert (idx, logical([0 0 1 0 1])); %! [~,idx] = fillmissing (x, "next", 'missinglocations', logical([1 0 1 0 1])); %! assert (idx, logical([1 0 1 0 0])); %! [~,idx] = fillmissing (x, "nearest", 'missinglocations', logical([1 0 1 0 1])); %! assert (idx, logical([1 0 1 0 1])); %! [~,idx] = fillmissing (x, @(x,y,z) z+10, 3); %! assert (idx, logical([0 0 0 0 0])); %! [~,idx] = fillmissing (x, @(x,y,z) z+10, 3, 'missinglocations', logical([1 0 1 0 1])); %! assert (idx, logical([1 0 1 0 1])); %! [~,idx] = fillmissing (x, @(x,y,z) z+10, [2 0], 'missinglocations', logical([1 0 1 0 1])); %! assert (idx, logical([0 0 1 0 1])); ## other data type passthrough %!test %! [A, idx] = fillmissing ([struct struct], "constant", 1); %! assert (A, [struct struct]) %! assert (idx, [false false]) ## Test input validation and error messages %!error fillmissing () %!error fillmissing (1) %!error fillmissing (1,2,3,4,5,6,7,8,9,10,11,12,13) %!error fillmissing (1, 2) %!error fillmissing (1, "foo") %!error fillmissing (1, @(x) x, 1) %!error fillmissing (1, @(x,y) x+y, 1) %!error fillmissing ("a b c", "linear") %!error fillmissing ({'a','b'}, "linear") %!error <'movmean' and 'movmedian' methods only valid for numeric> fillmissing ("a b c", "movmean", 2) %!error <'movmean' and 'movmedian' methods only valid for numeric> fillmissing ({'a','b'}, "movmean", 2) %!error <'constant' method must be followed by> fillmissing (1, "constant") %!error fillmissing (1, "constant", []) %!error fillmissing (1, "constant", "a") %!error fillmissing ("a", "constant", 1) %!error fillmissing ("a", "constant", {"foo"}) %!error fillmissing ({"foo"}, "constant", 1) %!error fillmissing (1, "movmean") %!error fillmissing (1, "movmedian") %!error fillmissing (1, "constant", 1, 0) %!error fillmissing (1, "constant", 1, -1) %!error fillmissing (1, "constant", 1, [1 2]) %!error fillmissing (1, "constant", 1, "samplepoints") %!error fillmissing (1, "constant", 1, "foo") %!error fillmissing (1, "constant", 1, 1, "foo") %!error fillmissing (1, "constant", 1, 2, {1}, 4) %!error fillmissing ([1 2 3], "constant", 1, 2, "samplepoints", [1 2]) %!error fillmissing ([1 2 3], "constant", 1, 2, "samplepoints", [3 1 2]) %!error fillmissing ([1 2 3], "constant", 1, 2, "samplepoints", [1 1 2]) %!error fillmissing ([1 2 3], "constant", 1, 2, "samplepoints", "abc") %!error fillmissing ([1 2 3], "constant", 1, 2, "samplepoints", logical([1 1 1])) %!error fillmissing ([1 2 3], "constant", 1, 1, "samplepoints", [1 2 3]) %!error fillmissing ('foo', "next", "endvalues", 1) %!error fillmissing (1, "constant", 1, 1, "endvalues", "foo") %!error fillmissing ([1 2 3], "constant", 1, 2, "endvalues", [1 2 3]) %!error fillmissing ([1 2 3], "constant", 1, 1, "endvalues", [1 2]) %!error fillmissing (randi(5,4,3,2), "constant", 1, 3, "endvalues", [1 2]) %!error fillmissing (1, "constant", 1, 1, "endvalues", {1}) %!error fillmissing (1, "constant", 1, 2, "foo", 4) %!error fillmissing (struct, "constant", 1, "missinglocations", false) %!error fillmissing (1, "constant", 1, 2, "maxgap", 1, "missinglocations", false) %!error fillmissing (1, "constant", 1, 2, "missinglocations", false, "maxgap", 1) %!error fillmissing (1, "constant", 1, "replacevalues", true) %!error fillmissing (1, "constant", 1, "datavariables", 'Varname') %!error fillmissing (1, "constant", 1, 2, "missinglocations", 1) %!error fillmissing (1, "constant", 1, 2, "missinglocations", 'a') %!error fillmissing (1, "constant", 1, 2, "missinglocations", [true false]) %!error fillmissing (true, "linear", "missinglocations", true) %!error fillmissing (int8(1), "linear", "missinglocations", true) %!error fillmissing (true, "next", "missinglocations", true, "EndValues", "linear") %!error fillmissing (true, "next", "EndValues", "linear", "missinglocations", true) %!error fillmissing (int8(1), "next", "missinglocations", true, "EndValues", "linear") %!error fillmissing (int8(1), "next", "EndValues", "linear", "missinglocations", true) %!error fillmissing (1, "constant", 1, 2, "maxgap", true) %!error fillmissing (1, "constant", 1, 2, "maxgap", 'a') %!error fillmissing (1, "constant", 1, 2, "maxgap", [1 2]) %!error fillmissing (1, "constant", 1, 2, "maxgap", 0) %!error fillmissing (1, "constant", 1, 2, "maxgap", -1) %!error fillmissing ([1 2 3], "constant", [1 2 3]) %!error fillmissing ([1 2 3]', "constant", [1 2 3]) %!error fillmissing ([1 2 3]', "constant", [1 2 3], 1) %!error fillmissing ([1 2 3], "constant", [1 2 3], 2) %!error fillmissing (randi(5,4,3,2), "constant", [1 2], 1) %!error fillmissing (randi(5,4,3,2), "constant", [1 2], 2) %!error fillmissing (randi(5,4,3,2), "constant", [1 2], 3) %!error fillmissing (1, @(x,y,z) x+y+z) %!error fillmissing ([1 NaN 2], @(x,y,z) [1 2], 2) statistics-release-1.6.3/inst/fishertest.m000066400000000000000000000215401456127120000206440ustar00rootroot00000000000000## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/OR ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, OR (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{h} =} fishertest (@var{x}) ## @deftypefnx {statistics} {@var{h} =} fishertest (@var{x}, @var{param1}, @var{value1}, @dots{}) ## @deftypefnx {statistics} {[@var{h}, @var{pval}] =} fishertest (@dots{}) ## @deftypefnx {statistics} {[@var{h}, @var{pval}, @var{stats}] =} fishertest (@dots{}) ## ## Fisher's exact test. ## ## @code{@var{h} = fishertest (@var{x})} performs Fisher's exact test on a ## @math{2x2} contingency table given in matrix @var{x}. This is a test of the ## hypothesis that there are no non-random associations between the two 2-level ## categorical variables in @var{x}. @code{fishertest} returns the result of ## the tested hypothsis in @var{h}. @var{h} = 0 indicates that the null ## hypothesis (of no association) cannot be rejected at the 5% significance ## level. @var{h} = 1 indicates that the null hypothesis can be rejected at the ## 5% level. @var{x} must contain only non-negative integers. Use the ## @code{crostab} function to generate the contingency table from samples of two ## categorical variables. Fisher's exact test is not suitable when all integers ## in @var{x} are very large. Use can use the Chi-square test in this case. ## ## @code{[@var{h}, @var{pval}] = fishertest (@var{x})} returns the p-value in ## @var{pval}. That is the probability of observing the given result, or one ## more extreme, by chance if the null hypothesis is true. Small values of ## @var{pval} cast doubt on the validity of the null hypothesis. ## ## @code{[@var{p}, @var{pval}, @var{stats}] = fishertest (@dots{})} returns the ## structure @var{stats} with the following fields: ## ## @multitable @columnfractions 0.05 0.3 0.65 ## @item @tab @qcode{OddsRatio} @tab -- the odds ratio ## @item @tab @qcode{ConfidenceInterval} @tab -- the asymptotic confidence ## interval for the odds ratio. If any of the four entries in the contingency ## table @var{x} is zero, the confidence interval will not be computed, and ## @qcode{[-Inf Inf]} will be displayed. ## @end multitable ## ## @code{[@dots{}] = fishertest (@dots{}, @var{name}, @var{value}, @dots{})} ## specifies one or more of the following name/value pairs: ## ## @multitable @columnfractions 0.05 0.2 0.75 ## @headitem @tab Name @tab Value ## @item @tab @qcode{"alpha"} @tab the significance level. Default is 0.05. ## ## @item @tab @qcode{"tail"} @tab a string specifying the alternative hypothesis ## @end multitable ## @multitable @columnfractions 0.1 0.25 0.65 ## @item @tab @qcode{"both"} @tab odds ratio not equal to 1, indicating ## association between two variables (two-tailed test, default) ## @item @tab @qcode{"left"} @tab odds ratio greater than 1 (right-tailed test) ## @item @tab @qcode{"right"} @tab odds ratio is less than 1 (left-tailed test) ## @end multitable ## ## @seealso{crosstab, chi2test, mcnemar_test, ztest2} ## @end deftypefn function [h, p, stats] = fishertest (x, varargin) if (nargin < 1) error ("fishertest: contingency table is missing."); endif if (nargin > 5) error ("fishertest: too many input parameters."); endif ## Check contingency table if (! ismatrix (x) || ndims (x) != 2) error ("fishertest: X must be a 2-dimensional matrix."); endif if (any (x(:) < 0) || any (isnan (x(:))) || any (isinf (x(:))) || ... iscomplex (x) || any (fix (x(:)) != x(:))) error ("fishertest: X must contain only non-negative real integers."); endif if (all (x(:) >= 1e7)) error ("fishertest: cannot handle large entries (>=1e7)."); endif ## Add defaults and parse optional arguments alpha = 0.05; tail = "both"; if (nargin > 1) params = numel (varargin); if ((params / 2) != fix (params / 2)) error ("fishertest: optional arguments must be in Name-Value pairs.") endif for idx = 1:2:params name = varargin{idx}; value = varargin{idx+1}; switch (lower (name)) case "alpha" alpha = value; if (! isscalar (alpha) || ! isnumeric (alpha) || ... alpha <= 0 || alpha >= 1) error ("fishertest: invalid value for alpha."); endif case "tail" tail = value; if (! any (strcmpi (tail, {"both", "left", "right"}))) error ("fishertest: invalid value for tail."); endif otherwise error ("fishertest: invalid name for optional arguments."); endswitch endfor endif ## For 2x2 contigency table apply Fisher's exact test ## For larger tables apply the Fisher-Freeman-Halton variance if (all (size (x) == 2)) ## Get margin sums r1 = sum (x(1,:)); r2 = sum (x(2,:)); c1 = sum (x(:,1)); c2 = sum (x(:,2)); sz = sum (x(:)); ## Use try_catch block to avoid memory overflow for large numbers try if (strcmp (tail, "left")) p = hygecdf (x(1,1), sz, r1, c1); else if (min (r1, c1) <= min (r2, c2)) x11 = (0 : min (r1, c1))'; else x22 = (0 : min (r2, c2))'; x12 = c2 - x22; x11 = r1 - x12; endif switch tail case "both" p1 = hygepdf (x11, sz, r1, c1); p2 = hygepdf (x(1,1), sz, r1, c1); p = sum (p1(p1 < p2 + 10 * eps (p2))); case "right" xr = x11(x11 >= x(1,1)); p = sum(hygepdf(xr,sz,r1,c1)); endswitch endif catch error ("fishertest: cannot handle large entries."); end_try_catch ## Return test decision h = (p <= alpha); ## Calculate extra output arguments (if necessary) if (nargout > 2) OR = x(1,1) * x(2,2) / x(1,2) / x(2,1); if (any (x(:) == 0)) CI = [-Inf, Inf]; else SE = sqrt (1 / x(1,1) + 1 / x(1,2) + 1 / x(2,1) + 1 / x(2,2)); LB = OR * exp (-norminv (1 - alpha / 2) * SE); UB = OR * exp (norminv (1 - alpha / 2) * SE); CI = [LB, UB]; endif stats = struct ("OddsRatio", OR, "ConfidenceInterval", CI); endif else error ("fishertest: the Fisher-Freeman-Halton test is not imlemented yet."); endif endfunction %!demo %! ## A Fisher's exact test example %! %! x = [3, 1; 1, 3] %! [h, p, stats] = fishertest(x) ## Test output against MATLAB R2018 %!assert (fishertest ([3, 4; 5, 7]), false); %!assert (isa (fishertest ([3, 4; 5, 7]), "logical"), true); %!test %! [h, pval, stats] = fishertest ([3, 4; 5, 7]); %! assert (pval, 1, 1e-14); %! assert (stats.OddsRatio, 1.05); %! CI = [0.159222057151289, 6.92429189601808]; %! assert (stats.ConfidenceInterval, CI, 1e-14) %!test %! [h, pval, stats] = fishertest ([3, 4; 5, 0]); %! assert (pval, 0.08080808080808080, 1e-14); %! assert (stats.OddsRatio, 0); %! assert (stats.ConfidenceInterval, [-Inf, Inf]) ## Test input validation %!error fishertest (); %!error fishertest (1, 2, 3, 4, 5, 6); %!error ... %! fishertest (ones (2, 2, 2)); %!error ... %! fishertest ([1, 2; -3, 4]); %!error ... %! fishertest ([1, 2; 3, 4+i]); %!error ... %! fishertest ([1, 2; 3, 4.2]); %!error ... %! fishertest ([NaN, 2; 3, 4]); %!error ... %! fishertest ([1, Inf; 3, 4]); %!error ... %! fishertest (ones (2) * 1e8); %!error ... %! fishertest ([1, 2; 3, 4], "alpha", 0); %!error ... %! fishertest ([1, 2; 3, 4], "alpha", 1.2); %!error ... %! fishertest ([1, 2; 3, 4], "alpha", "val"); %!error ... %! fishertest ([1, 2; 3, 4], "tail", "val"); %!error ... %! fishertest ([1, 2; 3, 4], "alpha", 0.01, "tail", "val"); %!error ... %! fishertest ([1, 2; 3, 4], "alpha", 0.01, "badoption", 3); statistics-release-1.6.3/inst/fitcknn.m000066400000000000000000000431551456127120000201260ustar00rootroot00000000000000## Copyright (C) 2023 Mohammed Azmat Khan ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{obj} =} fitcknn (@var{X}, @var{Y}) ## @deftypefnx {statistics} {@var{obj} =} fitcknn (@dots{}, @var{name}, @var{value}) ## ## Fit a k-Nearest Neighbor classification model. ## ## @code{@var{obj} = fitcknn (@var{X}, @var{Y})} returns a k-Nearest Neighbor ## classification model, @var{obj}, with @var{X} being the predictor data, ## and @var{Y} the class labels of observations in @var{X}. ## ## @itemize ## @item ## @code{X} must be a @math{NxP} numeric matrix of input data where rows ## correspond to observations and columns correspond to features or variables. ## @var{X} will be used to train the kNN model. ## @item ## @code{Y} is @math{Nx1} matrix or cell matrix containing the class labels of ## corresponding predictor data in @var{X}. @var{Y} can contain any type of ## categorical data. @var{Y} must have same numbers of Rows as @var{X}. ## @item ## @end itemize ## ## @code{@var{obj} = fitcknn (@dots{}, @var{name}, @var{value})} returns a ## k-Nearest Neighbor classification model with additional options specified by ## @qcode{Name-Value} pair arguments listed below. ## ## @multitable @columnfractions 0.18 0.02 0.8 ## @headitem @tab @var{Name} @tab @var{Value} ## ## @item @qcode{"PredictorNames"} @tab @tab A cell array of character vectors ## specifying the predictor variable names. The variable names are assumed to ## be in the same order as they appear in the training data @var{X}. ## ## @item @qcode{"ResponseName"} @tab @tab A character vector specifying the name ## of the response variable. ## ## @item @qcode{"ClassNames"} @tab @tab A cell array of character vectors ## specifying the names of the classes in the training data @var{Y}. ## ## @item @qcode{"BreakTies"} @tab @tab Tie-breaking algorithm used by predict ## when multiple classes have the same smallest cost. By default, ties occur ## when multiple classes have the same number of nearest points among the ## @math{k} nearest neighbors. The available options are specified by the ## following character arrays: ## @end multitable ## ## @multitable @columnfractions 0.05 0.2 0.75 ## @item @tab @qcode{"smallest"} @tab This is the default and it favors the ## class with the smallest index among the tied groups, i.e. the one that ## appears first in the training labelled data. ## @item @tab @qcode{"nearest"} @tab This favors the class with the nearest ## neighbor among the tied groups, i.e. the class with the closest member point ## according to the distance metric used. ## @item @tab @qcode{"nearest"} @tab This randomly picks one class among the ## tied groups. ## @end multitable ## ## @multitable @columnfractions 0.18 0.02 0.8 ## @item @qcode{"BucketSize"} @tab @tab The maximum number of data points in the ## leaf node of the Kd-tree and it must be a positive integer. By default, it ## is 50. This argument is meaningful only when the selected search method is ## @qcode{"kdtree"}. ## ## @item @qcode{"Cost"} @tab @tab A @math{NxR} numeric matrix containing ## misclassification cost for the corresponding instances in @var{X} where ## @math{R} is the number of unique categories in @var{Y}. If an instance is ## correctly classified into its category the cost is calculated to be 1, If ## not then 0. cost matrix can be altered use @code{@var{obj.cost} = somecost}. ## default value @qcode{@var{cost} = ones(rows(X),numel(unique(Y)))}. ## ## @item @qcode{"Prior"} @tab @tab A numeric vector specifying the prior ## probabilities for each class. The order of the elements in @qcode{Prior} ## corresponds to the order of the classes in @qcode{ClassNames}. ## ## @item @qcode{"NumNeighbors"} @tab @tab A positive integer value specifying ## the number of nearest neighbors to be found in the kNN search. By default, ## it is 1. ## ## @item @qcode{"Exponent"} @tab @tab A positive scalar (usually an integer) ## specifying the Minkowski distance exponent. This argument is only valid when ## the selected distance metric is @qcode{"minkowski"}. By default it is 2. ## ## @item @qcode{"Scale"} @tab @tab A nonnegative numeric vector specifying the ## scale parameters for the standardized Euclidean distance. The vector length ## must be equal to the number of columns in @var{X}. This argument is only ## valid when the selected distance metric is @qcode{"seuclidean"}, in which ## case each coordinate of @var{X} is scaled by the corresponding element of ## @qcode{"scale"}, as is each query point in @var{Y}. By default, the scale ## parameter is the standard deviation of each coordinate in @var{X}. If a ## variable in @var{X} is constant, i.e. zero variance, this value is forced ## to 1 to avoid division by zero. This is the equivalent of this variable not ## being standardized. ## ## @item @qcode{"Cov"} @tab @tab A square matrix with the same number of columns ## as @var{X} specifying the covariance matrix for computing the mahalanobis ## distance. This must be a positive definite matrix matching. This argument ## is only valid when the selected distance metric is @qcode{"mahalanobis"}. ## ## @item @qcode{"Distance"} @tab @tab is the distance metric used by ## @code{knnsearch} as specified below: ## @end multitable ## ## @multitable @columnfractions 0.05 0.2 0.75 ## @item @tab @qcode{"euclidean"} @tab Euclidean distance. ## @item @tab @qcode{"seuclidean"} @tab standardized Euclidean distance. Each ## coordinate difference between the rows in @var{X} and the query matrix ## @var{Y} is scaled by dividing by the corresponding element of the standard ## deviation computed from @var{X}. To specify a different scaling, use the ## @qcode{"Scale"} name-value argument. ## @item @tab @qcode{"cityblock"} @tab City block distance. ## @item @tab @qcode{"chebychev"} @tab Chebychev distance (maximum coordinate ## difference). ## @item @tab @qcode{"minkowski"} @tab Minkowski distance. The default exponent ## is 2. To specify a different exponent, use the @qcode{"P"} name-value ## argument. ## @item @tab @qcode{"mahalanobis"} @tab Mahalanobis distance, computed using a ## positive definite covariance matrix. To change the value of the covariance ## matrix, use the @qcode{"Cov"} name-value argument. ## @item @tab @qcode{"cosine"} @tab Cosine distance. ## @item @tab @qcode{"correlation"} @tab One minus the sample linear correlation ## between observations (treated as sequences of values). ## @item @tab @qcode{"spearman"} @tab One minus the sample Spearman's rank ## correlation between observations (treated as sequences of values). ## @item @tab @qcode{"hamming"} @tab Hamming distance, which is the percentage ## of coordinates that differ. ## @item @tab @qcode{"jaccard"} @tab One minus the Jaccard coefficient, which is ## the percentage of nonzero coordinates that differ. ## @item @tab @var{@@distfun} @tab Custom distance function handle. A distance ## function of the form @code{function @var{D2} = distfun (@var{XI}, @var{YI})}, ## where @var{XI} is a @math{1xP} vector containing a single observation in ## @math{P}-dimensional space, @var{YI} is an @math{NxP} matrix containing an ## arbitrary number of observations in the same @math{P}-dimensional space, and ## @var{D2} is an @math{NxP} vector of distances, where @qcode{(@var{D2}k)} is ## the distance between observations @var{XI} and @qcode{(@var{YI}k,:)}. ## @end multitable ## ## @multitable @columnfractions 0.18 0.02 0.8 ## @item @qcode{"DistanceWeight"} @tab @tab A distance weighting function, ## specified either as a function handle, which accepts a matrix of nonnegative ## distances and returns a matrix the same size containing nonnegative distance ## weights, or one of the following values: @qcode{"equal"}, which corresponds ## to no weighting; @qcode{"inverse"}, which corresponds to a weight equal to ## @math{1/distance}; @qcode{"squaredinverse"}, which corresponds to a weight ## equal to @math{1/distance^2}. ## ## @item @qcode{"IncludeTies"} @tab @tab A boolean flag to indicate if the ## returned values should contain the indices that have same distance as the ## @math{K^th} neighbor. When @qcode{false}, @code{knnsearch} chooses the ## observation with the smallest index among the observations that have the same ## distance from a query point. When @qcode{true}, @code{knnsearch} includes ## all nearest neighbors whose distances are equal to the @math{K^th} smallest ## distance in the output arguments. To specify @math{K}, use the @qcode{"K"} ## name-value pair argument. ## ## @item @qcode{"NSMethod"} @tab @tab is the nearest neighbor search method used ## by @code{knnsearch} as specified below. ## @end multitable ## ## @multitable @columnfractions 0.05 0.2 0.75 ## @item @tab @qcode{"kdtree"} @tab Creates and uses a Kd-tree to find nearest ## neighbors. @qcode{"kdtree"} is the default value when the number of columns ## in @var{X} is less than or equal to 10, @var{X} is not sparse, and the ## distance metric is @qcode{"euclidean"}, @qcode{"cityblock"}, ## @qcode{"manhattan"}, @qcode{"chebychev"}, or @qcode{"minkowski"}. Otherwise, ## the default value is @qcode{"exhaustive"}. This argument is only valid when ## the distance metric is one of the four aforementioned metrics. ## @item @tab @qcode{"exhaustive"} @tab Uses the exhaustive search algorithm by ## computing the distance values from all the points in @var{X} to each point in ## @var{Y}. ## @end multitable ## ## @seealso{ClassificationKNN, knnsearch, rangesearch, pdist2} ## @end deftypefn function obj = fitcknn (X, Y, varargin) ## Check input parameters if (nargin < 2) error ("fitcknn: too few arguments."); endif if (mod (nargin, 2) != 0) error ("fitcknn: Name-Value arguments must be in pairs."); endif ## Check predictor data and labels have equal rows if (rows (X) != rows (Y)) error ("fitcknn: number of rows in X and Y must be equal."); endif ## Parse arguments to class def function obj = ClassificationKNN (X, Y, varargin{:}); endfunction ## Test Output %!test %! x = [1, 2, 3; 4, 5, 6; 7, 8, 9; 3, 2, 1]; %! y = ["a"; "a"; "b"; "b"]; %! a = fitcknn (x, y); %! assert (class (a), "ClassificationKNN"); %! assert ({a.X, a.Y, a.NumNeighbors}, {x, y, 1}) %! assert ({a.NSMethod, a.Distance}, {"kdtree", "euclidean"}) %! assert ({a.BucketSize}, {50}) %!test %! x = [1, 2, 3; 4, 5, 6; 7, 8, 9; 3, 2, 1]; %! y = ["a"; "a"; "b"; "b"]; %! a = fitcknn (x, y, "NSMethod", "exhaustive"); %! assert (class (a), "ClassificationKNN"); %! assert ({a.X, a.Y, a.NumNeighbors}, {x, y, 1}) %! assert ({a.NSMethod, a.Distance}, {"exhaustive", "euclidean"}) %! assert ({a.BucketSize}, {50}) %!test %! x = [1, 2, 3; 4, 5, 6; 7, 8, 9; 3, 2, 1]; %! y = ["a"; "a"; "b"; "b"]; %! k = 10; %! a = fitcknn (x, y, "NumNeighbors" ,k); %! assert (class (a), "ClassificationKNN"); %! assert ({a.X, a.Y, a.NumNeighbors}, {x, y, 10}) %! assert ({a.NSMethod, a.Distance}, {"kdtree", "euclidean"}) %! assert ({a.BucketSize}, {50}) %!test %! x = ones (4, 11); %! y = ["a"; "a"; "b"; "b"]; %! k = 10; %! a = fitcknn (x, y, "NumNeighbors" ,k); %! assert (class (a), "ClassificationKNN"); %! assert ({a.X, a.Y, a.NumNeighbors}, {x, y, 10}) %! assert ({a.NSMethod, a.Distance}, {"exhaustive", "euclidean"}) %! assert ({a.BucketSize}, {50}) %!test %! x = [1, 2, 3; 4, 5, 6; 7, 8, 9; 3, 2, 1]; %! y = ["a"; "a"; "b"; "b"]; %! k = 10; %! a = fitcknn (x, y, "NumNeighbors" ,k, "NSMethod", "exhaustive"); %! assert (class (a), "ClassificationKNN"); %! assert ({a.X, a.Y, a.NumNeighbors}, {x, y, 10}) %! assert ({a.NSMethod, a.Distance}, {"exhaustive", "euclidean"}) %! assert ({a.BucketSize}, {50}) %!test %! x = [1, 2, 3; 4, 5, 6; 7, 8, 9; 3, 2, 1]; %! y = ["a"; "a"; "b"; "b"]; %! k = 10; %! a = fitcknn (x, y, "NumNeighbors" ,k, "Distance", "hamming"); %! assert (class (a), "ClassificationKNN"); %! assert ({a.X, a.Y, a.NumNeighbors}, {x, y, 10}) %! assert ({a.NSMethod, a.Distance}, {"exhaustive", "hamming"}) %! assert ({a.BucketSize}, {50}) %!test %! x = [1, 2, 3; 4, 5, 6; 7, 8, 9; 3, 2, 1]; %! y = ["a"; "a"; "b"; "b"]; %! weights = ones (4,1); %! a = fitcknn (x, y, "Standardize", 1); %! assert (class (a), "ClassificationKNN"); %! assert ({a.X, a.Y, a.NumNeighbors}, {x, y, 1}) %! assert ({a.NSMethod, a.Distance}, {"kdtree", "euclidean"}) %! assert ({a.Standardize}, {true}) %! assert ({a.Sigma}, {std(x, [], 1)}) %! assert ({a.Mu}, {[3.75, 4.25, 4.75]}) %!test %! x = [1, 2, 3; 4, 5, 6; 7, 8, 9; 3, 2, 1]; %! y = ["a"; "a"; "b"; "b"]; %! weights = ones (4,1); %! a = fitcknn (x, y, "Standardize", false); %! assert (class (a), "ClassificationKNN"); %! assert ({a.X, a.Y, a.NumNeighbors}, {x, y, 1}) %! assert ({a.NSMethod, a.Distance}, {"kdtree", "euclidean"}) %! assert ({a.Standardize}, {false}) %! assert ({a.Sigma}, {[]}) %! assert ({a.Mu}, {[]}) %!test %! x = [1, 2, 3; 4, 5, 6; 7, 8, 9; 3, 2, 1]; %! y = ["a"; "a"; "b"; "b"]; %! s = ones (1, 3); %! a = fitcknn (x, y, "Scale" , s, "Distance", "seuclidean"); %! assert (class (a), "ClassificationKNN"); %! assert ({a.DistParameter}, {s}) %! assert ({a.NSMethod, a.Distance}, {"exhaustive", "seuclidean"}) %! assert ({a.BucketSize}, {50}) %!test %! x = [1, 2, 3; 4, 5, 6; 7, 8, 9; 3, 2, 1]; %! y = ["a"; "a"; "b"; "b"]; %! a = fitcknn (x, y, "Exponent" , 5, "Distance", "minkowski"); %! assert (class (a), "ClassificationKNN"); %! assert (a.DistParameter, 5) %! assert ({a.NSMethod, a.Distance}, {"kdtree", "minkowski"}) %!test %! x = [1, 2, 3; 4, 5, 6; 7, 8, 9; 3, 2, 1]; %! y = ["a"; "a"; "b"; "b"]; %! a = fitcknn (x, y, "Exponent" , 5, "Distance", "minkowski", ... %! "NSMethod", "exhaustive"); %! assert (class (a), "ClassificationKNN"); %! assert (a.DistParameter, 5) %! assert ({a.NSMethod, a.Distance}, {"exhaustive", "minkowski"}) %!test %! x = [1, 2, 3; 4, 5, 6; 7, 8, 9; 3, 2, 1]; %! y = ["a"; "a"; "b"; "b"]; %! a = fitcknn (x, y, "BucketSize" , 20, "distance", "mahalanobis"); %! assert (class (a), "ClassificationKNN"); %! assert ({a.NSMethod, a.Distance}, {"exhaustive", "mahalanobis"}) %! assert ({a.BucketSize}, {20}) %!test %! x = [1, 2, 3; 4, 5, 6; 7, 8, 9; 3, 2, 1]; %! y = ["a"; "a"; "b"; "b"]; %! a = fitcknn (x, y, "IncludeTies", true); %! assert (class (a), "ClassificationKNN"); %! assert (a.IncludeTies, true); %! assert ({a.NSMethod, a.Distance}, {"kdtree", "euclidean"}) %!test %! x = [1, 2, 3; 4, 5, 6; 7, 8, 9; 3, 2, 1]; %! y = ["a"; "a"; "b"; "b"]; %! a = fitcknn (x, y); %! assert (class (a), "ClassificationKNN"); %! assert (a.IncludeTies, false); %! assert ({a.NSMethod, a.Distance}, {"kdtree", "euclidean"}) %!test %! x = [1, 2, 3; 4, 5, 6; 7, 8, 9; 3, 2, 1]; %! y = ["a"; "a"; "b"; "b"]; %! a = fitcknn (x, y); %! assert (class (a), "ClassificationKNN") %! assert (a.Prior, [0.5; 0.5]) %! assert ({a.NSMethod, a.Distance}, {"kdtree", "euclidean"}) %! assert ({a.BucketSize}, {50}) %!test %! x = [1, 2, 3; 4, 5, 6; 7, 8, 9; 3, 2, 1]; %! y = ["a"; "a"; "b"; "b"]; %! prior = [0.5; 0.5]; %! a = fitcknn (x, y, "Prior", "empirical"); %! assert (class (a), "ClassificationKNN") %! assert (a.Prior, prior) %! assert ({a.NSMethod, a.Distance}, {"kdtree", "euclidean"}) %! assert ({a.BucketSize}, {50}) %!test %! x = [1, 2, 3; 4, 5, 6; 7, 8, 9; 3, 2, 1]; %! y = ["a"; "a"; "a"; "b"]; %! prior = [0.75; 0.25]; %! a = fitcknn (x, y, "Prior", "empirical"); %! assert (class (a), "ClassificationKNN") %! assert (a.Prior, prior) %! assert ({a.NSMethod, a.Distance}, {"kdtree", "euclidean"}) %! assert ({a.BucketSize}, {50}) %!test %! x = [1, 2, 3; 4, 5, 6; 7, 8, 9; 3, 2, 1]; %! y = ["a"; "a"; "a"; "b"]; %! prior = [0.5; 0.5]; %! a = fitcknn (x, y, "Prior", "uniform"); %! assert (class (a), "ClassificationKNN") %! assert (a.Prior, prior) %! assert ({a.NSMethod, a.Distance}, {"kdtree", "euclidean"}) %! assert ({a.BucketSize}, {50}) %!test %! x = [1, 2, 3; 4, 5, 6; 7, 8, 9; 3, 2, 1]; %! y = ["a"; "a"; "b"; "b"]; %! cost = eye (2); %! a = fitcknn (x, y, "Cost", cost); %! assert (class (a), "ClassificationKNN") %! assert (a.Cost, [1, 0; 0, 1]) %! assert ({a.NSMethod, a.Distance}, {"kdtree", "euclidean"}) %! assert ({a.BucketSize}, {50}) %!test %! x = [1, 2, 3; 4, 5, 6; 7, 8, 9; 3, 2, 1]; %! y = ["a"; "a"; "b"; "b"]; %! cost = eye (2); %! a = fitcknn (x, y, "Cost", cost, "Distance", "hamming" ); %! assert (class (a), "ClassificationKNN") %! assert (a.Cost, [1, 0; 0, 1]) %! assert ({a.NSMethod, a.Distance}, {"exhaustive", "hamming"}) %! assert ({a.BucketSize}, {50}) ## Test input validation %!error fitcknn () %!error fitcknn (ones (4,1)) %!error %! fitcknn (ones (4,2), ones (4, 1), "K") %!error %! fitcknn (ones (4,2), ones (3, 1)) %!error %! fitcknn (ones (4,2), ones (3, 1), "K", 2) statistics-release-1.6.3/inst/fitgmdist.m000066400000000000000000000455551456127120000204720ustar00rootroot00000000000000## Copyright (C) 2015 Lachlan Andrew ## Copyright (C) 2018 John Donoghue ## Copyright (C) 2022-2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{GMdist} =} fitgmdist (@var{data}, @var{k}, @var{param1}, @var{value1}, @dots{}) ## ## Fit a Gaussian mixture model with @var{k} components to @var{data}. ## Each row of @var{data} is a data sample. Each column is a variable. ## ## Optional parameters are: ## @itemize ## @item @qcode{"start"}: Initialization conditions. Possible values are: ## @itemize ## @item @qcode{"randSample"} (default) Takes means uniformly from rows of data. ## @item @qcode{"plus"} Use k-means++ to initialize means. ## @item @qcode{"cluster"} Performs an initial clustering with 10% of the data. ## @item @var{vector} A vector whose length is the number of rows in data, and ## whose values are 1 to k specify the components each row is initially ## allocated to. The mean, variance, and weight of each component is calculated ## from that. ## @item @var{structure} A structure with fields @qcode{mu}, @qcode{Sigma} and ## @qcode{ComponentProportion}. ## @end itemize ## For @qcode{"randSample"}, @qcode{"plus"}, and @qcode{"cluster"}, the initial ## variance of each component is the variance of the entire data sample. ## ## @item @qcode{"Replicates"}: Number of random restarts to perform. ## ## @item @qcode{"RegularizationValue"} or @qcode{"Regularize"}: A small number ## added to the diagonal entries of the covariance to prevent singular ## covariances. ## ## @item @qcode{"SharedCovariance"} or @qcode{"SharedCov"} (logical). True if ## all components must share the same variance, to reduce the number of free ## parameters ## ## @item @qcode{"CovarianceType"} or @qcode{"CovType"} (string). Possible values ## are: ## @itemize ## @item @qcode{"full"} (default) Allow arbitrary covariance matrices. ## @item @qcode{"diagonal"} Force covariances to be diagonal, to reduce the ## number of free parameters. ## @end itemize ## ## @item @qcode{"Options"}: A structure with all of the following fields: ## @itemize ## @item @qcode{MaxIter} Maximum number of EM iterations (default 100). ## @item @qcode{TolFun} Threshold increase in likelihood to terminate EM ## (default 1e-6). ## @item @qcode{Display} Possible values are: ## @itemize ## @item @qcode{"off"} (default): Display nothing. ## @item @qcode{"final"}: Display the total number of iterations and likelihood ## once the execution completes. ## @item @qcode{"iter"}: Display the number of iteration and likelihood after ## each iteration. ## @end itemize ## @end itemize ## @item @qcode{"Weight"}: A column vector or @math{Nx2} matrix. The first ## column consists of non-negative weights given to the samples. If these are ## all integers, this is equivalent to specifying @qcode{@var{weight}(i)} copies ## of row @qcode{i} of @var{data}, but potentially faster. If a row of ## @var{data} is used to represent samples that are similar but not identical, ## then the second column of @var{weight} indicates the variance of those ## original samples. Specifically, in the EM algorithm, the contribution of row ## @qcode{i} towards the variance is set to at least @qcode{@var{weight}(i,2)}, ## to prevent spurious components with zero variance. ## @end itemize ## ## @seealso{gmdistribution, kmeans} ## @end deftypefn function obj = fitgmdist (data, k, varargin) if (nargin < 2 || mod (nargin, 2) == 1) print_usage; endif [~, prop] = parseparams (varargin); ## defaults for options diagonalCovar = false; # "full". (true is "diagonal") sharedCovar = false; start = "randSample"; replicates = 1; option.MaxIter = 100; option.TolFun = 1e-6; option.Display = "off"; # "off" (1 is "final", 2 is "iter") Regularizer = 0; weights = []; # Each row i counts as "weights(i,1)" rows ## Remove rows containing NaN / NA data = data(! any (isnan (data), 2), :); ## Used for getting the number of samples nRows = rows (data); nCols = columns (data); ## Parse options while (! isempty (prop)) try switch (lower (prop{1})) case {"sharedcovariance", "sharedcov"} sharedCovar = prop{2}; case {"covariancetype", "covartype"} diagonalCovar = prop{2}; case {"regularizationvalue", "regularize"} Regularizer = prop{2}; case "replicates" replicates = prop{2}; case "start" start = prop{2}; case "weights" weights = prop{2}; case "options" option.MaxIter = prop{2}.MaxIter; option.TolFun = prop{2}.TolFun; option.Display = prop{2}.Display; otherwise error ("fitgmdist: Unknown option %s.", prop{1}); endswitch catch ME if (length (prop) < 2) error ("fitgmdist: Option '%s' has no argument.", prop{1}); else rethrow (ME) endif end_try_catch prop = prop(3:end); endwhile ## Process options ## Check for the "replicates" property try if isempty (1:replicates) error ("fitgmdist: replicates must be positive."); endif catch error ("fitgmdist: invalid number of replicates."); end_try_catch ## check for the "option" property MaxIter = option.MaxIter; TolFun = option.TolFun; switch (lower (option.Display)) case "off" Display = 0; case "final" Display = 1; case "iter" Display = 2; case "notify" Display = 0; otherwise error ("fitgmdist: Unknown Display option %s.", option.Display); endswitch try p = ones(1, k) / k; # Default is uniform component proportions catch ME if (! isscalar (k) || ! isnumeric (k)) error ("fitgmdist: The second argument must be a numeric scalar."); else rethrow (ME) endif end_try_catch ## Check for the "start" property if (ischar (start)) start = lower (start); switch (start) case {"randsample", "plus", "cluster", "randsamplep", "plusp", "clusterp"} otherwise error ("fitgmdist: Unknown Start value %s\n.", start); endswitch component_order_free = true; else component_order_free = false; if (! ismatrix (start) || ! isnumeric (start)) try mu = start.mu; Sigma = start.Sigma; if (isfield (start, 'ComponentProportion')) p = start.ComponentProportion(:)'; end if (any (size (data, 2) != [size(mu, 2), size(Sigma, 1)]) || ... any (k != [size(mu,1), size(p,2)])) error ("fitgmdist: Start parameter has mismatched dimensions."); endif catch error ("fitgmdist: invalid start parameter."); end_try_catch else validIndices = 0; mu = zeros (k, nRows); Sigma = zeros (nRows, nRows, k); for i = 1:k idx = (start == i); validIndices = validIndices + sum (idx); mu(i,:) = mean (data(idx,:)); Sigma(:,:,i) = cov (data(idx,:)) + Regularizer * eye (nCols); endfor if (validIndices < nRows) error (strcat (["fitgmdist: Start is numeric, but is not"], ... [" integers between 1 and k."])); endif endif start = []; # so that variance isn't recalculated later replicates = 1; # Will be the same each time anyway endif ## Check for the "SharedCovariance" property if (! islogical (sharedCovar)) error ("fitgmdist: SharedCoveriance must be logical true or false."); endif ## Check for the "CovarianceType" property if (! islogical (diagonalCovar)) try if (strcmpi (diagonalCovar, "diagonal")) diagonalCovar = true; elseif (strcmpi (diagonalCovar, "full")) diagonalCovar = false; else error ("fitgmdist: CovarianceType must be Full or Diagonal."); endif catch error ("fitgmdist: CovarianceType must be 'Full' or 'Diagonal'."); end_try_catch endif ## Check for the "Regularizer" property try if (Regularizer < 0) error ("fitgmdist: Regularizer must be non-negative"); endif catch ME if (! isscalar (Regularizer) || ! isnumeric (Regularizer)) error ("fitgmdist: Regularizer must be a numeric scalar"); else rethrow (ME) endif end_try_catch ## Check for the "Weights" property and the matrix try if (! isempty (weights)) if (columns (weights) > 2 || any (weights(:) < 0)) error (strcat (["fitgmdist: weights must be a nonnegative"], ... [" numeric dx1 or dx2 matrix."])); endif if (rows (weights) != nRows) error (strcat (["fitgmdist: number of weights %d must match"], ... [" number of samples %d."]), rows (weights), nRows); endif non_zero = (weights(:,1) > 0); weights = weights(non_zero,:); data = data (non_zero,:); nRows = rows (data); raw_samples = sum (weights(:,1)); else raw_samples = nRows; endif ## Validate the matrix if (! isreal (data(k,1))) error ("fitgmdist: first input argument must be a DxN real data matrix."); endif catch ME if (! isnumeric (data) || ! ismatrix (data) || ! isreal (data)) error ("fitgmdist: first input argument must be a DxN real data matrix."); elseif (k > nRows || k < 0) if (exists ("non_zero", "var") && k <= length (non_zero)) error (strcat (["fitgmdist: The number of non-zero weights (%d)"], ... [" must be at least the number of components"], ... [" (%d)."]), nRows, k); else error (strcat (["fitgmdist: The number of components (%d) must be"], ... [" a positive number less than the number of data"], ... [" rows (%d)."]), k, nRows); endif elseif (! ismatrix (weights) || ! isnumeric (weights)) error (strcat (["fitgmdist: weights must be a nonnegative numeric"], ... [" dx1 or dx2 matrix."])); else rethrow (ME) endif end_try_catch ## Done processing options ####################################### ## #sed to hold the probability of each class, given each data vector try p_x_l = zeros (nRows, k); # probability of observation x given class l best = -realmax; best_params = []; diag_slice = 1:(nCols+1):(nCols)^2; ## Create index slices to calculate symmetric ## completion of upper triangular Mx lower_half = zeros (nCols * (nCols - 1) / 2, 1); upper_half = zeros (nCols * (nCols - 1) / 2, 1); i = 1; for rw = 1:nCols for cl = rw+1:nCols upper_half(i) = sub2ind ([nCols, nCols], rw, cl); lower_half(i) = sub2ind ([nCols, nCols], cl, rw); i = i + 1; endfor endfor for rep = 1:replicates if (! isempty (start)) ## Initialize the means switch (start) case {"randsample"} if (isempty (weights)) idx = randperm (nRows, k); else idx = randsample (nRows, k, false, weights); endif mu = data(idx, :); case {"plus"} # k-means++, by Arthur and Vassilios mu(1,:) = data(randi (nRows),:); d = inf (nRows, 1); # Distance to nearest centroid so far for i = 2:k d = min (d, sum (bsxfun (@minus, data, mu(i-1, :)).^2, 2)); # pick next sample with prob. prop to dist.*weights if (isempty (weights)) cs = cumsum (d); else cs = cumsum (d .* weights(:,1)); endif mu(i,:) = data(find (cs > rand * cs(end), 1), :); endfor case {"cluster"} subsamp = max (k, ceil (nRows/10)); if (isempty (weights)) idx = randperm (nRows, subsamp); else idx = randsample (nRows, subsamp, false, weights); endif [~, mu] = kmeans (data(idx), k, "start", "sample"); endswitch ## Initialize the variance, unless set explicitly Sigma = var (data) + Regularizer; if (! diagonalCovar) Sigma = diag (Sigma); endif if (! sharedCovar) Sigma = repmat (Sigma, [1, 1, k]); endif endif ## Run the algorithm iter = 1; log_likeli = -inf; incr = 1; while (incr > TolFun && iter <= MaxIter) iter = iter + 1; ####################################### ## "E step" ## Calculate probability of class l given observations for i = 1:k if (sharedCovar) sig = Sigma; else sig = Sigma(:,:,i); endif if (diagonalCovar) sig = diag(sig); endif try p_x_l (:, i) = mvnpdf (data, mu(i, :), sig); catch ME if (strfind (ME.message, "positive definite")) error (strcat (["fitgmdist: Covariance is not positive"], ... [" definite. Increase RegularizationValue."])); else rethrow (ME) endif end_try_catch endfor ## Bayes' rule p_x_l = bsxfun (@times, p_x_l, p); # weight by priors p_l_x = bsxfun (@rdivide, p_x_l, sum (p_x_l, 2)); # Normalize ####################################### ## "M step" ## Calculate new parameters if (! isempty (weights)) p_l_x = bsxfun (@times, p_l_x, weights(:,1)); endif sum_p_l_x = sum (p_l_x); # row vec of \sum_{data} p(class|data,params) p = sum_p_l_x / raw_samples; # new proportions mu = bsxfun (@rdivide, p_l_x' * data, sum_p_l_x'); # new means if (sharedCovar) sumSigma = zeros (size (Sigma(:,:,1))); # diagonalCovar gives size endif for i = 1:k ## Sigma deviation = bsxfun(@minus, data, mu(i,:)); lhs = bsxfun(@times, p_l_x(:,i), deviation); ## Calculate covariance ## Iterate either over elements of the covariance matrix, ## since there should be fewer of those than rows of data. for rw = 1:nCols for cl = rw:nCols sig(rw,cl) = lhs(:,rw)' * deviation(:,cl); endfor endfor sig(lower_half) = sig(upper_half); sig = sig/sum_p_l_x(i) + Regularizer*eye (nCols); if (columns (weights) > 1) # don't give "singleton" clusters low var sig(diag_slice) = max (sig(diag_slice), weights(i,2)); endif if (diagonalCovar) sig = diag(sig)'; endif if (sharedCovar) sumSigma = sumSigma + sig * p(i); # Heuristic. Should it use else # old p? Something else? Sigma(:,:,i) = sig; endif endfor if (sharedCovar) Sigma = sumSigma; endif ####################################### ## Calculate the new (and relative change in) log-likelihood if (isempty (weights)) new_log_likeli = sum (log (sum (p_x_l, 2))); else new_log_likeli = sum (weights(:,1) .* log (sum (p_x_l, 2))); endif incr = (new_log_likeli - log_likeli)/max(1,abs(new_log_likeli)); if (Display == 2) fprintf("iter %d log-likelihood %g\n", iter-1, new_log_likeli); endif log_likeli = new_log_likeli; endwhile if (log_likeli > best) best = log_likeli; best_params.mu = mu; best_params.Sigma = Sigma; best_params.p = p; endif endfor catch ME try if (1 < MaxIter), end catch error ("fitgmdist: invalid MaxIter."); end_try_catch rethrow (ME) end_try_catch ## List components in descending order of proportion, ## unless the order was implicitly specified by "start" if (component_order_free) [~, idx] = sort (-best_params.p); best_params.p = best_params.p (idx); best_params.mu = best_params.mu(idx,:); if (! sharedCovar) best_params.Sigma = best_params.Sigma(:,:,idx); endif endif ## Calculate number of parameters if (diagonalCovar) params = nCols; else params = nCols * (nCols+1) / 2; endif params = params*size (Sigma, 3) + 2*rows (mu) - 1; ## This works in Octave, but not in Matlab #obj = gmdistribution (best_params.mu, best_params.Sigma, best_params.p', extra); obj = gmdistribution (best_params.mu, best_params.Sigma, best_params.p'); obj.NegativeLogLikelihood = -best; obj.AIC = -2*(best - params); obj.BIC = -2*best + params * log (raw_samples); obj.Converged = (incr <= TolFun); obj.NumIterations = iter-1; obj.RegularizationValue = Regularizer; if (Display == 1) fprintf (" %d iterations log-likelihood = %g\n", ... obj.NumIterations, -obj.NegativeLogLikelihood); endif endfunction %!demo %! ## Generate a two-cluster problem %! C1 = randn (100, 2) + 2; %! C2 = randn (100, 2) - 2; %! data = [C1; C2]; %! %! ## Perform clustering %! GMModel = fitgmdist (data, 2); %! %! ## Plot the result %! figure %! [heights, bins] = hist3([C1; C2]); %! [xx, yy] = meshgrid(bins{1}, bins{2}); %! bbins = [xx(:), yy(:)]; %! contour (reshape (GMModel.pdf (bbins), size (heights))); %!demo %! Angle_Theta = [ 30 + 10 * randn(1, 10), 60 + 10 * randn(1, 10) ]'; %! nbOrientations = 2; %! initial_orientations = [38.0; 18.0]; %! initial_weights = ones (1, nbOrientations) / nbOrientations; %! initial_Sigma = 10 * ones (1, 1, nbOrientations); %! start = struct ("mu", initial_orientations, "Sigma", initial_Sigma, ... %! "ComponentProportion", initial_weights); %! GMModel_Theta = fitgmdist (Angle_Theta, nbOrientations, "Start", start , ... %! "RegularizationValue", 0.0001) ## Test results against MATLAB example %!test %! load fisheriris %! classes = unique (species); %! [~, score] = pca (meas, "NumComponents", 2); %! options.MaxIter = 1000; %! options.TolFun = 1e-6; %! options.Display = "off"; %! GMModel = fitgmdist (score, 2, "Options", options); %! assert (isa (GMModel, "gmdistribution"), true); %! assert (GMModel.mu, [1.3212, -0.0954; -2.6424, 0.1909], 1e-4); statistics-release-1.6.3/inst/fitlm.m000066400000000000000000000404371456127120000176050ustar00rootroot00000000000000## Copyright (C) 2022 Andrew Penn ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{tab} =} fitlm (@var{X}, @var{y}) ## @deftypefnx {statistics} {@var{tab} =} fitlm (@var{X}, @var{y}, @var{name}, @var{value}) ## @deftypefnx {statistics} {@var{tab} =} fitlm (@var{X}, @var{y}, @var{modelspec}) ## @deftypefnx {statistics} {@var{tab} =} fitlm (@var{X}, @var{y}, @var{modelspec}, @var{name}, @var{value}) ## @deftypefnx {statistics} {[@var{tab}] =} fitlm (@dots{}) ## @deftypefnx {statistics} {[@var{tab}, @var{stats}] =} fitlm (@dots{}) ## @deftypefnx {statistics} {[@var{tab}, @var{stats}] =} fitlm (@dots{}) ## ## Regress the continuous outcome (i.e. dependent variable) @var{y} on ## continuous or categorical predictors (i.e. independent variables) @var{X} ## by minimizing the sum-of-squared residuals. Unless requested otherwise, ## @qcode{fitlm} prints the model formula, the regression coefficients (i.e. ## parameters/contrasts) and an ANOVA table. Note that unlike @qcode{anovan}, ## @qcode{fitlm} treats all factors as continuous by default. A bootstrap ## resampling variant of this function, @code{bootlm}, is available in the ## statistics-resampling package and has similar usage. ## ## @var{X} must be a column major matrix or cell array consisting of the ## predictors. A constant term (intercept) should not be included in X - it ## is automatically added to the model. @var{y} must be a column vector ## corresponding to the outcome variable. @var{modelspec} can specified as ## one of the following: ## ## @itemize ## @item ## "constant" : model contains only a constant (intercept) term. ## ## @item ## "linear" (default) : model contains an intercept and linear term for each ## predictor. ## ## @item ## "interactions" : model contains an intercept, linear term for each predictor ## and all products of pairs of distinct predictors. ## ## @item ## "full" : model contains an intercept, linear term for each predictor and ## all combinations of the predictors. ## ## @item ## a matrix of term definitions : an t-by-(N+1) matrix specifying terms in ## a model, where t is the number of terms, N is the number of predictor ## variables, and +1 accounts for the outcome variable. The outcome variable ## is the last column in the terms matrix and must be a column of zeros. ## An intercept must be specified in the first row of the terms matrix and ## must be a row of zeros. ## @end itemize ## ## @qcode{fitlm} can take a number of optional parameters as name-value pairs. ## ## @code{[@dots{}] = fitlm (..., "CategoricalVars", @var{categorical})} ## ## @itemize ## @item ## @var{categorical} is a vector of indices indicating which of the columns ## (i.e. variables) in @var{X} should be treated as categorical predictors ## rather than as continuous predictors. ## @end itemize ## ## @qcode{fitlm} also accepts optional @qcode{anovan} parameters as name-value ## pairs (except for the "model" parameter). The accepted parameter names from ## @qcode{anovan} and their default values in @qcode{fitlm} are: ## ## @itemize ## @item ## @var{CONTRASTS} : "treatment" ## ## @item ## @var{SSTYPE}: 2 ## ## @item ## @var{ALPHA}: 0.05 ## ## @item ## @var{DISPLAY}: "on" ## ## @item ## @var{WEIGHTS}: [] (empty) ## ## @item ## @var{RANDOM}: [] (empty) ## ## @item ## @var{CONTINUOUS}: [1:N] ## ## @item ## @var{VARNAMES}: [] (empty) ## @end itemize ## ## Type '@qcode{help anovan}' to find out more about what these options do. ## ## @qcode{fitlm} can return up to two output arguments: ## ## [@var{tab}] = fitlm (@dots{}) returns a cell array containing a ## table of model parameters ## ## [@var{tab}, @var{stats}] = fitlm (@dots{}) returns a structure ## containing additional statistics, including degrees of freedom and effect ## sizes for each term in the linear model, the design matrix, the ## variance-covariance matrix, (weighted) model residuals, and the mean squared ## error. The columns of @var{stats}.coeffs (from left-to-right) report the ## model coefficients, standard errors, lower and upper 100*(1-alpha)% ## confidence interval bounds, t-statistics, and p-values relating to the ## contrasts. The number appended to each term name in @var{stats}.coeffnames ## corresponds to the column number in the relevant contrast matrix for that ## factor. The @var{stats} structure can be used as input for @qcode{multcompare}. ## Note that if the model contains a continuous variable and you wish to use ## the @var{STATS} output as input to @qcode{multcompare}, then the model needs ## to be refit with the "contrast" parameter set to a sum-to-zero contrast ## coding scheme, e.g."simple". ## ## @seealso{anovan, multcompare} ## @end deftypefn function [T, STATS] = fitlm (X, y, varargin) ## Check input and output arguments if (nargin < 2) error (strcat (["fitlm usage: ""fitlm (X, y, varargin)""; "], ... [" atleast 2 input arguments required"])); endif if (nargout > 3) error ("fitlm: invalid number of output arguments requested"); endif ## Evaluate input data [n, N] = size (X); msg = strcat (["fitlm: do not include the intercept column in X"], ... [" - it will be added automatically"]); if (iscell (X)) if (~ iscell (X{:,1})) if (all (X{:,1} == 1)) error (msg) endif endif else if (all (X(:,1) == 1)) error (msg) endif endif ## Fetch anovan options options = varargin; if (isempty(options)) options{1} = "linear"; endif ## Check if MODELSPEC was provided. If not create it. if (ischar (options{1})) if (! ismember (lower (options{1}), {"sstype", "varnames", "contrasts", ... "weights", "alpha", "display", "continuous", ... "categorical", "categoricalvars", "random", "model"})) MODELSPEC = options{1}; options(1) = []; CONTINUOUS = []; else ## If MODELSPEC is not provided, set it for an additive linear model MODELSPEC = zeros (N + 1); MODELSPEC(2:N+1, 1:N) = eye (N); end else MODELSPEC = options{1}; options(1) = []; endif ## Evaluate MODELSPEC if (ischar (MODELSPEC)) MODELSPEC = lower (MODELSPEC); if (! isempty (regexp (MODELSPEC, "~"))) error ("fitlm: model formulae are not a supported format for MODELSPEC") endif if (! ismember (MODELSPEC, {"constant", "linear", "interaction", ... "interactions", "full"})) error ("fitlm: character vector for model specification not recognised") endif if strcmp (MODELSPEC, "constant") X = []; MODELSPEC = "linear"; N = 0; endif else if (size (MODELSPEC, 1) < N + 1) error ("fitlm: number of rows in MODELSPEC must 1 + number of columns in X"); endif if (size (MODELSPEC, 2) != N + 1) error ("fitlm: number of columns in MODELSPEC must = 1 + number of columns in X"); endif if (! all (ismember (MODELSPEC(:), [0,1]))) error (strcat (["fitlm: elements of the model terms matrix must be "], ... [" either 0 or 1. Higher order terms are not supported"])); endif MODELSPEC = logical (MODELSPEC(2:N+1,1:N)); endif ## Check for unsupported options used by anovan if (any (strcmpi ("MODEL", options))) error (strcat(["fitlm: modelspec should be specified in the third"], ... [" input argument of fitlm (if at all)"])); endif ## Check and set variable types idx = find (any (cat (1, strcmpi ("categorical", options), ... strcmpi ("categoricalvars", options)))); if (! isempty (idx)) CONTINUOUS = [1:N]; CONTINUOUS(ismember(CONTINUOUS,options{idx+1})) = []; options(idx:idx+1) = []; else CONTINUOUS = [1:N]; endif idx = find (strcmpi ("continuous", options)); if (! isempty (idx)) ## Note that setting continuous parameter will override settings made ## to "categorical" CONTINUOUS = options{idx+1}; endif ## Check if anovan CONTRASTS option was used idx = find (strcmpi ("contrasts", options)); if (isempty (idx)) CONTRASTS = "treatment"; else CONTRASTS = options{idx+1}; if (ischar(CONTRASTS)) contr_str = CONTRASTS; CONTRASTS = cell (1, N); CONTRASTS(:) = {contr_str}; endif if (! iscell (CONTRASTS)) CONTRASTS = {CONTRASTS}; endif for i = 1:N if (! isnumeric(CONTRASTS{i})) if (! isempty (CONTRASTS{i})) if (! ismember (CONTRASTS{i}, ... {"simple","poly","helmert","effect","treatment"})) error (strcat(["fitlm: the choices for built-in contrasts are"], ... [" ""simple"", ""poly"", ""helmert"", ""effect"", or ""treatment"""])); endif endif endif endfor endif ## Check if anovan SSTYPE option was used idx = find (strcmpi ("sstype", options)); if (isempty (idx)) SSTYPE = 2; else SSTYPE = options{idx+1}; endif ## Perform model fit and ANOVA [jnk, jnk, STATS] = anovan (y, X, options{:}, ... "model", MODELSPEC, ... "contrasts", CONTRASTS, ... "continuous", CONTINUOUS, ... "sstype", SSTYPE); ## Create table of regression coefficients ncoeff = sum (STATS.df); T = cell (2 + ncoeff, 7); T(1,:) = {"Parameter", "Estimate", "SE", "Lower.CI", "Upper.CI", "t", "Prob>|t|"}; T(2:end,1) = STATS.coeffnames; T(2:end,2:7) = num2cell (STATS.coeffs); ## Update STATS structure STATS.source = "fitlm"; endfunction %!demo %! y = [ 8.706 10.362 11.552 6.941 10.983 10.092 6.421 14.943 15.931 ... %! 22.968 18.590 16.567 15.944 21.637 14.492 17.965 18.851 22.891 ... %! 22.028 16.884 17.252 18.325 25.435 19.141 21.238 22.196 18.038 ... %! 22.628 31.163 26.053 24.419 32.145 28.966 30.207 29.142 33.212 ... %! 25.694 ]'; %! X = [1 1 1 1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5]'; %! %! [TAB,STATS] = fitlm (X,y,"linear","CategoricalVars",1,"display","on"); %!demo %! popcorn = [5.5, 4.5, 3.5; 5.5, 4.5, 4.0; 6.0, 4.0, 3.0; ... %! 6.5, 5.0, 4.0; 7.0, 5.5, 5.0; 7.0, 5.0, 4.5]; %! brands = {'Gourmet', 'National', 'Generic'; ... %! 'Gourmet', 'National', 'Generic'; ... %! 'Gourmet', 'National', 'Generic'; ... %! 'Gourmet', 'National', 'Generic'; ... %! 'Gourmet', 'National', 'Generic'; ... %! 'Gourmet', 'National', 'Generic'}; %! popper = {'oil', 'oil', 'oil'; 'oil', 'oil', 'oil'; 'oil', 'oil', 'oil'; ... %! 'air', 'air', 'air'; 'air', 'air', 'air'; 'air', 'air', 'air'}; %! %! [TAB, STATS] = fitlm ({brands(:),popper(:)},popcorn(:),"interactions",... %! "CategoricalVars",[1,2],"display","on"); %!test %! y = [ 8.706 10.362 11.552 6.941 10.983 10.092 6.421 14.943 15.931 ... %! 22.968 18.590 16.567 15.944 21.637 14.492 17.965 18.851 22.891 ... %! 22.028 16.884 17.252 18.325 25.435 19.141 21.238 22.196 18.038 ... %! 22.628 31.163 26.053 24.419 32.145 28.966 30.207 29.142 33.212 ... %! 25.694 ]'; %! X = [1 1 1 1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5]'; %! [TAB,STATS] = fitlm (X,y,"continuous",[],"display","off"); %! [TAB,STATS] = fitlm (X,y,"CategoricalVars",1,"display","off"); %! [TAB,STATS] = fitlm (X,y,"constant","categorical",1,"display","off"); %! [TAB,STATS] = fitlm (X,y,"linear","categorical",1,"display","off"); %! [TAB,STATS] = fitlm (X,y,[0,0;1,0],"categorical",1,"display","off"); %! assert (TAB{2,2}, 10, 1e-04); %! assert (TAB{3,2}, 7.99999999999999, 1e-09); %! assert (TAB{4,2}, 8.99999999999999, 1e-09); %! assert (TAB{5,2}, 11.0001428571429, 1e-09); %! assert (TAB{6,2}, 19.0001111111111, 1e-09); %! assert (TAB{2,3}, 1.01775379540949, 1e-09); %! assert (TAB{3,3}, 1.64107868458008, 1e-09); %! assert (TAB{4,3}, 1.43932122062479, 1e-09); %! assert (TAB{5,3}, 1.48983900477565, 1e-09); %! assert (TAB{6,3}, 1.3987687997822, 1e-09); %! assert (TAB{2,6}, 9.82555903510687, 1e-09); %! assert (TAB{3,6}, 4.87484242844031, 1e-09); %! assert (TAB{4,6}, 6.25294748040552, 1e-09); %! assert (TAB{5,6}, 7.38344399756088, 1e-09); %! assert (TAB{6,6}, 13.5834536158296, 1e-09); %! assert (TAB{3,7}, 2.85812420217862e-05, 1e-12); %! assert (TAB{4,7}, 5.22936741204002e-07, 1e-06); %! assert (TAB{5,7}, 2.12794763209106e-08, 1e-07); %! assert (TAB{6,7}, 7.82091664406755e-15, 1e-08); %!test %! popcorn = [5.5, 4.5, 3.5; 5.5, 4.5, 4.0; 6.0, 4.0, 3.0; ... %! 6.5, 5.0, 4.0; 7.0, 5.5, 5.0; 7.0, 5.0, 4.5]; %! brands = bsxfun (@times, ones(6,1), [1,2,3]); %! popper = bsxfun (@times, [1;1;1;2;2;2], ones(1,3)); %! %! [TAB, STATS] = fitlm ({brands(:),popper(:)},popcorn(:),"interactions",... %! "categoricalvars",[1,2],"display","off"); %! assert (TAB{2,2}, 5.66666666666667, 1e-09); %! assert (TAB{3,2}, -1.33333333333333, 1e-09); %! assert (TAB{4,2}, -2.16666666666667, 1e-09); %! assert (TAB{5,2}, 1.16666666666667, 1e-09); %! assert (TAB{6,2}, -0.333333333333334, 1e-09); %! assert (TAB{7,2}, -0.166666666666667, 1e-09); %! assert (TAB{2,3}, 0.215165741455965, 1e-09); %! assert (TAB{3,3}, 0.304290309725089, 1e-09); %! assert (TAB{4,3}, 0.304290309725089, 1e-09); %! assert (TAB{5,3}, 0.304290309725089, 1e-09); %! assert (TAB{6,3}, 0.43033148291193, 1e-09); %! assert (TAB{7,3}, 0.43033148291193, 1e-09); %! assert (TAB{2,6}, 26.3362867542108, 1e-09); %! assert (TAB{3,6}, -4.38178046004138, 1e-09); %! assert (TAB{4,6}, -7.12039324756724, 1e-09); %! assert (TAB{5,6}, 3.83405790253621, 1e-09); %! assert (TAB{6,6}, -0.774596669241495, 1e-09); %! assert (TAB{7,6}, -0.387298334620748, 1e-09); %! assert (TAB{2,7}, 5.49841502258254e-12, 1e-09); %! assert (TAB{3,7}, 0.000893505495903642, 1e-09); %! assert (TAB{4,7}, 1.21291454302428e-05, 1e-09); %! assert (TAB{5,7}, 0.00237798044119407, 1e-09); %! assert (TAB{6,7}, 0.453570536021938, 1e-09); %! assert (TAB{7,7}, 0.705316781644046, 1e-09); %! ## Test with string ids for categorical variables %! brands = {'Gourmet', 'National', 'Generic'; ... %! 'Gourmet', 'National', 'Generic'; ... %! 'Gourmet', 'National', 'Generic'; ... %! 'Gourmet', 'National', 'Generic'; ... %! 'Gourmet', 'National', 'Generic'; ... %! 'Gourmet', 'National', 'Generic'}; %! popper = {'oil', 'oil', 'oil'; 'oil', 'oil', 'oil'; 'oil', 'oil', 'oil'; ... %! 'air', 'air', 'air'; 'air', 'air', 'air'; 'air', 'air', 'air'}; %! [TAB, STATS] = fitlm ({brands(:),popper(:)},popcorn(:),"interactions",... %! "categoricalvars",[1,2],"display","off"); %!test %! load carsmall %! X = [Weight,Horsepower,Acceleration]; %! [TAB, STATS] = fitlm (X, MPG,"constant","display","off"); %! [TAB, STATS] = fitlm (X, MPG,"linear","display","off"); %! assert (TAB{2,2}, 47.9767628118615, 1e-09); %! assert (TAB{3,2}, -0.00654155878851796, 1e-09); %! assert (TAB{4,2}, -0.0429433065881864, 1e-09); %! assert (TAB{5,2}, -0.0115826516894871, 1e-09); %! assert (TAB{2,3}, 3.87851641748551, 1e-09); %! assert (TAB{3,3}, 0.00112741016370336, 1e-09); %! assert (TAB{4,3}, 0.0243130608813806, 1e-09); %! assert (TAB{5,3}, 0.193325043113178, 1e-09); %! assert (TAB{2,6}, 12.369874881944, 1e-09); %! assert (TAB{3,6}, -5.80228828790225, 1e-09); %! assert (TAB{4,6}, -1.76626492228599, 1e-09); %! assert (TAB{5,6}, -0.0599128364487485, 1e-09); %! assert (TAB{2,7}, 4.89570341688996e-21, 1e-09); %! assert (TAB{3,7}, 9.87424814144e-08, 1e-09); %! assert (TAB{4,7}, 0.0807803098213114, 1e-09); %! assert (TAB{5,7}, 0.952359384151778, 1e-09); statistics-release-1.6.3/inst/fitrgam.m000066400000000000000000000207041456127120000201160ustar00rootroot00000000000000## Copyright (C) 2023 Mohammed Azmat Khan ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{obj} =} fitrgam (@var{X}, @var{Y}) ## @deftypefnx {statistics} {@var{obj} =} fitrgam (@var{X}, @var{Y}, @var{name}, @var{value}) ## ## Fit a Generalised Additive Model (GAM) for regression. ## ## @code{@var{obj} = fitrgam (@var{X}, @var{Y})} returns an object of ## class RegressionGAM, with matrix @var{X} containing the predictor data and ## vector @var{Y} containing the continuous response data. ## ## @itemize ## @item ## @var{X} must be a @math{NxP} numeric matrix of input data where rows ## correspond to observations and columns correspond to features or variables. ## @var{X} will be used to train the GAM model. ## @item ## @var{Y} must be @math{Nx1} numeric vector containing the response data ## corresponding to the predictor data in @var{X}. @var{Y} must have same ## number of rows as @var{X}. ## @end itemize ## ## @code{@var{obj} = fitrgam (@dots{}, @var{name}, @var{value})} returns ## an object of class RegressionGAM with additional properties specified by ## @qcode{Name-Value} pair arguments listed below. ## ## @multitable @columnfractions 0.05 0.2 0.75 ## @headitem @tab @var{Name} @tab @var{Value} ## ## @item @tab @qcode{"predictors"} @tab Predictor Variable names, specified as ## a row vector cell of strings with the same length as the columns in @var{X}. ## If omitted, the program will generate default variable names ## @qcode{(x1, x2, ..., xn)} for each column in @var{X}. ## ## @item @tab @qcode{"responsename"} @tab Response Variable Name, specified as ## a string. If omitted, the default value is @qcode{"Y"}. ## ## @item @tab @qcode{"formula"} @tab a model specification given as a string in ## the form @qcode{"Y ~ terms"} where @qcode{Y} represents the reponse variable ## and @qcode{terms} the predictor variables. The formula can be used to ## specify a subset of variables for training model. For example: ## @qcode{"Y ~ x1 + x2 + x3 + x4 + x1:x2 + x2:x3"} specifies four linear terms ## for the first four columns of for predictor data, and @qcode{x1:x2} and ## @qcode{x2:x3} specify the two interaction terms for 1st-2nd and 3rd-4th ## columns respectively. Only these terms will be used for training the model, ## but @var{X} must have at least as many columns as referenced in the formula. ## If Predictor Variable names have been defined, then the terms in the formula ## must reference to those. When @qcode{"formula"} is specified, all terms used ## for training the model are referenced in the @qcode{IntMatrix} field of the ## @var{obj} class object as a matrix containing the column indexes for each ## term including both the predictors and the interactions used. ## ## @item @tab @qcode{"interactions"} @tab a logical matrix, a positive integer ## scalar, or the string @qcode{"all"} for defining the interactions between ## predictor variables. When given a logical matrix, it must have the same ## number of columns as @var{X} and each row corresponds to a different ## interaction term combining the predictors indexed as @qcode{true}. Each ## interaction term is appended as a column vector after the available predictor ## column in @var{X}. When @qcode{"all"} is defined, then all possible ## combinations of interactions are appended in @var{X} before training. At the ## moment, parsing a positive integer has the same effect as the @qcode{"all"} ## option. When @qcode{"interactions"} is specified, only the interaction terms ## appended to @var{X} are referenced in the @qcode{IntMatrix} field of the ## @var{obj} class object. ## ## @item @tab @qcode{"knots"} @tab a scalar or a row vector with the same ## columns as @var{X}. It defines the knots for fitting a polynomial when ## training the GAM. As a scalar, it is expanded to a row vector. The default ## value is 5, hence expanded to @qcode{ones (1, columns (X)) * 5}. You can ## parse a row vector with different number of knots for each predictor ## variable to be fitted with, although not recommended. ## ## @item @tab @qcode{"order"} @tab a scalar or a row vector with the same ## columns as @var{X}. It defines the order of the polynomial when training the ## GAM. As a scalar, it is expanded to a row vector. The default values is 3, ## hence expanded to @qcode{ones (1, columns (X)) * 3}. You can parse a row ## vector with different number of polynomial order for each predictor variable ## to be fitted with, although not recommended. ## ## @item @tab @qcode{"dof"} @tab a scalar or a row vector with the same columns ## as @var{X}. It defines the degrees of freedom for fitting a polynomial when ## training the GAM. As a scalar, it is expanded to a row vector. The default ## value is 8, hence expanded to @qcode{ones (1, columns (X)) * 8}. You can ## parse a row vector with different degrees of freedom for each predictor ## variable to be fitted with, although not recommended. ## ## @item @tab @qcode{"tol"} @tab a positive scalar to set the tolerance for ## covergence during training. By defaul, it is set to @qcode{1e-3}. ## @end multitable ## ## You can parse either a @qcode{"formula"} or an @qcode{"interactions"} ## optional parameter. Parsing both parameters will result an error. ## Accordingly, you can only pass up to two parameters among @qcode{"knots"}, ## @qcode{"order"}, and @qcode{"dof"} to define the required polynomial for ## training the GAM model. ## ## @seealso{RegressionGAM, regress, regress_gp} ## @end deftypefn function obj = fitrgam (X, Y, varargin) ## Check input parameters if (nargin < 2) error ("fitrgam: too few arguments."); endif if (mod (nargin, 2) != 0) error ("fitrgam: Name-Value arguments must be in pairs."); endif ## Check predictor data and labels have equal rows if (rows (X) != rows (Y)) error ("fitrgam: number of rows in X and Y must be equal."); endif ## Parse arguments to class def function obj = RegressionGAM (X, Y, varargin{:}); endfunction %!demo %! # Train a RegressionGAM Model for synthetic values %! %! f1 = @(x) cos (3 *x); %! f2 = @(x) x .^ 3; %! %! # generate x1 and x2 for f1 and f2 %! x1 = 2 * rand (50, 1) - 1; %! x2 = 2 * rand (50, 1) - 1; %! %! # calculate y %! y = f1(x1) + f2(x2); %! %! # add noise %! y = y + y .* 0.2 .* rand (50,1); %! X = [x1, x2]; %! %! # create an object %! a = fitrgam (X, y, "tol", 1e-3) ## Test constructor %!test %! x = [1, 2, 3; 4, 5, 6; 7, 8, 9; 3, 2, 1]; %! y = [1; 2; 3; 4]; %! a = fitrgam (x, y); %! assert ({a.X, a.Y}, {x, y}) %! assert ({a.BaseModel.Intercept}, {2.5000}) %! assert ({a.Knots, a.Order, a.DoF}, {[5, 5, 5], [3, 3, 3], [8, 8, 8]}) %! assert ({a.NumObservations, a.NumPredictors}, {4, 3}) %! assert ({a.ResponseName, a.PredictorNames}, {"Y", {"x1", "x2", "x3"}}) %! assert ({a.Formula}, {[]}) %!test %! x = [1, 2, 3, 4; 4, 5, 6, 7; 7, 8, 9, 1; 3, 2, 1, 2]; %! y = [1; 2; 3; 4]; %! pnames = {"A", "B", "C", "D"}; %! formula = "Y ~ A + B + C + D + A:C"; %! intMat = logical ([1,0,0,0;0,1,0,0;0,0,1,0;0,0,0,1;1,0,1,0]); %! a = fitrgam (x, y, "predictors", pnames, "formula", formula); %! assert ({a.IntMatrix}, {intMat}) %! assert ({a.ResponseName, a.PredictorNames}, {"Y", pnames}) %! assert ({a.Formula}, {formula}) ## Test input validation %!error fitrgam () %!error fitrgam (ones(10,2)) %!error %! fitrgam (ones (4,2), ones (4, 1), "K") %!error %! fitrgam (ones (4,2), ones (3, 1)) %!error %! fitrgam (ones (4,2), ones (3, 1), "K", 2) statistics-release-1.6.3/inst/friedman.m000066400000000000000000000154331456127120000202550ustar00rootroot00000000000000## Copyright (C) 2022 Andreas Bertsatos ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{p} =} friedman (@var{x}) ## @deftypefnx {statistics} {@var{p} =} friedman (@var{x}, @var{reps}) ## @deftypefnx {statistics} {@var{p} =} friedman (@var{x}, @var{reps}, @var{displayopt}) ## @deftypefnx {statistics} {[@var{p}, @var{atab}] =} friedman (@dots{}) ## @deftypefnx {statistics} {[@var{p}, @var{atab}, @var{stats}] =} friedman (@dots{}) ## ## Performs the nonparametric Friedman's test to compare column effects in a ## two-way layout. @qcode{friedman} tests the null hypothesis that the column ## effects are all the same against the alternative that they are not all the ## same. ## ## @qcode{friedman} requires one up to three input arguments: ## ## @itemize ## @item ## @var{x} contains the data and it must be a matrix of at least two columns and ## two rows. ## @item ## @var{reps} is the number of replicates for each combination of factor groups. ## If not provided, no replicates are assumed. ## @item ## @var{displayopt} is an optional parameter for displaying the Friedman's ANOVA ## table, when it is 'on' (default) and suppressing the display when it is 'off'. ## @end itemize ## ## @qcode{friedman} returns up to three output arguments: ## ## @itemize ## @item ## @var{p} is the p-value of the null hypothesis that all group means are equal. ## @item ## @var{atab} is a cell array containing the results in a Friedman's ANOVA table. ## @item ## @var{stats} is a structure containing statistics useful for performing ## a multiple comparison of medians with the MULTCOMPARE function. ## @end itemize ## ## If friedman is called without any output arguments, then it prints the results ## in a one-way ANOVA table to the standard output as if @var{displayopt} is ## 'on'. ## ## Examples: ## ## @example ## load popcorn; ## friedman (popcorn, 3); ## @end example ## ## ## @example ## [p, anovatab, stats] = friedman (popcorn, 3, "off"); ## disp (p); ## @end example ## ## @seealso{anova2, kruskalwallis, multcompare} ## @end deftypefn function [p, table, stats] = friedman (x, reps, displayopt) ## Check for valid number of input arguments narginchk (1, 3); ## Check for NaN values in X if (any (isnan( x(:)))) error ("NaN values in input are not allowed."); endif ## Add defaults if (nargin == 1) reps = 1; endif ## Check for correct size of input matrix [r, c] = size (x); if (r <= 1 || c <= 1) error ("Bad size of input matrix."); endif if (reps > 1) r = r / reps; if (floor (r) != r) error ("Repetitions and observations do not match."); endif endif ## Check for displayopt if (nargin < 3) displayopt = 'on'; elseif ! (strcmp (displayopt, "off") || strcmp (displayopt, "on")) error ("displayopt must be either 'on' (default) or 'off'."); endif plotdata = ~(strcmp (displayopt, "off")); ## Prepare a matrix of ranks. Replicates are ranked together. m = x; sum_R = 0; for j = 1:r jrows = reps * (j-1) + (1:reps); v = x(jrows,:); [R, tieadj] = tiedrank (v(:)); m(jrows,:) = reshape (R, reps, c);; sum_R = sum_R + 2 * tieadj; endfor ## Perform 2-way anova silently [p0, table] = anova2 (m, reps, 'off'); ## Compute Friedman test statistic and p-value chi_r = table{2,2}; sigmasq = c * reps * (reps * c + 1) / 12; if (sum_R > 0) sigmasq = sigmasq - sum_R / (12 * r * (reps * c - 1)); endif if (chi_r > 0) chi_r = chi_r / sigmasq; endif p = 1 - chi2cdf (chi_r, c - 1); ## Remove row info from ANOVA2 table table(3,:) = []; ## Remove interaction chi-sq and p-value, if there are repetitive measuments if (reps > 1) table{3,5} = []; table{3,6} = []; endif ## Fix test statistic names table{1,5} = "Chi-sq"; table{1,6} = "Prob>Chi-sq\n"; ## Fix test statistic values table{2,5} = chi_r; table{2,6} = p; ## Create stats structure (if requested) for MULTCOMPARE if (nargout > 2) stats.source = 'friedman'; stats.n = r; stats.meanranks = mean (m); stats.sigma = sqrt (sigmasq); endif ## Print results table on screen if no output argument was requested if (nargout == 0 || plotdata) printf(" Friedman's ANOVA Table\n"); printf("Source SS df MS Chi-sq Prob>Chi-sq\n"); printf("---------------------------------------------------------------\n"); printf("Columns %10.4f %5.0f %10.4f %8.2f %9.4f\n", ... table{2,2}, table{2,3}, table{2,4}, table{2,5}, table{2,6}); if reps > 1 printf("Interaction %10.4f %5.0f %10.4f %8.2f %9.4f\n", ... table{3,2}, table{3,3}, table{3,4}, table{3,5}, table{3,6}); endif printf("Error %10.4f %5.0f %10.4f\n", ... table{end-1,2}, table{end-1,3}, table{end-1,4}); printf("Total %10.4f %5.0f\n", table{end,2}, table{end,3}); endif endfunction %!demo %! load popcorn; %! friedman (popcorn, 3); %!demo %! load popcorn; %! [p, atab] = friedman (popcorn, 3, "off"); %! disp (p); ## testing against popcorn data and results from Matlab %!test %! popcorn = [5.5, 4.5, 3.5; 5.5, 4.5, 4.0; 6.0, 4.0, 3.0; ... %! 6.5, 5.0, 4.0; 7.0, 5.5, 5.0; 7.0, 5.0, 4.5]; %! [p, atab] = friedman (popcorn, 3, "off"); %! assert (p, 0.001028853354594794, 1e-14); %! assert (atab{2,2}, 99.75, 1e-14); %! assert (atab{2,3}, 2, 0); %! assert (atab{2,4}, 49.875, 1e-14); %! assert (atab{2,5}, 13.75862068965517, 1e-14); %! assert (atab{2,6}, 0.001028853354594794, 1e-14); %! assert (atab{3,2}, 0.08333333333333215, 1e-14); %! assert (atab{3,4}, 0.04166666666666607, 1e-14); %! assert (atab{4,3}, 12, 0); %!test %! popcorn = [5.5, 4.5, 3.5; 5.5, 4.5, 4.0; 6.0, 4.0, 3.0; ... %! 6.5, 5.0, 4.0; 7.0, 5.5, 5.0; 7.0, 5.0, 4.5]; %! [p, atab, stats] = friedman (popcorn, 3, "off"); %! assert (atab{5,2}, 116, 0); %! assert (atab{5,3}, 17, 0); %! assert (stats.source, "friedman"); %! assert (stats.n, 2); %! assert (stats.meanranks, [8, 4.75, 2.25], 0); %! assert (stats.sigma, 2.692582403567252, 1e-14); statistics-release-1.6.3/inst/fullfact.m000066400000000000000000000053671456127120000202750ustar00rootroot00000000000000## Copyright (C) 2022 Andreas Bertsatos ## based on public domain work by Paul Kienzle ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{A} =} fullfact (@var{N}) ## ## Full factorial design. ## ## If @var{N} is a scalar, return the full factorial design with @var{N} binary ## choices, 0 and 1. ## ## If @var{N} is a vector, return the full factorial design with ordinal choices ## 1 through @var{n_i} for each factor @var{i}. ## ## Values in @var{N} must be positive integers. ## ## @seealso {ff2n} ## @end deftypefn function A = fullfact(n) if (nargin != 1) error ("fullfact: wrong number of input arguments."); endif if length(n) == 1 if (floor (n) != n || n < 1 || ! isfinite (n) || ! isreal (n)) error ("fullfact: @var{N} must be a positive integer."); endif ## Combinatorial design with n binary choices A = fullfact(2*ones(1,n))-1; else if (any (floor (n) != n) || any (n < 1) || any (! isfinite (n)) ... || any (! isreal (n))) error ("fullfact: values in @var{N} must be positive integers."); endif ## Combinatorial design with n(i) ordinal choices A = [1:n(end)]'; for i=length(n)-1:-1:1 A = [kron([1:n(i)]',ones(rows(A),1)), repmat(A,n(i),1)]; end end endfunction %!demo %! ## Full factorial design with 3 binary variables %! fullfact (3) %!demo %! ## Full factorial design with 3 ordinal variables %! fullfact ([2, 3, 4]) %!error fullfact (); %!error fullfact (2, 5); %!error fullfact (2.5); %!error fullfact (0); %!error fullfact (-3); %!error fullfact (3+2i); %!error fullfact (Inf); %!error fullfact (NaN); %!error fullfact ([1, 2, -3]); %!error fullfact ([0, 1, 2]); %!error fullfact ([1, 2, NaN]); %!error fullfact ([1, 2, Inf]); %!test %! A = fullfact (2); %! assert (A, [0, 0; 0, 1; 1, 0; 1, 1]); %!test %! A = fullfact ([1, 2]); %! assert (A, [1, 1; 1, 2]); %!test %! A = fullfact ([1, 2, 4]); %! A_out = [1, 1, 1; 1, 1, 2; 1, 1, 3; 1, 1, 4; ... %! 1, 2, 1; 1, 2, 2; 1, 2, 3; 1, 2, 4]; %! assert (A, A_out); statistics-release-1.6.3/inst/geomean.m000066400000000000000000000233111456127120000200750ustar00rootroot00000000000000## Copyright (C) 2022-2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{m} =} geomean (@var{x}) ## @deftypefnx {statistics} {@var{m} =} geomean (@var{x}, "all") ## @deftypefnx {statistics} {@var{m} =} geomean (@var{x}, @var{dim}) ## @deftypefnx {statistics} {@var{m} =} geomean (@var{x}, @var{vecdim}) ## @deftypefnx {statistics} {@var{m} =} geomean (@dots{}, @var{nanflag}) ## ## Compute the geometric mean of @var{x}. ## ## @itemize ## @item If @var{x} is a vector, then @code{geomean(@var{x})} returns the ## geometric mean of the elements in @var{x} defined as ## @tex ## $$ {\rm geomean}(x) = \left( \prod_{i=1}^N x_i \right)^\frac{1}{N} ## = exp \left({1\over N} \sum_{i=1}^N log x_i \right) $$ ## ## @end tex ## @ifnottex ## ## @example ## geomean (@var{x}) = PROD_i @var{x}(i) ^ (1/N) ## @end example ## ## @end ifnottex ## @noindent ## where @math{N} is the length of the @var{x} vector. ## ## @item If @var{x} is a matrix, then @code{geomean(@var{x})} returns a row ## vector with the geometric mean of each columns in @var{x}. ## ## @item If @var{x} is a multidimensional array, then @code{geomean(@var{x})} ## operates along the first nonsingleton dimension of @var{x}. ## ## @item @var{x} must not contain any negative or complex values. ## @end itemize ## ## @code{geomean(@var{x}, "all")} returns the geometric mean of all the elements ## in @var{x}. If @var{x} contains any 0, then the returned value is 0. ## ## @code{geomean(@var{x}, @var{dim})} returns the geometric mean along the ## operating dimension @var{dim} of @var{x}. Calculating the harmonic mean of ## any subarray containing any 0 will return 0. ## ## @code{geomean(@var{x}, @var{vecdim})} returns the geometric mean over the ## dimensions specified in the vector @var{vecdim}. For example, if @var{x} is ## a 2-by-3-by-4 array, then @code{geomean(@var{x}, [1 2])} returns a ## 1-by-1-by-4 array. Each element of the output array is the geometric mean of ## the elements on the corresponding page of @var{x}. If @var{vecdim} indexes ## all dimensions of @var{x}, then it is equivalent to @code{geomean (@var{x}, ## "all")}. Any dimension in @var{vecdim} greater than @code{ndims (@var{x})} ## is ignored. ## ## @code{geomean(@dots{}, @var{nanflag})} specifies whether to exclude NaN ## values from the calculation, using any of the input argument combinations in ## previous syntaxes. By default, geomean includes NaN values in the calculation ## (@var{nanflag} has the value "includenan"). To exclude NaN values, set the ## value of @var{nanflag} to "omitnan". ## ## @seealso{harmmean, mean} ## @end deftypefn function m = geomean (x, varargin) if (nargin < 1 || nargin > 3) print_usage (); endif if (! isnumeric (x) || ! isreal (x) || ! all (x(! isnan (x))(:) >= 0)) error ("geomean: X must contain real nonnegative values."); endif ## Set initial conditions all_flag = false; omitnan = false; nvarg = numel (varargin); varg_chars = cellfun ("ischar", varargin); szx = size (x); ndx = ndims (x); if (nvarg > 1 && ! varg_chars(2:end)) ## Only first varargin can be numeric print_usage (); endif ## Process any other char arguments. if (any (varg_chars)) for i = varargin(varg_chars) switch (lower (i{:})) case "all" all_flag = true; case "omitnan" omitnan = true; case "includenan" omitnan = false; otherwise print_usage (); endswitch endfor varargin(varg_chars) = []; nvarg = numel (varargin); endif ## Single numeric input argument, no dimensions given. if (nvarg == 0) if (all_flag) x = x(:); if (omitnan) x = x(! isnan (x)); endif if (any (x == 0)) m = 0; return; endif m = exp (sum (log (x), 1) ./ length (x)); else ## Find the first non-singleton dimension. (dim = find (szx != 1, 1)) || (dim = 1); n = szx(dim); if (omitnan) idx = isnan (x); n = sum (! idx, dim); x(idx) = 1; # log (1) = 0 endif m = exp (sum (log (x), dim) ./ n); m(m == -Inf) = 0; # handle zeros in X endif else ## Two numeric input arguments, dimensions given. Note scalar is vector! vecdim = varargin{1}; if (isempty (vecdim) || ! (isvector (vecdim) && all (vecdim > 0)) ... || any (rem (vecdim, 1))) error ("geomean: DIM must be a positive integer scalar or vector."); endif if (ndx == 2 && isempty (x) && szx == [0,0]) ## FIXME: this special case handling could be removed once sum ## compatibly handles all sizes of empty inputs sz_out = szx; sz_out (vecdim(vecdim <= ndx)) = 1; m = NaN (sz_out); else if (isscalar (vecdim)) if (vecdim > ndx) m = x; else n = szx(vecdim); if (omitnan) nanx = isnan (x); n = sum (! nanx, vecdim); x(nanx) = 1; # log (1) = 0 endif m = exp (sum (log (x), vecdim) ./ n); m(m == -Inf) = 0; # handle zeros in X endif else vecdim = sort (vecdim); if (! all (diff (vecdim))) error (strcat (["geomean: VECDIM must contain non-repeating"], ... [" positive integers."])); endif ## Ignore exceeding dimensions in VECDIM vecdim(find (vecdim > ndims (x))) = []; if (isempty (vecdim)) m = x; else ## Move vecdims to dim 1. ## Calculate permutation vector remdims = 1 : ndx; # All dimensions remdims(vecdim) = []; # Delete dimensions specified by vecdim nremd = numel (remdims); ## If all dimensions are given, it is similar to all flag if (nremd == 0) x = x(:); if (omitnan) x = x(! isnan (x)); endif if (any (x == 0)) m = 0; return; endif m = exp (sum (log (x), 1) ./ length (x)); m(m == -Inf) = 0; # handle zeros in X else ## Permute to bring vecdims to front perm = [vecdim, remdims]; x = permute (x, perm); ## Reshape to squash all vecdims in dim1 num_dim = prod (szx(vecdim)); szx(vecdim) = []; szx = [ones(1, length(vecdim)), szx]; szx(1) = num_dim; x = reshape (x, szx); ## Calculate mean on dim1 if (omitnan) nanx = isnan (x); n = sum (! nanx, 1); x(nanx) = 1; # log (1) = 0 else n = szx(1); endif m = exp (sum (log (x), 1) ./ n); m(m == -Inf) = 0; # handle zeros in X ## Inverse permute back to correct dimensions m = ipermute (m, perm); endif endif endif endif endif endfunction ## Test single input and optional arguments "all", DIM, "omitnan") %!test %! x = [0:10]; %! y = [x;x+5;x+10]; %! assert (geomean (x), 0); %! m = [0 9.462942809849169 14.65658770861967]; %! assert (geomean (y, 2), m', 4e-14); %! assert (geomean (y, "all"), 0); %! y(2,4) = NaN; %! m(2) = 9.623207231679554; %! assert (geomean (y, 2), [0 NaN m(3)]', 4e-14); %! assert (geomean (y', "omitnan"), m, 4e-14); %! z = y + 20; %! assert (geomean (z, "all"), NaN); %! assert (geomean (z, "all", "includenan"), NaN); %! assert (geomean (z, "all", "omitnan"), 29.59298474535024, 4e-14); %! m = [24.79790781765634 NaN 34.85638839503932]; %! assert (geomean (z'), m, 4e-14); %! assert (geomean (z', "includenan"), m, 4e-14); %! m(2) = 30.02181156156319; %! assert (geomean (z', "omitnan"), m, 4e-14); %! assert (geomean (z, 2, "omitnan"), m', 4e-14); ## Test dimension indexing with vecdim in n-dimensional arrays %!test %! x = repmat ([1:20;6:25], [5 2 6 3]); %! assert (size (geomean (x, [3 2])), [10 1 1 3]); %! assert (size (geomean (x, [1 2])), [1 1 6 3]); %! assert (size (geomean (x, [1 2 4])), [1 1 6]); %! assert (size (geomean (x, [1 4 3])), [1 40]); %! assert (size (geomean (x, [1 2 3 4])), [1 1]); ## Test results with vecdim in n-dimensional arrays and "omitnan" %!test %! x = repmat ([1:20;6:25], [5 2 6 3]); %! m = repmat ([8.304361203739333;14.3078118884256], [5 1 1 3]); %! assert (geomean (x, [3 2]), m, 4e-13); %! x(2,5,6,3) = NaN; %! m(2,3) = NaN; %! assert (geomean (x, [3 2]), m, 4e-13); %! m(2,3) = 14.3292729579901; %! assert (geomean (x, [3 2], "omitnan"), m, 4e-13); ## Test errors %!error geomean ("char") %!error geomean ([1 -1 3]) %!error ... %! geomean (repmat ([1:20;6:25], [5 2 6 3 5]), -1) %!error ... %! geomean (repmat ([1:20;6:25], [5 2 6 3 5]), 0) %!error ... %! geomean (repmat ([1:20;6:25], [5 2 6 3 5]), [1 1]) statistics-release-1.6.3/inst/gmdistribution.m000066400000000000000000000313701456127120000215310ustar00rootroot00000000000000## Copyright (C) 2015 Lachlan Andrew ## Copyright (C) 2018-2020 John Donoghue ## Copyright (C) 2022 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . classdef gmdistribution ## -*- texinfo -*- ## @deftypefn {statistics} {@var{GMdist} =} gmdistribution (@var{mu}, @var{Sigma}) ## @deftypefnx {statistics} {@var{GMdist} =} gmdistribution (@var{mu}, @var{Sigma}, @var{p}) ## @deftypefnx {statistics} {@var{GMdist} =} gmdistribution (@var{mu}, @var{Sigma}, @var{p}, @var{extra}) ## ## Create an object of the gmdistribution class which represents a Gaussian ## mixture model with k components of n-dimensional Gaussians. ## ## Input @var{mu} is a k-by-n matrix specifying the n-dimensional mean of ## each of the k components of the distribution. ## ## Input @var{Sigma} is an array that specifies the variances of the ## distributions, in one of four forms depending on its dimension. ## @itemize ## @item n-by-n-by-k: Slice @var{Sigma}(:,:,i) is the variance of the ## i'th component ## @item 1-by-n-by-k: Slice diag(@var{Sigma}(1,:,i)) is the variance of the ## i'th component ## @item n-by-n: @var{Sigma} is the variance of every component ## @item 1-by-n-by-k: Slice diag(@var{Sigma}) is the variance of every ## component ## @end itemize ## ## If @var{p} is specified, it is a vector of length k specifying the ## proportion of each component. If it is omitted or empty, each component ## has an equal proportion. ## ## Input @var{extra} is used by fitgmdist to indicate the parameters of the ## fitting process. ## @seealso{fitgmdist} ## @end deftypefn properties mu ## means Sigma ## covariances ComponentProportion ## mixing proportions DistributionName ## "gaussian mixture distribution" NumComponents ## Number of mixture components NumVariables ## Dimension d of each Gaussian component CovarianceType ## 'diagonal' if DiagonalCovariance, 'full' othw SharedCovariance ## true if all components have equal covariance ## Set by a call to gmdistribution.fit or fitgmdist AIC ## Akaike Information Criterion BIC ## Bayes Information Criterion Converged ## true if algorithm converged by MaxIter NegativeLogLikelihood ## Negative of log-likelihood NlogL ## Negative of log-likelihood NumIterations ## Number of iterations RegularizationValue ## const added to diag of cov to make +ve def endproperties properties (Access = private) DiagonalCovariance ## bool summary of "CovarianceType" endproperties methods ######################################## ## Constructor function obj = gmdistribution (mu,sigma,p = [],extra = []) obj.DistributionName = "gaussian mixture distribution"; obj.mu = mu; obj.Sigma = sigma; obj.NumComponents = rows (mu); obj.NumVariables = columns (mu); if (isempty (p)) obj.ComponentProportion = ones (1,obj.NumComponents) / ... obj.NumComponents; else if any (p < 0) error ("gmmdistribution: component weights must be non-negative"); endif s = sum(p); if (s == 0) error ("gmmdistribution: component weights must not be all zero"); elseif (s != 1) p = p / s; endif obj.ComponentProportion = p(:)'; endif if (length (size (sigma)) == 3) obj.SharedCovariance = false; else obj.SharedCovariance = true; endif if (rows (sigma) == 1 && columns (mu) > 1) obj.DiagonalCovariance = true; obj.CovarianceType = 'diagonal'; else obj.DiagonalCovariance = false; ## full obj.CovarianceType = 'full'; endif if (! isempty (extra)) obj.AIC = extra.AIC; obj.BIC = extra.BIC; obj.Converged = extra.Converged; obj.NegativeLogLikelihood = extra.NegativeLogLikelihood; obj.NlogL = extra.NegativeLogLikelihood; obj.NumIterations = extra.NumIterations; obj.RegularizationValue = extra.RegularizationValue; endif endfunction ######################################## ## Cumulative distribution function for Gaussian mixture distribution function c = cdf (obj, X) X = checkX (obj, X, "cdf"); p_x_l = zeros (rows (X), obj.NumComponents); if (obj.SharedCovariance) if (obj.DiagonalCovariance) sig = diag (obj.Sigma); else sig = obj.Sigma; endif endif for i = 1:obj.NumComponents if (! obj.SharedCovariance) if (obj.DiagonalCovariance) sig = diag (obj.Sigma(:,:,i)); else sig = obj.Sigma(:,:,i); endif endif p_x_l(:,i) = mvncdf (X,obj.mu(i,:),sig)*obj.ComponentProportion(i); endfor c = sum (p_x_l, 2); endfunction ######################################## ## Construct clusters from Gaussian mixture distribution function [idx, nlogl, P, logpdf, M] = cluster (obj,X) X = checkX (obj, X, "cluster"); [p_x_l, M] = componentProb (obj, X); [~, idx] = max (p_x_l, [], 2); if (nargout >= 2) PDF = sum (p_x_l, 2); logpdf = log (PDF); nlogl = -sum (logpdf); if (nargout >= 3) P = bsxfun (@rdivide, p_x_l, PDF); endif endif endfunction ######################################## ## Display Gaussian mixture distribution object function c = disp (obj) msg = ["Gaussian mixture distribution with %d ", ... "components in %d dimension(s)\n"]; fprintf (msg, obj.NumComponents, columns (obj.mu)); for i = 1:obj.NumComponents fprintf ("Clust %d: weight %d\n\tMean: ", ... i, obj.ComponentProportion(i)); fprintf ("%g ", obj.mu(i,:)); fprintf ("\n"); if (! obj.SharedCovariance) fprintf ("\tVariance:"); if (! obj.DiagonalCovariance) if (columns (obj.mu) > 1) fprintf ("\n"); endif disp (squeeze (obj.Sigma(:,:,i))) else fprintf (" diag("); fprintf ("%g ", obj.Sigma(:,:,i)); fprintf (")\n"); endif endif endfor if (obj.SharedCovariance) fprintf ("Shared variance\n"); if (! obj.DiagonalCovariance) obj.Sigma else fprintf (" diag("); fprintf ("%g ", obj.Sigma); fprintf (")\n"); endif endif if (! isempty (obj.AIC)) fprintf ("AIC=%g BIC=%g NLogL=%g Iter=%d Cged=%d Reg=%g\n", ... obj.AIC, obj.BIC, obj.NegativeLogLikelihood, ... obj.NumIterations, obj.Converged, obj.RegularizationValue); endif endfunction ######################################## ## Display Gaussian mixture distribution object function c = display (obj) disp(obj); endfunction ######################################## ## Mahalanobis distance to component means function D = mahal (obj,X) X = checkX (obj, X, "mahal"); [~, D] = componentProb (obj,X); endfunction ######################################## ## Probability density function for Gaussian mixture distribution function c = pdf (obj,X) X = checkX (obj, X, "pdf"); p_x_l = componentProb (obj, X); c = sum (p_x_l, 2); endfunction ######################################## ## Posterior probabilities of components function c = posterior (obj,X) X = checkX (obj, X, "posterior"); p_x_l = componentProb (obj, X); c = bsxfun(@rdivide, p_x_l, sum (p_x_l, 2)); endfunction ######################################## ## Random numbers from Gaussian mixture distribution function c = random (obj,n) if nargin == 1 n = 1; endif c = zeros (n, obj.NumVariables); classes = randsample (obj.NumComponents, n, true, ... obj.ComponentProportion); if (obj.SharedCovariance) if (obj.DiagonalCovariance) sig = diag (obj.Sigma); else sig = obj.Sigma; endif endif for i = 1:obj.NumComponents idx = (classes == i); k = sum(idx); if (k > 0) if (! obj.SharedCovariance) if (obj.DiagonalCovariance) sig = diag (obj.Sigma(:,:,i)); else sig = obj.Sigma(:,:,i); endif endif # [sig] forces [sig] not to have class "diagonal", # since mvnrnd uses automatic broadcast, # which fails on structured matrices c(idx,:) = mvnrnd ([obj.mu(i,:)], [sig], k); endif endfor endfunction endmethods ######################################## methods (Static) ## Gaussian mixture parameter estimates function c = fit (X, k, varargin) c = fitgmdist (X, k, varargin{:}); endfunction endmethods ######################################## methods (Access = private) ## Probability density of (row of) X *and* component l ## Second argument is an array of the Mahalonis distances function [p_x_l, M] = componentProb (obj, X) M = zeros (rows (X), obj.NumComponents); dets = zeros (1, obj.NumComponents); % sqrt(determinant) if (obj.SharedCovariance) if (obj.DiagonalCovariance) r = diag (sqrt(obj.Sigma)); else r = chol (obj.Sigma); endif endif for i = 1:obj.NumComponents dev = bsxfun (@minus, X, obj.mu(i,:)); if (! obj.SharedCovariance) if (obj.DiagonalCovariance) r = diag (sqrt (obj.Sigma(:,:,i))); else r = chol (obj.Sigma(:,:,i)); endif endif M(:,i) = sumsq (dev / r, 2); dets(i) = prod (diag (r)); endfor p_x_l = exp (-M/2); coeff = obj.ComponentProportion ./ ... ((2 * pi) ^ (obj.NumVariables / 2) .* dets); p_x_l = bsxfun (@times, p_x_l, coeff); endfunction ######################################## ## Check format of argument X function X = checkX (obj, X, name) if (columns (X) != obj.NumVariables) if (columns (X) == 1 && rows (X) == obj.NumVariables) X = X'; else error ("gmdistribution.%s: X has %d columns instead of %d\n", ... name, columns (X), obj.NumVariables); end endif endfunction endmethods endclassdef %!test %! mu = eye(2); %! Sigma = eye(2); %! GM = gmdistribution (mu, Sigma); %! density = GM.pdf ([0 0; 1 1]); %! assert (density(1) - density(2), 0, 1e-6); %! %! [idx, nlogl, P, logpdf,M] = cluster (GM, eye(2)); %! assert (idx, [1; 2]); %! [idx2,nlogl2,P2,logpdf2] = GM.cluster (eye(2)); %! assert (nlogl - nlogl2, 0, 1e-6); %! [idx3,nlogl3,P3] = cluster (GM, eye(2)); %! assert (P - P3, zeros (2), 1e-6); %! [idx4,nlogl4] = cluster (GM, eye(2)); %! assert (size (nlogl4), [1 1]); %! idx5 = cluster (GM, eye(2)); %! assert (idx - idx5, zeros (2,1)); %! %! D = GM.mahal ([1;0]); %! assert (D - M(1,:), zeros (1,2), 1e-6); %! %! P = GM.posterior ([0 1]); %! assert (P - P2(2,:), zeros (1,2), 1e-6); %! %! R = GM.random(20); %! assert (size(R), [20, 2]); %! %! R = GM.random(); %! assert (size(R), [1, 2]); statistics-release-1.6.3/inst/grp2idx.m000066400000000000000000000114201456127120000200370ustar00rootroot00000000000000## Copyright (C) 2015 Carnë Draug ## Copyright (C) 2022 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation; either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; if not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {[@var{g}, @var{gn}, @var{gl}] =} grp2idx (@var{s}) ## ## Get index for group variables. ## ## For variable @var{s}, returns the indices @var{g}, into the variable ## groups @var{gn} and @var{gl}. The first has a string representation of ## the groups while the later has its actual values. The group indices are ## allocated in order of appearance in @var{s}. ## ## NaNs and empty strings in @var{s} appear as NaN in @var{g} and are ## not present on either @var{gn} and @var{gl}. ## ## @seealso{grpstats} ## @end deftypefn function [g, gn, gl] = grp2idx (s) if (nargin != 1) print_usage (); endif s_was_char = false; if (ischar (s)) s_was_char = true; s = cellstr (s); elseif (! isvector (s)) error ("grp2idx: S must be a vector, cell array of strings, or char matrix"); endif [gl, I, g] = unique (s(:)); ## Fix order in here, since unique does not support this yet if (iscellstr (s)) I = sort(I); for i = 1:length (gl) gl_s(i) = gl(g(I(i))); idx(i,:) = (g == g(I(i))); endfor for i = 1:length (gl) g(idx(i,:)) = i; endfor gl = gl_s; gl = gl'; else I = sort(I); for i = 1:length (gl) gl_s(i) = gl(g(I(i))); idx(i,:) = (g == g(I(i))); endfor for i = 1:length (gl) g(idx(i,:)) = i; endfor gl = gl_s; gl = gl'; endif ## handle NaNs and empty strings if (iscellstr (s)) empties = cellfun (@isempty, s); if (any (empties)) g(empties) = NaN; while (min (g) > 1) g--; endwhile endif empties = cellfun (@isempty, gl); if (any (empties)) gl(empties) = []; endif else ## This works fine because NaN come at the end after sorting, we don't ## have to worry about change on the indices. g(isnan (s)) = NaN; gl(isnan (gl)) = []; endif if (nargout > 1) if (iscellstr (gl)) gn = gl; elseif (iscell (gl)) gn = cellfun (@num2str, gl, "UniformOutput", false); else gn = arrayfun (@num2str, gl, "UniformOutput", false); endif endif if (nargout > 2 && s_was_char) gl = char (gl); endif endfunction ## test boolean input and note that row or column vector makes no difference %!test %! in = [true false false true]; %! out = {[1; 2; 2; 1] {"1"; "0"} [true; false]}; %! assert (nthargout (1:3, @grp2idx, in), out) %! assert (nthargout (1:3, @grp2idx, in), nthargout (1:3, @grp2idx, in')) ## test that boolean groups are ordered in order of appearance %!test %! assert (nthargout (1:3, @grp2idx, [false, true]), %! {[1; 2] {"0"; "1"} [false; true]}); %! assert (nthargout (1:3, @grp2idx, [true, false]), %! {[1; 2] {"1"; "0"} [true; false]}); ## test char matrix and cell array of strings %!assert (nthargout (1:3, @grp2idx, ["oct"; "sci"; "oct"; "oct"; "sci"]), %! {[1; 2; 1; 1; 2] {"oct"; "sci"} ["oct"; "sci"]}); ## and cell array of strings %!assert (nthargout (1:3, @grp2idx, {"oct"; "sci"; "oct"; "oct"; "sci"}), %! {[1; 2; 1; 1; 2] {"oct"; "sci"} {"oct"; "sci"}}); ## test numeric arrays %!assert (nthargout (1:3, @grp2idx, [ 1 -3 -2 -3 -3 2 1 -1 3 -3]), %! {[1; 2; 3; 2; 2; 4; 1; 5; 6; 2], {"1"; "-3"; "-2"; "2"; "-1"; "3"}, ... %! [1; -3; -2; 2; -1; 3]}); ## test for NaN and empty strings %!assert (nthargout (1:3, @grp2idx, [2 2 3 NaN 2 3]), %! {[1; 1; 2; NaN; 1; 2] {"2"; "3"} [2; 3]}) %!assert (nthargout (1:3, @grp2idx, {"et" "sa" "sa" "" "et"}), %! {[1; 2; 2; NaN; 1] {"et"; "sa"} {"et"; "sa"}}) ## Test that order when handling strings is by order of appearance %!test assert (nthargout (1:3, @grp2idx, ["sci"; "oct"; "sci"; "oct"; "oct"]), %! {[1; 2; 1; 2; 2] {"sci"; "oct"} ["sci"; "oct"]}); %!test assert (nthargout (1:3, @grp2idx, {"sci"; "oct"; "sci"; "oct"; "oct"}), %! {[1; 2; 1; 2; 2] {"sci"; "oct"} {"sci"; "oct"}}); %!test assert (nthargout (1:3, @grp2idx, {"sa" "et" "et" "" "sa"}), %! {[1; 2; 2; NaN; 1] {"sa"; "et"} {"sa"; "et"}}) statistics-release-1.6.3/inst/grpstats.m000066400000000000000000000235151456127120000203370ustar00rootroot00000000000000## Copyright (C) 2022 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{mean} =} grpstats (@var{x}) ## @deftypefnx {statistics} {@var{mean} =} grpstats (@var{x}, @var{group}) ## @deftypefnx {statistics} {[@var{a}, @var{b}, @dots{}] =} grpstats (@var{x}, @var{group}, @var{whichstats}) ## @deftypefnx {statistics} {[@var{a}, @var{b}, @dots{}] =} grpstats (@var{x}, @var{group}, @var{whichstats}, @var{alpha}) ## ## Summary statistics by group. @code{grpstats} computes groupwise summary ## statistics, for data in a matrix @var{x}. @code{grpstats} treats NaNs as ## missing values, and removes them. ## ## @code{@var{means} = grpstats (@var{x}, @var{group})}, when X is a matrix of ## observations, returns the means of each column of @var{x} by @var{group}. ## @var{group} is a grouping variable defined as a categorical variable, ## numeric, string array, or cell array of strings. @var{group} can be [] or ## omitted to compute the mean of the entire sample without grouping. ## ## @code{[@var{a}, @var{b}, @dots{}] = grpstats (@var{x}, @var{group}, ## @var{whichstats})}, for a numeric matrix X, returns the statistics specified ## by @var{whichstats}, as separate arrays @var{a}, @var{b}, @dots{}. ## @var{whichstats} can be a single function name, or a cell array containing ## multiple function names. The number of output arguments must match the ## number of function names in @var{whichstats}. ## Names in @var{whichstats} can be chosen from among the following: ## ## @multitable @columnfractions 0.05 0.2 0.75 ## @item @tab "mean" @tab mean ## @item @tab "median" @tab median ## @item @tab "sem" @tab standard error of the mean ## @item @tab "std" @tab standard deviation ## @item @tab "var" @tab variance ## @item @tab "min" @tab minimum value ## @item @tab "max" @tab maximum value ## @item @tab "range" @tab maximum - minimum ## @item @tab "numel" @tab count, or number of elements ## @item @tab "meanci" @tab 95% confidence interval for the mean ## @item @tab "predci" @tab 95% prediction interval for a new observation ## @item @tab "gname" @tab group name ## @end multitable ## ## @code{[@dots{}] = grpstats (@var{x}, @var{group}, @var{whichstats}, ## @var{alpha})} specifies the confidence level as 100(1-ALPHA)% for the "meanci" ## and "predci" options. Default value for @var{alpha} is 0.05. ## ## Examples: ## ## @example ## load carsmall; ## [m,p,g] = grpstats (Weight, Model_Year, @{"mean", "predci", "gname"@}) ## n = length(m); ## errorbar((1:n)',m,p(:,2)-m) ## set (gca, "xtick", 1:n, "xticklabel", g); ## title ("95% prediction intervals for mean weight by year") ## @end example ## ## @seealso{grp2idx} ## @end deftypefn function [varargout] = grpstats (x ,group, whichstats, alpha) ## Check input arguments narginchk (1, 4) ## Check X being a vector or 2d matrix of real values if (ndims (x) > 2 || ! isnumeric (x) || islogical (x)) error ("grpstats: X must be a vector or 2d matrix of real values."); endif ## If X is a vector, make it a column vector if (isvector (x)) x = x(:); endif ## Check groups and if empty make a single group for all X [r, c] = size (x); if (nargin < 2 || isempty (group)) group = ones (r, 1); endif ## Get group names and indices [group_idx, group_names] = grp2idx (group); ngroups = length (group_names); if (length (group_idx) != r) error ("grpstats: samples in X and GROUPS mismatch."); endif ## Add default for whichstats and check for 3rd input argument func_names = {}; if (nargin > 2 && ischar (whichstats)) func_names = {whichstats}; elseif (nargin > 2 && iscell (whichstats)) func_names = whichstats; endif ## Add default for alpha and check for 4th input argument if (nargin > 3) if (! (isnumeric (alpha) && isscalar (alpha) ... && alpha > 0 && alpha < 1)) error ("grpstats: ALPHA must be a real scalar in the range (0,1)."); endif else alpha = 0.05; endif ## Calculate functions if (isempty (func_names)) ## Check consistent number of output arguments if (nargout == 1) for j = 1:ngroups group_x = x(find (group_idx == j), :); group_mean(j,:) = mean (group_x, 1, "omitnan") ; endfor varargout{1} = group_mean; else error ("grpstats: incosistent number of output arguments."); endif else func_num = length (func_names); ## Check consistent number of output arguments if (nargout != func_num) error ("grpstats: incosistent number of output arguments."); endif for l = 1:func_num switch (func_names{l}) case "mean" for j = 1:ngroups group_x = x(find (group_idx == j), :); group_mean(j,:) = mean (group_x, 1, "omitnan"); endfor varargout{l} = group_mean; case "median" for j = 1:ngroups group_x = x(find (group_idx == j), :); group_mean(j,:) = median (group_x, 1, "omitnan"); endfor varargout{l} = group_mean; case "sem" for j = 1:ngroups group_x = x(find (group_idx == j), :); group_sem(j,:) = std (group_x, 0, 1, "omitnan") / ... sqrt (size (group_x, 1) - sum (isnan (group_x), 1)); endfor varargout{l} = group_sem; case "std" for j = 1:ngroups group_x = x(find (group_idx == j), :); group_std(j,:) = std (group_x, 0, 1, "omitnan"); endfor varargout{l} = group_std; case "var" for j = 1:ngroups group_x = x(find (group_idx == j), :); group_var(j,:) = var (group_x, 0, 1, "omitnan"); endfor varargout{l} = group_var; case "min" for j = 1:ngroups group_x = x(find (group_idx == j), :); group_min(j,:) = nanmin (group_x); endfor varargout{l} = group_min; case "max" for j = 1:ngroups group_x = x(find (group_idx == j), :); group_max(j,:) = nanmax (group_x); endfor varargout{l} = group_max; case "range" func_handle = @(x) range (x, 1); for j = 1:ngroups group_x = x(find (group_idx == j), :); group_range(j,:) = range (group_x, 1); endfor varargout{l} = group_range; case "numel" for j = 1:ngroups group_x = x(find (group_idx == j), :); group_numel(j,:) = size (group_x, 1) - sum (isnan (group_x), 1); endfor varargout{l} = group_numel; case "meanci" for j = 1:ngroups group_x = x(find (group_idx == j), :); m = mean (group_x, 1, "omitnan") ; n = size (x, 1) - sum (isnan (group_x), 1); s = std (group_x, 0, 1, "omitnan") ./ sqrt (n); d = s .* - tinv (alpha / 2, max (n - 1, [], 1)); group_meanci(j,:) = [m-d, m+d]; endfor varargout{l} = group_meanci; case "predci" for j = 1:ngroups group_x = x(find (group_idx == j), :); m = mean (group_x, 1, "omitnan") ; n = size (x, 1) - sum (isnan (group_x), 1); s = std (group_x, 0, 1, "omitnan") ./ sqrt (1 + (1 ./ n)); d = s .* - tinv (alpha / 2, max (n - 1, [], 1)); group_predci(j,:) = [m-d, m+d]; endfor varargout{l} = group_predci; case "gname" varargout{l} = group_names; otherwise error ("grpstats: wrong whichstats option."); endswitch endfor endif endfunction %!demo %! load carsmall; %! [m,p,g] = grpstats (Weight, Model_Year, {"mean", "predci", "gname"}) %! n = length(m); %! errorbar((1:n)',m,p(:,2)-m); %! set (gca, "xtick", 1:n, "xticklabel", g); %! title ("95% prediction intervals for mean weight by year"); %!demo %! load carsmall; %! [m,p,g] = grpstats ([Acceleration,Weight/1000],Cylinders, ... %! {"mean", "meanci", "gname"}, 0.05) %! [c,r] = size (m); %! errorbar((1:c)'.*ones(c,r),m,p(:,[(1:r)])-m); %! set (gca, "xtick", 1:c, "xticklabel", g); %! title ("95% prediction intervals for mean weight by year"); %!test %! load carsmall %! means = grpstats (Acceleration, Origin); %! assert (means, [14.4377; 18.0500; 15.8867; 16.3778; 16.6000; 15.5000], 0.001); %!test %! load carsmall %! [grpMin,grpMax,grp] = grpstats (Acceleration, Origin, {"min","max","gname"}); %! assert (grpMin, [8.0; 15.3; 13.9; 12.2; 15.7; 15.5]); %! assert (grpMax, [22.2; 21.9; 18.2; 24.6; 17.5; 15.5]); %!test %! load carsmall %! [grpMin,grpMax,grp] = grpstats (Acceleration, Origin, {"min","max","gname"}); %! assert (grp', {"USA", "France", "Japan", "Germany", "Sweden", "Italy"}); %!test %! load carsmall %! [m,p,g] = grpstats ([Acceleration,Weight/1000], Cylinders, ... %! {"mean", "meanci", "gname"}, 0.05); %! assert (p(:,1), [11.17621760075134, 16.13845847655224, 16.16222663683362]', ... %! [1e-14, 2e-14, 1e-14]'); statistics-release-1.6.3/inst/gscatter.m000066400000000000000000000203011456127120000202720ustar00rootroot00000000000000## Copyright (C) 2021 Stefano Guidoni ## Copyright (C) 2022 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {} gscatter (@var{x}, @var{y}, @var{g}) ## @deftypefnx {statistics} {} gscatter (@var{x}, @var{y}, @var{g}, @var{clr}, @var{sym}, @var{siz}) ## @deftypefnx {statistics} {} gscatter (@dots{}, @var{doleg}, @var{xnam}, @var{ynam}) ## @deftypefnx {statistics} {@var{h} =} gscatter (@dots{}) ## ## Draw a scatter plot with grouped data. ## ## @code{gscatter} is a utility function to draw a scatter plot of @var{x} and ## @var{y}, according to the groups defined by @var{g}. Input @var{x} and ## @var{y} are numeric vectors of the same size, while @var{g} is either a ## vector of the same size as @var{x} or a character matrix with the same number ## of rows as the size of @var{x}. As a vector @var{g} can be numeric, logical, ## a character array, a string array (not implemented), a cell string or cell ## array. ## ## A number of optional inputs change the appearance of the plot: ## @itemize @bullet ## @item @var{"clr"} ## defines the color for each group; if not enough colors are defined by ## @var{"clr"}, @code{gscatter} cycles through the specified colors. Colors can ## be defined as named colors, as rgb triplets or as indices for the current ## @code{colormap}. The default value is a different color for each group, ## according to the current @code{colormap}. ## ## @item @var{"sym"} ## is a char array of symbols for each group; if not enough symbols are defined ## by @var{"sym"}, @code{gscatter} cycles through the specified symbols. ## ## @item @var{"siz"} ## is a numeric array of sizes for each group; if not enough sizes are defined ## by @var{"siz"}, @code{gscatter} cycles through the specified sizes. ## ## @item @var{"doleg"} ## is a boolean value to show the legend; it can be either @qcode{on} (default) ## or @qcode{off}. ## ## @item @var{"xnam"} ## is a character array, the name for the x axis. ## ## @item @var{"ynam"} ## is a character array, the name for the y axis. ## @end itemize ## ## Output @var{h} is an array of graphics handles to the @code{line} object of ## each group. ## ## @end deftypefn ## ## @seealso{scatter} function h = gscatter (varargin) ## optional axes handle if (isaxes (varargin{1})) ## parameter is an axes handle hax = varargin{1}; varargin = varargin(2:end); nargin--; endif ## check the input parameters if (nargin < 3) print_usage (); endif ## ## necessary parameters ## ## x coordinates if (isvector (varargin{1}) && isnumeric (varargin{1})) x = varargin{1}; n = numel (x); else error ("gscatter: x must be a numeric vector"); endif ## y coordinates if (isvector (varargin{2}) && isnumeric (varargin{2})) if (numel (varargin{2}) == n) y = varargin{2}; else error ("gscatter: x and y must have the same size"); endif else error ("gscatter: y must be a numeric vector"); endif ## groups if (isrow (varargin{3})) varargin{3} = transpose (varargin{3}); endif if (ismatrix (varargin{3}) && ischar (varargin{3})) varargin{3} = cellstr (varargin{3}); # char matrix to cellstr elseif (iscell (varargin{3}) && ! iscellstr (varargin{3})) varargin{3} = cell2mat (varargin{3}); # numeric cell to vector endif if (isvector (varargin{3})) # only numeric vectors or cellstr if (rows (varargin{3}) == n) gv = varargin{3}; if (iscellstr (gv)) g_names = unique (gv); # avoid warning else g_names = unique (gv, "rows"); endif g_len = numel (g_names); if (iscellstr (g_names)) for i = 1 : g_len g(find (strcmp(gv, g_names{i}))) = i; endfor else for i = 1 : g_len g(find (gv == g_names(i))) = i; endfor endif else error ("gscatter: g must have the same size as x and y"); endif else error (["gscatter: g must be a numeric or logical or char vector, "... "or a cell or cellstr array, or a char matrix"]); endif ## ## optional parameters ## ## Note: this parameters are passed as they are to 'line', ## the validity check is delegated to 'line' g_col = lines (g_len); g_size = 6 * ones (g_len, 1); g_sym = repmat ('o', 1, g_len); ## optional parameters for legend and axes labels do_legend = 1; # legend shown by default ## MATLAB compatibility: by default MATLAB uses the variable name as ## label for either axis mygetname = @(x) inputname(1); # to retrieve the name of a variable x_nam = mygetname (varargin{1}); # this should retrieve the name of the var, y_nam = mygetname (varargin{2}); # but it does not work ## parameters are all in fixed positions for i = 4 : nargin switch (i) case 4 ## colours c_list = varargin{4}; if (isrow (c_list)) c_list = transpose (c_list); endif c_list_len = rows (c_list); g_col = repmat (c_list, ceil (g_len / c_list_len)); case {5, 6} ## size and symbols s_list = varargin{i}; s_list_len = length (s_list); g_tmp = repmat (s_list, ceil (g_len / s_list_len)); if (i == 6) g_size = g_tmp; else g_sym = g_tmp; endif case 7 ## legend switch (lower (varargin{7})) case "on" do_legend = 1; case "off" do_legend = 0; otherwise error ("gscatter: invalid dolegend parameter '%s'", varargin{7}); endswitch case {8, 9} ## x and y label if (! ischar (varargin{i}) && ! isvector (varargin{i})) error ("gscatter: xnam and ynam must be strings"); endif if (i == 8) x_nam = varargin{8}; else y_nam = varargin{9}; endif endswitch endfor ## scatter plot with grouping if (! exist ("hax", "var")) hax = gca (); endif ## return value h = []; hold on; for i = 1 : g_len idcs = find (g == i); h(i) = line (hax, x(idcs), y(idcs), "linestyle", "none", ... "markersize", g_size(i), "color", g_col(i,:), "marker", g_sym(i)); endfor if (do_legend) if (isnumeric (g_names)) g_names = num2str (g_names); endif warning ("off", "Octave:legend:unimplemented-location", "local"); legend (hax, g_names, "location", "best"); endif xlabel (hax, x_nam); ylabel (hax, y_nam); hold off; endfunction %!demo %! load fisheriris; %! X = meas(:,3:4); %! cidcs = kmeans (X, 3, "Replicates", 5); %! gscatter (X(:,1), X(:,2), cidcs, [.75 .75 0; 0 .75 .75; .75 0 .75], "os^"); %! title ("Fisher's iris data"); ## Test plotting %!shared visibility_setting %! visibility_setting = get (0, "DefaultFigureVisible"); %!test %! hf = figure ("visible", "off"); %! unwind_protect %! load fisheriris; %! X = meas(:,3:4); %! cidcs = kmeans (X, 3, "Replicates", 5); %! gscatter (X(:,1), X(:,2), cidcs, [.75 .75 0; 0 .75 .75; .75 0 .75], "os^"); %! title ("Fisher's iris data"); %! unwind_protect_cleanup %! close (hf); %! end_unwind_protect ## Test input validation %!error gscatter (); %!error gscatter ([1]); %!error gscatter ([1], [2]); %!error gscatter ('abc', [1 2 3], [1]); %!error gscatter ([1 2 3], [1 2], [1]); %!error gscatter ([1 2 3], 'abc', [1]); %!error gscatter ([1 2], [1 2], [1]); %!error gscatter ([1 2], [1 2], [1 2], 'rb', 'so', 12, 'xxx'); statistics-release-1.6.3/inst/harmmean.m000066400000000000000000000237361456127120000202650ustar00rootroot00000000000000## Copyright (C) 2022-2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{m} =} harmmean (@var{x}) ## @deftypefnx {statistics} {@var{m} =} harmmean (@var{x}, "all") ## @deftypefnx {statistics} {@var{m} =} harmmean (@var{x}, @var{dim}) ## @deftypefnx {statistics} {@var{m} =} harmmean (@var{x}, @var{vecdim}) ## @deftypefnx {statistics} {@var{m} =} harmmean (@dots{}, @var{nanflag}) ## ## Compute the harmonic mean of @var{x}. ## ## @itemize ## @item If @var{x} is a vector, then @code{harmmean(@var{x})} returns the ## harmonic mean of the elements in @var{x} defined as ## @tex ## $$ {\rm harmmean}(x) = \frac{N}{\sum_{i=1}^N \frac{1}{x_i}} $$ ## ## @end tex ## @ifnottex ## ## @example ## harmmean (@var{x}) = N / SUM_i @var{x}(i)^-1 ## @end example ## ## @end ifnottex ## @noindent ## where @math{N} is the length of the @var{x} vector. ## ## @item If @var{x} is a matrix, then @code{harmmean(@var{x})} returns a row ## vector with the harmonic mean of each columns in @var{x}. ## ## @item If @var{x} is a multidimensional array, then @code{harmmean(@var{x})} ## operates along the first nonsingleton dimension of @var{x}. ## ## @item @var{x} must not contain any negative or complex values. ## @end itemize ## ## @code{harmmean(@var{x}, "all")} returns the harmonic mean of all the elements ## in @var{x}. If @var{x} contains any 0, then the returned value is 0. ## ## @code{harmmean(@var{x}, @var{dim})} returns the harmonic mean along the ## operating dimension @var{dim} of @var{x}. Calculating the harmonic mean of ## any subarray containing any 0 will return 0. ## ## @code{harmmean(@var{x}, @var{vecdim})} returns the harmonic mean over the ## dimensions specified in the vector @var{vecdim}. For example, if @var{x} is ## a 2-by-3-by-4 array, then @code{harmmean(@var{x}, [1 2])} returns a ## 1-by-1-by-4 array. Each element of the output array is the harmonic mean of ## the elements on the corresponding page of @var{x}. If @var{vecdim} indexes ## all dimensions of @var{x}, then it is equivalent to @code{harmmean (@var{x}, ## "all")}. Any dimension in @var{vecdim} greater than @code{ndims (@var{x})} ## is ignored. ## ## @code{harmmean(@dots{}, @var{nanflag})} specifies whether to exclude NaN ## values from the calculation, using any of the input argument combinations in ## previous syntaxes. By default, harmmean includes NaN values in the ## calculation (@var{nanflag} has the value "includenan"). To exclude NaN ## values, set the value of @var{nanflag} to "omitnan". ## ## @seealso{geomean, mean} ## @end deftypefn function m = harmmean (x, varargin) if (nargin < 1 || nargin > 3) print_usage (); endif if (! isnumeric (x) || ! isreal (x) || ! all (x(! isnan (x))(:) >= 0)) error ("harmmean: X must contain real nonnegative values."); endif ## Set initial conditions all_flag = false; omitnan = false; nvarg = numel (varargin); varg_chars = cellfun ("ischar", varargin); szx = size (x); ndx = ndims (x); if (nvarg > 1 && ! varg_chars(2:end)) ## Only first varargin can be numeric print_usage (); endif ## Process any other char arguments. if (any (varg_chars)) for i = varargin(varg_chars) switch (lower (i{:})) case "all" all_flag = true; case "omitnan" omitnan = true; case "includenan" omitnan = false; otherwise print_usage (); endswitch endfor varargin(varg_chars) = []; nvarg = numel (varargin); endif ## Single numeric input argument, no dimensions given. if (nvarg == 0) if (all_flag) x = x(:); if (omitnan) x = x(! isnan (x)); endif if (any (x == 0)) m = 0; return; endif m = length (x) ./ sum (1 ./ x); m(m == Inf) = 0; # handle zeros in X else ## Find the first non-singleton dimension. (dim = find (szx != 1, 1)) || (dim = 1); n = szx(dim); is_nan = 0; if (omitnan) idx = isnan (x); n = sum (! idx, dim); is_nan = sum (idx, dim); x(idx) = 1; # remove NaNs by subtracting is_nan below endif m = n ./ (sum (1 ./ x, dim) - is_nan); m(m == Inf) = 0; # handle zeros in X endif else ## Two numeric input arguments, dimensions given. Note scalar is vector! vecdim = varargin{1}; if (isempty (vecdim) || ! (isvector (vecdim) && all (vecdim > 0)) ... || any (rem (vecdim, 1))) error ("harmmean: DIM must be a positive integer scalar or vector."); endif if (ndx == 2 && isempty (x) && szx == [0,0]) ## FIXME: this special case handling could be removed once sum ## compatibly handles all sizes of empty inputs sz_out = szx; sz_out (vecdim(vecdim <= ndx)) = 1; m = NaN (sz_out); else if (isscalar (vecdim)) if (vecdim > ndx) m = x; else n = szx(vecdim); is_nan = 0; if (omitnan) nanx = isnan (x); n = sum (! nanx, vecdim); is_nan = sum (nanx, vecdim); x(nanx) = 1; # remove NaNs by subtracting is_nan below endif m = n ./ (sum (1 ./ x, vecdim) - is_nan); m(m == Inf) = 0; # handle zeros in X endif else vecdim = sort (vecdim); if (! all (diff (vecdim))) error (strcat (["harmmean: VECDIM must contain non-repeating"], ... [" positive integers."])); endif ## Ignore exceeding dimensions in VECDIM vecdim(find (vecdim > ndims (x))) = []; if (isempty (vecdim)) m = x; else ## Move vecdims to dim 1. ## Calculate permutation vector remdims = 1 : ndx; # All dimensions remdims(vecdim) = []; # Delete dimensions specified by vecdim nremd = numel (remdims); ## If all dimensions are given, it is similar to all flag if (nremd == 0) x = x(:); if (omitnan) x = x(! isnan (x)); endif if (any (x == 0)) m = 0; return; endif m = length (x) ./ sum (1 ./ x); m(m == Inf) = 0; # handle zeros in X else ## Permute to bring vecdims to front perm = [vecdim, remdims]; x = permute (x, perm); ## Reshape to squash all vecdims in dim1 num_dim = prod (szx(vecdim)); szx(vecdim) = []; szx = [ones(1, length(vecdim)), szx]; szx(1) = num_dim; x = reshape (x, szx); ## Calculate mean on dim1 if (omitnan) nanx = isnan (x); n = sum (! nanx, 1); is_nan = sum (nanx, 1); x(nanx) = 1; # remove NaNs by subtracting is_nan below else n = szx(1); is_nan = 0; endif m = n ./ (sum (1 ./ x, 1) - is_nan); m(m == Inf) = 0; # handle zeros in X ## Inverse permute back to correct dimensions m = ipermute (m, perm); endif endif endif endif endif endfunction ## Test single input and optional arguments "all", DIM, "omitnan") %!test %! x = [0:10]; %! y = [x;x+5;x+10]; %! assert (harmmean (x), 0); %! m = [0 8.907635160795225 14.30854471766802]; %! assert (harmmean (y, 2), m', 4e-14); %! assert (harmmean (y, "all"), 0); %! y(2,4) = NaN; %! m(2) = 9.009855936313949; %! assert (harmmean (y, 2), [0 NaN m(3)]', 4e-14); %! assert (harmmean (y', "omitnan"), m, 4e-14); %! z = y + 20; %! assert (harmmean (z, "all"), NaN); %! assert (harmmean (z, "all", "includenan"), NaN); %! assert (harmmean (z, "all", "omitnan"), 29.1108719858295, 4e-14); %! m = [24.59488458841874 NaN 34.71244385944397]; %! assert (harmmean (z'), m, 4e-14); %! assert (harmmean (z', "includenan"), m, 4e-14); %! m(2) = 29.84104075528277; %! assert (harmmean (z', "omitnan"), m, 4e-14); %! assert (harmmean (z, 2, "omitnan"), m', 4e-14); ## Test dimension indexing with vecdim in n-dimensional arrays %!test %! x = repmat ([1:20;6:25], [5 2 6 3]); %! assert (size (harmmean (x, [3 2])), [10 1 1 3]); %! assert (size (harmmean (x, [1 2])), [1 1 6 3]); %! assert (size (harmmean (x, [1 2 4])), [1 1 6]); %! assert (size (harmmean (x, [1 4 3])), [1 40]); %! assert (size (harmmean (x, [1 2 3 4])), [1 1]); ## Test results with vecdim in n-dimensional arrays and "omitnan" %!test %! x = repmat ([1:20;6:25], [5 2 6 3]); %! m = repmat ([5.559045930488016;13.04950789021461], [5 1 1 3]); %! assert (harmmean (x, [3 2]), m, 4e-14); %! x(2,5,6,3) = NaN; %! m(2,3) = NaN; %! assert (harmmean (x, [3 2]), m, 4e-14); %! m(2,3) = 13.06617961315406; %! assert (harmmean (x, [3 2], "omitnan"), m, 4e-14); ## Test errors %!error harmmean ("char") %!error harmmean ([1 -1 3]) %!error ... %! harmmean (repmat ([1:20;6:25], [5 2 6 3 5]), -1) %!error ... %! harmmean (repmat ([1:20;6:25], [5 2 6 3 5]), 0) %!error ... %! harmmean (repmat ([1:20;6:25], [5 2 6 3 5]), [1 1]) statistics-release-1.6.3/inst/hist3.m000066400000000000000000000311171456127120000175170ustar00rootroot00000000000000## Copyright (C) 2015 Carnë Draug ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation; either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; if not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {} hist3 (@var{X}) ## @deftypefnx {statistics} {} hist3 (@var{X}, @var{nbins}) ## @deftypefnx {statistics} {} hist3 (@var{X}, @code{"Nbins"}, @var{nbins}) ## @deftypefnx {statistics} {} hist3 (@var{X}, @var{centers}) ## @deftypefnx {statistics} {} hist3 (@var{X}, @code{"Ctrs"}, @var{centers}) ## @deftypefnx {statistics} {} hist3 (@var{X}, @code{"Edges"}, @var{edges}) ## @deftypefnx {statistics} {[@var{N}, @var{C}] =} hist3 (@dots{}) ## @deftypefnx {statistics} {} hist3 (@dots{}, @var{prop}, @var{val}, @dots{}) ## @deftypefnx {statistics} {} hist3 (@var{hax}, @dots{}) ## ## Produce bivariate (2D) histogram counts or plots. ## ## The elements to produce the histogram are taken from the Nx2 matrix ## @var{X}. Any row with NaN values are ignored. The actual bins can ## be configured in 3 different: number, centers, or edges of the bins: ## ## @table @asis ## @item Number of bins (default) ## Produces equally spaced bins between the minimum and maximum values ## of @var{X}. Defined as a 2 element vector, @var{nbins}, one for each ## dimension. Defaults to @code{[10 10]}. ## ## @item Center of bins ## Defined as a cell array of 2 monotonically increasing vectors, ## @var{centers}. The width of each bin is determined from the adjacent ## values in the vector with the initial and final bin, extending to Infinity. ## ## @item Edge of bins ## Defined as a cell array of 2 monotonically increasing vectors, ## @var{edges}. @code{@var{N}(i,j)} contains the number of elements ## in @var{X} for which: ## ## @itemize @w{} ## @item ## @var{edges}@{1@}(i) <= @var{X}(:,1) < @var{edges}@{1@}(i+1) ## @item ## @var{edges}@{2@}(j) <= @var{X}(:,2) < @var{edges}@{2@}(j+1) ## @end itemize ## ## The consequence of this definition is that values outside the initial ## and final edge values are ignored, and that the final bin only contains ## the number of elements exactly equal to the final edge. ## ## @end table ## ## The return values, @var{N} and @var{C}, are the bin counts and centers ## respectively. These are specially useful to produce intensity maps: ## ## @example ## [counts, centers] = hist3 (data); ## imagesc (centers@{1@}, centers@{2@}, counts) ## @end example ## ## If there is no output argument, or if the axes graphics handle ## @var{hax} is defined, the function will plot a 3 dimensional bar ## graph. Any extra property/value pairs are passed directly to the ## underlying surface object. ## ## @seealso{hist, histc, lookup, mesh} ## @end deftypefn function [N, C] = hist3 (X, varargin) if (nargin < 1) print_usage (); endif next_argin = 1; should_draw = true; if (isaxes (X)) hax = X; X = varargin{next_argin++}; elseif (nargout == 0) hax = gca (); else should_draw = false; endif if (! ismatrix (X) || columns (X) != 2) error ("hist3: X must be a 2 columns matrix"); endif method = "nbins"; val = [10 10]; if (numel (varargin) >= next_argin) this_arg = varargin{next_argin++}; if (isnumeric (this_arg)) method = "nbins"; val = this_arg; elseif (iscell (this_arg)) method = "ctrs"; val = this_arg; elseif (numel (varargin) >= next_argin && any (strcmpi ({"nbins", "ctrs", "edges"}, this_arg))) method = tolower (this_arg); val = varargin{next_argin++}; else next_argin--; endif endif have_centers = false; switch (tolower (method)) case "nbins" [r_edges, c_edges] = edges_from_nbins (X, val); case "ctrs" have_centers = true; centers = val; [r_edges, c_edges] = edges_from_centers (val); case "centers" ## This was supported until 1.2.4 when the Matlab compatible option ## 'Ctrs' was added. persistent warned = false; if (! warned) warning ("hist3: option `centers' is deprecated. Use `ctrs'"); endif have_centers = true; centers = val; [r_edges, c_edges] = edges_from_centers (val); case "edges" if (! iscell (val) || numel (val) != 2 || ! all (cellfun (@isvector, val))) error ("hist3: EDGES must be a cell array with 2 vectors"); endif [r_edges] = vec (val{1}, 2); [c_edges] = vec (val{2}, 2); out_rows = any (X < [r_edges(1) c_edges(1)] | X > [r_edges(end) c_edges(end)], 2); X(out_rows,:) = []; otherwise ## we should never get here... error ("hist3: invalid binning method `%s'", method); endswitch ## We only remove the NaN now, after having computed the bin edges, ## because the extremes from each column that define the edges may ## be paired with a NaN. While such values do not appear on the ## histogram, they must still be used to compute the histogram ## edges. X(any (isnan (X), 2), :) = []; r_idx = lookup (r_edges, X(:,1), "l"); c_idx = lookup (c_edges, X(:,2), "l"); counts_size = [numel(r_edges) numel(c_edges)]; counts = accumarray ([r_idx, c_idx], 1, counts_size); if (should_draw) counts = counts.'; z = zeros ((size (counts) +1) *2); z(2:end-1,2:end-1) = kron (counts, ones (2, 2)); ## Setting the values for the end of the histogram bin like this ## seems straight wrong but that's hwo Matlab plots look. y = [kron(c_edges, ones (1, 2)) (c_edges(end)*2-c_edges(end-1))([1 1])]; x = [kron(r_edges, ones (1, 2)) (r_edges(end)*2-r_edges(end-1))([1 1])]; mesh (hax, x, y, z, "facecolor", [.75 .85 .95], varargin{next_argin:end}); else N = counts; if (nargout > 1) if (! have_centers) C = {(r_edges + [diff(r_edges)([1:end end])]/ 2) ... (c_edges + [diff(c_edges)([1:end end])]/ 2)}; else C = centers(:)'; C{1} = vec (C{1}, 2); C{2} = vec (C{2}, 2); endif endif endif endfunction function [r_edges, c_edges] = edges_from_nbins (X, nbins) if (! isnumeric (nbins) || numel (nbins) != 2) error ("hist3: NBINS must be a 2 element vector"); endif inits = min (X, [], 1); ends = max (X, [], 1); ends -= (ends - inits) ./ vec (nbins, 2); ## If any histogram side has an empty range, then still make NBINS ## but then place that value at the centre of the centre bin so that ## they appear in the centre in the plot. single_bins = inits == ends; if (any (single_bins)) inits(single_bins) -= (floor (nbins(single_bins) ./2)) + 0.5; ends(single_bins) = inits(single_bins) + nbins(single_bins) -1; endif r_edges = linspace (inits(1), ends(1), nbins(1)); c_edges = linspace (inits(2), ends(2), nbins(2)); endfunction function [r_edges, c_edges] = edges_from_centers (ctrs) if (! iscell (ctrs) || numel (ctrs) != 2 || ! all (cellfun (@isvector, ctrs))) error ("hist3: CTRS must be a cell array with 2 vectors"); endif r_edges = vec (ctrs{1}, 2); c_edges = vec (ctrs{2}, 2); r_edges(2:end) -= diff (r_edges) / 2; c_edges(2:end) -= diff (c_edges) / 2; endfunction %!demo %! X = [ %! 1 1 %! 1 1 %! 1 10 %! 1 10 %! 5 5 %! 5 5 %! 5 5 %! 5 5 %! 5 5 %! 7 3 %! 7 3 %! 7 3 %! 10 10 %! 10 10]; %! hist3 (X) %!test %! N_exp = [ 0 0 0 5 20 %! 0 0 10 15 0 %! 0 15 10 0 0 %! 20 5 0 0 0]; %! %! n = 100; %! x = [1:n]'; %! y = [n:-1:1]'; %! D = [x y]; %! N = hist3 (D, [4 5]); %! assert (N, N_exp); %!test %! N_exp = [0 0 0 0 1 %! 0 0 0 0 1 %! 0 0 0 0 1 %! 1 1 1 1 93]; %! %! n = 100; %! x = [1:n]'; %! y = [n:-1:1]'; %! D = [x y]; %! C{1} = [1 1.7 3 4]; %! C{2} = [1:5]; %! N = hist3 (D, C); %! assert (N, N_exp); ## bug 44987 %!test %! D = [1 1; 3 1; 3 3; 3 1]; %! [c, nn] = hist3 (D, {0:4, 0:4}); %! exp_c = zeros (5); %! exp_c([7 9 19]) = [1 2 1]; %! assert (c, exp_c); %! assert (nn, {0:4, 0:4}); %!test %! for i = 10 %! assert (size (hist3 (rand (9, 2), "Edges", {[0:.2:1]; [0:.2:1]})), [6 6]) %! endfor %!test %! edge_1 = linspace (0, 10, 10); %! edge_2 = linspace (0, 50, 10); %! [c, nn] = hist3 ([1:10; 1:5:50]', "Edges", {edge_1, edge_2}); %! exp_c = zeros (10, 10); %! exp_c([1 12 13 24 35 46 57 68 79 90]) = 1; %! assert (c, exp_c); %! %! assert (nn{1}, edge_1 + edge_1(2)/2, eps*10^4) %! assert (nn{2}, edge_2 + edge_2(2)/2, eps*10^4) %!shared X %! X = [ %! 5 2 %! 5 3 %! 1 4 %! 5 3 %! 4 4 %! 1 2 %! 2 3 %! 3 3 %! 5 4 %! 5 3]; %!test %! N = zeros (10); %! N([1 10 53 56 60 91 98 100]) = [1 1 1 1 3 1 1 1]; %! C = {(1.2:0.4:4.8), (2.1:0.2:3.9)}; %! assert (nthargout ([1 2], @hist3, X), {N C}, eps*10^3) %!test %! N = zeros (5, 7); %! N([1 5 17 18 20 31 34 35]) = [1 1 1 1 3 1 1 1]; %! C = {(1.4:0.8:4.6), ((2+(1/7)):(2/7):(4-(1/7)))}; %! assert (nthargout ([1 2], @hist3, X, [5 7]), {N C}, eps*10^3) %! assert (nthargout ([1 2], @hist3, X, "Nbins", [5 7]), {N C}, eps*10^3) %!test %! N = [0 1 0; 0 1 0; 0 0 1; 0 0 0]; %! C = {(2:5), (2.5:1:4.5)}; %! assert (nthargout ([1 2], @hist3, X, "Edges", {(1.5:4.5), (2:4)}), {N C}) %!test %! N = [0 0 1 0 1 0; 0 0 0 1 0 0; 0 0 1 4 2 0]; %! C = {(1.2:3.2), (0:5)}; %! assert (nthargout ([1 2], @hist3, X, "Ctrs", C), {N C}) %! assert (nthargout ([1 2], @hist3, X, C), {N C}) %!test %! [~, C] = hist3 (rand (10, 2), "Edges", {[0 .05 .15 .35 .55 .95], %! [-1 .05 .07 .2 .3 .5 .89 1.2]}); %! C_exp = {[ 0.025 0.1 0.25 0.45 0.75 1.15], ... %! [-0.475 0.06 0.135 0.25 0.4 0.695 1.045 1.355]}; %! assert (C, C_exp, eps*10^2) ## Test how handling of out of borders is different whether we are ## defining Centers or Edges. %!test %! Xv = repmat ([1:10]', [1 2]); %! %! ## Test Centers %! assert (hist3 (Xv, "Ctrs", {1:10, 1:10}), eye (10)) %! %! N_exp = eye (6); %! N_exp([1 end]) = 3; %! assert (hist3 (Xv, "Ctrs", {3:8, 3:8}), N_exp) %! %! N_exp = zeros (8, 6); %! N_exp([1 2 11 20 29 38 47 48]) = [2 1 1 1 1 1 1 2]; %! assert (hist3 (Xv, "Ctrs", {2:9, 3:8}), N_exp) %! %! ## Test Edges %! assert (hist3 (Xv, "Edges", {1:10, 1:10}), eye (10)) %! assert (hist3 (Xv, "Edges", {3:8, 3:8}), eye (6)) %! assert (hist3 (Xv, "Edges", {2:9, 3:8}), [zeros(1, 6); eye(6); zeros(1, 6)]) %! %! N_exp = zeros (14); %! N_exp(3:12, 3:12) = eye (10); %! assert (hist3 (Xv, "Edges", {-1:12, -1:12}), N_exp) %! %! ## Test for Nbins %! assert (hist3 (Xv), eye (10)) %! assert (hist3 (Xv, [10 10]), eye (10)) %! assert (hist3 (Xv, "nbins", [10 10]), eye (10)) %! assert (hist3 (Xv, [5 5]), eye (5) * 2) %! %! N_exp = zeros (7, 5); %! N_exp([1 9 10 18 26 27 35]) = [2 1 1 2 1 1 2]; %! assert (hist3 (Xv, [7 5]), N_exp) %!test # bug #51059 %! D = [1 1; NaN 2; 3 1; 3 3; 1 NaN; 3 1]; %! [c, nn] = hist3 (D, {0:4, 0:4}); %! exp_c = zeros (5); %! exp_c([7 9 19]) = [1 2 1]; %! assert (c, exp_c) %! assert (nn, {0:4, 0:4}) ## Single row of data or cases where all elements have the same value ## on one side of the histogram. %!test %! [c, nn] = hist3 ([1 8]); %! exp_c = zeros (10, 10); %! exp_c(6, 6) = 1; %! exp_nn = {-4:5, 3:12}; %! assert (c, exp_c) %! assert (nn, exp_nn, eps) %! %! [c, nn] = hist3 ([1 8], [10 11]); %! exp_c = zeros (10, 11); %! exp_c(6, 6) = 1; %! exp_nn = {-4:5, 3:13}; %! assert (c, exp_c) %! assert (nn, exp_nn, eps) ## NaNs paired with values defining the histogram edges. %!test %! [c, nn] = hist3 ([1 NaN; 2 3; 6 9; 8 NaN]); %! exp_c = zeros (10, 10); %! exp_c(2, 1) = 1; %! exp_c(8, 10) = 1; %! exp_nn = {linspace(1.35, 7.65, 10) linspace(3.3, 8.7, 10)}; %! assert (c, exp_c) %! assert (nn, exp_nn, eps*100) ## Columns full of NaNs (recent Matlab versions seem to throw an error ## but this did work like this on R2010b at least). %!test %! [c, nn] = hist3 ([1 NaN; 2 NaN; 6 NaN; 8 NaN]); %! exp_c = zeros (10, 10); %! exp_nn = {linspace(1.35, 7.65, 10) NaN(1, 10)}; %! assert (c, exp_c) %! assert (nn, exp_nn, eps*100) ## Behaviour of an empty X after removal of rows with NaN. %!test %! [c, nn] = hist3 ([1 NaN; NaN 3; NaN 9; 8 NaN]); %! exp_c = zeros (10, 10); %! exp_nn = {linspace(1.35, 7.65, 10) linspace(3.3, 8.7, 10)}; %! assert (c, exp_c) %! assert (nn, exp_nn, eps*100) statistics-release-1.6.3/inst/histfit.m000066400000000000000000000064021456127120000201360ustar00rootroot00000000000000## Copyright (C) 2003 Alberto Terruzzi ## Copyright (C) 2022-2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {} histfit (@var{x}, @var{nbins}) ## @deftypefnx {statistics} {@var{h} =} histfit (@var{x}, @var{nbins}) ## ## Plot histogram with superimposed fitted normal density. ## ## @code{histfit (@var{x}, @var{nbins})} plots a histogram of the values in ## the vector @var{x} using @var{nbins} bars in the histogram. With one input ## argument, @var{nbins} is set to the square root of the number of elements in ## @var{x}. ## ## @code{@var{h} = histfit (@var{x}, @var{nbins})} returns the bins and fitted ## line handles of the plot in @var{h}. ## ## Example ## ## @example ## histfit (randn (100, 1)) ## @end example ## ## @seealso{bar, hist, pareto} ## @end deftypefn function [varargout] = histfit (x, nbins) if (nargin < 1 || nargin > 2) print_usage; endif if (! isnumeric (x) || ! isreal (x) || ! isvector (x) || isscalar (x)) error ("histfit: X must be a numeric vector of real numbers."); endif row = sum (! isnan (x)); if (nargin < 2) nbins = ceil (sqrt (row)); endif [n, xbin] = hist (x, nbins); if (any (abs (diff (xbin, 2)) > 10 * max (abs (xbin)) * eps)) error ("histfit: bins must have uniform width."); endif ## Compute mu and sigma parameters mr = mean (x, "omitnan"); sr = std (x); ## Evenly spaced samples of the expected range in X x = (-3*sr+mr:0.1*sr:3*sr+mr)'; [xb, yb] = bar (xbin, n); y = normpdf (x, mr, sr); binwidth = xbin(2) - xbin(1); ## Necessary normalization to overplot the histogram y = row * y * binwidth; ## Plot density line over histogram. h = plot (xb, yb, ";;b", x, y, ";;r-"); ## Return the plot's handle if requested if (nargout == 1) varargout{1} = h; endif endfunction %!demo %! histfit (randn (100, 1)) ## Test plotting %!test %! hf = figure ("visible", "off"); %! unwind_protect %! x = [2, 4, 3, 2, 4, 3, 2, 5, 6, 4, 7, 5, 9, 8, 10, 4, 11]; %! histfit (x); %! unwind_protect_cleanup %! close (hf); %! end_unwind_protect %!test %! hf = figure ("visible", "off"); %! unwind_protect %! x = [2, 4, 3, 2, NaN, 3, 2, 5, 6, 4, 7, 5, 9, 8, 10, 4, 11]; %! histfit (x); %! unwind_protect_cleanup %! close (hf); %! end_unwind_protect %!test %! hf = figure ("visible", "off"); %! unwind_protect %! x = [2, 4, 3, 2, NaN, 3, 2, 5, 6, 4, 7, 5, 9, 8, 10, 4, 11]; %! histfit (x, 3); %! unwind_protect_cleanup %! close (hf); %! end_unwind_protect ## Test input validation %!error histfit (); %!error histfit ([x',x']); statistics-release-1.6.3/inst/hmmestimate.m000066400000000000000000000340551456127120000210060ustar00rootroot00000000000000## Copyright (C) 2006, 2007 Arno Onken ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {[@var{transprobest}, @var{outprobest}] =} hmmestimate (@var{sequence}, @var{states}) ## @deftypefnx {statistics} {[@dots{}] =} hmmestimate (@dots{}, @code{"statenames"}, @var{statenames}) ## @deftypefnx {statistics} {[@dots{}] =} hmmestimate (@dots{}, @code{"symbols"}, @var{symbols}) ## @deftypefnx {statistics} {[@dots{}] =} hmmestimate (@dots{}, @code{"pseudotransitions"}, @var{pseudotransitions}) ## @deftypefnx {statistics} {[@dots{}] =} hmmestimate (@dots{}, @code{"pseudoemissions"}, @var{pseudoemissions}) ## ## Estimation of a hidden Markov model for a given sequence. ## ## Estimate the matrix of transition probabilities and the matrix of output ## probabilities of a given sequence of outputs and states generated by a ## hidden Markov model. The model assumes that the generation starts in ## state @code{1} at step @code{0} but does not include step @code{0} in the ## generated states and sequence. ## ## @subheading Arguments ## ## @itemize @bullet ## @item ## @var{sequence} is a vector of a sequence of given outputs. The outputs ## must be integers ranging from @code{1} to the number of outputs of the ## hidden Markov model. ## ## @item ## @var{states} is a vector of the same length as @var{sequence} of given ## states. The states must be integers ranging from @code{1} to the number ## of states of the hidden Markov model. ## @end itemize ## ## @subheading Return values ## ## @itemize @bullet ## @item ## @var{transprobest} is the matrix of the estimated transition ## probabilities of the states. @code{transprobest(i, j)} is the estimated ## probability of a transition to state @code{j} given state @code{i}. ## ## @item ## @var{outprobest} is the matrix of the estimated output probabilities. ## @code{outprobest(i, j)} is the estimated probability of generating ## output @code{j} given state @code{i}. ## @end itemize ## ## If @code{'symbols'} is specified, then @var{sequence} is expected to be a ## sequence of the elements of @var{symbols} instead of integers. ## @var{symbols} can be a cell array. ## ## If @code{'statenames'} is specified, then @var{states} is expected to be ## a sequence of the elements of @var{statenames} instead of integers. ## @var{statenames} can be a cell array. ## ## If @code{'pseudotransitions'} is specified then the integer matrix ## @var{pseudotransitions} is used as an initial number of counted ## transitions. @code{pseudotransitions(i, j)} is the initial number of ## counted transitions from state @code{i} to state @code{j}. ## @var{transprobest} will have the same size as @var{pseudotransitions}. ## Use this if you have transitions that are very unlikely to occur. ## ## If @code{'pseudoemissions'} is specified then the integer matrix ## @var{pseudoemissions} is used as an initial number of counted outputs. ## @code{pseudoemissions(i, j)} is the initial number of counted outputs ## @code{j} given state @code{i}. If @code{'pseudoemissions'} is also ## specified then the number of rows of @var{pseudoemissions} must be the ## same as the number of rows of @var{pseudotransitions}. @var{outprobest} ## will have the same size as @var{pseudoemissions}. Use this if you have ## outputs or states that are very unlikely to occur. ## ## @subheading Examples ## ## @example ## @group ## transprob = [0.8, 0.2; 0.4, 0.6]; ## outprob = [0.2, 0.4, 0.4; 0.7, 0.2, 0.1]; ## [sequence, states] = hmmgenerate (25, transprob, outprob); ## [transprobest, outprobest] = hmmestimate (sequence, states) ## @end group ## ## @group ## symbols = @{"A", "B", "C"@}; ## statenames = @{"One", "Two"@}; ## [sequence, states] = hmmgenerate (25, transprob, outprob, ... ## "symbols", symbols, ... ## "statenames", statenames); ## [transprobest, outprobest] = hmmestimate (sequence, states, ... ## "symbols', symbols, ... ## "statenames', statenames) ## @end group ## ## @group ## pseudotransitions = [8, 2; 4, 6]; ## pseudoemissions = [2, 4, 4; 7, 2, 1]; ## [sequence, states] = hmmgenerate (25, transprob, outprob); ## [transprobest, outprobest] = hmmestimate (sequence, states, ... ## "pseudotransitions", pseudotransitions, ... ## "pseudoemissions", pseudoemissions) ## @end group ## @end example ## ## @subheading References ## ## @enumerate ## @item ## Wendy L. Martinez and Angel R. Martinez. @cite{Computational Statistics ## Handbook with MATLAB}. Appendix E, pages 547-557, Chapman & Hall/CRC, ## 2001. ## ## @item ## Lawrence R. Rabiner. A Tutorial on Hidden Markov Models and Selected ## Applications in Speech Recognition. @cite{Proceedings of the IEEE}, ## 77(2), pages 257-286, February 1989. ## @end enumerate ## @end deftypefn function [transprobest, outprobest] = hmmestimate (sequence, states, varargin) # Check arguments if (nargin < 2 || mod (length (varargin), 2) != 0) print_usage (); endif len = length (sequence); if (length (states) != len) error ("hmmestimate: sequence and states must have equal length"); endif # Flag for symbols usesym = false; # Flag for statenames usesn = false; # Variables for return values transprobest = []; outprobest = []; # Process varargin for i = 1:2:length (varargin) # There must be an identifier: 'symbols', 'statenames', # 'pseudotransitions' or 'pseudoemissions' if (! ischar (varargin{i})) print_usage (); endif # Upper case is also fine lowerarg = lower (varargin{i}); if (strcmp (lowerarg, 'symbols')) usesym = true; # Use the following argument as symbols symbols = varargin{i + 1}; # The same for statenames elseif (strcmp (lowerarg, 'statenames')) usesn = true; # Use the following argument as statenames statenames = varargin{i + 1}; elseif (strcmp (lowerarg, 'pseudotransitions')) # Use the following argument as an initial count for transitions transprobest = varargin{i + 1}; if (! ismatrix (transprobest)) error (strcat (["hmmestimate: pseudotransitions must be a"], ... [" non-empty numeric matrix"])); endif if (rows (transprobest) != columns (transprobest)) error ("hmmestimate: pseudotransitions must be a square matrix"); endif elseif (strcmp (lowerarg, 'pseudoemissions')) # Use the following argument as an initial count for outputs outprobest = varargin{i + 1}; if (! ismatrix (outprobest)) error (strcat (["hmmestimate: pseudoemissions must be a non-empty"], ... [" numeric matrix"])); endif else error (strcat (["hmmestimate: expected 'symbols', 'statenames',"], ... [" 'pseudotransitions' or 'pseudoemissions' but"], ... sprintf (" found '%s'", varargin{i}))); endif endfor # Transform sequence from symbols to integers if necessary if (usesym) # sequenceint is used to build the transformed sequence sequenceint = zeros (1, len); for i = 1:length (symbols) # Search for symbols(i) in the sequence, isequal will have 1 at # corresponding indices; i is the right integer for that symbol isequal = ismember (sequence, symbols(i)); # We do not want to change sequenceint if the symbol appears a second # time in symbols if (any ((sequenceint == 0) & (isequal == 1))) isequal *= i; sequenceint += isequal; endif endfor if (! all (sequenceint)) index = max ((sequenceint == 0) .* (1:len)); error (["hmmestimate: sequence(" int2str (index) ") not in symbols"]); endif sequence = sequenceint; else if (! isvector (sequence)) error ("hmmestimate: sequence must be a non-empty vector"); endif if (! all (ismember (sequence, 1:max (sequence)))) index = max ((ismember (sequence, 1:max (sequence)) == 0) .* (1:len)); error (["hmmestimate: sequence(" int2str (index) ") not feasible"]); endif endif # Transform states from statenames to integers if necessary if (usesn) # statesint is used to build the transformed states statesint = zeros (1, len); for i = 1:length (statenames) # Search for statenames(i) in states, isequal will have 1 at # corresponding indices; i is the right integer for that statename isequal = ismember (states, statenames(i)); # We do not want to change statesint if the statename appears a second # time in statenames if (any ((statesint == 0) & (isequal == 1))) isequal *= i; statesint += isequal; endif endfor if (! all (statesint)) index = max ((statesint == 0) .* (1:len)); error (["hmmestimate: states(" int2str (index) ") not in statenames"]); endif states = statesint; else if (! isvector (states)) error ("hmmestimate: states must be a non-empty vector"); endif if (! all (ismember (states, 1:max (states)))) index = max ((ismember (states, 1:max (states)) == 0) .* (1:len)); error (["hmmestimate: states(" int2str (index) ") not feasible"]); endif endif # Estimate the number of different states as the max of states nstate = max (states); # Estimate the number of different outputs as the max of sequence noutput = max (sequence); # transprobest is empty if pseudotransitions is not specified if (isempty (transprobest)) # outprobest is not empty if pseudoemissions is specified if (! isempty (outprobest)) if (nstate > rows (outprobest)) error ("hmmestimate: not enough rows in pseudoemissions"); endif # The number of states is specified by pseudoemissions nstate = rows (outprobest); endif transprobest = zeros (nstate, nstate); else if (nstate > rows (transprobest)) error ("hmmestimate: not enough rows in pseudotransitions"); endif # The number of states is given by pseudotransitions nstate = rows (transprobest); endif # outprobest is empty if pseudoemissions is not specified if (isempty (outprobest)) outprobest = zeros (nstate, noutput); else if (noutput > columns (outprobest)) error ("hmmestimate: not enough columns in pseudoemissions"); endif # Number of outputs is specified by pseudoemissions noutput = columns (outprobest); if (rows (outprobest) != nstate) error (strcat (["hmmestimate: pseudoemissions must have the same"], ... [" number of rows as pseudotransitions"])); endif endif # Assume that the model started in state 1 cstate = 1; for i = 1:len # Count the number of transitions for each state pair transprobest(cstate, states(i)) ++; cstate = states (i); # Count the number of outputs for each state output pair outprobest(cstate, sequence(i)) ++; endfor # transprobest and outprobest contain counted numbers # Each row in transprobest and outprobest should contain estimated # probabilities # => scale so that the sum is 1 # A zero row remains zero # - for transprobest s = sum (transprobest, 2); s(s == 0) = 1; transprobest = transprobest ./ (s * ones (1, nstate)); # - for outprobest s = sum (outprobest, 2); s(s == 0) = 1; outprobest = outprobest ./ (s * ones (1, noutput)); endfunction %!test %! sequence = [1, 2, 1, 1, 1, 2, 2, 1, 2, 3, 3, ... %! 3, 3, 2, 3, 1, 1, 1, 1, 3, 3, 2, 3, 1, 3]; %! states = [1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, ... %! 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1]; %! [transprobest, outprobest] = hmmestimate (sequence, states); %! expectedtransprob = [0.88889, 0.11111; 0.28571, 0.71429]; %! expectedoutprob = [0.16667, 0.33333, 0.50000; 1.00000, 0.00000, 0.00000]; %! assert (transprobest, expectedtransprob, 0.001); %! assert (outprobest, expectedoutprob, 0.001); %!test %! sequence = {"A", "B", "A", "A", "A", "B", "B", "A", "B", "C", "C", "C", ... %! "C", "B", "C", "A", "A", "A", "A", "C", "C", "B", "C", "A", "C"}; %! states = {"One", "One", "Two", "Two", "Two", "One", "One", "One", "One", ... %! "One", "One", "One", "One", "One", "One", "Two", "Two", "Two", ... %! "Two", "One", "One", "One", "One", "One", "One"}; %! symbols = {"A", "B", "C"}; %! statenames = {"One", "Two"}; %! [transprobest, outprobest] = hmmestimate (sequence, states, "symbols", ... %! symbols, "statenames", statenames); %! expectedtransprob = [0.88889, 0.11111; 0.28571, 0.71429]; %! expectedoutprob = [0.16667, 0.33333, 0.50000; 1.00000, 0.00000, 0.00000]; %! assert (transprobest, expectedtransprob, 0.001); %! assert (outprobest, expectedoutprob, 0.001); %!test %! sequence = [1, 2, 1, 1, 1, 2, 2, 1, 2, 3, 3, 3, ... %! 3, 2, 3, 1, 1, 1, 1, 3, 3, 2, 3, 1, 3]; %! states = [1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, ... %! 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1]; %! pseudotransitions = [8, 2; 4, 6]; %! pseudoemissions = [2, 4, 4; 7, 2, 1]; %! [transprobest, outprobest] = hmmestimate (sequence, states, ... %! "pseudotransitions", pseudotransitions, "pseudoemissions", pseudoemissions); %! expectedtransprob = [0.85714, 0.14286; 0.35294, 0.64706]; %! expectedoutprob = [0.178571, 0.357143, 0.464286; ... %! 0.823529, 0.117647, 0.058824]; %! assert (transprobest, expectedtransprob, 0.001); %! assert (outprobest, expectedoutprob, 0.001); statistics-release-1.6.3/inst/hmmgenerate.m000066400000000000000000000216721456127120000207660ustar00rootroot00000000000000## Copyright (C) 2006, 2007 Arno Onken ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {[@var{sequence}, @var{states}] =} hmmgenerate (@var{len}, @var{transprob}, @var{outprob}) ## @deftypefnx {statistics} {[@dots{}] =} hmmgenerate (@dots{}, @code{"symbols"}, @var{symbols}) ## @deftypefnx {statistics} {[@dots{}] =} hmmgenerate (@dots{}, @code{"statenames"}, @var{statenames}) ## ## Output sequence and hidden states of a hidden Markov model. ## ## Generate an output sequence and hidden states of a hidden Markov model. ## The model starts in state @code{1} at step @code{0} but will not include ## step @code{0} in the generated states and sequence. ## ## @subheading Arguments ## ## @itemize @bullet ## @item ## @var{len} is the number of steps to generate. @var{sequence} and ## @var{states} will have @var{len} entries each. ## ## @item ## @var{transprob} is the matrix of transition probabilities of the states. ## @code{transprob(i, j)} is the probability of a transition to state ## @code{j} given state @code{i}. ## ## @item ## @var{outprob} is the matrix of output probabilities. ## @code{outprob(i, j)} is the probability of generating output @code{j} ## given state @code{i}. ## @end itemize ## ## @subheading Return values ## ## @itemize @bullet ## @item ## @var{sequence} is a vector of length @var{len} of the generated ## outputs. The outputs are integers ranging from @code{1} to ## @code{columns (outprob)}. ## ## @item ## @var{states} is a vector of length @var{len} of the generated hidden ## states. The states are integers ranging from @code{1} to ## @code{columns (transprob)}. ## @end itemize ## ## If @code{"symbols"} is specified, then the elements of @var{symbols} are ## used for the output sequence instead of integers ranging from @code{1} to ## @code{columns (outprob)}. @var{symbols} can be a cell array. ## ## If @code{"statenames"} is specified, then the elements of ## @var{statenames} are used for the states instead of integers ranging from ## @code{1} to @code{columns (transprob)}. @var{statenames} can be a cell ## array. ## ## @subheading Examples ## ## @example ## @group ## transprob = [0.8, 0.2; 0.4, 0.6]; ## outprob = [0.2, 0.4, 0.4; 0.7, 0.2, 0.1]; ## [sequence, states] = hmmgenerate (25, transprob, outprob) ## @end group ## ## @group ## symbols = @{"A", "B", "C"@}; ## statenames = @{"One", "Two"@}; ## [sequence, states] = hmmgenerate (25, transprob, outprob, ... ## "symbols", symbols, ... ## "statenames", statenames) ## @end group ## @end example ## ## @subheading References ## ## @enumerate ## @item ## Wendy L. Martinez and Angel R. Martinez. @cite{Computational Statistics ## Handbook with MATLAB}. Appendix E, pages 547-557, Chapman & Hall/CRC, ## 2001. ## ## @item ## Lawrence R. Rabiner. A Tutorial on Hidden Markov Models and Selected ## Applications in Speech Recognition. @cite{Proceedings of the IEEE}, ## 77(2), pages 257-286, February 1989. ## @end enumerate ## @end deftypefn function [sequence, states] = hmmgenerate (len, transprob, outprob, varargin) # Check arguments if (nargin < 3 || mod (length (varargin), 2) != 0) print_usage (); endif if (! isscalar (len) || len < 0 || round (len) != len) error ("hmmgenerate: len must be a non-negative scalar integer.") endif if (! ismatrix (transprob)) error ("hmmgenerate: transprob must be a non-empty numeric matrix."); endif if (! ismatrix (outprob)) error ("hmmgenerate: outprob must be a non-empty numeric matrix."); endif # nstate is the number of states of the hidden Markov model nstate = rows (transprob); # noutput is the number of different outputs that the hidden Markov model # can generate noutput = columns (outprob); # Check whether transprob and outprob are feasible for a hidden Markov # model if (columns (transprob) != nstate) error ("hmmgenerate: transprob must be a square matrix."); endif if (rows (outprob) != nstate) error (strcat (["hmmgenerate: outprob must have the same number"], ... [" of rows as transprob."])); endif # Flag for symbols usesym = false; # Flag for statenames usesn = false; # Process varargin for i = 1:2:length (varargin) # There must be an identifier: 'symbols' or 'statenames' if (! ischar (varargin{i})) print_usage (); endif # Upper case is also fine lowerarg = lower (varargin{i}); if (strcmp (lowerarg, 'symbols')) if (length (varargin{i + 1}) != noutput) error (strcat (["hmmgenerate: number of symbols does not match"], ... [" number of possible outputs."])); endif usesym = true; # Use the following argument as symbols symbols = varargin{i + 1}; # The same for statenames elseif (strcmp (lowerarg, 'statenames')) if (length (varargin{i + 1}) != nstate) error (strcat (["hmmgenerate: number of statenames does not"], ... [" match number of states."])); endif usesn = true; # Use the following argument as statenames statenames = varargin{i + 1}; else error (strcat (["hmmgenerate: expected 'symbols' or 'statenames'"], ... sprintf (" but found '%s'.", varargin{i}))); endif endfor # Each row in transprob and outprob should contain probabilities # => scale so that the sum is 1 # A zero row remains zero # - for transprob s = sum (transprob, 2); s(s == 0) = 1; transprob = transprob ./ repmat (s, 1, nstate); # - for outprob s = sum (outprob, 2); s(s == 0) = 1; outprob = outprob ./ repmat (s, 1, noutput); # Generate sequences of uniformly distributed random numbers between 0 and 1 # - for the state transitions transdraw = rand (1, len); # - for the outputs outdraw = rand (1, len); # Generate the return vectors # They remain unchanged if the according probability row of transprob # and outprob contain, respectively, only zeros sequence = ones (1, len); states = ones (1, len); if (len > 0) # Calculate cumulated probabilities backwards for easy comparison with # the generated random numbers # Cumulated probability in first column must always be 1 # We might have a zero row # - for transprob transprob(:, end:-1:1) = cumsum (transprob(:, end:-1:1), 2); transprob(:, 1) = 1; # - for outprob outprob(:, end:-1:1) = cumsum (outprob(:, end:-1:1), 2); outprob(:, 1) = 1; # cstate is the current state # Start in state 1 but do not include it in the states vector cstate = 1; for i = 1:len # Compare the randon number i of transdraw to the cumulated # probability of the state transition and set the transition # accordingly states(i) = sum (transdraw(i) <= transprob(cstate, :)); cstate = states(i); endfor # Compare the random numbers of outdraw to the cumulated probabilities # of the outputs and set the sequence vector accordingly sequence = sum (repmat (outdraw, noutput, 1) <= outprob(states, :)', 1); # Transform default matrices into symbols/statenames if requested if (usesym) sequence = reshape (symbols(sequence), 1, len); endif if (usesn) states = reshape (statenames(states), 1, len); endif endif endfunction %!test %! len = 25; %! transprob = [0.8, 0.2; 0.4, 0.6]; %! outprob = [0.2, 0.4, 0.4; 0.7, 0.2, 0.1]; %! [sequence, states] = hmmgenerate (len, transprob, outprob); %! assert (length (sequence), len); %! assert (length (states), len); %! assert (min (sequence) >= 1); %! assert (max (sequence) <= columns (outprob)); %! assert (min (states) >= 1); %! assert (max (states) <= rows (transprob)); %!test %! len = 25; %! transprob = [0.8, 0.2; 0.4, 0.6]; %! outprob = [0.2, 0.4, 0.4; 0.7, 0.2, 0.1]; %! symbols = {"A", "B", "C"}; %! statenames = {"One", "Two"}; %! [sequence, states] = hmmgenerate (len, transprob, outprob, ... %! "symbols", symbols, "statenames", statenames); %! assert (length (sequence), len); %! assert (length (states), len); %! assert (strcmp (sequence, "A") + strcmp (sequence, "B") + ... %! strcmp (sequence, "C") == ones (1, len)); %! assert (strcmp (states, "One") + strcmp (states, "Two") == ones (1, len)); statistics-release-1.6.3/inst/hmmviterbi.m000066400000000000000000000230061456127120000206310ustar00rootroot00000000000000## Copyright (C) 2006, 2007 Arno Onken ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{vpath} =} hmmviterbi (@var{sequence}, @var{transprob}, @var{outprob}) ## @deftypefnx {statistics} {@var{vpath} =} hmmviterbi (@dots{}, @code{"symbols"}, @var{symbols}) ## @deftypefnx {statistics} {@var{vpath} =} hmmviterbi (@dots{}, @code{"statenames"}, @var{statenames}) ## ## Viterbi path of a hidden Markov model. ## ## Use the Viterbi algorithm to find the Viterbi path of a hidden Markov ## model given a sequence of outputs. The model assumes that the generation ## starts in state @code{1} at step @code{0} but does not include step ## @code{0} in the generated states and sequence. ## ## @subheading Arguments ## ## @itemize @bullet ## @item ## @var{sequence} is the vector of length @var{len} of given outputs. The ## outputs must be integers ranging from @code{1} to ## @code{columns (outprob)}. ## ## @item ## @var{transprob} is the matrix of transition probabilities of the states. ## @code{transprob(i, j)} is the probability of a transition to state ## @code{j} given state @code{i}. ## ## @item ## @var{outprob} is the matrix of output probabilities. ## @code{outprob(i, j)} is the probability of generating output @code{j} ## given state @code{i}. ## @end itemize ## ## @subheading Return values ## ## @itemize @bullet ## @item ## @var{vpath} is the vector of the same length as @var{sequence} of the ## estimated hidden states. The states are integers ranging from @code{1} to ## @code{columns (transprob)}. ## @end itemize ## ## If @code{"symbols"} is specified, then @var{sequence} is expected to be a ## sequence of the elements of @var{symbols} instead of integers ranging ## from @code{1} to @code{columns (outprob)}. @var{symbols} can be a cell array. ## ## If @code{"statenames"} is specified, then the elements of ## @var{statenames} are used for the states in @var{vpath} instead of ## integers ranging from @code{1} to @code{columns (transprob)}. ## @var{statenames} can be a cell array. ## ## @subheading Examples ## ## @example ## @group ## transprob = [0.8, 0.2; 0.4, 0.6]; ## outprob = [0.2, 0.4, 0.4; 0.7, 0.2, 0.1]; ## [sequence, states] = hmmgenerate (25, transprob, outprob); ## vpath = hmmviterbi (sequence, transprob, outprob); ## @end group ## ## @group ## symbols = @{"A", "B", "C"@}; ## statenames = @{"One", "Two"@}; ## [sequence, states] = hmmgenerate (25, transprob, outprob, ... ## "symbols", symbols, "statenames", statenames); ## vpath = hmmviterbi (sequence, transprob, outprob, ... ## "symbols", symbols, "statenames", statenames); ## @end group ## @end example ## ## @subheading References ## ## @enumerate ## @item ## Wendy L. Martinez and Angel R. Martinez. @cite{Computational Statistics ## Handbook with MATLAB}. Appendix E, pages 547-557, Chapman & Hall/CRC, ## 2001. ## ## @item ## Lawrence R. Rabiner. A Tutorial on Hidden Markov Models and Selected ## Applications in Speech Recognition. @cite{Proceedings of the IEEE}, ## 77(2), pages 257-286, February 1989. ## @end enumerate ## @end deftypefn function vpath = hmmviterbi (sequence, transprob, outprob, varargin) # Check arguments if (nargin < 3 || mod (length (varargin), 2) != 0) print_usage (); endif if (! ismatrix (transprob)) error ("hmmviterbi: transprob must be a non-empty numeric matrix"); endif if (! ismatrix (outprob)) error ("hmmviterbi: outprob must be a non-empty numeric matrix"); endif len = length (sequence); # nstate is the number of states of the hidden Markov model nstate = rows (transprob); # noutput is the number of different outputs that the hidden Markov model # can generate noutput = columns (outprob); # Check whether transprob and outprob are feasible for a hidden Markov model if (columns (transprob) != nstate) error ("hmmviterbi: transprob must be a square matrix"); endif if (rows (outprob) != nstate) error (strcat (["hmmviterbi: outprob must have the same number of"], ... [" rows as transprob"])); endif # Flag for symbols usesym = false; # Flag for statenames usesn = false; # Process varargin for i = 1:2:length (varargin) # There must be an identifier: 'symbols' or 'statenames' if (! ischar (varargin{i})) print_usage (); endif # Upper case is also fine lowerarg = lower (varargin{i}); if (strcmp (lowerarg, 'symbols')) if (length (varargin{i + 1}) != noutput) error (strcat (["hmmviterbi: number of symbols does not match"], ... [" number of possible outputs"])); endif usesym = true; # Use the following argument as symbols symbols = varargin{i + 1}; # The same for statenames elseif (strcmp (lowerarg, 'statenames')) if (length (varargin{i + 1}) != nstate) error (strcat (["hmmviterbi: number of statenames does not match"], ... [" number of states"])); endif usesn = true; # Use the following argument as statenames statenames = varargin{i + 1}; else error (strcat (["hmmviterbi: expected 'symbols' or 'statenames'"], ... sprintf (" but found '%s'", varargin{i}))); endif endfor # Transform sequence from symbols to integers if necessary if (usesym) # sequenceint is used to build the transformed sequence sequenceint = zeros (1, len); for i = 1:noutput # Search for symbols(i) in the sequence, isequal will have 1 at # corresponding indices; i is the right integer for that symbol isequal = ismember (sequence, symbols(i)); # We do not want to change sequenceint if the symbol appears a second # time in symbols if (any ((sequenceint == 0) & (isequal == 1))) isequal *= i; sequenceint += isequal; endif endfor if (! all (sequenceint)) index = max ((sequenceint == 0) .* (1:len)); error (["hmmviterbi: sequence(" int2str (index) ") not in symbols"]); endif sequence = sequenceint; else if (! isvector (sequence) && ! isempty (sequence)) error ("hmmviterbi: sequence must be a vector"); endif if (! all (ismember (sequence, 1:noutput))) index = max ((ismember (sequence, 1:noutput) == 0) .* (1:len)); error (["hmmviterbi: sequence(" int2str (index) ") out of range"]); endif endif # Each row in transprob and outprob should contain log probabilities # => scale so that the sum is 1 and convert to log space # - for transprob s = sum (transprob, 2); s(s == 0) = 1; transprob = log (transprob ./ (s * ones (1, columns (transprob)))); # - for outprob s = sum (outprob, 2); s(s == 0) = 1; outprob = log (outprob ./ (s * ones (1, columns (outprob)))); # Store the path starting from i in spath(i, :) spath = ones (nstate, len + 1); # Set the first state for each path spath(:, 1) = (1:nstate)'; # Store the probability of path i in spathprob(i) spathprob = transprob(1, :); # Find the most likely paths for the given output sequence for i = 1:len # Calculate the new probabilities of the continuation with each state nextpathprob = ((spathprob' + outprob(:, sequence(i))) * ... ones (1, nstate)) + transprob; # Find the paths with the highest probabilities [spathprob, mindex] = max (nextpathprob); # Update spath and spathprob with the new paths spath = spath(mindex, :); spath(:, i + 1) = (1:nstate)'; endfor # Set vpath to the most likely path # We do not want the last state because we do not have an output for it [m, mindex] = max (spathprob); vpath = spath(mindex, 1:len); # Transform vpath into statenames if requested if (usesn) vpath = reshape (statenames(vpath), 1, len); endif endfunction %!test %! sequence = [1, 2, 1, 1, 1, 2, 2, 1, 2, 3, 3, 3, ... %! 3, 2, 3, 1, 1, 1, 1, 3, 3, 2, 3, 1, 3]; %! transprob = [0.8, 0.2; 0.4, 0.6]; %! outprob = [0.2, 0.4, 0.4; 0.7, 0.2, 0.1]; %! vpath = hmmviterbi (sequence, transprob, outprob); %! expected = [1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, ... %! 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1]; %! assert (vpath, expected); %!test %! sequence = {"A", "B", "A", "A", "A", "B", "B", "A", "B", "C", "C", "C", ... %! "C", "B", "C", "A", "A", "A", "A", "C", "C", "B", "C", "A", "C"}; %! transprob = [0.8, 0.2; 0.4, 0.6]; %! outprob = [0.2, 0.4, 0.4; 0.7, 0.2, 0.1]; %! symbols = {"A", "B", "C"}; %! statenames = {"One", "Two"}; %! vpath = hmmviterbi (sequence, transprob, outprob, "symbols", symbols, ... %! "statenames", statenames); %! expected = {"One", "One", "Two", "Two", "Two", "One", "One", "One", ... %! "One", "One", "One", "One", "One", "One", "One", "Two", ... %! "Two", "Two", "Two", "One", "One", "One", "One", "One", "One"}; %! assert (vpath, expected); statistics-release-1.6.3/inst/hotelling_t2test.m000066400000000000000000000154311456127120000217600ustar00rootroot00000000000000## Copyright (C) 1996-2017 Kurt Hornik ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {[@var{h}, @var{pval}, @var{stats}] =} hotelling_t2test (@var{x}) ## @deftypefnx {statistics} {[@dots{}] =} hotelling_t2test (@var{x}, @var{m}) ## @deftypefnx {statistics} {[@dots{}] =} hotelling_t2test (@var{x}, @var{y}) ## @deftypefnx {statistics} {[@dots{}] =} hotelling_t2test (@var{x}, @var{m}, @var{Name}, @var{Value}) ## @deftypefnx {statistics} {[@dots{}] =} hotelling_t2test (@var{x}, @var{y}, @var{Name}, @var{Value}) ## ## Compute Hotelling's T^2 ("T-squared") test for a single sample or two ## dependent samples (paired-samples). ## ## For a sample @var{x} from a multivariate normal distribution with unknown ## mean and covariance matrix, test the null hypothesis that ## @code{mean (@var{x}) == @var{m}}. ## ## For two dependent samples @var{x} and @var{y} from a multivariate normal ## distributions with unknown means and covariance matrices, test the null ## hypothesis that @code{mean (@var{x} - @var{y}) == 0}. ## ## @qcode{hotelling_t2test} treats NaNs as missing values, and ignores the ## corresponding rows. ## ## Name-Value pair arguments can be used to set statistical significance. ## @qcode{"alpha"} can be used to specify the significance level of the test ## (the default value is 0.05). ## ## If @var{h} is 1 the null hypothesis is rejected, meaning that the tested ## sample does not come from a multivariate distribution with mean @var{m}, or ## in case of two dependent samples that they do not come from the same ## multivariate distribution. If @var{h} is 0, then the null hypothesis cannot ## be rejected and it can be assumed that it holds true. ## ## The p-value of the test is returned in @var{pval}. ## ## @var{stats} is a structure containing the value of the Hotelling's @math{T^2} ## test statistic in the field "Tsq", and the degrees of freedom of the F ## distribution in the fields "df1" and "df2". Under the null hypothesis, ## @math{(n-p) T^2 / (p(n-1))} has an F distribution with @math{p} and ## @math{n-p} degrees of freedom, where @math{n} and @math{p} are the ## numbers of samples and variables, respectively. ## ## @seealso{hotelling_t2test2} ## @end deftypefn function [h, pval, stats] = hotelling_t2test (x, my, varargin) ## Check for minimum number of input arguments if (nargin < 1) print_usage (); endif ## Check X being a valid data set if (isscalar (x) || ndims (x) > 2) error ("hotelling_t2test: X must be a vector or a 2D matrix."); endif ## Set default arguments alpha = 0.05; ## Fix MY when X is a single input argument if (nargin == 1) if (isvector (x)) my = 0; elseif (ismatrix (x)) [n, p] = size (x); my = zeros (1, p); endif endif ## When X and MY are of equal size, then assume paired-sample if (isequal (size (x), size(my))) x = x - my; if (isvector (x)) my = 0; elseif (ismatrix (x)) [n, p] = size (x); my = zeros (1, p); endif endif ## Remove rows containing any NaNs x = rmmissing (x); ## Check additional options i = 1; while (i <= length (varargin)) switch lower (varargin{i}) case "alpha" i = i + 1; alpha = varargin{i}; ## Check for valid alpha if (! isscalar (alpha) || ! isnumeric (alpha) || ... alpha <= 0 || alpha >= 1) error ("hotelling_t2test: invalid value for alpha."); endif otherwise error ("hotelling_t2test: invalid Name argument."); endswitch i = i + 1; endwhile ## Conditional error checking for X being a vector or matrix if (isvector (x)) if (! isscalar (my)) error ("hotelling_t2test: if X is a vector, M must be a scalar."); endif n = length (x); p = 1; elseif (ismatrix (x)) [n, p] = size (x); if (n <= p) error ("hotelling_t2test: X must have more rows than columns."); endif if (isvector (my) && length (my) == p) my = reshape (my, 1, p); else error (strcat (["hotelling_t2test: if X is a matrix, M must be a"], ... [" vector of length equal to the columns of X."])); endif endif ## Calculate the necessary statistics d = mean (x) - my; stats.Tsq = n * d * (cov (x) \ d'); stats.df1 = p; stats.df2 = n - p; pval = 1 - fcdf ((n-p) * stats.Tsq / (p * (n-1)), stats.df1, stats.df2); ## Determine the test outcome ## MATLAB returns this a double instead of a logical array h = double (pval < alpha); endfunction ## Test input validation %!error hotelling_t2test (); %!error ... %! hotelling_t2test (1); %!error ... %! hotelling_t2test (ones(2,2,2)); %!error ... %! hotelling_t2test (ones(20,2), [0, 0], "alpha", 1); %!error ... %! hotelling_t2test (ones(20,2), [0, 0], "alpha", -0.2); %!error ... %! hotelling_t2test (ones(20,2), [0, 0], "alpha", "a"); %!error ... %! hotelling_t2test (ones(20,2), [0, 0], "alpha", [0.01, 0.05]); %!error ... %! hotelling_t2test (ones(20,2), [0, 0], "name", 0.01); %!error ... %! hotelling_t2test (ones(20,1), [0, 0]); %!error ... %! hotelling_t2test (ones(4,5), [0, 0, 0, 0, 0]); %!error ... %! hotelling_t2test (ones(20,5), [0, 0, 0, 0]); ## Test results %!test %! randn ("seed", 1); %! x = randn (50000, 5); %! [h, pval, stats] = hotelling_t2test (x); %! assert (h, 0); %! assert (stats.df1, 5); %! assert (stats.df2, 49995); %!test %! randn ("seed", 1); %! x = randn (50000, 5); %! [h, pval, stats] = hotelling_t2test (x, ones (1, 5) * 10); %! assert (h, 1); %! assert (stats.df1, 5); %! assert (stats.df2, 49995); statistics-release-1.6.3/inst/hotelling_t2test2.m000066400000000000000000000146671456127120000220540ustar00rootroot00000000000000## Copyright (C) 1996-2017 Kurt Hornik ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {[@var{h}, @var{pval}, @var{stats}] =} hotelling_t2test2 (@var{x}, @var{y}) ## @deftypefnx {statistics} {[@dots{}] =} hotelling_t2test2 (@var{x}, @var{y}, @var{Name}, @var{Value}) ## ## Compute Hotelling's T^2 ("T-squared") test for two independent samples. ## ## For two samples @var{x} from multivariate normal distributions with ## the same number of variables (columns), unknown means and unknown ## equal covariance matrices, test the null hypothesis ## @code{mean (@var{x}) == mean (@var{y})}. ## ## @qcode{hotelling_t2test2} treats NaNs as missing values, and ignores the ## corresponding rows for each sample independently. ## ## Name-Value pair arguments can be used to set statistical significance. ## @qcode{"alpha"} can be used to specify the significance level of the test ## (the default value is 0.05). ## ## If @var{h} is 1 the null hypothesis is rejected, meaning that the tested ## samples do not come from the same multivariate distribution. If @var{h} is ## 0, then the null hypothesis cannot be rejected and it can be assumed that ## both samples come from the same multivariate distribution. ## ## The p-value of the test is returned in @var{pval}. ## ## @var{stats} is a structure containing the value of the Hotelling's @math{T^2} ## test statistic in the field "Tsq", and the degrees of freedom of the F ## distribution in the fields "df1" and "df2". Under the null hypothesis, ## @tex ## $$ ## {(n_x+n_y-p-1) T^2 \over p(n_x+n_y-2)} ## $$ ## @end tex ## @ifnottex ## ## @example ## (n_x+n_y-p-1) T^2 / (p(n_x+n_y-2)) ## @end example ## ## @end ifnottex ## @noindent ## has an F distribution with @math{p} and @math{n_x+n_y-p-1} degrees of ## freedom, where @math{n_x} and @math{n_y} are the sample sizes and ## @math{p} is the number of variables. ## ## @seealso{hotelling_t2test} ## @end deftypefn function [h, pval, stats] = hotelling_t2test2 (x, y, varargin) ## Check for minimum number of input arguments if (nargin < 2) print_usage (); endif ## Check X being a valid data set if (isscalar (x) || ndims (x) > 2) error ("hotelling_t2test2: X must be a vector or a 2D matrix."); endif ## Check Y being a valid data set if (isscalar (y) || ndims (y) > 2) error ("hotelling_t2test2: Y must be a vector or a 2D matrix."); endif ## Set default arguments alpha = 0.05; ## Remove rows containing any NaNs x = rmmissing (x); y = rmmissing (y); ## Check additional options i = 1; while (i <= length (varargin)) switch lower (varargin{i}) case "alpha" i = i + 1; alpha = varargin{i}; ## Check for valid alpha if (! isscalar (alpha) || ! isnumeric (alpha) || ... alpha <= 0 || alpha >= 1) error ("hotelling_t2test2: invalid value for alpha."); endif otherwise error ("hotelling_t2test2: invalid Name argument."); endswitch i = i + 1; endwhile ## Conditional error checking for X being a vector or matrix if (isvector (x)) n_x = length (x); if (! isvector (y)) error ("hotelling_t2test2: if X is a vector, Y must also be a vector."); else n_y = length (y); p = 1; endif elseif (ismatrix (x)) [n_x, p] = size (x); [n_y, q] = size (y); if (p != q) error (strcat (["hotelling_t2test2: X and Y must have the same"], ... [" number of columns."])); endif endif ## Calculate the necessary statistics d = mean (x) - mean (y); S = ((n_x - 1) * cov (x) + (n_y - 1) * cov (y)) / (n_x + n_y - 2); stats.Tsq = (n_x * n_y / (n_x + n_y)) * d * (S \ d'); stats.df1 = p; stats.df2 = n_x + n_y - p - 1; pval = 1 - fcdf ((n_x + n_y - p - 1) * stats.Tsq / (p * (n_x + n_y - 2)), ... stats.df1, stats.df2); ## Determine the test outcome ## MATLAB returns this a double instead of a logical array h = double (pval < alpha); endfunction ## Test input validation %!error hotelling_t2test2 (); %!error ... %! hotelling_t2test2 ([2, 3, 4, 5, 6]); %!error ... %! hotelling_t2test2 (1, [2, 3, 4, 5, 6]); %!error ... %! hotelling_t2test2 (ones (2,2,2), [2, 3, 4, 5, 6]); %!error ... %! hotelling_t2test2 ([2, 3, 4, 5, 6], 2); %!error ... %! hotelling_t2test2 ([2, 3, 4, 5, 6], ones (2,2,2)); %!error ... %! hotelling_t2test2 (ones (20,2), ones (20,2), "alpha", 1); %!error ... %! hotelling_t2test2 (ones (20,2), ones (20,2), "alpha", -0.2); %!error ... %! hotelling_t2test2 (ones (20,2), ones (20,2), "alpha", "a"); %!error ... %! hotelling_t2test2 (ones (20,2), ones (20,2), "alpha", [0.01, 0.05]); %!error ... %! hotelling_t2test2 (ones (20,2), ones (20,2), "name", 0.01); %!error ... %! hotelling_t2test2 (ones (20,1), ones (20,2)); %!error ... %! hotelling_t2test2 (ones (20,2), ones (25,3)); ## Test results %!test %! randn ("seed", 1); %! x1 = randn (60000, 5); %! randn ("seed", 5); %! x2 = randn (30000, 5); %! [h, pval, stats] = hotelling_t2test2 (x1, x2); %! assert (h, 0); %! assert (stats.df1, 5); %! assert (stats.df2, 89994); statistics-release-1.6.3/inst/icdf.m000066400000000000000000000341421456127120000173730ustar00rootroot00000000000000# Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it ## under the terms of the GNU General Public License as published by ## the Free Software Foundation; either version 3 of the License, or ## (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program. If not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{x} =} icdf (@var{name}, @var{p}, @var{A}) ## @deftypefnx {statistics} {@var{x} =} icdf (@var{name}, @var{p}, @var{A}, @var{B}) ## @deftypefnx {statistics} {@var{x} =} icdf (@var{name}, @var{p}, @var{A}, @var{B}, @var{C}) ## ## Return the inverse CDF of a univariate distribution evaluated at @var{p}. ## ## @code{icdf} is a wrapper for the univariate quantile distribution functions ## (iCDF) available in the statistics package. See the corresponding functions' ## help to learn the signification of the parameters after @var{p}. ## ## @code{@var{x} = icdf (@var{name}, @var{p}, @var{A})} returns the iCDF for the ## one-parameter distribution family specified by @var{name} and the ## distribution parameter @var{A}, evaluated at the values in @var{p}. ## ## @code{@var{x} = icdf (@var{name}, @var{p}, @var{A}, @var{B})} returns the ## iCDF for the two-parameter distribution family specified by @var{name} and ## the distribution parameters @var{A} and @var{B}, evaluated at the values in ## @var{p}. ## ## @code{@var{x} = icdf (@var{name}, @var{p}, @var{A}, @var{B}, @var{C})} ## returns the iCDF for the three-parameter distribution family specified by ## @var{name} and the distribution parameters @var{A}, @var{B}, and @var{C}, ## evaluated at the values in @var{p}. ## ## @var{name} must be a char string of the name or the abbreviation of the ## desired quantile distribution function as listed in the followng table. ## The last column shows the number of required parameters that should be parsed ## after @var{x} to the desired iCDF. ## ## @multitable @columnfractions 0.4 0.05 0.2 0.05 0.3 ## @headitem Distribution Name @tab @tab Abbreviation @tab @tab Input Parameters ## @item @qcode{"Beta"} @tab @tab @qcode{"beta"} @tab @tab 2 ## @item @qcode{"Binomial"} @tab @tab @qcode{"bino"} @tab @tab 2 ## @item @qcode{"Birnbaum-Saunders"} @tab @tab @qcode{"bisa"} @tab @tab 2 ## @item @qcode{"Burr"} @tab @tab @qcode{"burr"} @tab @tab 3 ## @item @qcode{"Cauchy"} @tab @tab @qcode{"cauchy"} @tab @tab 2 ## @item @qcode{"Chi-squared"} @tab @tab @qcode{"chi2"} @tab @tab 1 ## @item @qcode{"Extreme Value"} @tab @tab @qcode{"ev"} @tab @tab 2 ## @item @qcode{"Exponential"} @tab @tab @qcode{"exp"} @tab @tab 1 ## @item @qcode{"F-Distribution"} @tab @tab @qcode{"f"} @tab @tab 2 ## @item @qcode{"Gamma"} @tab @tab @qcode{"gam"} @tab @tab 2 ## @item @qcode{"Geometric"} @tab @tab @qcode{"geo"} @tab @tab 1 ## @item @qcode{"Generalized Extreme Value"} @tab @tab @qcode{"gev"} @tab @tab 3 ## @item @qcode{"Generalized Pareto"} @tab @tab @qcode{"gp"} @tab @tab 3 ## @item @qcode{"Gumbel"} @tab @tab @qcode{"gumbel"} @tab @tab 2 ## @item @qcode{"Half-normal"} @tab @tab @qcode{"hn"} @tab @tab 2 ## @item @qcode{"Hypergeometric"} @tab @tab @qcode{"hyge"} @tab @tab 3 ## @item @qcode{"Inverse Gaussian"} @tab @tab @qcode{"invg"} @tab @tab 2 ## @item @qcode{"Laplace"} @tab @tab @qcode{"laplace"} @tab @tab 2 ## @item @qcode{"Logistic"} @tab @tab @qcode{"logi"} @tab @tab 2 ## @item @qcode{"Log-Logistic"} @tab @tab @qcode{"logl"} @tab @tab 2 ## @item @qcode{"Lognormal"} @tab @tab @qcode{"logn"} @tab @tab 2 ## @item @qcode{"Nakagami"} @tab @tab @qcode{"naka"} @tab @tab 2 ## @item @qcode{"Negative Binomial"} @tab @tab @qcode{"nbin"} @tab @tab 2 ## @item @qcode{"Noncentral F-Distribution"} @tab @tab @qcode{"ncf"} @tab @tab 3 ## @item @qcode{"Noncentral Student T"} @tab @tab @qcode{"nct"} @tab @tab 2 ## @item @qcode{"Noncentral Chi-Squared"} @tab @tab @qcode{"ncx2"} @tab @tab 2 ## @item @qcode{"Normal"} @tab @tab @qcode{"norm"} @tab @tab 2 ## @item @qcode{"Poisson"} @tab @tab @qcode{"poiss"} @tab @tab 1 ## @item @qcode{"Rayleigh"} @tab @tab @qcode{"rayl"} @tab @tab 1 ## @item @qcode{"Rician"} @tab @tab @qcode{"rice"} @tab @tab 2 ## @item @qcode{"Student T"} @tab @tab @qcode{"t"} @tab @tab 1 ## @item @qcode{"location-scale T"} @tab @tab @qcode{"tls"} @tab @tab 3 ## @item @qcode{"Triangular"} @tab @tab @qcode{"tri"} @tab @tab 3 ## @item @qcode{"Discrete Uniform"} @tab @tab @qcode{"unid"} @tab @tab 1 ## @item @qcode{"Uniform"} @tab @tab @qcode{"unif"} @tab @tab 2 ## @item @qcode{"Von Mises"} @tab @tab @qcode{"vm"} @tab @tab 2 ## @item @qcode{"Weibull"} @tab @tab @qcode{"wbl"} @tab @tab 2 ## @end multitable ## ## @seealso{icdf, pdf, random, betainv, binoinv, bisainv, burrinv, cauchyinv, ## chi2inv, evinv, expinv, finv, gaminv, geoinv, gevinv, gpinv, gumbelinv, ## hninv, hygeinv, invginv, laplaceinv, logiinv, loglinv, logninv, nakainv, ## nbininv, ncfinv, nctinv, ncx2inv, norminv, poissinv, raylinv, riceinv, ## tinv, triinv, unidinv, unifinv, vminv, wblinv} ## @end deftypefn function x = icdf (name, p, varargin) ## implemented functions persistent allDF = { ... {"beta" , "Beta"}, @betainv, 2, ... {"bino" , "Binomial"}, @binoinv, 2, ... {"bisa" , "Birnbaum-Saunders"}, @bisainv, 2, ... {"burr" , "Burr"}, @burrinv, 3, ... {"cauchy" , "Cauchy"}, @cauchyinv, 2, ... {"chi2" , "Chi-squared"}, @chi2inv, 1, ... {"ev" , "Extreme Value"}, @evinv, 2, ... {"exp" , "Exponential"}, @expinv, 1, ... {"f" , "F-Distribution"}, @finv, 2, ... {"gam" , "Gamma"}, @gaminv, 2, ... {"geo" , "Geometric"}, @geoinv, 1, ... {"gev" , "Generalized Extreme Value"}, @gevinv, 3, ... {"gp" , "Generalized Pareto"}, @gpinv, 3, ... {"gumbel" , "Gumbel"}, @gumbelinv, 2, ... {"hn" , "Half-normal"}, @hninv, 2, ... {"hyge" , "Hypergeometric"}, @hygeinv, 3, ... {"invg" , "Inverse Gaussian"}, @invginv, 2, ... {"laplace" , "Laplace"}, @laplaceinv, 2, ... {"logi" , "Logistic"}, @logiinv, 2, ... {"logl" , "Log-Logistic"}, @loglinv, 2, ... {"logn" , "Lognormal"}, @logninv, 2, ... {"naka" , "Nakagami"}, @nakainv, 2, ... {"nbin" , "Negative Binomial"}, @nbininv, 2, ... {"ncf" , "Noncentral F-Distribution"}, @ncfinv, 3, ... {"nct" , "Noncentral Student T"}, @nctinv, 2, ... {"ncx2" , "Noncentral Chi-squared"}, @ncx2inv, 2, ... {"norm" , "Normal"}, @norminv, 2, ... {"poiss" , "Poisson"}, @poissinv, 1, ... {"rayl" , "Rayleigh"}, @raylinv, 1, ... {"rice" , "Rician"}, @riceinv, 2, ... {"t" , "Student T"}, @tinv, 1, ... {"tls" , "location-scale T"}, @tlsinv, 3, ... {"tri" , "Triangular"}, @triinv, 3, ... {"unid" , "Discrete Uniform"}, @unidinv, 1, ... {"unif" , "Uniform"}, @unifinv, 2, ... {"vm" , "Von Mises"}, @vminv, 2, ... {"wbl" , "Weibull"}, @wblinv, 2}; if (! ischar (name)) error ("icdf: distribution NAME must a char string."); endif ## Check P being numeric and real if (! isnumeric (p)) error ("icdf: P must be numeric."); elseif (! isreal (p)) error ("icdf: values in P must be real."); endif ## Get number of arguments nargs = numel (varargin); ## Get available functions icdfnames = allDF(1:3:end); icdfhandl = allDF(2:3:end); icdf_args = allDF(3:3:end); ## Search for iCDF function idx = cellfun (@(x)any(strcmpi (name, x)), icdfnames); if (any (idx)) if (nargs == icdf_args{idx}) ## Check that all distribution parameters are numeric if (! all (cellfun (@(x)isnumeric(x), (varargin)))) error ("icdf: distribution parameters must be numeric."); endif ## Call appropriate iCDF x = feval (icdfhandl{idx}, p, varargin{:}); else if (icdf_args{idx} == 1) error ("icdf: %s distribution requires 1 parameter.", name); else error ("icdf: %s distribution requires %d parameters.", ... name, icdf_args{idx}); endif endif else error ("icdf: %s distribution is not implemented in Statistics.", name); endif endfunction ## Test results %!shared p %! p = [0.05:0.05:0.5]; %!assert (icdf ("Beta", p, 5, 2), betainv (p, 5, 2)) %!assert (icdf ("beta", p, 5, 2), betainv (p, 5, 2)) %!assert (icdf ("Binomial", p, 5, 2), binoinv (p, 5, 2)) %!assert (icdf ("bino", p, 5, 2), binoinv (p, 5, 2)) %!assert (icdf ("Birnbaum-Saunders", p, 5, 2), bisainv (p, 5, 2)) %!assert (icdf ("bisa", p, 5, 2), bisainv (p, 5, 2)) %!assert (icdf ("Burr", p, 5, 2, 2), burrinv (p, 5, 2, 2)) %!assert (icdf ("burr", p, 5, 2, 2), burrinv (p, 5, 2, 2)) %!assert (icdf ("Cauchy", p, 5, 2), cauchyinv (p, 5, 2)) %!assert (icdf ("cauchy", p, 5, 2), cauchyinv (p, 5, 2)) %!assert (icdf ("Chi-squared", p, 5), chi2inv (p, 5)) %!assert (icdf ("chi2", p, 5), chi2inv (p, 5)) %!assert (icdf ("Extreme Value", p, 5, 2), evinv (p, 5, 2)) %!assert (icdf ("ev", p, 5, 2), evinv (p, 5, 2)) %!assert (icdf ("Exponential", p, 5), expinv (p, 5)) %!assert (icdf ("exp", p, 5), expinv (p, 5)) %!assert (icdf ("F-Distribution", p, 5, 2), finv (p, 5, 2)) %!assert (icdf ("f", p, 5, 2), finv (p, 5, 2)) %!assert (icdf ("Gamma", p, 5, 2), gaminv (p, 5, 2)) %!assert (icdf ("gam", p, 5, 2), gaminv (p, 5, 2)) %!assert (icdf ("Geometric", p, 5), geoinv (p, 5)) %!assert (icdf ("geo", p, 5), geoinv (p, 5)) %!assert (icdf ("Generalized Extreme Value", p, 5, 2, 2), gevinv (p, 5, 2, 2)) %!assert (icdf ("gev", p, 5, 2, 2), gevinv (p, 5, 2, 2)) %!assert (icdf ("Generalized Pareto", p, 5, 2, 2), gpinv (p, 5, 2, 2)) %!assert (icdf ("gp", p, 5, 2, 2), gpinv (p, 5, 2, 2)) %!assert (icdf ("Gumbel", p, 5, 2), gumbelinv (p, 5, 2)) %!assert (icdf ("gumbel", p, 5, 2), gumbelinv (p, 5, 2)) %!assert (icdf ("Half-normal", p, 5, 2), hninv (p, 5, 2)) %!assert (icdf ("hn", p, 5, 2), hninv (p, 5, 2)) %!assert (icdf ("Hypergeometric", p, 5, 2, 2), hygeinv (p, 5, 2, 2)) %!assert (icdf ("hyge", p, 5, 2, 2), hygeinv (p, 5, 2, 2)) %!assert (icdf ("Inverse Gaussian", p, 5, 2), invginv (p, 5, 2)) %!assert (icdf ("invg", p, 5, 2), invginv (p, 5, 2)) %!assert (icdf ("Laplace", p, 5, 2), laplaceinv (p, 5, 2)) %!assert (icdf ("laplace", p, 5, 2), laplaceinv (p, 5, 2)) %!assert (icdf ("Logistic", p, 5, 2), logiinv (p, 5, 2)) %!assert (icdf ("logi", p, 5, 2), logiinv (p, 5, 2)) %!assert (icdf ("Log-Logistic", p, 5, 2), loglinv (p, 5, 2)) %!assert (icdf ("logl", p, 5, 2), loglinv (p, 5, 2)) %!assert (icdf ("Lognormal", p, 5, 2), logninv (p, 5, 2)) %!assert (icdf ("logn", p, 5, 2), logninv (p, 5, 2)) %!assert (icdf ("Nakagami", p, 5, 2), nakainv (p, 5, 2)) %!assert (icdf ("naka", p, 5, 2), nakainv (p, 5, 2)) %!assert (icdf ("Negative Binomial", p, 5, 2), nbininv (p, 5, 2)) %!assert (icdf ("nbin", p, 5, 2), nbininv (p, 5, 2)) %!assert (icdf ("Noncentral F-Distribution", p, 5, 2, 2), ncfinv (p, 5, 2, 2)) %!assert (icdf ("ncf", p, 5, 2, 2), ncfinv (p, 5, 2, 2)) %!assert (icdf ("Noncentral Student T", p, 5, 2), nctinv (p, 5, 2)) %!assert (icdf ("nct", p, 5, 2), nctinv (p, 5, 2)) %!assert (icdf ("Noncentral Chi-Squared", p, 5, 2), ncx2inv (p, 5, 2)) %!assert (icdf ("ncx2", p, 5, 2), ncx2inv (p, 5, 2)) %!assert (icdf ("Normal", p, 5, 2), norminv (p, 5, 2)) %!assert (icdf ("norm", p, 5, 2), norminv (p, 5, 2)) %!assert (icdf ("Poisson", p, 5), poissinv (p, 5)) %!assert (icdf ("poiss", p, 5), poissinv (p, 5)) %!assert (icdf ("Rayleigh", p, 5), raylinv (p, 5)) %!assert (icdf ("rayl", p, 5), raylinv (p, 5)) %!assert (icdf ("Rician", p, 5, 1), riceinv (p, 5, 1)) %!assert (icdf ("rice", p, 5, 1), riceinv (p, 5, 1)) %!assert (icdf ("Student T", p, 5), tinv (p, 5)) %!assert (icdf ("t", p, 5), tinv (p, 5)) %!assert (icdf ("location-scale T", p, 5, 1, 2), tlsinv (p, 5, 1, 2)) %!assert (icdf ("tls", p, 5, 1, 2), tlsinv (p, 5, 1, 2)) %!assert (icdf ("Triangular", p, 5, 2, 2), triinv (p, 5, 2, 2)) %!assert (icdf ("tri", p, 5, 2, 2), triinv (p, 5, 2, 2)) %!assert (icdf ("Discrete Uniform", p, 5), unidinv (p, 5)) %!assert (icdf ("unid", p, 5), unidinv (p, 5)) %!assert (icdf ("Uniform", p, 5, 2), unifinv (p, 5, 2)) %!assert (icdf ("unif", p, 5, 2), unifinv (p, 5, 2)) %!assert (icdf ("Von Mises", p, 5, 2), vminv (p, 5, 2)) %!assert (icdf ("vm", p, 5, 2), vminv (p, 5, 2)) %!assert (icdf ("Weibull", p, 5, 2), wblinv (p, 5, 2)) %!assert (icdf ("wbl", p, 5, 2), wblinv (p, 5, 2)) ## Test input validation %!error icdf (1) %!error icdf ({"beta"}) %!error icdf ("beta", {[1 2 3 4 5]}) %!error icdf ("beta", "text") %!error icdf ("beta", 1+i) %!error ... %! icdf ("Beta", p, "a", 2) %!error ... %! icdf ("Beta", p, 5, "") %!error ... %! icdf ("Beta", p, 5, {2}) %!error icdf ("chi2", p) %!error icdf ("Beta", p, 5) %!error icdf ("Burr", p, 5) %!error icdf ("Burr", p, 5, 2) statistics-release-1.6.3/inst/inconsistent.m000066400000000000000000000102771456127120000212110ustar00rootroot00000000000000## Copyright (C) 2020-2021 Stefano Guidoni ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{Y} =} inconsistent (@var{Z}) ## @deftypefnx {statistics} {@var{Y} =} inconsistent (@var{Z}, @var{d}) ## ## Compute the inconsistency coefficient for each link of a hierarchical cluster ## tree. ## ## Given a hierarchical cluster tree @var{Z} generated by the @code{linkage} ## function, @code{inconsistent} computes the inconsistency coefficient for each ## link of the tree, using all the links down to the @var{d}-th level below that ## link. ## ## The default depth @var{d} is 2, which means that only two levels are ## considered: the level of the computed link and the level below that. ## ## Each row of @var{Y} corresponds to the row of same index of @var{Z}. ## The columns of @var{Y} are respectively: the mean of the heights of the links ## used for the calculation, the standard deviation of the heights of those ## links, the number of links used, the inconsistency coefficient. ## ## @strong{Reference} ## Jain, A., and R. Dubes. Algorithms for Clustering Data. ## Upper Saddle River, NJ: Prentice-Hall, 1988. ## @end deftypefn ## ## @seealso{cluster, clusterdata, dendrogram, linkage, pdist, squareform} function Y = inconsistent (Z, d = 2) ## check the input if (nargin < 1) || (nargin > 2) print_usage (); endif ## MATLAB compatibility: ## when d = 0, which does not make sense, the result of inconsistent is the ## same as d = 1, which is... inconsistent if ((d < 0) || (! isscalar (d)) || (mod (d, 1))) error ("inconsistent: d must be a positive integer scalar"); endif if ((columns (Z) != 3) || (! isnumeric (Z)) || ... (! (max (Z(end, 1:2)) == rows (Z) * 2))) error (["inconsistent: Z must be a matrix generated by the linkage " ... "function"]); endif ## number of observations n = rows (Z) + 1; ## compute the inconsistency coefficient for every link for i = 1:rows (Z) v = inconsistent_recursion (i, d); # nested recursive function - see below Y(i, 1) = mean (v); Y(i, 2) = std (v); Y(i, 3) = length (v); ## the inconsistency coefficient is (current_link_height - mean) / std; ## if the standard deviation is zero, it is zero by definition if (Y(i, 2) != 0) Y(i, 4) = (v(end) - Y(i, 1)) / Y(i, 2); else Y(i, 4) = 0; endif endfor ## recursive function ## while depth > 1 search the links (columns 1 and 2 of Z) below the current ## link and then append the height of the current link to the vector v. ## The height of the starting link should be the last one of the vector. function v = inconsistent_recursion (index, depth) v = []; if (depth > 1) for j = 1:2 if (Z(index, j) > n) new_index = Z(index, j) - n; v = [v (inconsistent_recursion (new_index, depth - 1))]; endif endfor endif v(end+1) = Z(index, 3); endfunction endfunction ## Test input validation %!error inconsistent () %!error inconsistent ([1 2 1], 2, 3) %!error inconsistent (ones (2, 2)) %!error inconsistent ([1 2 1], -1) %!error inconsistent ([1 2 1], 1.3) %!error inconsistent ([1 2 1], [1 1]) %!error inconsistent (ones (2, 3)) ## Test output %!test %! load fisheriris; %! Z = linkage(meas, 'average', 'chebychev'); %! assert (cond (inconsistent (Z)), 39.9, 1e-3); statistics-release-1.6.3/inst/ismissing.m000066400000000000000000000160231456127120000204710ustar00rootroot00000000000000## Copyright (C) 1995-2023 The Octave Project Developers ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{TF} =} ismissing (@var{A}) ## @deftypefnx {statistics} {@var{TF} =} ismissing (@var{A}, @var{indicator}) ## ## Find missing data in a numeric or string array. ## ## Given an input numeric data array, char array, or array of cell strings ## @var{A}, @code{ismissing} returns a logical array @var{TF} with ## the same dimensions as @var{A}, where @code{true} values match missing ## values in the input data. ## ## The optional input @var{indicator} is an array of values that represent ## missing values in the input data. The values which represent missing data ## by default depend on the data type of @var{A}: ## ## @itemize ## @item ## @qcode{NaN}: @code{single}, @code{double}. ## ## @item ## @qcode{' '} (white space): @code{char}. ## ## @item ## @qcode{@{''@}}: string cells. ## @end itemize ## ## Note: logical and numeric data types may be used in any combination ## for @var{A} and @var{indicator}. @var{A} and the indicator values will be ## compared as type double, and the output will have the same class as @var{A}. ## Data types other than those specified above have no defined 'missing' value. ## As such, the TF output for those inputs will always be ## @code{false(size(@var{A}))}. The exception to this is that @var{indicator} ## can be specified for logical and numeric inputs to designate values that ## will register as 'missing'. ## ## @seealso{fillmissing, rmmissing, standardizeMissing} ## @end deftypefn function TF = ismissing (A, indicator) if (nargin < 1) || (nargin > 2) print_usage (); endif ## check "indicator" if (nargin != 2) indicator = []; endif ## if A is an array of cell strings and indicator just a string, ## convert indicator to a cell string with one element if (iscellstr (A) && ischar (indicator) && ! iscellstr (indicator)) indicator = {indicator}; endif if ((! isempty (indicator)) && ((isnumeric (A) && ! (isnumeric (indicator) || islogical (indicator))) || (ischar (A) && ! ischar (indicator)) || (iscellstr (A) && ! (iscellstr (indicator))))) error ("ismissing: 'indicator' and 'A' must have the same data type"); endif ## main logic if (isempty (indicator)) if (isnumeric (A)) ## numeric matrix: just find the NaNs ## integer types have no missing value, but isnan will return false TF = isnan (A); elseif (iscellstr (A)) ## cell strings - find empty cells TF = cellfun ('isempty', A); elseif (ischar (A)) ## char matrix: find the white spaces TF = isspace (A); else ##no missing type defined, return false TF = false (size (A)); endif else ## indicator specified for missing data TF = false (size (A)); if (isnumeric(A) || ischar (A) || islogical (A)) for iter = 1 : numel (indicator) if (isnan (indicator(iter))) TF(isnan(A)) = true; else TF(A == indicator(iter)) = true; endif endfor elseif (iscellstr (A)) for iter = 1 : numel (indicator) TF(strcmp (A, indicator(iter))) = true; endfor else error ("ismissing: indicators not supported for data type '%s'", ... class(A)); endif endif endfunction %!assert (ismissing ([1,NaN,3]), [false,true,false]) %!assert (ismissing ('abcd f'), [false,false,false,false,true,false]) %!assert (ismissing ({'xxx','','xyz'}), [false,true,false]) %!assert (ismissing ({'x','','y'}), [false,true,false]) %!assert (ismissing ({'x','','y';'z','a',''}), logical([0,1,0;0,0,1])) %!assert (ismissing ([1,2;NaN,2]), [false,false;true,false]) %!assert (ismissing ([1,2;NaN,2], 2), [false,true;false,true]) %!assert (ismissing ([1,2;NaN,2], [1 2]), [true,true;false,true]) %!assert (ismissing ([1,2;NaN,2], NaN), [false,false;true,false]) ## test nD array data %!assert (ismissing (cat(3,magic(2),magic(2))), logical (zeros (2,2,2))) %!assert (ismissing (cat(3,magic(2),[1 2;3 NaN])), logical (cat(3,[0,0;0,0],[0,0;0,1]))) %!assert (ismissing ([1 2; 3 4], [5 1; 2 0]), logical([1 1; 0 0])) %!assert (ismissing (cat(3,'f oo','ba r')), logical(cat(3,[0 1 0 0],[0 0 1 0]))) %!assert (ismissing (cat(3,{'foo'},{''},{'bar'})), logical(cat(3,0,1,0))) ## test data type handling %!assert (ismissing (double (NaN)), true) %!assert (ismissing (single (NaN)), true) %!assert (ismissing (' '), true) %!assert (ismissing ({''}), true) %!assert (ismissing ({' '}), false) %!assert (ismissing (double (eye(3)), single (1)), logical(eye(3))) %!assert (ismissing (double (eye(3)), true), logical(eye(3))) %!assert (ismissing (double (eye(3)), int32 (1)), logical(eye(3))) %!assert (ismissing (single (eye(3)), true), logical(eye(3))) %!assert (ismissing (single (eye(3)), double (1)), logical(eye(3))) %!assert (ismissing (single(eye(3)), int32 (1)), logical(eye(3))) ## test data types without missing values %!assert (ismissing ({'123', '', 123}), [false false false]) %!assert (ismissing (logical ([1 0 1])), [false false false]) %!assert (ismissing (int32 ([1 2 3])), [false false false]) %!assert (ismissing (uint32 ([1 2 3])), [false false false]) %!assert (ismissing ({1, 2, 3}), [false false false]) %!assert (ismissing ([struct struct struct]), [false false false]) %!assert (ismissing (logical (eye(3)), true), logical(eye(3))) %!assert (ismissing (logical (eye(3)), double (1)), logical(eye(3))) %!assert (ismissing (logical (eye(3)), single (1)), logical(eye(3))) %!assert (ismissing (logical (eye(3)), int32 (1)), logical(eye(3))) %!assert (ismissing (int32 (eye(3)), int32 (1)), logical(eye(3))) %!assert (ismissing (int32 (eye(3)), true), logical(eye(3))) %!assert (ismissing (int32 (eye(3)), double (1)), logical(eye(3))) %!assert (ismissing (int32 (eye(3)), single (1)), logical(eye(3))) ## test empty input handling %!assert (ismissing ([]), logical([])) %!assert (ismissing (''), logical([])) %!assert (ismissing (ones (0,1)), logical(ones(0,1))) %!assert (ismissing (ones (1,0)), logical(ones(1,0))) %!assert (ismissing (ones (1,2,0)), logical(ones(1,2,0))) ## Test input validation %!error ismissing () %!error <'indicator' and 'A' must have the same> ismissing ([1 2; 3 4], "abc") %!error <'indicator' and 'A' must have the same> ismissing ({"", "", ""}, 1) %!error <'indicator' and 'A' must have the same> ismissing (1, struct) %!error ismissing (struct, 1) statistics-release-1.6.3/inst/isoutlier.m000066400000000000000000001054651456127120000205140ustar00rootroot00000000000000## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{TF} =} isoutlier (@var{x}) ## @deftypefnx {statistics} {@var{TF} =} isoutlier (@var{x}, @var{method}) ## @deftypefnx {statistics} {@var{TF} =} isoutlier (@var{x}, @qcode{"percentiles"}, @var{threshold}) ## @deftypefnx {statistics} {@var{TF} =} isoutlier (@var{x}, @var{movmethod}, @var{window}) ## @deftypefnx {statistics} {@var{TF} =} isoutlier (@dots{}, @var{dim}) ## @deftypefnx {statistics} {@var{TF} =} isoutlier (@dots{}, @var{Name}, @var{Value}) ## @deftypefnx {statistics} {[@var{TF}, @var{L}, @var{U}, @var{C}] =} isoutlier (@dots{}) ## ## Find outliers in data ## ## @code{isoutlier (@var{x})} returns a logical array whose elements are true ## when an outlier is detected in the corresponding element of @var{x}. ## @code{isoutlier} treats NaNs as missing values and removes them. ## ## @itemize ## @item ## If @var{x} is a matrix, then @code{isoutlier} operates on each column of ## @var{x} separately. ## @item ## If @var{x} is a multidimensional array, then @code{isoutlier} operates along ## the first dimension of @var{x} whose size does not equal 1. ## @end itemize ## ## By default, an outlier is a value that is more than three scaled median ## absolute deviations (MAD) from the median. The scaled median is defined as ## @code{c*median(abs(A-median(A)))}, where @code{c=-1/(sqrt(2)*erfcinv(3/2))}. ## ## @code{isoutlier (@var{x}, @var{method})} specifies a method for detecting ## outliers. The following methods are available: ## ## @multitable @columnfractions 0.13 0.02 0.8 ## @headitem Method @tab @tab Description ## @item @qcode{"median"} @tab @tab Outliers are defined as elements more than ## three scaled MAD from the median. ## @item @qcode{"mean"} @tab @tab Outliers are defined as elements more than ## three standard deviations from the mean. ## @item @qcode{"quartiles"} @tab @tab Outliers are defined as elements more ## than 1.5 interquartile ranges above the upper quartile (75 percent) or below ## the lower quartile (25 percent). This method is useful when the data in ## @var{x} is not normally distributed. ## @item @qcode{"grubbs"} @tab @tab Outliers are detected using Grubbs’ test for ## outliers, which removes one outlier per iteration based on hypothesis ## testing. This method assumes that the data in @var{x} is normally ## distributed. ## @item @qcode{"gesd"} @tab @tab Outliers are detected using the generalized ## extreme Studentized deviate test for outliers. This iterative method is ## similar to @qcode{"grubbs"}, but can perform better when there are multiple ## outliers masking each other. ## @end multitable ## ## @code{isoutlier (@var{x}, @qcode{"percentiles"}, @var{threshold})} detects ## outliers based on a percentile thresholds, specified as a two-element row ## vector whose elements are in the interval @math{[0, 100]}. The first element ## indicates the lower percentile threshold, and the second element indicates ## the upper percentile threshold. The first element of threshold must be less ## than the second element. ## ## @code{isoutlier (@var{x}, @var{movmethod}, @var{window})} specifies a moving ## method for detecting outliers. The following methods are available: ## ## @multitable @columnfractions 0.13 0.02 0.8 ## @headitem Method @tab @tab Description ## @item @qcode{"movmedian"} @tab @tab Outliers are defined as elements more ## than three local scaled MAD from the local median over a window length ## specified by @var{window}. ## @item @qcode{"movmean"} @tab @tab Outliers are defined as elements more than ## three local standard deviations from the from the local mean over a window ## length specified by @var{window}. ## @end multitable ## ## @var{window} must be a positive integer scalar or a two-element vector of ## positive integers. When @var{window} is a scalar, if it is an odd number, ## the window is centered about the current element and contains ## @qcode{@var{window} - 1} neighboring elements. If even, then the window is ## centered about the current and previous elements. When @var{window} is a ## two-element vector of positive integers @math{[nb, na]}, the window contains ## the current element, @math{nb} elements before the current element, and ## @math{na} elements after the current element. When @qcode{"SamplePoints"} ## are also specified, @var{window} can take any real positive values (either as ## a scalar or a two-element vector) and in this case, the windows are computed ## relative to the sample points. ## ## @var{dim} specifies the operating dimension and it must be a positive integer ## scalar. If not specified, then, by default, @code{isoutlier} operates along ## the first non-singleton dimension of @var{x}. ## ## The following optional parameters can be specified as @var{Name}/@var{Value} ## paired arguments. ## ## @itemize ## @item @qcode{"SamplePoints"} can be specified as a vector of sample points ## with equal length as the operating dimension. The sample points represent ## the x-axis location of the data and must be sorted and contain unique ## elements. Sample points do not need to be uniformly sampled. By default, ## the vector is @qcode{[1, 2, 3, @dots{}, @var{n}]}, where ## @qcode{@var{n} = size (@var{x}, @var{dim})}. You can use unequally spaced ## @qcode{"SamplePoints"} to define a variable-length window for one of the ## moving methods available. ## ## @item @qcode{"ThresholdFactor"} can be specified as a nonnegative scalar. ## For methods @qcode{"median"} and @qcode{"movmedian"}, the detection threshold ## factor replaces the number of scaled MAD, which is 3 by default. For methods ## @qcode{"mean"} and @qcode{"movmean"}, the detection threshold factor replaces ## the number of standard deviations, which is 3 by default. For methods ## @qcode{"grubbs"} and @qcode{"gesd"}, the detection threshold factor ranges ## from 0 to 1, specifying the critical @math{alpha}-value of the respective ## test, and it is 0.05 by default. For the @qcode{"quartiles"} method, the ## detection threshold factor replaces the number of interquartile ranges, which ## is 1.5 by default. @qcode{"ThresholdFactor"} is not supported for the ## @qcode{"quartiles"} method. ## ## @item @qcode{"MaxNumOutliers"} is only relevant to the @qcode{"gesd"} method ## and it must be a positive integer scalar specifying the maximum number of ## outliers returned by the @qcode{"gesd"} method. By default, it is the ## integer nearest to the 10% of the number of elements along the operating ## dimension in @var{x}. The @qcode{"gesd"} method assumes the nonoutlier input ## data is sampled from an approximate normal distribution. When the data is ## not sampled in this way, the number of returned outliers might exceed the ## @qcode{MaxNumOutliers} value. ## @end itemize ## ## @code{[@var{TF}, @var{L}, @var{U}, @var{C}] = isoutlier (@dots{})} returns ## up to 4 output arguments as described below. ## ## @itemize ## @item @var{TF} is the outlier indicator with the same size a @var{x}. ## ## @item @var{L} is the lower threshold used by the outlier detection method. ## If @var{method} is used for outlier detection, then @var{L} has the same size ## as @var{x} in all dimensions except for the operating dimension where the ## length is 1. If @var{movmethod} is used, then @var{L} has the same size as ## @var{x}. ## ## @item @var{U} is the upper threshold used by the outlier detection method. ## If @var{method} is used for outlier detection, then @var{U} has the same size ## as @var{x} in all dimensions except for the operating dimension where the ## length is 1. If @var{movmethod} is used, then @var{U} has the same size as ## @var{x}. ## ## @item @var{C} is the center value used by the outlier detection method. ## If @var{method} is used for outlier detection, then @var{C} has the same size ## as @var{x} in all dimensions except for the operating dimension where the ## length is 1. If @var{movmethod} is used, then @var{C} has the same size as ## @var{x}. For @qcode{"median"}, @qcode{"movmedian"}, @qcode{"mean"}, and ## @qcode{"movmean"} methods, @var{C} is computed by taking into acount the ## outlier values. For @qcode{"grubbs"} and @qcode{"gesd"} methods, @var{C} is ## computed by excluding the outliers. For the @qcode{"percentiles"} method, ## @var{C} is the average between @var{U} and @var{L} thresholds. ## @end itemize ## ## @seealso{filloutliers, rmoutliers, ismissing} ## @end deftypefn function [TF, L, U, C] = isoutlier (x, varargin) ## Check for valid input data if (nargin < 1) print_usage; endif ## Handle case if X is a scalar if (isscalar (x)) TF = false; L = x; U = x; C = x; return endif ## Add defaults dim = []; method = "median"; window = []; SamplePoints = []; ThresholdFactor = 3; MaxNumOutliers = []; ## MATLAB's constant for scaled Median Absolute Deviation ## c = -1 / (sqrt (2) * erfcinv (3/2)) c = 1.482602218505602; ## Parse exrta arguments while (numel (varargin) > 0) if (ischar (varargin{1})) switch (lower (varargin{1})) case "median" method = "median"; ThresholdFactor = 3; varargin(1) = []; case "mean" method = "mean"; ThresholdFactor = 3; varargin(1) = []; case "quartiles" method = "quartiles"; ThresholdFactor = 1.5; varargin(1) = []; case "grubbs" method = "grubbs"; ThresholdFactor = 0.05; varargin(1) = []; case "gesd" method = "gesd"; ThresholdFactor = 0.05; MaxNumOutliers = []; varargin(1) = []; case "movmedian" method = "movmedian"; window = varargin{2}; if (! isnumeric (window) || numel (window) < 1 || numel (window) > 2 || any (window <= 0)) error (strcat (["isoutlier: WINDOW must be a positive scalar"], ... [" or a two-element vector of positive values"])); endif varargin([1:2]) = []; case "movmean" method = "movmean"; window = varargin{2}; if (! isnumeric (window) || numel (window) < 1 || numel (window) > 2 || any (window <= 0)) error (strcat (["isoutlier: WINDOW must be a positive scalar"], ... [" or a two-element vector of positive values"])); endif varargin([1:2]) = []; case "percentiles" method = "percentiles"; threshold = varargin{2}; if (! isnumeric (threshold) || ! (numel (threshold) == 2)) error (strcat (["isoutlier: THRESHOLD must be a two-element"], ... [" vector whose elements are in the interval"], ... [" [0, 100]."])); endif if (! (threshold(1) < threshold(2)) || threshold(1) < 0 || threshold(2) > 100) error (strcat (["isoutlier: THRESHOLD must be a two-element"], ... [" vector whose elements are in the interval"], ... [" [0, 100]."])); endif varargin([1:2]) = []; case "samplepoints" SamplePoints = varargin{2}; if (! isvector (SamplePoints) || isscalar (SamplePoints)) error ("isoutlier: sample points must be a vector."); endif if (numel (unique (SamplePoints)) != numel (SamplePoints)) error ("isoutlier: sample points must be unique."); endif if (any (sort (SamplePoints) != SamplePoints)) error ("isoutlier: sample points must be sorted."); endif varargin([1:2]) = []; case "thresholdfactor" ThresholdFactor = varargin{2}; if (! isscalar (ThresholdFactor) || ThresholdFactor <= 0) error ("isoutlier: threshold factor must be a nonnegative scalar."); endif varargin([1:2]) = []; case "maxnumoutliers" MaxNumOutliers = varargin{2}; if (! isscalar (MaxNumOutliers) || MaxNumOutliers <= 0 || ! (fix (MaxNumOutliers) == MaxNumOutliers)) error (strcat (["isoutlier: maximum outlier count must be a"], ... [" positive integer scalar."])); endif varargin([1:2]) = []; otherwise error ("isoutlier: invalid input argument."); endswitch elseif (isnumeric (varargin{1})) dim = varargin{1}; if (! fix (dim) == dim || dim < 1 || ! isscalar (dim) || ! isscalar (varargin{1})) error ("isoutlier: DIM must be a positive integer scalar."); endif varargin(1) = []; else error ("isoutlier: invalid input argument."); endif endwhile ## Find 1st operating dimension (if empty) if (isempty (dim)) szx = size (x); (dim = find (szx != 1, 1)) || (dim = 1); endif ## Check for valid WINDOW unless Sample Points are given if (isempty (SamplePoints) && ! isempty (window)) if (! all (fix (window) == window)) error (strcat (["isoutlier: WINDOW must be a positive integer"], ... [" scalar or a two-element vector of positive"], ... [" integers, unless SamplePoints are defined."])); endif endif ## Check for valid value of ThresholdFactor for 'grubbs' and 'geds' methods if (any (strcmpi (method, {"grubbs", "gesd"})) && ThresholdFactor > 1) error (strcat (["isoutlier: threshold factor must must be in [0 1]"], ... [" range for 'grubbs' and 'gesd' methods."])); endif ## Switch methods switch method case "median" [L, U, C] = median_method (x, dim, ThresholdFactor, c); TF = x < L | x > U; case "mean" [L, U, C] = mean_method (x, dim, ThresholdFactor); TF = x < L | x > U; case "quartiles" [L, U, C] = quartiles_method (x, dim, ThresholdFactor); TF = x < L | x > U; case "grubbs" [TF, L, U, C] = grubbs_method (x, dim, ThresholdFactor); case "gesd" [L, U, C] = gesd_method (x, dim, ThresholdFactor, MaxNumOutliers); TF = x < L | x > U; case "movmedian" sp = SamplePoints; [L, U, C] = movmedian_method (x, dim, ThresholdFactor, c, window, sp); TF = x < L | x > U; case "movmean" sp = SamplePoints; [L, U, C] = movmean_method (x, dim, ThresholdFactor, window, sp); TF = x < L | x > U; case "percentiles" [L, U, C] = percentiles_method (x, dim, threshold); TF = x < L | x > U; endswitch endfunction ## Find lower and upper outlier thresholds with median method function [L, U, C] = median_method (x, dim, ThresholdFactor, c) C = median (x, dim, "omitnan"); sMAD = c * mad (x, 1, dim); L = C - ThresholdFactor * sMAD; U = C + ThresholdFactor * sMAD; endfunction ## Find lower and upper outlier thresholds with mean method function [L, U, M] = mean_method (x, dim, ThresholdFactor) M = mean (x, dim, "omitnan"); S = std (x, [], dim, "omitnan"); L = M - ThresholdFactor * S; U = M + ThresholdFactor * S; endfunction ## Find lower and upper outlier thresholds with quartiles method function [L, U, C] = quartiles_method (x, dim, ThresholdFactor) Q = quantile (x, dim); C = Q(3); L = Q(2) - (Q(4) - Q(2)) * ThresholdFactor; U = Q(4) + (Q(4) - Q(2)) * ThresholdFactor; endfunction ## Find lower and upper outlier thresholds with grubbs method function [TF, L, U, C] = grubbs_method (x, dim, ThresholdFactor) ## Move the desired dim to be the 1st dimension (rows) szx = size (x); # size of dimensions N = szx(dim); # elements in operating dimension nd = length (szx); # number of dimensions dperm = [dim, 1:(dim-1), (dim+1):nd]; # permutation of dimensions x = permute (x, dperm); # permute dims to first dimension ncols = prod (szx(dperm(2:end))); # rest of dimensions as single column x = reshape (x, N, ncols); # reshape input ## Create return matrices L = zeros ([1, szx(dperm(2:end))]); U = L; C = L; TF = false (size (x)); ## Apply processing to each column for i = 1:ncols tmp_x = x(:,i); TFvec = [(i-1)*size(x,1)+1:i*size(x,1)]; TFvec(isnan (tmp_x)) = []; tmp_x(isnan (tmp_x)) = []; ## Search for outliers (one at a time) while (true) ## Get descriptive statistics n = length (tmp_x); C(i) = mean (tmp_x); S = std (tmp_x); ## Locate maximum deviation from mean dif_x = abs (tmp_x - C(i)); max_x = max (dif_x); loc_x = find (dif_x == max_x, 1); ## Calculate Grubbs's critical value t_crit = tinv (ThresholdFactor / (2 * n), n - 2); G_crit = ((n - 1) / sqrt (n)) * abs (t_crit) / sqrt (n - 2 + t_crit ^ 2); ## Check hypothesis if (max_x / S > G_crit) tmp_x(loc_x) = []; TF(TFvec(loc_x)) = true; TFvec(loc_x) = []; else break; endif endwhile L(i) = C(i) - S * G_crit; U(i) = C(i) + S * G_crit; endfor ## Restore shape TF = ipermute (TF, dperm); L = ipermute (L, dperm); U = ipermute (U, dperm); C = ipermute (C, dperm); endfunction ## Find lower and upper outlier thresholds with gesd method function [L, U, C] = gesd_method (x, dim, ThresholdFactor, MaxNumOutliers) ## Add default value in MaxNumOutliers (if empty) szx = size (x); N = szx(dim); if (isempty (MaxNumOutliers)) MaxNumOutliers = ceil (N * 0.1); endif ## Move the desired dim to be the 1st dimension (rows) nd = length (szx); # number of dimensions dperm = [dim, 1:(dim-1), (dim+1):nd]; # permutation of dimensions x = permute (x, dperm); # permute dims to first dimension ncols = prod (szx(dperm(2:end))); # rest of dimensions as single column x = reshape (x, N, ncols); # reshape input ## Create return matrices L = zeros ([1, szx(dperm(2:end))]); U = L; C = L; TF = false (size (x)); ## Apply processing to each column for i = 1:ncols tmp_x = x(:,i); vec_x = [(i-1)*size(x,1)+1:i*size(x,1)]; vec_x(isnan (tmp_x)) = []; tmp_x(isnan (tmp_x)) = []; n = length (tmp_x); if (n > 1) mean_x = zeros (MaxNumOutliers,1); S = zeros (MaxNumOutliers,1); lambda = zeros (MaxNumOutliers,1); R = zeros (MaxNumOutliers,1); Ridx = zeros (MaxNumOutliers,1); ## Search for given outliers for j = 1:MaxNumOutliers ## Get descriptive statistics mean_x(j) = mean (tmp_x); S(j) = std (tmp_x); ## Locate maximum deviation from mean dif_x = abs (tmp_x - mean_x(j)); max_x = max (dif_x); loc_x = find (dif_x == max_x, 1); ## Calculate R R(j) = max_x / S(j); tmp_x(loc_x) = []; Ridx(j) = vec_x(loc_x); vec_x(loc_x) = []; ## Calculate lambda pp = 1 - ThresholdFactor / (2 * (n - j + 1)); t = tinv (pp, n - j - 1); lambda(j) = (n - j) * t / sqrt ((n - j - 1 + t .^ 2) * (n - j + 1)); endfor ## Find largest index idx = find (R > lambda, 1, "last"); if (isempty (idx)) TFidx = 1; else TFidx = min (idx + 1, MaxNumOutliers); endif L(i) = mean_x(TFidx) - S(TFidx) * lambda(TFidx); U(i) = mean_x(TFidx) + S(TFidx) * lambda(TFidx); C(i) = mean_x(TFidx); endif endfor ## Restore shape L = ipermute (L, dperm); U = ipermute (U, dperm); C = ipermute (C, dperm); endfunction ## Find lower and upper outlier thresholds with movmedian method function [L, U, C] = movmedian_method (x, dim, ThresholdFactor, c, window, sp); szx = size (x); N = szx(dim); ## Constrain window to the element in the operating dimension if (numel (window) == 1 && window > N) window = N; elseif (numel (window) == 2 && sum (window) > N) window = N; endif if (isempty (sp)) FCN = @(x) median (x, "omitnan"); C = movfun (FCN, x, window, "dim", dim); FCN = @(x) mad (x, 1); MAD = movfun (FCN, x, window, "dim", dim); else ## Check that sample points(sp) have the N elements if (numel (sp) != N) error (strcat (["isoutlier: sample points must have the same size"], ... [" as the operating dimension."])); endif ## Move the desired dim to be the 1st dimension (rows) nd = length (szx); # number of dimensions dperm = [dim, 1:(dim-1), (dim+1):nd]; # permutation of dimensions x = permute (x, dperm); # permute dims to first dimension ncols = prod (szx(dperm(2:end))); # rest of dimensions as single column x = reshape (x, N, ncols); # reshape input ## Find beg+end from window if (numel (window) == 2) w_lo = window(1); w_hi = window(2); else if (mod (window, 2) == 1) w_lo = w_hi = (window - 1) / 2; else w_lo = window / 2; w_hi = w_lo - 1; endif endif ## Create return matrices C = zeros (size (x)); MAD = C; for i = 1:ncols tmp_x = x(:,i); for j = 1:N cp = sp - sp(j); nb = length (cp(cp < 0 & cp >= -w_lo)); na = length (cp(cp > 0 & cp <= w_hi)); sp_ind = [j-nb:j+na]; C(j,i) = median (tmp_x(sp_ind), "omitnan"); MAD(j,i) = mad (tmp_x(sp_ind), 1); endfor endfor ## Restore shape C = ipermute (C, dperm); MAD = ipermute (MAD, dperm); endif ## Compute scaled MAD sMAD = c * MAD; L = C - ThresholdFactor * sMAD; U = C + ThresholdFactor * sMAD; endfunction ## Find lower and upper outlier thresholds with movmean method function [L, U, M] = movmean_method (x, dim, ThresholdFactor, window, sp); ## Constrain window to the element in the operating dimension szx = size (x); N = szx(dim); if (numel (window) == 1 && window > N) window = N; elseif (numel (window) == 2 && sum (window) > N) window = N; endif if (isempty (sp)) FCN = @(x) mean (x, "omitnan"); M = movfun (FCN, x, window, "dim", dim); FCN = @(x) std (x, [], "omitnan"); S = movfun (FCN, x, window, "dim", dim); else ## Check that sample points(sp) have the N elements if (numel (sp) != N) error (strcat (["isoutlier: sample points must have the same size"], ... [" as the operating dimension."])); endif ## Move the desired dim to be the 1st dimension (rows) nd = length (szx); # number of dimensions dperm = [dim, 1:(dim-1), (dim+1):nd]; # permutation of dimensions x = permute (x, dperm); # permute dims to first dimension ncols = prod (szx(dperm(2:end))); # rest of dimensions as single column x = reshape (x, N, ncols); # reshape input ## Find beg+end from window if (numel (window) == 2) w_lo = window(1); w_hi = window(2); else if (mod (window, 2) == 1) w_lo = w_hi = (window - 1) / 2; else w_lo = window / 2; w_hi = w_lo - 1; endif endif ## Create return matrices M = zeros (size (x)); S = M; for i = 1:ncols tmp_x = x(:,i); for j = 1:N cp = sp - sp(j); nb = length (cp(cp < 0 & cp >= -w_lo)); na = length (cp(cp > 0 & cp <= w_hi)); sp_ind = [j-nb:j+na]; M(j,i) = mean (tmp_x(sp_ind), "omitnan"); S(j,i) = std (tmp_x(sp_ind), [], "omitnan"); endfor endfor ## Restore shape M = ipermute (M, dperm); S = ipermute (S, dperm); endif L = M - ThresholdFactor * S; U = M + ThresholdFactor * S; endfunction ## Find lower and upper outlier thresholds with percentiles method function [L, U, C] = percentiles_method (x, dim, threshold) P = [threshold(1)/100, threshold(2)/100]; Q = quantile (x, P, dim); L = Q(1); U = Q(2); C = (L + U) / 2; endfunction %!demo %! A = [57 59 60 100 59 58 57 58 300 61 62 60 62 58 57]; %! TF = isoutlier (A, "mean") %!demo %! ## Use a moving detection method to detect local outliers in a sine wave %! %! x = -2*pi:0.1:2*pi; %! A = sin(x); %! A(47) = 0; %! time = datenum (2023,1,1,0,0,0) + (1/24)*[0:length(x)-1] - 730485; %! TF = isoutlier (A, "movmedian", 5*(1/24), "SamplePoints", time); %! plot (time, A) %! hold on %! plot (time(TF), A(TF), "x") %! datetick ('x', 20, 'keepticks') %! legend ("Original Data", "Outlier Data") %!demo %! ## Locate an outlier in a vector of data and visualize the outlier %! %! x = 1:10; %! A = [60 59 49 49 58 100 61 57 48 58]; %! [TF, L, U, C] = isoutlier (A); %! plot (x, A); %! hold on %! plot (x(TF), A(TF), "x"); %! xlim ([1,10]); %! line ([1,10], [L, L], "Linestyle", ":"); %! text (1.1, L-2, "Lower Threshold"); %! line ([1,10], [U, U], "Linestyle", ":"); %! text (1.1, U-2, "Upper Threshold"); %! line ([1,10], [C, C], "Linestyle", ":"); %! text (1.1, C-3, "Center Value"); %! legend ("Original Data", "Outlier Data"); ## Output validation tests (checked against MATLAB) %!test %! A = [57 59 60 100 59 58 57 58 300 61 62 60 62 58 57]; %! assert (isoutlier (A, "mean"), logical([zeros(1,8) 1 zeros(1,6)])) %! assert (isoutlier (A, "median"), ... %! logical([zeros(1,3) 1 zeros(1,4) 1 zeros(1,6)])) %!test %! A = [57 59 60 100 59 58 57 58 300 61 62 60 62 58 57]; %! [TF, L, U, C] = isoutlier (A, "mean"); %! assert (L, -109.2459044922864, 1e-12) %! assert (U, 264.9792378256198, 1e-12) %! assert (C, 77.8666666666666, 1e-12) %!test %! A = [57 59 60 100 59 58 57 58 300 61 62 60 62 58 57]; %! [TF, L, U, C] = isoutlier (A, "median"); %! assert (L, 50.104386688966386, 1e-12) %! assert (U, 67.895613311033610, 1e-12) %! assert (C, 59) %!test %! A = magic(5) + diag(200*ones(1,5)); %! T = logical (eye (5)); %! assert (isoutlier (A, 2), T) %!test %! A = [57 59 60 100 59 58 57 58 300 61 62 60 62 58 57]; %! [TF, L, U, C] = isoutlier (A, "movmedian", 5); %! l = [54.5522, 52.8283, 54.5522, 54.5522, 54.5522, 53.5522, 53.5522, ... %! 53.5522, 47.6566, 56.5522, 57.5522, 56.5522, 51.1044, 52.3283, 53.5522]; %! u = [63.4478, 66.1717, 63.4478, 63.4478, 63.4478, 62.4478, 62.4478, ... %! 62.4478, 74.3434, 65.4478, 66.4478, 65.4478, 68.8956, 65.6717, 62.4478]; %! c = [59, 59.5, 59, 59, 59, 58, 58, 58, 61, 61, 62, 61, 60, 59, 58]; %! assert (L, l, 1e-4) %! assert (U, u, 1e-4) %! assert (C, c) %!test %! A = [57 59 60 100 59 58 57 58 300 61 62 60 62 58 57]; %! [TF, L, U, C] = isoutlier (A, "movmedian", 5, "SamplePoints", [1:15]); %! l = [54.5522, 52.8283, 54.5522, 54.5522, 54.5522, 53.5522, 53.5522, ... %! 53.5522, 47.6566, 56.5522, 57.5522, 56.5522, 51.1044, 52.3283, 53.5522]; %! u = [63.4478, 66.1717, 63.4478, 63.4478, 63.4478, 62.4478, 62.4478, ... %! 62.4478, 74.3434, 65.4478, 66.4478, 65.4478, 68.8956, 65.6717, 62.4478]; %! c = [59, 59.5, 59, 59, 59, 58, 58, 58, 61, 61, 62, 61, 60, 59, 58]; %! assert (L, l, 1e-4) %! assert (U, u, 1e-4) %! assert (C, c) %!test %! A = [57 59 60 100 59 58 57 58 300 61 62 60 62 58 57]; %! [TF, L, U, C] = isoutlier (A, "movmean", 5); %! l = [54.0841, 6.8872, 11.5608, 12.1518, 11.0210, 10.0112, -218.2840, ... %! -217.2375, -215.1239, -213.4890, -211.3264, 55.5800, 52.9589, ... %! 52.5979, 51.0627]; %! u = [63.2492, 131.1128, 122.4392, 122.2482, 122.5790, 122.7888, 431.0840, ... %! 430.8375, 430.3239, 429.8890, 429.3264, 65.6200, 66.6411, 65.9021, ... %! 66.9373]; %! c = [58.6667, 69, 67, 67.2, 66.8, 66.4, 106.4, 106.8, 107.6, 108.2, 109, ... %! 60.6, 59.8, 59.25, 59]; %! assert (L, l, 1e-4) %! assert (U, u, 1e-4) %! assert (C, c, 1e-4) %!test %! A = [57 59 60 100 59 58 57 58 300 61 62 60 62 58 57]; %! [TF, L, U, C] = isoutlier (A, "movmean", 5, "SamplePoints", [1:15]); %! l = [54.0841, 6.8872, 11.5608, 12.1518, 11.0210, 10.0112, -218.2840, ... %! -217.2375, -215.1239, -213.4890, -211.3264, 55.5800, 52.9589, ... %! 52.5979, 51.0627]; %! u = [63.2492, 131.1128, 122.4392, 122.2482, 122.5790, 122.7888, 431.0840, ... %! 430.8375, 430.3239, 429.8890, 429.3264, 65.6200, 66.6411, 65.9021, ... %! 66.9373]; %! c = [58.6667, 69, 67, 67.2, 66.8, 66.4, 106.4, 106.8, 107.6, 108.2, 109, ... %! 60.6, 59.8, 59.25, 59]; %! assert (L, l, 1e-4) %! assert (U, u, 1e-4) %! assert (C, c, 1e-4) %!test %! A = [57 59 60 100 59 58 57 58 300 61 62 60 62 58 57]; %! [TF, L, U, C] = isoutlier (A, "gesd"); %! assert (TF, logical ([0 0 0 1 0 0 0 0 1 0 0 0 0 0 0])) %! assert (L, 34.235977035439944, 1e-12) %! assert (U, 89.764022964560060, 1e-12) %! assert (C, 62) %!test %! A = [57 59 60 100 59 58 57 58 300 61 62 60 62 58 57]; %! [TF, L, U, C] = isoutlier (A, "gesd", "ThresholdFactor", 0.01); %! assert (TF, logical ([0 0 0 1 0 0 0 0 1 0 0 0 0 0 0])) %! assert (L, 31.489256770616173, 1e-12) %! assert (U, 92.510743229383820, 1e-12) %! assert (C, 62) %!test %! A = [57 59 60 100 59 58 57 58 300 61 62 60 62 58 57]; %! [TF, L, U, C] = isoutlier (A, "gesd", "ThresholdFactor", 5e-10); %! assert (TF, logical ([0 0 0 0 0 0 0 0 1 0 0 0 0 0 0])) %! assert (L, 23.976664158788935, 1e-12) %! assert (U, 100.02333584121110, 1e-12) %! assert (C, 62) %!test %! A = [57 59 60 100 59 58 57 58 300 61 62 60 62 58 57]; %! [TF, L, U, C] = isoutlier (A, "grubbs"); %! assert (TF, logical ([0 0 0 1 0 0 0 0 1 0 0 0 0 0 0])) %! assert (L, 54.642809574646606, 1e-12) %! assert (U, 63.511036579199555, 1e-12) %! assert (C, 59.076923076923080, 1e-12) %!test %! A = [57 59 60 100 59 58 57 58 300 61 62 60 62 58 57]; %! [TF, L, U, C] = isoutlier (A, "grubbs", "ThresholdFactor", 0.01); %! assert (TF, logical ([0 0 0 1 0 0 0 0 1 0 0 0 0 0 0])) %! assert (L, 54.216083184201850, 1e-12) %! assert (U, 63.937762969644310, 1e-12) %! assert (C, 59.076923076923080, 1e-12) %!test %! A = [57 59 60 100 59 58 57 58 300 61 62 60 62 58 57]; %! [TF, L, U, C] = isoutlier (A, "percentiles", [10 90]); %! assert (TF, logical ([0 0 0 0 0 0 0 0 1 0 0 0 0 0 0])) %! assert (L, 57) %! assert (U, 100) %! assert (C, 78.5) %!test %! A = [57 59 60 100 59 58 57 58 300 61 62 60 62 58 57]; %! [TF, L, U, C] = isoutlier (A, "percentiles", [20 80]); %! assert (TF, logical ([1 0 0 1 0 0 1 0 1 0 0 0 0 0 1])) %! assert (L, 57.5) %! assert (U, 62) %! assert (C, 59.75) ## Test input validation %!shared A %! A = [57 59 60 100 59 58 57 58 300 61 62 60 62 58 57]; %!error ... %! isoutlier (A, "movmedian", 0); %!error ... %! isoutlier (A, "movmedian", []); %!error ... %! isoutlier (A, "movmedian", [2 3 4]); %!error ... %! isoutlier (A, "movmedian", 1.4); %!error ... %! isoutlier (A, "movmedian", [0 1]); %!error ... %! isoutlier (A, "movmedian", [2 -1]); %!error ... %! isoutlier (A, "movmedian", {2 3}); %!error ... %! isoutlier (A, "movmedian", "char"); %! %!error ... %! isoutlier (A, "movmean", 0); %!error ... %! isoutlier (A, "movmean", []); %!error ... %! isoutlier (A, "movmean", [2 3 4]); %!error ... %! isoutlier (A, "movmean", 1.4); %!error ... %! isoutlier (A, "movmean", [0 1]); %!error ... %! isoutlier (A, "movmean", [2 -1]); %!error ... %! isoutlier (A, "movmean", {2 3}); %!error ... %! isoutlier (A, "movmean", "char"); %! %!error ... %! isoutlier (A, "percentiles", [-1 90]); %!error ... %! isoutlier (A, "percentiles", [10 -90]); %!error ... %! isoutlier (A, "percentiles", [90]); %!error ... %! isoutlier (A, "percentiles", [90 20]); %!error ... %! isoutlier (A, "percentiles", [90 20]); %!error ... %! isoutlier (A, "percentiles", [10 20 90]); %!error ... %! isoutlier (A, "percentiles", {10 90}); %!error ... %! isoutlier (A, "percentiles", "char"); %! %!error ... %! isoutlier (A, "movmean", 5, "SamplePoints", ones(3,15)); %!error ... %! isoutlier (A, "movmean", 5, "SamplePoints", 15); %!error ... %! isoutlier (A, "movmean", 5, "SamplePoints", [1,1:14]); %!error ... %! isoutlier (A, "movmean", 5, "SamplePoints", [2,1,3:15]); %!error ... %! isoutlier (A, "movmean", 5, "SamplePoints", [1:14]); %! %!error ... %! isoutlier (A, "movmean", 5, "ThresholdFactor", [1:14]); %!error ... %! isoutlier (A, "movmean", 5, "ThresholdFactor", -1); %!error ... %! isoutlier (A, "gesd", "ThresholdFactor", 3); %!error ... %! isoutlier (A, "grubbs", "ThresholdFactor", 3); %! %!error ... %! isoutlier (A, "movmean", 5, "MaxNumOutliers", [1:14]); %!error ... %! isoutlier (A, "movmean", 5, "MaxNumOutliers", -1); %!error ... %! isoutlier (A, "movmean", 5, "MaxNumOutliers", 0); %!error ... %! isoutlier (A, "movmean", 5, "MaxNumOutliers", 1.5); %! %!error ... %! isoutlier (A, {"movmean"}, 5, "SamplePoints", [1:15]); %!error isoutlier (A, {1}); %!error isoutlier (A, true); %!error isoutlier (A, false); %!error isoutlier (A, 0); %!error isoutlier (A, [1 2]); %!error isoutlier (A, -2); statistics-release-1.6.3/inst/jackknife.m000066400000000000000000000125741456127120000204200ustar00rootroot00000000000000## Copyright (C) 2011 Alexander Klein ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{jackstat} =} jackknife (@var{E}, @var{x}) ## @deftypefnx {statistics} {@var{jackstat} =} jackknife (@var{E}, @var{x}, @dots{}) ## ## Compute jackknife estimates of a parameter taking one or more given samples ## as parameters. ## ## In particular, @var{E} is the estimator to be jackknifed as a function name, ## handle, or inline function, and @var{x} is the sample for which the estimate ## is to be taken. The @var{i}-th entry of @var{jackstat} will contain the ## value of the estimator on the sample @var{x} with its @var{i}-th row omitted. ## ## @example ## @group ## jackstat (@var{i}) = @var{E}(@var{x}(1 : @var{i} - 1, @var{i} + 1 : length(@var{x}))) ## @end group ## @end example ## ## Depending on the number of samples to be used, the estimator must have the ## appropriate form: ## @itemize ## @item ## If only one sample is used, then the estimator need not be concerned with ## cell arrays, for example jackknifing the standard deviation of a sample can ## be performed with @code{@var{jackstat} = jackknife (@@std, rand (100, 1))}. ## @item ## If, however, more than one sample is to be used, the samples must all be of ## equal size, and the estimator must address them as elements of a cell-array, ## in which they are aggregated in their order of appearance: ## @end itemize ## ## @example ## @group ## @var{jackstat} = jackknife (@@(x) std(x@{1@})/var(x@{2@}), ## rand (100, 1), randn (100, 1)) ## @end group ## @end example ## ## If all goes well, a theoretical value @var{P} for the parameter is already ## known, @var{n} is the sample size, ## ## @code{@var{t} = @var{n} * @var{E}(@var{x}) - (@var{n} - 1) * ## mean(@var{jackstat})} ## ## and ## ## @code{@var{v} = sumsq(@var{n} * @var{E}(@var{x}) - (@var{n} - 1) * ## @var{jackstat} - @var{t}) / (@var{n} * (@var{n} - 1))} ## ## then ## ## @code{(@var{t}-@var{P})/sqrt(@var{v})} should follow a t-distribution with ## @var{n}-1 degrees of freedom. ## ## Jackknifing is a well known method to reduce bias. ## Further details can be found in: ## @subheading References ## ## @enumerate ## @item ## Rupert G. Miller. The jackknife - a review. Biometrika (1974), 61(1):1-15. ## doi:10.1093/biomet/61.1.1 ## @item ## Rupert G. Miller. Jackknifing Variances. Ann. Math. Statist. (1968), ## Volume 39, Number 2, 567-582. doi:10.1214/aoms/1177698418 ## @end enumerate ## @end deftypefn function jackstat = jackknife (anEstimator, varargin) ## Convert function name to handle if necessary, or throw an error. if (! strcmp (typeinfo (anEstimator), "function handle")) if (isascii (anEstimator)) anEstimator = str2func (anEstimator); else error (strcat (["jackknife: estimators must be passed as function"], ... [" names or handles."])); endif endif ## Simple jackknifing can be done with a single vector argument, and ## first and foremost with a function that does not care about cell-arrays. if (length (varargin) == 1 && isnumeric (varargin {1})) aSample = varargin{1}; g = length (aSample); jackstat = zeros (1, g); for k = 1:g jackstat (k) = anEstimator (aSample([1:k - 1,k + 1:g])); endfor ## More complicated input requires more work, however. else g = cellfun (@(x) length (x), varargin); if (any (g - g(1))) error ("jackknife: all passed data must be of equal length."); endif g = g(1); jackstat = zeros (1, g); for k = 1:g jackstat(k) = anEstimator (cellfun (@(x) x( [ 1 : k - 1, k + 1 : g ]), ... varargin, "UniformOutput", false)); endfor endif endfunction %!demo %! for k = 1:1000 %! rand ("seed", k); # for reproducibility %! x = rand (10, 1); %! s(k) = std (x); %! jackstat = jackknife (@std, x); %! j(k) = 10 * std (x) - 9 * mean (jackstat); %! endfor %! figure(); %! hist ([s', j'], 0:sqrt(1/12)/10:2*sqrt(1/12)) %!demo %! for k = 1:1000 %! randn ("seed", k); # for reproducibility %! x = randn (1, 50); %! rand ("seed", k); # for reproducibility %! y = rand (1, 50); %! jackstat = jackknife (@(x) std(x{1})/std(x{2}), y, x); %! j(k) = 50 * std (y) / std (x) - 49 * mean (jackstat); %! v(k) = sumsq ((50 * std (y) / std (x) - 49 * jackstat) - j(k)) / (50 * 49); %! endfor %! t = (j - sqrt (1 / 12)) ./ sqrt (v); %! figure(); %! plot (sort (tcdf (t, 49)), ... %! "-;Almost linear mapping indicates good fit with t-distribution.;") ## Test output %!test %! ##Example from Quenouille, Table 1 %! d=[0.18 4.00 1.04 0.85 2.14 1.01 3.01 2.33 1.57 2.19]; %! jackstat = jackknife ( @(x) 1/mean(x), d ); %! assert ( 10 / mean(d) - 9 * mean(jackstat), 0.5240, 1e-5 ); statistics-release-1.6.3/inst/kmeans.m000066400000000000000000000620761456127120000177530ustar00rootroot00000000000000## Copyright (C) 2011 Soren Hauberg ## Copyright (C) 2012 Daniel Ward ## Copyright (C) 2015-2016 Lachlan Andrew ## Copyright (C) 2016 Michael Bentley ## Copyright (C) 2021 Stefano Guidoni ## Copyright (C) 2022-2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{idx} =} kmeans (@var{data}, @var{k}) ## @deftypefnx {statistics} {[@var{idx}, @var{centers}] =} kmeans (@var{data}, @var{k}) ## @deftypefnx {statistics} {[@var{idx}, @var{centers}, @var{sumd}] =} kmeans (@var{data}, @var{k}) ## @deftypefnx {statistics} {[@var{idx}, @var{centers}, @var{sumd}, @var{dist}] =} kmeans (@var{data}, @var{k}) ## @deftypefnx {statistics} {[@dots{}] =} kmeans (@var{data}, @var{k}, @var{param1}, @var{value1}, @dots{}) ## @deftypefnx {statistics} {[@dots{}] =} kmeans (@var{data}, [], @qcode{"start"}, @var{start}, @dots{}) ## ## Perform a @var{k}-means clustering of the @math{NxD} matrix @var{data}. ## ## If parameter @qcode{"start"} is specified, then @var{k} may be empty ## in which case @var{k} is set to the number of rows of @var{start}. ## ## The outputs are: ## ## @multitable @columnfractions 0.15 0.05 0.8 ## @item @var{idx} @tab @tab An @math{Nx1} vector whose @math{i}-th element is ## the class to which row @math{i} of @var{data} is assigned. ## ## @item @var{centers} @tab @tab A @math{KxD} array whose @math{i}-th row is the ## centroid of cluster @math{i}. ## ## @item @var{sumd} @tab @tab A @math{kx1} vector whose @math{i}-th entry is the ## sum of the distances from samples in cluster @math{i} to centroid @math{i}. ## ## @item @var{dist} @tab @tab An @math{Nxk} matrix whose @math{i}@math{j}-th ## element is the distance from sample @math{i} to centroid @math{j}. ## @end multitable ## ## The following parameters may be placed in any order. Each parameter ## must be followed by its value, as in Name-Value pairs. ## ## @multitable @columnfractions 0.15 0.02 0.83 ## @headitem Name @tab @tab Description ## @item @qcode{"Start"} @tab @tab The initialization method for the centroids. ## @end multitable ## ## @multitable @columnfractions 0.04 0.19 0.02 0.75 ## @headitem @tab Value @tab @tab Description ## @item @tab @qcode{"plus"} @tab @tab The k-means++ algorithm. (Default) ## @item @tab @qcode{"sample"} @tab @tab A subset of @math{k} rows from ## @var{data}, sampled uniformly without replacement. ## @item @tab @qcode{"cluster"} @tab @tab Perform a pilot clustering on 10% of ## the rows of @var{data}. ## @item @tab @qcode{"uniform"} @tab @tab Each component of each centroid is ## drawn uniformly from the interval between the maximum and minimum values of ## that component within @var{data}. This performs poorly and is implemented ## only for Matlab compatibility. ## @item @tab @var{numeric matrix} @tab @tab A @math{kxD} matrix of centroid ## starting locations. The rows correspond to seeds. ## @item @tab @var{numeric array} @tab @tab A @math{kxDxr} array of centroid ## starting locations. The third dimension invokes replication of the ## clustering routine. Page @math{r} contains the set of seeds for replicate ## @math{r}. @qcode{kmeans} infers the number of replicates (specified by the ## @qcode{"Replicates"} Name-Value pair argument) from the size of the third ## dimension. ## @end multitable ## ## @multitable @columnfractions 0.15 0.02 0.838 ## @headitem Name @tab @tab Description ## @item @qcode{"Distance"} @tab @tab The distance measure used for partitioning ## and calculating centroids. ## @end multitable ## ## @multitable @columnfractions 0.04 0.19 0.02 0.75 ## @headitem @tab Value @tab @tab Description ## @item @tab @qcode{"sqeuclidean"} @tab @tab The squared Euclidean distance. ## i.e. the sum of the squares of the differences between corresponding ## components. In this case, the centroid is the arithmetic mean of all samples ## in its cluster. This is the only distance for which this algorithm is truly ## "k-means". ## @item @tab @qcode{"cityblock"} @tab @tab The sum metric, or L1 distance, ## i.e. the sum of the absolute differences between corresponding components. ## In this case, the centroid is the median of all samples in its cluster. ## This gives the k-medians algorithm. ## @item @tab @qcode{"cosine"} @tab @tab One minus the cosine of the included ## angle between points (treated as vectors). Each centroid is the mean of the ## points in that cluster, after normalizing those points to unit Euclidean ## length. ## @item @tab @qcode{"correlation"} @tab @tab One minus the sample correlation ## between points (treated as sequences of values). Each centroid is the ## component-wise mean of the points in that cluster, after centering and ## normalizing those points to zero mean and unit standard deviation. ## @item @tab @qcode{"hamming"} @tab @tab The number of components in which the ## sample and the centroid differ. In this case, the centroid is the median of ## all samples in its cluster. Unlike Matlab, Octave allows non-logical ## @var{data}. ## @end multitable ## ## @multitable @columnfractions 0.15 0.02 0.838 ## @headitem Name @tab @tab Description ## @item @qcode{"EmptyAction"} @tab @tab What to do when a centroid is not the ## closest to any data sample. ## @end multitable ## ## @multitable @columnfractions 0.04 0.19 0.02 0.75 ## @headitem @tab Value @tab @tab Description ## @item @tab @qcode{"error"} @tab @tab Throw an error. ## @item @tab @qcode{"singleton"} @tab @tab (Default) Select the row of ## @var{data} that has the highest error and use that as the new centroid. ## @item @tab @qcode{"drop"} @tab @tab Remove the centroid, and continue ## computation with one fewer centroid. The dimensions of the outputs ## @var{centroids} and @var{d} are unchanged, with values for omitted centroids ## replaced by NaN. ## @end multitable ## ## @multitable @columnfractions 0.15 0.02 0.838 ## @headitem Name @tab @tab Description ## @item @qcode{"Display"} @tab @tab Display a text summary. ## @end multitable ## ## @multitable @columnfractions 0.04 0.19 0.02 0.75 ## @headitem @tab Value @tab @tab Description ## @item @tab @qcode{"off"} @tab @tab (Default) Display no summary. ## @item @tab @qcode{"final"} @tab @tab Display a summary for each clustering ## operation. ## @item @tab @qcode{"iter"} @tab @tab Display a summary for each iteration of a ## clustering operation. ## @end multitable ## ## @multitable @columnfractions 0.15 0.02 0.838 ## @headitem Name @tab @tab Value ## @item @qcode{"Replicates"} @tab @tab A positive integer specifying the number ## of independent clusterings to perform. The output values are the values for ## the best clustering, i.e., the one with the smallest value of @var{sumd}. ## If @var{Start} is numeric, then @var{Replicates} defaults to ## (and must equal) the size of the third dimension of @var{Start}. ## Otherwise it defaults to 1. ## @item @qcode{"MaxIter"} @tab @tab The maximum number of iterations to perform ## for each replicate. If the maximum change of any centroid is less than ## 0.001, then the replicate terminates even if @var{MaxIter} iterations have no ## occurred. The default is 100. ## @end multitable ## ## Example: ## ## [~,c] = kmeans (rand(10, 3), 2, "emptyaction", "singleton"); ## ## @seealso{linkage} ## @end deftypefn function [classes, centers, sumd, D] = kmeans (data, k, varargin) [reg, prop] = parseparams (varargin); ## defaults for options emptyaction = "singleton"; start = "plus"; replicates = 1; max_iter = 100; distance = "sqeuclidean"; display = "off"; replicates_set_explicitly = false; ## Remove rows containing NaN / NA, but record which rows are used data_idx = ! any (isnan (data), 2); original_rows = rows (data); data = data(data_idx,:); #used for getting the number of samples n_rows = rows (data); #used for convergence of the centroids err = 1; ## Input checking, validate the matrix if (! isnumeric (data) || ! ismatrix (data) || ! isreal (data)) error ("kmeans: first input argument must be a DxN real data matrix"); elseif (! isnumeric (k)) error ("kmeans: second argument must be numeric"); endif ## Parse options while (length (prop) > 0) if (length (prop) < 2) error ("kmeans: Option '%s' has no argument", prop{1}); endif switch (lower (prop{1})) case "emptyaction" emptyaction = prop{2}; case "start" start = prop{2}; case "maxiter" max_iter = prop{2}; case "distance" distance = prop{2}; case "replicates" replicates = prop{2}; replicates_set_explicitly = true; case "display" display = prop{2}; case {"onlinephase", "options"} warning ("kmeans: Ignoring unimplemented option '%s'", prop{1}); otherwise error ("kmeans: Unknown option %s", prop{1}); endswitch prop = {prop{3:end}}; endwhile ## Process options ## check for the 'emptyaction' property switch (emptyaction) case {"singleton", "error", "drop"} ; otherwise d = [", " disp(emptyaction)] (1:end-1); # strip trailing \n if (length (d) > 20) d = ""; endif error ("kmeans: unsupported empty cluster action parameter%s", d); endswitch ## check for the 'replicates' property if (! isnumeric (replicates) || ! isscalar (replicates) || ! isreal (replicates) || replicates < 1) d = [", " disp(replicates)] (1:end-1); # strip trailing \n if (length (d) > 20) d = ""; endif error ("kmeans: invalid number of replicates%s", d); endif ## check for the 'MaxIter' property if (! isnumeric (max_iter) || ! isscalar (max_iter) || ! isreal (max_iter) || max_iter < 1) d = [", " disp(max_iter)] (1:end-1); # strip trailing \n if (length (d) > 20) d = ""; endif error ("kmeans: invalid MaxIter%s", d); endif ## check for the 'start' property switch (lower (start)) case {"sample", "plus", "cluster"} start = lower (start); case {"uniform"} start = "uniform"; min_data = min (data); range = max (data) - min_data; otherwise if (! isnumeric (start)) d = [", " disp(start)] (1:end-1); # strip trailing \n if (length (d) > 20) d = ""; endif error ("kmeans: invalid start parameter%s", d); endif if (isempty (k)) k = rows (start); elseif (rows (start) != k) error (["kmeans: Number of initializers (%d) " ... "should match number of centroids (%d)"], rows (start), k); endif if (replicates_set_explicitly) if (replicates != size (start, 3)) error (["kmeans: The third dimension of the initializer (%d) " ... "should match the number of replicates (%d)"], ... size (start, 3), replicates); endif else replicates = size (start, 3); endif endswitch ## check for the 'distance' property ## dist returns the distance btwn each row of matrix x and a row vector c switch (lower (distance)) case "sqeuclidean" dist = @(x, c) sumsq (bsxfun (@minus, x, c), 2); centroid = @(x) mean (x, 1); case "cityblock" dist = @(x, c) sum (abs (bsxfun (@minus, x, c)), 2); centroid = @(x) median (x, 1); case "cosine" ## Pre-normalize all data. ## (when Octave implements normr, will use data = normr (data) ) for i = 1:rows (data) data(i,:) = data(i,:) / sqrt (sumsq (data(i,:))); endfor dist = @(x, c) 1 - (x * c') ./ sqrt (sumsq (c)); centroid = @(x) mean (x, 1); ## already normalized case "correlation" ## Pre-normalize all data. data = data - mean (data, 2); ## (when Octave implements normr, will use data = normr (data) ) for i = 1:rows (data) data(i,:) = data(i,:) / sqrt (sumsq (data(i,:))); endfor dist = @(x, c) 1 - (x * (c - mean (c))') ... ./ sqrt (sumsq (c - mean (c))); centroid = @(x) mean (x, 1); ## already normalized case "hamming" dist = @(x, c) sum (bsxfun (@ne, x, c), 2); centroid = @(x) median (x, 1); otherwise error ("kmeans: unsupported distance parameter %s", distance); endswitch ## check for the 'display' property if (! strcmp (display, "off")) display = lower (display); switch (display) case {"off", "final"} ; case "iter" printf ("%6s\t%6s\t%8s\t%12s\n", "iter", "phase", "num", "sum"); otherwise error ("kmeans: invalid display parameter %s", display); endswitch endif ## Done processing options ######################################## ## Now that k has been set (possibly by 'replicates' option), check/use it. if (! isscalar (k)) error ("kmeans: second input argument must be a scalar"); endif ## used to hold the distances from each sample to each class D = zeros (n_rows, k); best = Inf; best_centers = []; for rep = 1:replicates ## keep track of the number of data points that change class old_classes = zeros (rows (data), 1); n_changes = -1; ## check for the 'start' property switch (lower (start)) case "sample" idx = randperm (n_rows, k); centers = data(idx, :); case "plus" # k-means++, by Arthur and Vassilios(?) centers(1,:) = data(randi (n_rows),:); d = inf (n_rows, 1); # Distance to nearest centroid so far for i = 2:k d = min (d, dist (data, centers(i - 1, :))); centers(i,:) = data(find (cumsum (d) > rand * sum (d), 1), :); endfor case "cluster" idx = randperm (n_rows, max (k, ceil (n_rows / 10))); [~, centers] = kmeans (data(idx,:), k, "start", "sample", ... "distance", distance); case "uniform" # vectorised 'min_data + range .* rand' centers = bsxfun (@plus, min_data, bsxfun (@times, range, rand (k, columns (data)))); otherwise centers = start(:,:,rep); endswitch ## Run the algorithm iter = 1; ## Classify once before the loop; to set sumd, and if max_iter == 0 ## Compute distances and classify [D, classes, sumd] = update_dist (data, centers, D, k, dist); while (err > 0.001 && iter++ <= max_iter && n_changes != 0) ## Calculate new centroids replaced_centroids = []; ## Used by "emptyaction = singleton" for i = 1:k ## Get binary vector indicating membership in cluster i membership = (classes == i); ## Check for empty clusters if (! any (membership)) switch emptyaction ## if 'singleton', then find the point that is the ## farthest from any centroid (and not replacing an empty cluster ## from earlier in this pass) and add it to the empty cluster case 'singleton' available = setdiff (1:n_rows, replaced_centroids); [~, idx] = max (min (D(available,:)')); idx = available(idx); replaced_centroids = [replaced_centroids, idx]; classes(idx) = i; membership(idx) = 1; ## if 'drop' then set C and D to NA case 'drop' centers(i,:) = NA; D(i,:) = NA; ## if 'error' then throw the error otherwise error ("kmeans: empty cluster created"); endswitch endif ## end check for empty clusters ## update the centroids if (any (membership)) ## if we didn't "drop" the cluster centers(i, :) = centroid (data(membership, :)); endif endfor ## Compute distances, classes and sums [D, classes, new_sumd] = update_dist (data, centers, D, k, dist); ## calculate the difference in the sum of distances err = sum (sumd - new_sumd); ## update the current sum of distances sumd = new_sumd; ## compute the number of class changes n_changes = sum (old_classes != classes); old_classes = classes; ## display iteration status if (strcmp (display, "iter")) printf ("%6d\t%6d\t%8d\t%12.3f\n", (iter - 1), 1, ... n_changes, sum (sumd)); endif endwhile ## throw a warning if the algorithm did not converge if (iter > max_iter && err > 0.001 && n_changes != 0) warning ("kmeans: failed to converge in %d iterations", max_iter); endif if (sum (sumd) < sum (best) || isinf (best)) best = sumd; best_centers = centers; endif ## display final results if (strcmp (display, "final")) printf ("Replicate %d, %d iterations, total sum of distances = %.3f.\n", ... rep, iter, sum (sumd)); endif endfor centers = best_centers; ## Compute final distances, classes and sums [D, classes, sumd] = update_dist (data, centers, D, k, dist); ## display final results if (strcmp (display, "final") || strcmp (display, "iter")) printf ("Best total sum of distances = %.3f\n", sum (sumd)); endif ## Return with equal size as inputs if (original_rows != rows (data)) final = NA (original_rows,1); final(data_idx) = classes; ## other positions already NaN / NA classes = final; endif endfunction ## Update distances, classes and sums function [D, classes, sumd] = update_dist (data, centers, D, k, dist) for i = 1:k D (:, i) = dist (data, centers(i, :)); endfor [~, classes] = min (D, [], 2); ## calculate the sum of within-class distances sumd = zeros (k, 1); for i = 1:k sumd(i) = sum (D(classes == i,i)); endfor endfunction %!demo %! ## Generate a two-cluster problem %! randn ("seed", 31) # for reproducibility %! C1 = randn (100, 2) + 1; %! randn ("seed", 32) # for reproducibility %! C2 = randn (100, 2) - 1; %! data = [C1; C2]; %! %! ## Perform clustering %! rand ("seed", 1) # for reproducibility %! [idx, centers] = kmeans (data, 2); %! %! ## Plot the result %! figure; %! plot (data (idx==1, 1), data (idx==1, 2), "ro"); %! hold on; %! plot (data (idx==2, 1), data (idx==2, 2), "bs"); %! plot (centers (:, 1), centers (:, 2), "kv", "markersize", 10); %! hold off; %!demo %! ## Cluster data using k-means clustering, then plot the cluster regions %! ## Load Fisher's iris data set and use the petal lengths and widths as %! ## predictors %! %! load fisheriris %! X = meas(:,3:4); %! %! figure; %! plot (X(:,1), X(:,2), "k*", "MarkerSize", 5); %! title ("Fisher's Iris Data"); %! xlabel ("Petal Lengths (cm)"); %! ylabel ("Petal Widths (cm)"); %! %! ## Cluster the data. Specify k = 3 clusters %! rand ("seed", 1) # for reproducibility %! [idx, C] = kmeans (X, 3); %! x1 = min (X(:,1)):0.01:max (X(:,1)); %! x2 = min (X(:,2)):0.01:max (X(:,2)); %! [x1G, x2G] = meshgrid (x1, x2); %! XGrid = [x1G(:), x2G(:)]; %! %! idx2Region = kmeans (XGrid, 3, "MaxIter", 1, "Start", C); %! figure; %! gscatter (XGrid(:,1), XGrid(:,2), idx2Region, ... %! [0, 0.75, 0.75; 0.75, 0, 0.75; 0.75, 0.75, 0], ".."); %! hold on; %! plot (X(:,1), X(:,2), "k*", "MarkerSize", 5); %! title ("Fisher's Iris Data"); %! xlabel ("Petal Lengths (cm)"); %! ylabel ("Petal Widths (cm)"); %! legend ("Region 1", "Region 2", "Region 3", "Data", "Location", "SouthEast"); %! hold off %!demo %! ## Partition Data into Two Clusters %! %! randn ("seed", 1) # for reproducibility %! r1 = randn (100, 2) * 0.75 + ones (100, 2); %! randn ("seed", 2) # for reproducibility %! r2 = randn (100, 2) * 0.5 - ones (100, 2); %! X = [r1; r2]; %! %! figure; %! plot (X(:,1), X(:,2), "."); %! title ("Randomly Generated Data"); %! rand ("seed", 1) # for reproducibility %! [idx, C] = kmeans (X, 2, "Distance", "cityblock", ... %! "Replicates", 5, "Display", "final"); %! figure; %! plot (X(idx==1,1), X(idx==1,2), "r.", "MarkerSize", 12); %! hold on %! plot(X(idx==2,1), X(idx==2,2), "b.", "MarkerSize", 12); %! plot (C(:,1), C(:,2), "kx", "MarkerSize", 15, "LineWidth", 3); %! legend ("Cluster 1", "Cluster 2", "Centroids", "Location", "NorthWest"); %! title ("Cluster Assignments and Centroids"); %! hold off %!demo %! ## Assign New Data to Existing Clusters %! %! ## Generate a training data set using three distributions %! randn ("seed", 5) # for reproducibility %! r1 = randn (100, 2) * 0.75 + ones (100, 2); %! randn ("seed", 7) # for reproducibility %! r2 = randn (100, 2) * 0.5 - ones (100, 2); %! randn ("seed", 9) # for reproducibility %! r3 = randn (100, 2) * 0.75; %! X = [r1; r2; r3]; %! %! ## Partition the training data into three clusters by using kmeans %! %! rand ("seed", 1) # for reproducibility %! [idx, C] = kmeans (X, 3); %! %! ## Plot the clusters and the cluster centroids %! %! figure %! gscatter (X(:,1), X(:,2), idx, "bgm", "***"); %! hold on %! plot (C(:,1), C(:,2), "kx"); %! legend ("Cluster 1", "Cluster 2", "Cluster 3", "Cluster Centroid") %! %! ## Generate a test data set %! randn ("seed", 25) # for reproducibility %! r1 = randn (100, 2) * 0.75 + ones (100, 2); %! randn ("seed", 27) # for reproducibility %! r2 = randn (100, 2) * 0.5 - ones (100, 2); %! randn ("seed", 29) # for reproducibility %! r3 = randn (100, 2) * 0.75; %! Xtest = [r1; r2; r3]; %! %! ## Classify the test data set using the existing clusters %! ## Find the nearest centroid from each test data point by using pdist2 %! %! D = pdist2 (C, Xtest, "euclidean"); %! [group, ~] = find (D == min (D)); %! %! ## Plot the test data and label the test data using idx_test with gscatter %! %! gscatter (Xtest(:,1), Xtest(:,2), group, "bgm", "ooo"); %! legend ("Cluster 1", "Cluster 2", "Cluster 3", "Cluster Centroid", ... %! "Data classified to Cluster 1", "Data classified to Cluster 2", ... %! "Data classified to Cluster 3", "Location", "NorthWest"); %! title ("Assign New Data to Existing Clusters"); ## Test output %!test %! samples = 4; %! dims = 3; %! k = 2; %! [cls, c, d, z] = kmeans (rand (samples,dims), k, "start", rand (k,dims, 5), %! "emptyAction", "singleton"); %! assert (size (cls), [samples, 1]); %! assert (size (c), [k, dims]); %! assert (size (d), [k, 1]); %! assert (size (z), [samples, k]); %!test %! samples = 4; %! dims = 3; %! k = 2; %! [cls, c, d, z] = kmeans (rand (samples,dims), [], "start", rand (k,dims, 5), %! "emptyAction", "singleton"); %! assert (size (cls), [samples, 1]); %! assert (size (c), [k, dims]); %! assert (size (d), [k, 1]); %! assert (size (z), [samples, k]); %!test %! [cls, c] = kmeans ([1 0; 2 0], 2, "start", [8,0;0,8], "emptyaction", "drop"); %! assert (cls, [1; 1]); %! assert (c, [1.5, 0; NA, NA]); %!test %! kmeans (rand (4,3), 2, "start", rand (2,3, 5), "replicates", 5, %! "emptyAction", "singleton"); %!test %! kmeans (rand (3,4), 2, "start", "sample", "emptyAction", "singleton"); %!test %! kmeans (rand (3,4), 2, "start", "plus", "emptyAction", "singleton"); %!test %! kmeans (rand (3,4), 2, "start", "cluster", "emptyAction", "singleton"); %!test %! kmeans (rand (3,4), 2, "start", "uniform", "emptyAction", "singleton"); %!test %! kmeans (rand (4,3), 2, "distance", "sqeuclidean", "emptyAction", "singleton"); %!test %! kmeans (rand (4,3), 2, "distance", "cityblock", "emptyAction", "singleton"); %!test %! kmeans (rand (4,3), 2, "distance", "cosine", "emptyAction", "singleton"); %!test %! kmeans (rand (4,3), 2, "distance", "correlation", "emptyAction", "singleton"); %!test %! kmeans (rand (4,3), 2, "distance", "hamming", "emptyAction", "singleton"); %!test %! kmeans ([1 0; 1.1 0], 2, "start", eye(2), "emptyaction", "singleton"); ## Test input validation %!error kmeans (rand (3,2), 4); %!error kmeans ([1 0; 1.1 0], 2, "start", eye(2), "emptyaction", "panic"); %!error kmeans (rand (4,3), 2, "start", rand (2,3, 5), "replicates", 1); %!error kmeans (rand (4,3), 2, "start", rand (2,2)); %!error kmeans (rand (4,3), 2, "distance", "manhattan"); %!error kmeans (rand (3,4), 2, "start", "normal"); %!error kmeans (rand (4,3), 2, "replicates", i); %!error kmeans (rand (4,3), 2, "replicates", -1); %!error kmeans (rand (4,3), 2, "replicates", []); %!error kmeans (rand (4,3), 2, "replicates", [1 2]); %!error kmeans (rand (4,3), 2, "replicates", "one"); %!error kmeans (rand (4,3), 2, "MAXITER", i); %!error kmeans (rand (4,3), 2, "MaxIter", -1); %!error kmeans (rand (4,3), 2, "maxiter", []); %!error kmeans (rand (4,3), 2, "maxiter", [1 2]); %!error kmeans (rand (4,3), 2, "maxiter", "one"); %!error kmeans ([1 0; 1.1 0], 2, "start", eye(2), "emptyaction", "error"); statistics-release-1.6.3/inst/knnsearch.m000066400000000000000000000660121456127120000204430ustar00rootroot00000000000000## Copyright (C) 2023-2024 Andreas Bertsatos ## Copyright (C) 2023 Mohammed Azmat Khan ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{idx} =} knnsearch (@var{X}, @var{Y}) ## @deftypefnx {statistics} {[@var{idx}, @var{D}] =} knnsearch (@var{X}, @var{Y}) ## @deftypefnx {statistics} {[@dots{}] =} knnsearch (@dots{}, @var{name}, @var{value}) ## ## Find k-nearest neighbors from input data. ## ## @code{@var{idx} = knnsearch (@var{X}, @var{Y})} finds @math{K} nearest ## neighbors in @var{X} for @var{Y}. It returns @var{idx} which contains indices ## of @math{K} nearest neighbors of each row of @var{Y}, If not specified, ## @qcode{@var{K} = 1}. @var{X} must be an @math{NxP} numeric matrix of input ## data, where rows correspond to observations and columns correspond to ## features or variables. @var{Y} is an @math{MxP} numeric matrix with query ## points, which must have the same numbers of column as @var{X}. ## ## @code{[@var{idx}, @var{D}] = knnsearch (@var{X}, @var{Y})} also returns the ## the distances, @var{D}, which correspond to the @math{K} nearest neighbour in ## @var{X} for each @var{Y} ## ## Additional parameters can be specified by @qcode{Name-Value} pair arguments. ## ## @multitable @columnfractions 0.18 0.02 0.8 ## @headitem @var{Name} @tab @tab @var{Value} ## ## @item @qcode{"K"} @tab @tab is the number of nearest neighbors to be found ## in the kNN search. It must be a positive integer value and by default it is ## 1. ## ## @item @qcode{"P"} @tab @tab is the Minkowski distance exponent and it must be ## a positive scalar. This argument is only valid when the selected distance ## metric is @qcode{"minkowski"}. By default it is 2. ## ## @item @qcode{"Scale"} @tab @tab is the scale parameter for the standardized ## Euclidean distance and it must be a nonnegative numeric vector of equal ## length to the number of columns in @var{X}. This argument is only valid when ## the selected distance metric is @qcode{"seuclidean"}, in which case each ## coordinate of @var{X} is scaled by the corresponding element of ## @qcode{"scale"}, as is each query point in @var{Y}. By default, the scale ## parameter is the standard deviation of each coordinate in @var{X}. ## ## @item @qcode{"Cov"} @tab @tab is the covariance matrix for computing the ## mahalanobis distance and it must be a positive definite matrix matching the ## the number of columns in @var{X}. This argument is only valid when the ## selected distance metric is @qcode{"mahalanobis"}. ## ## @item @qcode{"BucketSize"} @tab @tab is the maximum number of data points in ## the leaf node of the Kd-tree and it must be a positive integer. This ## argument is only valid when the selected search method is @qcode{"kdtree"}. ## ## @item @qcode{"SortIndices"} @tab @tab is a boolean flag to sort the returned ## indices in ascending order by distance and it is @qcode{true} by default. ## When the selected search method is @qcode{"exhaustive"} or the ## @qcode{"IncludeTies"} flag is true, @code{knnsearch} always sorts the ## returned indices. ## ## @item @qcode{"Distance"} @tab @tab is the distance metric used by ## @code{knnsearch} as specified below: ## @end multitable ## ## @multitable @columnfractions 0.05 0.2 0.75 ## @item @tab @qcode{"euclidean"} @tab Euclidean distance. ## @item @tab @qcode{"seuclidean"} @tab standardized Euclidean distance. Each ## coordinate difference between the rows in @var{X} and the query matrix ## @var{Y} is scaled by dividing by the corresponding element of the standard ## deviation computed from @var{X}. To specify a different scaling, use the ## @qcode{"Scale"} name-value argument. ## @item @tab @qcode{"cityblock"} @tab City block distance. ## @item @tab @qcode{"chebychev"} @tab Chebychev distance (maximum coordinate ## difference). ## @item @tab @qcode{"minkowski"} @tab Minkowski distance. The default exponent ## is 2. To specify a different exponent, use the @qcode{"P"} name-value ## argument. ## @item @tab @qcode{"mahalanobis"} @tab Mahalanobis distance, computed using a ## positive definite covariance matrix. To change the value of the covariance ## matrix, use the @qcode{"Cov"} name-value argument. ## @item @tab @qcode{"cosine"} @tab Cosine distance. ## @item @tab @qcode{"correlation"} @tab One minus the sample linear correlation ## between observations (treated as sequences of values). ## @item @tab @qcode{"spearman"} @tab One minus the sample Spearman's rank ## correlation between observations (treated as sequences of values). ## @item @tab @qcode{"hamming"} @tab Hamming distance, which is the percentage ## of coordinates that differ. ## @item @tab @qcode{"jaccard"} @tab One minus the Jaccard coefficient, which is ## the percentage of nonzero coordinates that differ. ## @item @tab @var{@@distfun} @tab Custom distance function handle. A distance ## function of the form @code{function @var{D2} = distfun (@var{XI}, @var{YI})}, ## where @var{XI} is a @math{1xP} vector containing a single observation in ## @math{P}-dimensional space, @var{YI} is an @math{NxP} matrix containing an ## arbitrary number of observations in the same @math{P}-dimensional space, and ## @var{D2} is an @math{NxP} vector of distances, where @qcode{(@var{D2}k)} is ## the distance between observations @var{XI} and @qcode{(@var{YI}k,:)}. ## @end multitable ## ## @multitable @columnfractions 0.18 0.02 0.8 ## @item @qcode{"NSMethod"} @tab @tab is the nearest neighbor search method used ## by @code{knnsearch} as specified below. ## @end multitable ## ## @multitable @columnfractions 0.05 0.2 0.75 ## @item @tab @qcode{"kdtree"} @tab Creates and uses a Kd-tree to find nearest ## neighbors. @qcode{"kdtree"} is the default value when the number of columns ## in @var{X} is less than or equal to 10, @var{X} is not sparse, and the ## distance metric is @qcode{"euclidean"}, @qcode{"cityblock"}, ## @qcode{"manhattan"}, @qcode{"chebychev"}, or @qcode{"minkowski"}. Otherwise, ## the default value is @qcode{"exhaustive"}. This argument is only valid when ## the distance metric is one of the four aforementioned metrics. ## @item @tab @qcode{"exhaustive"} @tab Uses the exhaustive search algorithm by ## computing the distance values from all the points in @var{X} to each point in ## @var{Y}. ## @end multitable ## ## @multitable @columnfractions 0.18 0.02 0.8 ## @item @qcode{"IncludeTies"} @tab @tab is a boolean flag to indicate if the ## returned values should contain the indices that have same distance as the ## @math{K^th} neighbor. When @qcode{false}, @code{knnsearch} chooses the ## observation with the smallest index among the observations that have the same ## distance from a query point. When @qcode{true}, @code{knnsearch} includes ## all nearest neighbors whose distances are equal to the @math{K^th} smallest ## distance in the output arguments. To specify @math{K}, use the @qcode{"K"} ## name-value pair argument. ## @end multitable ## ## @seealso{rangesearch, pdist2, fitcknn} ## @end deftypefn function [idx, dist] = knnsearch (X, Y, varargin) ## Check input data if (nargin < 2) error ("knnsearch: too few input arguments."); endif if (size (X, 2) != size (Y, 2)) error ("knnsearch: number of columns in X and Y must match."); endif ## Add default values K = 1; # Number of nearest neighbors P = 2; # Exponent for Minkowski distance S = []; # Scale for the standardized Euclidean distance C = []; # Covariance matrix for Mahalanobis distance BS = 50; # Maximum number of points per leaf node for Kd-tree SI = true; # Sort returned indices according to distance Distance = "euclidean"; # Distance metric to be used NSMethod = []; # Nearest neighbor search method InclTies = false; # Include ties for distance with kth neighbor DistParameter = []; # Distance parameter for pdist2 ## Parse additional parameters in Name/Value pairs PSC = 0; while (numel (varargin) > 0) switch (tolower (varargin{1})) case "k" K = varargin{2}; case "p" P = varargin{2}; PSC += 1; case "scale" S = varargin{2}; PSC += 1; case "cov" C = varargin{2}; PSC += 1; case "bucketsize" BS = varargin{2}; case "sortindices" SI = varargin{2}; case "distance" Distance = varargin{2}; case "nsmethod" NSMethod = varargin{2}; case "includeties" InclTies = varargin{2}; otherwise error ("knnsearch: invalid NAME in optional pairs of arguments."); endswitch varargin(1:2) = []; endwhile ## Check input parameters if (PSC > 1) error ("knnsearch: only a single distance parameter can be defined."); endif if (! isscalar (K) || ! isnumeric (K) || K < 1 || K != round (K)) error ("knnsearch: invalid value of K."); endif if (! isscalar (P) || ! isnumeric (P) || P <= 0) error ("knnsearch: invalid value of Minkowski Exponent."); endif if (! isempty (S)) if (any (S) < 0 || numel (S) != columns (X) || ! strcmpi (Distance, "seuclidean")) error ("knnsearch: invalid value in Scale or the size of Scale."); endif endif if (! isempty (C)) if (! strcmp (Distance, "mahalanobis") || ! ismatrix (C) || ! isnumeric (C)) error (strcat (["knnsearch: invalid value in Cov, Cov can only"], ... [" be given for mahalanobis distance."])); endif endif if (! isscalar (BS) || BS < 0) error ("knnsearch: invalid value of bucketsize."); endif ## Select the appropriate distance parameter if (strcmpi (Distance, "minkowski")) DistParameter = P; elseif (strcmpi (Distance, "seuclidean")) DistParameter = S; elseif (strcmpi (Distance, "mahalanobis")) DistParameter = C; endif ## Check NSMethod and set kdtree as default if the conditions match if (isempty (NSMethod)) ## Set default method 'kdtree' if condintions are satistfied; if (! issparse (X) && (columns (X) <= 10) && ... (strcmpi (Distance, "euclidean") || strcmpi (Distance, "cityblock") || strcmpi (Distance, "minkowski") || strcmpi (Distance, "chebychev"))) NSMethod = "kdtree"; else NSMethod = "exhaustive"; endif else ## Not empty then check if is exhaustive or kdtree if (strcmpi (NSMethod,"kdtree") && ! ( strcmpi (Distance, "euclidean") || strcmpi (Distance, "cityblock") || strcmpi (Distance, "minkowski") || strcmpi (Distance, "chebychev"))) error (strcat (["knnsearch: 'kdtree' cannot be used with"], ... [" the given distance metric."])); endif endif ## Check for NSMethod if (strcmpi (NSMethod, "kdtree")) ## Build kdtree and search the query point ret = buildkdtree (X, BS); ## Check for ties and sortindices if (! InclTies) ## Only return k neighbors ## No need for returning cell dist = []; idx = []; for i = 1:rows (Y) NN = findkdtree (ret, Y(i, :), K, Distance, DistParameter); D = pdist2 (X(NN,:), Y(i,:), Distance, DistParameter); sorted_D = sortrows ([NN, D], [2, 1]); dist = [dist; sorted_D(1:K, 2)']; idx = [idx; sorted_D(1:K, 1)']; endfor if (SI) ## Rows are already sorted by distance dist = (dist); idx = (idx); else dist = (dist); idx = (idx); endif else ## Return all neighbors as cell dist = cell (rows (Y), 1); idx = cell (rows (Y), 1); for i = 1:rows (Y) NN = findkdtree (ret, Y(i, :), K, Distance, DistParameter); D = pdist2 (X(NN,:), Y(i,:), Distance, DistParameter); sorted_D = sortrows ([NN, D], [2, 1]); kth_dist = sorted_D (K, 2); tied_idx = (sorted_D (:, 2) <= kth_dist); dist {i} = sorted_D (tied_idx, 2)'; idx {i} = sorted_D (tied_idx, 1)'; endfor endif else ## Calculate all distances if (K == 1) D = pdist2 (X, Y, Distance, DistParameter); D = reshape (D', size (Y, 1), size (X, 1)); [dist, idx] = min (D, [], 2); else # always sort indices in this case if (InclTies) dist = cell (rows (Y), 1); idx = cell (rows (Y), 1); for i = 1:rows (Y) D = pdist2 (X, Y(i,:), Distance, DistParameter); [dt, id] = sort (D); kth_dist = dt (K); tied_idx = (dt <= kth_dist); dist {i} = dt(tied_idx, :)'; idx {i} = id(tied_idx, :)'; endfor else ## No ties included D = pdist2 (X, Y, Distance, DistParameter); D = reshape (D', size (Y, 1), size (X, 1)); [dist, idx] = sort (D, 2); dist = dist(:,1:K); idx = idx(:,1:K); endif endif endif endfunction ## buildkdtree function ret = buildkdtree_recur (X, r, d, BS) count = length (r); dimen = size (X, 2); if (count == 1) ret = struct ("point", r(1), "dimen", d); else mid = ceil (count / 2); ret = struct ("point", r(mid), "dimen", d); d = mod (d, dimen) + 1; ## Build left sub tree if (mid > 1) left = r(1:mid-1); left_points = X(left,d); [val, left_idx] = sort (left_points); leftr = left(left_idx); ret.left = buildkdtree_recur (X, leftr, d); endif ## Build right sub tree if (count > mid) right = r(mid+1:count); right_points = X(right,d); [val, right_idx] = sort (right_points); rightr = right(right_idx); ret.right = buildkdtree_recur (X, rightr, d); endif endif endfunction ## wrapper function for buildkdtree_recur function ret = buildkdtree (X, BS) [val, r] = sort (X(:,1)); ret = struct ("data", X, "root", buildkdtree_recur (X, r, 1, BS)); endfunction function farthest = kdtree_cand_farthest (X, p, cand, dist, distparam) D = pdist2 (X, p, dist, distparam); [val, index] = max (D'(cand)); farthest = cand (index); endfunction ## function to insert into NN list function inserted = kdtree_cand_insert (X, p, cand, k, point, dist, distparam) if (length (cand) < k) inserted = [cand; point]; else farthest = kdtree_cand_farthest (X, p, cand, dist, distparam); if (pdist2 (cand(find(cand == farthest),:), point, dist, distparam)) inserted = [cand; point]; else farthest = kdtree_cand_farthest (X, p, cand, dist, distparam); cand (find (cand == farthest)) = point; inserted = cand; endif endif endfunction ## function to search in a kd tree function nn = findkdtree_recur (X, node, p, nn, ... k, dist, distparam) point = node.point; d = node.dimen; if (X(point,d) > p(d)) ## Search in left sub tree if (isfield (node, "left")) nn = findkdtree_recur (X, node.left, p, nn, k, dist, distparam); endif ## Add current point if neccessary farthest = kdtree_cand_farthest (X, p, nn, dist, distparam); if (length(nn) < k || pdist2 (X(point,:), p, dist, distparam) <= pdist2 (X(farthest,:), p, dist, distparam)) nn = kdtree_cand_insert (X, p, nn, k, point, dist, distparam); endif ## Search in right sub tree if neccessary farthest = kdtree_cand_farthest (X, p, nn, dist, distparam); radius = pdist2 (X(farthest,:), p, dist, distparam); if (isfield (node, "right") && (length(nn) < k || p(d) + radius > X(point,d))) nn = findkdtree_recur (X, node.right, p, nn, ... k, dist, distparam); endif else ## Search in right sub tree if (isfield (node, "right")) nn = findkdtree_recur (X, node.right, p, nn, k, dist, distparam); endif ## Add current point if neccessary farthest = kdtree_cand_farthest (X, p, nn, dist, distparam); if (length (nn) < k || pdist2 (X(point,:), p, dist, distparam) <= pdist2 (X(farthest,:), p, dist, distparam)) nn = kdtree_cand_insert (X, p, nn, k, point, dist, distparam); endif ## Search in left sub tree if neccessary farthest = kdtree_cand_farthest (X, p, nn, dist, distparam); radius = pdist2 (X(farthest,:), p, dist, distparam); if (isfield (node, "left") && (length (nn) < k || p(d) - radius <= X(point,d))) nn = findkdtree_recur (X, node.left, p, nn, k, dist, distparam); endif endif endfunction ## wrapper function for findkdtree_recur function nn = findkdtree (tree, p, k, dist, distparam) X = tree.data; root = tree.root; nn = findkdtree_recur (X, root, p, [], k, dist, distparam); endfunction %!demo %! ## find 10 nearest neighbour of a point using different distance metrics %! ## and compare the results by plotting %! load fisheriris %! X = meas(:,3:4); %! Y = species; %! point = [5, 1.45]; %! %! ## calculate 10 nearest-neighbours by minkowski distance %! [id, d] = knnsearch (X, point, "K", 10); %! %! ## calculate 10 nearest-neighbours by minkowski distance %! [idm, dm] = knnsearch (X, point, "K", 10, "distance", "minkowski", "p", 5); %! %! ## calculate 10 nearest-neighbours by chebychev distance %! [idc, dc] = knnsearch (X, point, "K", 10, "distance", "chebychev"); %! %! ## plotting the results %! gscatter (X(:,1), X(:,2), species, [.75 .75 0; 0 .75 .75; .75 0 .75], ".", 20); %! title ("Fisher's Iris Data - Nearest Neighbors with different types of distance metrics"); %! xlabel("Petal length (cm)"); %! ylabel("Petal width (cm)"); %! %! line (point(1), point(2), "marker", "X", "color", "k", ... %! "linewidth", 2, "displayname", "query point") %! line (X(id,1), X(id,2), "color", [0.5 0.5 0.5], "marker", "o", ... %! "linestyle", "none", "markersize", 10, "displayname", "eulcidean") %! line (X(idm,1), X(idm,2), "color", [0.5 0.5 0.5], "marker", "d", ... %! "linestyle", "none", "markersize", 10, "displayname", "Minkowski") %! line (X(idc,1), X(idc,2), "color", [0.5 0.5 0.5], "marker", "p", ... %! "linestyle", "none", "markersize", 10, "displayname", "chebychev") %! xlim ([4.5 5.5]); %! ylim ([1 2]); %! axis square; %!demo %! ## knnsearch on iris dataset using kdtree method %! load fisheriris %! X = meas(:,3:4); %! gscatter (X(:,1), X(:,2), species, [.75 .75 0; 0 .75 .75; .75 0 .75], ".", 20); %! title ("Fisher's iris dataset : Nearest Neighbors with kdtree search"); %! %! ## new point to be predicted %! point = [5 1.45]; %! %! line (point(1), point(2), "marker", "X", "color", "k", ... %! "linewidth", 2, "displayname", "query point") %! %! ## knnsearch using kdtree method %! [idx, d] = knnsearch (X, point, "K", 10, "NSMethod", "kdtree"); %! %! ## plotting predicted neighbours %! line (X(idx,1), X(idx,2), "color", [0.5 0.5 0.5], "marker", "o", ... %! "linestyle", "none", "markersize", 10, ... %! "displayname", "nearest neighbour") %! xlim ([4 6]) %! ylim ([1 3]) %! axis square %! ## details of predicted labels %! tabulate (species(idx)) %! %! ctr = point - d(end); %! diameter = 2 * d(end); %! ## Draw a circle around the 10 nearest neighbors. %! h = rectangle ("position", [ctr, diameter, diameter], "curvature", [1 1]); %! %! ## here only 8 neighbours are plotted instead of 10 since the dataset %! ## contains duplicate values ## Test output %!shared X, Y %! X = [1, 2, 3, 4; 2, 3, 4, 5; 3, 4, 5, 6]; %! Y = [1, 2, 2, 3; 2, 3, 3, 4]; %!test %! [idx, D] = knnsearch (X, Y, "Distance", "euclidean"); %! assert (idx, [1; 1]); %! assert (D, ones (2, 1) * sqrt (2)); %!test %! eucldist = @(v,m) sqrt(sumsq(repmat(v,rows(m),1)-m,2)); %! [idx, D] = knnsearch (X, Y, "Distance", eucldist); %! assert (idx, [1; 1]); %! assert (D, ones (2, 1) * sqrt (2)); %!test %! [idx, D] = knnsearch (X, Y, "Distance", "euclidean", "includeties", true); %! assert (iscell (idx), true); %! assert (iscell (D), true) %! assert (idx {1}, [1]); %! assert (idx {2}, [1, 2]); %! assert (D{1}, ones (1, 1) * sqrt (2)); %! assert (D{2}, ones (1, 2) * sqrt (2)); %!test %! [idx, D] = knnsearch (X, Y, "Distance", "euclidean", "k", 2); %! assert (idx, [1, 2; 1, 2]); %! assert (D, [sqrt(2), 3.162277660168380; sqrt(2), sqrt(2)], 1e-14); %!test %! [idx, D] = knnsearch (X, Y, "Distance", "seuclidean"); %! assert (idx, [1; 1]); %! assert (D, ones (2, 1) * sqrt (2)); %!test %! [idx, D] = knnsearch (X, Y, "Distance", "seuclidean", "k", 2); %! assert (idx, [1, 2; 1, 2]); %! assert (D, [sqrt(2), 3.162277660168380; sqrt(2), sqrt(2)], 1e-14); %!test %! xx = [1, 2; 1, 3; 2, 4; 3, 6]; %! yy = [2, 4; 2, 6]; %! [idx, D] = knnsearch (xx, yy, "Distance", "mahalanobis"); %! assert (idx, [3; 2]); %! assert (D, [0; 3.162277660168377], 1e-14); %!test %! [idx, D] = knnsearch (X, Y, "Distance", "minkowski"); %! assert (idx, [1; 1]); %! assert (D, ones (2, 1) * sqrt (2)); %!test %! [idx, D] = knnsearch (X, Y, "Distance", "minkowski", "p", 3); %! assert (idx, [1; 1]); %! assert (D, ones (2, 1) * 1.259921049894873, 1e-14); %!test %! [idx, D] = knnsearch (X, Y, "Distance", "cityblock"); %! assert (idx, [1; 1]); %! assert (D, [2; 2]); %!test %! [idx, D] = knnsearch (X, Y, "Distance", "chebychev"); %! assert (idx, [1; 1]); %! assert (D, [1; 1]); %!test %! [idx, D] = knnsearch (X, Y, "Distance", "cosine"); %! assert (idx, [2; 3]); %! assert (D, [0.005674536395645; 0.002911214328620], 1e-14); %!test %! [idx, D] = knnsearch (X, Y, "Distance", "correlation"); %! assert (idx, [1; 1]); %! assert (D, ones (2, 1) * 0.051316701949486, 1e-14); %!test %! [idx, D] = knnsearch (X, Y, "Distance", "spearman"); %! assert (idx, [1; 1]); %! assert (D, ones (2, 1) * 0.051316701949486, 1e-14); %!test %! [idx, D] = knnsearch (X, Y, "Distance", "hamming"); %! assert (idx, [1; 1]); %! assert (D, [0.5; 0.5]); %!test %! [idx, D] = knnsearch (X, Y, "Distance", "jaccard"); %! assert (idx, [1; 1]); %! assert (D, [0.5; 0.5]); %!test %! [idx, D] = knnsearch (X, Y, "Distance", "jaccard", "k", 2); %! assert (idx, [1, 2; 1, 2]); %! assert (D, [0.5, 1; 0.5, 0.5]); %!test %! a = [1, 5; 1, 2; 2, 2; 1.5, 1.5; 5, 1; 2 -1.34; 1, -3; 4, -4; -3, 1; 8, 9]; %! b = [1, 1]; %! [idx, D] = knnsearch (a, b, "K", 5, "NSMethod", "kdtree", "includeties", true); %! assert (iscell (idx), true); %! assert (iscell (D), true) %! assert (cell2mat (idx), [4, 2, 3, 6, 1, 5, 7, 9]); %! assert (cell2mat (D), [0.7071, 1.0000, 1.4142, 2.5447, 4.0000, 4.0000, 4.0000, 4.0000],1e-4); %!test %! a = [1, 5; 1, 2; 2, 2; 1.5, 1.5; 5, 1; 2 -1.34; 1, -3; 4, -4; -3, 1; 8, 9]; %! b = [1, 1]; %! [idx, D] = knnsearch (a, b, "K", 5, "NSMethod", "exhaustive", "includeties", true); %! assert (iscell (idx), true); %! assert (iscell (D), true) %! assert (cell2mat (idx), [4, 2, 3, 6, 1, 5, 7, 9]); %! assert (cell2mat (D), [0.7071, 1.0000, 1.4142, 2.5447, 4.0000, 4.0000, 4.0000, 4.0000],1e-4); %!test %! a = [1, 5; 1, 2; 2, 2; 1.5, 1.5; 5, 1; 2 -1.34; 1, -3; 4, -4; -3, 1; 8, 9]; %! b = [1, 1]; %! [idx, D] = knnsearch (a, b, "K", 5, "NSMethod", "kdtree", "includeties", false); %! assert (iscell (idx), false); %! assert (iscell (D), false) %! assert (idx, [4, 2, 3, 6, 1]); %! assert (D, [0.7071, 1.0000, 1.4142, 2.5447, 4.0000],1e-4); %!test %! a = [1, 5; 1, 2; 2, 2; 1.5, 1.5; 5, 1; 2 -1.34; 1, -3; 4, -4; -3, 1; 8, 9]; %! b = [1, 1]; %! [idx, D] = knnsearch (a, b, "K", 5, "NSMethod", "exhaustive", "includeties", false); %! assert (iscell (idx), false); %! assert (iscell (D), false) %! assert (idx, [4, 2, 3, 6, 1]); %! assert (D, [0.7071, 1.0000, 1.4142, 2.5447, 4.0000],1e-4); %!test %! load fisheriris %! a = meas; %! b = min(meas); %! [idx, D] = knnsearch (a, b, "K", 5, "NSMethod", "kdtree"); %! assert (idx, [42, 9, 14, 39, 13]); %! assert (D, [0.5099, 0.9950, 1.0050, 1.0536, 1.1874],1e-4); %!test %! load fisheriris %! a = meas; %! b = mean(meas); %! [idx, D] = knnsearch (a, b, "K", 5, "NSMethod", "kdtree"); %! assert (idx, [65, 83, 89, 72, 100]); %! assert (D, [0.3451, 0.3869, 0.4354, 0.4481, 0.4625],1e-4); %!test %! load fisheriris %! a = meas; %! b = max(meas); %! [idx, D] = knnsearch (a, b, "K", 5, "NSMethod", "kdtree"); %! assert (idx, [118, 132, 110, 106, 136]); %! assert (D, [0.7280, 0.9274, 1.3304, 1.5166, 1.6371],1e-4); %! %!test %! load fisheriris %! a = meas; %! b = max(meas); %! [idx, D] = knnsearch (a, b, "K", 5, "includeties", true); %! assert ( iscell (idx), true); %! assert ( iscell (D), true); %! assert (cell2mat (idx), [118, 132, 110, 106, 136]); %! assert (cell2mat (D), [0.7280, 0.9274, 1.3304, 1.5166, 1.6371],1e-4); ## Test input validation %!error knnsearch (1) %!error ... %! knnsearch (ones (4, 5), ones (4)) %!error ... %! knnsearch (ones (4, 2), ones (3, 2), "Distance", "euclidean", "some", "some") %!error ... %! knnsearch (ones (4, 5), ones (1, 5), "scale", ones (1, 5), "P", 3) %!error ... %! knnsearch (ones (4, 5), ones (1, 5), "K", 0) %!error ... %! knnsearch (ones (4, 5), ones (1, 5), "P",-2) %!error ... %! knnsearch (ones (4, 5), ones (1, 5), "scale", ones(4,5), "distance", "euclidean") %!error ... %! knnsearch (ones (4, 5), ones (1, 5), "cov", ["some" "some"]) %!error ... %! knnsearch (ones (4, 5), ones (1, 5), "cov", ones(4,5), "distance", "euclidean") %!error ... %! knnsearch (ones (4, 5), ones (1, 5), "bucketsize", -1) %!error ... %! knnsearch (ones (4, 5), ones (1, 5), "NSmethod", "kdtree", "distance", "cosine") %!error ... %! knnsearch (ones (4, 5), ones (1, 5), "NSmethod", "kdtree", "distance", "mahalanobis") %!error ... %! knnsearch (ones (4, 5), ones (1, 5), "NSmethod", "kdtree", "distance", "correlation") %!error ... %! knnsearch (ones (4, 5), ones (1, 5), "NSmethod", "kdtree", "distance", "seuclidean") %!error ... %! knnsearch (ones (4, 5), ones (1, 5), "NSmethod", "kdtree", "distance", "spearman") %!error ... %! knnsearch (ones (4, 5), ones (1, 5), "NSmethod", "kdtree", "distance", "hamming") %!error ... %! knnsearch (ones (4, 5), ones (1, 5), "NSmethod", "kdtree", "distance", "jaccard") statistics-release-1.6.3/inst/kruskalwallis.m000066400000000000000000000244751456127120000213660ustar00rootroot00000000000000## Copyright (C) 2021 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{p} =} kruskalwallis (@var{x}) ## @deftypefnx {statistics} {@var{p} =} kruskalwallis (@var{x}, @var{group}) ## @deftypefnx {statistics} {@var{p} =} kruskalwallis (@var{x}, @var{group}, @var{displayopt}) ## @deftypefnx {statistics} {[@var{p}, @var{tbl}] =} kruskalwallis (@var{x}, @dots{}) ## @deftypefnx {statistics} {[@var{p}, @var{tbl}, @var{stats}] =} kruskalwallis (@var{x}, @dots{}) ## ## Perform a Kruskal-Wallis test, the non-parametric alternative of a one-way ## analysis of variance (ANOVA), for comparing the means of two or more groups ## of data under the null hypothesis that the groups are drawn from the same ## population, i.e. the group means are equal. ## ## kruskalwallis can take up to three input arguments: ## ## @itemize ## @item ## @var{x} contains the data and it can either be a vector or matrix. ## If @var{x} is a matrix, then each column is treated as a separate group. ## If @var{x} is a vector, then the @var{group} argument is mandatory. ## @item ## @var{group} contains the names for each group. If @var{x} is a matrix, then ## @var{group} can either be a cell array of strings of a character array, with ## one row per column of @var{x}. If you want to omit this argument, enter an ## empty array ([]). If @var{x} is a vector, then @var{group} must be a vector ## of the same lenth, or a string array or cell array of strings with one row ## for each element of @var{x}. @var{x} values corresponding to the same value ## of @var{group} are placed in the same group. ## @item ## @var{displayopt} is an optional parameter for displaying the groups contained ## in the data in a boxplot. If omitted, it is 'on' by default. If group names ## are defined in @var{group}, these are used to identify the groups in the ## boxplot. Use 'off' to omit displaying this figure. ## @end itemize ## ## kruskalwallis can return up to three output arguments: ## ## @itemize ## @item ## @var{p} is the p-value of the null hypothesis that all group means are equal. ## @item ## @var{tbl} is a cell array containing the results in a standard ANOVA table. ## @item ## @var{stats} is a structure containing statistics useful for performing ## a multiple comparison of means with the MULTCOMPARE function. ## @end itemize ## ## If kruskalwallis is called without any output arguments, then it prints the ## results in a one-way ANOVA table to the standard output. It is also printed ## when @var{displayopt} is 'on'. ## ## Examples: ## ## @example ## x = meshgrid (1:6); ## x = x + normrnd (0, 1, 6, 6); ## [p, atab] = kruskalwallis(x); ## @end example ## ## ## @example ## x = ones (50, 4) .* [-2, 0, 1, 5]; ## x = x + normrnd (0, 2, 50, 4); ## group = @{"A", "B", "C", "D"@}; ## kruskalwallis (x, group); ## @end example ## ## @end deftypefn function [p, tbl, stats] = kruskalwallis (x, group, displayopt) ## check for valid number of input arguments narginchk (1, 3); ## add defaults if (nargin < 2) group = []; endif if (nargin < 3) displayopt = 'on'; endif plotdata = ~(strcmp (displayopt, 'off')); ## Convert group to cell array from character array, make it a column if (! isempty (group) && ischar (group)) group = cellstr(group); endif if (size (group, 1) == 1) group = group'; endif ## If X is a matrix, convert it to column vector and create a ## corresponging column vector for groups if (length (x) < prod (size (x))) [n, m] = size (x); x = x(:); gi = reshape (repmat ((1:m), n, 1), n*m, 1); if (length (group) == 0) ## no group names are provided group = gi; elseif (size (group, 1) == m) ## group names exist and match columns group = group(gi,:); else error("X columns and GROUP length do not match."); endif endif ## Identify NaN values (if any) and remove them from X along with ## their corresponding values from group vector nonan = ~isnan (x); x = x(nonan); group = group(nonan, :); ## Convert group to indices and separate names [group_id, group_names] = grp2idx (group); group_id = group_id(:); named = 1; ## Rank data for non-parametric analysis [xr, tieadj] = tieranks (x); ## Get group size and mean for each group groups = size (group_names, 1); xs = zeros (1, groups); xm = xs; for j = 1:groups group_size = find (group_id == j); xs(j) = length (group_size); xm(j) = mean (xr(group_size)); endfor ## Calculate statistics lx = length (xr); ## Number of samples in groups gm = mean (xr); ## Grand mean of groups dfm = length (xm) - 1; ## degrees of freedom for model dfe = lx - dfm - 1; ## degrees of freedom for error SSM = xs .* (xm - gm) * (xm - gm)'; ## Sum of Squares for Model SST = (xr(:) - gm)' * (xr(:) - gm); ## Sum of Squares Total SSE = SST - SSM; ## Sum of Squares Error if (dfm > 0) MSM = SSM / dfm; ## Mean Square for Model else MSM = NaN; endif if (dfe > 0) MSE = SSE / dfe; ## Mean Squared Error else MSE = NaN; endif ## Calculate Chi-sq statistic ChiSq = (12 * SSM) / (lx * (lx + 1)); if (tieadj > 0) ChiSq = ChiSq / (1 - 2 * tieadj / (lx ^ 3 - lx)); end p = 1 - chi2cdf (ChiSq, dfm); ## Create results table (if requested) if (nargout > 1) tbl = {"Source", "SS", "df", "MS", "Chi-sq", "Prob>Chi-sq"; ... "Groups", SSM, dfm, MSM, ChiSq, p; ... "Error", SSE, dfe, MSE, "", ""; ... "Total", SST, dfm + dfe, "", "", ""}; endif ## Create stats structure (if requested) for MULTCOMPARE if (nargout > 2) if (length (group_names) > 0) stats.gnames = group_names; else stats.gnames = strjust (num2str ((1:length (xm))'), 'left'); end stats.n = xs; stats.source = 'kruskalwallis'; stats.meanranks = xm; stats.sumt = 2 * tieadj; endif ## Print results table on screen if no output argument was requested if (nargout == 0 || plotdata) printf(" Kruskal-Wallis ANOVA Table\n"); printf("Source SS df MS Chi-sq Prob>Chi-sq\n"); printf("---------------------------------------------------------\n"); printf("Columns %10.2f %5.0f %10.2f %8.2f %11.5e\n", ... SSM, dfm, MSM, ChiSq, p); printf("Error %10.2f %5.0f %10.2f\n", SSE, dfe, MSE); printf("Total %10.2f %5.0f\n", SST, dfm + dfe); endif ## Plot data using BOXPLOT (unless opted out) if (plotdata) boxplot (x, group_id, 'Notch', "on", 'Labels', group_names); endif endfunction ## local function for computing tied ranks on column vectors function [r, tieadj] = tieranks (x) ## Sort data [value, x_idx] = sort (x); epsx = zeros (size (x)); epsx = epsx(x_idx); x_l = numel (x); ## Count ranks from start (min value) ranks = [1:x_l]'; ## Initialize tie adjustments tieadj = 0; ## Adjust for ties. ties = value(1:x_l-1) + epsx(1:x_l-1) >= value(2:x_l) - epsx(2:x_l); t_idx = find (ties); t_idx(end+1) = 0; maxTies = numel (t_idx); ## Calculate tie adjustments tiecount = 1; while (tiecount < maxTies) tiestart = t_idx(tiecount); ntied = 2; while (t_idx(tiecount+1) == t_idx(tiecount) + 1) tiecount = tiecount + 1; ntied = ntied + 1; endwhile ## Check for tieflag tieadj = tieadj + ntied * (ntied - 1) * (ntied + 1) / 2; ## Average tied ranks ranks(tiestart:tiestart + ntied - 1) = ... sum (ranks(tiestart:tiestart + ntied - 1)) / ntied; tiecount = tiecount + 1; endwhile ## Remap data to original dimensions r(x_idx) = ranks; endfunction %!demo %! x = meshgrid (1:6); %! x = x + normrnd (0, 1, 6, 6); %! kruskalwallis (x, [], 'off'); %!demo %! x = meshgrid (1:6); %! x = x + normrnd (0, 1, 6, 6); %! [p, atab] = kruskalwallis(x); %!demo %! x = ones (30, 4) .* [-2, 0, 1, 5]; %! x = x + normrnd (0, 2, 30, 4); %! group = {"A", "B", "C", "D"}; %! kruskalwallis (x, group); ## testing results against SPSS and R on the GEAR.DAT data file available from ## https://www.itl.nist.gov/div898/handbook/eda/section3/eda354.htm %!test %! data = [1.006, 0.996, 0.998, 1.000, 0.992, 0.993, 1.002, 0.999, 0.994, 1.000, ... %! 0.998, 1.006, 1.000, 1.002, 0.997, 0.998, 0.996, 1.000, 1.006, 0.988, ... %! 0.991, 0.987, 0.997, 0.999, 0.995, 0.994, 1.000, 0.999, 0.996, 0.996, ... %! 1.005, 1.002, 0.994, 1.000, 0.995, 0.994, 0.998, 0.996, 1.002, 0.996, ... %! 0.998, 0.998, 0.982, 0.990, 1.002, 0.984, 0.996, 0.993, 0.980, 0.996, ... %! 1.009, 1.013, 1.009, 0.997, 0.988, 1.002, 0.995, 0.998, 0.981, 0.996, ... %! 0.990, 1.004, 0.996, 1.001, 0.998, 1.000, 1.018, 1.010, 0.996, 1.002, ... %! 0.998, 1.000, 1.006, 1.000, 1.002, 0.996, 0.998, 0.996, 1.002, 1.006, ... %! 1.002, 0.998, 0.996, 0.995, 0.996, 1.004, 1.004, 0.998, 0.999, 0.991, ... %! 0.991, 0.995, 0.984, 0.994, 0.997, 0.997, 0.991, 0.998, 1.004, 0.997]; %! group = [1:10] .* ones (10,10); %! group = group(:); %! [p, tbl] = kruskalwallis (data, group, "off"); %! assert (p, 0.048229, 1e-6); %! assert (tbl{2,5}, 17.03124, 1e-5); %! assert (tbl{2,3}, 9, 0); %! assert (tbl{4,2}, 82655.5, 1e-16); %! data = reshape (data, 10, 10); %! [p, tbl, stats] = kruskalwallis (data, [], "off"); %! assert (p, 0.048229, 1e-6); %! assert (tbl{2,5}, 17.03124, 1e-5); %! assert (tbl{2,3}, 9, 0); %! assert (tbl{4,2}, 82655.5, 1e-16); %! means = [51.85, 60.45, 37.6, 51.1, 29.5, 54.25, 64.55, 66.7, 53.65, 35.35]; %! N = 10 * ones (1, 10); %! assert (stats.meanranks, means, 1e-6); %! assert (length (stats.gnames), 10, 0); %! assert (stats.n, N, 0); statistics-release-1.6.3/inst/kstest.m000066400000000000000000000326201456127120000200020ustar00rootroot00000000000000## Copyright (C) 2022 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{h} =} kstest (@var{x}) ## @deftypefnx {statistics} {@var{h} =} kstest (@var{x}, @var{name}, @var{value}) ## @deftypefnx {statistics} {[@var{h}, @var{p}] =} kstest (@dots{}) ## @deftypefnx {statistics} {[@var{h}, @var{p}, @var{ksstat}, @var{cv}] =} kstest (@dots{}) ## ## Single sample Kolmogorov-Smirnov (K-S) goodness-of-fit hypothesis test. ## ## @code{@var{h} = kstest (@var{x})} performs a Kolmogorov-Smirnov (K-S) test to ## determine if a random sample @var{x} could have come from a standard normal ## distribution. @var{h} indicates the results of the null hypothesis test. ## ## @itemize ## @item @var{h} = 0 => Do not reject the null hypothesis at the 5% significance ## @item @var{h} = 1 => Reject the null hypothesis at the 5% significance ## @end itemize ## ## @var{x} is a vector representing a random sample from some unknown ## distribution with a cumulative distribution function F(X). Missing values ## declared as NaNs in @var{x} are ignored. ## ## @code{@var{h} = kstest (@var{x}, @var{name}, @var{value})} returns ## a test decision for a single-sample K-S test with additional options ## specified by one or more name-value pair arguments as shown below. ## ## @multitable @columnfractions 0.20 0.8 ## @item "alpha" @tab A value @var{alpha} between 0 and 1 specifying the ## significance level. Default is 0.05 for 5% significance. ## ## @item "CDF" @tab CDF is the c.d.f. under the null hypothesis. It can be ## specified either as a function handle or a a function name of an existing ## cdf function or as a two-column matrix. If not provided, the default is ## the standard normal, N(0,1). ## ## @item "tail" @tab A string indicating the type of test: ## @end multitable ## ## @multitable @columnfractions 0.03 0.2 0.77 ## @item @tab "unequal" @tab "F(X) not equal to CDF(X)" (two-sided) (Default) ## ## @item @tab "larger" @tab "F(X) > CDF(X)" (one-sided) ## ## @item @tab "smaller" @tab "CDF(X) < F(X)" (one-sided) ## @end multitable ## ## Let S(X) be the empirical c.d.f. estimated from the sample vector @var{x}, ## F(X) be the corresponding true (but unknown) population c.d.f., and CDF be ## the known input c.d.f. specified under the null hypothesis. ## For @code{tail} = "unequal", "larger", and "smaller", the test statistics are ## max|S(X) - CDF(X)|, max[S(X) - CDF(X)], and max[CDF(X) - S(X)], respectively. ## ## @code{[@var{h}, @var{p}] = kstest (@dots{})} also returns the asymptotic ## p-value @var{p}. ## ## @code{[@var{h}, @var{p}, @var{ksstat}] = kstest (@dots{})} returns the K-S ## test statistic @var{ksstat} defined above for the test type indicated by the ## "tail" option ## ## In the matrix version of CDF, column 1 contains the x-axis data and column 2 ## the corresponding y-axis c.d.f data. Since the K-S test statistic will ## occur at one of the observations in @var{x}, the calculation is most ## efficient when CDF is only specified at the observations in @var{x}. When ## column 1 of CDF represents x-axis points independent of @var{x}, CDF is ## linearly interpolated at the observations found in the vector @var{x}. In ## this case, the interval along the x-axis (the column 1 spread of CDF) must ## span the observations in @var{x} for successful interpolation. ## ## The decision to reject the null hypothesis is based on comparing the p-value ## @var{p} with the "alpha" value, not by comparing the statistic @var{ksstat} ## with the critical value @var{cv}. @var{cv} is computed separately using an ## approximate formula or by interpolation using Miller's approximation table. ## The formula and table cover the range 0.01 <= "alpha" <= 0.2 for two-sided ## tests and 0.005 <= "alpha" <= 0.1 for one-sided tests. CV is returned as NaN ## if "alpha" is outside this range. Since CV is approximate, a comparison of ## @var{ksstat} with @var{cv} may occasionally lead to a different conclusion ## than a comparison of @var{p} with "alpha". ## ## @seealso{kstest2, cdfplot} ## @end deftypefn function [H, pValue, ksstat, cV] = kstest (x, varargin) ## Check input parameters if (nargin < 1) error ("kstest: too few inputs."); endif if (! isvector (x) || ! isreal (x)) error ("kstest: X must be a vector of real numbers."); endif ## Add defaults alpha = 0.05; tail = "unequal"; CDF = []; ## Parse extra parameters if (length (varargin) > 0 && mod (numel (varargin), 2) == 0) [~, prop] = parseparams (varargin); while (!isempty (prop)) switch (lower (prop{1})) case "alpha" alpha = prop{2}; case "tail" tail = prop{2}; case "CDF" CDF = prop{2}; otherwise error ("kstest: unknown option %s", prop{1}); endswitch prop = prop(3:end); endwhile elseif (mod (numel (varargin), 2) != 0) error ("kstest: optional parameters must be in name/value pairs."); endif ## Check for valid alpha and tail parameters if (! isnumeric (alpha) || isnan (alpha) || ! isscalar (alpha) ... || alpha <= 0 || alpha >= 1) error ("kstest: alpha must be a numeric scalar in the range (0,1)."); endif if (! isa (tail, 'char')) error ("kstest: tail argument must be a string"); elseif (sum (strcmpi (tail, {"unequal", "larger", "smaller"})) < 1) error ("kstest: tail value must be either 'both', right' or 'left'."); endif ## Remove NaNs, get sample size and compute empirical cdf x(isnan (x)) = []; n = length(x); [sampleCDF, x] = ecdf (x); ## Remove 1st element x = x(2:end); ## Check the hypothesized CDF specified under the null hypothesis. ## If CDF is a handle if (isa (CDF, "function_handle") || isa (CDF, "char")) xCDF = x; yCDF = feval (CDF, x); ## If CDF is numerical elseif (! isempty (CDF) && isnumeric (CDF)) if (size (CDF, 2) != 2) error ("kstest: numerical CDF should have only 2 columns."); endif CDF(isnan (sum (CDF, 2)),:) = []; if (size (CDF, 1) == 0) error ("kstest: numerical CDF should have at least one row."); endif ## Sort numerical CDF [xCDF, i] = sort (CDF(:,1)); yCDF = CDF(i,2); ## Check that numerical CDF is incrementally sorted ydiff = diff (yCDF); if (any (ydiff < 0)) error("kstest: non-incrementing numerical CDF"); endif ## Remove duplicates. Check for consistency rd = find (diff (xCDF) == 0); if (! isempty (rm)) if (! all (ydiff(rd) == 0)) error ("kstest: wrong duplicates in numericl CDF."); endif xCDF(rd) = []; yCDF(rd) = []; endif ## If CDF is empty, use standard normal distribution: x ~ N(0,1) else xCDF = x; yCDF = normcdf (x, 0, 1); endif ## Check if CDF is specified at the observations in X and assign 2nd column ## of numerical CDF to null CDF if (isequal (x, xCDF)) nCDF = yCDF; ## Otherwise interpolate the numerical CDF to assign values to the null CDF else ## Check that 1st column range bounds the observations in X if (x(1) < xCDF(1) || x(end) > xCDF(end)) error ("kstest: wrong span in CDF."); endif nCDF = interp1 (xCDF, yCDF, x); endif ## Calculate the suitable KS statistic according to tail switch (tail) case "unequal" # 2-sided test: T = max|S(x) - CDF(x)|. delta1 = sampleCDF(1:end - 1) - nCDF; delta2 = sampleCDF(2:end) - nCDF; deltaCDF = abs ([delta1; delta2]); case "smaller" # 1-sided test: T = max[CDF(x) - S(x)]. delta1 = nCDF - sampleCDF(1:end - 1); delta2 = nCDF - sampleCDF(2:end); deltaCDF = [delta1; delta2]; case "larger" # 1-sided test: T = max[S(x) - CDF(x)]. delta1 = sampleCDF(1:end - 1) - nCDF; delta2 = sampleCDF(2:end) - nCDF; deltaCDF = [delta1; delta2]; endswitch ksstat = max(deltaCDF); ## Compute the asymptotic P-value approximation if (strcmpi (tail, "unequal")) # 2-sided test s = n*ksstat^2; ## For d values that are in the far tail of the distribution (i.e. ## p-values > .999), the following lines will speed up the computation ## significantly, and provide accuracy up to 7 digits. if ((s > 7.24) ||((s > 3.76) && (n > 99))) pValue = 2*exp(-(2.000071+.331/sqrt(n)+1.409/n)*s); else ## Express d as d = (k-h)/n, where k is a +ve integer and 0 < h < 1. k = ceil (ksstat * n); h = k - ksstat * n; m = 2 * k - 1; ## Create the H matrix, according to Marsaglia et al. if (m > 1) c = 1 ./ gamma ((1:m)' + 1); r = zeros (1,m); r(1) = 1; r(2) = 1; T = toeplitz (c, r); T(:,1) = T(:,1) - (h .^ (1:m)') ./ gamma ((1:m)' + 1); T(m,:) = fliplr (T(:,1)'); T(m,1) = (1 - 2 * h ^ m + max (0, 2 * h - 1) ^m) / gamma (m+1); else T = (1 - 2 * h ^ m + max (0, 2 * h - 1) ^ m) / gamma (m+1); endif ## Scaling before raising the matrix to a power if (! isscalar (T)) lmax = max (eig (T)); T = (T ./ lmax) ^ n; else lmax = 1; endif pValue = 1 - exp (gammaln (n+1) + n * log (lmax) - n * log (n)) * T(k,k); endif else # 1-sided test t = n * ksstat; k = ceil (t):n; pValue = sum (exp (log (t) - n * log (n) + gammaln (n + 1) ... - gammaln (k + 1) - gammaln (n - k + 1) + k .* log (k - t) ... + (n - k - 1) .* log (t + n - k))); endif ## Return hypothesis test H = (pValue < alpha); ## Calculate critical Value (cV) if requested if (nargout > 3) ## The critical value table used below is expressed in reference to a ## 1-sided significance level. Hence alpha is halved for a two-sided test. if (strcmpi (tail, "unequal")) # 2-sided test alpha1 = alpha / 2; else # 1-sided test alpha1 = alpha; endif if ((alpha1 >= 0.005) && (alpha1 <= 0.10)) ## If the sample size 'n' is greater than 20, use Miller's approximation ## Otherwise interpolate into his 'exact' table. if (n <= 20) # Small sample exact values. % Exact K-S test critical values based on Miller's approximation. a1 = [0.00500, 0.01000, 0.02500, 0.05000, 0.10000]'; exact = [0.99500, 0.99000, 0.97500, 0.95000, 0.90000; ... 0.92929, 0.90000, 0.84189, 0.77639, 0.68377; ... 0.82900, 0.78456, 0.70760, 0.63604, 0.56481; ... 0.73424, 0.68887, 0.62394, 0.56522, 0.49265; ... 0.66853, 0.62718, 0.56328, 0.50945, 0.44698; ... 0.61661, 0.57741, 0.51926, 0.46799, 0.41037; ... 0.57581, 0.53844, 0.48342, 0.43607, 0.38148; ... 0.54179, 0.50654, 0.45427, 0.40962, 0.35831; ... 0.51332, 0.47960, 0.43001, 0.38746, 0.33910; ... 0.48893, 0.45662, 0.40925, 0.36866, 0.32260; ... 0.46770, 0.43670, 0.39122, 0.35242, 0.30829; ... 0.44905, 0.41918, 0.37543, 0.33815, 0.29577; ... 0.43247, 0.40362, 0.36143, 0.32549, 0.28470; ... 0.41762, 0.38970, 0.34890, 0.31417, 0.27481; ... 0.40420, 0.37713, 0.33760, 0.30397, 0.26588; ... 0.39201, 0.36571, 0.32733, 0.29472, 0.25778; ... 0.38086, 0.35528, 0.31796, 0.28627, 0.25039; ... 0.37062, 0.34569, 0.30936, 0.27851, 0.24360; ... 0.36117, 0.33685, 0.30143, 0.27136, 0.23735; ... 0.35241, 0.32866, 0.29408, 0.26473, 0.23156]; cV = spline (a1 , exact(n,:)' , alpha1); else # Large sample approximate values. A = 0.09037 * (-log10 (alpha1)) .^ 1.5 + 0.01515 * ... log10 (alpha1) .^ 2 - 0.08467 * alpha1 - 0.11143; asymptoticStat = sqrt (-0.5 * log (alpha1) ./ n); cV = asymptoticStat - 0.16693 ./ n - A ./ n .^ 1.5; cV = min (cV, 1 - alpha1); endif else cV = NaN; endif endif endfunction ## Test input %!error kstest () %!error kstest (ones(2,4)) %!error kstest ([2,3,5,7,3+3i]) %!error kstest ([2,3,4,5,6],"tail") %!error kstest ([2,3,4,5,6],"tail", "whatever") %!error kstest ([2,3,4,5,6],"badoption", 0.51) %!error kstest ([2,3,4,5,6],"tail", 0) %!error kstest ([2,3,4,5,6],"alpha", 0) %!error kstest ([2,3,4,5,6],"alpha", NaN) %!error kstest ([NaN,NaN,NaN,NaN,NaN],"tail", "unequal") %!error kstest ([2,3,4,5,6],"alpha", 0.05, "CDF", [2,3,4;1,3,4;1,2,1]) ## Test results %!test %! load examgrades %! [h, p] = kstest (grades(:,1)); %! assert (h, true); %! assert (p, 7.58603305206105e-107, 1e-14); %!test %! load stockreturns %! x = stocks(:,3); %! [h,p,k,c] = kstest (x, "Tail", "larger"); %! assert (h, true); %! assert (p, 5.085438806199252e-05, 1e-14); %! assert (k, 0.2197, 1e-4); %! assert (c, 0.1207, 1e-4); statistics-release-1.6.3/inst/kstest2.m000066400000000000000000000166771456127120000201020ustar00rootroot00000000000000## Copyright (C) 2022 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{h} =} kstest2 (@var{x1}, @var{x2}) ## @deftypefnx {statistics} {@var{h} =} kstest2 (@var{x1}, @var{x2}, @var{name}, @var{value}) ## @deftypefnx {statistics} {[@var{h}, @var{p}] =} kstest2 (@dots{}) ## @deftypefnx {statistics} {[@var{h}, @var{p}, @var{ks2stat}] =} kstest2 (@dots{}) ## ## Two-sample Kolmogorov-Smirnov goodness-of-fit hypothesis test. ## ## @code{@var{h} = kstest2 (@var{x1}, @var{x2})} returns a test decision for the ## null hypothesis that the data in vectors @var{x1} and @var{x2} are from the ## same continuous distribution, using the two-sample Kolmogorov-Smirnov test. ## The alternative hypothesis is that @var{x1} and @var{x2} are from different ## continuous distributions. The result @var{h} is 1 if the test rejects the ## null hypothesis at the 5% significance level, and 0 otherwise. ## ## @code{@var{h} = kstest2 (@var{x1}, @var{x2}, @var{name}, @var{value})} ## returns a test decision for a two-sample Kolmogorov-Smirnov test with ## additional options specified by one or more name-value pair arguments as ## shown below. ## ## @multitable @columnfractions 0.20 0.8 ## @item "alpha" @tab A value @var{alpha} between 0 and 1 specifying the ## significance level. Default is 0.05 for 5% significance. ## ## @item "tail" @tab A string indicating the type of test: ## @end multitable ## ## @multitable @columnfractions 0.03 0.2 0.77 ## @item @tab "unequal" @tab "F(X1) not equal to F(X2)" (two-sided) [Default] ## ## @item @tab "larger" @tab "F(X1) > F(X2)" (one-sided) ## ## @item @tab "smaller" @tab "F(X1) < F(X2)" (one-sided) ## @end multitable ## ## The two-sided test uses the maximum absolute difference between the cdfs of ## the distributions of the two data vectors. The test statistic is ## @code{D* = max(|F1(x) - F2(x)|)}, where F1(x) is the proportion of @var{x1} ## values less or equal to x and F2(x) is the proportion of @var{x2} values less ## than or equal to x. The one-sided test uses the actual value of the ## difference between the cdfs of the distributions of the two data vectors ## rather than the absolute value. The test statistic is ## @code{D* = max(F1(x) - F2(x))} or @code{D* = max(F2(x) - F1(x))} for ## @code{tail} = "larger" or "smaller", respectively. ## ## @code{[@var{h}, @var{p}] = kstest2 (@dots{})} also returns the ## asymptotic p-value @var{p}. ## ## @code{[@var{h}, @var{p}, @var{ks2stat}] = kstest2 (@dots{})} also returns ## the Kolmogorov-Smirnov test statistic @var{ks2stat} defined above for the ## test type indicated by @code{tail}. ## ## @seealso{kstest, cdfplot} ## @end deftypefn function [H, pValue, ks2stat] = kstest2 (x1, x2, varargin) ## Check input parameters if nargin < 2 error ("kstest2: Too few inputs."); endif if ! isvector (x1) || ! isreal (x1) || ! isvector (x2) || ! isreal (x2) error ("kstest2: X1 and X2 must be vectors of real numbers."); endif ## Add defaults alpha = 0.05; tail = "unequal"; ## Parse extra parameters if nargin > 2 && mod (numel (varargin), 2) == 0 [~, prop] = parseparams (varargin); while (!isempty (prop)) switch (lower (prop{1})) case "alpha" alpha = prop{2}; case "tail" tail = prop{2}; otherwise error ("kstest2: Unknown option %s", prop{1}); endswitch prop = prop(3:end); endwhile elseif nargin > 2 error ("kstest2: optional parameters must be in name/value pairs."); endif ## Check for valid alpha and tail parameters if (! isnumeric (alpha) || isnan (alpha) || ! isscalar (alpha) ... || alpha <= 0 || alpha >= 1) error ("kstest2: alpha must be a numeric scalar in the range (0,1)."); endif if ! isa (tail, 'char') error ("kstest2: tail argument must be a string"); elseif sum (strcmpi (tail, {"unequal", "larger", "smaller"})) < 1 error ("kstest2: tail value must be either 'both', right' or 'left'."); endif ## Make x1 and x2 column vectors x1 = x1(:); x2 = x2(:); ## Remove missing values (NaN) x1(isnan (x1)) = []; x2(isnan (x2)) = []; ## Check for remaining data in both vectors if isempty (x1) error ("kstest2: Not enough data in X1"); elseif isempty (x2) error ("kstest2: Not enough data in X2"); endif ## Calculate F1(x) and F2(x) binEdges = [-inf; sort([x1;x2]); inf]; binCounts1 = histc (x1 , binEdges, 1); binCounts2 = histc (x2 , binEdges, 1); sumCounts1 = cumsum (binCounts1) ./ sum (binCounts1); sumCounts2 = cumsum (binCounts2) ./ sum (binCounts2); sampleCDF1 = sumCounts1(1:end - 1); sampleCDF2 = sumCounts2(1:end - 1); ## Calculate the suitable KS statistic according to tail switch tail case "unequal" # 2-sided test: T = max|F1(x) - F2(x)|. deltaCDF = abs (sampleCDF1 - sampleCDF2); case "smaller" # 1-sided test: T = max[F2(x) - F1(x)]. deltaCDF = sampleCDF2 - sampleCDF1; case "larger" # 1-sided test: T = max[F1(x) - F2(x)]. deltaCDF = sampleCDF1 - sampleCDF2; endswitch ks2stat = max (deltaCDF); ## Compute the asymptotic P-value approximation n_x1 = length(x1); n_x2 = length(x2); n = n_x1 * n_x2 /(n_x1 + n_x2); lambda = max ((sqrt (n) + 0.12 + 0.11 / sqrt (n)) * ks2stat, 0); if strcmpi (tail, "unequal") # 2-sided test v = [1:101]; pValue = 2 * sum ((-1) .^ (v-1) .* exp (-2 * lambda * lambda * v .^ 2)); pValue = min (max (pValue, 0), 1); else # 1-sided test pValue = exp(-2 * lambda * lambda); endif ## Return hypothesis test H = (alpha >= pValue); endfunction ## Test input %!error kstest2 ([1,2,3,4,5,5]) %!error kstest2 (ones(2,4), [1,2,3,4,5,5]) %!error kstest2 ([2,3,5,7,3+3i], [1,2,3,4,5,5]) %!error kstest2 ([2,3,4,5,6],[3;5;7;8;7;6;5],"tail") %!error kstest2 ([2,3,4,5,6],[3;5;7;8;7;6;5],"tail", "whatever") %!error kstest2 ([2,3,4,5,6],[3;5;7;8;7;6;5],"badoption", 0.51) %!error kstest2 ([2,3,4,5,6],[3;5;7;8;7;6;5],"tail", 0) %!error kstest2 ([2,3,4,5,6],[3;5;7;8;7;6;5],"alpha", 0) %!error kstest2 ([2,3,4,5,6],[3;5;7;8;7;6;5],"alpha", NaN) %!error kstest2 ([NaN,NaN,NaN,NaN,NaN],[3;5;7;8;7;6;5],"tail", "unequal") ## Test results %!test %! load examgrades %! [h, p] = kstest2 (grades(:,1), grades(:,2)); %! assert (h, false); %! assert (p, 0.1222791870137312, 1e-14); %!test %! load examgrades %! [h, p] = kstest2 (grades(:,1), grades(:,2), "tail", "larger"); %! assert (h, false); %! assert (p, 0.1844421391011258, 1e-14); %!test %! load examgrades %! [h, p] = kstest2 (grades(:,1), grades(:,2), "tail", "smaller"); %! assert (h, false); %! assert (p, 0.06115357930171663, 1e-14); %!test %! load examgrades %! [h, p] = kstest2 (grades(:,1), grades(:,2), "tail", "smaller", "alpha", 0.1); %! assert (h, true); %! assert (p, 0.06115357930171663, 1e-14); statistics-release-1.6.3/inst/levene_test.m000066400000000000000000000271731456127120000210110ustar00rootroot00000000000000## Copyright (C) 2022 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{h} =} levene_test (@var{x}) ## @deftypefnx {statistics} {@var{h} =} levene_test (@var{x}, @var{group}) ## @deftypefnx {statistics} {@var{h} =} levene_test (@var{x}, @var{alpha}) ## @deftypefnx {statistics} {@var{h} =} levene_test (@var{x}, @var{testtype}) ## @deftypefnx {statistics} {@var{h} =} levene_test (@var{x}, @var{group}, @var{alpha}) ## @deftypefnx {statistics} {@var{h} =} levene_test (@var{x}, @var{group}, @var{testtype}) ## @deftypefnx {statistics} {@var{h} =} levene_test (@var{x}, @var{group}, @var{alpha}, @var{testtype}) ## @deftypefnx {statistics} {[@var{h}, @var{pval}] =} levene_test (@dots{}) ## @deftypefnx {statistics} {[@var{h}, @var{pval}, @var{W}] =} levene_test (@dots{}) ## @deftypefnx {statistics} {[@var{h}, @var{pval}, @var{W}, @var{df}] =} levene_test (@dots{}) ## ## Perform a Levene's test for the homogeneity of variances. ## ## Under the null hypothesis of equal variances, the test statistic @var{W} ## approximately follows an F distribution with @var{df} degrees of ## freedom being a vector ([k-1, N-k]). ## ## The p-value (1 minus the CDF of this distribution at @var{W}) is returned in ## @var{pval}. @var{h} = 1 if the null hypothesis is rejected at the ## significance level of @var{alpha}. Otherwise @var{h} = 0. ## ## Input Arguments: ## ## @itemize ## @item ## @var{x} contains the data and it can either be a vector or matrix. ## If @var{x} is a matrix, then each column is treated as a separate group. ## If @var{x} is a vector, then the @var{group} argument is mandatory. ## NaN values are omitted. ## ## @item ## @var{group} contains the names for each group. If @var{x} is a vector, then ## @var{group} must be a vector of the same length, or a string array or cell ## array of strings with one row for each element of @var{x}. @var{x} values ## corresponding to the same value of @var{group} are placed in the same group. ## If @var{x} is a matrix, then @var{group} can either be a cell array of ## strings of a character array, with one row per column of @var{x} in the same ## way it is used in @code{anova1} function. If @var{x} is a matrix, then ## @var{group} can be omitted either by entering an empty array ([]) or by ## parsing only @var{alpha} as a second argument (if required to change its ## default value). ## ## @item ## @var{alpha} is the statistical significance value at which the null ## hypothesis is rejected. Its default value is 0.05 and it can be parsed ## either as a second argument (when @var{group} is omitted) or as a third ## argument. ## ## @item ## @var{testtype} is a string determining the type of Levene's test. By default ## it is set to "absolute", but the user can also parse "quadratic" in order to ## perform Levene's Quadratic test for equal variances or "median" in order to ## to perform the Brown-Forsythe's test. These options determine how the Z_ij ## values are computed. If an invalid name is parsed for @var{testtype}, then ## the Levene's Absolute test is performed. ## @end itemize ## ## @seealso{bartlett_test, vartest2, vartestn} ## @end deftypefn function [h, pval, W, df] = levene_test (x, varargin) ## Check for valid number of input arguments if (nargin < 1 || nargin > 4) error ("levene_test: invalid number of input arguments."); endif ## Add defaults group = []; alpha = 0.05; ttype = "absolute"; ## Check for 2nd argument being ALPHA, GROUP, or TESTTYPE if (nargin > 1) if (isscalar (varargin{1}) && isnumeric (varargin{1}) ... && numel (varargin{1}) == 1) alpha = varargin{1}; ## Check for valid alpha value if (alpha <= 0 || alpha >= 1) error ("levene_test: wrong value for alpha."); endif elseif (any (strcmpi (varargin{1}, {"absolute", "quadratic", "median"}))) ttype = varargin{1}; elseif (isvector (varargin{1}) && numel (varargin{1} > 1)) if ((size (x, 2) == 1 && size (x, 1) == numel (varargin{1})) || ... (size (x, 2) > 1 && size (x, 2) == numel (varargin{1}))) group = varargin{1}; else error ("levene_test: GROUP and X mismatch."); endif elseif (isempty (varargin{1})) ## Do nothing else error ("levene_test: invalid second input argument."); endif endif ## Check for 3rd argument if (nargin > 2) if (isscalar (varargin{2}) && isnumeric (varargin{2}) ... && numel (varargin{2} == 1)) alpha = varargin{2}; ## Check for valid alpha value if (alpha <= 0 || alpha >= 1) error ("levene_test: wrong value for alpha."); endif elseif (any (strcmpi (varargin{2}, {"absolute", "quadratic", "median"}))) ttype = varargin{2}; else error ("levene_test: invalid third input argument."); endif endif ## Check for 3rd argument if (nargin > 3) if (any (strcmpi (varargin{3}, {"absolute", "quadratic", "median"}))) ttype = varargin{3}; else error ("levene_test: invalid option for TESTTYPE as 4th argument."); endif endif ## Convert group to cell array from character array, make it a column if (! isempty (group) && ischar (group)) group = cellstr (group); endif if (size (group, 1) == 1) group = group'; endif ## If x is a matrix, convert it to column vector and create a ## corresponging column vector for groups if (length (x) < prod (size (x))) [n, m] = size (x); x = x(:); gi = reshape (repmat ((1:m), n, 1), n*m, 1); if (length (group) == 0) ## no group names are provided group = gi; elseif (size (group, 1) == m) ## group names exist and match columns group = group(gi,:); else error ("levene_test: columns in X and GROUP length do not match."); endif endif ## Check that x and group are the same size if (! all (numel (x) == numel (group))) error (srtcat (["levene_test: GROUP must be a vector with the same"], ... [" number of rows as x."])); endif ## Identify NaN values (if any) and remove them from X along with ## their corresponding values from group vector nonan = ! isnan (x); x = x(nonan); group = group(nonan, :); ## Convert group to indices and separate names [group_id, group_names] = grp2idx (group); group_id = group_id(:); ## Get sample size (N_i), mean (Y_i), median (Y_I) and sample values (Y_ij) ## for groups with more than one sample groups = size (group_names, 1); rgroup = []; N_i = zeros (1, groups); Y_i = N_i; Y_I = N_i; for k = 1:groups group_size = find (group_id == k); if (length (group_size) > 1) N_i(k) = length (group_size); Y_i(k) = mean (x(group_size)); Y_I(k) = median (x(group_size)); Y_ij{k} = x(group_size); else warning (strcat (sprintf ("levene_test: GROUP %s has a single", ... group_names{k}), [" sample and is not included in the test.\n"])); rgroup = [rgroup, k]; N_i(k) = 1; Y_i(k) = x(group_size); Y_I(k) = x(group_size); Y_ij{k} = x(group_size); endif endfor ## Remove groups with a single sample if (! isempty (rgroup)) N_i(rgroup) = []; Y_i(rgroup) = []; Y_I(rgroup) = []; Y_ij(rgroup) = []; k = k - numel (rgroup); endif ## Compute Z_ij for "absolute" or "quadratic" Levene's test switch (lower (ttype)) case "absolute" for i = 1:k Z_ij{i} = abs (Y_ij{i} - Y_i(i)); endfor case "quadratic" for i = 1:k Z_ij{i} = sqrt ((Y_ij{i} - Y_i(i)) .^ 2); endfor case "median" for i = 1:k Z_ij{i} = abs (Y_ij{i} - Y_I(i)); endfor endswitch ## Compute Z_i and Z_ Z_ = []; for i = 1:k Z_i(i) = mean (Z_ij{i}); Z_ = [Z_; Z_ij{i}(:)]; endfor Z_ = mean (Z_); ## Compute total sample size (N) N = sum (N_i); ## Calculate W statistic. termA = (N - k) / (k - 1); termB = sum (N_i .* ((Z_i - Z_) .^ 2)); termC = 0; for i = 1:k termC += sum ((Z_ij{i} - Z_i(i)) .^ 2); endfor W = termA * (termB / termC); ## Calculate p-value from the chi-square distribution pval = 1 - fcdf (W, k - 1, N - k); ## Save dfs df = [k-1, N-k]; ## Determine the test outcome h = double (pval < alpha); endfunction ## Test input validation %!error levene_test () %!error ... %! levene_test (1, 2, 3, 4, 5); %!error levene_test (randn (50, 2), 0); %!error ... %! levene_test (randn (50, 2), [1, 2, 3]); %!error ... %! levene_test (randn (50, 1), ones (55, 1)); %!error ... %! levene_test (randn (50, 1), ones (50, 2)); %!error ... %! levene_test (randn (50, 2), [], 1.2); %!error ... %! levene_test (randn (50, 2), "some_string"); %!error ... %! levene_test (randn (50, 2), [], "alpha"); %!error ... %! levene_test (randn (50, 1), [ones(25, 1); 2*ones(25, 1)], 1.2); %!error ... %! levene_test (randn (50, 1), [ones(25, 1); 2*ones(25, 1)], "err"); %!error ... %! levene_test (randn (50, 1), [ones(25, 1); 2*ones(25, 1)], 0.05, "type"); %!warning ... %! levene_test (randn (50, 1), [ones(24, 1); 2*ones(25, 1); 3]); ## Test results %!test %! load examgrades %! [h, pval, W, df] = levene_test (grades); %! assert (h, 1); %! assert (pval, 9.523239714592791e-07, 1e-14); %! assert (W, 8.59529, 1e-5); %! assert (df, [4, 595]); %!test %! load examgrades %! [h, pval, W, df] = levene_test (grades, [], "quadratic"); %! assert (h, 1); %! assert (pval, 9.523239714592791e-07, 1e-14); %! assert (W, 8.59529, 1e-5); %! assert (df, [4, 595]); %!test %! load examgrades %! [h, pval, W, df] = levene_test (grades, [], "median"); %! assert (h, 1); %! assert (pval, 1.312093241723211e-06, 1e-14); %! assert (W, 8.415969, 1e-6); %! assert (df, [4, 595]); %!test %! load examgrades %! [h, pval, W, df] = levene_test (grades(:,[1:3])); %! assert (h, 1); %! assert (pval, 0.004349390980463497, 1e-14); %! assert (W, 5.52139, 1e-5); %! assert (df, [2, 357]); %!test %! load examgrades %! [h, pval, W, df] = levene_test (grades(:,[1:3]), "median"); %! assert (h, 1); %! assert (pval, 0.004355216763951453, 1e-14); %! assert (W, 5.52001, 1e-5); %! assert (df, [2, 357]); %!test %! load examgrades %! [h, pval, W, df] = levene_test (grades(:,[3,4]), "quadratic"); %! assert (h, 0); %! assert (pval, 0.1807494957440653, 2e-14); %! assert (W, 1.80200, 1e-5); %! assert (df, [1, 238]); %!test %! load examgrades %! [h, pval, W, df] = levene_test (grades(:,[3,4]), "median"); %! assert (h, 0); %! assert (pval, 0.1978225622063785, 2e-14); %! assert (W, 1.66768, 1e-5); %! assert (df, [1, 238]); statistics-release-1.6.3/inst/linkage.m000066400000000000000000000253101456127120000200750ustar00rootroot00000000000000## Copyright (C) 2008 Francesco Potortì ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{y} =} linkage (@var{d}) ## @deftypefnx {statistics} {@var{y} =} linkage (@var{d}, @var{method}) ## @deftypefnx {statistics} {@var{y} =} linkage (@var{x}) ## @deftypefnx {statistics} {@var{y} =} linkage (@var{x}, @var{method}) ## @deftypefnx {statistics} {@var{y} =} linkage (@var{x}, @var{method}, @var{metric}) ## @deftypefnx {statistics} {@var{y} =} linkage (@var{x}, @var{method}, @var{arglist}) ## ## Produce a hierarchical clustering dendrogram. ## ## @var{d} is the dissimilarity matrix relative to n observations, ## formatted as a @math{(n-1)*n/2}x1 vector as produced by @code{pdist}. ## Alternatively, @var{x} contains data formatted for input to ## @code{pdist}, @var{metric} is a metric for @code{pdist} and ## @var{arglist} is a cell array containing arguments that are passed to ## @code{pdist}. ## ## @code{linkage} starts by putting each observation into a singleton ## cluster and numbering those from 1 to n. Then it merges two ## clusters, chosen according to @var{method}, to create a new cluster ## numbered n+1, and so on until all observations are grouped into ## a single cluster numbered 2(n-1). Row k of the ## (m-1)x3 output matrix relates to cluster n+k: the first ## two columns are the numbers of the two component clusters and column ## 3 contains their distance. ## ## @var{method} defines the way the distance between two clusters is ## computed and how they are recomputed when two clusters are merged: ## ## @table @samp ## @item "single" (default) ## Distance between two clusters is the minimum distance between two ## elements belonging each to one cluster. Produces a cluster tree ## known as minimum spanning tree. ## ## @item "complete" ## Furthest distance between two elements belonging each to one cluster. ## ## @item "average" ## Unweighted pair group method with averaging (UPGMA). ## The mean distance between all pair of elements each belonging to one ## cluster. ## ## @item "weighted" ## Weighted pair group method with averaging (WPGMA). ## When two clusters A and B are joined together, the new distance to a ## cluster C is the mean between distances A-C and B-C. ## ## @item "centroid" ## Unweighted Pair-Group Method using Centroids (UPGMC). ## Assumes Euclidean metric. The distance between cluster centroids, ## each centroid being the center of mass of a cluster. ## ## @item "median" ## Weighted pair-group method using centroids (WPGMC). ## Assumes Euclidean metric. Distance between cluster centroids. When ## two clusters are joined together, the new centroid is the midpoint ## between the joined centroids. ## ## @item "ward" ## Ward's sum of squared deviations about the group mean (ESS). ## Also known as minimum variance or inner squared distance. ## Assumes Euclidean metric. How much the moment of inertia of the ## merged cluster exceeds the sum of those of the individual clusters. ## @end table ## ## @strong{Reference} ## Ward, J. H. Hierarchical Grouping to Optimize an Objective Function ## J. Am. Statist. Assoc. 1963, 58, 236-244, ## @url{http://iv.slis.indiana.edu/sw/data/ward.pdf}. ## ## @seealso{pdist,squareform} ## @end deftypefn function dgram = linkage (d, method = "single", distarg, savememory) ## check the input if (nargin == 4) && (strcmpi (savememory, "savememory")) warning ("Octave:linkage_savemem", ... "linkage: option 'savememory' not implemented"); elseif (nargin < 1) || (nargin > 3) print_usage (); endif if (isempty (d)) error ("linkage: d cannot be empty"); endif methods = struct ... ("name", { "single"; "complete"; "average"; "weighted"; "centroid"; "median"; "ward" }, "distfunc", {(@(x) min(x)) # single (@(x) max(x)) # complete (@(x,i,j,w) sum(diag(w([i,j]))*x)/sum(w([i,j]))) # average (@(x) mean(x)) # weighted (@massdist) # centroid (@(x,i) massdist(x,i)) # median (@inertialdist) # ward }); mask = strcmp (lower (method), {methods.name}); if (! any (mask)) error ("linkage: %s: unknown method", method); endif dist = {methods.distfunc}{mask}; if (nargin >= 3 && ! isvector (d)) if (ischar (distarg)) d = pdist (d, distarg); elseif (iscell (distarg)) d = pdist (d, distarg{:}); else print_usage (); endif elseif (nargin < 3) if (! isvector (d)) d = pdist (d); endif else print_usage (); endif d = squareform (d, "tomatrix"); # dissimilarity NxN matrix n = rows (d); # the number of observations diagidx = sub2ind ([n,n], 1:n, 1:n); # indices of diagonal elements d(diagidx) = Inf; # consider a cluster as far from itself ## For equal-distance nodes, the order in which clusters are ## merged is arbitrary. Rotating the initial matrix produces an ## ordering similar to Matlab's. cname = n:-1:1; # cluster names in d d = rot90 (d, 2); # exchange low and high cluster numbers weight = ones (1, n); # cluster weights dgram = zeros (n-1, 3); # clusters from n+1 to 2*n-1 for cluster = n+1 : 2*n-1 ## Find the two nearest clusters [m midx] = min (d(:)); [r, c] = ind2sub (size (d), midx); ## Here is the new cluster dgram(cluster-n, :) = [cname(r) cname(c) d(r, c)]; ## Put it in place of the first one and remove the second cname(r) = cluster; cname(c) = []; ## Compute the new distances. ## (Octave-7+ needs switch stmt to avoid 'called with too many inputs' err.) switch find (mask) case {1, 2, 4} # 1 arg newd = dist (d([r c], :)); case {3, 5, 7} # 4 args newd = dist (d([r c], :), r, c, weight); case 6 # 2 args newd = dist (d([r c], :), r); otherwise endswitch newd(r) = Inf; # Take care of the diagonal element ## Put distances in place of the first ones, remove the second ones d(r,:) = newd; d(:,r) = newd'; d(c,:) = []; d(:,c) = []; ## The new weight is the sum of the components' weights weight(r) += weight(c); weight(c) = []; endfor ## Sort the cluster numbers, as Matlab does dgram(:,1:2) = sort (dgram(:,1:2), 2); ## Check that distances are monotonically increasing if (any (diff (dgram(:,3)) < 0)) warning ("Octave:clustering", "linkage: cluster distances do not monotonically increase\n\ you should probably use a method different from \"%s\"", method); endif endfunction ## Take two row vectors, which are the Euclidean distances of clusters I ## and J from the others. Column I of second row contains the distance ## between clusters I and J. The centre of gravity of the new cluster ## is on the segment joining the old ones. W are the weights of all ## clusters. Use the law of cosines to find the distances of the new ## cluster from all the others. function y = massdist (x, i, j, w) x .^= 2; # Squared Euclidean distances if (nargin == 2) # Median distance qi = 0.5; # Equal weights ("weighted") else # Centroid distance qi = 1 / (1 + w(j) / w(i)); # Proportional weights ("unweighted") endif y = sqrt (qi * x(1, :) + (1 - qi) * (x(2, :) - qi * x(2, i))); endfunction ## Take two row vectors, which are the inertial distances of clusters I ## and J from the others. Column I of second row contains the inertial ## distance between clusters I and J. The centre of gravity of the new ## cluster K is on the segment joining I and J. W are the weights of ## all clusters. Convert inertial to Euclidean distances, then use the ## law of cosines to find the Euclidean distances of K from all the ## other clusters, convert them back to inertial distances and return ## them. function y = inertialdist (x, i, j, w) wi = w(i); # The cluster wj = w(j); # weights. s = [wi + w; # Sum of weights for wj + w]; # all cluster pairs. p = [wi * w; # Product of weights for wj * w]; # all cluster pairs. x = x.^2 .* s ./ p; # Convert inertial dist. to squared Eucl. sij = wi + wj; # Sum of weights of I and J qi = wi / sij; # Normalise the weight of I ## Squared Euclidean distances between all clusters and new cluster K x = qi * x(1, :) + (1 - qi) * (x(2, :) - qi * x(2, i)); y = sqrt (x * sij .* w ./ (sij + w)); # convert Eucl. dist. to inertial endfunction %!shared x, t %! x = reshape (mod (magic (6),5), [], 3); %! t = 1e-6; %!assert (cond (linkage (pdist (x))), 34.119045, t); %!assert (cond (linkage (pdist (x), "complete")), 21.793345, t); %!assert (cond (linkage (pdist (x), "average")), 27.045012, t); %!assert (cond (linkage (pdist (x), "weighted")), 27.412889, t); %! lastwarn(); # Clear last warning before the test %!warning linkage (pdist (x), "centroid"); %!test %! warning off Octave:clustering %! assert (cond (linkage (pdist (x), "centroid")), 27.457477, t); %! warning on Octave:clustering %!warning linkage (pdist (x), "median"); %!test %! warning off Octave:clustering %! assert (cond (linkage (pdist (x), "median")), 27.683325, t); %! warning on Octave:clustering %!assert (cond (linkage (pdist (x), "ward")), 17.195198, t); %!assert (cond (linkage (x, "ward", "euclidean")), 17.195198, t); %!assert (cond (linkage (x, "ward", {"euclidean"})), 17.195198, t); %!assert (cond (linkage (x, "ward", {"minkowski", 2})), 17.195198, t); statistics-release-1.6.3/inst/logistic_regression.m000066400000000000000000000257021456127120000225450ustar00rootroot00000000000000## Copyright (C) 1995-2017 Kurt Hornik ## Copyright (C) 2022 Andrew Penn ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {[@var{intercept}, @var{slope}, @var{dev}, @var{dl}, @var{d2l}, @var{P}, @var{stats}] =} logistic_regression (@var{y}, @var{x}, @var{print}, @var{intercept}, @var{slope}) ## ## Perform ordinal logistic regression. ## ## Suppose @var{y} takes values in k ordered categories, and let ## @code{P_i (@var{x})} be the cumulative probability that @var{y} ## falls in one of the first i categories given the covariate ## @var{x}. Then ## ## @example ## [@var{intercept}, @var{slope}] = logistic_regression (@var{y}, @var{x}) ## @end example ## ## @noindent ## fits the model ## ## @example ## logit (P_i (@var{x})) = @var{x} * @var{slope} + @var{intercept}_i, i = 1 @dots{} k-1 ## @end example ## ## The number of ordinal categories, k, is taken to be the number ## of distinct values of @code{round (@var{y})}. If k equals 2, ## @var{y} is binary and the model is ordinary logistic regression. The ## matrix @var{x} is assumed to have full column rank. ## ## Given @var{y} only, @code{@var{intercept} = logistic_regression (@var{y})} ## fits the model with baseline logit odds only. ## ## The full form is ## ## @example ## @group ## [@var{intercept}, @var{slope}, @var{dev}, @var{dl}, @var{d2l}, @var{P}, @var{stats}] ## = logistic_regression (@var{y}, @var{x}, @var{print}, @var{intercept}, @var{slope}) ## @end group ## @end example ## ## @noindent ## in which all output arguments and all input arguments except @var{y} ## are optional. ## ## Setting @var{print} to 1 requests summary information about the fitted ## model to be displayed. Setting @var{print} to 2 requests information ## about convergence at each iteration. Other values request no ## information to be displayed. The input arguments @var{intercept} and ## @var{slope} give initial estimates for @var{intercept} and @var{slope}. ## ## The returned value @var{dev} holds minus twice the log-likelihood. ## ## The returned values @var{dl} and @var{d2l} are the vector of first ## and the matrix of second derivatives of the log-likelihood with ## respect to @var{intercept} and @var{slope}. ## ## @var{P} holds estimates for the conditional distribution of @var{y} ## given @var{x}. ## ## @var{stats} returns a structure that contains the following fields: ## @itemize ## @item ## "intercept": intercept coefficients ## @item ## "slope": slope coefficients ## @item ## "coeff": regression coefficients (intercepts and slops) ## @item ## "covb": estimated covariance matrix for coefficients (coeff) ## @item ## "coeffcorr": correlation matrix for coeff ## @item ## "se": standard errors of the coeff ## @item ## "z": z statistics for coeff ## @item ## "pval": p-values for coeff ## @end itemize ## @end deftypefn function [intercept, slope, dev, dl, d2l, P, stats] = logistic_regression (y, x, print, intercept, slope) ## check input y = round (y(:)); if (nargin < 2) x = zeros (length (y), 0); endif; xymissing = (isnan (y) | any (isnan (x), 2)); y(xymissing) = []; x(xymissing,:) = []; [my, ny] = size (y); [mx, nx] = size (x); if (mx != my) error ("logistic_regression: X and Y must have the same number of observations"); endif ## initial calculations tol = 1e-12; incr = 10; decr = 2; ymin = min (y); ymax = max (y); yrange = ymax - ymin; z = (y * ones (1, yrange)) == ((y * 0 + 1) * (ymin : (ymax - 1))); z1 = (y * ones (1, yrange)) == ((y * 0 + 1) * ((ymin + 1) : ymax)); z = z(:, any (z)); z1 = z1(:, any(z1)); [mz, nz] = size (z); ## starting values if (nargin < 3) print = 0; endif; if (nargin < 4) g = cumsum (sum (z))' ./ my; intercept = log (g ./ (1 - g)); endif; if (nargin < 5) slope = zeros (nx, 1); endif; tb = [intercept; slope]; ## likelihood and derivatives at starting values [g, g1, p, dev] = logistic_regression_likelihood (y, x, tb, z, z1); [dl, d2l] = logistic_regression_derivatives (x, z, z1, g, g1, p); epsilon = std (vec (d2l)) / 1000; ## maximize likelihood using Levenberg modified Newton's method iter = 0; while (abs (dl' * (d2l \ dl) / length (dl)) > tol) iter += 1; tbold = tb; devold = dev; tb = tbold - d2l \ dl; [g, g1, p, dev] = logistic_regression_likelihood (y, x, tb, z, z1); if ((dev - devold) / (dl' * (tb - tbold)) < 0) epsilon /= decr; else while ((dev - devold) / (dl' * (tb - tbold)) > 0) epsilon *= incr; if (epsilon > 1e+15) error ("logistic_regression: epsilon too large"); endif tb = tbold - (d2l - epsilon * eye (size (d2l))) \ dl; [g, g1, p, dev] = logistic_regression_likelihood (y, x, tb, z, z1); disp ("epsilon"); disp (epsilon); endwhile endif [dl, d2l] = logistic_regression_derivatives (x, z, z1, g, g1, p); if (print == 2) disp ("Iteration"); disp (iter); disp ("Deviance"); disp (dev); disp ("First derivative"); disp (dl'); disp ("Eigenvalues of second derivative"); disp (eig (d2l)'); endif endwhile ## tidy up output intercept = tb(1 : nz, 1); slope = tb((nz + 1) : (nz + nx), 1); cov = inv (-d2l); se = sqrt (diag (cov)); if (nargout > 5) ## Compute predicted probabilities (P) if (nx > 0) e = ((x * slope) * ones (1, nz)) + ((y * 0 + 1) * intercept'); else e = (y * 0 + 1) * intercept'; endif P = diff ([(y * 0), (exp (e) ./ (1 + exp (e))), (y * 0 + 1)]')'; endif if (nargout > 6) ## Create stats structure dfe = mx - nx - 1; zstat = tb ./ se; coeffcorr = cov2corr (cov); resid = y - P(:,2); stats = struct ("intercept", intercept, ... "slope", slope, ... "coeff", tb, ... "cov", cov, ... "coeffcorr", coeffcorr, ... "se", se, ... "z", zstat, ... "pval", 2 * normcdf (-abs (zstat))); endif if (print >= 1) printf ("\n"); printf ("Logistic Regression Results:\n"); printf ("\n"); printf ("Number of Iterations: %d\n", iter); printf ("Deviance: %f\n", dev); printf ("Parameter Estimates:\n"); printf (" Intercept S.E.\n"); for i = 1 : nz printf (" %8.4f %8.4f\n", tb (i), se (i)); endfor if (nx > 0) printf (" Slope S.E.\n"); for i = (nz + 1) : (nz + nx) printf (" %8.4f %8.4f\n", tb (i), se (i)); endfor endif endif endfunction function [g, g1, p, dev] = logistic_regression_likelihood (y, x, slope, z, z1) ## Calculate the likelihood for the ordinal logistic regression model. e = exp ([z, x] * slope); e1 = exp ([z1, x] * slope); g = e ./ (1 + e); g1 = e1 ./ (1 + e1); g = max (y == max (y), g); g1 = min (y > min (y), g1); p = g - g1; dev = -2 * sum (log (p)); endfunction function [dl, d2l] = logistic_regression_derivatives (x, z, z1, g, g1, p) ## Calculate derivatives of the log-likelihood for ordinal logistic regression ## first derivative v = g .* (1 - g) ./ p; v1 = g1 .* (1 - g1) ./ p; dlogp = [(diag (v) * z - diag (v1) * z1), (diag (v - v1) * x)]; dl = sum (dlogp)'; ## second derivative w = v .* (1 - 2 * g); w1 = v1 .* (1 - 2 * g1); d2l = [z, x]' * diag (w) * [z, x] - [z1, x]' * diag (w1) * [z1, x] ... - dlogp' * dlogp; endfunction function R = cov2corr (vcov) ## Convert covariance matrix to correlation matrix sed = sqrt (diag (vcov)); R = vcov ./ (sed * sed'); R = (R + R') / 2; # This step ensures that the matrix is positive definite endfunction %!test %! # Output compared to following MATLAB commands %! # [B, DEV, STATS] = mnrfit(X,Y+1,'model','ordinal'); %! # P = mnrval(B,X) %! X = [1.489381332449196, 1.1534152241851305; ... %! 1.8110085304863965, 0.9449666896938425; ... %! -0.04453299665130296, 0.34278203449678646; ... %! -0.36616019468850347, 1.130254275908322; ... %! 0.15339143291005095, -0.7921044310668951; ... %! -1.6031878794469698, -1.8343471035233376; ... %! -0.14349521143198166, -0.6762996896828459; ... %! -0.4403818557740143, -0.7921044310668951; ... %! -0.7372685001160434, -0.027793137932169563; ... %! -0.11875465773681024, 0.5512305689880763]; %! Y = [1,1,1,1,1,0,0,0,0,0]'; %! [INTERCEPT, SLOPE, DEV, DL, D2L, P] = logistic_regression (Y, X, false); #%! assert (DEV, 5.680728861124, 1e-05); #%! assert (INTERCEPT(1), -1.10999599948243, 1e-05); #%! assert (SLOPE(1), -9.12480634225699, 1e-05); #%! assert (SLOPE(2), -2.18746124517476, 1e-05); #%! assert (corr(P(:,1),Y), -0.786673288976468, 1e-05); %!test %! # Output compared to following MATLAB commands %! # [B, DEV, STATS] = mnrfit(X,Y+1,'model','ordinal'); %! load carbig %! X = [Acceleration Displacement Horsepower Weight]; %! miles = [1,1,1,1,1,1,1,1,1,1,NaN,NaN,NaN,NaN,NaN,1,1,NaN,1,1,2,2,1,2,2,2, ... %! 2,2,2,2,2,1,1,1,1,2,2,2,2,NaN,2,1,1,2,1,1,1,1,1,1,1,1,1,2,2,1,2, ... %! 2,3,3,3,3,2,2,2,2,2,2,2,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,2,2,2,2,2, ... %! 2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,1,1,1,1,1,2,2,2,1,2,2, ... %! 2,1,1,3,2,2,2,1,2,2,1,2,2,2,1,3,2,3,2,1,1,1,1,1,1,1,1,3,2,2,3,3, ... %! 2,2,2,2,2,3,2,1,1,1,1,1,1,1,1,1,1,1,2,2,1,3,2,2,2,2,2,2,1,3,2,2, ... %! 2,2,2,3,2,2,2,2,2,1,1,1,1,2,2,2,2,3,2,3,3,2,1,1,1,3,3,2,2,2,1,2, ... %! 2,1,1,1,1,1,3,3,3,2,3,1,1,1,1,1,2,2,1,1,1,1,1,3,2,2,2,3,3,3,3,2, ... %! 2,2,4,3,3,4,3,2,2,2,2,2,2,2,2,2,2,2,1,1,2,1,1,1,3,2,2,3,2,2,2,2, ... %! 2,1,2,1,3,3,2,2,2,2,2,1,1,1,1,1,1,2,1,3,3,3,2,2,2,2,2,3,3,3,3,2, ... %! 2,2,3,4,3,3,3,2,2,2,2,3,3,3,3,3,4,2,4,4,4,3,3,4,4,3,3,3,2,3,2,3, ... %! 2,2,2,2,3,4,4,3,3,3,3,3,3,3,3,3,3,3,3,3,3,2,NaN,3,2,2,2,2,2,1,2, ... %! 2,3,3,3,2,2,2,3,3,3,3,3,3,3,3,3,3,3,2,3,2,2,3,3,2,2,4,3,2,3]'; %! [INTERCEPT, SLOPE, DEV, DL, D2L, P] = logistic_regression (miles, X, false); %! assert (DEV, 433.197174495549, 1e-05); %! assert (INTERCEPT(1), -16.6895155618903, 1e-05); %! assert (INTERCEPT(2), -11.7207818178493, 1e-05); %! assert (INTERCEPT(3), -8.0605768506075, 1e-05); %! assert (SLOPE(1), 0.104762463756714, 1e-05); %! assert (SLOPE(2), 0.0103357623191891, 1e-05); %! assert (SLOPE(3), 0.0645199313242276, 1e-05); %! assert (SLOPE(4), 0.00166377028388103, 1e-05); statistics-release-1.6.3/inst/logit.m000066400000000000000000000027251456127120000176060ustar00rootroot00000000000000## Copyright (C) 1995-2017 Kurt Hornik ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{x} =} logit (@var{p}) ## ## Compute the logit for each value of @var{p} ## ## The logit is defined as ## @tex ## $$ {\rm logit}(p) = \log\Big({p \over 1-p}\Big) $$ ## @end tex ## @ifnottex ## ## @example ## logit (@var{p}) = log (@var{p} / (1-@var{p})) ## @end example ## ## @end ifnottex ## @seealso{probit, logicdf} ## @end deftypefn function x = logit (p) if (nargin != 1) print_usage (); endif x = logiinv (p, 0, 1); endfunction %!test %! p = [0.01:0.01:0.99]; %! assert (logit (p), log (p ./ (1-p)), 25*eps); %!assert (logit ([-1, 0, 0.5, 1, 2]), [NaN, -Inf, 0, +Inf, NaN]) ## Test input validation %!error logit () %!error logit (1, 2) statistics-release-1.6.3/inst/mahal.m000066400000000000000000000051701456127120000175470ustar00rootroot00000000000000## Copyright (C) 2015 Lachlan Andrew ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it ## under the terms of the GNU General Public License as published by ## the Free Software Foundation; either version 3 of the License, or (at ## your option) any later version. ## ## This program, is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{d} =} mahal (@var{y}, @var{x}) ## ## Mahalanobis' D-square distance. ## ## Return the Mahalanobis' D-square distance of the points in ## @var{y} from the distribution implied by points @var{x}. ## ## Specifically, it uses a Cholesky decomposition to set ## ## @example ## answer(i) = (@var{y}(i,:) - mean (@var{x})) * inv (A) * (@var{y}(i,:)-mean (@var{x}))' ## @end example ## ## where A is the covariance of @var{x}. ## ## The data @var{x} and @var{y} must have the same number of components ## (columns), but may have a different number of observations (rows). ## ## @end deftypefn function retval = mahal (y, x) if (nargin != 2) print_usage (); endif if (! (isnumeric (x) || islogical (x)) || ! (isnumeric (y) || islogical (y))) error ("mahal: X and Y must be numeric matrices or vectors"); endif if (! ismatrix (x) || ! ismatrix (y)) error ("mahal: X and Y must be 2-D matrices or vectors"); endif [xr, xc] = size (x); [yr, yc] = size (y); if (xc != yc) error ("mahal: X and Y must have the same number of columns"); endif if (isinteger (x)) x = double (x); endif xm = mean (x, 1); ## Center data by subtracting mean of x x = bsxfun (@minus, x, xm); y = bsxfun (@minus, y, xm); w = (x' * x) / (xr - 1); retval = sumsq (y / chol (w), 2); endfunction ## Test input validation %!error mahal () %!error mahal (1, 2, 3) %!error mahal ("A", "B") %!error mahal ([1, 2], ["A", "B"]) %!error mahal (ones (2, 2, 2)) %!error mahal (ones (2, 2), ones (2, 2, 2)) %!error mahal (ones (2, 2), ones (2, 3)) %!test %! X = [1 0; 0 1; 1 1; 0 0]; %! assert (mahal (X, X), [1.5; 1.5; 1.5; 1.5], 10*eps) %! assert (mahal (X, X+1), [7.5; 7.5; 1.5; 13.5], 10*eps) %!assert (mahal ([true; true], [false; true]), [0.5; 0.5], eps) statistics-release-1.6.3/inst/manova1.m000066400000000000000000000215461456127120000200340ustar00rootroot00000000000000## Copyright (C) 2022 Andreas Bertsatos ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{d} =} manova1 (@var{x}, @var{group}) ## @deftypefnx {statistics} {@var{d} =} manova1 (@var{x}, @var{group}, @var{alpha}) ## @deftypefnx {statistics} {[@var{d}, @var{p}] =} manova1 (@dots{}) ## @deftypefnx {statistics} {[@var{d}, @var{p}, @var{stats}] =} manova1 (@dots{}) ## ## One-way multivariate analysis of variance (MANOVA). ## ## @code{@var{d} = manova1 (@var{x}, @var{group}, @var{alpha})} performs a ## one-way MANOVA for comparing the mean vectors of two or more groups of ## multivariate data. ## ## @var{x} is a matrix with each row representing a multivariate observation, ## and each column representing a variable. ## ## @var{group} is a numeric vector, string array, or cell array of strings with ## the same number of rows as @var{x}. @var{x} values are in the same group if ## they correspond to the same value of GROUP. ## ## @var{alpha} is the scalar significance level and is 0.05 by default. ## ## @var{d} is an estimate of the dimension of the group means. It is the ## smallest dimension such that a test of the hypothesis that the means lie on ## a space of that dimension is not rejected. If @var{d} = 0 for example, we ## cannot reject the hypothesis that the means are the same. If @var{d} = 1, we ## reject the hypothesis that the means are the same but we cannot reject the ## hypothesis that they lie on a line. ## ## @code{[@var{d}, @var{p}] = manova1 (@dots{})} returns P, a vector of p-values ## for testing the null hypothesis that the mean vectors of the groups lie on ## various dimensions. P(1) is the p-value for a test of dimension 0, P(2) for ## dimension 1, etc. ## ## @code{[@var{d}, @var{p}, @var{stats}] = manova1 (@dots{})} returns a STATS ## structure with the following fields: ## ## @multitable @columnfractions 0.05 0.2 0.75 ## @item @tab "W" @tab within-group sum of squares and products matrix ## @item @tab "B" @tab between-group sum of squares and products matrix ## @item @tab "T" @tab total sum of squares and products matrix ## @item @tab "dfW" @tab degrees of freedom for WSSP matrix ## @item @tab "dfB" @tab degrees of freedom for BSSP matrix ## @item @tab "dfT" @tab degrees of freedom for TSSP matrix ## @item @tab "lambda" @tab value of Wilk's lambda (the test statistic) ## @item @tab "chisq" @tab transformation of lambda to a chi-square distribution ## @item @tab "chisqdf" @tab degrees of freedom for chisq ## @item @tab "eigenval" @tab eigenvalues of (WSSP^-1) * BSSP ## @item @tab "eigenvec" @tab eigenvectors of (WSSP^-1) * BSSP; these are the ## coefficients for canonical variables, and they are scaled so the within-group ## variance of C is 1 ## @item @tab "canon" @tab canonical variables, equal to XC*eigenvec, where XC ## is X with columns centered by subtracting their means ## @item @tab "mdist" @tab Mahalanobis distance from each point to its group mean ## @item @tab "gmdist" @tab Mahalanobis distances between each pair of group means ## @item @tab "gnames" @tab Group names ## @end multitable ## ## The canonical variables C have the property that C(:,1) is the linear ## combination of the @var{x} columns that has the maximum separation between ## groups, C(:,2) has the maximum separation subject to it being orthogonal to ## C(:,1), and so on. ## ## @end deftypefn function [d, p, stats] = manova1 (x, group, alpha) ## Check input arguments narginchk(2,3) nargoutchk(1,3) ## Validate alpha value if parsed or add default if (nargin > 2) if (length (alpha) > 1 || ! isreal (alpha)) error ("manova1: Alpha must be a real scalar."); elseif (alpha <= 0 || alpha >= 1) error ("manova1: Alpha must be in the range (0,1)."); endif else alpha = 0.05; endif ## Convert group to cell array from character array if (ischar (group)) group = cellstr (group); endif ## Make group a column if (size (group, 1) == 1) group = group'; endif ## Check for equal size in samples between groups and data if (size (group, 1) != size (x, 1)) error("manova1: Samples in X and groups mismatch."); endif ## Remove samples (rows) in X and GROUP if there are missing values in X no_nan = (sum (isnan (x), 2) == 0); x = x(no_nan, :); group = group(no_nan, :); is_nan = ! no_nan; ## Get group names and indices [group_idx, group_names] = grp2idx (group); ngroups = length (group_names); ## Remove NaN values from updated GROUP no_nan = ! isnan (group_idx); if (! all (no_nan)) group_idx = group_idx(no_nan); x = x(no_nan,: ); is_nan(! is_nan) = ! no_nan; endif ## Get number of samples and variables [nsample, nvar] = size(x); realgroups = ismember(1:ngroups, group_idx); nrgroups = sum(realgroups); ## Calculate Total Sum of Squares and Products matrix xm = mean (x); x = x - xm; TSSP = x' * x; ## Calculate Within-samples Sum of Squares and Products matrix WSSP = zeros (size (TSSP)); for j = 1:ngroups row = find (group_idx == j); ## Only meaningful for groups with more than one samples if (length (row) > 1) group_x = x(row, :); group_x = group_x - mean (group_x); WSSP = WSSP + group_x' * group_x; endif endfor ## Calculate Between-samples Sum of Squares and Products matrix BSSP = TSSP - WSSP; ## Instead of simply computing `eig (BSSP / WSSP)` we use Matlab's technique ## with chol to insure v' * WSSP * v = I is met [R, p] = chol (WSSP); if (p > 0) error("manova1: Cannot factorize WSSP."); endif S = R' \ BSSP / R; ## Remove asymmetry caused by roundoff S = (S + S') / 2; [vv, ed] = eig (S); v = R \ vv; ## Sort in descending order [e,ei] = sort (diag (ed)); ## Check for valid eigevalues if (min(e) <= -1) error ("manova1: wrong value in eigenvector: singular sum of squares."); endif ## Compute Barlett's statistic for each dimension dims = 0:(min (nrgroups - 1, nvar) - 1); lambda = flipud (1 ./ cumprod (e + 1)); lambda = lambda(1 + dims); chistat = -(nsample - 1 - (nrgroups + nvar) / 2) .* log (lambda); chisqdf = ((nvar - dims) .* (nrgroups - 1 - dims))'; pp = 1 - chi2cdf (chistat, chisqdf); ## Get dimension where we can reject the null hypothesis d = dims(pp>alpha); if (length(d) > 0) d = d(1); else d = max(dims) + 1; end ## Create extra outputs as necessary if (nargout > 1) p = pp; endif if (nargout > 2) stats.W = WSSP; stats.B = BSSP; stats.T = TSSP; stats.dfW = nsample - nrgroups; stats.dfB = nrgroups - 1; stats.dfT = nsample - 1; stats.lambda = lambda; stats.chisq = chistat; stats.chisqdf = chisqdf; ## Reorder to increasing stats.eigenval = flipud(e); ## Flip so that it is in order of increasing eigenvalues v = v(:, flipud (ei)); ## Re-scale eigenvectors so the within-group variance is 1 vs = diag((v' * WSSP * v))' ./ (nsample - nrgroups); vs(vs<=0) = 1; v = v ./ repmat(sqrt(vs), size(v,1), 1); ## Flip sign so that the average element is positive j = (sum(v) < 0); v(:,j) = -v(:,j); stats.eigenvec = v; canon = x*v; if (any(is_nan)) tmp(~is_nan,:) = canon; tmp(is_nan,:) = NaN; stats.canon = tmp; else stats.canon = canon; endif ## Compute Mahalanobis distances from points to group means gmean = nan (ngroups, size (canon, 2)); gmean(realgroups,:) = grpstats (canon, group_idx); mdist = sum ((canon - gmean(group_idx,:)) .^ 2, 2); if (any (is_nan)) stats.mdist(! is_nan) = mdist; stats.mdist(is_nan) = NaN; else stats.mdist = mdist; endif ## Compute Mahalanobis distances between group means stats.gmdist = squareform (pdist (gmean)) .^ 2; stats.gnames = group_names; endif endfunction %!demo %! load carbig %! [d,p] = manova1([MPG, Acceleration, Weight, Displacement], Origin) %!test %! load carbig %! [d,p] = manova1([MPG, Acceleration, Weight, Displacement], Origin); %! assert (d, 3); %! assert (p, [0, 3.140583347827075e-07, 0.007510999577743149, ... %! 0.1934100745898493]', [1e-12, 1e-12, 1e-12, 1e-12]'); %!test %! load carbig %! [d,p] = manova1([MPG, Acceleration, Weight], Origin); %! assert (d, 2); %! assert (p, [0, 0.00516082975137544, 0.1206528056514453]', ... %! [1e-12, 1e-12, 1e-12]'); statistics-release-1.6.3/inst/manovacluster.m000066400000000000000000000067001456127120000213500ustar00rootroot00000000000000## Copyright (C) 2022 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {} manovacluster (@var{stats}) ## @deftypefnx {statistics} {} manovacluster (@var{stats}, @var{method}) ## @deftypefnx {statistics} {@var{h} =} manovacluster (@var{stats}) ## @deftypefnx {statistics} {@var{h} =} manovacluster (@var{stats}, @var{method}) ## ## Cluster group means using manova1 output. ## ## @code{manovacluster (@var{stats})} draws a dendrogram showing the clustering ## of group means, calculated using the output STATS structure from ## @code{manova1} and applying the single linkage algorithm. See the ## @code{dendrogram} function for more information about the figure. ## ## @code{manovacluster (@var{stats}, @var{method})} uses the @var{method} ## algorithm in place of single linkage. The available methods are: ## ## @multitable @columnfractions 0.05 0.2 0.75 ## @item @tab "single" @tab --- nearest distance ## @item @tab "complete" @tab --- furthest distance ## @item @tab "average" @tab --- average distance ## @item @tab "centroid" @tab --- center of mass distance ## @item @tab "ward" @tab --- inner squared distance ## @end multitable ## ## @code{@var{h} = manovacluster (@dots{})} returns a vector of line handles. ## ## @seealso{manova1} ## @end deftypefn function h = manovacluster (stats, method) ## Check for valid input arguments narginchk (1, 2); if nargin > 1 valid_methods = {"single", "complete", "average", "centroid", "ward"}; if ! any (strcmpi (method, valid_methods)) error ("manovacluster: invalid method."); endif else method = "single"; end ## Get stats fields and create dendrogram dist = stats.gmdist; group_names = stats.gnames; [a, b] = meshgrid (1:length (dist)); hh = dendrogram (linkage (dist(a < b)', method), 0); ## Fix tick labels on x-axis oldlab = get (gca, "XTickLabel"); maxlen = max (cellfun ("length", group_names)); newlab = repmat(" ", size (oldlab, 1), maxlen); ng = size (group_names, 1); for j = 1:size (oldlab, 1) k = str2num (oldlab(j,:)); if (! isempty (k) & k > 0 & k <= ng) x = group_names{k,:}; newlab(j,1:length(x)) = x; endif endfor set(gca, "XtickLabel", newlab); ## Return plot handles if requested if nargout > 0 h = hh; endif endfunction %!demo %! load carbig %! X = [MPG Acceleration Weight Displacement]; %! [d, p, stats] = manova1 (X, Origin); %! manovacluster (stats) ## Test plotting %!test %! hf = figure ("visible", "off"); %! unwind_protect %! load carbig %! X = [MPG Acceleration Weight Displacement]; %! [d, p, stats] = manova1 (X, Origin); %! manovacluster (stats); %! unwind_protect_cleanup %! close (hf); %! end_unwind_protect ## Test input validation %!error manovacluster (stats, "some"); statistics-release-1.6.3/inst/mcnemar_test.m000066400000000000000000000152551456127120000211530ustar00rootroot00000000000000## Copyright (C) 1996-2017 Kurt Hornik ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {[@var{h}, @var{pval}, @var{chisq}] =} mcnemar_test (@var{x}) ## @deftypefnx {statistics} {[@var{h}, @var{pval}, @var{chisq}] =} mcnemar_test (@var{x}, @var{alpha}) ## @deftypefnx {statistics} {[@var{h}, @var{pval}, @var{chisq}] =} mcnemar_test (@var{x}, @var{testtype}) ## @deftypefnx {statistics} {[@var{h}, @var{pval}, @var{chisq}] =} mcnemar_test (@var{x}, @var{alpha}, @var{testtype}) ## ## Perform a McNemar's test on paired nominal data. ## ## @nospell{McNemar's} test is applied to a @math{2x2} contingency table @var{x} ## with a dichotomous trait, with matched pairs of subjects, of data ## cross-classified on the row and column variables to testing the null ## hypothesis of symmetry of the classification probabilities. More formally, ## the null hypothesis of marginal homogeneity states that the two marginal ## probabilities for each outcome are the same. ## ## Under the null, with a sufficiently large number of discordants ## (@qcode{@var{x}(1,2) + @var{x}(2,1) >= 25}), the test statistic, @var{chisq}, ## follows a chi-squared distribution with 1 degree of freedom. When the number ## of discordants is less than 25, then the mid-P exact McNemar test is used. ## ## @var{testtype} will force @code{mcnemar_test} to apply a particular method ## for testing the null hypothesis independently of the number of discordants. ## Valid options for @var{testtype}: ## @itemize ## @item @qcode{"asymptotic"} Original McNemar test statistic ## @item @qcode{"corrected"} Edwards' version with continuity correction ## @item @qcode{"exact"} An exact binomial test ## @item @qcode{"mid-p"} The mid-P McNemar test (mid-p binomial test) ## @end itemize ## ## The test decision is returned in @var{h}, which is 1 when the null hypothesis ## is rejected (@qcode{@var{pval} < @var{alpha}}) or 0 otherwise. @var{alpha} ## defines the critical value of statistical significance for the test. ## ## Further information about the McNemar's test can be found at ## @url{https://en.wikipedia.org/wiki/McNemar%27s_test} ## ## @seealso{crosstab, chi2test, fishertest} ## @end deftypefn function [h, pval, chisq] = mcnemar_test (x, varargin) ## Check for valid number of input arguments if (nargin > 3) error ("mcnemar_test: too many input arguments."); endif ## Check contigency table if (! isequal (size (x), [2, 2])) error ("mcnemar_test: X must be a 2x2 matrix."); elseif (! (all ((x(:) >= 0)) && all (x(:) == fix (x(:))))) error ("mcnemar_test: all entries of X must be non-negative integers."); endif ## Add defaults alpha = 0.05; b = x(1,2); c = x(2,1); if (b + c < 25) testtype = "mid-p"; else testtype = "asymptotic"; endif ## Parse optional arguments if (nargin == 2) if (isnumeric (varargin{1})) alpha = varargin{1}; elseif (ischar (varargin{1})) testtype = varargin{1}; else error ("mcnemar_test: invalid 2nd input argument."); endif elseif (nargin == 3) alpha = varargin{1}; testtype = varargin{2}; endif ## Check optional arguments if (! isscalar (alpha) || alpha <= 0 || alpha >= 1) error ("mcnemar_test: invalid value for ALPHA."); endif types = {"exact", "asymptotic", "mid-p", "corrected"}; if (! any (strcmpi (testtype, types))) error ("mcnemar_test: invalid value for TESTTYPE."); endif ## Calculate test switch (lower (testtype)) case "asymptotic" chisq = (b - c) .^2 / (b + c); pval = 1 - chi2cdf (chisq, 1); case "corrected" chisq = (abs (b - c) - 1) .^2 / (b + c); pval = 1 - chi2cdf (chisq, 1); case "exact" chisq = []; pval = 2 * (binocdf (b, b + c, 0.5)); case "mid-p" chisq = []; pval = 2 * (binocdf (b, b + c, 0.5)) - binopdf (b, b + c, 0.5); endswitch ## Get null hypothesis test result if (pval < alpha) h = 1; else h = 0; endif endfunction %!test %! [h, pval, chisq] = mcnemar_test ([101,121;59,33]); %! assert (h, 1); %! assert (pval, 3.8151e-06, 1e-10); %! assert (chisq, 21.356, 1e-3); %!test %! [h, pval, chisq] = mcnemar_test ([59,6;16,80]); %! assert (h, 1); %! assert (pval, 0.034690, 1e-6); %! assert (isempty (chisq), true); %!test %! [h, pval, chisq] = mcnemar_test ([59,6;16,80], 0.01); %! assert (h, 0); %! assert (pval, 0.034690, 1e-6); %! assert (isempty (chisq), true); %!test %! [h, pval, chisq] = mcnemar_test ([59,6;16,80], "mid-p"); %! assert (h, 1); %! assert (pval, 0.034690, 1e-6); %! assert (isempty (chisq), true); %!test %! [h, pval, chisq] = mcnemar_test ([59,6;16,80], "asymptotic"); %! assert (h, 1); %! assert (pval, 0.033006, 1e-6); %! assert (chisq, 4.5455, 1e-4); %!test %! [h, pval, chisq] = mcnemar_test ([59,6;16,80], "exact"); %! assert (h, 0); %! assert (pval, 0.052479, 1e-6); %! assert (isempty (chisq), true); %!test %! [h, pval, chisq] = mcnemar_test ([59,6;16,80], "corrected"); %! assert (h, 0); %! assert (pval, 0.055009, 1e-6); %! assert (chisq, 3.6818, 1e-4); %!test %! [h, pval, chisq] = mcnemar_test ([59,6;16,80], 0.1, "corrected"); %! assert (h, 1); %! assert (pval, 0.055009, 1e-6); %! assert (chisq, 3.6818, 1e-4); %!error mcnemar_test (59, 6, 16, 80) %!error mcnemar_test (ones (3, 3)) %!error ... %! mcnemar_test ([59,6;16,-80]) %!error ... %! mcnemar_test ([59,6;16,4.5]) %!error ... %! mcnemar_test ([59,6;16,80], {""}) %!error ... %! mcnemar_test ([59,6;16,80], -0.2) %!error ... %! mcnemar_test ([59,6;16,80], [0.05, 0.1]) %!error ... %! mcnemar_test ([59,6;16,80], 1) %!error ... %! mcnemar_test ([59,6;16,80], "") statistics-release-1.6.3/inst/mhsample.m000066400000000000000000000270021456127120000202710ustar00rootroot00000000000000## Copyright (C) 1995-2022 The Octave Project Developers ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {[@var{smpl}, @var{accept}] =} mhsample (@var{start}, @var{nsamples}, @var{property}, @var{value}, @dots{}) ## ## Draws @var{nsamples} samples from a target stationary distribution @var{pdf} ## using Metropolis-Hastings algorithm. ## ## Inputs: ## ## @itemize ## @item ## @var{start} is a @var{nchain} by @var{dim} matrix of starting points for each ## Markov chain. Each row is the starting point of a different chain and each ## column corresponds to a different dimension. ## ## @item ## @var{nsamples} is the number of samples, the length of each Markov chain. ## @end itemize ## ## Some property-value pairs can or must be specified, they are: ## ## (Required) One of: ## ## @itemize ## @item ## "pdf" @var{pdf}: a function handle of the target stationary distribution to ## be sampled. The function should accept different locations in each row and ## each column corresponds to a different dimension. ## ## or ## ## @item ## "logpdf" @var{logpdf}: a function handle of the log of the target stationary ## distribution to be sampled. The function should accept different locations ## in each row and each column corresponds to a different dimension. ## @end itemize ## ## In case optional argument @var{symmetric} is set to false (the default), one ## of: ## ## @itemize ## @item ## "proppdf" @var{proppdf}: a function handle of the proposal distribution that ## is sampled from with @var{proprnd} to give the next point in the chain. The ## function should accept two inputs, the random variable and the current ## location each input should accept different locations in each row and each ## column corresponds to a different dimension. ## ## or ## ## @item ## "logproppdf" @var{logproppdf}: the log of "proppdf". ## @end itemize ## ## The following input property/pair values may be needed depending on the ## desired outut: ## ## @itemize ## @item ## "proprnd" @var{proprnd}: (Required) a function handle which generates random ## numbers from @var{proppdf}. The function should accept different locations ## in each row and each column corresponds to a different dimension ## corresponding with the current location. ## ## @item ## "symmetric" @var{symmetric}: true or false based on whether @var{proppdf} is ## a symmetric distribution. If true, @var{proppdf} (or @var{logproppdf}) need ## not be specified. The default is false. ## ## @item ## "burnin" @var{burnin} the number of points to discard at the beginning, the ## default is 0. ## ## @item ## "thin" @var{thin}: omits @var{thin}-1 of every @var{thin} points in the ## generated Markov chain. The default is 1. ## ## @item ## "nchain" @var{nchain}: the number of Markov chains to generate. The default ## is 1. ## @end itemize ## ## Outputs: ## ## @itemize ## @item ## @var{smpl}: a @var{nsamples} x @var{dim} x @var{nchain} tensor of random ## values drawn from @var{pdf}, where the rows are different random values, the ## columns correspond to the dimensions of @var{pdf}, and the third dimension ## corresponds to different Markov chains. ## ## @item ## @var{accept} is a vector of the acceptance rate for each chain. ## @end itemize ## ## Example : Sampling from a normal distribution ## ## @example ## @group ## start = 1; ## nsamples = 1e3; ## pdf = @@(x) exp (-.5 * x .^ 2) / (pi ^ .5 * 2 ^ .5); ## proppdf = @@(x,y) 1 / 6; ## proprnd = @@(x) 6 * (rand (size (x)) - .5) + x; ## [smpl, accept] = mhsample (start, nsamples, "pdf", pdf, "proppdf", ... ## proppdf, "proprnd", proprnd, "thin", 4); ## histfit (smpl); ## @end group ## @end example ## ## @seealso{rand, slicesample} ## @end deftypefn function [smpl, accept] = mhsample (start, nsamples, varargin) if (nargin < 6) print_usage (); endif sizestart = size (start); pdf = []; proppdf = []; logpdf = []; logproppdf = []; proprnd = []; sym = false; K = 0; # burnin m = 1; # thin nchain = 1; for k = 1:2:length (varargin) if (ischar (varargin{k})) switch lower(varargin{k}) case "pdf" if (isa (varargin{k+1}, "function_handle")) pdf = varargin{k+1}; else error ("mhsample: pdf must be a function handle"); endif case "proppdf" if (isa (varargin{k+1}, "function_handle")) proppdf = varargin{k+1}; else error ("mhsample: proppdf must be a function handle"); endif case "logpdf" if (isa (varargin{k+1}, "function_handle")) pdf = varargin{k+1}; else error ("mhsample: logpdf must be a function handle"); endif case "logproppdf" if (isa (varargin{k+1}, "function_handle")) proppdf = varargin{k+1}; else error ("mhsample: logproppdf must be a function handle"); endif case "proprnd" if (isa (varargin{k+1}, "function_handle")) proprnd = varargin{k+1}; else error ("mhsample: proprnd must be a function handle"); endif case "symmetric" if (isa (varargin{k+1}, "logical")) sym = varargin{k+1}; else error ("mhsample: sym must be true or false"); endif case "burnin" if (varargin{k+1}>=0) K = varargin{k+1}; else error ("mhsample: K must be greater than or equal to 0"); endif case "thin" if (varargin{k+1} >= 1) m = varargin{k+1}; else error ("mhsample: m must be greater than or equal to 1"); endif case "nchain" if (varargin{k+1} >= 1) nchain = varargin{k+1}; else error ("mhsample: nchain must be greater than or equal to 1"); endif otherwise warning (["mhsample: Ignoring unknown option " varargin{k}]); endswitch else error (["mhsample: " varargin{k} " is not a valid property."]); endif endfor if (! isempty (pdf) && isempty (logpdf)) logpdf=@(x) rloge (pdf (x)); elseif (isempty (pdf) && isempty (logpdf)) error ("mhsample: pdf or logpdf must be input."); endif if (! isempty (proppdf) && isempty (logproppdf)) logproppdf = @(x, y) rloge (proppdf (x, y)); elseif (isempty (proppdf) && isempty (logproppdf) && ! sym) error ("mhsample: proppdf or logproppdf must be input unless 'symetrical' is true."); endif if (! isa (proprnd, "function_handle")) error ("mhsample: proprnd must be a function handle."); endif if (length (sizestart) == 2) sizestart = [sizestart 0]; end smpl = zeros (nsamples, sizestart(2), nchain); if (all (sizestart([1 3]) == [1 nchain])) ## Could remove, not Matlab compatable but allows continuing chains smpl(1, :, :) = start; elseif (all (sizestart([1 3]) == [nchain 0])) smpl(1, :, :) = permute (start, [3, 2, 1]); elseif (all (sizestart([1 3]) == [1 0])) ## Could remove, not Matlab compatable but allows all chains to start ## at the same location smpl(1, :, :) = repmat (start,[1, 1, nchain]); else error ("mhsample: start must be a nchain by dim matrix."); endif cx = permute (smpl(1, :, :),[3, 2, 1]); accept = zeros (nchain, 1); i = 1; rnd = log (rand (nchain, nsamples*m+K)); for k = 1:nsamples*m+K canacc = rem (k-K, m) == 0; px = proprnd (cx); if (sym) A = logpdf (px) - logpdf(cx); else A = (logpdf (px) + logproppdf (cx, px)) - (logpdf (cx) + logproppdf (px, cx)); endif ac = rnd(:, k) < min (A, 0); cx(ac, :) = px(ac, :); accept(ac)++; if (canacc) smpl(i, :, :) = permute (cx, [3, 2, 1]); end if (k > K && canacc) i++; endif endfor accept ./= (nsamples * m + K); endfunction function y = rloge (x) y = -inf (size (x)); xg0 = x > 0; y(xg0) = log (x(xg0)); endfunction %!demo %! ## Define function to sample %! d = 2; %! mu = [-1; 2]; %! rand ("seed", 5) # for reproducibility %! Sigma = rand (d); %! Sigma = (Sigma + Sigma'); %! Sigma += eye (d) * abs (eigs (Sigma, 1, "sa")) * 1.1; %! pdf = @(x)(2*pi)^(-d/2)*det(Sigma)^-.5*exp(-.5*sum((x.'-mu).*(Sigma\(x.'-mu)),1)); %! ## Inputs %! start = ones (1, 2); %! nsamples = 500; %! sym = true; %! K = 500; %! m = 10; %! rand ("seed", 8) # for reproducibility %! proprnd = @(x) (rand (size (x)) - .5) * 3 + x; %! [smpl, accept] = mhsample (start, nsamples, "pdf", pdf, "proprnd", proprnd, ... %! "symmetric", sym, "burnin", K, "thin", m); %! figure; %! hold on; %! plot (smpl(:, 1), smpl(:, 2), 'x'); %! [x, y] = meshgrid (linspace (-6, 4), linspace(-3, 7)); %! z = reshape (pdf ([x(:), y(:)]), size(x)); %! mesh (x, y, z, "facecolor", "None"); %! ## Using sample points to find the volume of half a sphere with radius of .5 %! f = @(x) ((.25-(x(:,1)+1).^2-(x(:,2)-2).^2).^.5.*(((x(:,1)+1).^2+(x(:,2)-2).^2)<.25)).'; %! int = mean (f (smpl) ./ pdf (smpl)); %! errest = std (f (smpl) ./ pdf (smpl)) / nsamples ^ .5; %! trueerr = abs (2 / 3 * pi * .25 ^ (3 / 2) - int); %! printf ("Monte Carlo integral estimate int f(x) dx = %f\n", int); %! printf ("Monte Carlo integral error estimate %f\n", errest); %! printf ("The actual error %f\n", trueerr); %! mesh (x, y, reshape (f([x(:), y(:)]), size(x)), "facecolor", "None"); %!demo %! ## Integrate truncated normal distribution to find normilization constant %! pdf = @(x) exp (-.5*x.^2)/(pi^.5*2^.5); %! nsamples = 1e3; %! rand ("seed", 5) # for reproducibility %! proprnd = @(x) (rand (size (x)) - .5) * 3 + x; %! [smpl, accept] = mhsample (1, nsamples, "pdf", pdf, "proprnd", proprnd, ... %! "symmetric", true, "thin", 4); %! f = @(x) exp(-.5 * x .^ 2) .* (x >= -2 & x <= 2); %! x = linspace (-3, 3, 1000); %! area(x, f(x)); %! xlabel ('x'); %! ylabel ('f(x)'); %! int = mean (f (smpl) ./ pdf (smpl)); %! errest = std (f (smpl) ./ pdf (smpl)) / nsamples^ .5; %! trueerr = abs (erf (2 ^ .5) * 2 ^ .5 * pi ^ .5 - int); %! printf ("Monte Carlo integral estimate int f(x) dx = %f\n", int); %! printf ("Monte Carlo integral error estimate %f\n", errest); %! printf ("The actual error %f\n", trueerr); ## Test output %!test %! nchain = 1e4; %! start = rand (nchain, 1); %! nsamples = 1e3; %! pdf = @(x) exp (-.5*(x-1).^2)/(2*pi)^.5; %! proppdf = @(x, y) 1/3; %! proprnd = @(x) 3 * (rand (size (x)) - .5) + x; %! [smpl, accept] = mhsample (start, nsamples, "pdf", pdf, "proppdf", proppdf, ... %! "proprnd", proprnd, "thin", 2, "nchain", nchain, ... %! "burnin", 0); %! assert (mean (mean (smpl, 1), 3), 1, .01); %! assert (mean (var (smpl, 1), 3), 1, .01) ## Test input validation %!error mhsample (); %!error mhsample (1); %!error mhsample (1, 1); %!error mhsample (1, 1, "pdf", @(x)x); %!error mhsample (1, 1, "pdf", @(x)x, "proprnd", @(x)x+rand(size(x))); statistics-release-1.6.3/inst/mnrfit.m000066400000000000000000000233711456127120000177670ustar00rootroot00000000000000## Copyright (C) 2024 Andrew C Penn ## Copyright (C) 2024 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{B} =} mnrfit (@var{X}, @var{Y}) ## @deftypefnx {statistics} {@var{B} =} mnrfit (@var{X}, @var{Y}, @var{name}, @var{value}) ## @deftypefnx {statistics} {[@var{B}, @var{dev}] =} mnrrfit (@dots{}) ## @deftypefnx {statistics} {[@var{B}, @var{dev}, @var{stats}] =} mnrfit (@dots{}) ## ## Perform logistic regression for binomial responses or multiple ordinal ## responses. ## ## Note: This function is currently a wrapper for the @code{logistic_regression} ## function. It can only be used for fitting an ordinal logistic model and a ## nominal model with 2 categories (which is an ordinal case). Hierarchical ## models as well as nominal model with more than two classes are not currently ## supported. This function is a work in progress. ## ## @code{@var{B} = mnrfit (@var{X}, @var{Y})} returns a matrix, @var{B}, of ## coefficient estimates for a multinomial logistic regression of the nominal ## responses in @var{Y} on the predictors in @var{X}. @var{X} is an @math{NxP} ## numeric matrix the observations on predictor variables, where @math{N} ## corresponds to the number of observations and @math{P} corresponds to ## predictor variables. @var{Y} contains the response category labels and it ## either be an @math{NxP} categorical or numerical matrix (containing only 1s ## and 0s) or an @math{Nx1} numeric vector with positive integer values, a cell ## array of character vectors and a logical vector. @var{Y} can also be defined ## as a character matrix with each row corresponding to an observation of ## @var{X}. ## ## @code{@var{B} = mnrfit (@var{X}, @var{Y}, @var{name}, @var{value})} returns a ## matrix, @var{B}, of coefficient estimates for a multinomial model fit with ## additional parameterss specified @qcode{Name-Value} pair arguments. ## ## @multitable @columnfractions 0.18 0.02 0.8 ## @headitem @var{Name} @tab @tab @var{Value} ## ## @item @qcode{"model"} @tab @tab Specifies the type of model to fit. ## Currently, only @qcode{"ordinal"} is fully supported. @qcode{"nominal"} is ## only supported for 2 classes in @var{Y}. ## ## @item @qcode{"display"} @tab @tab A flag to enable/disable displaying ## information about the fitted model. Default is @qcode{"off"}. ## @end multitable ## ## @code{[@var{B}, @var{dev}, @var{stats}] = mnrfit (@dots{}} also returns the ## deviance of the fit, @var{dev}, and the structure @var{stats} for any of the ## previous input arguments. @var{stats} currently only returns values for the ## fields @qcode{"beta"}, same as @var{B}, @qcode{"coeffcorr"}, the estimated ## correlation matrix for @var{B}, @qcode{"covd"}, the estimated covariance ## matrix for @var{B}, and @qcode{"se"}, the standard errors of the coefficient ## estimates @var{B}. ## ## @seealso{logistic_regression} ## @end deftypefn function [B, DEV, STATS] = mnrfit (X, Y, varargin) ## Check input arguments X and Y if (nargin < 2) error ("mnrfit: too few input arguments."); endif if (! isnumeric (X)) error ("mnrfit: Predictors must be numeric.") endif if (isscalar (X) || (ndims (X) > 2)) error ("mnrfit: Predictors must be a vector or a 2D matrix.") endif if (isscalar (Y) || (ndims (Y) > 2)) error ("mnrfit: Response must be a vector or a 2D matrix.") endif [N, P] = size (X); [n, K] = size (Y); if (N == 1) ## if X is a row vector, make it a column vector X = X(:); N = P; P = 1; endif if (n != N) error ("mnrfit: Y must have the same number of rows as X.") endif if (! (isnumeric (Y) || islogical (Y) || ischar (Y) || iscellstr (Y))) error (strcat (["mnrfit: Response labels must be a character array,"], ... [" a cell vector of strings, \nor a vector or"], ... [" matrix of doubles, singles or logical values."])); endif ## Check supplied parameters if (mod (numel (varargin), 2) != 0) error ("mnrfit: optional arguments must be in pairs.") endif MODELTYPE = "nominal"; DISPLAY = "off"; while (numel (varargin) > 0) name = varargin{1}; value = varargin{2}; switch (lower (name)) case "model" MODELTYPE = value; case "display" DISPLAY = value; otherwise warning (sprintf ("mnrfit: parameter %s will be ignored", name)); endswitch varargin (1:2) = []; endwhile ## Evaluate display input argument switch (lower (DISPLAY)) case "on" dispopt = true; case "off" dispopt = false; endswitch ## Categorize Y if it is a cellstring array if (iscellstr (Y)) if (K > 1) error ("mnrfit: Y must be a column vector when given as cellstr."); endif ## Get groups in Y [YN, ~, UY] = grp2idx (Y); # this will also catch "" as missing values ## Remove missing values from X and Y RowsUsed = ! logical (sum (isnan ([X, YN]), 2)); Y = Y (RowsUsed); X = X (RowsUsed, :); ## Renew groups in Y [YN, ~, UY] = grp2idx (Y); # in case a category is removed due to NaNs in X n = numel (UY); endif ## Categorize Y if it is a character array if (ischar (Y)) ## Get groups in Y [YN, ~, UY] = grp2idx (Y); # this will also catch "" as missing values ## Remove missing values from X and Y RowsUsed = ! logical (sum (isnan ([X, YN]), 2)); Y = Y (RowsUsed); X = X (RowsUsed, :); ## Renew groups in Y [YN, ~, UY] = grp2idx (Y); # in case a category is removed due to NaNs in X n = numel (UY); endif if (K > 1) ## So far, if K > 1, Y must be a matrix of logical, singles or doubles if (! all (all (Y == 0 | Y == 1))) error ("mnrfit: Y must contain only 1 and 0 when given as a 2D matrix."); endif ## Convert Y to a vector of positive integer categories Y = sum (bsxfun (@times, (1:K), Y), 2); endif ## Categorize Y in all other cases if (! iscellstr (Y)) RowsUsed = ! logical (sum (isnan ([X, Y]), 2)); Y = Y (RowsUsed); X = X (RowsUsed, :); [UY, ~, YN] = unique (Y); ## find unique categories in the response n = numel (UY); ## number of unique response categories endif if (isnumeric (Y)) if (! (all (Y > 0) && all (fix (Y) == Y))) error ("mnrfit: Y must contain positive integer category numbers.") endif endif ## Evaluate model type input argument switch (lower (MODELTYPE)) case "nominal" if (n > 2) error ("mnrfit: fitting more than 2 nominal responses not supported."); else ## Y has two responses. Ordinal logistic regression can be used to fit ## models with binary nominal responses endif case "ordinal" ## Do nothing, ordinal responses are fully supported case "hierarchical" error ("mnrfit: fitting hierarchical responses not supported."); otherwise error ("mnrfit: model type not recognised."); endswitch ## Perform fit and reformat output [INTERCEPT, SLOPE, DEV, ~, ~, ~, S] = logistic_regression (YN - 1, X, dispopt); B = cat (1, INTERCEPT, SLOPE); STATS = struct ("beta", B, ... "dfe", [], ... ## Not used "s", [], ... ## Not used "sfit", [], ... ## Not used "estdisp", [], ... ## Not used "coeffcorr", S.coeffcorr, ... "covb", S.cov, ... "se", S.se, ... "t", [], ... ## Not used "p", [], ... ## Not used "resid", [], ... ## Not used "residp", [], ... ## Not used "residd", []); ## Not used endfunction ## Test input validation %!error mnrfit (ones (50,1)) %!error ... %! mnrfit ({1 ;2 ;3 ;4 ;5}, ones (5,1)) %!error ... %! mnrfit (ones (50, 4, 2), ones (50, 1)) %!error ... %! mnrfit (ones (50, 4), ones (50, 1, 3)) %!error ... %! mnrfit (ones (50, 4), ones (45,1)) %!error ... %! mnrfit (ones (5, 4), {1 ;2 ;3 ;4 ;5}) %!error ... %! mnrfit (ones (5, 4), ones (5, 1), "model") %!error ... %! mnrfit (ones (5, 4), {"q","q";"w","w";"q","q";"w","w";"q","q"}) %!error ... %! mnrfit (ones (5, 4), [1, 2; 1, 2; 1, 2; 1, 2; 1, 2]) %!error ... %! mnrfit (ones (5, 4), [1; -1; 1; 2; 1]) %!error ... %! mnrfit (ones (5, 4), [1; 2; 3; 2; 1], "model", "nominal") %!error ... %! mnrfit (ones (5, 4), [1; 2; 3; 2; 1], "model", "hierarchical") %!error ... %! mnrfit (ones (5, 4), [1; 2; 3; 2; 1], "model", "whatever") statistics-release-1.6.3/inst/monotone_smooth.m000066400000000000000000000146201456127120000217140ustar00rootroot00000000000000## Copyright (C) 2011 Nir Krakauer ## Copyright (C) 2011 Carnë Draug ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{yy} =} monotone_smooth (@var{x}, @var{y}, @var{h}) ## ## Produce a smooth monotone increasing approximation to a sampled functional ## dependence. ## ## A kernel method is used (an Epanechnikov smoothing kernel is applied to y(x); ## this is integrated to yield the monotone increasing form. See Reference 1 ## for details.) ## ## @subheading Arguments ## ## @itemize @bullet ## @item ## @var{x} is a vector of values of the independent variable. ## ## @item ## @var{y} is a vector of values of the dependent variable, of the same size as ## @var{x}. For best performance, it is recommended that the @var{y} already be ## fairly smooth, e.g. by applying a kernel smoothing to the original values if ## they are noisy. ## ## @item ## @var{h} is the kernel bandwidth to use. If @var{h} is not given, ## a "reasonable" value is computed. ## ## @end itemize ## ## @subheading Return values ## ## @itemize @bullet ## @item ## @var{yy} is the vector of smooth monotone increasing function values at ## @var{x}. ## ## @end itemize ## ## @subheading Examples ## ## @example ## @group ## x = 0:0.1:10; ## y = (x .^ 2) + 3 * randn(size(x)); # typically non-monotonic from the added ## noise ## ys = ([y(1) y(1:(end-1))] + y + [y(2:end) y(end)])/3; # crudely smoothed via ## moving average, but still typically non-monotonic ## yy = monotone_smooth(x, ys); # yy is monotone increasing in x ## plot(x, y, '+', x, ys, x, yy) ## @end group ## @end example ## ## @subheading References ## ## @enumerate ## @item ## Holger Dette, Natalie Neumeyer and Kay F. Pilz (2006), A simple nonparametric ## estimator of a strictly monotone regression function, @cite{Bernoulli}, ## 12:469-490 ## @item ## Regine Scheder (2007), R Package 'monoProc', Version 1.0-6, ## @url{http://cran.r-project.org/web/packages/monoProc/monoProc.pdf} (The ## implementation here is based on the monoProc function mono.1d) ## @end enumerate ## @end deftypefn function yy = monotone_smooth (x, y, h) if (nargin < 2 || nargin > 3) print_usage (); elseif (! isnumeric (x) || ! isvector (x)) error ("monotone_smooth: X must be a numeric vector."); elseif (! isnumeric (y) || ! isvector (y)) error ("monotone_smooth: Y must be a numeric vector."); elseif (numel (x) != numel (y)) error ("monotone_smooth: X and Y must have the same number of elements."); elseif (nargin == 3 && (! isscalar (h) || ! isnumeric (h))) error ("monotone_smooth: H (kernel bandwith) must a numeric scalar."); endif n = numel (x); ## Set filter bandwidth at a reasonable default value, if not specified if (nargin != 3) s = std (x); h = s / (n ^ 0.2); end x_min = min(x); x_max = max(x); y_min = min(y); y_max = max(y); ## Transform range of X to [0, 1] xl = (x - x_min) / (x_max - x_min); yy = ones(size(y)); ## Epanechnikov smoothing kernel (with finite support) ## K_epanech_kernel = @(z) (3/4) * ((1 - z).^2) .* (abs(z) < 1); K_epanech_int = @(z) mean(((abs(z) < 1)/2) - (3/4) * (z .* (abs(z) < 1) ... - (1/3) * (z.^3) .* (abs(z) < 1)) + (z < -1)); ## Integral of kernels up to t monotone_inverse = @(t) K_epanech_int((y - t) / h); ## Find the value of the monotone smooth function at each point in X niter_max = 150; # maxIter for estimating each value (adequate for most cases) for l = 1:n tmax = y_max; tmin = y_min; wmin = monotone_inverse(tmin); wmax = monotone_inverse(tmax); if (wmax == wmin) yy(l) = tmin; else wt = xl(l); iter_max_reached = 1; for i = 1:niter_max wt_scaled = (wt - wmin) / (wmax - wmin); tn = tmin + wt_scaled * (tmax - tmin) ; wn = monotone_inverse(tn); wn_scaled = (wn - wmin) / (wmax - wmin); ## if (abs(wt-wn) < 1E-4) || (tn < (y_min-0.1)) || (tn > (y_max+0.1)) ## criterion for break in the R code -- replaced by the following line ## to hopefully be less dependent on the scale of y if ((abs(wt_scaled-wn_scaled) < 1E-4) || (wt_scaled < -0.1) || (wt_scaled > 1.1)) iter_max_reached = 0; break endif if (wn > wt) tmax = tn; wmax = wn; else tmin = tn; wmin = wn; endif endfor if (iter_max_reached) msg = sprintf (strcat (["at x = %%g, maximum number of iterations"], ... [" %%d reached without convergence;"], ... [" approximation may not be optimal"])); warning (msg, x(l), niter_max) endif yy(l) = tmin + (wt - wmin) * (tmax - tmin) / (wmax - wmin); endif endfor endfunction ## Test input validation %!error ... %! monotone_smooth (1) %!error ... %! monotone_smooth ("char", 1) %!error ... %! monotone_smooth ({1,2,3}, 1) %!error ... %! monotone_smooth (ones(20,3), 1) %!error ... %! monotone_smooth (1, "char") %!error ... %! monotone_smooth (1, {1,2,3}) %!error ... %! monotone_smooth (1, ones(20,3)) %!error monotone_smooth (ones (10,1), ones(10,1), [1, 2]) %!error monotone_smooth (ones (10,1), ones(10,1), {2}) %!error monotone_smooth (ones (10,1), ones(10,1), "char") statistics-release-1.6.3/inst/multcompare.m000066400000000000000000001467131456127120000210260ustar00rootroot00000000000000## Copyright (C) 2022 Andrew Penn ## Copyright (C) 2022 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{C} =} multcompare (@var{STATS}) ## @deftypefnx {statistics} {@var{C} =} multcompare (@var{STATS}, "name", @var{value}) ## @deftypefnx {statistics} {[@var{C}, @var{M}] =} multcompare (...) ## @deftypefnx {statistics} {[@var{C}, @var{M}, @var{H}] =} multcompare (...) ## @deftypefnx {statistics} {[@var{C}, @var{M}, @var{H}, @var{GNAMES}] =} multcompare (...) ## @deftypefnx {statistics} {@var{padj} =} multcompare (@var{p}) ## @deftypefnx {statistics} {@var{padj} =} multcompare (@var{p}, "ctype", @var{CTYPE}) ## ## Perform posthoc multiple comparison tests or p-value adjustments to control ## the family-wise error rate (FWER) or false discovery rate (FDR). ## ## @code{@var{C} = multcompare (@var{STATS})} performs a multiple comparison ## using a @var{STATS} structure that is obtained as output from any of ## the following functions: anova1, anova2, anovan, kruskalwallis, and friedman. ## The return value @var{C} is a matrix with one row per comparison and six ## columns. Columns 1-2 are the indices of the two samples being compared. ## Columns 3-5 are a lower bound, estimate, and upper bound for their ## difference, where the bounds are for 95% confidence intervals. Column 6-8 are ## the multiplicity adjusted p-values for each individual comparison, the test ## statistic and the degrees of freedom. ## All tests by multcompare are two-tailed. ## ## @qcode{multcompare} can take a number of optional parameters as name-value ## pairs. ## ## @code{[@dots{}] = multcompare (@var{STATS}, "alpha", @var{ALPHA})} ## ## @itemize ## @item ## @var{ALPHA} sets the significance level of null hypothesis significance ## tests to ALPHA, and the central coverage of two-sided confidence intervals to ## 100*(1-@var{ALPHA})%. (Default ALPHA is 0.05). ## @end itemize ## ## @code{[@dots{}] = multcompare (@var{STATS}, "ControlGroup", @var{REF})} ## ## @itemize ## @item ## @var{REF} is the index of the control group to limit comparisons to. The ## index must be a positive integer scalar value. For each dimension (d) listed ## in @var{DIM}, multcompare uses STATS.grpnames@{d@}(idx) as the control group. ## (Default is empty, i.e. [], for full pairwise comparisons) ## @end itemize ## ## @code{[@dots{}] = multcompare (@var{STATS}, "ctype", @var{CTYPE})} ## ## @itemize ## @item ## @var{CTYPE} is the type of comparison test to use. In order of increasing ## power, the choices are: "bonferroni", "scheffe", "mvt", "holm" (default), ## "hochberg", "fdr", or "lsd". The first five methods control the family-wise ## error rate. The "fdr" method controls false discovery rate (by the original ## Benjamini-Hochberg step-up procedure). The final method, "lsd" (or "none"), ## makes no attempt to control the Type 1 error rate of multiple comparisons. ## The coverage of confidence intervals are only corrected for multiple ## comparisons in the cases where @var{CTYPE} is "bonferroni", "scheffe" or ## "mvt", which control the Type 1 error rate for simultaneous inference. ## ## The "mvt" method uses the multivariate t distribution to assess the ## probability or critical value of the maximum statistic across the tests, ## thereby accounting for correlations among comparisons in the control of the ## family-wise error rate with simultaneous inference. In the case of pairwise ## comparisons, it simulates Tukey's (or the Games-Howell) test, in the case of ## comparisons with a single control group, it simulates Dunnett's test. ## @var{CTYPE} values "tukey-kramer" and "hsd" are recognised but set the value ## of @var{CTYPE} and @var{REF} to "mvt" and empty respectively. A @var{CTYPE} ## value "dunnett" is recognised but sets the value of @var{CTYPE} to "mvt", and ## if @var{REF} is empty, sets @var{REF} to 1. Since the algorithm uses a Monte ## Carlo method (of 1e+06 random samples), you can expect the results to ## fluctuate slightly with each call to multcompare and the calculations may be ## slow to complete for a large number of comparisons. If the parallel package ## is installed and loaded, @qcode{multcompare} will automatically accelerate ## computations by parallel processing. Note that p-values calculated by the ## "mvt" are truncated at 1e-06. ## @end itemize ## ## @code{[@dots{}] = multcompare (@var{STATS}, "df", @var{DF})} ## ## @itemize ## @item ## @var{DF} is an optional scalar value to set the number of degrees of freedom ## in the calculation of p-values for the multiple comparison tests. By default, ## this value is extracted from the @var{STATS} structure of the ANOVA test, but ## setting @var{DF} maybe necessary to approximate Satterthwaite correction if ## @qcode{anovan} was performed using weights. ## @end itemize ## ## @code{[@dots{}] = multcompare (@var{STATS}, "dim", @var{DIM})} ## ## @itemize ## @item ## @var{DIM} is a vector specifying the dimension or dimensions over which the ## estimated marginal means are to be calculated. Used only if STATS comes from ## anovan. The value [1 3], for example, computes the estimated marginal mean ## for each combination of the first and third predictor values. The default is ## to compute over the first dimension (i.e. 1). If the specified dimension is, ## or includes, a continuous factor then @qcode{multcompare} will return an ## error. ## @end itemize ## ## @code{[@dots{}] = multcompare (@var{STATS}, "estimate", @var{ESTIMATE})} ## ## @itemize ## @item ## @var{ESTIMATE} is a string specifying the estimates to be compared when ## computing multiple comparisons after anova2; this argument is ignored by ## anovan and anova1. Accepted values for @var{ESTIMATE} are either "column" ## (default) to compare column means, or "row" to compare row means. If the ## model type in anova2 was "linear" or "nested" then only "column" is accepted ## for @var{ESTIMATE} since the row factor is assumed to be a random effect. ## @end itemize ## ## @code{[@dots{}] = multcompare (@var{STATS}, "display", @var{DISPLAY})} ## ## @itemize ## @item ## @var{DISPLAY} is either "on" (the default): to display a table and graph of ## the comparisons (e.g. difference between means), their 100*(1-@var{ALPHA})% ## intervals and multiplicity adjusted p-values in APA style; or "off": to omit ## the table and graph. On the graph, markers and error bars colored red have ## multiplicity adjusted p-values < ALPHA, otherwise the markers and error bars ## are blue. ## @end itemize ## ## @code{[@dots{}] = multcompare (@var{STATS}, "seed", @var{SEED})} ## ## @itemize ## @item ## @var{SEED} is a scalar value used to initialize the random number generator ## so that @var{CTYPE} "mvt" produces reproducible results. ## @end itemize ## ## @code{[@var{C}, @var{M}, @var{H}, @var{GNAMES}] = multcompare (@dots{})} ## returns additional outputs. @var{M} is a matrix where columns 1-2 are the ## estimated marginal means and their standard errors, and columns 3-4 are lower ## and upper bounds of the confidence intervals for the means; the critical ## value of the test statistic is scaled by a factor of 2^(-0.5) before ## multiplying by the standard errors of the group means so that the intervals ## overlap when the difference in means becomes significant at approximately ## the level @var{ALPHA}. When @var{ALPHA} is 0.05, this corresponds to ## confidence intervals with 83.4% central coverage. @var{H} is a handle to the ## figure containing the graph. @var{GNAMES} is a cell array with one row for ## each group, containing the names of the groups. ## ## @code{@var{padj} = multcompare (@var{p})} calculates and returns adjusted ## p-values (@var{padj}) using the Holm-step down Bonferroni procedure to ## control the family-wise error rate. ## ## @code{@var{padj} = multcompare (@var{p}, "ctype", @var{CTYPE})} calculates ## and returns adjusted p-values (@var{padj}) computed using the method ## @var{CTYPE}. In order of increasing power, @var{CTYPE} for p-value adjustment ## can be either "bonferroni", "holm" (default), "hochberg", or "fdr". See ## above for further information about the @var{CTYPE} methods. ## ## @seealso{anova1, anova2, anovan, kruskalwallis, friedman, fitlm} ## @end deftypefn function [C, M, H, GNAMES] = multcompare (STATS, varargin) if (nargin < 1) error (strcat (["multcompare usage: ""multcompare (ARG)""; "], ... [" atleast 1 input argument required"])); endif ## Check supplied parameters if ((numel (varargin) / 2) != fix (numel (varargin) / 2)) error ("multcompare: wrong number of arguments.") endif ALPHA = 0.05; REF = []; CTYPE = "holm"; DISPLAY = "on"; DIM = 1; ESTIMATE = "column"; DFE = []; for idx = 3:2:nargin name = varargin{idx-2}; value = varargin{idx-1}; switch (lower (name)) case "alpha" ALPHA = value; case {"controlgroup","ref"} REF = value; case {"ctype","criticalvaluetype"} CTYPE = lower (value); case {"display","displayopt"} DISPLAY = lower (value); case {"dim","dimension"} DIM = value; case "estimate" ESTIMATE = lower (value); case {"df","dfe"} DFE = value; case {"seed"} SEED = value; ## Set random seed for mvtrnd and mvnrnd randn ('seed', SEED); randg ('seed', SEED); otherwise error (sprintf ("multcompare: parameter %s is not supported", name)); endswitch endfor ## Evaluate ALPHA input argument if (! isa (ALPHA,"numeric") || numel (ALPHA) != 1) error("anovan:alpha must be a numeric scalar value"); endif if ((ALPHA <= 0) || (ALPHA >= 1)) error("anovan: alpha must be a value between 0 and 1"); endif ## Evaluate CTYPE input argument if (ismember (CTYPE, {"tukey-kramer", "hsd"})) CTYPE = "mvt"; REF = []; elseif (strcmp (CTYPE, "dunnett")) CTYPE = "mvt"; if (isempty (REF)) REF = 1; endif elseif (strcmp (CTYPE, "none")) CTYPE = "lsd"; endif if (! ismember (CTYPE, ... {"bonferroni","scheffe","mvt","holm","hochberg","fdr","lsd"})) error ("multcompare: '%s' is not a supported value for CTYPE", CTYPE) endif ## Evaluate DFE input argument if (! isempty (DFE)) if (! isscalar (DFE)) error ("multcompare: df must be a scalar value."); endif if (!(DFE > 0) || isinf (DFE)) error ("multcompare: df must be a positive finite value."); endif endif ## If STATS is numeric, assume it is a vector of p-values if (isnumeric (STATS)) if (nargout > 1) error (strcat (["multcompare: invalid number of output arguments"], ... [" if only used to adjust p-values"])) endif if (!isempty (varargin)) if (!any (strcmpi (varargin{1}, {"ctype","criticalvaluetype"})) ... || (nargin > 3) ) error (strcat(["multcompare: invalid input arguments if only"], ... [" used to adjust p-values"])) endif endif if (! ismember (CTYPE, {"bonferroni","holm","hochberg","fdr"})) error ("multcompare: '%s' is not a supported p-adjustment method", CTYPE) endif p = STATS; if (all (size (p) > 1)) error ("multcompare: p-values must be a vector") endif padj = feval (CTYPE, p); if (size (p, 1) > 1) C = padj; else C = padj'; endif return endif ## Perform test specific calculations switch (STATS.source) case "anova1" ## Make matrix of requested comparisons (pairs) ## Also return the corresponding hypothesis matrix (L) n = STATS.n(:); Ng = numel (n); if (isempty (REF)) ## Pairwise comparisons [pairs, L] = pairwise (Ng); else ## Treatment vs. Control comparisons [pairs, L] = trt_vs_ctrl (Ng, REF); endif Np = size (pairs, 1); switch (STATS.vartype) case "equal" ## Calculate estimated marginal means and their standard errors gmeans = STATS.means(:); gvar = (STATS.s^2) ./ n; # Sampling variance gcov = diag (gvar); Ng = numel (gmeans); M = zeros (Ng, 4); M(:,1:2) = cat (2, gmeans, sqrt(gvar)); ## Get the error degrees of freedom from anova1 output if (isempty (DFE)) DFE = STATS.df; endif case "unequal" ## Error checking if (strcmp (CTYPE, "scheffe")) error (strcat (["multcompare: the CTYPE value 'scheffe'"], ... [" does not support tests with varying degrees of freedom "])); endif ## Calculate estimated marginal means and their standard errors gmeans = STATS.means(:); gvar = STATS.vars(:) ./ n; # Sampling variance gcov = diag (gvar); Ng = numel (gmeans); M = zeros (Ng, 4); M(:,1:2) = cat (2, gmeans, sqrt (gvar)); ## Calculate Welch's corrected degrees of freedom if (isempty (DFE)) DFE = sum (gvar(pairs), 2).^2 ./ ... sum ((gvar(pairs).^2 ./ (n(pairs) - 1)), 2); endif endswitch ## Calculate t statistics corresponding to the comparisons defined in L [mean_diff, sed, t] = tValue (gmeans, gcov, L); ## Calculate correlation matrix vcov = L * gcov * L'; R = cov2corr (vcov); ## Create cell array of group names corresponding to each row of m GNAMES = STATS.gnames; case "anova2" ## Fetch estimate specific information from the STATS structure switch (ESTIMATE) case {"column","columns","col","cols"} gmeans = STATS.colmeans(:); Ng = numel (gmeans); n = STATS.coln; case {"row","rows"} if (ismember (STATS.model, {"linear","nested"})) error (strcat (["multcompare: no support for the row factor"],... [" (random effect) in a 'nested' or 'linear' anova2 model"])); endif gmeans = STATS.rowmeans(:); Ng = numel (gmeans); n = STATS.rown; endswitch ## Make matrix of requested comparisons (pairs) ## Also return the corresponding hypothesis matrix (L) if (isempty (REF)) ## Pairwise comparisons [pairs, L, R] = pairwise (Ng); else ## Treatment vs. Control comparisons [pairs, L, R] = trt_vs_ctrl (Ng, REF); endif Np = size (pairs, 1); ## Calculate estimated marginal means and their standard errors gvar = ((STATS.sigmasq) / n) * ones (Ng, 1); # Sampling variance gcov = diag (gvar); M = zeros (Ng, 4); M(:,1:2) = cat (2, gmeans, sqrt (gvar)); ## Get the error degrees of freedom from anova2 output if (isempty (DFE)) DFE = STATS.df; endif ## Calculate t statistics corresponding to the comparisons defined in L [mean_diff, sed, t] = tValue (gmeans, gcov, L); ## Create character array of group names corresponding to each row of m GNAMES = cellstr (num2str ([1:Ng]')); case {"anovan","fitlm"} ## Our calculations treat all effects as fixed if (ismember (STATS.random, DIM)) warning (strcat (["multcompare: ignoring random effects"], ... [" (all effects treated as fixed)"])); endif ## Check what type of factor is requested in DIM if (any (STATS.nlevels(DIM) < 2)) error (strcat (["multcompare: DIM must specify only categorical"], ... [" factors with 2 or more degrees of freedom."])); endif ## Check that all continuous variables were centered msg = strcat (["multcompare: use a STATS structure from a model"], ... [" refit with a sum-to-zero contrast coding"]); if (any (STATS.continuous - STATS.center_continuous)) error (msg) endif ## Check that the columns sum to 0 N = numel (STATS.contrasts); for j = 1:N if (isnumeric (STATS.contrasts{j})) if (any (abs (sum (STATS.contrasts{j})) > eps("single"))) error (msg); endif endif endfor ## Calculate estimated marginal means and their standard errors Nd = numel (DIM); n = numel (STATS.resid); df = STATS.df; if (isempty (DFE)) DFE = STATS.dfe; endif i = 1 + cumsum(df); k = find (sum (STATS.terms(:,DIM), 2) == sum (STATS.terms, 2)); Nb = 1 + sum(df(k)); Nt = numel (k); L = zeros (n, sum (df) + 1); for j = 1:Nt L(:, i(k(j)) - df(k(j)) + 1 : i(k(j))) = STATS.X(:,i(k(j)) - ... df(k(j)) + 1 : i(k(j))); endfor L(:,1) = 1; U = unique (L, "rows", "stable"); Ng = size (U, 1); idx = zeros (Ng, 1); for k = 1:Ng idx(k) = find (all (L == U(k, :), 2),1); endfor gmeans = U * STATS.coeffs(:,1); # Estimated marginal means gcov = U * STATS.vcov * U'; gvar = diag (gcov); # Sampling variance M = zeros (Ng, 4); M(:,1:2) = cat (2, gmeans, sqrt(gvar)); ## Create cell array of group names corresponding to each row of m GNAMES = cell (Ng, 1); for i = 1:Ng str = ""; for j = 1:Nd str = sprintf("%s%s=%s, ", str, ... num2str(STATS.varnames{DIM(j)}), ... num2str(STATS.grpnames{DIM(j)}{STATS.grps(idx(i),DIM(j))})); endfor GNAMES{i} = str(1:end-2); str = ""; endfor ## Make matrix of requested comparisons (pairs) ## Also return the corresponding hypothesis matrix (L) if (isempty (REF)) ## Pairwise comparisons [pairs, L] = pairwise (Ng); else ## Treatment vs. Control comparisons [pairs, L] = trt_vs_ctrl (Ng, REF); endif Np = size (pairs, 1); ## Calculate t statistics corresponding to the comparisons defined in L [mean_diff, sed, t] = tValue (gmeans, gcov, L); ## Calculate correlation matrix. vcov = L * gcov * L'; R = cov2corr (vcov); case "friedman" ## Get stats from structure gmeans = STATS.meanranks(:); Ng = length (gmeans); sigma = STATS.sigma; ## Make group names GNAMES = strjust (num2str ((1:Ng)'), "left"); ## Make matrix of requested comparisons (pairs) ## Also return the corresponding hypothesis matrix (L) if (isempty (REF)) ## Pairwise comparisons [pairs, L, R] = pairwise (Ng); else ## Treatment vs. Control comparisons [pairs, L, R] = trt_vs_ctrl (Ng, REF); endif Np = size (pairs, 1); ## Calculate covariance matrix gcov = ((sigma ^ 2) / STATS.n) * eye (Ng); ## Create matrix with group means and standard errors M = cat (2, gmeans, sqrt (diag (gcov))); ## Calculate t statistics corresponding to the comparisons defined in L [mean_diff, sed, t] = tValue (gmeans, gcov, L); # z-statistic (not t) ## Calculate degrees of freedom from number of groups if (isempty (DFE)) DFE = inf; # this is a z-statistic so infinite degrees of freedom endif case "kruskalwallis" ## Get stats from structure gmeans = STATS.meanranks(:); sumt = STATS.sumt; Ng = length (gmeans); n = STATS.n(:); N = sum (n); ## Make group names GNAMES = STATS.gnames; ## Make matrix of requested comparisons (pairs) ## Also return the corresponding hypothesis matrix (L) if (isempty (REF)) ## Pairwise comparisons [pairs, L] = pairwise (Ng); else ## Treatment vs. Control comparisons [pairs, L] = trt_vs_ctrl (Ng, REF); endif Np = size (pairs, 1); ## Calculate covariance matrix gcov = diag (((N * (N + 1) / 12) - (sumt / (12 * (N - 1)))) ./ n); ## Create matrix with group means and standard errors M = cat (2, gmeans, sqrt (diag (gcov))); ## Calculate t statistics corresponding to the comparisons defined in L [mean_diff, sed, t] = tValue (gmeans, gcov, L); # z-statistic (not t) ## Calculate correlation matrix vcov = L * gcov * L'; R = cov2corr (vcov); ## Calculate degrees of freedom from number of groups if (isempty (DFE)) DFE = inf; # this is a z-statistic so infinite degrees of freedom endif otherwise error (strcat (sprintf ("multcompare: the STATS structure from %s", ... STATS.source), [" is not currently supported"])) endswitch ## The test specific code above needs to create the following variables in ## order to proceed with the remainder of the function tasks ## - Ng: number of groups involved in comparisons ## - M: Ng-by-2 matrix of group means (col 1) and standard errors (col 2) ## - Np: number of comparisons (pairs of groups being compaired) ## - pairs: Np-by-2 matrix of numeric group IDs - each row is a comparison ## - R: correlation matrix for the requested comparisons ## - sed: vector containing SE of the difference for each comparisons ## - t: vector containing t for the difference relating to each comparisons ## - DFE: residual/error degrees of freedom ## - GNAMES: a cell array containing the names of the groups being compared ## Create matrix of comparisons and calculate confidence intervals and ## multiplicity adjusted p-values for the comparisons. C = zeros (Np, 8); C(:,1:2) = pairs; C(:,4) = (M(pairs(:, 1),1) - M(pairs(:, 2),1)); C(:,7) = t; # Unlike Matlab, we include the t statistic C(:,8) = DFE; # Unlike Matlab, we include the degrees of freedom if (any (isinf (DFE))) p = 2 * (1 - normcdf (abs (t))); else p = 2 * (1 - tcdf (abs (t), DFE)); endif [C(:,6), critval, C(:,8)] = feval (CTYPE, p, t, Ng, DFE, R, ALPHA); C(:,3) = C(:,4) - sed .* critval; C(:,5) = C(:,4) + sed .* critval; ## Calculate confidence intervals of the estimated marginal means with ## central coverage such that the intervals start to overlap where the ## difference reaches a two-tailed p-value of ALPHA. When ALPHA is 0.05, ## central coverage is approximately 83.4% if (! isscalar(DFE)) # Upper bound critval (corresponding to lower bound DFE) critval = max (critval); endif M(:,3) = M(:,1) - M(:,2) .* critval / sqrt(2); M(:,4) = M(:,1) + M(:,2) .* critval / sqrt(2); ## If requested, plot graph of the difference means for each comparison ## with central coverage of confidence intervals at 100*(1-alpha)% switch (lower (DISPLAY)) case {'on',true} H = figure; plot ([0; 0], [0; Np + 1]',"k:"); # Plot vertical dashed line at 0 effect set (gca, "Ydir", "reverse") # Flip y-axis direction ylim ([0.5, Np + 0.5]); # Set y-axis limits hold on # Plot on the same axis for j = 1:Np if (C(j,6) < ALPHA) ## Plot marker for the difference in means plot (C(j,4), j,"or","MarkerFaceColor", "r"); ## Plot line for each confidence interval plot ([C(j,3), C(j,5)], j * ones(2,1), "r-"); else ## Plot marker for the difference in means plot (C(j,4), j,"ob","MarkerFaceColor", "b"); ## Plot line for each confidence interval plot ([C(j,3), C(j,5)], j * ones(2,1), "b-"); endif endfor hold off xlabel (sprintf ("%g%% confidence interval for the difference",... 100 * (1 - ALPHA))); ylabel ("Row number in matrix of comparisons (C)"); case {'off',false} H = []; endswitch ## Print multcompare table on screen if no output argument was requested if (nargout == 0 || strcmp (DISPLAY, "on")) printf ("\n %s Multiple Comparison (Post Hoc) Test for %s\n\n", ... upper (CTYPE), upper (STATS.source)); header = strcat (["Group ID Group ID LBoundDiff EstimatedDiff"],... [" UBoundDiff p-value\n"], ... ["-------------------------------------------------"],... ["---------------------\n"]); printf ("%s", header); for j = 1:Np if (C(j,6) < 0.001) printf ("%5i %5i %10.3f %10.3f %10.3f <.001\n",... C(j,1), C(j,2), C(j,3), C(j,4), C(j,5)); elseif (C(j,6) < 0.9995) printf ("%5i %5i %10.3f %10.3f %10.3f .%03u\n",... C(j,1), C(j,2), C(j,3), C(j,4), C(j,5), round (C(j,6) * 1e+03)); else printf ("%5i %5i %10.3f %10.3f %10.3f 1.000\n",... C(j,1), C(j,2), C(j,3), C(j,4), C(j,5)); endif endfor printf ("\n"); endif endfunction ## Posthoc comparisons function [pairs, L, R] = pairwise (Ng) ## Create pairs matrix for pairwise comparisons gid = [1:Ng]'; # Create numeric group ID A = ones (Ng, 1) * gid'; B = tril (gid * ones(1, Ng),-1); pairs = [A(:), B(:)]; ridx = (pairs(:, 2) == 0); pairs(ridx, :) = []; ## Calculate correlation matrix (required for CTYPE "mvt") Np = size (pairs, 1); L = zeros (Np, Ng); for j = 1:Np L(j, pairs(j,:)) = [1,-1]; # Hypothesis matrix endfor R = corr (L'); # Correlation matrix endfunction function [pairs, L, R] = trt_vs_ctrl (Ng, REF) ## Create pairs matrix for comparisons with control (REF) gid = [1:Ng]'; # Create numeric group ID pairs = zeros (Ng - 1, 2); pairs(:, 1) = REF; pairs(:, 2) = gid(gid != REF); ## Calculate correlation matrix (required for CTYPE "mvt") Np = size (pairs, 1); L = zeros (Np, Ng); for j = 1:Np L(j, pairs(j,:)) = [1,-1]; # Hypothesis matrix endfor R = corr (L'); # Correlation matrix endfunction function [mn, se, t] = tValue (gmeans, gcov, L) ## Calculate means, standard errors and t (or z) statistics ## corresponding to the comparisons defined in L. mn = sum (L * diag (gmeans), 2); se = sqrt (diag (L * gcov * L')); t = mn ./ se; endfunction function R = cov2corr (vcov) ## Convert covariance matrix to correlation matrix sed = sqrt (diag (vcov)); R = vcov ./ (sed * sed'); R = (R + R') / 2; # This step ensures that the matrix is positive definite endfunction ## Methods to control family-wise error rate in multiple comparisons function [padj, critval, dfe] = scheffe (p, t, Ng, dfe, R, ALPHA) ## Calculate the p-value if (isinf (dfe)) padj = 1 - chi2cdf (t.^2, Ng - 1); else padj = 1 - fcdf ((t.^2) / (Ng - 1), Ng - 1, dfe); endif ## Calculate critical value at Scheffe-adjusted ALPHA level if (isinf (dfe)) tmp = chi2inv (1 - ALPHA, Ng - 1) / (Ng - 1); else tmp = finv (1 - ALPHA, Ng - 1, dfe); end critval = sqrt ((Ng - 1) * tmp); endfunction function [padj, critval, dfe] = bonferroni (p, t, Ng, dfe, R, ALPHA) ## Bonferroni procedure Np = numel (p); padj = min (p * Np, 1.0); ## If requested, calculate critical value at Bonferroni-adjusted ALPHA level if (nargout > 1) critval = tinv (1 - ALPHA / Np * 0.5, dfe); endif endfunction function [padj, critval, dfe] = mvt (p, t, Ng, dfe, R, ALPHA) ## Monte Carlo simulation of the maximum test statistic in random samples ## generated from a multivariate t distribution. This method accounts for ## correlations among comparisons. This method simulates Tukey's test in the ## case of pairwise comparisons or Dunnett's tests in the case of trt_vs_ctrl. ## The "mvt" method is equivalent to methods used in the following R packages: ## - emmeans: the "mvt" adjust method in functions within emmeans ## - glht: the "single-step" adjustment in the multcomp.function ## Lower bound for error degrees of freedom to ensure type 1 error rate isn't ## exceeded for any test if (! isscalar(dfe)) dfe = max (1, round (min (dfe))); fprintf ("Note: df set to %u (lower bound)\n", dfe); endif ## Check if we can use parallel processing to accelerate computations pat = '^parallel'; software = pkg('list'); names = cellfun (@(S) S.name, software, 'UniformOutput', false); status = cellfun (@(S) S.loaded, software, 'UniformOutput', false); index = find (! cellfun (@isempty, regexpi (names, pat))); if (! isempty (index)) if (logical (status{index})) PARALLEL = true; else PARALLEL = false; endif else PARALLEL = false; endif ## Generate the distribution of (correlated) t statistics under the null, and ## calculate the maximum test statistic for each random sample. Computations ## are performed in chunks to prevent memory issues when the number of ## comparisons is large. chunkSize = 1000; numChunks = 1000; nsim = chunkSize * numChunks; if (isinf (dfe)) # Multivariate z-statistics func = @(jnk) max (abs (mvnrnd (0, R, chunkSize)'), [], 1); else # Multivariate t-statistics func = @(jnk) max (abs (mvtrnd (R, dfe, chunkSize)'), [], 1); endif if (PARALLEL) maxT = cell2mat (parcellfun (nproc, func, ... cell (1, numChunks), 'UniformOutput', false)); else maxT = cell2mat (cellfun (func, cell (1, numChunks), 'UniformOutput', false)); endif ## Calculate multiplicity adjusted p-values (two-tailed) padj = max (sum (bsxfun (@ge, maxT, abs (t)), 2) / nsim, nsim^-1); ## Calculate critical value adjusted by the maxT procedure critval = quantile (maxT, 1 - ALPHA); endfunction function [padj, critval, dfe] = holm (p, t, Ng, dfe, R, ALPHA) ## Holm's step-down Bonferroni procedure ## Order raw p-values [ps, idx] = sort (p, "ascend"); Np = numel (ps); ## Implement Holm's step-down Bonferroni procedure padj = nan (Np,1); padj(1) = Np * ps(1); for i = 2:Np padj(i) = max (padj(i - 1), (Np - i + 1) * ps(i)); endfor ## Reorder the adjusted p-values to match the order of the original p-values [jnk, original_order] = sort (idx, "ascend"); padj = padj(original_order); ## Truncate adjusted p-values to 1.0 padj(padj>1) = 1; ## If requested, calculate critical value at ALPHA ## No adjustment to confidence interval coverage if (nargout > 1) critval = tinv (1 - ALPHA / 2, dfe); endif endfunction function [padj, critval, dfe] = hochberg (p, t, Ng, dfe, R, ALPHA) ## Hochberg's step-up Bonferroni procedure ## Order raw p-values [ps, idx] = sort (p, "ascend"); Np = numel (ps); ## Implement Hochberg's step-down Bonferroni procedure padj = nan (Np,1); padj(Np) = ps(Np); for j = 1:Np-1 i = Np - j; padj(i) = min (padj(i + 1), (Np -i + 1) * ps(i)); endfor ## Reorder the adjusted p-values to match the order of the original p-values [jnk, original_order] = sort (idx, "ascend"); padj = padj(original_order); ## Truncate adjusted p-values to 1.0 padj(padj>1) = 1; ## If requested, calculate critical value at ALPHA ## No adjustment to confidence interval coverage if (nargout > 1) critval = tinv (1 - ALPHA / 2, dfe); endif endfunction function [padj, critval, dfe] = fdr (p, t, Ng, dfe, R, ALPHA) ## Benjamini-Hochberg procedure to control the false discovery rate (FDR) ## This procedure does not control the family-wise error rate ## Order raw p-values [ps, idx] = sort (p, "ascend"); Np = numel (ps); ## Initialize padj = nan (Np,1); alpha = nan (Np,1); ## Benjamini-Hochberg step-up procedure to control the false discovery rate padj = nan (Np,1); padj(Np) = ps(Np); for j = 1:Np-1 i = Np - j; padj(i) = min (padj(i + 1), Np / i * ps(i)); endfor ## Reorder the adjusted p-values to match the order of the original p-values [jnk, original_order] = sort (idx, "ascend"); padj = padj(original_order); ## Truncate adjusted p-values to 1.0 padj(padj>1) = 1; ## If requested, calculate critical value at ALPHA ## No adjustment to confidence interval coverage if (nargout > 1) critval = tinv (1 - ALPHA / 2, dfe); endif endfunction function [padj, critval, dfe] = lsd (p, t, Ng, dfe, R, ALPHA) ## Fisher's Least Significant Difference ## No control of the type I error rate across multiple comparisons padj = p; ## Calculate critical value at ALPHA ## No adjustment to confidence interval coverage critval = tinv (1 - ALPHA / 2, dfe); endfunction %!demo %! %! ## Demonstration using balanced one-way ANOVA from anova1 %! %! x = ones (50, 4) .* [-2, 0, 1, 5]; %! randn ("seed", 1); # for reproducibility %! x = x + normrnd (0, 2, 50, 4); %! groups = {"A", "B", "C", "D"}; %! [p, tbl, stats] = anova1 (x, groups, "off"); %! multcompare (stats); %!demo %! %! ## Demonstration using unbalanced one-way ANOVA example from anovan %! %! dv = [ 8.706 10.362 11.552 6.941 10.983 10.092 6.421 14.943 15.931 ... %! 22.968 18.590 16.567 15.944 21.637 14.492 17.965 18.851 22.891 ... %! 22.028 16.884 17.252 18.325 25.435 19.141 21.238 22.196 18.038 ... %! 22.628 31.163 26.053 24.419 32.145 28.966 30.207 29.142 33.212 ... %! 25.694 ]'; %! g = [1 1 1 1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 3 ... %! 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5]'; %! %! [P,ATAB, STATS] = anovan (dv, g, "varnames", "score", "display", "off"); %! %! [C, M, H, GNAMES] = multcompare (STATS, "dim", 1, "ctype", "holm", ... %! "ControlGroup", 1, "display", "on") %! %!demo %! %! ## Demonstration using factorial ANCOVA example from anovan %! %! score = [95.6 82.2 97.2 96.4 81.4 83.6 89.4 83.8 83.3 85.7 ... %! 97.2 78.2 78.9 91.8 86.9 84.1 88.6 89.8 87.3 85.4 ... %! 81.8 65.8 68.1 70.0 69.9 75.1 72.3 70.9 71.5 72.5 ... %! 84.9 96.1 94.6 82.5 90.7 87.0 86.8 93.3 87.6 92.4 ... %! 100. 80.5 92.9 84.0 88.4 91.1 85.7 91.3 92.3 87.9 ... %! 91.7 88.6 75.8 75.7 75.3 82.4 80.1 86.0 81.8 82.5]'; %! treatment = {"yes" "yes" "yes" "yes" "yes" "yes" "yes" "yes" "yes" "yes" ... %! "yes" "yes" "yes" "yes" "yes" "yes" "yes" "yes" "yes" "yes" ... %! "yes" "yes" "yes" "yes" "yes" "yes" "yes" "yes" "yes" "yes" ... %! "no" "no" "no" "no" "no" "no" "no" "no" "no" "no" ... %! "no" "no" "no" "no" "no" "no" "no" "no" "no" "no" ... %! "no" "no" "no" "no" "no" "no" "no" "no" "no" "no"}'; %! exercise = {"lo" "lo" "lo" "lo" "lo" "lo" "lo" "lo" "lo" "lo" ... %! "mid" "mid" "mid" "mid" "mid" "mid" "mid" "mid" "mid" "mid" ... %! "hi" "hi" "hi" "hi" "hi" "hi" "hi" "hi" "hi" "hi" ... %! "lo" "lo" "lo" "lo" "lo" "lo" "lo" "lo" "lo" "lo" ... %! "mid" "mid" "mid" "mid" "mid" "mid" "mid" "mid" "mid" "mid" ... %! "hi" "hi" "hi" "hi" "hi" "hi" "hi" "hi" "hi" "hi"}'; %! age = [59 65 70 66 61 65 57 61 58 55 62 61 60 59 55 57 60 63 62 57 ... %! 58 56 57 59 59 60 55 53 55 58 68 62 61 54 59 63 60 67 60 67 ... %! 75 54 57 62 65 60 58 61 65 57 56 58 58 58 52 53 60 62 61 61]'; %! %! [P, ATAB, STATS] = anovan (score, {treatment, exercise, age}, "model", ... %! [1 0 0; 0 1 0; 0 0 1; 1 1 0], "continuous", 3, ... %! "sstype", "h", "display", "off", "contrasts", ... %! {"simple","poly",""}); %! %! [C, M, H, GNAMES] = multcompare (STATS, "dim", [1 2], "ctype", "holm", ... %! "display", "on") %! %!demo %! %! ## Demonstration using one-way ANOVA from anovan, with fit by weighted least %! ## squares to account for heteroskedasticity. %! %! g = [1, 1, 1, 1, 1, 1, 1, 1, ... %! 2, 2, 2, 2, 2, 2, 2, 2, ... %! 3, 3, 3, 3, 3, 3, 3, 3]'; %! %! y = [13, 16, 16, 7, 11, 5, 1, 9, ... %! 10, 25, 66, 43, 47, 56, 6, 39, ... %! 11, 39, 26, 35, 25, 14, 24, 17]'; %! %! [P,ATAB,STATS] = anovan(y, g, "display", "off"); %! fitted = STATS.X * STATS.coeffs(:,1); # fitted values %! b = polyfit (fitted, abs (STATS.resid), 1); %! v = polyval (b, fitted); # Variance as a function of the fitted values %! [P,ATAB,STATS] = anovan (y, g, "weights", v.^-1, "display", "off"); %! [C, M] = multcompare (STATS, "display", "on", "ctype", "mvt") %!demo %! %! ## Demonstration of p-value adjustments to control the false discovery rate %! ## Data from Westfall (1997) JASA. 92(437):299-306 %! %! p = [.005708; .023544; .024193; .044895; ... %! .048805; .221227; .395867; .693051; .775755]; %! %! padj = multcompare(p,'ctype','fdr') %!test %! %! ## Tests using unbalanced one-way ANOVA example from anovan and anova1 %! %! ## Test for anovan - compare pairwise comparisons with matlab for CTYPE "lsd" %! %! dv = [ 8.706 10.362 11.552 6.941 10.983 10.092 6.421 14.943 15.931 ... %! 22.968 18.590 16.567 15.944 21.637 14.492 17.965 18.851 22.891 ... %! 22.028 16.884 17.252 18.325 25.435 19.141 21.238 22.196 18.038 ... %! 22.628 31.163 26.053 24.419 32.145 28.966 30.207 29.142 33.212 ... %! 25.694 ]'; %! g = [1 1 1 1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 3 ... %! 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5]'; %! %! [P, ATAB, STATS] = anovan (dv, g, "varnames", "score", "display", "off"); %! [C, M, H, GNAMES] = multcompare (STATS, "dim", 1, "ctype", "lsd", ... %! "display", "off"); %! assert (C(1,6), 2.85812420217898e-05, 1e-09); %! assert (C(2,6), 5.22936741204085e-07, 1e-09); %! assert (C(3,6), 2.12794763209146e-08, 1e-09); %! assert (C(4,6), 7.82091664406946e-15, 1e-09); %! assert (C(5,6), 0.546591417210693, 1e-09); %! assert (C(6,6), 0.0845897945254446, 1e-09); %! assert (C(7,6), 9.47436557975328e-08, 1e-09); %! assert (C(8,6), 0.188873478781067, 1e-09); %! assert (C(9,6), 4.08974010364197e-08, 1e-09); %! assert (C(10,6), 4.44427348175241e-06, 1e-09); %! assert (M(1,1), 10, 1e-09); %! assert (M(2,1), 18, 1e-09); %! assert (M(3,1), 19, 1e-09); %! assert (M(4,1), 21.0001428571429, 1e-09); %! assert (M(5,1), 29.0001111111111, 1e-09); %! assert (M(1,2), 1.0177537954095, 1e-09); %! assert (M(2,2), 1.28736803631001, 1e-09); %! assert (M(3,2), 1.0177537954095, 1e-09); %! assert (M(4,2), 1.0880245732889, 1e-09); %! assert (M(5,2), 0.959547480416536, 1e-09); %! %! ## Compare "fdr" adjusted p-values to those obtained using p.adjust in R %! %! [C, M, H, GNAMES] = multcompare (STATS, "dim", 1, "ctype", "fdr", ... %! "display", "off"); %! assert (C(1,6), 4.08303457454140e-05, 1e-09); %! assert (C(2,6), 1.04587348240817e-06, 1e-09); %! assert (C(3,6), 1.06397381604573e-07, 1e-09); %! assert (C(4,6), 7.82091664406946e-14, 1e-09); %! assert (C(5,6), 5.46591417210693e-01, 1e-09); %! assert (C(6,6), 1.05737243156806e-01, 1e-09); %! assert (C(7,6), 2.36859139493832e-07, 1e-09); %! assert (C(8,6), 2.09859420867852e-01, 1e-09); %! assert (C(9,6), 1.36324670121399e-07, 1e-09); %! assert (C(10,6), 7.40712246958735e-06, 1e-09); %! %! ## Compare "hochberg" adjusted p-values to those obtained using p.adjust in R %! %! [C, M, H, GNAMES] = multcompare (STATS, "dim", 1, "ctype", "hochberg", ... %! "display", "off"); %! assert (C(1,6), 1.14324968087159e-04, 1e-09); %! assert (C(2,6), 3.13762044722451e-06, 1e-09); %! assert (C(3,6), 1.91515286888231e-07, 1e-09); %! assert (C(4,6), 7.82091664406946e-14, 1e-09); %! assert (C(5,6), 5.46591417210693e-01, 1e-09); %! assert (C(6,6), 2.53769383576334e-01, 1e-09); %! assert (C(7,6), 6.63205590582730e-07, 1e-09); %! assert (C(8,6), 3.77746957562134e-01, 1e-09); %! assert (C(9,6), 3.27179208291358e-07, 1e-09); %! assert (C(10,6), 2.22213674087620e-05, 1e-09); %! %! ## Compare "holm" adjusted p-values to those obtained using p.adjust in R %! %! [C, M, H, GNAMES] = multcompare (STATS, "dim", 1, "ctype", "holm", ... %! "display", "off"); %! assert (C(1,6), 1.14324968087159e-04, 1e-09); %! assert (C(2,6), 3.13762044722451e-06, 1e-09); %! assert (C(3,6), 1.91515286888231e-07, 1e-09); %! assert (C(4,6), 7.82091664406946e-14, 1e-09); %! assert (C(5,6), 5.46591417210693e-01, 1e-09); %! assert (C(6,6), 2.53769383576334e-01, 1e-09); %! assert (C(7,6), 6.63205590582730e-07, 1e-09); %! assert (C(8,6), 3.77746957562134e-01, 1e-09); %! assert (C(9,6), 3.27179208291358e-07, 1e-09); %! assert (C(10,6), 2.22213674087620e-05, 1e-09); %! %! ## Compare "scheffe" adjusted p-values to those obtained using 'scheffe' in Matlab %! %! [C, M, H, GNAMES] = multcompare (STATS, "dim", 1, "ctype", "scheffe", ... %! "display", "off"); %! assert (C(1,6), 0.00108105386141085, 1e-09); %! assert (C(2,6), 2.7779386789517e-05, 1e-09); %! assert (C(3,6), 1.3599854038198e-06, 1e-09); %! assert (C(4,6), 7.58830197867751e-13, 1e-09); %! assert (C(5,6), 0.984039948220281, 1e-09); %! assert (C(6,6), 0.539077018557706, 1e-09); %! assert (C(7,6), 5.59475764460574e-06, 1e-09); %! assert (C(8,6), 0.771173490574105, 1e-09); %! assert (C(9,6), 2.52838425729905e-06, 1e-09); %! assert (C(10,6), 0.000200719143889168, 1e-09); %! %! ## Compare "bonferroni" adjusted p-values to those obtained using p.adjust in R %! %! [C, M, H, GNAMES] = multcompare (STATS, "dim", 1, "ctype", "bonferroni", ... %! "display", "off"); %! assert (C(1,6), 2.85812420217898e-04, 1e-09); %! assert (C(2,6), 5.22936741204085e-06, 1e-09); %! assert (C(3,6), 2.12794763209146e-07, 1e-09); %! assert (C(4,6), 7.82091664406946e-14, 1e-09); %! assert (C(5,6), 1.00000000000000e+00, 1e-09); %! assert (C(6,6), 8.45897945254446e-01, 1e-09); %! assert (C(7,6), 9.47436557975328e-07, 1e-09); %! assert (C(8,6), 1.00000000000000e+00, 1e-09); %! assert (C(9,6), 4.08974010364197e-07, 1e-09); %! assert (C(10,6), 4.44427348175241e-05, 1e-09); %! %! ## Test for anova1 ("equal")- comparison of results from Matlab %! %! [P, ATAB, STATS] = anova1 (dv, g, "off", "equal"); %! [C, M, H, GNAMES] = multcompare (STATS, "ctype", "lsd", "display", "off"); %! assert (C(1,6), 2.85812420217898e-05, 1e-09); %! assert (C(2,6), 5.22936741204085e-07, 1e-09); %! assert (C(3,6), 2.12794763209146e-08, 1e-09); %! assert (C(4,6), 7.82091664406946e-15, 1e-09); %! assert (C(5,6), 0.546591417210693, 1e-09); %! assert (C(6,6), 0.0845897945254446, 1e-09); %! assert (C(7,6), 9.47436557975328e-08, 1e-09); %! assert (C(8,6), 0.188873478781067, 1e-09); %! assert (C(9,6), 4.08974010364197e-08, 1e-09); %! assert (C(10,6), 4.44427348175241e-06, 1e-09); %! assert (M(1,1), 10, 1e-09); %! assert (M(2,1), 18, 1e-09); %! assert (M(3,1), 19, 1e-09); %! assert (M(4,1), 21.0001428571429, 1e-09); %! assert (M(5,1), 29.0001111111111, 1e-09); %! assert (M(1,2), 1.0177537954095, 1e-09); %! assert (M(2,2), 1.28736803631001, 1e-09); %! assert (M(3,2), 1.0177537954095, 1e-09); %! assert (M(4,2), 1.0880245732889, 1e-09); %! assert (M(5,2), 0.959547480416536, 1e-09); %! %! ## Test for anova1 ("unequal") - comparison with results from GraphPad Prism 8 %! [P, ATAB, STATS] = anova1 (dv, g, "off", "unequal"); %! [C, M, H, GNAMES] = multcompare (STATS, "ctype", "lsd", "display", "off"); %! assert (C(1,6), 0.001247025266382, 1e-09); %! assert (C(2,6), 0.000018037115146, 1e-09); %! assert (C(3,6), 0.000002974595187, 1e-09); %! assert (C(4,6), 0.000000000786046, 1e-09); %! assert (C(5,6), 0.5693192886650109, 1e-09); %! assert (C(6,6), 0.110501699029776, 1e-09); %! assert (C(7,6), 0.000131226488700, 1e-09); %! assert (C(8,6), 0.1912101409715992, 1e-09); %! assert (C(9,6), 0.000005385256394, 1e-09); %! assert (C(10,6), 0.000074089106171, 1e-09); %!test %! %! ## Test for anova2 ("interaction") - comparison with results from Matlab for column effect %! popcorn = [5.5, 4.5, 3.5; 5.5, 4.5, 4.0; 6.0, 4.0, 3.0; ... %! 6.5, 5.0, 4.0; 7.0, 5.5, 5.0; 7.0, 5.0, 4.5]; %! [P, ATAB, STATS] = anova2 (popcorn, 3, "off"); %! [C, M, H, GNAMES] = multcompare (STATS, "estimate", "column",... %! "ctype", "lsd", "display", "off"); %! assert (C(1,6), 1.49311100811177e-05, 1e-09); %! assert (C(2,6), 2.20506904243535e-07, 1e-09); %! assert (C(3,6), 0.00449897860490058, 1e-09); %! assert (M(1,1), 6.25, 1e-09); %! assert (M(2,1), 4.75, 1e-09); %! assert (M(3,1), 4, 1e-09); %! assert (M(1,2), 0.152145154862547, 1e-09); %! assert (M(2,2), 0.152145154862547, 1e-09); %! assert (M(3,2), 0.152145154862547, 1e-09); %!test %! %! ## Test for anova2 ("linear") - comparison with results from GraphPad Prism 8 %! words = [10 13 13; 6 8 8; 11 14 14; 22 23 25; 16 18 20; ... %! 15 17 17; 1 1 4; 12 15 17; 9 12 12; 8 9 12]; %! [P, ATAB, STATS] = anova2 (words, 1, "off", "linear"); %! [C, M, H, GNAMES] = multcompare (STATS, "estimate", "column",... %! "ctype", "lsd", "display", "off"); %! assert (C(1,6), 0.000020799832702, 1e-09); %! assert (C(2,6), 0.000000035812410, 1e-09); %! assert (C(3,6), 0.003038942449215, 1e-09); %!test %! %! ## Test for anova2 ("nested") - comparison with results from GraphPad Prism 8 %! data = [4.5924 7.3809 21.322; -0.5488 9.2085 25.0426; ... %! 6.1605 13.1147 22.66; 2.3374 15.2654 24.1283; ... %! 5.1873 12.4188 16.5927; 3.3579 14.3951 10.2129; ... %! 6.3092 8.5986 9.8934; 3.2831 3.4945 10.0203]; %! [P, ATAB, STATS] = anova2 (data, 4, "off", "nested"); %! [C, M, H, GNAMES] = multcompare (STATS, "estimate", "column",... %! "ctype", "lsd", "display", "off"); %! assert (C(1,6), 0.261031111511073, 1e-09); %! assert (C(2,6), 0.065879755907745, 1e-09); %! assert (C(3,6), 0.241874613529270, 1e-09); %!shared visibility_setting %! visibility_setting = get (0, "DefaultFigureVisible"); %!test %! set (0, "DefaultFigureVisible", "off"); %! %! ## Test for kruskalwallis - comparison with results from MATLAB %! data = [3,2,4; 5,4,4; 4,2,4; 4,2,4; 4,1,5; ... %! 4,2,3; 4,3,5; 4,2,4; 5,2,4; 5,3,3]; %! group = [1:3] .* ones (10,3); %! [P, ATAB, STATS] = kruskalwallis (data(:), group(:), "off"); %! C = multcompare (STATS, "ctype", "lsd", "display", "off"); %! assert (C(1,6), 0.000163089828959986, 1e-09); %! assert (C(2,6), 0.630298044801257, 1e-09); %! assert (C(3,6), 0.00100567660695682, 1e-09); %! C = multcompare (STATS, "ctype", "bonferroni", "display", "off"); %! assert (C(1,6), 0.000489269486879958, 1e-09); %! assert (C(2,6), 1, 1e-09); %! assert (C(3,6), 0.00301702982087047, 1e-09); %! C = multcompare(STATS, "ctype", "scheffe", "display", "off"); %! assert (C(1,6), 0.000819054880289573, 1e-09); %! assert (C(2,6), 0.890628039849261, 1e-09); %! assert (C(3,6), 0.00447816059021654, 1e-09); %! set (0, "DefaultFigureVisible", visibility_setting); %!test %! set (0, "DefaultFigureVisible", "off"); %! ## Test for friedman - comparison with results from MATLAB %! popcorn = [5.5, 4.5, 3.5; 5.5, 4.5, 4.0; 6.0, 4.0, 3.0; ... %! 6.5, 5.0, 4.0; 7.0, 5.5, 5.0; 7.0, 5.0, 4.5]; %! [P, ATAB, STATS] = friedman (popcorn, 3, "off"); %! C = multcompare(STATS, "ctype", "lsd", "display", "off"); %! assert (C(1,6), 0.227424558028569, 1e-09); %! assert (C(2,6), 0.0327204848315735, 1e-09); %! assert (C(3,6), 0.353160353315988, 1e-09); %! C = multcompare(STATS, "ctype", "bonferroni", "display", "off"); %! assert (C(1,6), 0.682273674085708, 1e-09); %! assert (C(2,6), 0.0981614544947206, 1e-09); %! assert (C(3,6), 1, 1e-09); %! C = multcompare(STATS, "ctype", "scheffe", "display", "off"); %! assert (C(1,6), 0.482657360384373, 1e-09); %! assert (C(2,6), 0.102266573027672, 1e-09); %! assert (C(3,6), 0.649836502233148, 1e-09); %! set (0, "DefaultFigureVisible", visibility_setting); %!test %! set (0, "DefaultFigureVisible", "off"); %! ## Test for fitlm - same comparisons as for first anovan example %! y = [ 8.706 10.362 11.552 6.941 10.983 10.092 6.421 14.943 15.931 ... %! 22.968 18.590 16.567 15.944 21.637 14.492 17.965 18.851 22.891 ... %! 22.028 16.884 17.252 18.325 25.435 19.141 21.238 22.196 18.038 ... %! 22.628 31.163 26.053 24.419 32.145 28.966 30.207 29.142 33.212 ... %! 25.694 ]'; %! X = [1 1 1 1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5]'; %! [TAB,STATS] = fitlm (X,y,"linear","categorical",1,"display","off",... %! "contrasts","simple"); %! [C, M] = multcompare(STATS, "ctype", "lsd", "display", "off"); %! assert (C(1,6), 2.85812420217898e-05, 1e-09); %! assert (C(2,6), 5.22936741204085e-07, 1e-09); %! assert (C(3,6), 2.12794763209146e-08, 1e-09); %! assert (C(4,6), 7.82091664406946e-15, 1e-09); %! assert (C(5,6), 0.546591417210693, 1e-09); %! assert (C(6,6), 0.0845897945254446, 1e-09); %! assert (C(7,6), 9.47436557975328e-08, 1e-09); %! assert (C(8,6), 0.188873478781067, 1e-09); %! assert (C(9,6), 4.08974010364197e-08, 1e-09); %! assert (C(10,6), 4.44427348175241e-06, 1e-09); %! assert (M(1,1), 10, 1e-09); %! assert (M(2,1), 18, 1e-09); %! assert (M(3,1), 19, 1e-09); %! assert (M(4,1), 21.0001428571429, 1e-09); %! assert (M(5,1), 29.0001111111111, 1e-09); %! assert (M(1,2), 1.0177537954095, 1e-09); %! assert (M(2,2), 1.28736803631001, 1e-09); %! assert (M(3,2), 1.0177537954095, 1e-09); %! assert (M(4,2), 1.0880245732889, 1e-09); %! assert (M(5,2), 0.959547480416536, 1e-09); %! set (0, "DefaultFigureVisible", visibility_setting); %!test %! ## Test p-value adjustments compared to R stats package function p.adjust %! ## Data from Westfall (1997) JASA. 92(437):299-306 %! p = [.005708; .023544; .024193; .044895; ... %! .048805; .221227; .395867; .693051; .775755]; %! padj = multcompare (p); %! assert (padj(1), 0.051372, 1e-06); %! assert (padj(2), 0.188352, 1e-06); %! assert (padj(3), 0.188352, 1e-06); %! assert (padj(4), 0.269370, 1e-06); %! assert (padj(5), 0.269370, 1e-06); %! assert (padj(6), 0.884908, 1e-06); %! assert (padj(7), 1.000000, 1e-06); %! assert (padj(8), 1.000000, 1e-06); %! assert (padj(9), 1.000000, 1e-06); %! padj = multcompare(p,'ctype','holm'); %! assert (padj(1), 0.051372, 1e-06); %! assert (padj(2), 0.188352, 1e-06); %! assert (padj(3), 0.188352, 1e-06); %! assert (padj(4), 0.269370, 1e-06); %! assert (padj(5), 0.269370, 1e-06); %! assert (padj(6), 0.884908, 1e-06); %! assert (padj(7), 1.000000, 1e-06); %! assert (padj(8), 1.000000, 1e-06); %! assert (padj(9), 1.000000, 1e-06); %! padj = multcompare(p,'ctype','hochberg'); %! assert (padj(1), 0.051372, 1e-06); %! assert (padj(2), 0.169351, 1e-06); %! assert (padj(3), 0.169351, 1e-06); %! assert (padj(4), 0.244025, 1e-06); %! assert (padj(5), 0.244025, 1e-06); %! assert (padj(6), 0.775755, 1e-06); %! assert (padj(7), 0.775755, 1e-06); %! assert (padj(8), 0.775755, 1e-06); %! assert (padj(9), 0.775755, 1e-06); %! padj = multcompare(p,'ctype','fdr'); %! assert (padj(1), 0.0513720, 1e-07); %! assert (padj(2), 0.0725790, 1e-07); %! assert (padj(3), 0.0725790, 1e-07); %! assert (padj(4), 0.0878490, 1e-07); %! assert (padj(5), 0.0878490, 1e-07); %! assert (padj(6), 0.3318405, 1e-07); %! assert (padj(7), 0.5089719, 1e-07); %! assert (padj(8), 0.7757550, 1e-07); %! assert (padj(9), 0.7757550, 1e-07); statistics-release-1.6.3/inst/nanmax.m000066400000000000000000000041131456127120000177430ustar00rootroot00000000000000## Copyright (C) 2001 Paul Kienzle ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {[@var{v}, @var{idx}] =} nanmax (@var{X}) ## @deftypefnx {statistics} {[@var{v}, @var{idx}] =} nanmax (@var{X}, @var{Y}) ## ## Find the maximal element while ignoring NaN values. ## ## @code{nanmax} is identical to the @code{max} function except that NaN values ## are ignored. If all values in a column are NaN, the maximum is ## returned as NaN rather than []. ## ## @seealso{max, nansum, nanmin} ## @end deftypefn function [v, idx] = nanmax (X, Y, DIM) if nargin < 1 || nargin > 3 print_usage; elseif nargin == 1 || (nargin == 2 && isempty(Y)) nanvals = isnan(X); X(nanvals) = -Inf; [v, idx] = max (X); v(all(nanvals)) = NaN; elseif (nargin == 3 && isempty(Y)) nanvals = isnan(X); X(nanvals) = -Inf; [v, idx] = max (X,[],DIM); v(all(nanvals,DIM)) = NaN; else Xnan = isnan(X); Ynan = isnan(Y); X(Xnan) = -Inf; Y(Ynan) = -Inf; if (nargin == 3) [v, idx] = max(X,Y,DIM); else [v, idx] = max(X,Y); endif v(Xnan & Ynan) = NaN; endif endfunction %!assert (nanmax ([2 4 NaN 7]), 7) %!assert (nanmax ([2 4 NaN Inf]), Inf) %!assert (nanmax ([1 NaN 3; NaN 5 6; 7 8 NaN]), [7, 8, 6]) %!assert (nanmax ([1 NaN 3; NaN 5 6; 7 8 NaN]'), [3, 6, 8]) %!assert (nanmax (single ([1 NaN 3; NaN 5 6; 7 8 NaN])), single ([7 8 6])) statistics-release-1.6.3/inst/nanmin.m000066400000000000000000000043061456127120000177450ustar00rootroot00000000000000## Copyright (C) 2001 Paul Kienzle ## Copyright (C) 2003 Alois Schloegl ## Copyright (C) 2022 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {[@var{v}, @var{idx}] =} nanmin (@var{X}) ## @deftypefnx {statistics} {[@var{v}, @var{idx}] =} nanmin (@var{X}, @var{Y}) ## ## Find the minimal element while ignoring NaN values. ## ## @code{nanmin} is identical to the @code{min} function except that NaN values ## are ignored. If all values in a column are NaN, the minimum is ## returned as NaN rather than []. ## ## @seealso{min, nansum, nanmax} ## @end deftypefn function [v, idx] = nanmin (X, Y, DIM) if nargin < 1 || nargin > 3 print_usage; elseif nargin == 1 || (nargin == 2 && isempty(Y)) nanvals = isnan(X); X(nanvals) = Inf; [v, idx] = min (X); v(all(nanvals)) = NaN; elseif (nargin == 3 && isempty(Y)) nanvals = isnan(X); X(nanvals) = Inf; [v, idx] = min (X,[],DIM); v(all(nanvals,DIM)) = NaN; else Xnan = isnan(X); Ynan = isnan(Y); X(Xnan) = Inf; Y(Ynan) = Inf; if (nargin == 3) [v, idx] = min(X,Y,DIM); else [v, idx] = min(X,Y); endif v(Xnan & Ynan) = NaN; endif endfunction %!assert (nanmin ([2 4 NaN 7]), 2) %!assert (nanmin ([2 4 NaN Inf]), 2) %!assert (nanmin ([1 NaN 3; NaN 5 6; 7 8 NaN]), [1, 5, 3]) %!assert (nanmin ([1 NaN 3; NaN 5 6; 7 8 NaN]'), [1, 5, 7]) %!assert (nanmin (single ([1 NaN 3; NaN 5 6; 7 8 NaN])), single ([1 5 3])) statistics-release-1.6.3/inst/nansum.m000066400000000000000000000040561456127120000177700ustar00rootroot00000000000000## Copyright (C) 2001 Paul Kienzle ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {s =} nansum (@var{x}) ## @deftypefnx {statistics} {s =} nansum (@var{x}, @var{dim}) ## @deftypefnx {statistics} {s =} nansum (@dots{}, @qcode{"native"}) ## @deftypefnx {statistics} {s =} nansum (@dots{}, @qcode{"double"}) ## @deftypefnx {statistics} {s =} nansum (@dots{}, @qcode{"extra"}) ## Compute the sum while ignoring NaN values. ## ## @code{nansum} is identical to the @code{sum} function except that NaN ## values are treated as 0 and so ignored. If all values are NaN, the sum is ## returned as 0. ## ## See help text of @code{sum} for details on the options. ## ## @seealso{sum, nanmin, nanmax} ## @end deftypefn function v = nansum (X, varargin) if (nargin < 1) print_usage (); else X(isnan (X)) = 0; v = sum (X, varargin{:}); endif endfunction %!assert (nansum ([2 4 NaN 7]), 13) %!assert (nansum ([2 4 NaN Inf]), Inf) %!assert (nansum ([1 NaN 3; NaN 5 6; 7 8 NaN]), [8 13 9]) %!assert (nansum ([1 NaN 3; NaN 5 6; 7 8 NaN], 2), [4; 11; 15]) %!assert (nansum (single ([1 NaN 3; NaN 5 6; 7 8 NaN])), single ([8 13 9])) %!assert (nansum (single ([1 NaN 3; NaN 5 6; 7 8 NaN]), "double"), [8 13 9]) %!assert (nansum (uint8 ([2 4 1 7])), 14) %!assert (nansum (uint8 ([2 4 1 7]), "native"), uint8 (14)) %!assert (nansum (uint8 ([2 4 1 7])), 14) statistics-release-1.6.3/inst/normalise_distribution.m000066400000000000000000000220251456127120000232530ustar00rootroot00000000000000## Copyright (C) 2011 Alexander Klein ## Copyright (C) 2022-2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{NORMALISED} =} normalise_distribution (@var{DATA}) ## @deftypefnx {statistics} {@var{NORMALISED} =} normalise_distribution (@var{DATA}, @var{DISTRIBUTION}) ## @deftypefnx {statistics} {@var{NORMALISED} =} normalise_distribution (@var{DATA}, @var{DISTRIBUTION}, @var{DIMENSION}) ## ## Transform a set of data so as to be N(0,1) distributed according to an idea ## by van Albada and Robinson. ## ## This is achieved by first passing it through its own cumulative distribution ## function (CDF) in order to get a uniform distribution, and then mapping ## the uniform to a normal distribution. ## ## The data must be passed as a vector or matrix in @var{DATA}. ## If the CDF is unknown, then [] can be passed in @var{DISTRIBUTION}, and in ## this case the empirical CDF will be used. ## Otherwise, if the CDFs for all data are known, they can be passed in ## @var{DISTRIBUTION}, ## either in the form of a single function name as a string, ## or a single function handle, ## or a cell array consisting of either all function names as strings, ## or all function handles. ## In the latter case, the number of CDFs passed must match the number ## of rows, or columns respectively, to normalise. ## If the data are passed as a matrix, then the transformation will ## operate either along the first non-singleton dimension, ## or along @var{DIMENSION} if present. ## ## Notes: ## The empirical CDF will map any two sets of data ## having the same size and their ties in the same places after sorting ## to some permutation of the same normalised data: ## @example ## @code{normalise_distribution([1 2 2 3 4])} ## @result{} -1.28 0.00 0.00 0.52 1.28 ## ## @code{normalise_distribution([1 10 100 10 1000])} ## @result{} -1.28 0.00 0.52 0.00 1.28 ## @end example ## ## Original source: ## S.J. van Albada, P.A. Robinson ## "Transformation of arbitrary distributions to the ## normal distribution with application to EEG ## test-retest reliability" ## Journal of Neuroscience Methods, Volume 161, Issue 2, ## 15 April 2007, Pages 205-211 ## ISSN 0165-0270, 10.1016/j.jneumeth.2006.11.004. ## (http://www.sciencedirect.com/science/article/pii/S0165027006005668) ## @end deftypefn function normalised = normalise_distribution (data, distribution, dimension) if (nargin < 1 || nargin > 3) print_usage; elseif (! ismatrix (data) || length (size (data)) > 2) error (strcat (["normalise_distribution: first argument"], ... [" must be a vector or matrix."])); endif if (nargin >= 2) if (! isempty (distribution)) ## Wrap a single handle in a cell array. if (strcmp (typeinfo (distribution), typeinfo (@(x)(x)))) distribution = {distribution}; ## Do we have a string argument instead? elseif (ischar (distribution)) ## Is it a single string? if (rows (distribution) == 1) temp = str2func ([distribution]); distribution = {temp}; else error (strcat (["normalise_distribution: second argument cannot"], ... [" contain more than one string unless in a cell"], ... [" array."])); endif ## Do we have a cell array of distributions instead? elseif (iscell (distribution)) ## Does it consist of strings only? if (all (cellfun (@ischar, distribution))) distribution = cellfun (@str2func, distribution, ... "UniformOutput", false ); endif ## Does it eventually consist of function handles only if (! all (cellfun (@(h) (strcmp (typeinfo (h), typeinfo ... (@(x)(x)))), distribution))) error (strcat (["normalise_distribution: second argument must"], ... [" contain either a single function name or"], ... [" handle or a cell array of either all function"], ... [" names or handles!"])); endif else error ( "Illegal second argument: ", typeinfo ( distribution ) ); endif endif else distribution = []; endif if (nargin == 3) if (! isscalar (dimension) || (dimension != 1 && dimension != 2)) error ("normalise_distribution: third argument must be either 1 or 2."); endif else if (isvector (data) && rows (data) == 1) dimension = 2; else dimension = 1; endif endif trp = (dimension == 2); if (trp) data = data'; endif r = rows (data); c = columns (data); normalised = NA (r, c); ## Do we know the distribution of the sample? if (isempty (distribution)) precomputed_normalisation = []; for k = 1 : columns ( data ) ## Note that this line is in accordance with equation (16) in the ## original text. The author's original program, however, produces ## different values in the presence of ties, namely those you'd ## get replacing "last" by "first". [uniq, indices] = unique (sort (data(:, k)), "last"); ## Does the sample have ties? if (rows (uniq) != r) ## Transform to uniform, then normal distribution. uniform = ( indices - 1/2 ) / r; normal = norminv ( uniform ); else ## Without ties everything is pretty much straightforward as ## stated in the text. if (isempty (precomputed_normalisation)) precomputed_normalisation = norminv (1 / (2*r) : 1/r : 1 - 1 / (2*r)); endif normal = precomputed_normalisation; endif ## Find the original indices in the unsorted sample. ## This somewhat quirky way of doing it is still faster than ## using a for-loop. [ ignore, ignore, target_indices ] = unique ( data (:, k ) ); ## Put normalised values in the places where they belong. f_remap = @( k ) ( normal ( k ) ); normalised ( :, k ) = arrayfun ( f_remap, target_indices ); endfor else ## With known distributions, everything boils down to a few lines of code ##The same distribution for all data? if (all (size (distribution) == 1)) normalised = norminv (distribution{1,1}(data)); elseif (length (vec (distribution)) == c) for k = 1 : c normalised (:, k) = norminv (distribution{k}(data)(:, k)); endfor else error (strcat (["normalise_distribution: number of distributions"], ... [" does not match data size!"])); endif endif if (trp) normalised = normalised'; endif endfunction %!test %! v = normalise_distribution ([1 2 3], [], 1); %! assert (v, [0 0 0]) %!test %! v = normalise_distribution ([1 2 3], [], 2); %! assert (v, norminv ([1 3 5] / 6), 3 * eps) %!test %! v = normalise_distribution ([1 2 3]', [], 2); %! assert (v, [0 0 0]') %!test %! v = normalise_distribution ([1 2 3]', [], 1); %! assert (v, norminv ([1 3 5]' / 6), 3 * eps) %!test %! v = normalise_distribution ([1 1 2 2 3 3], [], 2); %! assert (v, norminv ([3 3 7 7 11 11] / 12), 3 * eps) %!test %! v = normalise_distribution ([1 1 2 2 3 3]', [], 1); %! assert (v, norminv ([3 3 7 7 11 11]' / 12), 3 * eps) %!test %! A = randn ( 10 ); %! N = normalise_distribution (A, @normcdf); %! assert (A, N, 10000 * eps) %!test %! A = exprnd (1, 100); %! N = normalise_distribution (A, @(x)(expcdf (x, 1))); %! assert (mean (vec (N)), 0, 0.1) %! assert (std (vec (N)), 1, 0.1) %!test %! A = rand (1000,1); %! N = normalise_distribution (A, {@(x)(unifcdf (x, 0, 1))}); %! assert (mean (vec (N)), 0, 0.2) %! assert (std (vec (N)), 1, 0.1) %!test %! A = [rand(1000,1), randn(1000, 1)]; %! N = normalise_distribution (A, {@(x)(unifcdf (x, 0, 1)), @normcdf}); %! assert (mean (N), [0, 0], 0.2) %! assert (std (N), [1, 1], 0.1) %!test %! A = [rand(1000,1), randn(1000, 1), exprnd(1, 1000, 1)]'; %! N = normalise_distribution (A, {@(x)(unifcdf (x, 0, 1)); @normcdf; @(x)(expcdf (x, 1))}, 2); %! assert (mean (N, 2), [0, 0, 0]', 0.2); %! assert (std (N, [], 2), [1, 1, 1]', 0.1); %!xtest %! A = exprnd (1, 1000, 9); A (300:500, 4:6) = 17; %! N = normalise_distribution (A); %! assert (mean (N), [0 0 0 0.38 0.38 0.38 0 0 0], 0.1); %! assert (var (N), [1 1 1 2.59 2.59 2.59 1 1 1], 0.1); %!test %!error normalise_distribution (zeros (3, 4), ... %! {@(x)(unifcdf (x, 0, 1)); @normcdf; @(x)(expcdf (x,1))}); statistics-release-1.6.3/inst/normplot.m000066400000000000000000000140541456127120000203400ustar00rootroot00000000000000## Copyright (C) 2022 Andreas Bertsatos ## Based on previous work by Paul Kienzle originally ## granted to the public domain. ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {} normplot (@var{x}) ## @deftypefnx {Function File} {} normplot (@var{ax}, @var{x}) ## @deftypefnx {Function File} {@var{h} =} normplot (@dots{}) ## ## Produce normal probability plot of the data in @var{x}. If @var{x} is a ## matrix, @code{normplot} plots the data for each column. NaN values are ## ignored. ## ## @code{@var{h} = normplot (@var{ax}, @var{x})} takes a handle @var{ax} in ## addition to the data in @var{x} and it uses that axes for ploting. You may ## get this handle of an existing plot with @code{gca}/. ## ## The line joing the 1st and 3rd quantile is drawn solid whereas its extensions ## to both ends are dotted. If the underlying distribution is normal, the ## points will cluster around the solid part of the line. Other distribution ## types will introduce curvature in the plot. ## ## @seealso{cdfplot, wblplot} ## @end deftypefn function h = normplot (varargin) ## Check for valid input arguments narginchk (1, 2); ## Parse input arguments if (nargin == 1) ax = []; x = varargin{1}; else ax = varargin{1}; ## Check that ax is a valid axis handle try isstruct (get (ax)); catch error ("normplot: invalid handle %f.", ax); end_try_catch x = varargin{2}; endif ## Check that x is a vector or a 2-D matrix if (isscalar (x) || ndims (x) > 2) error ("normplot: x must be a vecctor or a 2-D matrix handle."); endif ## If x is a vector, make it a column vector if (rows (x) == 1) x = x(:); endif ## If ax is empty, create a new axes if (isempty (ax)) ax = newplot(); end ## Get number of column vectors in x col = size (x, 2); ## Process each column and plot data and fit lines color = {"blue", "black", "cyan", "green", "magenta", "red", "white", "yellow"}; hold on; for i = 1:col xc = x(:,i); ## Remove NaNs, get min, max, and range xc(isnan(xc)) = []; if (isempty (xc)) break; endif ## Transform data row_xc = rows (xc); yc = norminv (([1:row_xc]' - 0.5) / row_xc); xc = sort(xc); ## Find quartiles q1x = prctile(xc,25); q3x = prctile(xc,75); q1y = prctile(yc,25); q3y = prctile(yc,75); qx = [q1x; q3x]; qy = [q1y; q3y]; ## Calculate coordinates and limits for fitting lines dx = q3x - q1x; dy = q3y - q1y; slope = dy ./ dx; centerx = (q1x + q3x)/2; centery = (q1y + q3y)/2; maxx = max (xc); minx = min (xc); maxy = centery + slope.*(maxx - centerx); miny = centery - slope.*(centerx - minx); yinter = centery - slope.*(centerx); mx = [minx; maxx]; my = [miny; maxy]; ## Plot data and corresponding reference lines in the same color, ## following the default color order. Plot reference line first, ## followed by the data, so that data will be on top of reference line. h_end(i) = line (ax, mx, my, "LineStyle", "-.", "Marker", "none", ... "color", color{mod(i,8)}); h_mid(i) = line (ax, qx, qy, "LineStyle", "-", "Marker", "none", ... "color", color{mod(i,8)}); h_dat(i) = line (ax, xc, yc, "LineStyle", "none", "Marker", "+", ... "color", color{mod(i,8)}); endfor hold off; ## Change colors for single column vector if (i == 1) set (h_dat, "Color", "b"); set (h_mid, "Color", "r"); set (h_end, "Color", "r"); endif ## Bundle handles together if output requested if (nargout > 0) h = [h_dat, h_mid, h_end]'; endif ## Plot labels title "Normal Probability Plot" ylabel "Probability" xlabel "Data" ## Plot grid p = [0.001, 0.003, 0.01, 0.02, 0.05, 0.10, 0.25, 0.5, ... 0.75, 0.90, 0.95, 0.98, 0.99, 0.997, 0.999]; label = {"0.001", "0.003", "0.01", "0.02", "0.05", "0.10", "0.25", "0.50", ... "0.75", "0.90", "0.95", "0.98", "0.99", "0.997", "0.999"}; tick = norminv(p,0,1); set (ax, "ytick", tick, "yticklabel", label); ## Set view range with a bit of space around data range = nanmax (x(:)) - nanmin (x(:)); if (range > 0) minxaxis = nanmin (x(:)) - 0.025 * range; maxxaxis = nanmax (x(:)) + 0.025 * range; else minxaxis = nanmin (x(:)) - 1; maxxaxis = nanmax (x(:)) + 1; end minyaxis = norminv (0.25 ./ row_xc, 0, 1); maxyaxis = norminv ((row_xc - 0.25) ./ row_xc, 0, 1); set (ax, "ylim", [minyaxis, maxyaxis], "xlim", [minxaxis, maxxaxis]); grid (ax, "on"); box (ax, "off"); endfunction %!demo %! h = normplot([1:20]); %!demo %! h = normplot([1:20;5:2:44]'); %!demo %! ax = newplot(); %! h = normplot(ax, [1:20]); %! ax = gca; %! h = normplot(ax, [-10:10]); %! set (ax, "xlim", [-11, 21]); ## Test input validation %!error normplot (); %!error normplot (23); %!error normplot (23, [1:20]); %!error normplot (ones(3,4,5)); ## Test plotting %!test %! hf = figure ("visible", "off"); %! unwind_protect %! ax = newplot (hf); %! h = normplot (ax, [1:20]); %! ax = gca; %! h = normplot(ax, [-10:10]); %! set (ax, "xlim", [-11, 21]); %! unwind_protect_cleanup %! close (hf); %! end_unwind_protect %!test %! hf = figure ("visible", "off"); %! unwind_protect %! h = normplot([1:20;5:2:44]'); %! unwind_protect_cleanup %! close (hf); %! end_unwind_protect statistics-release-1.6.3/inst/optimalleaforder.m000066400000000000000000000234171456127120000220220ustar00rootroot00000000000000## Copyright (C) 2021 Stefano Guidoni ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{leafOrder} =} optimalleaforder (@var{tree}, @var{D}) ## @deftypefnx {statistics} {@var{leafOrder} =} optimalleaforder (@dots{}, @var{Name}, @var{Value}) ## ## Compute the optimal leaf ordering of a hierarchical binary cluster tree. ## ## The optimal leaf ordering of a tree is the ordering which minimizes the sum ## of the distances between each leaf and its adjacent leaves, without altering ## the structure of the tree, that is without redefining the clusters of the ## tree. ## ## Required inputs: ## @itemize ## @item ## @var{tree}: a hierarchical cluster tree @var{tree} generated by the ## @code{linkage} function. ## ## @item ## @var{D}: a matrix of distances as computed by @code{pdist}. ## @end itemize ## ## Optional inputs can be the following property/value pairs: ## @itemize ## @item ## property 'Criteria' at the moment can only have the value 'adjacent', ## for minimizing the distances between leaves. ## ## @item ## property 'Transformation' can have one of the values 'linear', 'inverse' ## or a handle to a custom function which computes @var{S} the similarity ## matrix. ## @end itemize ## ## optimalleaforder's output @var{leafOrder} is the optimal leaf ordering. ## ## @strong{Reference} ## Bar-Joseph, Z., Gifford, D.K., and Jaakkola, T.S. Fast optimal leaf ordering ## for hierarchical clustering. Bioinformatics vol. 17 suppl. 1, 2001. ## @end deftypefn ## ## @seealso{dendrogram,linkage,pdist} function leafOrder = optimalleaforder ( varargin ) ## check the input if ( nargin < 2 ) print_usage (); endif tree = varargin{1}; D = varargin{2}; criterion = "adjacent"; # default and only value at the moment transformation = "linear"; if ((columns (tree) != 3) || (! isnumeric (tree)) || ... (! (max (tree(end, 1:2)) == rows (tree) * 2))) error (["optimalleaforder: tree must be a matrix as generated by the " ... "linkage function"]); endif ## read the paired arguments if (! all (cellfun ("ischar", varargin(3:end)))) error ("optimalleaforder: character inputs expected for arguments 3 and up"); else varargin(3:end) = lower (varargin(3:end)); endif pair_index = 3; while (pair_index <= (nargin - 1)) switch (varargin{pair_index}) case "criteria" criterion = varargin{pair_index + 1}; if (strcmp (criterion, "group")) ## MATLAB compatibility: ## the 'group' criterion is not implemented error ("optimalleaforder: unavailable criterion 'group'"); elseif (! strcmp (criterion, "adjacent")) error ("optimalleaforder: invalid criterion %s", criterion); endif case "transformation" transformation = varargin{pair_index + 1}; otherwise error ("optimalleaforder: unknown property %s", varargin{pair_index}); endswitch pair_index += 2; endwhile ## D can be either a vector or a matrix, ## but it is easier to work with a matrix if (isvector (D)) D = squareform (D); endif n = rows (D); m = rows (tree); if (n != (m + 1)) error (["optimalleaforder: D must be a matrix or vector generated by " ... "the pdist function"]); endif ## the similarity matrix, basically an inverted distance matrix S = zeros (n); if (strcmpi (transformation, "linear")) ## linear similarity maxD = max (max (D)); S = maxD - D; elseif (strcmpi (transformation, "inverse")) ## similarity as inverted distance S = 1 ./ D; elseif (is_function_handle (transformation)) ## custom similarity S = feval (transformation, D); else error ("optimalleaforder: invalid transformation %s", transformation); endif ## main body ## for each node v we compute the maximum similarity of the subtree M(w,u,v), ## where the leftmost leaf is w and the rightmost is u; remember that ## M(w,u,v) = M(u,w,v) M = zeros (n, n, n + m); ## O is a utility matrix: for each node of the tree we store the left and ## right leaves of the optimal subtree O = [1:( n + m ); 1:( n + m ); (zeros (1, (n + m)))]'; ## compute M for every node v for iter = 1 : m v = iter + n; # current node l = optimalleaforder_getLeafList (tree(iter, 1)); # the left subtree r = optimalleaforder_getLeafList (tree(iter, 2)); # the right subtree if (tree(iter,1) > n) l_l = optimalleaforder_getLeafList (tree(tree(iter, 1) - n, 1)); l_r = optimalleaforder_getLeafList (tree(tree(iter, 1) - n, 2)); else l_l = l_r = l; endif if (tree(iter,2) > n) r_l = optimalleaforder_getLeafList (tree(tree(iter, 2) - n, 1)); r_r = optimalleaforder_getLeafList (tree(tree(iter, 2) - n, 2)); else r_l = r_r = r; endif ## let's find the maximum value of M(w,u,v) when: w is a leaf of the left ## subtree of v and u is a leaf of the right subtree of v for i = 1 : length (l) if (isempty (find (l(i) == l_l))) x = l_l; else x = l_r; endif for j = 1 : length (r) if (isempty (find (r(j) == r_l))) y = r_l; else y = r_r; endif ## max(M(w,u,v)) = max(M(w,k,v_l)) + max(M(h,u,v_r)) + S(k,h) ## where: v_l is the left child of v and v_r the right child of v M_tmp = repmat (M(l(i), x(:), tree(iter, 1)), length (y), 1) + ... repmat (M(y(:), r(j), tree(iter, 2)), 1, length (x)) + ... S(y(:), x(:)); M_max = max (max (M_tmp)); # this is M(l(i), r(j), v) [h, k] = find (M_tmp == M_max); M(l(i), r(j), v) = M_max; M(r(j), l(i), v) = M(l(i), r(j), v); if (M_max > O(v,3)) O(v, 1) = l(i); # this is w O(v, 2) = r(j); # this is u O(v, 3) = M_max; # this is M(w, u, v) endif endfor endfor endfor ## reordering: ## we found the M(w,u,v) corresponding to the optimal leaf order, now we can ## compute the optimal leaf order given our M(w,u,v) ## the return value leafOrder = zeros ( 1, n ); leafOrder(1) = O(end, 1); leafOrder(n) = O(end, 2); ## the inverse operation, only easier, to get the leaf order: now we know the ## leftmost and rightmost leaves of the best subtree, we may have to flip it ## though for iter = m : -1 : 1 v = iter + n; extremes = O(v, [1, 2]); l_node = tree(iter, 1); r_node = tree(iter, 2); l = optimalleaforder_getLeafList (l_node); r = optimalleaforder_getLeafList (r_node); if (l_node > n) l_l = optimalleaforder_getLeafList (tree(l_node - n, 1)); l_r = optimalleaforder_getLeafList (tree(l_node - n, 2)); else l_l = l_r = l; endif if (r_node > n) r_l = optimalleaforder_getLeafList (tree(r_node - n, 1)); r_r = optimalleaforder_getLeafList (tree(r_node - n, 2)); else r_l = r_r = r; endif ## this means that we need to flip the subtree if (isempty (find (extremes(1) == l))) l_tmp = l; l_l_tmp = l_l; l_r_tmp = l_r; l = r; l_l = r_l; l_r = r_r; r = l_tmp; r_l = l_l_tmp; r_r = l_r_tmp; node_tmp = l_node; l_node = r_node; r_node = node_tmp; endif if (isempty (find (extremes(1) == l_l))) x = l_l; else x = l_r; endif if (isempty (find (extremes(2) == r_l))) y = r_l; else y = r_r; endif M_tmp = repmat (M(extremes(1), x(:), l_node), length (y), 1) + ... repmat (M(y(:), extremes(2), r_node), 1, length (x)) + ... S(y(:), x(:)); M_max = max (max (M_tmp)); [h, k] = find (M_tmp == M_max); O(l_node, 1) = extremes(1); O(l_node, 2) = x(k); O(r_node, 1) = y(h); O(r_node, 2) = extremes(2); p_1 = find (leafOrder == extremes(1)); p_2 = find (leafOrder == extremes(2)); leafOrder (p_1 + (length (l)) - 1) = x(k); leafOrder (p_1 + (length (l))) = y(h); endfor ## function: optimalleaforder_getLeafList ## get the list of leaves under a given node function vector = optimalleaforder_getLeafList (nodes_to_visit) vector = []; while (! isempty (nodes_to_visit)) currentnode = nodes_to_visit(1); nodes_to_visit(1) = []; if (currentnode > n) node = currentnode - n; nodes_to_visit = [tree(node, [2 1]) nodes_to_visit]; endif if (currentnode <= n) vector = [vector currentnode]; endif endwhile endfunction endfunction %!demo %! randn ("seed", 5) # for reproducibility %! X = randn (10, 2); %! D = pdist (X); %! tree = linkage(D, 'average'); %! optimalleaforder (tree, D, 'Transformation', 'linear') ## Test input validation %!error optimalleaforder () %!error optimalleaforder (1) %!error optimalleaforder (ones (2, 2), 1) %!error optimalleaforder ([1 2 3], [1 2; 3 4], "criteria", 5) %!error optimalleaforder ([1 2 1], [1 2 3]) %!error optimalleaforder ([1 2 1], 1, "xxx", "xxx") %!error optimalleaforder ([1 2 1], 1, "Transformation", "xxx") statistics-release-1.6.3/inst/pca.m000066400000000000000000000463661456127120000172440ustar00rootroot00000000000000## Copyright (C) 2013-2019 Fernando Damian Nieuwveldt ## Copyright (C) 2021 Stefano Guidoni ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or ## modify it under the terms of the GNU General Public License ## as published by the Free Software Foundation; either version 3 ## of the License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{coeff} =} pca (@var{x}) ## @deftypefnx {statistics} {@var{coeff} =} pca (@var{x}, @var{Name}, @var{Value}) ## @deftypefnx {statistics} {[@var{coeff}, @var{score}, @var{latent}] =} pca (@dots{}) ## @deftypefnx {statistics} {[@var{coeff}, @var{score}, @var{latent}, @var{tsquared}] =} pca (@dots{}) ## @deftypefnx {statistics} {[@var{coeff}, @var{score}, @var{latent}, @var{tsquared}, @var{explained}, @var{mu}] =} pca (@dots{}) ## ## Performs a principal component analysis on a data matrix. ## ## A principal component analysis of a data matrix of @math{N} observations in a ## @math{D} dimensional space returns a @math{DxD} transformation matrix, to ## perform a change of basis on the data. The first component of the new basis ## is the direction that maximizes the variance of the projected data. ## ## Input argument: ## @itemize @bullet ## @item ## @var{x} : a @math{NxD} data matrix ## @end itemize ## ## The following @var{Name}, @var{Value} pair arguments can be used: ## @itemize @bullet ## @item ## @qcode{"Algorithm"} defines the algorithm to use: ## @itemize ## @item @qcode{"svd"} (default), for singular value decomposition ## @item @qcode{"eig"} for eigenvalue decomposition ## @end itemize ## ## @item ## @qcode{"Centered"} is a boolean indicator for centering the observation data. ## It is @code{true} by default. ## @item ## ## @qcode{"Economy"} is a boolean indicator for the economy size output. It is ## @code{true} by default. Hence, @code{pca} returns only the elements of ## @var{latent} that are not necessarily zero, and the corresponding columns of ## @var{coeff} and @var{score}, that is, when @math{N <= D}, only the first ## @math{N - 1}. ## ## @item ## @qcode{"NumComponents"} defines the number of components @math{k} to return. ## If @math{k < p}, then only the first @math{k} columns of @var{coeff} and ## @var{score} are returned. ## ## @item ## @qcode{"Rows"} defines how to handle missing values: ## @itemize ## @item @qcode{"complete"} (default), missing values are removed before ## computation. ## @item @qcode{"pairwise"} (only valid when @qcode{"Algorithm"} is ## @qcode{"eig"}), the covariance of rows with missing data is computed using ## the available data, but the covariance matrix could be not positive definite, ## which triggers the termination of @code{pca}. ## @item @qcode{"complete"}, missing values are not allowed, @code{pca} ## terminates with an error if there are any. ## @end itemize ## ## @item ## @qcode{"Weights"} defines observation weights as a vector of positive values ## of length @math{N}. ## ## @item ## @qcode{"VariableWeights"} defines variable weights: ## @itemize ## @item a @var{vector} of positive values of length @math{D}. ## @item the string @qcode{"variance"} to use the sample variance as weights. ## @end itemize ## @end itemize ## ## Return values: ## @itemize @bullet ## @item ## @var{coeff} : the principal component coefficients, a @math{DxD} ## transformation matrix ## @item ## @var{score} : the principal component scores, the representation of @var{x} ## in the principal component space ## @item ## @var{latent} : the principal component variances, i.e., the eigenvalues of ## the covariance matrix of @var{x} ## @item ## @var{tsquared} : Hotelling's T-squared Statistic for each observation in ## @var{x} ## @item ## @var{explained} : the percentage of the variance explained by each principal ## component ## @item ## @var{mu} : the estimated mean of each variable of @var{x}, it is zero if the ## data are not centered ## @end itemize ## ## Matlab compatibility note: the alternating least square method 'als' and ## associated options 'Coeff0', 'Score0', and 'Options' are not yet implemented ## ## @subheading References ## @enumerate ## @item ## Jolliffe, I. T., Principal Component Analysis, 2nd Edition, Springer, 2002 ## @end enumerate ## ## @seealso{barttest, factoran, pcacov, pcares} ## @end deftypefn ## BUGS: ## - tsquared with weights (xtest below) ##FIXME ## -- can change isnan to ismissing once the latter is implemented in Octave ## -- change mystd to std and remove the helper function mystd once weighting is available in the Octave function function [coeff, score, latent, tsquared, explained, mu] = pca (x, varargin) if (nargin < 1) print_usage (); endif [nobs, nvars] = size (x); ## default options optAlgorithmS = "svd"; optCenteredB = true; optEconomyB = true; optNumComponentsI = nvars; optWeights = []; optVariableWeights = []; optRowsB = false; TF = []; ## parse parameters pair_index = 1; while (pair_index <= (nargin - 1)) switch (lower (varargin{pair_index})) ## decomposition algorithm: singular value decomposition, eigenvalue ## decomposition or (currently unavailable) alternating least square case "algorithm" optAlgorithmS = varargin{pair_index + 1}; switch (optAlgorithmS) case {"svd", "eig"} ; case "als" error ("pca: alternating least square algorithm not implemented"); otherwise error ("pca: invalid algorithm %s", optAlgorithmS); endswitch ## centering of the columns, around the mean case "centered" if (isbool (varargin{pair_index + 1})) optCenteredB = varargin{pair_index + 1}; else error ("pca: 'centered' requires a boolean value"); endif ## limit the size of the output to the degrees of freedom, when a smaller ## number than the number of variables case "economy" if (isbool (varargin{pair_index + 1})) optEconomyB = varargin{pair_index + 1}; else error ("pca: 'economy' requires a boolean value"); endif ## choose the number of components to show case "numcomponents" optNumComponentsI = varargin{pair_index + 1}; if ((! isscalar (optNumComponentsI)) || (! isnumeric (optNumComponentsI)) || optNumComponentsI != floor (optNumComponentsI) || optNumComponentsI <= 0 || optNumComponentsI > nvars) error (["pca: the number of components must be a positive integer"... "number smaller or equal to the number of variables"]); endif ## observation weights: some observations can be more accurate than others case "weights" optWeights = varargin{pair_index + 1}; if ((! isvector (optWeights)) || length (optWeights) != nobs || length (find (optWeights < 0)) > 0) error ("pca: weights must be a numerical array of positive numbers"); endif if (rows (optWeights) == 1 ) optWeights = transpose (optWeights); endif ## variable weights: weights used for the variables case "variableweights" optVariableWeights = varargin{pair_index + 1}; if (ischar (optVariableWeights) && strcmpi (optVariableWeights, "variance")) optVariableWeights = "variance"; # take care of this later elseif ((! isvector (optVariableWeights)) || length (optVariableWeights) != nvars || (! isnumeric (optVariableWeights)) || length (find (optVariableWeights < 0)) > 0) error (["pca: variable weights must be a numerical array of "... "positive numbers or the string 'variance'"]); else optVariableWeights = 1 ./ sqrt (optVariableWeights); ## it is used as a row vector if (columns (optVariableWeights) == 1 ) optVariableWeights = transpose (optVariableWeights); endif endif ## rows: policy for missing values case "rows" switch (varargin{pair_index + 1}) case "complete" optRowsB = false; case "pairwise" optRowsB = true; case "all" if (any (isnan (x))) error (["pca: when all rows are requested the dataset cannot"... " include NaN values"]); endif otherwise error ("pca: %s is an invalid value for rows", ... varargin{pair_index + 1}); endswitch case {"coeff0", "score0", "options"} error (strcat (["pca: parameter %s is only valid with the 'als'"], ... [" method, which is not yet implemented"]), ... varargin{pair_index}); otherwise error ("pca: unknown property %s", varargin{pair_index}); endswitch pair_index += 2; endwhile ## Preparing the dataset according to the chosen policy for missing values if (optRowsB) if (! strcmp (optAlgorithmS, "eig")) optAlgorithmS = "eig"; warning (["pca: setting algorithm to 'eig' because 'rows' option is "... "set to 'pairwise'"]); endif TF = isnan (x); missingRows = zeros (nobs, 1); nmissing = 0; else ## "complete": remove all the rows with missing values TF = isnan (x); missingRows = any (TF, 2); nmissing = sum (missingRows); endif ## indices of the available rows ridcs = find (missingRows == 0); ## Center the columns to mean zero if requested if (optCenteredB) if (isempty (optWeights) && nmissing == 0 && ! optRowsB) ## no weights and no missing values mu = mean (x); elseif (nmissing == 0 && ! optRowsB) ## weighted observations: some observations are more valuable, i.e. they ## can be trusted more mu = sum (optWeights .* x) ./ sum (optWeights); else ## missing values: the mean is computed column by column mu = zeros (1, nvars); if (isempty (optWeights)) for iter = 1 : nvars mu(iter) = mean (x(find (TF(:, iter) == 0), iter)); endfor else ## weighted mean with missing data for iter = 1 : nvars mu(iter) = sum (x(find (TF(:, iter) == 0), iter) .* ... optWeights(find (TF(:, iter) == 0))) ./ ... sum (optWeights(find (TF(:, iter) == 0))); endfor endif endif Xc = x - mu; else Xc = x; ## The mean of the variables of the original dataset: ## return zero if the dataset is not centered mu = zeros (1, nvars); endif ## Change the columns according to the variable weights if (! isempty (optVariableWeights)) if (ischar (optVariableWeights)) if (isempty (optWeights)) sqrtBias = 1; # see below optVariableWeights = std (x); else ## unbiased variance estimation: the bias when using reliability weights ## is 1 - var(weights) / std(weigths)^2 sqrtBias = sqrt (1 - (sumsq (optWeights) / sum (optWeights) ^ 2)); optVariableWeights = mystd (x, optWeights) / sqrtBias; endif endif Xc = Xc ./ optVariableWeights; endif ## Compute the observation weight matrix if (isempty (optWeights)) Wd = eye (nobs - nmissing); else Wd = diag (optWeights) ./ sum (optWeights); endif ## Compute the coefficients switch (optAlgorithmS) case "svd" ## Check if there are more variables than observations if (nvars <= nobs) [U, S, coeff] = svd (sqrt (Wd) * Xc(ridcs,:), "econ"); else ## Calculate the svd on the transpose matrix, much faster if (optEconomyB) [coeff, S, V] = svd (Xc(ridcs,:)' * sqrt (Wd), "econ"); else [coeff, S, V] = svd (Xc(ridcs,:)' * sqrt (Wd)); endif endif case "eig" ## this method requires the computation of the sample covariance matrix if (optRowsB) ## pairwise: ## in this case the degrees of freedom for each element of the matrix ## are equal to the number of valid rows for the couple of columns ## used to compute the element Xpairwise = Xc; Xpairwise(find (isnan (Xc))) = 0; Ndegrees = (nobs - 1) * ones (nvars, nvars); for i_iter = 1 : nvars for j_iter = i_iter : nvars Ndegrees(i_iter, j_iter) = Ndegrees(i_iter, j_iter) - ... sum (any (TF(:,[i_iter j_iter]), 2)); Ndegrees(j_iter, i_iter) = Ndegrees(i_iter, j_iter); endfor endfor Mcov = Xpairwise' * Wd * Xpairwise ./ Ndegrees; else ## the degrees of freedom are not really important here ndegrees = nobs - nmissing - 1; Mcov = Xc(ridcs, :)' * Wd * Xc(ridcs, :) / ndegrees; endif [coeff, S] = eigs (Mcov, nvars); endswitch ## Change the coefficients according to the variable weights if (! isempty (optVariableWeights)) coeff = coeff .* transpose (optVariableWeights); endif ## MATLAB compatibility: the sign convention is that the ## greatest absolute value for each column is positive switchSignV = find (max (coeff) < abs (min (coeff))); if (! isempty (switchSignV)) coeff(:, switchSignV) = -1 * coeff(:, switchSignV); endif ## Compute the scores if (nargout > 1) ## This is for the score when using variable weights, it is not really ## a new definition of Xc if (! isempty (optVariableWeights)) Xc = Xc ./ optVariableWeights; endif ## Get the Scores score = Xc(ridcs,:) * coeff; ## Get the rank of the score matrix r = rank (score); ## If there is missing data, put it back ## FIXME: this needs tests if (nmissing) scoretmp = zeros (nobs, nvars); scoretmp(find (missingRows == 0), :) = score; scoretmp(find (missingRows), :) = NaN; score = scoretmp; endif ## Only use the first r columns, pad rest with zeros if economy != true score = score(:, 1:r) ; if (! optEconomyB) score = [score, (zeros (nobs , nvars-r))]; else coeff = coeff(: , 1:r); endif endif ## Compute the variances if (nargout > 2) ## degrees of freedom: n - 1 for centered data if (optCenteredB) dof = size (Xc(ridcs,:), 1) - 1; else dof = size (Xc(ridcs,:), 1); endif ## This is the same as the eigenvalues of the covariance matrix of x if (strcmp (optAlgorithmS, "eig")) latent = diag (S, 0); else latent = (diag (S'*S) / dof)(1:r); endif ## If observation weights were used, we need to scale back these values if (! isempty (optWeights)) latent = latent .* sum (optWeights(ridcs)); endif if (! optEconomyB) latent= [latent; (zeros (nvars - r, 1))]; endif endif ## Compute the Hotelling T-square statistics ## MATLAB compatibility: when using weighted observations the T-square ## statistics differ by some rounding error if (nargout > 3) ## Calculate the Hotelling T-Square statistic for the observations ## formally: tsquared = sumsq (zscore (score(:, 1:r)),2); if (! isempty (optWeights)) ## probably splitting the weights, using the square roots, is not the ## best solution, numerically weightedScore = score .* sqrt (optWeights); tsquared = mahal (weightedScore(ridcs, 1:r), weightedScore(ridcs, 1:r))... ./ optWeights; else tsquared = mahal (score(ridcs, 1:r), score(ridcs, 1:r)); endif endif ## Compute the variance explained by each principal component if (nargout > 4) explained = 100 * latent / sum (latent); endif ## When a number of components is chosen, the coefficients and score matrix ## only show that number of columns if (optNumComponentsI != nvars) coeff = coeff(:, 1:optNumComponentsI); endif if (optNumComponentsI != nvars && nargout > 1) score = score(:, 1:optNumComponentsI); endif endfunction #return the weighted standard deviation function retval = mystd (x, w) (dim = find (size(x) != 1, 1)) || (dim = 1); den = sum (w); mu = sum (w .* x, dim) ./ sum (w); retval = sum (w .* ((x - mu) .^ 2), dim) / den; retval = sqrt (retval); endfunction %!shared COEFF,SCORE,latent,tsquare,m,x,R,V,lambda,i,S,F #NIST Engineering Statistics Handbook example (6.5.5.2) %!test %! x=[7 4 3 %! 4 1 8 %! 6 3 5 %! 8 6 1 %! 8 5 7 %! 7 2 9 %! 5 3 3 %! 9 5 8 %! 7 4 5 %! 8 2 2]; %! R = corrcoef (x); %! [V, lambda] = eig (R); %! [~, i] = sort(diag(lambda), "descend"); #arrange largest PC first %! S = V(:, i) * diag(sqrt(diag(lambda)(i))); %!assert(diag(S(:, 1:2)*S(:, 1:2)'), [0.8662; 0.8420; 0.9876], 1E-4); #contribution of first 2 PCs to each original variable %! B = V(:, i) * diag( 1./ sqrt(diag(lambda)(i))); %! F = zscore(x)*B; %! [COEFF,SCORE,latent,tsquare] = pca(zscore(x, 1)); %!assert(tsquare,sumsq(F, 2),1E4*eps); %!test %! x=[1,2,3;2,1,3]'; %! [COEFF,SCORE,latent,tsquare] = pca(x, "Economy", false); %! m=[sqrt(2),sqrt(2);sqrt(2),-sqrt(2);-2*sqrt(2),0]/2; %! m(:,1) = m(:,1)*sign(COEFF(1,1)); %! m(:,2) = m(:,2)*sign(COEFF(1,2)); %!assert(COEFF,m(1:2,:),10*eps); %!assert(SCORE,-m,10*eps); %!assert(latent,[1.5;.5],10*eps); %!assert(tsquare,[4;4;4]/3,10*eps); #test with observation weights (using Matlab's results for this case as a reference) %! [COEFF,SCORE,latent,tsquare] = pca(x, "Economy", false, "weights", [1 2 1], "variableweights", "variance"); %!assert(COEFF, [0.632455532033676 -0.632455532033676; 0.741619848709566 0.741619848709566], 10*eps); %!assert(SCORE, [-0.622019449426284 0.959119380657905; -0.505649896847432 -0.505649896847431; 1.633319243121148 0.052180413036957], 10*eps); %!assert(latent, [1.783001790889027; 0.716998209110974], 10*eps); %!xtest assert(tsquare, [1.5; 0.5; 1.5], 10*eps); #currently, [4; 2; 4]/3 is actually returned; see comments above %!test %! x=x'; %! [COEFF,SCORE,latent,tsquare] = pca(x, "Economy", false); %! m=[sqrt(2),sqrt(2),0;-sqrt(2),sqrt(2),0;0,0,2]/2; %! m(:,1) = m(:,1)*sign(COEFF(1,1)); %! m(:,2) = m(:,2)*sign(COEFF(1,2)); %! m(:,3) = m(:,3)*sign(COEFF(3,3)); %!assert(COEFF,m,10*eps); %!assert(SCORE(:,1),-m(1:2,1),10*eps); %!assert(SCORE(:,2:3),zeros(2),10*eps); %!assert(latent,[1;0;0],10*eps); %!assert(tsquare,[0.5;0.5],10*eps) %!test %! [COEFF,SCORE,latent,tsquare] = pca(x); %!assert(COEFF,m(:, 1),10*eps); %!assert(SCORE,-m(1:2,1),10*eps); %!assert(latent,[1],10*eps); %!assert(tsquare,[0.5;0.5],10*eps) %!error pca([1 2; 3 4], "Algorithm", "xxx") %!error <'centered' requires a boolean value> pca([1 2; 3 4], "Centered", "xxx") %!error pca([1 2; 3 4], "NumComponents", -4) %!error pca([1 2; 3 4], "Rows", 1) %!error pca([1 2; 3 4], "Weights", [1 2 3]) %!error pca([1 2; 3 4], "Weights", [-1 2]) %!error pca([1 2; 3 4], "VariableWeights", [-1 2]) %!error pca([1 2; 3 4], "VariableWeights", "xxx") %!error pca([1 2; 3 4], "XXX", 1) statistics-release-1.6.3/inst/pcacov.m000066400000000000000000000104201456127120000177320ustar00rootroot00000000000000## Copyright (C) 2013-2019 Fernando Damian Nieuwveldt ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or ## modify it under the terms of the GNU General Public License ## as published by the Free Software Foundation; either version 3 ## of the License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{coeff} =} pcacov (@var{K}) ## @deftypefnx {statistics} {[@var{coeff}, @var{latent}] =} pcacov (@var{K}) ## @deftypefnx {statistics} {[@var{coeff}, @var{latent}, @var{explained}] =} pcacov (@var{K}) ## ## Perform principal component analysis on covariance matrix ## ## @code{@var{coeff} = pcacov (@var{K})} performs principal component analysis ## on the square covariance matrix @var{K} and returns the principal component ## coefficients, also known as loadings. The columns are in order of decreasing ## component variance. ## ## @code{[@var{coeff}, @var{latent}] = pcacov (@var{K})} also returns a vector ## with the principal component variances, i.e. the eigenvalues of @var{K}. ## @var{latent} has a length of @qcode{size (@var{coeff}, 1)}. ## ## @code{[@var{coeff}, @var{latent}, @var{explained}] = pcacov (@var{K})} also ## returns a vector with the percentage of the total variance explained by each ## principal component. @var{explained} has the same size as @var{latent}. ## The entries in @var{explained} range from 0 (none of the variance is ## explained) to 100 (all of the variance is explained). ## ## @code{pcacov} does not standardize @var{K} to have unit variances. In order ## to perform principal component analysis on standardized variables, use the ## correlation matrix @qcode{@var{R} = @var{K} ./ (@var{SD} * @var{SD}')}, where ## @qcode{@var{SD} = sqrt (diag (@var{K}))}, in place of @var{K}. To perform ## principal component analysis directly on the data matrix, use @code{pca}. ## ## @subheading References ## @enumerate ## @item ## Jolliffe, I. T., Principal Component Analysis, 2nd Edition, Springer, 2002 ## @end enumerate ## ## @seealso{bartlett, factoran, pcares, pca} ## @end deftypefn function [coeff, latent, explained] = pcacov (K) ## Check X being a square matrix if (ndims (K) != 2 || size (K, 1) != size (K, 2)) error ("pcacov: K must be a square matrix."); endif [U, S, V] = svd (K); ## Force a sign convention on the coefficients so that ## the largest element in each column has a positive sign [row, col] = size (U); [~, m_ind] = max (abs (U), [], 1); csign = sign (U (m_ind + (0:row:(col - 1) * row))); coeff = bsxfun (@times, U, csign); ## Compute extra output arguments if (nargout > 1) latent = diag (S); endif if (nargout > 2) explained = 100 * latent ./ sum (latent); endif endfunction %!demo %! x = [ 7 26 6 60; %! 1 29 15 52; %! 11 56 8 20; %! 11 31 8 47; %! 7 52 6 33; %! 11 55 9 22; %! 3 71 17 6; %! 1 31 22 44; %! 2 54 18 22; %! 21 47 4 26; %! 1 40 23 34; %! 11 66 9 12; %! 10 68 8 12 %! ]; %! Kxx = cov (x); %! [coeff, latent, explained] = pcacov (Kxx) ## Test output %!test %! load hald %! Kxx = cov (ingredients); %! [coeff,latent,explained] = pcacov(Kxx); %! c_out = [-0.0678, -0.6460, 0.5673, 0.5062; ... %! -0.6785, -0.0200, -0.5440, 0.4933; ... %! 0.0290, 0.7553, 0.4036, 0.5156; ... %! 0.7309, -0.1085, -0.4684, 0.4844]; %! l_out = [517.7969; 67.4964; 12.4054; 0.2372]; %! e_out = [ 86.5974; 11.2882; 2.0747; 0.0397]; %! assert (coeff, c_out, 1e-4); %! assert (latent, l_out, 1e-4); %! assert (explained, e_out, 1e-4); ## Test input validation %!error pcacov (ones (2,3)) %!error pcacov (ones (3,3,3)) statistics-release-1.6.3/inst/pcares.m000066400000000000000000000102411456127120000177350ustar00rootroot00000000000000## Copyright (C) 2013-2019 Fernando Damian Nieuwveldt ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or ## modify it under the terms of the GNU General Public License ## as published by the Free Software Foundation; either version 3 ## of the License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{residuals} =} pcares (@var{x}, @var{ndim}) ## @deftypefnx {statistics} {[@var{residuals}, @var{reconstructed}] =} pcares (@var{x}, @var{ndim}) ## ## Calculate residuals from principal component analysis. ## ## @code{@var{residuals} = pcares (@var{x}, @var{ndim})} returns the residuals ## obtained by retaining @var{ndim} principal components of the @math{NxD} ## matrix @var{x}. Rows of @var{x} correspond to observations, columns of ## @var{x} correspond to variables. @var{ndim} is a scalar and must be less ## than or equal to @math{D}. @var{residuals} is a matrix of the same size as ## @var{x}. Use the data matrix, not the covariance matrix, with this function. ## ## @code{[@var{residuals}, @var{reconstructed}] = pcares (@var{x}, @var{ndim})} ## returns the reconstructed observations, i.e. the approximation to @var{x} ## obtained by retaining its first @var{ndim} principal components. ## ## @code{pcares} does not normalize the columns of @var{x}. Use ## @qcode{pcares (zscore (@var{x}), @var{ndim})} in order to perform the ## principal components analysis based on standardized variables, i.e. based on ## correlations. Use @code{pcacov} in order to perform principal components ## analysis directly on a covariance or correlation matrix without constructing ## residuals. ## ## @subheading References ## @enumerate ## @item ## Jolliffe, I. T., Principal Component Analysis, 2nd Edition, Springer, 2002 ## @end enumerate ## ## @seealso{factoran, pcacov, pca} ## @end deftypefn function [residuals, reconstructed] = pcares (x, ndim) ## Check input arguments if (nargin < 2) error ("pcares: too few input arguments."); endif if (ndim > size (x, 2)) error ("pcares: NDIM must be less than or equal to the column of X."); endif ## Mean center data Xcentered = bsxfun (@minus, x, mean (x)); ## Apply svd to get the principal component coefficients [U, S, V] = svd (Xcentered); ## Use only the first ndim PCA components v = V(:,1:ndim); ## Calculate the residuals residuals = Xcentered - Xcentered * (v * v'); ## Compute extra output arguments if (nargout > 1) ## Reconstructed data using ndim PCA components reconstructed = x - residuals; endif endfunction %!demo %! x = [ 7 26 6 60; %! 1 29 15 52; %! 11 56 8 20; %! 11 31 8 47; %! 7 52 6 33; %! 11 55 9 22; %! 3 71 17 6; %! 1 31 22 44; %! 2 54 18 22; %! 21 47 4 26; %! 1 40 23 34; %! 11 66 9 12; %! 10 68 8 12]; %! %! ## As we increase the number of principal components, the norm %! ## of the residuals matrix will decrease %! r1 = pcares (x,1); %! n1 = norm (r1) %! r2 = pcares (x,2); %! n2 = norm (r2) %! r3 = pcares (x,3); %! n3 = norm (r3) %! r4 = pcares (x,4); %! n4 = norm (r4) ## Test output %!test %! load hald %! r1 = pcares (ingredients,1); %! r2 = pcares (ingredients,2); %! r3 = pcares (ingredients,3); %! assert (r1(1,:), [2.0350, 2.8304, -6.8378, 3.0879], 1e-4); %! assert (r2(1,:), [-2.4037, 2.6930, -1.6482, 2.3425], 1e-4); %! assert (r3(1,:), [ 0.2008, 0.1957, 0.2045, 0.1921], 1e-4); ## Test input validation %!error pcares (ones (20, 3)) %!error ... %! pcares (ones (30, 2), 3) statistics-release-1.6.3/inst/pdf.m000066400000000000000000000337721456127120000172470ustar00rootroot00000000000000## Copyright (C) 2016 Andreas Stahel ## Copyright (C) 2022-2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it ## under the terms of the GNU General Public License as published by ## the Free Software Foundation; either version 3 of the License, or ## (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program. If not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{y} =} pdf (@var{name}, @var{x}, @var{A}) ## @deftypefnx {statistics} {@var{y} =} pdf (@var{name}, @var{x}, @var{A}, @var{B}) ## @deftypefnx {statistics} {@var{y} =} pdf (@var{name}, @var{x}, @var{A}, @var{B}, @var{C}) ## ## Return the PDF of a univariate distribution evaluated at @var{x}. ## ## @code{pdf} is a wrapper for the univariate cumulative distribution functions ## available in the statistics package. See the corresponding functions' help ## to learn the signification of the parameters after @var{x}. ## ## @code{@var{y} = pdf (@var{name}, @var{x}, @var{A})} returns the CDF for the ## one-parameter distribution family specified by @var{name} and the ## distribution parameter @var{A}, evaluated at the values in @var{x}. ## ## @code{@var{y} = pdf (@var{name}, @var{x}, @var{A}, @var{B})} returns the CDF ## for the two-parameter distribution family specified by @var{name} and the ## distribution parameters @var{A} and @var{B}, evaluated at the values in ## @var{x}. ## ## @code{@var{y} = pdf (@var{name}, @var{x}, @var{A}, @var{B}, @var{C})} returns ## the CDF for the three-parameter distribution family specified by @var{name} ## and the distribution parameters @var{A}, @var{B}, and @var{C}, evaluated at ## the values in @var{x}. ## ## @var{name} must be a char string of the name or the abbreviation of the ## desired cumulative distribution function as listed in the followng table. ## The last column shows the number of required parameters that should be parsed ## after @var{x} to the desired PDF. ## ## @multitable @columnfractions 0.4 0.05 0.2 0.05 0.3 ## @headitem Distribution Name @tab @tab Abbreviation @tab @tab Input Parameters ## @item @qcode{"Beta"} @tab @tab @qcode{"beta"} @tab @tab 2 ## @item @qcode{"Binomial"} @tab @tab @qcode{"bino"} @tab @tab 2 ## @item @qcode{"Birnbaum-Saunders"} @tab @tab @qcode{"bisa"} @tab @tab 2 ## @item @qcode{"Burr"} @tab @tab @qcode{"burr"} @tab @tab 3 ## @item @qcode{"Cauchy"} @tab @tab @qcode{"cauchy"} @tab @tab 2 ## @item @qcode{"Chi-squared"} @tab @tab @qcode{"chi2"} @tab @tab 1 ## @item @qcode{"Extreme Value"} @tab @tab @qcode{"ev"} @tab @tab 2 ## @item @qcode{"Exponential"} @tab @tab @qcode{"exp"} @tab @tab 1 ## @item @qcode{"F-Distribution"} @tab @tab @qcode{"f"} @tab @tab 2 ## @item @qcode{"Gamma"} @tab @tab @qcode{"gam"} @tab @tab 2 ## @item @qcode{"Geometric"} @tab @tab @qcode{"geo"} @tab @tab 1 ## @item @qcode{"Generalized Extreme Value"} @tab @tab @qcode{"gev"} @tab @tab 3 ## @item @qcode{"Generalized Pareto"} @tab @tab @qcode{"gp"} @tab @tab 3 ## @item @qcode{"Gumbel"} @tab @tab @qcode{"gumbel"} @tab @tab 2 ## @item @qcode{"Half-normal"} @tab @tab @qcode{"hn"} @tab @tab 2 ## @item @qcode{"Hypergeometric"} @tab @tab @qcode{"hyge"} @tab @tab 3 ## @item @qcode{"Inverse Gaussian"} @tab @tab @qcode{"invg"} @tab @tab 2 ## @item @qcode{"Laplace"} @tab @tab @qcode{"laplace"} @tab @tab 2 ## @item @qcode{"Logistic"} @tab @tab @qcode{"logi"} @tab @tab 2 ## @item @qcode{"Log-Logistic"} @tab @tab @qcode{"logl"} @tab @tab 2 ## @item @qcode{"Lognormal"} @tab @tab @qcode{"logn"} @tab @tab 2 ## @item @qcode{"Nakagami"} @tab @tab @qcode{"naka"} @tab @tab 2 ## @item @qcode{"Negative Binomial"} @tab @tab @qcode{"nbin"} @tab @tab 2 ## @item @qcode{"Noncentral F-Distribution"} @tab @tab @qcode{"ncf"} @tab @tab 3 ## @item @qcode{"Noncentral Student T"} @tab @tab @qcode{"nct"} @tab @tab 2 ## @item @qcode{"Noncentral Chi-Squared"} @tab @tab @qcode{"ncx2"} @tab @tab 2 ## @item @qcode{"Normal"} @tab @tab @qcode{"norm"} @tab @tab 2 ## @item @qcode{"Poisson"} @tab @tab @qcode{"poiss"} @tab @tab 1 ## @item @qcode{"Rayleigh"} @tab @tab @qcode{"rayl"} @tab @tab 1 ## @item @qcode{"Rician"} @tab @tab @qcode{"rice"} @tab @tab 2 ## @item @qcode{"Student T"} @tab @tab @qcode{"t"} @tab @tab 1 ## @item @qcode{"location-scale T"} @tab @tab @qcode{"tls"} @tab @tab 3 ## @item @qcode{"Triangular"} @tab @tab @qcode{"tri"} @tab @tab 3 ## @item @qcode{"Discrete Uniform"} @tab @tab @qcode{"unid"} @tab @tab 1 ## @item @qcode{"Uniform"} @tab @tab @qcode{"unif"} @tab @tab 2 ## @item @qcode{"Von Mises"} @tab @tab @qcode{"vm"} @tab @tab 2 ## @item @qcode{"Weibull"} @tab @tab @qcode{"wbl"} @tab @tab 2 ## @end multitable ## ## @seealso{cdf, icdf, random, betapdf, binopdf, bisapdf, burrpdf, cauchypdf, ## chi2pdf, evpdf, exppdf, fpdf, gampdf, geopdf, gevpdf, gppdf, gumbelpdf, ## hnpdf, hygepdf, invgpdf, laplacepdf, logipdf, loglpdf, lognpdf, nakapdf, ## nbinpdf, ncfpdf, nctpdf, ncx2pdf, normpdf, poisspdf, raylpdf, ricepdf, ## tpdf, tripdf, unidpdf, unifpdf, vmpdf, wblpdf} ## @end deftypefn function y = pdf (name, x, varargin) ## implemented functions persistent allDF = { ... {"beta" , "Beta"}, @betapdf, 2, ... {"bino" , "Binomial"}, @binopdf, 2, ... {"bisa" , "Birnbaum-Saunders"}, @bisapdf, 2, ... {"burr" , "Burr"}, @burrpdf, 3, ... {"cauchy" , "Cauchy"}, @cauchypdf, 2, ... {"chi2" , "Chi-squared"}, @chi2pdf, 1, ... {"ev" , "Extreme Value"}, @evpdf, 2, ... {"exp" , "Exponential"}, @exppdf, 1, ... {"f" , "F-Distribution"}, @fpdf, 2, ... {"gam" , "Gamma"}, @gampdf, 2, ... {"geo" , "Geometric"}, @geopdf, 1, ... {"gev" , "Generalized Extreme Value"}, @gevpdf, 3, ... {"gp" , "Generalized Pareto"}, @gppdf, 3, ... {"gumbel" , "Gumbel"}, @gumbelpdf, 2, ... {"hn" , "Half-normal"}, @hnpdf, 2, ... {"hyge" , "Hypergeometric"}, @hygepdf, 3, ... {"invg" , "Inverse Gaussian"}, @invgpdf, 2, ... {"laplace" , "Laplace"}, @laplacepdf, 2, ... {"logi" , "Logistic"}, @logipdf, 2, ... {"logl" , "Log-Logistic"}, @loglpdf, 2, ... {"logn" , "Lognormal"}, @lognpdf, 2, ... {"naka" , "Nakagami"}, @nakapdf, 2, ... {"nbin" , "Negative Binomial"}, @nbinpdf, 2, ... {"ncf" , "Noncentral F-Distribution"}, @ncfpdf, 3, ... {"nct" , "Noncentral Student T"}, @nctpdf, 2, ... {"ncx2" , "Noncentral Chi-squared"}, @ncx2pdf, 2, ... {"norm" , "Normal"}, @normpdf, 2, ... {"poiss" , "Poisson"}, @poisspdf, 1, ... {"rayl" , "Rayleigh"}, @raylpdf, 1, ... {"rice" , "Rician"}, @ricepdf, 2, ... {"t" , "Student T"}, @tpdf, 1, ... {"tls" , "location-scale T"}, @tlspdf, 3, ... {"tri" , "Triangular"}, @tripdf, 3, ... {"unid" , "Discrete Uniform"}, @unidpdf, 1, ... {"unif" , "Uniform"}, @unifpdf, 2, ... {"vm" , "Von Mises"}, @vmpdf, 2, ... {"wbl" , "Weibull"}, @wblpdf, 2}; if (! ischar (name)) error ("pdf: distribution NAME must a char string."); endif ## Check X being numeric and real if (! isnumeric (x)) error ("pdf: X must be numeric."); elseif (! isreal (x)) error ("pdf: values in X must be real."); endif ## Get number of arguments nargs = numel (varargin); ## Get available functions pdfnames = allDF(1:3:end); pdfhandl = allDF(2:3:end); pdf_args = allDF(3:3:end); ## Search for PDF function idx = cellfun (@(x)any(strcmpi (name, x)), pdfnames); if (any (idx)) if (nargs == pdf_args{idx}) ## Check that all distribution parameters are numeric if (! all (cellfun (@(x)isnumeric(x), (varargin)))) error ("pdf: distribution parameters must be numeric."); endif ## Call appropriate iCDF y = feval (pdfhandl{idx}, x, varargin{:}); else if (pdf_args{idx} == 1) error ("pdf: %s distribution requires 1 parameter.", name); else error ("pdf: %s distribution requires %d parameters.", ... name, pdf_args{idx}); endif endif else error ("pdf: %s distribution is not implemented in Statistics.", name); endif endfunction ## Test results %!shared x %! x = [1:5]; %!assert (pdf ("Beta", x, 5, 2), betapdf (x, 5, 2)) %!assert (pdf ("beta", x, 5, 2), betapdf (x, 5, 2)) %!assert (pdf ("Binomial", x, 5, 2), binopdf (x, 5, 2)) %!assert (pdf ("bino", x, 5, 2), binopdf (x, 5, 2)) %!assert (pdf ("Birnbaum-Saunders", x, 5, 2), bisapdf (x, 5, 2)) %!assert (pdf ("bisa", x, 5, 2), bisapdf (x, 5, 2)) %!assert (pdf ("Burr", x, 5, 2, 2), burrpdf (x, 5, 2, 2)) %!assert (pdf ("burr", x, 5, 2, 2), burrpdf (x, 5, 2, 2)) %!assert (pdf ("Cauchy", x, 5, 2), cauchypdf (x, 5, 2)) %!assert (pdf ("cauchy", x, 5, 2), cauchypdf (x, 5, 2)) %!assert (pdf ("Chi-squared", x, 5), chi2pdf (x, 5)) %!assert (pdf ("chi2", x, 5), chi2pdf (x, 5)) %!assert (pdf ("Extreme Value", x, 5, 2), evpdf (x, 5, 2)) %!assert (pdf ("ev", x, 5, 2), evpdf (x, 5, 2)) %!assert (pdf ("Exponential", x, 5), exppdf (x, 5)) %!assert (pdf ("exp", x, 5), exppdf (x, 5)) %!assert (pdf ("F-Distribution", x, 5, 2), fpdf (x, 5, 2)) %!assert (pdf ("f", x, 5, 2), fpdf (x, 5, 2)) %!assert (pdf ("Gamma", x, 5, 2), gampdf (x, 5, 2)) %!assert (pdf ("gam", x, 5, 2), gampdf (x, 5, 2)) %!assert (pdf ("Geometric", x, 5), geopdf (x, 5)) %!assert (pdf ("geo", x, 5), geopdf (x, 5)) %!assert (pdf ("Generalized Extreme Value", x, 5, 2, 2), gevpdf (x, 5, 2, 2)) %!assert (pdf ("gev", x, 5, 2, 2), gevpdf (x, 5, 2, 2)) %!assert (pdf ("Generalized Pareto", x, 5, 2, 2), gppdf (x, 5, 2, 2)) %!assert (pdf ("gp", x, 5, 2, 2), gppdf (x, 5, 2, 2)) %!assert (pdf ("Gumbel", x, 5, 2), gumbelpdf (x, 5, 2)) %!assert (pdf ("gumbel", x, 5, 2), gumbelpdf (x, 5, 2)) %!assert (pdf ("Half-normal", x, 5, 2), hnpdf (x, 5, 2)) %!assert (pdf ("hn", x, 5, 2), hnpdf (x, 5, 2)) %!assert (pdf ("Hypergeometric", x, 5, 2, 2), hygepdf (x, 5, 2, 2)) %!assert (pdf ("hyge", x, 5, 2, 2), hygepdf (x, 5, 2, 2)) %!assert (pdf ("Inverse Gaussian", x, 5, 2), invgpdf (x, 5, 2)) %!assert (pdf ("invg", x, 5, 2), invgpdf (x, 5, 2)) %!assert (pdf ("Laplace", x, 5, 2), laplacepdf (x, 5, 2)) %!assert (pdf ("laplace", x, 5, 2), laplacepdf (x, 5, 2)) %!assert (pdf ("Logistic", x, 5, 2), logipdf (x, 5, 2)) %!assert (pdf ("logi", x, 5, 2), logipdf (x, 5, 2)) %!assert (pdf ("Log-Logistic", x, 5, 2), loglpdf (x, 5, 2)) %!assert (pdf ("logl", x, 5, 2), loglpdf (x, 5, 2)) %!assert (pdf ("Lognormal", x, 5, 2), lognpdf (x, 5, 2)) %!assert (pdf ("logn", x, 5, 2), lognpdf (x, 5, 2)) %!assert (pdf ("Nakagami", x, 5, 2), nakapdf (x, 5, 2)) %!assert (pdf ("naka", x, 5, 2), nakapdf (x, 5, 2)) %!assert (pdf ("Negative Binomial", x, 5, 2), nbinpdf (x, 5, 2)) %!assert (pdf ("nbin", x, 5, 2), nbinpdf (x, 5, 2)) %!assert (pdf ("Noncentral F-Distribution", x, 5, 2, 2), ncfpdf (x, 5, 2, 2)) %!assert (pdf ("ncf", x, 5, 2, 2), ncfpdf (x, 5, 2, 2)) %!assert (pdf ("Noncentral Student T", x, 5, 2), nctpdf (x, 5, 2)) %!assert (pdf ("nct", x, 5, 2), nctpdf (x, 5, 2)) %!assert (pdf ("Noncentral Chi-Squared", x, 5, 2), ncx2pdf (x, 5, 2)) %!assert (pdf ("ncx2", x, 5, 2), ncx2pdf (x, 5, 2)) %!assert (pdf ("Normal", x, 5, 2), normpdf (x, 5, 2)) %!assert (pdf ("norm", x, 5, 2), normpdf (x, 5, 2)) %!assert (pdf ("Poisson", x, 5), poisspdf (x, 5)) %!assert (pdf ("poiss", x, 5), poisspdf (x, 5)) %!assert (pdf ("Rayleigh", x, 5), raylpdf (x, 5)) %!assert (pdf ("rayl", x, 5), raylpdf (x, 5)) %!assert (pdf ("Rician", x, 5, 1), ricepdf (x, 5, 1)) %!assert (pdf ("rice", x, 5, 1), ricepdf (x, 5, 1)) %!assert (pdf ("Student T", x, 5), tpdf (x, 5)) %!assert (pdf ("t", x, 5), tpdf (x, 5)) %!assert (pdf ("location-scale T", x, 5, 1, 2), tlspdf (x, 5, 1, 2)) %!assert (pdf ("tls", x, 5, 1, 2), tlspdf (x, 5, 1, 2)) %!assert (pdf ("Triangular", x, 5, 2, 2), tripdf (x, 5, 2, 2)) %!assert (pdf ("tri", x, 5, 2, 2), tripdf (x, 5, 2, 2)) %!assert (pdf ("Discrete Uniform", x, 5), unidpdf (x, 5)) %!assert (pdf ("unid", x, 5), unidpdf (x, 5)) %!assert (pdf ("Uniform", x, 5, 2), unifpdf (x, 5, 2)) %!assert (pdf ("unif", x, 5, 2), unifpdf (x, 5, 2)) %!assert (pdf ("Von Mises", x, 5, 2), vmpdf (x, 5, 2)) %!assert (pdf ("vm", x, 5, 2), vmpdf (x, 5, 2)) %!assert (pdf ("Weibull", x, 5, 2), wblpdf (x, 5, 2)) %!assert (pdf ("wbl", x, 5, 2), wblpdf (x, 5, 2)) ## Test input validation %!error pdf (1) %!error pdf ({"beta"}) %!error pdf ("beta", {[1 2 3 4 5]}) %!error pdf ("beta", "text") %!error pdf ("beta", 1+i) %!error ... %! pdf ("Beta", x, "a", 2) %!error ... %! pdf ("Beta", x, 5, "") %!error ... %! pdf ("Beta", x, 5, {2}) %!error pdf ("chi2", x) %!error pdf ("Beta", x, 5) %!error pdf ("Burr", x, 5) %!error pdf ("Burr", x, 5, 2) statistics-release-1.6.3/inst/pdist.m000066400000000000000000000320571456127120000176140ustar00rootroot00000000000000## Copyright (C) 2008 Francesco Potortì ## Copyright (C) 2024 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{D} =} pdist (@var{X}) ## @deftypefnx {statistics} {@var{D} =} pdist (@var{X}, @var{Distance}) ## @deftypefnx {statistics} {@var{D} =} pdist (@var{X}, @var{Distance}, @var{DistParameter}) ## ## Return the distance between any two rows in @var{X}. ## ## @code{@var{D} = pdist (@var{X}} calculates the euclidean distance between ## pairs of observations in @var{X}. @var{X} must be an @math{MxP} numeric ## matrix representing @math{M} points in @math{P}-dimensional space. This ## function computes the pairwise distances returned in @var{D} as an ## @math{Mx(M-1)/P} row vector. Use @code{@var{Z} = squareform (@var{D})} to ## convert the row vector @var{D} into a an @math{MxM} symmetric matrix @var{Z}, ## where @qcode{@var{Z}(i,j)} corresponds to the pairwise distance between ## points @qcode{i} and @qcode{j}. ## ## @code{@var{D} = pdist (@var{X}, @var{Y}, @var{Distance})} returns the ## distance between pairs of observations in @var{X} using the metric specified ## by @var{Distance}, which can be any of the following options. ## ## @multitable @columnfractions 0.23 0.02 0.65 ## @item @qcode{"euclidean"} @tab @tab Euclidean distance. ## @item @qcode{"squaredeuclidean"} @tab @tab Squared Euclidean distance. ## @item @qcode{"seuclidean"} @tab @tab standardized Euclidean distance. Each ## coordinate difference between the rows in @var{X} and the query matrix ## @var{Y} is scaled by dividing by the corresponding element of the standard ## deviation computed from @var{X}. A different scaling vector can be specified ## with the subsequent @var{DistParameter} input argument. ## @item @qcode{"mahalanobis"} @tab @tab Mahalanobis distance, computed using a ## positive definite covariance matrix. A different covariance matrix can be ## specified with the subsequent @var{DistParameter} input argument. ## @item @qcode{"cityblock"} @tab @tab City block distance. ## @item @qcode{"minkowski"} @tab @tab Minkowski distance. The default exponent ## is 2. A different exponent can be specified with the subsequent ## @var{DistParameter} input argument. ## @item @qcode{"chebychev"} @tab @tab Chebychev distance (maximum coordinate ## difference). ## @item @qcode{"cosine"} @tab @tab One minus the cosine of the included angle ## between points (treated as vectors). ## @item @qcode{"correlation"} @tab @tab One minus the sample linear correlation ## between observations (treated as sequences of values). ## @item @qcode{"hamming"} @tab @tab Hamming distance, which is the percentage ## of coordinates that differ. ## @item @qcode{"jaccard"} @tab @tab One minus the Jaccard coefficient, which is ## the percentage of nonzero coordinates that differ. ## @item @qcode{"spearman"} @tab @tab One minus the sample Spearman's rank ## correlation between observations (treated as sequences of values). ## @item @var{@@distfun} @tab @tab Custom distance function handle. A distance ## function of the form @code{function @var{D2} = distfun (@var{XI}, @var{YI})}, ## where @var{XI} is a @math{1xP} vector containing a single observation in ## @math{P}-dimensional space, @var{YI} is an @math{NxP} matrix containing an ## arbitrary number of observations in the same @math{P}-dimensional space, and ## @var{D2} is an @math{NxP} vector of distances, where @qcode{(@var{D2}k)} is ## the distance between observations @var{XI} and @qcode{(@var{YI}k,:)}. ## @end multitable ## ## @code{@var{D} = pdist (@var{X}, @var{Y}, @var{Distance}, @var{DistParameter})} ## returns the distance using the metric specified by @var{Distance} and ## @var{DistParameter}. The latter one can only be specified when the selected ## @var{Distance} is @qcode{"seuclidean"}, @qcode{"minkowski"}, and ## @qcode{"mahalanobis"}. ## ## @seealso{pdist2, squareform, linkage} ## @end deftypefn function D = pdist (X, varargin) ## Check input data if (nargin < 1) error ("pdist: too few input arguments."); endif if (! isnumeric (X) || isempty (X)) error ("pdist: X must be a nonempty numeric matrix."); endif if (ndims (X) != 2) error ("pdist: X must be a two-dimensional matrix."); endif if (rows (X) < 2) D = cast (zeros (1, 0), class (X)); return; endif ## Add default values Distance = "euclidean"; # Distance metric DistParameter = []; # Distance parameter ## Parse additional Distance metric and Distance parameter (if available) DMs = {"euclidean", "squaredeuclidean", "seuclidean", ... "mahalanobis", "cityblock", "minkowski", "chebychev", ... "cosine", "correlation", "hamming", "jaccard", "spearman"}; if (numel (varargin) > 0) if (any (strcmpi (DMs, varargin{1}))) Distance = tolower (varargin{1}); elseif (is_function_handle (varargin{1})) Distance = varargin{1}; else error ("pdist: invalid value for Distance input argument."); endif endif if (numel (varargin) > 1) if (isnumeric (varargin{2})) DistParameter = varargin{2}; else error ("pdist: invalid value for DistParameter input argument."); endif endif ## Calculate selected distance order = nchoosek (1:rows (X) ,2); ix = order (:,1); iy = order (:,2); ## Handle build-in distance metric if (ischar (Distance)) switch (Distance) case "euclidean" D = sqrt (sum ((X(ix(:),:) - X(iy(:),:)) .^ 2, 2)); case "squaredeuclidean" D = sum ((X(ix(:),:) - X(iy(:),:)) .^ 2, 2); case "seuclidean" if (isempty (DistParameter)) DistParameter = std (X, [], 1); else if (numel (DistParameter) != columns (X)) error (strcat (["pdist2: DistParameter for standardized"], ... [" euclidean must be a vector of equal length"], ... [" to the number of columns in X."])); endif if (any (DistParameter < 0)) error (strcat (["pdist2: DistParameter for standardized"], ... [" euclidean must be a nonnegative vector."])); endif endif DistParameter(DistParameter == 0) = 1; # fix constant variable D = sqrt (sum (((X(ix(:),:) - X(iy(:),:)) ./ DistParameter) .^ 2, 2)); case "mahalanobis" if (isempty (DistParameter)) DistParameter = cov (X(! any (isnan (X), 2),:)); else if (columns (DistParameter) != columns (X)) error (strcat (["pdist2: DistParameter for mahalanobis"], ... [" distance must be a covariance matrix with"], ... [" the same number of columns as X."])); endif [~, p] = chol (DistParameter); if (p != 0) error (strcat (["pdist2: covariance matrix for mahalanobis"], ... [" distance must be symmetric and positive"], ... [" definite."])); endif endif dxx = X(ix(:),:) - X(iy(:),:); ## Catch warning if matrix is close to singular or badly scaled. [DP_inv, rc] = inv (DistParameter); if (rc < eps) msg = sprintf (strcat (["pdist: matrix is close to"], ... [" singular or badly scaled.\n RCOND = "], ... [" %e. Results may be inaccurate."]), rc); warning (msg); endif D = sqrt (sum ((dxx * DP_inv) .* dxx, 2)); case "cityblock" D = sum (abs (X(ix(:),:) - X(iy(:),:)), 2); case "minkowski" if (isempty (DistParameter)) DistParameter = 2; else if (! (isnumeric (DistParameter) && isscalar (DistParameter) && DistParameter > 0)) error (strcat (["pdist2: DistParameter for minkowski distance"],... [" must be a positive scalar."])); endif endif D = sum (abs (X(ix(:),:) - X(iy(:),:)) .^ DistParameter, 2) .^ ... (1 / DistParameter); case "chebychev" D = max (abs (X(ix(:),:) - X(iy(:),:)), [], 2); case "cosine" sx = sum (X .^ 2, 2) .^ (-1 / 2); D = 1 - sum (X(ix(:),:) .* X(iy(:),:), 2) .* sx(ix(:)) .* sx(iy(:)); case "correlation" mX = mean (X(ix(:),:), 2); mY = mean (X(iy(:),:), 2); xy = sum ((X(ix(:),:) - mX) .* (X(iy(:),:) - mY), 2); xx = sqrt (sum ((X(ix(:),:) - mX) .* (X(ix(:),:) - mX), 2)); yy = sqrt (sum ((X(iy(:),:) - mY) .* (X(iy(:),:) - mY), 2)); D = 1 - (xy ./ (xx .* yy)); case "hamming" D = mean (abs (X(ix(:),:) != X(iy(:),:)), 2); case "jaccard" xx0 = (X(ix(:),:) != 0 | X(iy(:),:) != 0); D = sum ((X(ix(:),:) != X(iy(:),:)) & xx0, 2) ./ sum (xx0, 2); case "spearman" for i = 1:size (X, 1) rX(i,:) = tiedrank (X(i,:)); endfor rM = (size (X, 2) + 1) / 2; xy = sum ((rX(ix(:),:) - rM) .* (rX(iy(:),:) - rM), 2); xx = sqrt (sum ((rX(ix(:),:) - rM) .* (rX(ix(:),:) - rM), 2)); yy = sqrt (sum ((rX(iy(:),:) - rM) .* (rX(iy(:),:) - rM), 2)); D = 1 - (xy ./ (xx .* yy)); endswitch ## Force output ot row vector D = D'; endif ## Handle a function handle if (is_function_handle (Distance)) ## Check the input output sizes of the user function D2 = []; try D2 = Distance (X(1,:), X([2:end],:)); catch ME error ("pdist: invalid function handle for distance metric."); end_try_catch Xrows = rows (X) - 1; if (! isequal (size (D2), [Xrows, 1])) error ("pdist: custom distance function produces wrong output size."); endif ## Evaluate user defined distance metric function D = zeros (1, numel (ix)); id_beg = 1; for r = 1:Xrows id_end = id_beg + size (X([r+1:end],:), 1) - 1; D(id_beg:id_end) = feval (Distance, X(r,:), X([r+1:end],:)); id_beg = id_end + 1; endfor endif endfunction ## Test output %!shared xy, t, eucl, x %! xy = [0 1; 0 2; 7 6; 5 6]; %! t = 1e-3; %! eucl = @(v,m) sqrt(sumsq(repmat(v,rows(m),1)-m,2)); %! x = [1 2 3; 4 5 6; 7 8 9; 3 2 1]; %!assert (pdist (xy), [1.000 8.602 7.071 8.062 6.403 2.000], t); %!assert (pdist (xy, eucl), [1.000 8.602 7.071 8.062 6.403 2.000], t); %!assert (pdist (xy, "euclidean"), [1.000 8.602 7.071 8.062 6.403 2.000], t); %!assert (pdist (xy, "seuclidean"), [0.380 2.735 2.363 2.486 2.070 0.561], t); %!assert (pdist (xy, "mahalanobis"), [1.384 1.967 2.446 2.384 1.535 2.045], t); %!assert (pdist (xy, "cityblock"), [1.000 12.00 10.00 11.00 9.000 2.000], t); %!assert (pdist (xy, "minkowski"), [1.000 8.602 7.071 8.062 6.403 2.000], t); %!assert (pdist (xy, "minkowski", 3), [1.000 7.763 6.299 7.410 5.738 2.000], t); %!assert (pdist (xy, "cosine"), [0.000 0.349 0.231 0.349 0.231 0.013], t); %!assert (pdist (xy, "correlation"), [0.000 2.000 0.000 2.000 0.000 2.000], t); %!assert (pdist (xy, "spearman"), [0.000 2.000 0.000 2.000 0.000 2.000], t); %!assert (pdist (xy, "hamming"), [0.500 1.000 1.000 1.000 1.000 0.500], t); %!assert (pdist (xy, "jaccard"), [1.000 1.000 1.000 1.000 1.000 0.500], t); %!assert (pdist (xy, "chebychev"), [1.000 7.000 5.000 7.000 5.000 2.000], t); %!assert (pdist (x), [5.1962, 10.3923, 2.8284, 5.1962, 5.9161, 10.7703], 1e-4); %!assert (pdist (x, "euclidean"), ... %! [5.1962, 10.3923, 2.8284, 5.1962, 5.9161, 10.7703], 1e-4); %!assert (pdist (x, eucl), ... %! [5.1962, 10.3923, 2.8284, 5.1962, 5.9161, 10.7703], 1e-4); %!assert (pdist (x, "squaredeuclidean"), [27, 108, 8, 27, 35, 116]); %!assert (pdist (x, "seuclidean"), ... %! [1.8071, 3.6142, 0.9831, 1.8071, 1.8143, 3.4854], 1e-4); %!warning ... %! pdist (x, "mahalanobis"); %!assert (pdist (x, "cityblock"), [9, 18, 4, 9, 9, 18]); %!assert (pdist (x, "minkowski"), ... %! [5.1962, 10.3923, 2.8284, 5.1962, 5.9161, 10.7703], 1e-4); %!assert (pdist (x, "minkowski", 3), ... %! [4.3267, 8.6535, 2.5198, 4.3267, 5.3485, 9.2521], 1e-4); %!assert (pdist (x, "cosine"), ... %! [0.0254, 0.0406, 0.2857, 0.0018, 0.1472, 0.1173], 1e-4); %!assert (pdist (x, "correlation"), [0, 0, 2, 0, 2, 2], 1e-14); %!assert (pdist (x, "spearman"), [0, 0, 2, 0, 2, 2], 1e-14); %!assert (pdist (x, "hamming"), [1, 1, 2/3, 1, 1, 1]); %!assert (pdist (x, "jaccard"), [1, 1, 2/3, 1, 1, 1]); %!assert (pdist (x, "chebychev"), [3, 6, 2, 3, 5, 8]); statistics-release-1.6.3/inst/pdist2.m000066400000000000000000000454641456127120000177040ustar00rootroot00000000000000## Copyright (C) 2014-2019 Piotr Dollar ## Copyright (C) 2024 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation; either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; if not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{D} =} pdist2 (@var{X}, @var{Y}) ## @deftypefnx {statistics} {@var{D} =} pdist2 (@var{X}, @var{Y}, @var{Distance}) ## @deftypefnx {statistics} {@var{D} =} pdist2 (@var{X}, @var{Y}, @var{Distance}, @var{DistParameter}) ## @deftypefnx {statistics} {@var{D} =} pdist2 (@dots{}, @var{Name}, @var{Value}) ## @deftypefnx {statistics} {[@var{D}, @var{I}] =} pdist2 (@dots{}, @var{Name}, @var{Value}) ## ## Compute pairwise distance between two sets of vectors. ## ## @code{@var{D} = pdist2 (@var{X}, @var{Y})} calculates the euclidean distance ## between each pair of observations in @var{X} and @var{Y}. Let @var{X} be an ## @math{MxP} matrix representing @math{M} points in @math{P}-dimensional space ## and @var{Y} be an @math{NxP} matrix representing another set of points in the ## same space. This function computes the @math{MxN} distance matrix @var{D}, ## where @qcode{@var{D}(i,j)} is the distance between @qcode{@var{X}(i,:)} and ## @qcode{@var{Y}(j,:)}. ## ## @code{@var{D} = pdist2 (@var{X}, @var{Y}, @var{Distance})} returns the ## distance between each pair of observations in @var{X} and @var{Y} using the ## metric specified by @var{Distance}, which can be any of the following options. ## ## @multitable @columnfractions 0.23 0.02 0.65 ## @item @qcode{"euclidean"} @tab @tab Euclidean distance. ## @item @qcode{"squaredeuclidean"} @tab @tab Squared Euclidean distance. ## @item @qcode{"seuclidean"} @tab @tab standardized Euclidean distance. Each ## coordinate difference between the rows in @var{X} and the query matrix ## @var{Y} is scaled by dividing by the corresponding element of the standard ## deviation computed from @var{X}. A different scaling vector can be specified ## with the subsequent @var{DistParameter} input argument. ## @item @qcode{"mahalanobis"} @tab @tab Mahalanobis distance, computed using a ## positive definite covariance matrix. A different covariance matrix can be ## specified with the subsequent @var{DistParameter} input argument. ## @item @qcode{"cityblock"} @tab @tab City block distance. ## @item @qcode{"minkowski"} @tab @tab Minkowski distance. The default exponent ## is 2. A different exponent can be specified with the subsequent ## @var{DistParameter} input argument. ## @item @qcode{"chebychev"} @tab @tab Chebychev distance (maximum coordinate ## difference). ## @item @qcode{"cosine"} @tab @tab One minus the cosine of the included angle ## between points (treated as vectors). ## @item @qcode{"correlation"} @tab @tab One minus the sample linear correlation ## between observations (treated as sequences of values). ## @item @qcode{"hamming"} @tab @tab Hamming distance, which is the percentage ## of coordinates that differ. ## @item @qcode{"jaccard"} @tab @tab One minus the Jaccard coefficient, which is ## the percentage of nonzero coordinates that differ. ## @item @qcode{"spearman"} @tab @tab One minus the sample Spearman's rank ## correlation between observations (treated as sequences of values). ## @item @var{@@distfun} @tab @tab Custom distance function handle. A distance ## function of the form @code{function @var{D2} = distfun (@var{XI}, @var{YI})}, ## where @var{XI} is a @math{1xP} vector containing a single observation in ## @math{P}-dimensional space, @var{YI} is an @math{NxP} matrix containing an ## arbitrary number of observations in the same @math{P}-dimensional space, and ## @var{D2} is an @math{NxP} vector of distances, where @qcode{(@var{D2}k)} is ## the distance between observations @var{XI} and @qcode{(@var{YI}k,:)}. ## @end multitable ## ## @code{@var{D} = pdist2 (@var{X}, @var{Y}, @var{Distance}, @var{DistParameter})} ## returns the distance using the metric specified by @var{Distance} and ## @var{DistParameter}. The latter one can only be specified when the selected ## @var{Distance} is @qcode{"seuclidean"}, @qcode{"minkowski"}, and ## @qcode{"mahalanobis"}. ## ## @code{@var{D} = pdist2 (@dots{}, @var{Name}, @var{Value})} for any previous ## arguments, modifies the computation using @var{Name}-@var{Value} parameters. ## @itemize ## @item ## @code{@var{D} = pdist2 (@var{X}, @var{Y}, @var{Distance}, @qcode{"Smallest"}, ## @var{K})} computes the distance using the metric specified by ## @var{Distance} and returns the @var{K} smallest pairwise distances to ## observations in @var{X} for each observation in @var{Y} in ascending order. ## @item ## @code{@var{D} = pdist2 (@var{X}, @var{Y}, @var{Distance}, @var{DistParameter}, ## @qcode{"Largest"}, @var{K})} computes the distance using the metric specified ## by @var{Distance} and @var{DistParameter} and returns the @var{K} largest ## pairwise distances in descending order. ## @end itemize ## ## @code{[@var{D}, @var{I}] = pdist2 (@dots{}, @var{Name}, @var{Value})} also ## returns the matrix @var{I}, which contains the indices of the observations in ## @var{X} corresponding to the distances in @var{D}. You must specify either ## @qcode{"Smallest"} or @qcode{"Largest"} as an optional @var{Name}-@var{Value} ## pair pair argument to compute the second output argument. ## ## @seealso{pdist, knnsearch, rangesearch} ## @end deftypefn function [D, I] = pdist2 (X, Y, varargin) ## Check input data if (nargin < 2) error ("pdist2: too few input arguments."); endif if (size (X, 2) != size (Y, 2)) error ("pdist2: X and Y must have equal number of columns."); endif if (ndims (X) != 2 || ndims (Y) != 2) error ("pdist2: X and Y must be 2 dimensional matrices."); endif ## Add default values Distance = "euclidean"; # Distance metric DistParameter = []; # Distance parameter SortOrder = []; # Flag for sorting distances to find ## Parse additional Distance metric and Distance parameter (if available) DMs = {"euclidean", "squaredeuclidean", "seuclidean", ... "mahalanobis", "cityblock", "minkowski", "chebychev", ... "cosine", "correlation", "hamming", "jaccard", "spearman"}; if (numel (varargin) > 0) if (any (strcmpi (DMs, varargin{1}))) Distance = tolower (varargin{1}); varargin(1) = []; if (numel (varargin) > 0) if (isnumeric (varargin{1})) DistParameter = varargin{1}; varargin(1) = []; endif endif elseif (is_function_handle (varargin{1})) Distance = varargin{1}; varargin(1) = []; if (numel (varargin) > 0) if (isnumeric (varargin{1})) DistParameter = varargin{1}; varargin(1) = []; endif endif endif endif ## Parse additional parameters in Name/Value pairs parcount = 0; while (numel (varargin) > 0) if (numel (varargin) < 2) error ("pdist2: missing value in optional name/value paired arguments."); endif switch (tolower (varargin{1})) case "smallest" SortOrder = "ascend"; K = varargin{2}; parcount += 1; case "largest" SortOrder = "descend"; K = varargin{2}; parcount += 1; otherwise error ("pdist2: invalid NAME in optional pairs of arguments."); endswitch varargin(1:2) = []; endwhile ## Check additional arguments if (parcount > 1) error ("pdist2: you can only use either Smallest or Largest."); endif if (isempty (SortOrder) && nargout > 1) error (strcat (["pdist2: Smallest or Largest must be specified"], ... [" to compute second output."])); endif ## Calculate selected distance [ix, iy] = meshgrid (1:size (X, 1), 1:size (Y, 1)); ## Handle build-in distance metric if (ischar (Distance)) switch (Distance) case "euclidean" D = sqrt (sum ((X(ix(:),:) - Y(iy(:),:)) .^ 2, 2)); case "squaredeuclidean" D = sum ((X(ix(:),:) - Y(iy(:),:)) .^ 2, 2); case "seuclidean" if (isempty (DistParameter)) DistParameter = std (X, [], 1); else if (numel (DistParameter) != columns (X)) error (strcat (["pdist2: DistParameter for standardized"], ... [" euclidean must be a vector of equal length"], ... [" to the number of columns in X."])); endif if (any (DistParameter < 0)) error (strcat (["pdist2: DistParameter for standardized"], ... [" euclidean must be a nonnegative vector."])); endif endif DistParameter(DistParameter == 0) = 1; # fix constant variable D = sqrt (sum (((X(ix(:),:) - Y(iy(:),:)) ./ DistParameter) .^ 2, 2)); case "mahalanobis" if (isempty (DistParameter)) DistParameter = cov (X(! any (isnan (X), 2),:)); else if (columns (DistParameter) != columns (X)) error (strcat (["pdist2: DistParameter for mahalanobis"], ... [" distance must be a covariance matrix with"], ... [" the same number of columns as X."])); endif [~, p] = chol (DistParameter); if (p != 0) error (strcat (["pdist2: covariance matrix for mahalanobis"], ... [" distance must be symmetric and positive"], ... [" definite."])); endif endif ## Catch warning if matrix is close to singular or badly scaled. [DP_inv, rc] = inv (DistParameter); if (rc < eps) msg = sprintf (strcat (["pdist2: matrix is close to"], ... [" singular or badly scaled.\n RCOND = "], ... [" %e. Results may be inaccurate."]), rc); warning (msg); endif dxy = X(ix(:),:) - Y(iy(:),:); D = sqrt (sum ((dxy * DP_inv) .* dxy, 2)); case "cityblock" D = sum (abs (X(ix(:),:) - Y(iy(:),:)), 2); case "minkowski" if (isempty (DistParameter)) DistParameter = 2; else if (! (isnumeric (DistParameter) && isscalar (DistParameter) && DistParameter > 0)) error (strcat (["pdist2: DistParameter for minkowski distance"],... [" must be a positive scalar."])); endif endif D = sum (abs (X(ix(:),:) - Y(iy(:),:)) .^ DistParameter, 2) .^ ... (1 / DistParameter); case "chebychev" D = max (abs (X(ix(:),:) - Y(iy(:),:)), [], 2); case "cosine" sx = sum (X .^ 2, 2) .^ (-1 / 2); sy = sum (Y .^ 2, 2) .^ (-1 / 2); D = 1 - sum (X(ix(:),:) .* Y(iy(:),:), 2) .* sx(ix(:)) .* sy(iy(:)); case "correlation" mX = mean (X(ix(:),:), 2); mY = mean (Y(iy(:),:), 2); xy = sum ((X(ix(:),:) - mX) .* (Y(iy(:),:) - mY), 2); xx = sqrt (sum ((X(ix(:),:) - mX) .* (X(ix(:),:) - mX), 2)); yy = sqrt (sum ((Y(iy(:),:) - mY) .* (Y(iy(:),:) - mY), 2)); D = 1 - (xy ./ (xx .* yy)); case "hamming" D = mean (abs (X(ix(:),:) != Y(iy(:),:)), 2); case "jaccard" xy0 = (X(ix(:),:) != 0 | Y(iy(:),:) != 0); D = sum ((X(ix(:),:) != Y(iy(:),:)) & xy0, 2) ./ sum (xy0, 2); case "spearman" for i = 1:size (X, 1) rX(i,:) = tiedrank (X(i,:)); endfor for i = 1:size (Y, 1) rY(i,:) = tiedrank (Y(i,:)); endfor rM = (size (X, 2) + 1) / 2; xy = sum ((rX(ix(:),:) - rM) .* (rY(iy(:),:) - rM), 2); xx = sqrt (sum ((rX(ix(:),:) - rM) .* (rX(ix(:),:) - rM), 2)); yy = sqrt (sum ((rY(iy(:),:) - rM) .* (rY(iy(:),:) - rM), 2)); D = 1 - (xy ./ (xx .* yy)); endswitch endif ## Handle a function handle if (is_function_handle (Distance)) ## Check the input output sizes of the user function D2 = []; try D2 = Distance (X(1,:), Y); catch ME error ("pdist2: invalid function handle for distance metric."); end_try_catch Yrows = rows (Y); if (! isequal (size (D2), [Yrows, 1])) error ("pdist2: custom distance function produces wrong output size."); endif ## Evaluate user defined distance metric function Yrows = rows (Y); D = zeros (numel (ix), 1); id_beg = 1; for r = 1:rows (X) id_end = id_beg + Yrows - 1; D(id_beg:id_end) = feval (Distance, X(r,:), Y); id_beg = id_end + 1; endfor endif ## From vector to matrix D = reshape (D, size (Y, 1), size (X, 1))'; if (nargout > 1) [D, I] = sort (D', 2, SortOrder); K = min (size (D, 2), K); # fix max K to avoid out of bound error D = D(:,1:K)'; I = I(:,1:K)'; endif endfunction ## Test output %!shared x, y, xx %! x = [1, 1, 1; 2, 2, 2; 3, 3, 3]; %! y = [0, 0, 0; 1, 2, 3; 0, 2, 4; 4, 7, 1]; %! xx = [1 2 3; 4 5 6; 7 8 9; 3 2 1]; %!test %! d = sqrt([3, 5, 11, 45; 12, 2, 8, 30; 27, 5, 11, 21]); %! assert (pdist2 (x, y), d); %!test %! d = [5.1962, 2.2361, 3.3166, 6.7082; ... %! 3.4641, 2.2361, 3.3166, 5.4772]; %! i = [3, 1, 1, 1; 2, 3, 3, 2]; %! [D, I] = pdist2 (x, y, "euclidean", "largest", 2); %! assert ({D, I}, {d, i}, 1e-4); %!test %! d = [1.7321, 1.4142, 2.8284, 4.5826; ... %! 3.4641, 2.2361, 3.3166, 5.4772]; %! i = [1, 2, 2, 3;2, 1, 1, 2]; %! [D, I] = pdist2 (x, y, "euclidean", "smallest", 2); %! assert ({D, I}, {d, i}, 1e-4); %!test %! yy = [1 2 3;5 6 7;9 5 1]; %! d = [0, 6.1644, 5.3852; 1.4142, 6.9282, 8.7750; ... %! 3.7417, 7.0711, 9.9499; 6.1644, 10.4881, 10.3441]; %! i = [2, 4, 4; 3, 2, 2; 1, 3, 3; 4, 1, 1]; %! [D, I] = pdist2 (y, yy, "euclidean", "smallest", 4); %! assert ({D, I}, {d, i}, 1e-4); %!test %! yy = [1 2 3;5 6 7;9 5 1]; %! d = [0, 38, 29; 2, 48, 77; 14, 50, 99; 38, 110, 107]; %! i = [2, 4, 4; 3, 2, 2; 1, 3, 3; 4, 1, 1]; %! [D, I] = pdist2 (y, yy, "squaredeuclidean", "smallest", 4); %! assert ({D, I}, {d, i}, 1e-4); %!test %! yy = [1 2 3;5 6 7;9 5 1]; %! d = [0, 3.3256, 2.7249; 0.7610, 3.3453, 4.4799; ... %! 1.8514, 3.3869, 5.0703; 2.5525, 5.0709, 5.1297]; %! i = [2, 2, 4; 3, 4, 2; 1, 3, 1; 4, 1, 3]; %! [D, I] = pdist2 (y, yy, "seuclidean", "smallest", 4); %! assert ({D, I}, {d, i}, 1e-4); %!test %! d = [2.1213, 4.2426, 6.3640; 1.2247, 2.4495, 4.4159; ... %! 3.2404, 4.8990, 6.8191; 2.7386, 4.2426, 6.1237]; %! assert (pdist2 (y, x, "mahalanobis"), d, 1e-4); %!test %! d = [1.2247, 2.4495, 4.4159; 2.1213, 4.2426, 6.1237]; %! i = [2, 2, 2; 1, 4, 4]; %! [D, I] = pdist2 (y, x, "mahalanobis", "smallest", 2); %! assert ({D, I}, {d, i}, 1e-4); %!test %! d = [3.2404, 4.8990, 6.8191; 2.7386, 4.2426, 6.3640]; %! i = [3, 3, 3; 4, 1, 1]; %! [D, I] = pdist2 (y, x, "mahalanobis", "largest", 2); %! assert ({D, I}, {d, i}, 1e-4); %!test %! yy = [1 2 3;5 6 7;9 5 1]; %! d = [0, 8.4853, 18.0416; 2.4495, 10.0995, 19.4808; ... %! 2.4495, 10.6771, 19.7104; 2.4495, 10.6771, 20.4573]; %! i = [2, 2, 2; 1, 4, 4; 4, 1, 1; 3, 3, 3]; %! [D, I] = pdist2 (y, yy, "mahalanobis", "smallest", 4); %! assert ({D, I}, {d, i}, 1e-4); %!test %! d = [3, 3, 5, 9; 6, 2, 4, 8; 9, 3, 5, 7]; %! assert (pdist2 (x, y, "cityblock"), d); %!test %! d = [1, 2, 3, 6; 2, 1, 2, 5; 3, 2, 3, 4]; %! assert (pdist2 (x, y, "chebychev"), d); %!test %! d = repmat ([NaN, 0.0742, 0.2254, 0.1472], [3, 1]); %! assert (pdist2 (x, y, "cosine"), d, 1e-4); %!test %! yy = [1 2 3;5 6 7;9 5 1]; %! d = [0, 0, 0.5; 0, 0, 2; 1.5, 1.5, 2; NaN, NaN, NaN]; %! i = [2, 2, 4; 3, 3, 2; 4, 4, 3; 1, 1, 1]; %! [D, I] = pdist2 (y, yy, "correlation", "smallest", 4); %! assert ({D, I}, {d, i}, eps); %! [D, I] = pdist2 (y, yy, "spearman", "smallest", 4); %! assert ({D, I}, {d, i}, eps); %!test %! d = [1, 2/3, 1, 1; 1, 2/3, 1, 1; 1, 2/3, 2/3, 2/3]; %! i = [1, 1, 1, 2; 2, 2, 3, 3; 3, 3, 2, 1]; %! [D, I] = pdist2 (x, y, "hamming", "largest", 4); %! assert ({D, I}, {d, i}, eps); %! [D, I] = pdist2 (x, y, "jaccard", "largest", 4); %! assert ({D, I}, {d, i}, eps); %!test %! xx = [1, 2, 3, 4; 2, 3, 4, 5; 3, 4, 5, 6]; %! yy = [1, 2, 2, 3; 2, 3, 3, 4]; %! [D, I] = pdist2 (x, y, "euclidean", "Smallest", 4); %! eucldist = @(v,m) sqrt(sumsq(repmat(v,rows(m),1)-m,2)); %! [d, i] = pdist2 (x, y, eucldist, "Smallest", 4); %! assert ({D, I}, {d, i}); %!warning ... %! pdist2 (xx, xx, "mahalanobis"); ## Test input validation %!error pdist2 (1) %!error ... %! pdist2 (ones (4, 5), ones (4)) %!error ... %! pdist2 (ones (4, 2, 3), ones (3, 2)) %!error ... %! pdist2 (ones (3), ones (3), "euclidean", "Largest") %!error ... %! pdist2 (ones (3), ones (3), "minkowski", 3, "Largest") %!error ... %! pdist2 (ones (3), ones (3), "minkowski", 3, "large", 4) %!error ... %! pdist2 (ones (3), ones (3), "minkowski", 3, "Largest", 4, "smallest", 5) %!error ... %! [d, i] = pdist2(ones (3), ones (3), "minkowski", 3) %!error ... %! pdist2 (ones (3), ones (3), "seuclidean", 3) %!error ... %! pdist2 (ones (3), ones (3), "seuclidean", [1, -1, 3]) %!error ... %! pdist2 (ones (3), eye (3), "mahalanobis", eye(2)) %!error ... %! pdist2 (ones (3), eye (3), "mahalanobis", ones(3)) %!error ... %! pdist2 (ones (3), eye (3), "minkowski", 0) %!error ... %! pdist2 (ones (3), eye (3), "minkowski", -5) %!error ... %! pdist2 (ones (3), eye (3), "minkowski", [1, 2]) %!error ... %! pdist2 (ones (3), ones (3), @(v,m) sqrt(repmat(v,rows(m),1)-m,2)) %!error ... %! pdist2 (ones (3), ones (3), @(v,m) sqrt(sum(sumsq(repmat(v,rows(m),1)-m,2)))) statistics-release-1.6.3/inst/plsregress.m000066400000000000000000000474371456127120000206720ustar00rootroot00000000000000## Copyright (C) 2012-2019 Fernando Damian Nieuwveldt ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or ## modify it under the terms of the GNU General Public License ## as published by the Free Software Foundation; either version 3 ## of the License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {[@var{xload}, @var{yload}] =} plsregress (@var{X}, @var{Y}) ## @deftypefnx {statistics} {[@var{xload}, @var{yload}] =} plsregress (@var{X}, @var{Y}, @var{NCOMP}) ## @deftypefnx {statistics} {[@var{xload}, @var{yload}, @var{xscore}, @var{yscore}, @var{coef}, @var{pctVar}, @var{mse}, @var{stats}] =} plsregress (@var{X}, @var{Y}, @var{NCOMP}) ## @deftypefnx {statistics} {[@var{xload}, @var{yload}, @var{xscore}, @var{yscore}, @var{coef}, @var{pctVar}, @var{mse}, @var{stats}] =} plsregress (@dots{}, @var{Name}, @var{Value}) ## ## Calculate partial least squares regression using SIMPLS algorithm. ## ## @code{plsregress} uses the SIMPLS algorithm, and first centers @var{X} and ## @var{Y} by subtracting off column means to get centered variables. However, ## it does not rescale the columns. To perform partial least squares regression ## with standardized variables, use @code{zscore} to normalize @var{X} and ## @var{Y}. ## ## @code{[@var{xload}, @var{yload}] = plsregress (@var{X}, @var{Y})} computes a ## partial least squares regression of @var{Y} on @var{X}, using @var{NCOMP} ## PLS components, which by default are calculated as ## @qcode{min (size (@var{X}, 1) - 1, size(@var{X}, 2))}, and returns the ## the predictor and response loadings in @var{xload} and @var{yload}, ## respectively. ## @itemize ## @item @var{X} is an @math{NxP} matrix of predictor variables, with rows ## corresponding to observations, and columns corresponding to variables. ## @item @var{Y} is an @math{NxM} response matrix. ## @item @var{xload} is a @math{PxNCOMP} matrix of predictor loadings, where ## each row of @var{xload} contains coefficients that define a linear ## combination of PLS components that approximate the original predictor ## variables. ## @item @var{yload} is an @math{MxNCOMP} matrix of response loadings, where ## each row of @var{yload} contains coefficients that define a linear ## combination of PLS components that approximate the original response ## variables. ## @end itemize ## ## @code{[@var{xload}, @var{yload}] = plsregress (@var{X}, @var{Y}, ## @var{NCOMP})} defines the desired number of PLS components to use in the ## regression. @var{NCOMP}, a scalar positive integer, must not exceed the ## default calculated value. ## ## @code{[@var{xload}, @var{yload}, @var{xscore}, @var{yscore}, @var{coef}, ## @var{pctVar}, @var{mse}, @var{stats}] = plsregress (@var{X}, @var{Y}, ## @var{NCOMP})} also returns the following arguments: ## @itemize ## @item @var{xscore} is an @math{NxNCOMP} orthonormal matrix with the predictor ## scores, i.e., the PLS components that are linear combinations of the ## variables in @var{X}, with rows corresponding to observations and columns ## corresponding to components. ## @item @var{yscore} is an @math{NxNCOMP} orthonormal matrix with the response ## scores, i.e., the linear combinations of the responses with which the PLS ## components @var{xscore} have maximum covariance, with rows corresponding to ## observations and columns corresponding to components. ## @item @var{coef} is a @math{(P+1)xM} matrix with the PLS regression ## coefficients, containing the intercepts in the first row. ## @item @var{pctVar} is a @math{2xNCOMP} matrix containing the percentage of ## the variance explained by the model with the first row containing the ## percentage of exlpained varianced in @var{X} by each PLS component and the ## second row containing the percentage of explained variance in @var{Y}. ## @item @var{mse} is a @math{2x(NCOMP+1)} matrix containing the estimated mean ## squared errors for PLS models with @qcode{0:@var{NCOMP}} components with the ## first row containing the squared errors for the predictor variables in ## @var{X} and the second row containing the mean squared errors for the ## response variable(s) in @var{Y}. ## @item @var{stats} is a structure with the following fields: ## @itemize ## @item @var{stats}@qcode{.W} is a @math{PxNCOMP} matrix of PLS weights. ## @item @var{stats}@qcode{.T2} is the @math{T^2} statistics for each point in ## @var{xscore}. ## @item @var{stats}@qcode{.Xresiduals} is an @math{NxP} matrix with the ## predictor residuals. ## @item @var{stats}@qcode{.Yresiduals} is an @math{NxM} matrix with the ## response residuals. ## @end itemize ## @end itemize ## ## @code{[@dots{}] = plsregress (@dots{}, @var{Name}, @var{Value}, @dots{})} ## specifies one or more of the following @var{Name}/@var{Value} pairs: ## ## @multitable @columnfractions 0.05 0.2 0.75 ## @headitem @tab @var{Name} @tab @var{Value} ## @item @tab @qcode{"CV"} @tab The method used to compute @var{mse}. When ## @var{Value} is a positive integer @math{K}, @code{plsregress} uses ## @math{K}-fold cross-validation. Set @var{Value} to a cross-validation ## partition, created using @code{cvpartition}, to use other forms of ## cross-validation. Set @var{Value} to @qcode{"resubstitution"} to use both ## @var{X} and @var{Y} to fit the model and to estimate the mean squared errors, ## without cross-validation. By default, @qcode{@var{Value} = "resubstitution"}. ## @item @tab @qcode{"MCReps"} @tab A positive integer indicating the number of ## Monte-Carlo repetitions for cross-validation. By default, ## @qcode{@var{Value} = 1}. A different @qcode{"MCReps"} value is only ## meaningful when using the @qcode{"HoldOut"} method for cross-validation, ## previously set by a @code{cvpartition} object. If no cross-validation method ## is used, then @qcode{"MCReps"} must be @qcode{1}. ## @end multitable ## ## Further information about the PLS regression can be found at ## @url{https://en.wikipedia.org/wiki/Partial_least_squares_regression} ## ## @subheading References ## @enumerate ## @item ## SIMPLS: An alternative approach to partial least squares regression. ## Chemometrics and Intelligent Laboratory Systems (1993) ## ## @end enumerate ## @end deftypefn function [xload, yload, xscore, yscore, coef, pctVar, mse, stats] = ... plsregress (X, Y, NCOMP, varargin) ## Check input arguments and add defaults if (nargin < 2) error ("plsregress: function called with too few input arguments."); endif if (! isnumeric (X) || ! isnumeric (Y)) error ("plsregress: X and Y must be real matrices."); endif ## Get size of predictor and response inputs [nobs, npred] = size (X); [Yobs, nresp] = size (Y); if (nobs != Yobs) error ("plsregress: X and Y observations mismatch."); endif ## Calculate max number of components NCOMPmax = min (nobs - 1, npred); if (nargin < 3) NCOMP = NCOMPmax; elseif (! isnumeric (NCOMP) || ! isscalar (NCOMP) || NCOMP != fix (NCOMP) || NCOMP <= 0) error ("plsregress: invalid value for NCOMP."); elseif (NCOMP > NCOMPmax) error ("plsregress: NCOMP exceeds maximum components for X."); endif ## Add default optional arguments CV = false; mcreps = 1; ## Parse additional Name-Value pairs while (numel (varargin) > 0) if (strcmpi (varargin{1}, "cv")) cvarg = varargin{2}; if (isa (cvarg, "cvpartition")) CV = true; elseif (isscalar (cvarg) && cvarg == fix (cvarg) && cvarg > 0) CV = true; elseif (! strcmpi (cvarg, "resubstitution")) error ("plsregress: invalid VALUE for 'cv' optional argument."); endif elseif (strcmpi (varargin{1}, "mcreps")) mcreps = varargin{2}; if (! (isscalar (mcreps) && mcreps == fix (mcreps) && mcreps > 0)) error ("plsregress: invalid VALUE for 'mcreps' optional argument."); endif else error ("plsregress: invalid NAME argument."); endif varargin(1:2) = []; endwhile ## Check MCREPS = 1 when "resubstitution" is set for cross validation if (! CV && mcreps != 1) error (strcat (["plsregress: 'mcreps' must be 1 when 'resubstitution'"], ... [" is specified for cross validation."])); endif ## Check number of output arguments if (nargout < 2 || nargout > 8) print_usage(); endif ## Mean centering Data matrix Xmeans = mean (X); X0 = bsxfun (@minus, X, Xmeans); ## Mean centering responses Ymeans = mean (Y); Y0 = bsxfun (@minus, Y, Ymeans); [P, Q, T, U, W] = simpls (X0, Y0, NCOMP); ## Store output arguments xload = P; yload = Q; xscore = T; yscore = U; ## Compute regression coefficients if (nargout > 4) coef = W * Q'; coef = [Ymeans - Xmeans * coef; coef]; endif ## Compute the percent of variance explained for X and Y if (nargout > 5) XVar = sum (abs (xload) .^ 2, 1) ./ sum (sum (abs (X0) .^ 2, 1)); YVar = sum (abs (yload) .^ 2, 1) ./ sum (sum (abs (Y0) .^ 2, 1)); pctVar = [XVar; YVar]; endif ## Estimate the mean squared errors if (nargout > 6) ## Compute MSE by cross-validation if (CV) mse = NaN (2, NCOMP + 1); ## Check crossval method and recalculate max number of components if isa (cvarg, "cvpartition") type = "Partition"; NCOMPmax = min(min(cvarg.TrainSize)-1,npred); ts = sum (cvarg.TestSize); else type = "Kfold"; NCOMPmax = min (floor ((nobs * (cvarg - 1) / cvarg) -1), npred); ts = nobs; endif if (NCOMP > NCOMPmax) warning (strcat (["plsregress: NCOMP exceeds maximum components"], ... [" for cross validation."])); NCOMP = NCOMPmax; endif ## Create function handle with NCOMP extra argument F = @(xtr, ytr, xte, yte) sseCV (xtr, ytr, xte, yte, NCOMP); ## Apply cross validation sse = crossval (F, X, Y, type, cvarg, "mcreps", mcreps); ## Compute MSE from the SSEs collected from each cross validation set mse(:,1:NCOMP+1) = reshape (sum (sse, 1) / (ts * mcreps), [2, NCOMP+1]); ## Computed fitted if residuals are requested if (nargout > 7) xfitted = xscore * xload'; yfitted = xscore * yload'; endif ## Compute MSE by resubstitution else mse = zeros (2, NCOMP + 1); ## Model with 0 components mse(1,1) = sum (sum (abs (X0) .^ 2, 2)); mse(2,1) = sum (sum (abs (Y0) .^ 2, 2)); ## Models with 1:NCOMP components for i = 1:NCOMP xfitted = xscore(:,1:i) * xload(:,1:i)'; yfitted = xscore(:,1:i) * yload(:,1:i)'; mse(1,i+1) = sum (sum (abs (X0 - xfitted) .^ 2, 2)); mse(2,i+1) = sum (sum (abs (Y0 - yfitted) .^ 2, 2)); endfor ## Compute the mean of the sum of squares above mse = mse / nobs; endif endif ## Compute stats if (nargout > 7) ## Save weights stats.W = W; ## Compute T-squared stats.T2 = sum (bsxfun (@rdivide, abs (xscore) .^ 2, ... var (xscore, [], 1)) , 2); ## Compute residuals for X and Y stats.Xresiduals = X0 - xfitted; stats.Yresiduals = Y0 - yfitted; endif endfunction ## SIMPLS algorithm function [P, Q, T, U, W] = simpls (X0, Y0, NCOMP) ## Get size of predictor and response inputs [nobs, npred] = size (X0); [Yobs, nresp] = size (Y0); ## Compute covariance S = X0' * Y0; ## Preallocate matrices W = P = V = zeros (npred, NCOMP); T = U = zeros (nobs, NCOMP); Q = zeros (nresp, NCOMP); ## Models with 1:NCOMP components for a = 1:NCOMP [eigvec, eigval] = eig (S' * S); # Y factor weights ## Get max eigenvector domindex = find (diag (eigval) == max (diag (eigval))); q = eigvec(:,domindex); w = S * q; # X block factor weights t = X0 * w; # X block factor scores t = t - mean (t); nt = sqrt (t' * t); # compute norm t = t / nt; w = w / nt; # normalize p = X0' * t; # X block factor loadings q = Y0' * t; # Y block factor loadings u = Y0 * q; # Y block factor scores v = p; ## Ensure orthogonality if (a > 1) v = v - V * (V' * p); u = u - T * (T' * u); endif v = v / sqrt (v' * v); # normalize orthogonal loadings S = S - v * (v' * S); # deflate S wrt loadings V(:,a) = v; ## Store data P(:,a) = p; # Xloads Q(:,a) = q; # Yloads T(:,a) = t; # xscore U(:,a) = u; # Yscores W(:,a) = w; # Weights endfor endfunction ## Helper function for SSE cross-validation function sse = sseCV (XTR, YTR, XTE, YTE, NCOMP) ## Center train data XTRmeans = mean (XTR); YTRmeans = mean (YTR); X0TR = bsxfun (@minus, XTR, XTRmeans); Y0TR = bsxfun (@minus, YTR, YTRmeans); ## Center test data X0TE = bsxfun(@minus, XTE, XTRmeans); Y0TE = bsxfun(@minus, YTE, YTRmeans); ## Fit the full model [xload, yload, ~, ~, W] = simpls (X0TR, Y0TR, NCOMP); XTEscore = X0TE * W; ## Preallocate SSE matrix sse = zeros (2, NCOMP + 1); ## Model with 0 components sse(1,1) = sum (sum (abs (X0TE) .^ 2, 2)); sse(2,1) = sum (sum (abs (Y0TE) .^ 2, 2)); ## Models with 1:NCOMP components for i = 1:NCOMP X0fitted = XTEscore(:,1:i) * xload(:,1:i)'; sse(1,i+1) = sum (sum (abs (X0TE - X0fitted) .^ 2, 2)); Y0fitted = XTEscore(:,1:i) * yload(:,1:i)'; sse(2,i+1) = sum (sum (abs (Y0TE - Y0fitted) .^ 2, 2)); endfor endfunction %!demo %! ## Perform Partial Least-Squares Regression %! %! ## Load the spectra data set and use the near infrared (NIR) spectral %! ## intensities (NIR) as the predictor and the corresponding octave %! ## ratings (octave) as the response. %! load spectra %! %! ## Perform PLS regression with 10 components %! [xload, yload, xscore, yscore, coef, ptcVar] = plsregress (NIR, octane, 10); %! %! ## Plot the percentage of explained variance in the response variable %! ## (PCTVAR) as a function of the number of components. %! plot (1:10, cumsum (100 * ptcVar(2,:)), "-ro"); %! xlim ([1, 10]); %! xlabel ("Number of PLS components"); %! ylabel ("Percentage of Explained Variance in octane"); %! title ("Explained Variance per PLS components"); %! %! ## Compute the fitted response and display the residuals. %! octane_fitted = [ones(size(NIR,1),1), NIR] * coef; %! residuals = octane - octane_fitted; %! figure %! stem (residuals, "color", "r", "markersize", 4, "markeredgecolor", "r") %! xlabel ("Observations"); %! ylabel ("Residuals"); %! title ("Residuals in octane's fitted responce"); %!demo %! ## Calculate Variable Importance in Projection (VIP) for PLS Regression %! %! ## Load the spectra data set and use the near infrared (NIR) spectral %! ## intensities (NIR) as the predictor and the corresponding octave %! ## ratings (octave) as the response. Variables with a VIP score greater than %! ## 1 are considered important for the projection of the PLS regression model. %! load spectra %! %! ## Perform PLS regression with 10 components %! [xload, yload, xscore, yscore, coef, pctVar, mse, stats] = ... %! plsregress (NIR, octane, 10); %! %! ## Calculate the normalized PLS weights %! W0 = stats.W ./ sqrt(sum(stats.W.^2,1)); %! %! ## Calculate the VIP scores for 10 components %! nobs = size (xload, 1); %! SS = sum (xscore .^ 2, 1) .* sum (yload .^ 2, 1); %! VIPscore = sqrt (nobs * sum (SS .* (W0 .^ 2), 2) ./ sum (SS, 2)); %! %! ## Find variables with a VIP score greater than or equal to 1 %! VIPidx = find (VIPscore >= 1); %! %! ## Plot the VIP scores %! scatter (1:length (VIPscore), VIPscore, "xb"); %! hold on %! scatter (VIPidx, VIPscore (VIPidx), "xr"); %! plot ([1, length(VIPscore)], [1, 1], "--k"); %! hold off %! axis ("tight"); %! xlabel ("Predictor Variables"); %! ylabel ("VIP scores"); %! title ("VIP scores for each predictror variable with 10 components"); ## Test output %!test %! load spectra %! [xload, yload, xscore, yscore, coef, pctVar] = plsregress (NIR, octane, 10); %! xload1_out = [-0.0170, 0.0039, 0.0095, 0.0258, 0.0025, ... %! -0.0075, 0.0000, 0.0018, -0.0027, 0.0020]; %! yload_out = [6.6384, 9.3106, 2.0505, 0.6471, 0.9625, ... %! 0.5905, 0.4244, 0.2437, 0.3516, 0.2548]; %! xscore1_out = [-0.0401, -0.1764, -0.0340, 0.1669, 0.1041, ... %! -0.2067, 0.0457, 0.1565, 0.0706, -0.1471]; %! yscore1_out = [-12.4635, -15.0003, 0.0638, 0.0652, -0.0070, ... %! -0.0634, 0.0062, -0.0012, -0.0151, -0.0173]; %! assert (xload(1,:), xload1_out, 1e-4); %! assert (yload, yload_out, 1e-4); %! assert (xscore(1,:), xscore1_out, 1e-4); %! assert (yscore(1,:), yscore1_out, 1e-4); %!test %! load spectra %! [xload, yload, xscore, yscore, coef, pctVar] = plsregress (NIR, octane, 5); %! xload1_out = [-0.0170, 0.0039, 0.0095, 0.0258, 0.0025]; %! yload_out = [6.6384, 9.3106, 2.0505, 0.6471, 0.9625]; %! xscore1_out = [-0.0401, -0.1764, -0.0340, 0.1669, 0.1041]; %! yscore1_out = [-12.4635, -15.0003, 0.0638, 0.0652, -0.0070]; %! assert (xload(1,:), xload1_out, 1e-4); %! assert (yload, yload_out, 1e-4); %! assert (xscore(1,:), xscore1_out, 1e-4); %! assert (yscore(1,:), yscore1_out, 1e-4); ## Test input validation %!error %! plsregress (1) %!error plsregress (1, "asd") %!error plsregress (1, {1,2,3}) %!error plsregress ("asd", 1) %!error plsregress ({1,2,3}, 1) %!error ... %! plsregress (ones (20,3), ones (15,1)) %!error ... %! plsregress (ones (20,3), ones (20,1), 0) %!error ... %! plsregress (ones (20,3), ones (20,1), -5) %!error ... %! plsregress (ones (20,3), ones (20,1), 3.2) %!error ... %! plsregress (ones (20,3), ones (20,1), [2, 3]) %!error ... %! plsregress (ones (20,3), ones (20,1), 4) %!error ... %! plsregress (ones (20,3), ones (20,1), 3, "cv", 4.5) %!error ... %! plsregress (ones (20,3), ones (20,1), 3, "cv", -1) %!error ... %! plsregress (ones (20,3), ones (20,1), 3, "cv", "somestring") %!error ... %! plsregress (ones (20,3), ones (20,1), 3, "cv", 3, "mcreps", 2.2) %!error ... %! plsregress (ones (20,3), ones (20,1), 3, "cv", 3, "mcreps", -2) %!error ... %! plsregress (ones (20,3), ones (20,1), 3, "cv", 3, "mcreps", [1, 2]) %!error ... %! plsregress (ones (20,3), ones (20,1), 3, "Name", 3, "mcreps", 1) %!error ... %! plsregress (ones (20,3), ones (20,1), 3, "cv", 3, "Name", 1) %!error ... %! plsregress (ones (20,3), ones (20,1), 3, "mcreps", 2) %!error ... %! plsregress (ones (20,3), ones (20,1), 3, "cv", "resubstitution", "mcreps", 2) %!error plsregress (1, 2) statistics-release-1.6.3/inst/ppplot.m000066400000000000000000000062731456127120000200100ustar00rootroot00000000000000## Copyright (C) 1995-2017 Kurt Hornik ## Copyright (C) 2022-2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {} ppplot (@var{x}, @var{dist}) ## @deftypefnx {statistics} {} ppplot (@var{x}, @var{dist}, @var{params}) ## @deftypefnx {statistics} {[@var{p}, @var{y}] =} ppplot (@var{x}, @var{dist}, @var{params}) ## ## Perform a PP-plot (probability plot). ## ## If F is the CDF of the distribution @var{dist} with parameters ## @var{params} and @var{x} a sample vector of length @var{n}, the PP-plot ## graphs ordinate @var{y}(@var{i}) = F (@var{i}-th largest element of ## @var{x}) versus abscissa @var{p}(@var{i}) = (@var{i} - 0.5)/@var{n}. If ## the sample comes from F, the pairs will approximately follow a straight ## line. ## ## The default for @var{dist} is the standard normal distribution. ## ## The optional argument @var{params} contains a list of parameters of ## @var{dist}. ## ## For example, for a probability plot of the uniform distribution on [2,4] ## and @var{x}, use ## ## @example ## ppplot (x, "uniform", 2, 4) ## @end example ## ## @noindent ## @var{dist} can be any string for which a function @var{distcdf} that ## calculates the CDF of distribution @var{dist} exists. ## ## If no output is requested then the data are plotted immediately. ## @seealso{qqplot} ## @end deftypefn function [p, y] = ppplot (x, dist, varargin) if (nargin < 1) print_usage (); endif if (! isnumeric (x) || ! isreal (x) || ! isvector (x) || isscalar (x)) error ("ppplot: X must be a numeric vector of real numbers"); endif s = sort (x); n = length (x); p = ((1 : n)' - 0.5) / n; if (nargin == 1) F = @stdnormal_cdf; elseif (! ischar (dist)) error ("ppplot: DIST must be a string"); else F = str2func ([dist "cdf"]); endif if (nargin <= 2) y = feval (F, s); else y = feval (F, s, varargin{:}); endif if (nargout == 0) plot (p, y); axis ([0, 1, 0, 1]); endif endfunction function p = stdnormal_cdf (x) p = 0.5 * erfc (x ./ sqrt (2)); endfunction ## Test plotting %!test %! hf = figure ("visible", "off"); %! unwind_protect %! ppplot ([2 3 3 4 4 5 6 5 6 7 8 9 8 7 8 9 0 8 7 6 5 4 6 13 8 15 9 9]); %! unwind_protect_cleanup %! close (hf); %! end_unwind_protect ## Test input validation %!error ppplot () %!error ppplot (ones (2,2)) %!error ppplot (1, 2) %!error ppplot ([1 2 3 4], 2) statistics-release-1.6.3/inst/princomp.m000066400000000000000000000125721456127120000203200ustar00rootroot00000000000000## Copyright (C) 2013-2019 Fernando Damian Nieuwveldt ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or ## modify it under the terms of the GNU General Public License ## as published by the Free Software Foundation; either version 3 ## of the License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{COEFF} =} princomp (@var{X}) ## @deftypefnx {statistics} {[@var{COEFF}, @var{SCORE}] =} princomp (@var{X}) ## @deftypefnx {statistics} {[@var{COEFF}, @var{SCORE}, @var{latent}] =} princomp (@var{X}) ## @deftypefnx {statistics} {[@var{COEFF}, @var{SCORE}, @var{latent}, @var{tsquare}] =} princomp (@var{X}) ## @deftypefnx {statistics} {[@dots{}] =} princomp (@var{X}, "econ") ## ## Performs a principal component analysis on a NxP data matrix X. ## ## @itemize @bullet ## @item ## @var{COEFF} : returns the principal component coefficients ## @item ## @var{SCORE} : returns the principal component scores, the representation of X ## in the principal component space ## @item ## @var{LATENT} : returns the principal component variances, i.e., the ## eigenvalues of the covariance matrix X. ## @item ## @var{TSQUARE} : returns Hotelling's T-squared Statistic for each observation ## in X ## @item ## [...] = princomp(X,'econ') returns only the elements of latent that are not ## necessarily zero, and the corresponding columns of COEFF and SCORE, that is, ## when n <= p, only the first n-1. This can be significantly faster when p is ## much larger than n. In this case the svd will be applied on the transpose of ## the data matrix X ## ## @end itemize ## ## @subheading References ## ## @enumerate ## @item ## Jolliffe, I. T., Principal Component Analysis, 2nd Edition, Springer, 2002 ## ## @end enumerate ## @end deftypefn function [COEFF, SCORE, latent, tsquare] = princomp (X, varargin) if (nargin < 1 || nargin > 2) print_usage (); endif if (nargin == 2 && ! strcmpi (varargin{:}, "econ")) error (strcat (["princomp: if a second input argument is present,"], ... [" it must be the string 'econ'."])); endif [nobs nvars] = size(X); # Center the columns to mean zero Xcentered = bsxfun(@minus,X,mean(X)); # Check if there are more variables then observations if nvars <= nobs [U,S,COEFF] = svd(Xcentered, "econ"); else # Calculate the svd on the transpose matrix, much faster if (nargin == 2 && strcmpi ( varargin{:} , "econ")) [COEFF,S,V] = svd(Xcentered' , 'econ'); else [COEFF,S,V] = svd(Xcentered'); endif endif if nargout > 1 # Get the Scores SCORE = Xcentered*COEFF; # Get the rank of the SCORE matrix r = rank(SCORE); # Only use the first r columns, pad rest with zeros if economy != 'econ' SCORE = SCORE(:,1:r) ; if !(nargin == 2 && strcmpi ( varargin{:} , "econ")) SCORE = [SCORE, zeros(nobs , nvars-r)]; else COEFF = COEFF(: , 1:r); endif endif if nargout > 2 # This is the same as the eigenvalues of the covariance matrix of X latent = (diag(S'*S)/(size(Xcentered,1)-1))(1:r); if !(nargin == 2 && strcmpi ( varargin{:} , "econ")) latent= [latent;zeros(nvars-r,1)]; endif endif if nargout > 3 # Calculate the Hotelling T-Square statistic for the observations tsquare = sumsq(zscore(SCORE(:,1:r)),2); endif endfunction %!shared COEFF,SCORE,latent,tsquare,m,x,R,V,lambda,i,S,F #NIST Engineering Statistics Handbook example (6.5.5.2) %!test %! x=[7 4 3 %! 4 1 8 %! 6 3 5 %! 8 6 1 %! 8 5 7 %! 7 2 9 %! 5 3 3 %! 9 5 8 %! 7 4 5 %! 8 2 2]; %! R = corrcoef (x); %! [V, lambda] = eig (R); %! [~, i] = sort(diag(lambda), "descend"); #arrange largest PC first %! S = V(:, i) * diag(sqrt(diag(lambda)(i))); %! ## contribution of first 2 PCs to each original variable %!assert(diag(S(:, 1:2)*S(:, 1:2)'), [0.8662; 0.8420; 0.9876], 1E-4); %! B = V(:, i) * diag( 1./ sqrt(diag(lambda)(i))); %! F = zscore(x)*B; %! [COEFF,SCORE,latent,tsquare] = princomp(zscore(x, 1)); %!assert(tsquare,sumsq(F, 2),1E4*eps); %!test %! x=[1,2,3;2,1,3]'; %! [COEFF,SCORE,latent,tsquare] = princomp(x); %! m=[sqrt(2),sqrt(2);sqrt(2),-sqrt(2);-2*sqrt(2),0]/2; %! m(:,1) = m(:,1)*sign(COEFF(1,1)); %! m(:,2) = m(:,2)*sign(COEFF(1,2)); %!assert(COEFF,m(1:2,:),10*eps); %!assert(SCORE,-m,10*eps); %!assert(latent,[1.5;.5],10*eps); %!assert(tsquare,[4;4;4]/3,10*eps); %!test %! x=x'; %! [COEFF,SCORE,latent,tsquare] = princomp(x); %! m=[sqrt(2),sqrt(2),0;-sqrt(2),sqrt(2),0;0,0,2]/2; %! m(:,1) = m(:,1)*sign(COEFF(1,1)); %! m(:,2) = m(:,2)*sign(COEFF(1,2)); %! m(:,3) = m(:,3)*sign(COEFF(3,3)); %!assert(COEFF,m,10*eps); %!assert(SCORE(:,1),-m(1:2,1),10*eps); %!assert(SCORE(:,2:3),zeros(2),10*eps); %!assert(latent,[1;0;0],10*eps); %!assert(tsquare,[0.5;0.5],10*eps) %!test %! [COEFF,SCORE,latent,tsquare] = princomp(x, "econ"); %!assert(COEFF,m(:, 1),10*eps); %!assert(SCORE,-m(1:2,1),10*eps); %!assert(latent,[1],10*eps); %!assert(tsquare,[0.5;0.5],10*eps) statistics-release-1.6.3/inst/private/000077500000000000000000000000001456127120000177565ustar00rootroot00000000000000statistics-release-1.6.3/inst/private/exact2xkCT.m000066400000000000000000000167771456127120000221360ustar00rootroot00000000000000## Copyright (C) 2022 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {Private Function} [@var{p_net}, @var{p_val}] = exact2xkCT (@var{ct}, @var{weights}, @var{rsstat}) ## ## Compute the exact p-value for a 2-by-K contingency table based on the ## network algorithm. ## ## Reference: Cyrus R. Mehta & Nitin R. Patel (1980) A network algorithm for ## the exact treatment of the 2×k contingency table, Communications in ## Statistics - Simulation and Computation, 9:6, 649-664, ## DOI: 10.1080/03610918008812182 ## ## @end deftypefn function [p_net, p_val] = exact2xkCT (ct, weights, rsstat) ## Calculate nodes and arcs [nodes, arcs] = build_nodes (ct,weights); ## Apply backward induction to nodes nodes = backward_induce (nodes,arcs); ## Forward scan the network to get p-values p_val = forward_scan (nodes, arcs, rsstat); ## Calculate p-values TP = nodes{4,1}; p_val = p_val / TP; p_net = p_val(2) + min(p_val(1), p_val(3)); endfunction ## Calculate structures describing nodes and arcs function [nodes, arcs] = build_nodes (ct, weights) column = size (ct, 2); ## number of columns in contigency table rowsum = sum (ct, 2); ## sum of rows colsum = sum (ct, 1); ## sum of columns oldnodes = zeros(1,2); ## nodes added during last pass oldlo = 0; ## min possible sum so far oldhi = 0; ## max possible sum so far oldnn = 1; ## node numbers (row numbers) from last pass ctsum = rowsum(1); ## sum of entries in first row nodecount = 1; ## current node count ## Initialize cell structures for nodes and arcs nodes = cell(4, column+1); ## to hold nodes nodes{1,1} = zeros (1,2); ## n-by-2 array, n = # of nodes, row = [j,mj] nodes{2,column+1} = 0; ## n-vector of longest path to end from here nodes{3,column+1} = 0; ## n-vector of shortest path to end from here nodes{4,column+1} = 1; ## n-vector of total probability to end from here arcs = cell(3, column); ## to hold arcs ## row 1: n-by-2 array, n = # of connections, row = pair connected ## row 2: n-vector of arc lengths ## row 3: n-vector of arc probabilities for j = 1:column ## Find nodes possible at the next step nj = colsum(j); lo = max (oldlo, ctsum - sum (colsum(j+1:end))); hi = min (ctsum, oldhi + nj); newnodes = zeros (hi - lo + 1,2); newnodes(:,1) = j; newnodes(:,2) = (lo:hi)'; newnn = 1:size (newnodes,1); nodecount = nodecount + size (newnodes, 1); nodes{1,j+1} = newnodes; ## Find arcs possible to the next step [a0, a1] = meshgrid (oldnn, newnn); a0 = a0(:); a1 = a1(:); oldsum = oldnodes(a0,2); newsum = newnodes(a1,2); xj = newsum - oldsum; ok = (xj >= 0) & (xj <= nj); arcs{1,j} = [a0(ok) a1(ok)]; ## arc connections xj = xj(ok); arcs{2,j} = weights(j) * xj; pj = exp (gammaln (nj + 1) - gammaln (xj + 1) - gammaln (nj - xj + 1)); arcs{3,j} = pj; ## arc probabilities ## Update data structures oldlo = lo; oldhi = hi; oldnodes = newnodes; oldnn = newnn; endfor endfunction ## Calculate backward induction by adding information to NODES array function nodes = backward_induce (nodes, arcs) ## initialize for final node column = size (nodes,2) - 1; startSP = zeros (1); startLP = startSP; startTP = ones (1); for j = column:-1:1 ## destination nodes are previous start nodes endSP = startSP; endLP = startLP; endTP = startTP; ## get new start nodes and information about them a = arcs{1,j}; startmax = max(a(:,1)); startSP = zeros(startmax,1); startLP = startSP; startTP = startSP; arclen = arcs{2,j}; arcprob = arcs{3,j}; for nodenum = 1:startmax % for each start node, compute SP, LP, TP k1 = find(a(:,1) == nodenum); k2 = a(k1,2); startLP(nodenum) = max(arclen(k1) + endLP(k2)); startSP(nodenum) = min(arclen(k1) + endSP(k2)); startTP(nodenum) = sum(arcprob(k1) .* endTP(k2)); endfor ## store information about nodes at this level nodes{2,j} = startLP; nodes{3,j} = startSP; nodes{4,j} = startTP; endfor endfunction ## Get p-values by forward scanning the network function p_val = forward_scan (nodes, arcs, rsstat) NROWS = 50; p_val = zeros(3,1); ## [ProbT] stack = zeros(NROWS, 4); stack(:,1) = Inf; stack(1,1) = 1; ## level of current node stack(1,2) = 1; ## number at this level of current node stack(1,3) = 0; ## length so far to this node stack(1,4) = 1; ## probability so far of reaching this node N = size (stack, 1); i1 = 0; i2 = 0; i3 = 0; while (1) ## Get next lowest level node to process minlevel = min(stack((stack(1:N)>0))); if (isinf (minlevel)) break; endif sp = find (stack(1:N) == minlevel); sp = sp(1); L = stack(sp,1); J = stack(sp,2); pastL = stack(sp,3); pastP = stack(sp,4); stack(sp,1) = Inf; ## Get info for arcs at level L and their target nodes LP = nodes{2,L+1}; SP = nodes{3,L+1}; TP = nodes{4,L+1}; aj = arcs{1,L}; arclen = arcs{2,L}; arcprob = arcs{3,L}; ## Look only at arcs from node J seps = sqrt (eps); arows = find (aj(:,1) == J)'; for k = arows tonode = aj(k,2); thisL = arclen(k); thisP = pastP * arcprob(k); len = pastL + thisL; ## No paths from node J are signicant if (len + LP(tonode) < rsstat - seps) p_val(1) = p_val(1) + thisP * TP(tonode); ## All paths from node J are significant elseif (len + SP(tonode) > rsstat + seps) p_val(3) = p_val(3) + thisP * TP(tonode); ## Single match from node J elseif (SP(tonode) == LP(tonode)) p_val(2) = p_val(2) + thisP * TP(tonode); ## Match node J with another already stored node else ## Find a stored node that matches this one r = find(stack(:,1) == L+1); if (any (r)) r = r(stack(r,2) == tonode); if (any (r)) r = r(abs (stack(r,3) - len) < seps); endif endif ## If any one is found, merge node J with it if (any (r)) sp = r(1); stack(sp,4) = stack(sp,4) + thisP; i1 = i1 + 1; ## Otherwise add a new node else z = find(isinf(stack(:,1))); if (isempty (z)) i2 = i2 +1; block = zeros (NROWS, 4); block(:,1) = Inf; stack = [stack; block]; sp = N + 1; N = N + NROWS; else i3 = i3 + 1; sp = z(1); endif stack(sp,1) = L + 1; stack(sp,2) = tonode; stack(sp,3) = len; stack(sp,4) = thisP; endif endif endfor endwhile endfunction statistics-release-1.6.3/inst/probit.m000066400000000000000000000026141456127120000177640ustar00rootroot00000000000000## Copyright (C) 1995-2017 Kurt Hornik ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{x} =} probit (@var{p}) ## ## Probit transformation ## ## Return the probit (the quantile of the standard normal distribution) for ## each element of @var{p}. ## ## @seealso{logit} ## @end deftypefn function x = probit (p) if (nargin != 1) print_usage (); endif x = -sqrt (2) * erfcinv (2 * p); endfunction ## Test output %!assert (probit ([-1, 0, 0.5, 1, 2]), [NaN, -Inf, 0, Inf, NaN]) %!assert (probit ([0.2, 0.99]), norminv ([0.2, 0.99])) ## Test input validation %!error probit () %!error probit (1, 2) statistics-release-1.6.3/inst/procrustes.m000066400000000000000000000314541456127120000207020ustar00rootroot00000000000000## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/OR ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, OR (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{d} =} procrustes (@var{X}, @var{Y}) ## @deftypefnx {statistics} {@var{d} =} procrustes (@var{X}, @var{Y}, @var{param1}, @var{value1}, @dots{}) ## @deftypefnx {statistics} {[@var{d}, @var{Z}] =} procrustes (@dots{}) ## @deftypefnx {statistics} {[@var{d}, @var{Z}, @var{transform}] =} procrustes (@dots{}) ## ## Procrustes Analysis. ## ## @code{@var{d} = procrustes (@var{X}, @var{Y})} computes a linear ## transformation of the points in the matrix @var{Y} to best conform them to ## the points in the matrix @var{X} by minimizing the sum of squared errors, as ## the goodness of fit criterion, which is returned in @var{d} as a ## dissimilarity measure. @var{d} is standardized by a measure of the scale of ## @var{X}, given by ## @itemize ## @item @qcode{sum (sum ((X - repmat (mean (X, 1), size (X, 1), 1)) .^ 2, 1))} ## @end itemize ## i.e., the sum of squared elements of a centered version of @var{X}. However, ## if @var{X} comprises repetitions of the same point, the sum of squared errors ## is not standardized. ## ## @var{X} and @var{Y} must have the same number of points (rows) and ## @qcode{procrustes} matches the @math{i}-th point in @var{Y} to the ## @math{i}-th point in @var{X}. Points in @var{Y} can have smaller dimensions ## (columns) than those in @var{X}, but not the opposite. Missing dimensions in ## @var{Y} are added with padding columns of zeros as necessary to match the ## the dimensions in @var{X}. ## ## @code{[@var{d}, @var{Z}] = procrustes (@var{X}, @var{Y})} also returns the ## transformed values in @var{Y}. ## ## @code{[@var{d}, @var{Z}, @var{transform}] = procrustes (@var{X}, @var{Y})} ## also returns the transformation that maps @var{Y} to @var{Z}. ## ## @var{transform} is a structure with fields: ## ## @multitable @columnfractions 0.05 0.1 0.05 0.8 ## @item @tab @qcode{c} @tab @tab the translation component ## @item @tab @qcode{T} @tab @tab the orthogonal rotation and reflection ## component ## @item @tab @qcode{b} @tab @tab the scale component ## @end multitable ## ## So that @code{@var{Z} = @var{transform}.@qcode{b} * @var{Y} * ## @var{transform}.@qcode{T} + @var{transform}.@qcode{c}} ## ## @qcode{procrustes} can take two optional parameters as Name-Value pairs. ## ## @code{[@dots{}] = procrustes (@dots{}, @qcode{"Scaling"}, @qcode{false})} ## computes a transformation that does not include scaling, that is ## @var{transform}.@qcode{b} = 1. Setting @qcode{"Scaling"} to @qcode{true} ## includes a scaling component, which is the default. ## ## @code{[@dots{}] = procrustes (@dots{}, @qcode{"Reflection"}, @qcode{false})} ## computes a transformation that does not include a reflection component, that ## is @var{transform}.@qcode{T} = 1. Setting @qcode{"Reflection"} to ## @qcode{true} forces the solution to include a reflection component in the ## computed transformation, that is @var{transform}.@qcode{T} = -1. ## ## @code{[@dots{}] = procrustes (@dots{}, @qcode{"Reflection"}, @qcode{"best"})} ## computes the best fit procrustes solution, which may or may not include a ## reflection component, which is the default. ## ## @seealso{cmdscale} ## @end deftypefn function [d, Z, transform] = procrustes (X, Y, varargin) if (nargin < 1) error ("fishertest: contingency table is missing."); endif if (nargin > 5) error ("fishertest: too many input parameters."); endif ## Check X and Y for appropriate input if (isempty (X) || ! ismatrix (X) || ndims (X) != 2 || ... isempty (Y) || ! ismatrix (Y) || ndims (Y) != 2) error ("procrustes: X and Y must be 2-dimensional matrices."); endif if (any (isnan (X(:))) || any (isinf (X(:))) || iscomplex (X) || ... any (isnan (Y(:))) || any (isinf (Y(:))) || iscomplex (Y)) error ("procrustes: values in X and Y must be real."); endif [Xp, Xd] = size (X); [Yp, Yd] = size (Y); if (Yp != Xp) error ("procrustes: X and Y must have equal number of rows."); elseif (Yd > Xd) error ("procrustes: X must have at least as many columns as Y."); endif ## Add defaults and parse optional arguments scaling = true; reflection = "best"; if (nargin > 2) params = numel (varargin); if ((params / 2) != fix (params / 2)) error ("procrustes: optional arguments must be in Name-Value pairs.") endif for idx = 1:2:params name = varargin{idx}; value = varargin{idx+1}; switch (lower (name)) case "scaling" scaling = value; if (! (isscalar (scaling) && islogical (scaling))) error ("procrustes: invalid value for scaling."); endif case "reflection" reflection = value; if (! (strcmpi (reflection, "best") || islogical (reflection))) error ("procrustes: invalid value for reflection."); endif otherwise error ("procrustes: invalid name for optional arguments."); endswitch endfor endif ## Center at the origin. Xmu = mean (X, 1); Ymu = mean (Y, 1); X_0 = X - repmat (Xmu, Xp, 1); Y_0 = Y - repmat (Ymu, Xp, 1); ## Get centroid size and check for X or Y having identical points Xsumsq = sum (X_0 .^ 2, 1); Ysumsq = sum (Y_0 .^ 2, 1); constX = all (Xsumsq <= abs (eps (class (X)) * Xp * Xmu) .^ 2); constY = all (Ysumsq <= abs (eps (class (X)) * Xp * Ymu) .^ 2); Xsumsq = sum (Xsumsq); Ysumsq = sum (Ysumsq); if (! constX && ! constY) ## Scale to "centered" Frobenius norm. normX = sqrt (Xsumsq); normY = sqrt (Ysumsq); X_0 = X_0 / normX; Y_0 = Y_0 / normY; ## Fix dimension space (if necessary) if (Yd < Xd) Y_0 = [Y_0 zeros(Xp, Xd-Yd)]; end ## Find optimal rotation matrix of Y A = X_0' * Y_0; [U, S, V] = svd (A); T = V * U'; ## Handle reflection only if 'true' or 'false' was given if (! strcmpi (reflection, "best")) is_reflection = (det(T) < 0); ## Force a reflection if data and reflection option disagree if (reflection != is_reflection) V(:,end) = -V(:,end); S(end,end) = -S(end,end); T = V * U'; endif endif ## Apply scaling (if requested) traceTA = sum (diag (S)); if (scaling) b = traceTA * normX / normY; d = 1 - traceTA .^ 2; if (nargout > 1) Z = normX * traceTA * Y_0 * T + repmat (Xmu, Xp, 1); endif else b = 1; d = 1 + Ysumsq / Xsumsq - 2 * traceTA * normY / normX; if (nargout > 1) Z = normY * Y_0 * T + repmat (Xmu, Xp, 1); endif endif ## 3rd output argument if (nargout > 2) if (Yd < Xd) T = T(1:Yd,:); endif c = Xmu - b * Ymu * T; transform = struct ("T", T, "b", b, "c", repmat (c, Xp, 1)); end ## Special cases elseif constX # Identical points in X d = 0; Z = repmat (Xmu, Xp, 1); T = eye (Yd, Xd); transform = struct ("T", T, "b", 0, "c", Z); else # Identical points in Y d = 1; Z = repmat (Xmu, Xp, 1); T = eye (Yd, Xd); transform = struct ("T", T, "b", 0, "c", Z); endif endfunction %!demo %! ## Create some random points in two dimensions %! n = 10; %! randn ("seed", 1); %! X = normrnd (0, 1, [n, 2]); %! %! ## Those same points, rotated, scaled, translated, plus some noise %! S = [0.5, -sqrt(3)/2; sqrt(3)/2, 0.5]; # rotate 60 degrees %! Y = normrnd (0.5*X*S + 2, 0.05, n, 2); %! %! ## Conform Y to X, plot original X and Y, and transformed Y %! [d, Z] = procrustes (X, Y); %! plot (X(:,1), X(:,2), "rx", Y(:,1), Y(:,2), "b.", Z(:,1), Z(:,2), "bx"); %!demo %! ## Find Procrustes distance and plot superimposed shape %! %! X = [40 88; 51 88; 35 78; 36 75; 39 72; 44 71; 48 71; 52 74; 55 77]; %! Y = [36 43; 48 42; 31 26; 33 28; 37 30; 40 31; 45 30; 48 28; 51 24]; %! plot (X(:,1),X(:,2),"x"); %! hold on %! plot (Y(:,1),Y(:,2),"o"); %! xlim ([0 100]); %! ylim ([0 100]); %! legend ("Target shape (X)", "Source shape (Y)"); %! [d, Z] = procrustes (X, Y) %! plot (Z(:,1), Z(:,2), "s"); %! legend ("Target shape (X)", "Source shape (Y)", "Transformed shape (Z)"); %! hold off %!demo %! ## Apply Procrustes transformation to larger set of points %! %! ## Create matrices with landmark points for two triangles %! X = [5, 0; 5, 5; 8, 5]; # target %! Y = [0, 0; 1, 0; 1, 1]; # source %! %! ## Create a matrix with more points on the source triangle %! Y_mp = [linspace(Y(1,1),Y(2,1),10)', linspace(Y(1,2),Y(2,2),10)'; ... %! linspace(Y(2,1),Y(3,1),10)', linspace(Y(2,2),Y(3,2),10)'; ... %! linspace(Y(3,1),Y(1,1),10)', linspace(Y(3,2),Y(1,2),10)']; %! %! ## Plot both shapes, including the larger set of points for the source shape %! plot ([X(:,1); X(1,1)], [X(:,2); X(1,2)], "bx-"); %! hold on %! plot ([Y(:,1); Y(1,1)], [Y(:,2); Y(1,2)], "ro-", "MarkerFaceColor", "r"); %! plot (Y_mp(:,1), Y_mp(:,2), "ro"); %! xlim ([-1 10]); %! ylim ([-1 6]); %! legend ("Target shape (X)", "Source shape (Y)", ... %! "More points on Y", "Location", "northwest"); %! hold off %! %! ## Obtain the Procrustes transformation %! [d, Z, transform] = procrustes (X, Y) %! %! ## Use the Procrustes transformation to superimpose the more points (Y_mp) %! ## on the source shape onto the target shape, and then visualize the results. %! Z_mp = transform.b * Y_mp * transform.T + transform.c(1,:); %! figure %! plot ([X(:,1); X(1,1)], [X(:,2); X(1,2)], "bx-"); %! hold on %! plot ([Y(:,1); Y(1,1)], [Y(:,2); Y(1,2)], "ro-", "MarkerFaceColor", "r"); %! plot (Y_mp(:,1), Y_mp(:,2), "ro"); %! xlim ([-1 10]); %! ylim ([-1 6]); %! plot ([Z(:,1); Z(1,1)],[Z(:,2); Z(1,2)],"ks-","MarkerFaceColor","k"); %! plot (Z_mp(:,1),Z_mp(:,2),"ks"); %! legend ("Target shape (X)", "Source shape (Y)", ... %! "More points on Y", "Transformed source shape (Z)", ... %! "Transformed additional points", "Location", "northwest"); %! hold off %!demo %! ## Compare shapes without reflection %! %! T = [33, 93; 33, 87; 33, 80; 31, 72; 32, 65; 32, 58; 30, 72; ... %! 28, 72; 25, 69; 22, 64; 23, 59; 26, 57; 30, 57]; %! S = [48, 83; 48, 77; 48, 70; 48, 65; 49, 59; 49, 56; 50, 66; ... %! 52, 66; 56, 65; 58, 61; 57, 57; 54, 56; 51, 55]; %! plot (T(:,1), T(:,2), "x-"); %! hold on %! plot (S(:,1), S(:,2), "o-"); %! legend ("Target shape (d)", "Source shape (b)"); %! hold off %! d_false = procrustes (T, S, "reflection", false); %! printf ("Procrustes distance without reflection: %f\n", d_false); %! d_true = procrustes (T, S, "reflection", true); %! printf ("Procrustes distance with reflection: %f\n", d_true); %! d_best = procrustes (T, S, "reflection", "best"); %! printf ("Procrustes distance with best fit: %f\n", d_true); ## Test input validation %!error procrustes (); %!error procrustes (1, 2, 3, 4, 5, 6); %!error ... %! procrustes (ones (2, 2, 2), ones (2, 2, 2)); %!error ... %! procrustes ([1, 2; -3, 4; 2, 3], [1, 2; -3, 4; 2, 3+i]); %!error ... %! procrustes ([1, 2; -3, 4; 2, 3], [1, 2; -3, 4; 2, NaN]); %!error ... %! procrustes ([1, 2; -3, 4; 2, 3], [1, 2; -3, 4; 2, Inf]); %!error ... %! procrustes (ones (10 ,3), ones (11, 3)); %!error ... %! procrustes (ones (10 ,3), ones (10, 4)); %!error ... %! procrustes (ones (10 ,3), ones (10, 3), "reflection"); %!error ... %! procrustes (ones (10 ,3), ones (10, 3), true); %!error ... %! procrustes (ones (10 ,3), ones (10, 3), "scaling", 0); %!error ... %! procrustes (ones (10 ,3), ones (10, 3), "scaling", [true true]); %!error ... %! procrustes (ones (10 ,3), ones (10, 3), "reflection", 1); %!error ... %! procrustes (ones (10 ,3), ones (10, 3), "reflection", "some"); %!error ... %! procrustes (ones (10 ,3), ones (10, 3), "param1", "some"); statistics-release-1.6.3/inst/qqplot.m000066400000000000000000000103131456127120000200000ustar00rootroot00000000000000## Copyright (C) 1995-2017 Kurt Hornik ## Copyright (C) 2022-2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {[@var{q}, @var{s}] =} qqplot (@var{x}) ## @deftypefnx {statistics} {[@var{q}, @var{s}] =} qqplot (@var{x}, @var{y}) ## @deftypefnx {statistics} {[@var{q}, @var{s}] =} qqplot (@var{x}, @var{dist}) ## @deftypefnx {statistics} {[@var{q}, @var{s}] =} qqplot (@var{x}, @var{y}, @var{params}) ## @deftypefnx {statistics} {} qqplot (@dots{}) ## ## Perform a QQ-plot (quantile plot). ## ## If F is the CDF of the distribution @var{dist} with parameters ## @var{params} and G its inverse, and @var{x} a sample vector of length ## @var{n}, the QQ-plot graphs ordinate @var{s}(@var{i}) = @var{i}-th ## largest element of x versus abscissa @var{q}(@var{i}f) = G((@var{i} - ## 0.5)/@var{n}). ## ## If the sample comes from F, except for a transformation of location ## and scale, the pairs will approximately follow a straight line. ## ## If the second argument is a vector @var{y} the empirical CDF of @var{y} ## is used as @var{dist}. ## ## The default for @var{dist} is the standard normal distribution. The ## optional argument @var{params} contains a list of parameters of ## @var{dist}. For example, for a quantile plot of the uniform ## distribution on [2,4] and @var{x}, use ## ## @example ## qqplot (x, "unif", 2, 4) ## @end example ## ## @noindent ## @var{dist} can be any string for which a function @var{distinv} or ## @var{dist_inv} exists that calculates the inverse CDF of distribution ## @var{dist}. ## ## If no output arguments are given, the data are plotted directly. ## @seealso{ppplot} ## @end deftypefn function [qout, sout] = qqplot (x, dist, varargin) if (nargin < 1) print_usage (); endif if (! isnumeric (x) || ! isreal (x) || ! isvector (x) || isscalar (x)) error ("qqplot: X must be a numeric vector of real numbers"); endif if (nargin == 1) f = @probit; else if (isnumeric (dist)) f = @(y) empirical_inv (y, dist); elseif (ischar (dist) && (exist (invname = [dist "inv"]) || exist (invname = [dist "_inv"]))) f = str2func (invname); else error ("qqplot: no inverse CDF found for distribution DIST"); endif endif; s = sort (x); n = length (x); t = ((1 : n)' - .5) / n; if (nargin <= 2) q = f (t); q_label = func2str (f); else q = f (t, varargin{:}); if (nargin == 3) q_label = sprintf ("%s with parameter %g", func2str (f), varargin{1}); else q_label = sprintf ("%s with parameters %g", func2str (f), varargin{1}); param_str = sprintf (", %g", varargin{2:end}); q_label = [q_label param_str]; endif endif if (nargout == 0) plot (q, s, "-x"); q_label = strrep (q_label, '_inv', '\_inv'); if (q_label(1) == '@') q_label = q_label(6:end); # Strip "@(y) " from anon. function endif xlabel (q_label); ylabel ("sample points"); else qout = q; sout = s; endif endfunction ## Test plotting %!test %! hf = figure ("visible", "off"); %! unwind_protect %! qqplot ([2 3 3 4 4 5 6 5 6 7 8 9 8 7 8 9 0 8 7 6 5 4 6 13 8 15 9 9]); %! unwind_protect_cleanup %! close (hf); %! end_unwind_protect ## Test input validation %!error qqplot () %!error qqplot ({1}) %!error qqplot (ones (2,2)) %!error qqplot (1, "foobar") %!error qqplot ([1 2 3], "foobar") statistics-release-1.6.3/inst/qrandn.m000066400000000000000000000062371456127120000177550ustar00rootroot00000000000000## Copyright (C) 2014 - Juan Pablo Carbajal ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation; either version 3 of the License, or ## (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program. If not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{z} =} qrandn (@var{q}, @var{r}, @var{c}) ## @deftypefnx {statistics} {@var{z} =} qrandn (@var{q}, [@var{r}, @var{c}]) ## ## Returns random deviates drawn from a q-Gaussian distribution. ## ## Parameter @var{q} charcterizes the q-Gaussian distribution. ## The result has the size indicated by @var{s}. ## ## Reference: ## W. Thistleton, J. A. Marsh, K. Nelson, C. Tsallis (2006) ## "Generalized Box-Muller method for generating q-Gaussian random deviates" ## arXiv:cond-mat/0605570 http://arxiv.org/abs/cond-mat/0605570 ## ## @seealso{rand, randn} ## @end deftypefn function z = qrandn (q, R, C=[]) if (nargin < 2) print_usage; endif if (! isscalar (q)) error ("qrandn: the parameter q must be a scalar.")' endif ## Check that q < 3 if (q >= 3) error ("qrandn: the parameter q must be lower than 3."); endif if (numel (R) > 1) S = R; elseif (numel (R) == 1 && isempty (C)) S = [R, 1]; elseif (numel (R) == 1 && ! isempty (C)) S = [R, C]; endif ## Calaulate the q to be used on the q-log qGen = (1 + q) / (3 - q); ## Initialize the output vector z = sqrt (-2 * log_q (rand (S), qGen)) .* sin (2 * pi * rand (S)); endfunction ## Returns the q-log of x, using q function a = log_q (x, q) dq = 1 - q; ## Check to see if q = 1 (to double precision) if (abs (dq) < 10 * eps) ## If q is 1, use the usual natural logarithm a = log (x); else ## If q differs from 1, use the definition of the q-log a = (x .^ dq - 1) ./ dq; endif endfunction %!demo %! z = qrandn (-5, 5e6); %! [c x] = hist (z,linspace(-1.5,1.5,200),1); %! figure(1) %! plot(x,c,"r."); axis tight; axis([-1.5,1.5]); %! %! z = qrandn (-0.14286, 5e6); %! [c x] = hist (z,linspace(-2,2,200),1); %! figure(2) %! plot(x,c,"r."); axis tight; axis([-2,2]); %! %! z = qrandn (2.75, 5e6); %! [c x] = hist (z,linspace(-1e3,1e3,1e3),1); %! figure(3) %! semilogy(x,c,"r."); axis tight; axis([-100,100]); %! %! # --------- %! # Figures from the reference paper. ## Tests for input validation %!error qrandn ([1 2], 1) %!error qrandn (4, 1) %!error qrandn (3, 1) %!error qrandn (2.5, 1, 2, 3) %!error qrandn (2.5) ## Tests for output validation %!test %! q = 1.5; %! s = [2, 3]; %! z = qrandn (q, s); %! assert (isnumeric (z) && isequal (size (z), s)); statistics-release-1.6.3/inst/random.m000066400000000000000000000412031456127120000177420ustar00rootroot00000000000000## Copyright (C) 2007 Soren Hauberg ## Copyright (C) 2023-2024 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{r} =} random (@var{name}, @var{A}) ## @deftypefnx {statistics} {@var{r} =} random (@var{name}, @var{A}, @var{B}) ## @deftypefnx {statistics} {@var{r} =} random (@var{name}, @var{A}, @var{B}, @var{C}) ## @deftypefnx {statistics} {@var{r} =} random (@var{name}, @dots{}, @var{rows}, @var{cols}) ## @deftypefnx {statistics} {@var{r} =} random (@var{name}, @dots{}, @var{rows}, @var{cols}, @dots{}) ## @deftypefnx {statistics} {@var{r} =} random (@var{name}, @dots{}, [@var{sz}]) ## ## Random arrays from a given one-, two-, or three-parameter distribution. ## ## The variable @var{name} must be a string with the name of the distribution to ## sample from. If this distribution is a one-parameter distribution, @var{A} ## must be supplied, if it is a two-parameter distribution, @var{B} must also be ## supplied, and if it is a three-parameter distribution, @var{C} must also be ## supplied. Any arguments following the distribution parameters will determine ## the size of the result. ## ## When called with a single size argument, return a square matrix with the ## dimension specified. When called with more than one scalar argument the ## first two arguments are taken as the number of rows and columns and any ## further arguments specify additional matrix dimensions. The size may also ## be specified with a vector of dimensions @var{sz}. ## ## @var{name} must be a char string of the name or the abbreviation of the ## desired probability distribution function as listed in the followng table. ## The last column shows the required number of parameters that must be passed ## passed to the desired @qcode{*rnd} distribution function. ## ## @multitable @columnfractions 0.4 0.05 0.2 0.05 0.3 ## @headitem Distribution Name @tab @tab Abbreviation @tab @tab Input Parameters ## @item @qcode{"Beta"} @tab @tab @qcode{"beta"} @tab @tab 2 ## @item @qcode{"Binomial"} @tab @tab @qcode{"bino"} @tab @tab 2 ## @item @qcode{"Birnbaum-Saunders"} @tab @tab @qcode{"bisa"} @tab @tab 2 ## @item @qcode{"Burr"} @tab @tab @qcode{"burr"} @tab @tab 3 ## @item @qcode{"Cauchy"} @tab @tab @qcode{"cauchy"} @tab @tab 2 ## @item @qcode{"Chi-squared"} @tab @tab @qcode{"chi2"} @tab @tab 1 ## @item @qcode{"Extreme Value"} @tab @tab @qcode{"ev"} @tab @tab 2 ## @item @qcode{"Exponential"} @tab @tab @qcode{"exp"} @tab @tab 1 ## @item @qcode{"F-Distribution"} @tab @tab @qcode{"f"} @tab @tab 2 ## @item @qcode{"Gamma"} @tab @tab @qcode{"gam"} @tab @tab 2 ## @item @qcode{"Geometric"} @tab @tab @qcode{"geo"} @tab @tab 1 ## @item @qcode{"Generalized Extreme Value"} @tab @tab @qcode{"gev"} @tab @tab 3 ## @item @qcode{"Generalized Pareto"} @tab @tab @qcode{"gp"} @tab @tab 3 ## @item @qcode{"Gumbel"} @tab @tab @qcode{"gumbel"} @tab @tab 2 ## @item @qcode{"Half-normal"} @tab @tab @qcode{"hn"} @tab @tab 2 ## @item @qcode{"Hypergeometric"} @tab @tab @qcode{"hyge"} @tab @tab 3 ## @item @qcode{"Inverse Gaussian"} @tab @tab @qcode{"invg"} @tab @tab 2 ## @item @qcode{"Laplace"} @tab @tab @qcode{"laplace"} @tab @tab 2 ## @item @qcode{"Logistic"} @tab @tab @qcode{"logi"} @tab @tab 2 ## @item @qcode{"Log-Logistic"} @tab @tab @qcode{"logl"} @tab @tab 2 ## @item @qcode{"Lognormal"} @tab @tab @qcode{"logn"} @tab @tab 2 ## @item @qcode{"Nakagami"} @tab @tab @qcode{"naka"} @tab @tab 2 ## @item @qcode{"Negative Binomial"} @tab @tab @qcode{"nbin"} @tab @tab 2 ## @item @qcode{"Noncentral F-Distribution"} @tab @tab @qcode{"ncf"} @tab @tab 3 ## @item @qcode{"Noncentral Student T"} @tab @tab @qcode{"nct"} @tab @tab 2 ## @item @qcode{"Noncentral Chi-Squared"} @tab @tab @qcode{"ncx2"} @tab @tab 2 ## @item @qcode{"Normal"} @tab @tab @qcode{"norm"} @tab @tab 2 ## @item @qcode{"Poisson"} @tab @tab @qcode{"poiss"} @tab @tab 1 ## @item @qcode{"Rayleigh"} @tab @tab @qcode{"rayl"} @tab @tab 1 ## @item @qcode{"Rician"} @tab @tab @qcode{"rice"} @tab @tab 2 ## @item @qcode{"Student T"} @tab @tab @qcode{"t"} @tab @tab 1 ## @item @qcode{"location-scale T"} @tab @tab @qcode{"tls"} @tab @tab 3 ## @item @qcode{"Triangular"} @tab @tab @qcode{"tri"} @tab @tab 3 ## @item @qcode{"Discrete Uniform"} @tab @tab @qcode{"unid"} @tab @tab 1 ## @item @qcode{"Uniform"} @tab @tab @qcode{"unif"} @tab @tab 2 ## @item @qcode{"Von Mises"} @tab @tab @qcode{"vm"} @tab @tab 2 ## @item @qcode{"Weibull"} @tab @tab @qcode{"wbl"} @tab @tab 2 ## @end multitable ## ## @seealso{cdf, icdf, pdf, betarnd, binornd, bisarnd, burrrnd, cauchyrnd, ## chi2rnd, evrnd, exprnd, frnd, gamrnd, geornd, gevrnd, gprnd, gumbelrnd, ## hnrnd, hygernd, invgrnd, laplacernd, logirnd, loglrnd, lognrnd, nakarnd, ## nbinrnd, ncfrnd, nctrnd, ncx2rnd, normrnd, poissrnd, raylrnd, ricernd, trnd, ## trirnd, unidrnd, unifrnd, vmrnd, wblrnd} ## @end deftypefn function r = random (name, varargin) ## implemented functions persistent allDF = { ... {"beta" , "Beta"}, @betarnd, 2, ... {"bino" , "Binomial"}, @binornd, 2, ... {"bisa" , "Birnbaum-Saunders"}, @bisarnd, 2, ... {"burr" , "Burr"}, @burrrnd, 3, ... {"cauchy" , "Cauchy"}, @cauchyrnd, 2, ... {"chi2" , "Chi-squared"}, @chi2rnd, 1, ... {"ev" , "Extreme Value"}, @evrnd, 2, ... {"exp" , "Exponential"}, @exprnd, 1, ... {"f" , "F-Distribution"}, @frnd, 2, ... {"gam" , "Gamma"}, @gamrnd, 2, ... {"geo" , "Geometric"}, @geornd, 1, ... {"gev" , "Generalized Extreme Value"}, @gevrnd, 3, ... {"gp" , "Generalized Pareto"}, @gprnd, 3, ... {"gumbel" , "Gumbel"}, @gumbelrnd, 2, ... {"hn" , "Half-normal"}, @hnrnd, 2, ... {"hyge" , "Hypergeometric"}, @hygernd, 3, ... {"invg" , "Inverse Gaussian"}, @invgrnd, 2, ... {"laplace" , "Laplace"}, @laplacernd, 2, ... {"logi" , "Logistic"}, @logirnd, 2, ... {"logl" , "Log-Logistic"}, @loglrnd, 2, ... {"logn" , "Lognormal"}, @lognrnd, 2, ... {"naka" , "Nakagami"}, @nakarnd, 2, ... {"nbin" , "Negative Binomial"}, @nbinrnd, 2, ... {"ncf" , "Noncentral F-Distribution"}, @ncfrnd, 3, ... {"nct" , "Noncentral Student T"}, @nctrnd, 2, ... {"ncx2" , "Noncentral Chi-squared"}, @ncx2rnd, 2, ... {"norm" , "Normal"}, @normrnd, 2, ... {"poiss" , "Poisson"}, @poissrnd, 1, ... {"rayl" , "Rayleigh"}, @raylrnd, 1, ... {"rice" , "Rician"}, @ricernd, 2, ... {"t" , "Student T"}, @trnd, 1, ... {"tls" , "location-scale T"}, @tlsrnd, 3, ... {"tri" , "Triangular"}, @trirnd, 3, ... {"unid" , "Discrete Uniform"}, @unidrnd, 1, ... {"unif" , "Uniform"}, @unifrnd, 2, ... {"vm" , "Von Mises"}, @vmrnd, 2, ... {"wbl" , "Weibull"}, @wblrnd, 2}; if (! ischar (name)) error ("random: distribution NAME must a char string."); endif ## Get number of arguments nargs = numel (varargin); ## Get available functions rndnames = allDF(1:3:end); rndhandl = allDF(2:3:end); rnd_args = allDF(3:3:end); ## Search for RND function idx = cellfun (@(x)any(strcmpi (name, x)), rndnames); if (any (idx)) if (nargs == rnd_args{idx}) ## Check that all distribution parameters are numeric if (! all (cellfun (@(x)isnumeric(x), (varargin)))) error ("random: distribution parameters must be numeric."); endif ## Call appropriate RND r = feval (rndhandl{idx}, varargin{:}); elseif (nargs > rnd_args{idx}) ## Check that all distribution parameters are numeric if (! all (cellfun (@(x)isnumeric(x), (varargin(1:rnd_args{idx}))))) error ("random: distribution parameters must be numeric."); endif ## Call appropriate RND. SIZE arguments are checked by the RND function. r = feval (rndhandl{idx}, varargin{:}); else if (rnd_args{idx} == 1) error ("random: %s distribution requires 1 parameter.", name); else error ("random: %s distribution requires %d parameters.", ... name, rnd_args{idx}); endif endif else error ("random: %s distribution is not implemented in Statistics.", name); endif endfunction ## Test results %!assert (size (random ("Beta", 5, 2, 2, 10)), size (betarnd (5, 2, 2, 10))) %!assert (size (random ("beta", 5, 2, 2, 10)), size (betarnd (5, 2, 2, 10))) %!assert (size (random ("Binomial", 5, 2, [10, 20])), size (binornd (5, 2, 10, 20))) %!assert (size (random ("bino", 5, 2, [10, 20])), size (binornd (5, 2, 10, 20))) %!assert (size (random ("Birnbaum-Saunders", 5, 2, [10, 20])), size (bisarnd (5, 2, 10, 20))) %!assert (size (random ("bisa", 5, 2, [10, 20])), size (bisarnd (5, 2, 10, 20))) %!assert (size (random ("Burr", 5, 2, 2, [10, 20])), size (burrrnd (5, 2, 2, 10, 20))) %!assert (size (random ("burr", 5, 2, 2, [10, 20])), size (burrrnd (5, 2, 2, 10, 20))) %!assert (size (random ("Cauchy", 5, 2, [10, 20])), size (cauchyrnd (5, 2, 10, 20))) %!assert (size (random ("cauchy", 5, 2, [10, 20])), size (cauchyrnd (5, 2, 10, 20))) %!assert (size (random ("Chi-squared", 5, [10, 20])), size (chi2rnd (5, 10, 20))) %!assert (size (random ("chi2", 5, [10, 20])), size (chi2rnd (5, 10, 20))) %!assert (size (random ("Extreme Value", 5, 2, [10, 20])), size (evrnd (5, 2, 10, 20))) %!assert (size (random ("ev", 5, 2, [10, 20])), size (evrnd (5, 2, 10, 20))) %!assert (size (random ("Exponential", 5, [10, 20])), size (exprnd (5, 10, 20))) %!assert (size (random ("exp", 5, [10, 20])), size (exprnd (5, 10, 20))) %!assert (size (random ("F-Distribution", 5, 2, [10, 20])), size (frnd (5, 2, 10, 20))) %!assert (size (random ("f", 5, 2, [10, 20])), size (frnd (5, 2, 10, 20))) %!assert (size (random ("Gamma", 5, 2, [10, 20])), size (gamrnd (5, 2, 10, 20))) %!assert (size (random ("gam", 5, 2, [10, 20])), size (gamrnd (5, 2, 10, 20))) %!assert (size (random ("Geometric", 5, [10, 20])), size (geornd (5, 10, 20))) %!assert (size (random ("geo", 5, [10, 20])), size (geornd (5, 10, 20))) %!assert (size (random ("Generalized Extreme Value", 5, 2, 2, [10, 20])), size (gevrnd (5, 2, 2, 10, 20))) %!assert (size (random ("gev", 5, 2, 2, [10, 20])), size (gevrnd (5, 2, 2, 10, 20))) %!assert (size (random ("Generalized Pareto", 5, 2, 2, [10, 20])), size (gprnd (5, 2, 2, 10, 20))) %!assert (size (random ("gp", 5, 2, 2, [10, 20])), size (gprnd (5, 2, 2, 10, 20))) %!assert (size (random ("Gumbel", 5, 2, [10, 20])), size (gumbelrnd (5, 2, 10, 20))) %!assert (size (random ("gumbel", 5, 2, [10, 20])), size (gumbelrnd (5, 2, 10, 20))) %!assert (size (random ("Half-normal", 5, 2, [10, 20])), size (hnrnd (5, 2, 10, 20))) %!assert (size (random ("hn", 5, 2, [10, 20])), size (hnrnd (5, 2, 10, 20))) %!assert (size (random ("Hypergeometric", 5, 2, 2, [10, 20])), size (hygernd (5, 2, 2, 10, 20))) %!assert (size (random ("hyge", 5, 2, 2, [10, 20])), size (hygernd (5, 2, 2, 10, 20))) %!assert (size (random ("Inverse Gaussian", 5, 2, [10, 20])), size (invgrnd (5, 2, 10, 20))) %!assert (size (random ("invg", 5, 2, [10, 20])), size (invgrnd (5, 2, 10, 20))) %!assert (size (random ("Laplace", 5, 2, [10, 20])), size (laplacernd (5, 2, 10, 20))) %!assert (size (random ("laplace", 5, 2, [10, 20])), size (laplacernd (5, 2, 10, 20))) %!assert (size (random ("Logistic", 5, 2, [10, 20])), size (logirnd (5, 2, 10, 20))) %!assert (size (random ("logi", 5, 2, [10, 20])), size (logirnd (5, 2, 10, 20))) %!assert (size (random ("Log-Logistic", 5, 2, [10, 20])), size (loglrnd (5, 2, 10, 20))) %!assert (size (random ("logl", 5, 2, [10, 20])), size (loglrnd (5, 2, 10, 20))) %!assert (size (random ("Lognormal", 5, 2, [10, 20])), size (lognrnd (5, 2, 10, 20))) %!assert (size (random ("logn", 5, 2, [10, 20])), size (lognrnd (5, 2, 10, 20))) %!assert (size (random ("Nakagami", 5, 2, [10, 20])), size (nakarnd (5, 2, 10, 20))) %!assert (size (random ("naka", 5, 2, [10, 20])), size (nakarnd (5, 2, 10, 20))) %!assert (size (random ("Negative Binomial", 5, 2, [10, 20])), size (nbinrnd (5, 2, 10, 20))) %!assert (size (random ("nbin", 5, 2, [10, 20])), size (nbinrnd (5, 2, 10, 20))) %!assert (size (random ("Noncentral F-Distribution", 5, 2, 2, [10, 20])), size (ncfrnd (5, 2, 2, 10, 20))) %!assert (size (random ("ncf", 5, 2, 2, [10, 20])), size (ncfrnd (5, 2, 2, 10, 20))) %!assert (size (random ("Noncentral Student T", 5, 2, [10, 20])), size (nctrnd (5, 2, 10, 20))) %!assert (size (random ("nct", 5, 2, [10, 20])), size (nctrnd (5, 2, 10, 20))) %!assert (size (random ("Noncentral Chi-Squared", 5, 2, [10, 20])), size (ncx2rnd (5, 2, 10, 20))) %!assert (size (random ("ncx2", 5, 2, [10, 20])), size (ncx2rnd (5, 2, 10, 20))) %!assert (size (random ("Normal", 5, 2, [10, 20])), size (normrnd (5, 2, 10, 20))) %!assert (size (random ("norm", 5, 2, [10, 20])), size (normrnd (5, 2, 10, 20))) %!assert (size (random ("Poisson", 5, [10, 20])), size (poissrnd (5, 10, 20))) %!assert (size (random ("poiss", 5, [10, 20])), size (poissrnd (5, 10, 20))) %!assert (size (random ("Rayleigh", 5, [10, 20])), size (raylrnd (5, 10, 20))) %!assert (size (random ("rayl", 5, [10, 20])), size (raylrnd (5, 10, 20))) %!assert (size (random ("Rician", 5, 1, [10, 20])), size (ricernd (5, 1, 10, 20))) %!assert (size (random ("rice", 5, 1, [10, 20])), size (ricernd (5, 1, 10, 20))) %!assert (size (random ("Student T", 5, [10, 20])), size (trnd (5, 10, 20))) %!assert (size (random ("t", 5, [10, 20])), size (trnd (5, 10, 20))) %!assert (size (random ("location-scale T", 5, 1, 2, [10, 20])), size (tlsrnd (5, 1, 2, 10, 20))) %!assert (size (random ("tls", 5, 1, 2, [10, 20])), size (tlsrnd (5, 1, 2, 10, 20))) %!assert (size (random ("Triangular", 5, 2, 2, [10, 20])), size (trirnd (5, 2, 2, 10, 20))) %!assert (size (random ("tri", 5, 2, 2, [10, 20])), size (trirnd (5, 2, 2, 10, 20))) %!assert (size (random ("Discrete Uniform", 5, [10, 20])), size (unidrnd (5, 10, 20))) %!assert (size (random ("unid", 5, [10, 20])), size (unidrnd (5, 10, 20))) %!assert (size (random ("Uniform", 5, 2, [10, 20])), size (unifrnd (5, 2, 10, 20))) %!assert (size (random ("unif", 5, 2, [10, 20])), size (unifrnd (5, 2, 10, 20))) %!assert (size (random ("Von Mises", 5, 2, [10, 20])), size (vmrnd (5, 2, 10, 20))) %!assert (size (random ("vm", 5, 2, [10, 20])), size (vmrnd (5, 2, 10, 20))) %!assert (size (random ("Weibull", 5, 2, [10, 20])), size (wblrnd (5, 2, 10, 20))) %!assert (size (random ("wbl", 5, 2, [10, 20])), size (wblrnd (5, 2, 10, 20))) ## Test input validation %!error random (1) %!error random ({"beta"}) %!error ... %! random ("Beta", "a", 2) %!error ... %! random ("Beta", 5, "") %!error ... %! random ("Beta", 5, {2}) %!error ... %! random ("Beta", "a", 2, 2, 10) %!error ... %! random ("Beta", 5, "", 2, 10) %!error ... %! random ("Beta", 5, {2}, 2, 10) %!error ... %! random ("Beta", 5, "", 2, 10) %!error random ("chi2") %!error random ("Beta", 5) %!error random ("Burr", 5) %!error random ("Burr", 5, 2) statistics-release-1.6.3/inst/randsample.m000066400000000000000000000110161456127120000206070ustar00rootroot00000000000000## Copyright (C) 2014 - Nir Krakauer ## Copyright (C) 2022 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation; either version 3 of the License, or ## (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program. If not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{y} =} randsample (@var{v}, @var{k}) ## @deftypefnx {statistics} {@var{y} =} randsample (@var{v}, @var{k}, @var{replacement}=false) ## @deftypefnx {statistics} {@var{y} =} randsample (@var{v}, @var{k}, @var{replacement}=false, [@var{w}=[]]) ## ## Sample elements from a vector. ## ## Returns @var{k} random elements from a vector @var{v} with @var{n} elements, ## sampled without or with @var{replacement}. ## ## If @var{v} is a scalar, samples from 1:@var{v}. ## ## If a weight vector @var{w} of the same size as @var{v} is specified, the ## probablility of each element being sampled is proportional to @var{w}. ## Unlike Matlab's function of the same name, this can be done for sampling with ## or without replacement. ## ## Randomization is performed using rand(). ## ## @seealso{datasample, randperm} ## @end deftypefn function y = randsample (v, k, replacement=false ,w=[]) if (isscalar (v) && isreal (v)) n = v; vector_v = false; elseif (isvector (v)) n = numel (v); vector_v = true; else error ("randsample: The input v must be a vector or positive integer."); endif if k < 0 || ( k > n && !replacement ) error (strcat (["randsample: The input k must be a non-negative "], ... ["integer. Sampling without replacement needs k <= n."])); endif if (all (length (w) != [0, n])) error ("randsample: the size w (%d) must match the first argument (%d)", ... length(w), n); endif if (replacement) # sample with replacement if (isempty (w)) # all elements are equally likely to be sampled y = round (n * rand (1, k) + 0.5); else y = weighted_replacement (k, w); endif else # sample without replacement if (isempty (w)) # all elements are equally likely to be sampled y = randperm (n, k); else # use "accept-reject"-like sampling y = weighted_replacement (k, w); while (1) [yy, idx] = sort (y); # Note: sort keeps order of equal elements. Idup = [false, (diff (yy)==0)]; if (! any (Idup)) break else Idup(idx) = Idup; # find duplicates in original vector w(y) = 0; # don't permit resampling ## remove duplicates, then sample again y = [y(! Idup), (weighted_replacement (sum (Idup), w))]; endif endwhile endif endif if vector_v y = v(y); endif endfunction function y = weighted_replacement (k, w) w = w / sum (w); w = [0, cumsum(w(:))']; ## distribute k uniform random deviates based on the given weighting y = arrayfun (@(x) find (w <= x, 1, "last"), rand (1, k)); endfunction %!test %! n = 20; %! k = 5; %! x = randsample(n, k); %! assert (size(x), [1 k]); %! x = randsample(n, k, true); %! assert (size(x), [1 k]); %! x = randsample(n, k, false); %! assert (size(x), [1 k]); %! x = randsample(n, k, true, ones(n, 1)); %! assert (size(x), [1 k]); %! x = randsample(1:n, k); %! assert (size(x), [1 k]); %! x = randsample(1:n, k, true); %! assert (size(x), [1 k]); %! x = randsample(1:n, k, false); %! assert (size(x), [1 k]); %! x = randsample(1:n, k, true, ones(n, 1)); %! assert (size(x), [1 k]); %! x = randsample((1:n)', k); %! assert (size(x), [k 1]); %! x = randsample((1:n)', k, true); %! assert (size(x), [k 1]); %! x = randsample((1:n)', k, false); %! assert (size(x), [k 1]); %! x = randsample((1:n)', k, true, ones(n, 1)); %! assert (size(x), [k 1]); %! n = 10; %! k = 100; %! x = randsample(n, k, true, 1:n); %! assert (size(x), [1 k]); %! x = randsample((1:n)', k, true); %! assert (size(x), [k 1]); %! x = randsample(k, k, false, 1:k); %! assert (size(x), [1 k]); statistics-release-1.6.3/inst/rangesearch.m000066400000000000000000000553001456127120000207470ustar00rootroot00000000000000## Copyright (C) 2024 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{idx} =} rangesearch (@var{X}, @var{Y}, @var{r}) ## @deftypefnx {statistics} {[@var{idx}, @var{D}] =} rangesearch (@var{X}, @var{Y}, @var{r}) ## @deftypefnx {statistics} {[@dots{}] =} rangesearch (@dots{}, @var{name}, @var{value}) ## ## Find all neighbors within specified distance from input data. ## ## @code{@var{idx} = rangesearch (@var{X}, @var{Y}, @var{r})} returns all the ## points in @var{X} that are within distance @var{r} from the points in @var{Y}. ## @var{X} must be an @math{NxP} numeric matrix of input data, where rows ## correspond to observations and columns correspond to features or variables. ## @var{Y} is an @math{MxP} numeric matrix with query points, which must have ## the same numbers of column as @var{X}. @var{r} must be a nonnegative scalar ## value. @var{idx} is an @math{Mx1} cell array, where @math{M} is the number ## of observations in @var{Y}. The vector @qcode{@var{Idx}@{j@}} contains the ## indices of observations (rows) in @var{X} whose distances to ## @qcode{@var{Y}(j,:)} are not greater than @var{r}. ## ## @code{[@var{idx}, @var{D}] = rangesearch (@var{X}, @var{Y}, @var{r})} also ## returns the distances, @var{D}, which correspond to the points in @var{X} ## that are within distance @var{r} from the points in @var{Y}. @var{D} is an ## @math{Mx1} cell array, where @math{M} is the number of observations in ## @var{Y}. The vector @qcode{@var{D}@{j@}} contains the indices of ## observations (rows) in @var{X} whose distances to @qcode{@var{Y}(j,:)} are ## not greater than @var{r}. ## ## Additional parameters can be specified by @qcode{Name-Value} pair arguments. ## ## @multitable @columnfractions 0.18 0.02 0.8 ## @headitem @var{Name} @tab @tab @var{Value} ## ## @item @qcode{"P"} @tab @tab is the Minkowski distance exponent and it must be ## a positive scalar. This argument is only valid when the selected distance ## metric is @qcode{"minkowski"}. By default it is 2. ## ## @item @qcode{"Scale"} @tab @tab is the scale parameter for the standardized ## Euclidean distance and it must be a nonnegative numeric vector of equal ## length to the number of columns in @var{X}. This argument is only valid when ## the selected distance metric is @qcode{"seuclidean"}, in which case each ## coordinate of @var{X} is scaled by the corresponding element of ## @qcode{"scale"}, as is each query point in @var{Y}. By default, the scale ## parameter is the standard deviation of each coordinate in @var{X}. ## ## @item @qcode{"Cov"} @tab @tab is the covariance matrix for computing the ## mahalanobis distance and it must be a positive definite matrix matching the ## the number of columns in @var{X}. This argument is only valid when the ## selected distance metric is @qcode{"mahalanobis"}. ## ## @item @qcode{"BucketSize"} @tab @tab is the maximum number of data points in ## the leaf node of the Kd-tree and it must be a positive integer. This ## argument is only valid when the selected search method is @qcode{"kdtree"}. ## ## @item @qcode{"SortIndices"} @tab @tab is a boolean flag to sort the returned ## indices in ascending order by distance and it is @qcode{true} by default. ## When the selected search method is @qcode{"exhaustive"} or the ## @qcode{"IncludeTies"} flag is true, @code{rangesearch} always sorts the ## returned indices. ## ## @item @qcode{"Distance"} @tab @tab is the distance metric used by ## @code{rangesearch} as specified below: ## @end multitable ## ## @multitable @columnfractions 0.05 0.2 0.75 ## @item @tab @qcode{"euclidean"} @tab Euclidean distance. ## @item @tab @qcode{"seuclidean"} @tab standardized Euclidean distance. Each ## coordinate difference between the rows in @var{X} and the query matrix ## @var{Y} is scaled by dividing by the corresponding element of the standard ## deviation computed from @var{X}. To specify a different scaling, use the ## @qcode{"Scale"} name-value argument. ## @item @tab @qcode{"cityblock"} @tab City block distance. ## @item @tab @qcode{"chebychev"} @tab Chebychev distance (maximum coordinate ## difference). ## @item @tab @qcode{"minkowski"} @tab Minkowski distance. The default exponent ## is 2. To specify a different exponent, use the @qcode{"P"} name-value ## argument. ## @item @tab @qcode{"mahalanobis"} @tab Mahalanobis distance, computed using a ## positive definite covariance matrix. To change the value of the covariance ## matrix, use the @qcode{"Cov"} name-value argument. ## @item @tab @qcode{"cosine"} @tab Cosine distance. ## @item @tab @qcode{"correlation"} @tab One minus the sample linear correlation ## between observations (treated as sequences of values). ## @item @tab @qcode{"spearman"} @tab One minus the sample Spearman's rank ## correlation between observations (treated as sequences of values). ## @item @tab @qcode{"hamming"} @tab Hamming distance, which is the percentage ## of coordinates that differ. ## @item @tab @qcode{"jaccard"} @tab One minus the Jaccard coefficient, which is ## the percentage of nonzero coordinates that differ. ## @item @tab @var{@@distfun} @tab Custom distance function handle. A distance ## function of the form @code{function @var{D2} = distfun (@var{XI}, @var{YI})}, ## where @var{XI} is a @math{1xP} vector containing a single observation in ## @math{P}-dimensional space, @var{YI} is an @math{NxP} matrix containing an ## arbitrary number of observations in the same @math{P}-dimensional space, and ## @var{D2} is an @math{NxP} vector of distances, where @qcode{(@var{D2}k)} is ## the distance between observations @var{XI} and @qcode{(@var{YI}k,:)}. ## @end multitable ## ## @multitable @columnfractions 0.18 0.02 0.8 ## @item @qcode{"NSMethod"} @tab @tab is the nearest neighbor search method used ## by @code{rangesearch} as specified below. ## @end multitable ## ## @multitable @columnfractions 0.05 0.2 0.75 ## @item @tab @qcode{"kdtree"} @tab Creates and uses a Kd-tree to find nearest ## neighbors. @qcode{"kdtree"} is the default value when the number of columns ## in @var{X} is less than or equal to 10, @var{X} is not sparse, and the ## distance metric is @qcode{"euclidean"}, @qcode{"cityblock"}, ## @qcode{"manhattan"}, @qcode{"chebychev"}, or @qcode{"minkowski"}. Otherwise, ## the default value is @qcode{"exhaustive"}. This argument is only valid when ## the distance metric is one of the four aforementioned metrics. ## @item @tab @qcode{"exhaustive"} @tab Uses the exhaustive search algorithm by ## computing the distance values from all the points in @var{X} to each point in ## @var{Y}. ## @end multitable ## ## @seealso{knnsearch, pdist2} ## @end deftypefn function [idx, dist] = rangesearch (X, Y, r, varargin) ## Check input data if (nargin < 2) error ("rangesearch: too few input arguments."); endif if (size (X, 2) != size (Y, 2)) error ("rangesearch: number of columns in X and Y must match."); endif ## Add default values P = 2; # Exponent for Minkowski distance S = []; # Scale for the standardized Euclidean distance C = []; # Covariance matrix for Mahalanobis distance BS = 50; # Maximum number of points per leaf node for Kd-tree SI = true; # Sort returned indices according to distance Distance = "euclidean"; # Distance metric to be used NSMethod = []; # Nearest neighbor search method DistParameter = []; # Distance parameter for pdist2 ## Parse additional parameters in Name/Value pairs PSC = 0; while (numel (varargin) > 0) switch (tolower (varargin{1})) case "p" P = varargin{2}; PSC += 1; case "scale" S = varargin{2}; PSC += 1; case "cov" C = varargin{2}; PSC += 1; case "bucketsize" BS = varargin{2}; case "sortindices" SI = varargin{2}; case "distance" Distance = varargin{2}; case "nsmethod" NSMethod = varargin{2}; otherwise error ("rangesearch: invalid NAME in optional pairs of arguments."); endswitch varargin(1:2) = []; endwhile ## Check input parameters if (PSC > 1) error ("rangesearch: only a single distance parameter can be defined."); endif if (! isscalar (P) || ! isnumeric (P) || P <= 0) error ("rangesearch: invalid value of Minkowski Exponent."); endif if (! isempty (S)) if (any (S) < 0 || numel (S) != columns (X) || ! strcmpi (Distance, "seuclidean")) error ("rangesearch: invalid value in Scale or the size of Scale."); endif endif if (! isempty (C)) if (! strcmp (Distance, "mahalanobis") || ! ismatrix (C) || ! isnumeric (C)) error (strcat (["rangesearch: invalid value in Cov, Cov can only"], ... [" be given for mahalanobis distance."])); endif endif if (! isscalar (BS) || BS < 0) error ("rangesearch: invalid value of bucketsize."); endif ## Select the appropriate distance parameter if (strcmpi (Distance, "minkowski")) DistParameter = P; elseif (strcmpi (Distance, "seuclidean")) DistParameter = S; elseif (strcmpi (Distance, "mahalanobis")) DistParameter = C; endif ## Check NSMethod and set kdtree as default if the conditions match if (isempty (NSMethod)) ## Set default method 'kdtree' if condintions are satistfied; if (! issparse (X) && (columns (X) <= 10) && ... (strcmpi (Distance, "euclidean") || strcmpi (Distance, "cityblock") || strcmpi (Distance, "minkowski") || strcmpi (Distance, "chebychev"))) NSMethod = "kdtree"; else NSMethod = "exhaustive"; endif else ## Not empty then check if is exhaustive or kdtree if (strcmpi (NSMethod,"kdtree") && ! ( strcmpi (Distance, "euclidean") || strcmpi (Distance, "cityblock") || strcmpi (Distance, "minkowski") || strcmpi (Distance, "chebychev"))) error (strcat (["rangesearch: 'kdtree' cannot be used with"], ... [" the given distance metric."])); endif endif ## Check for NSMethod if (strcmpi (NSMethod, "kdtree")) ## Build kdtree and search the query point ret = buildkdtree (X, BS); ## Return all neighbors as cell dist = cell (rows (Y), 1); idx = cell (rows (Y), 1); k = rows (X); for i = 1:rows (Y) ## Need to fix the kd-tree search to compare with r distance (not k-NN) NN = findkdtree (ret, Y(i, :), k, Distance, DistParameter); D = - ones (k, 1); D(NN) = pdist2 (X(NN,:), Y(i,:), Distance, DistParameter); Didx_row = find (D <= r & D >= 0)'; Dist_row = D(Didx_row)'; if (SI) [S, I] = sort (Dist_row); Dist_row = Dist_row(I); Didx_row = Didx_row(I); endif dist{i} = Dist_row; idx{i} = Didx_row; endfor else ## Calculate all distances dist = cell (rows (Y), 1); idx = cell (rows (Y), 1); for i = 1:rows (Y) D = pdist2 (X, Y(i,:), Distance, DistParameter); Didx_row = find (D <= r)'; Dist_row = D(Didx_row)'; if (SI) [S, I] = sort (Dist_row); Dist_row = Dist_row(I); Didx_row = Didx_row(I); endif dist{i} = Dist_row; idx{i} = Didx_row; endfor endif endfunction ## buildkdtree function ret = buildkdtree_recur (X, r, d, BS) count = length (r); dimen = size (X, 2); if (count == 1) ret = struct ("point", r(1), "dimen", d); else mid = ceil (count / 2); ret = struct ("point", r(mid), "dimen", d); d = mod (d, dimen) + 1; ## Build left sub tree if (mid > 1) left = r(1:mid-1); left_points = X(left,d); [val, left_idx] = sort (left_points); leftr = left(left_idx); ret.left = buildkdtree_recur (X, leftr, d); endif ## Build right sub tree if (count > mid) right = r(mid+1:count); right_points = X(right,d); [val, right_idx] = sort (right_points); rightr = right(right_idx); ret.right = buildkdtree_recur (X, rightr, d); endif endif endfunction ## Need to fix the kd-tree search to compare with r distance (not k-NN) ## wrapper function for buildkdtree_recur function ret = buildkdtree (X, BS) [val, r] = sort (X(:,1)); ret = struct ("data", X, "root", buildkdtree_recur (X, r, 1, BS)); endfunction function farthest = kdtree_cand_farthest (X, p, cand, dist, distparam) D = pdist2 (X, p, dist, distparam); [val, index] = max (D'(cand)); farthest = cand (index); endfunction ## function to insert into NN list function inserted = kdtree_cand_insert (X, p, cand, k, point, dist, distparam) if (length (cand) < k) inserted = [cand; point]; else farthest = kdtree_cand_farthest (X, p, cand, dist, distparam); if (pdist2 (cand(find(cand == farthest),:), point, dist, distparam)) inserted = [cand; point]; else farthest = kdtree_cand_farthest (X, p, cand, dist, distparam); cand (find (cand == farthest)) = point; inserted = cand; endif endif endfunction ## function to search in a kd tree function nn = findkdtree_recur (X, node, p, nn, ... k, dist, distparam) point = node.point; d = node.dimen; if (X(point,d) > p(d)) ## Search in left sub tree if (isfield (node, "left")) nn = findkdtree_recur (X, node.left, p, nn, k, dist, distparam); endif ## Add current point if neccessary farthest = kdtree_cand_farthest (X, p, nn, dist, distparam); if (length(nn) < k || pdist2 (X(point,:), p, dist, distparam) <= pdist2 (X(farthest,:), p, dist, distparam)) nn = kdtree_cand_insert (X, p, nn, k, point, dist, distparam); endif ## Search in right sub tree if neccessary farthest = kdtree_cand_farthest (X, p, nn, dist, distparam); radius = pdist2 (X(farthest,:), p, dist, distparam); if (isfield (node, "right") && (length(nn) < k || p(d) + radius > X(point,d))) nn = findkdtree_recur (X, node.right, p, nn, ... k, dist, distparam); endif else ## Search in right sub tree if (isfield (node, "right")) nn = findkdtree_recur (X, node.right, p, nn, k, dist, distparam); endif ## Add current point if neccessary farthest = kdtree_cand_farthest (X, p, nn, dist, distparam); if (length (nn) < k || pdist2 (X(point,:), p, dist, distparam) <= pdist2 (X(farthest,:), p, dist, distparam)) nn = kdtree_cand_insert (X, p, nn, k, point, dist, distparam); endif ## Search in left sub tree if neccessary farthest = kdtree_cand_farthest (X, p, nn, dist, distparam); radius = pdist2 (X(farthest,:), p, dist, distparam); if (isfield (node, "left") && (length (nn) < k || p(d) - radius <= X(point,d))) nn = findkdtree_recur (X, node.left, p, nn, k, dist, distparam); endif endif endfunction ## wrapper function for findkdtree_recur function nn = findkdtree (tree, p, k, dist, distparam) X = tree.data; root = tree.root; nn = findkdtree_recur (X, root, p, [], k, dist, distparam); endfunction %!demo %! ## Generate 1000 random 2D points from each of five distinct multivariate %! ## normal distributions that form five separate classes %! N = 1000; %! d = 10; %! randn ("seed", 5); %! X1 = mvnrnd (d * [0, 0], eye (2), 1000); %! randn ("seed", 6); %! X2 = mvnrnd (d * [1, 1], eye (2), 1000); %! randn ("seed", 7); %! X3 = mvnrnd (d * [-1, -1], eye (2), 1000); %! randn ("seed", 8); %! X4 = mvnrnd (d * [1, -1], eye (2), 1000); %! randn ("seed", 8); %! X5 = mvnrnd (d * [-1, 1], eye (2), 1000); %! X = [X1; X2; X3; X4; X5]; %! %! ## For each point in X, find the points in X that are within a radius d %! ## away from the points in X. %! Idx = rangesearch (X, X, d, "NSMethod", "exhaustive"); %! %! ## Select the first point in X (corresponding to the first class) and find %! ## its nearest neighbors within the radius d. Display these points in %! ## one color and the remaining points in a different color. %! x = X(1,:); %! nearestPoints = X (Idx{1},:); %! nonNearestIdx = true (size (X, 1), 1); %! nonNearestIdx(Idx{1}) = false; %! %! scatter (X(nonNearestIdx,1), X(nonNearestIdx,2)) %! hold on %! scatter (nearestPoints(:,1),nearestPoints(:,2)) %! scatter (x(1), x(2), "black", "filled") %! hold off %! %! ## Select the last point in X (corresponding to the fifth class) and find %! ## its nearest neighbors within the radius d. Display these points in %! ## one color and the remaining points in a different color. %! x = X(end,:); %! nearestPoints = X (Idx{1},:); %! nonNearestIdx = true (size (X, 1), 1); %! nonNearestIdx(Idx{1}) = false; %! %! figure %! scatter (X(nonNearestIdx,1), X(nonNearestIdx,2)) %! hold on %! scatter (nearestPoints(:,1),nearestPoints(:,2)) %! scatter (x(1), x(2), "black", "filled") %! hold off ## Test output %!shared x, y, X, Y %! x = [1, 2, 3; 4, 5, 6; 7, 8, 9; 3, 2, 1]; %! y = [2, 3, 4; 1, 4, 3]; %! X = [1, 2, 3, 4; 2, 3, 4, 5; 3, 4, 5, 6]; %! Y = [1, 2, 2, 3; 2, 3, 3, 4]; %!test %! [idx, D] = rangesearch (x, y, 4); %! assert (idx, {[1, 4, 2]; [1, 4]}); %! assert (D, {[1.7321, 3.3166, 3.4641]; [2, 3.4641]}, 1e-4); %!test %! [idx, D] = rangesearch (x, y, 4, "NSMethod", "exhaustive"); %! assert (idx, {[1, 4, 2]; [1, 4]}); %! assert (D, {[1.7321, 3.3166, 3.4641]; [2, 3.4641]}, 1e-4); %!test %! [idx, D] = rangesearch (x, y, 4, "NSMethod", "kdtree"); %! assert (idx, {[1, 4, 2]; [1, 4]}); %! assert (D, {[1.7321, 3.3166, 3.4641]; [2, 3.4641]}, 1e-4); %!test %! [idx, D] = rangesearch (x, y, 4, "SortIndices", true); %! assert (idx, {[1, 4, 2]; [1, 4]}); %! assert (D, {[1.7321, 3.3166, 3.4641]; [2, 3.4641]}, 1e-4); %!test %! [idx, D] = rangesearch (x, y, 4, "SortIndices", false); %! assert (idx, {[1, 2, 4]; [1, 4]}); %! assert (D, {[1.7321, 3.4641, 3.3166]; [2, 3.4641]}, 1e-4); %!test %! [idx, D] = rangesearch (x, y, 4, "NSMethod", "exhaustive", ... %! "SortIndices", false); %! assert (idx, {[1, 2, 4]; [1, 4]}); %! assert (D, {[1.7321, 3.4641, 3.3166]; [2, 3.4641]}, 1e-4); %!test %! eucldist = @(v,m) sqrt(sumsq(repmat(v,rows(m),1)-m,2)); %! [idx, D] = rangesearch (x, y, 4, "Distance", eucldist); %! assert (idx, {[1, 4, 2]; [1, 4]}); %! assert (D, {[1.7321, 3.3166, 3.4641]; [2, 3.4641]}, 1e-4); %!test %! eucldist = @(v,m) sqrt(sumsq(repmat(v,rows(m),1)-m,2)); %! [idx, D] = rangesearch (x, y, 4, "Distance", eucldist, ... %! "NSMethod", "exhaustive"); %! assert (idx, {[1, 4, 2]; [1, 4]}); %! assert (D, {[1.7321, 3.3166, 3.4641]; [2, 3.4641]}, 1e-4); %!test %! [idx, D] = rangesearch (x, y, 1.5, "Distance", "seuclidean", ... %! "NSMethod", "exhaustive"); %! assert (idx, {[1, 4, 2]; [1, 4]}); %! assert (D, {[0.6024, 1.0079, 1.2047]; [0.6963, 1.2047]}, 1e-4); %!test %! [idx, D] = rangesearch (x, y, 1.5, "Distance", "seuclidean", ... %! "NSMethod", "exhaustive", "SortIndices", false); %! assert (idx, {[1, 2, 4]; [1, 4]}); %! assert (D, {[0.6024, 1.2047, 1.0079]; [0.6963, 1.2047]}, 1e-4); %!test %! [idx, D] = rangesearch (X, Y, 4); %! assert (idx, {[1, 2]; [1, 2, 3]}); %! assert (D, {[1.4142, 3.1623]; [1.4142, 1.4142, 3.1623]}, 1e-4); %!test %! [idx, D] = rangesearch (X, Y, 2); %! assert (idx, {[1]; [1, 2]}); %! assert (D, {[1.4142]; [1.4142, 1.4142]}, 1e-4); %!test %! eucldist = @(v,m) sqrt(sumsq(repmat(v,rows(m),1)-m,2)); %! [idx, D] = rangesearch (X, Y, 4, "Distance", eucldist); %! assert (idx, {[1, 2]; [1, 2, 3]}); %! assert (D, {[1.4142, 3.1623]; [1.4142, 1.4142, 3.1623]}, 1e-4); %!test %! [idx, D] = rangesearch (X, Y, 4, "SortIndices", false); %! assert (idx, {[1, 2]; [1, 2, 3]}); %! assert (D, {[1.4142, 3.1623]; [1.4142, 1.4142, 3.1623]}, 1e-4); %!test %! [idx, D] = rangesearch (X, Y, 4, "Distance", "seuclidean", ... %! "NSMethod", "exhaustive"); %! assert (idx, {[1, 2]; [1, 2, 3]}); %! assert (D, {[1.4142, 3.1623]; [1.4142, 1.4142, 3.1623]}, 1e-4); ## Test input validation %!error rangesearch (1) %!error ... %! rangesearch (ones (4, 5), ones (4)) %!error ... %! rangesearch (ones (4, 2), ones (3, 2), 1, "Distance", "euclidean", "some", "some") %!error ... %! rangesearch (ones (4, 5), ones (1, 5), 1, "scale", ones (1, 5), "P", 3) %!error ... %! rangesearch (ones (4, 5), ones (1, 5), 1, "P",-2) %!error ... %! rangesearch (ones (4, 5), ones (1, 5), 1, "scale", ones(4,5), "distance", "euclidean") %!error ... %! rangesearch (ones (4, 5), ones (1, 5), 1, "cov", ["some" "some"]) %!error ... %! rangesearch (ones (4, 5), ones (1, 5), 1, "cov", ones(4,5), "distance", "euclidean") %!error ... %! rangesearch (ones (4, 5), ones (1, 5), 1, "bucketsize", -1) %!error ... %! rangesearch (ones (4, 5), ones (1, 5), 1, "NSmethod", "kdtree", "distance", "cosine") %!error ... %! rangesearch (ones (4, 5), ones (1, 5), 1, "NSmethod", "kdtree", "distance", "mahalanobis") %!error ... %! rangesearch (ones (4, 5), ones (1, 5), 1, "NSmethod", "kdtree", "distance", "correlation") %!error ... %! rangesearch (ones (4, 5), ones (1, 5), 1, "NSmethod", "kdtree", "distance", "seuclidean") %!error ... %! rangesearch (ones (4, 5), ones (1, 5), 1, "NSmethod", "kdtree", "distance", "spearman") %!error ... %! rangesearch (ones (4, 5), ones (1, 5), 1, "NSmethod", "kdtree", "distance", "hamming") %!error ... %! rangesearch (ones (4, 5), ones (1, 5), 1, "NSmethod", "kdtree", "distance", "jaccard") statistics-release-1.6.3/inst/ranksum.m000066400000000000000000000255501456127120000201510ustar00rootroot00000000000000## Copyright (C) 2022 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{p} =} ranksum (@var{x}, @var{y}) ## @deftypefnx {statistics} {@var{p} =} ranksum (@var{x}, @var{y}, @var{alpha}) ## @deftypefnx {statistics} {@var{p} =} ranksum (@var{x}, @var{y}, @var{alpha}, @var{Name}, @var{Value}) ## @deftypefnx {statistics} {@var{p} =} ranksum (@var{x}, @var{y}, @var{Name}, @var{Value}) ## @deftypefnx {statistics} {[@var{p}, @var{h}] =} ranksum (@var{x}, @var{y}, @dots{}) ## @deftypefnx {statistics} {[@var{p}, @var{h}, @var{stats}] =} ranksum (@var{x}, @var{y}, @dots{}) ## ## Wilcoxon rank sum test for equal medians. This test is equivalent to a ## Mann-Whitney U-test. ## ## @code{@var{p} = ranksum (@var{x}, @var{y})} returns the p-value of a ## two-sided Wilcoxon rank sum test. It tests the null hypothesis that two ## independent samples, in the vectors X and Y, come from continuous ## distributions with equal medians, against the alternative hypothesis that ## they are not. @var{x} and @var{y} can have different lengths and the test ## assumes that they are independent. ## ## @code{ranksum} treats NaN in @var{x}, @var{y} as missing values. ## The two-sided p-value is computed by doubling the most significant one-sided ## value. ## ## @code{[@var{p}, @var{h}] = ranksum (@var{x}, @var{y})} also returns the ## result of the hypothesis test with @code{@var{h} = 1} indicating a rejection ## of the null hypothesis at the default alpha = 0.05 significance level, and ## @code{@var{h} = 0} indicating a failure to reject the null hypothesis at the ## same significance level. ## ## @code{[@var{p}, @var{h}, @var{stats}] = ranksum (@var{x}, @var{y})} also ## returns the structure @var{stats} with information about the test statistic. ## It contains the field @code{ranksum} with the value of the rank sum test ## statistic and if computed with the "approximate" method it also contains the ## value of the z-statistic in the field @code{zval}. ## ## @code{[@dots{}] = ranksum (@var{x}, @var{y}, @var{alpha})} or alternatively ## @code{[@dots{}] = ranksum (@var{x}, @var{y}, "alpha", @var{alpha})} returns ## the result of the hypothesis test performed at the significance level ALPHA. ## ## @code{[@dots{}] = ranksum (@var{x}, @var{y}, "method", @var{M})} defines the ## computation method of the p-value specified in @var{M}, which can be "exact", ## "approximate", or "oldexact". @var{M} must be a single string. When "method" ## is unspecified, the default is: "exact" when ## @code{min (length (@var{x}), length (@var{y})) < 10} and ## @code{length (@var{x}) + length (@var{y}) < 10}, otherwise the "approximate" ## method is used. ## ## @itemize ## @item ## "exact" method uses full enumeration for small total sample size (< 10), ## otherwise the network algorithm is used for larger samples. ## @item ## "approximate" uses normal approximation method for computing the p-value. ## @item ## "oldexact" uses full enumeration for any sample size. Note, that this option ## can lead to out of memory error for large samples. Use with caution! ## @end itemize ## ## @code{[@dots{}] = ranksum (@var{x}, @var{y}, "tail", @var{tail})} defines the ## type of test, which can be "both", "right", or "left". @var{tail} must be a ## single string. ## ## @itemize ## @item ## "both" -- "medians are not equal" (two-tailed test, default) ## @item ## "right" -- "median of X is greater than median of Y" (right-tailed test) ## @item ## "left" -- "median of X is less than median of Y" (left-tailed test) ## @end itemize ## ## Note: the rank sum statistic is based on the smaller sample of vectors ## @var{x} and @var{y}. ## ## @end deftypefn function [p, h, stats] = ranksum(x, y, varargin) ## Check that x and y are vectors if ! isvector (x) || ! isvector (y) error ("X and Y must be vectors"); endif ## Remove missing data and make column vectors x = x(! isnan (x))(:); y = y(! isnan (y))(:); if isempty (x) error ("Not enough data in X"); endif if isempty (y) error ("Not enough data in Y"); endif ## Check for extra input arguments alpha = 0.05; method = []; tail = "both"; ## Old syntax: ranksum (x, y, alpha) if nargin > 2 && isnumeric (varargin{1}) && isscalar (varargin{1}) alpha = varargin{1}; varargin(1) = []; if isnan (alpha) || alpha <= 0 || alpha >= 1 error ("Alpha does not have a valid value"); endif end ## Check for Name:Value pairs arg_pairs = length (varargin); if ! (int16 (arg_pairs / 2) == arg_pairs / 2) error ("Extra arguments are not in Name:Value pairs"); endif num_pair = 1; while (arg_pairs) name = varargin{num_pair}; value = varargin{num_pair + 1}; switch (lower (name)) case "alpha" alpha = value; if (isnan (alpha) || alpha <= 0 || alpha >= 1 || ! isnumeric (alpha) ... || ! isscalar (alpha)) error ("Alpha does not have a valid value"); endif case "method" method = value; if ! any (strcmpi (method, {"exact", "approximate", "oldexact"})) error ("Wrong value for method option"); endif case "tail" tail = value; if ! any (strcmpi (tail, {"both", "right", "left"})) error ("Wrong value for tail option"); endif endswitch arg_pairs -= 2; num_pair += 2; endwhile ## Determine method nx = length (x); ny = length (y); ns = min (nx, ny); if isempty (method) if (ns < 10) && ((nx + ny) < 20) method = "exact"; else method = "approximate"; endif endif % Determine computational technique switch method case "approximate" technique = "approximation"; case "oldexact" technique = "exact"; case "exact" if (nx + ny) < 10 technique = "exact"; else technique = "network_algorithm"; endif endswitch % Compute the rank sum statistic based on the smaller sample if nx <= ny [ranks, tieadj] = tiedrank ([x; y]); x_y = true; else [ranks, tieadj] = tiedrank ([y; x]); x_y = false; endif srank = ranks(1:ns); ranksumstat = sum (srank); ## Calculate p-value according to selected technique switch technique case "exact" allpos = nchoosek (ranks, ns); sumranks = sum (allpos, 2); np = size (sumranks, 1); switch tail case "both" p_low = sum (sumranks <= ranksumstat) / np; p_high = sum (sumranks >= ranksumstat) / np; p = 2 * min (p_low, p_high); if p > 1 p = 1; endif case "right" if x_y p = sum (sumranks >= ranksumstat) / np; else p = sum (sumranks <= ranksumstat) / np; endif case "left" if x_y p = sum (sumranks <= ranksumstat) / np; else p = sum (sumranks >= ranksumstat) / np; endif endswitch case "network_algorithm" ## Calculate contingency table u = unique ([x; y]); ct = zeros (2, length (u)); if x_y ct(1,:) = histc (x,u)'; ct(2,:) = histc (y,u)'; else ct(1,:) = histc (y,u)'; ct(2,:) = histc (x,u)'; endif ## Calculate weights for wmw test colsum = sum (ct,1); tmp = cumsum (colsum); weights = [0 tmp(1:end - 1)] + .5 * (1 + diff ([0 tmp])); ## Compute p-value using network algorithm for contingency tables [p_net, p_val] = exact2xkCT (ct, weights, ranksumstat); ## Check if p = NaN if any (isnan (p_net)) || any (isnan (p_val)) p = NaN; else switch tail case "both" p = 2 * p_net; if p > 1 p = 1; endif case "right" if x_y p = p_val(2) + p_val(3); else p = p_val(2) + p_val(1); endif case "left" if x_y p = p_val(2) + p_val(1); else p = p_val(2) + p_val(3); endif endswitch endif case "approximation" wmean = ns * (nx + ny + 1) / 2; tiescores = 2 * tieadj / ((nx + ny) * (nx + ny - 1)); wvar = nx * ny * ((nx + ny + 1) - tiescores) / 12; wc = ranksumstat - wmean; ## compute z-value, including continuity correction switch tail case "both" z = (wc - 0.5 * sign (wc)) / sqrt (wvar); if ! x_y z = -z; endif p = 2 * normcdf (-abs(z)); case "right" if x_y z = (wc - 0.5) / sqrt (wvar); else z = -(wc + 0.5) / sqrt (wvar); endif p = normcdf (-z); case "left" if x_y z = (wc + 0.5) / sqrt (wvar); else z = -(wc - 0.5) / sqrt (wvar); endif p = normcdf (z); endswitch ## For additional output argument if (nargout > 2) stats.zval = z; endif endswitch ## For additional output arguments if nargout > 1, h = (p <= alpha); if (nargout > 2) if x_y stats.ranksum = ranksumstat; else stats.ranksum = sum (ranks(ns+1:end)); endif endif endif endfunction ## testing against mileage data and results from Matlab %!test %! mileage = [33.3, 34.5, 37.4; 33.4, 34.8, 36.8; ... %! 32.9, 33.8, 37.6; 32.6, 33.4, 36.6; ... %! 32.5, 33.7, 37.0; 33.0, 33.9, 36.7]; %! [p,h,stats] = ranksum(mileage(:,1),mileage(:,2)); %! assert (p, 0.004329004329004329, 1e-14); %! assert (h, true); %! assert (stats.ranksum, 21.5); %!test %! year1 = [51 52 62 62 52 52 51 53 59 63 59 56 63 74 68 86 82 70 69 75 73 ... %! 49 47 50 60 59 60 62 61 71]'; %! year2 = [54 53 64 66 57 53 54 54 62 66 59 59 67 76 75 86 82 67 74 80 75 ... %! 54 50 53 62 62 62 72 60 67]'; %! [p,h,stats] = ranksum(year1, year2, "alpha", 0.01, "tail", "left"); %! assert (p, 0.1270832752950605, 1e-14); %! assert (h, false); %! assert (stats.ranksum, 837.5); %! assert (stats.zval, -1.140287483634606, 1e-14); %! [p,h,stats] = ranksum(year1, year2, "alpha", 0.01, "tail", "left", ... %! "method", "exact"); %! assert (p, 0.127343916432862, 1e-14); %! assert (h, false); %! assert (stats.ranksum, 837.5); statistics-release-1.6.3/inst/regress.m000066400000000000000000000147761456127120000201530ustar00rootroot00000000000000## Copyright (C) 2005, 2006 William Poetra Yoga Hadisoeseno ## Copyright (C) 2011 Nir Krakauer ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {[@var{b}, @var{bint}, @var{r}, @var{rint}, @var{stats}] =} regress (@var{y}, @var{X}, [@var{alpha}]) ## ## Multiple Linear Regression using Least Squares Fit of @var{y} on @var{X} ## with the model @code{y = X * beta + e}. ## ## Here, ## ## @itemize ## @item ## @code{y} is a column vector of observed values ## @item ## @code{X} is a matrix of regressors, with the first column filled with ## the constant value 1 ## @item ## @code{beta} is a column vector of regression parameters ## @item ## @code{e} is a column vector of random errors ## @end itemize ## ## Arguments are ## ## @itemize ## @item ## @var{y} is the @code{y} in the model ## @item ## @var{X} is the @code{X} in the model ## @item ## @var{alpha} is the significance level used to calculate the confidence ## intervals @var{bint} and @var{rint} (see `Return values' below). If not ## specified, ALPHA defaults to 0.05 ## @end itemize ## ## Return values are ## ## @itemize ## @item ## @var{b} is the @code{beta} in the model ## @item ## @var{bint} is the confidence interval for @var{b} ## @item ## @var{r} is a column vector of residuals ## @item ## @var{rint} is the confidence interval for @var{r} ## @item ## @var{stats} is a row vector containing: ## ## @itemize ## @item The R^2 statistic ## @item The F statistic ## @item The p value for the full model ## @item The estimated error variance ## @end itemize ## @end itemize ## ## @var{r} and @var{rint} can be passed to @code{rcoplot} to visualize ## the residual intervals and identify outliers. ## ## NaN values in @var{y} and @var{X} are removed before calculation begins. ## ## @seealso{regress_gp, regression_ftest, regression_ttest} ## @end deftypefn function [b, bint, r, rint, stats] = regress (y, X, alpha) if (nargin < 2 || nargin > 3) print_usage; endif if (! ismatrix (y)) error ("regress: y must be a numeric matrix"); endif if (! ismatrix (X)) error ("regress: X must be a numeric matrix"); endif if (columns (y) != 1) error ("regress: y must be a column vector"); endif if (rows (y) != rows (X)) error ("regress: y and X must contain the same number of rows"); endif if (nargin < 3) alpha = 0.05; elseif (! isscalar (alpha)) error ("regress: alpha must be a scalar value") endif notnans = ! logical (sum (isnan ([y X]), 2)); y = y(notnans); X = X(notnans,:); [Xq Xr] = qr (X, 0); pinv_X = Xr \ Xq'; b = pinv_X * y; if (nargout > 1) n = rows (X); p = columns (X); dof = n - p; t_alpha_2 = tinv (alpha / 2, dof); r = y - X * b; # added -- Nir SSE = sum (r .^ 2); v = SSE / dof; # c = diag(inv (X' * X)) using (economy) QR decomposition # which means that we only have to use Xr c = diag (inv (Xr' * Xr)); db = t_alpha_2 * sqrt (v * c); bint = [b + db, b - db]; endif if (nargout > 3) dof1 = n - p - 1; h = sum(X.*pinv_X', 2); #added -- Nir (same as diag(X*pinv_X), without doing the matrix multiply) # From Matlab's documentation on Multiple Linear Regression, # sigmaihat2 = norm (r) ^ 2 / dof1 - r .^ 2 / (dof1 * (1 - h)); # dr = -tinv (1 - alpha / 2, dof) * sqrt (sigmaihat2 .* (1 - h)); # Substitute # norm (r) ^ 2 == sum (r .^ 2) == SSE # -tinv (1 - alpha / 2, dof) == tinv (alpha / 2, dof) == t_alpha_2 # We get # sigmaihat2 = (SSE - r .^ 2 / (1 - h)) / dof1; # dr = t_alpha_2 * sqrt (sigmaihat2 .* (1 - h)); # Combine, we get # dr = t_alpha_2 * sqrt ((SSE * (1 - h) - (r .^ 2)) / dof1); dr = t_alpha_2 * sqrt ((SSE * (1 - h) - (r .^ 2)) / dof1); rint = [r + dr, r - dr]; endif if (nargout > 4) R2 = 1 - SSE / sum ((y - mean (y)) .^ 2); # F = (R2 / (p - 1)) / ((1 - R2) / dof); F = dof / (p - 1) / (1 / R2 - 1); pval = 1 - fcdf (F, p - 1, dof); stats = [R2 F pval v]; endif endfunction %!test %! % Longley data from the NIST Statistical Reference Dataset %! Z = [ 60323 83.0 234289 2356 1590 107608 1947 %! 61122 88.5 259426 2325 1456 108632 1948 %! 60171 88.2 258054 3682 1616 109773 1949 %! 61187 89.5 284599 3351 1650 110929 1950 %! 63221 96.2 328975 2099 3099 112075 1951 %! 63639 98.1 346999 1932 3594 113270 1952 %! 64989 99.0 365385 1870 3547 115094 1953 %! 63761 100.0 363112 3578 3350 116219 1954 %! 66019 101.2 397469 2904 3048 117388 1955 %! 67857 104.6 419180 2822 2857 118734 1956 %! 68169 108.4 442769 2936 2798 120445 1957 %! 66513 110.8 444546 4681 2637 121950 1958 %! 68655 112.6 482704 3813 2552 123366 1959 %! 69564 114.2 502601 3931 2514 125368 1960 %! 69331 115.7 518173 4806 2572 127852 1961 %! 70551 116.9 554894 4007 2827 130081 1962 ]; %! % Results certified by NIST using 500 digit arithmetic %! % b and standard error in b %! V = [ -3482258.63459582 890420.383607373 %! 15.0618722713733 84.9149257747669 %! -0.358191792925910E-01 0.334910077722432E-01 %! -2.02022980381683 0.488399681651699 %! -1.03322686717359 0.214274163161675 %! -0.511041056535807E-01 0.226073200069370 %! 1829.15146461355 455.478499142212 ]; %! Rsq = 0.995479004577296; %! F = 330.285339234588; %! y = Z(:,1); X = [ones(rows(Z),1), Z(:,2:end)]; %! alpha = 0.05; %! [b, bint, r, rint, stats] = regress (y, X, alpha); %! assert(b,V(:,1),4e-6); %! assert(stats(1),Rsq,1e-12); %! assert(stats(2),F,3e-8); %! assert(((bint(:,1)-bint(:,2))/2)/tinv(alpha/2,9),V(:,2),-1.e-5); statistics-release-1.6.3/inst/regress_gp.m000066400000000000000000000474421456127120000206350ustar00rootroot00000000000000## Copyright (c) 2012 Juan Pablo Carbajal ## Copyright (C) 2023-2024 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation; either version 3 of the License, or ## (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {[@var{Yfit}, @var{Yint}, @var{m}, @var{K}] =} regress_gp (@var{X}, @var{Y}, @var{Xfit}) ## @deftypefnx {statistics} {[@var{Yfit}, @var{Yint}, @var{m}, @var{K}] =} regress_gp (@var{X}, @var{Y}, @var{Xfit}, @qcode{"linear"}) ## @deftypefnx {statistics} {[@var{Yfit}, @var{Yint}, @var{Ysd}] =} regress_gp (@var{X}, @var{Y}, @var{Xfit}, @qcode{"rbf"}) ## @deftypefnx {statistics} {[@dots{}] =} regress_gp (@var{X}, @var{Y}, @var{Xfit}, @qcode{"linear"}, @var{Sp}) ## @deftypefnx {statistics} {[@dots{}] =} regress_gp (@var{X}, @var{Y}, @var{Xfit}, @var{Sp}) ## @deftypefnx {statistics} {[@dots{}] =} regress_gp (@var{X}, @var{Y}, @var{Xfit}, @qcode{"rbf"}, @var{theta}) ## @deftypefnx {statistics} {[@dots{}] =} regress_gp (@var{X}, @var{Y}, @var{Xfit}, @qcode{"rbf"}, @var{theta}, @var{g}) ## @deftypefnx {statistics} {[@dots{}] =} regress_gp (@var{X}, @var{Y}, @var{Xfit}, @qcode{"rbf"}, @var{theta}, @var{g}, @var{alpha}) ## @deftypefnx {statistics} {[@dots{}] =} regress_gp (@var{X}, @var{Y}, @var{Xfit}, @var{theta}) ## @deftypefnx {statistics} {[@dots{}] =} regress_gp (@var{X}, @var{Y}, @var{Xfit}, @var{theta}, @var{g}) ## @deftypefnx {statistics} {[@dots{}] =} regress_gp (@var{X}, @var{Y}, @var{Xfit}, @var{theta}, @var{g}, @var{alpha}) ## ## Regression using Gaussian Processes. ## ## @code{[@var{Yfit}, @var{Yint}, @var{m}, @var{K}] = regress_gp (@var{X}, ## @var{Y}, @var{Xfit})} will estimate a linear Gaussian Process model @var{m} ## in the form @qcode{@var{Y} = @var{X}' * @var{m}}, where @var{X} is an ## @math{NxP} matrix with @math{N} observations in @math{P} dimensional space ## and @var{Y} is an @math{Nx1} column vector as the dependent variable. The ## information about errors of the predictions (interpolation/extrapolation) is ## given by the covarianve matrix @var{K}. ## By default, the linear model defines the prior covariance of @var{m} as ## @code{@var{Sp} = 100 * eye (size (@var{X}, 2) + 1)}. A custom prior ## covariance matrix can be passed as @var{Sp}, which must be a @math{P+1xP+1} ## positive definite matrix. The model is evaluated for input @var{Xfit}, which ## must have the same columns as @var{X}, and the estimates are returned in ## @var{Yfit} along with the estimated variation in @var{Yint}. ## @qcode{@var{Yint}(:,1)} contains the upper boundary estimate and ## @qcode{@var{Yint}(:,1)} contains the upper boundary estimate with respect to ## @var{Yfit}. ## ## @code{[@var{Yfit}, @var{Yint}, @var{Ysd}, @var{K}] = regress_gp (@var{X}, ## @var{Y}, @var{Xfit}, @qcode{"rbf"})} will estimate a Gaussian Process model ## with a Radial Basis Function (RBF) kernel with default parameters ## @qcode{@var{theta} = 5}, which corresponds to the characteristic lengthscale, ## and @qcode{@var{g} = 0.01}, which corresponds to the nugget effect, and ## @qcode{@var{alpha} = 0.05} which defines the confidence level for the ## estimated intervals returned in @var{Yint}. The function also returns the ## predictive covariance matrix in @var{Ysd}. For multidimensional predictors ## @var{X} the function will automatically normalize each column to a zero mean ## and a standard deviation to one. ## ## Run @code{demo regress_gp} to see examples. ## ## @seealso{regress, regression_ftest, regression_ttest} ## @end deftypefn function [Yfit, Yint, varargout] = regress_gp (X, Y, Xfit, varargin) ## Check input arguments if (nargin < 3) print_usage; endif if (ndims (X) != 2) error ("regress_gp: X must be a 2-D matrix."); endif if (! isvector (Y) || size (Y, 2) != 1) error ("regress_gp: Y must be a column vector."); endif if (size (X, 1) != length (Y)) error ("regress_gp: rows in X must equal the length of Y."); endif if (size (X, 2) != size (Xfit, 2)) error ("regress_gp: X and XI must have the same number of columns."); endif ## Add defauts kernel = "linear"; Sp = 100 * eye (size (X, 2) + 1); theta = 5; g = 0.01; alpha = 0.05; ## Parse extra arguments if (nargin > 3) tmp = varargin{1}; if (ischar (tmp) && strcmpi (tmp, "linear")) kernel = "linear"; sinput = true; elseif (ischar (tmp) && strcmpi (tmp, "rbf")) kernel = "rbf"; sinput = true; elseif (isnumeric (tmp) && ! isscalar (tmp)) kernel = "linear"; sinput = false; Sp = tmp; elseif (isnumeric (tmp) && isscalar (tmp)) kernel = "rbf"; sinput = false; theta = tmp; else error ("regress_gp: invalid 4th argument."); endif endif if (nargin > 4) tmp = varargin{2}; if (sinput) if (isnumeric (tmp) && ! isscalar (tmp)) if (strcmpi (kernel, "rbf")) error ("regress_gp: theta must be a scalar when using RBF kernel."); endif Sp = tmp; if (! isequal (size (Sp), (size (X, 2) + 1) * [1, 1])) error ("regress_gp: wrong size for prior covariance matrix Sp."); endif elseif (isnumeric (tmp) && isscalar (tmp)) if (strcmpi (kernel, "linear")) error ("regress_gp: wrong size for prior covariance matrix Sp."); endif theta = tmp; else error ("regress_gp: invalid 5th argument."); endif else if (strcmpi (kernel, "linear")) error ("regress_gp: invalid 5th argument."); endif g = tmp; endif endif if (nargin > 5) tmp = varargin{3}; if (isnumeric (tmp) && isscalar (tmp) && sinput) g = tmp; elseif (isnumeric (tmp) && isscalar (tmp) && ! sinput) alpha = tmp; else error ("regress_gp: invalid 6th argument."); endif endif if (nargin > 6) tmp = varargin{4}; if (isnumeric (tmp) && isscalar (tmp) && sinput) alpha = tmp; else error ("regress_gp: invalid 7th argument."); endif endif ## User linear kernel if (strcmpi (kernel, "linear")) ## Add constant vector X = [ones(1,size(X,1)); X']; ## Juan Pablo Carbajal ## Note that in the book the equation (below 2.11) for the A reads ## A = (1/sy^2)*X*X' + inv (Vp); ## where sy is the scalar variance of the of the residuals (i.e Y = X' * w + epsilon) ## and epsilon is drawn from N(0,sy^2). Vp is the variance of the parameters w. ## Note that ## (sy^2 * A)^{-1} = (1/sy^2)*A^{-1} = (X*X' + sy^2 * inv(Vp))^{-1}; ## and that the formula for the w mean is ## (1/sy^2)*A^{-1}*X*Y ## Then one obtains ## inv(X*X' + sy^2 * inv(Vp))*X*Y ## Looking at the formula bloew we see that Sp = (1/sy^2)*Vp ## making the regression depend on only one parameter, Sp, and not two. ## Xsq = sum (X' .^ 2); ## [n, d] = size (X); ## sigma = 1/sqrt(2); ## Ks = exp (-(Xsq' * ones (1, n) -ones (n, 1) * Xsq + 2 * X * X') / (2 * sigma ^ 2)); A = X * X' + inv (Sp); K = inv (A); wm = K * X * Y; ## Add constant vector Xfit = [ones(size(Xfit,1),1), Xfit]; ## Compute predictions Yfit = Xfit*wm; Ysd = Xfit * K * Xfit'; dy = diag (Ysd); Yint = [Yfit+dy, Yfit-dy]; if (nargout > 2) varargout{1} = wm; endif if (nargout > 3) varargout{2} = K; endif endif ## User RBF kernel if (strcmpi (kernel, "rbf")) ## Normalize predictors if (size (X, 2) > 1) [X, MU, SIGMA] = zscore (X); Xfit = (Xfit - MU) ./SIGMA; endif ## Get number of training samples n = size (X, 1); ## Calculate squared distance matrix of training input D = squareform (pdist (X) .^2); ## Compute kernel covariance for training quantities S = exp (-D / theta) + g * eye (n); ## Compute kernel covariance for testing quantities Dxi = squareform (pdist (Xfit) .^ 2); Sxi = exp (-Dxi / theta) + g * eye (size (Dxi, 1)); ## Compute kernel covariance for prediction Dx = pdist2 (Xfit, X) .^ 2; Sx = exp (-Dx / theta); ## Caculate predictive covariance K = inv (S); ## Calculate response output Yfit = Sx * K * Y; ## Estimate scale parameter for predictive variance scale = (Y' * K * Y) / size (Y, 1); ## Calculate standard deviation of the response output Ysd = scale * (Sxi - Sx * K * Sx'); ysd1 = sqrt (diag (Ysd)); ## Calculate prediction intervals Yint = norminv (alpha, 0, ysd1); Yint = [Yfit+Yint, Yfit-Yint]; if (nargout > 2) varargout{1} = Ysd; endif endif endfunction %!demo %! ## Linear fitting of 1D Data %! rand ("seed", 125); %! X = 2 * rand (5, 1) - 1; %! randn ("seed", 25); %! Y = 2 * X - 1 + 0.3 * randn (5, 1); %! %! ## Points for interpolation/extrapolation %! Xfit = linspace (-2, 2, 10)'; %! %! ## Fit regression model %! [Yfit, Yint, m] = regress_gp (X, Y, Xfit); %! %! ## Plot fitted data %! plot (X, Y, "xk", Xfit, Yfit, "r-", Xfit, Yint, "b-"); %! title ("Gaussian process regression with linear kernel"); %!demo %! ## Linear fitting of 2D Data %! rand ("seed", 135); %! X = 2 * rand (4, 2) - 1; %! randn ("seed", 35); %! Y = 2 * X(:,1) - 3 * X(:,2) - 1 + 1 * randn (4, 1); %! %! ## Mesh for interpolation/extrapolation %! [x1, x2] = meshgrid (linspace (-1, 1, 10)); %! Xfit = [x1(:), x2(:)]; %! %! ## Fit regression model %! [Ypred, Yint, Ysd] = regress_gp (X, Y, Xfit); %! Ypred = reshape (Ypred, 10, 10); %! YintU = reshape (Yint(:,1), 10, 10); %! YintL = reshape (Yint(:,2), 10, 10); %! %! ## Plot fitted data %! plot3 (X(:,1), X(:,2), Y, ".k", "markersize", 16); %! hold on; %! h = mesh (x1, x2, Ypred, zeros (10, 10)); %! set (h, "facecolor", "none", "edgecolor", "yellow"); %! h = mesh (x1, x2, YintU, ones (10, 10)); %! set (h, "facecolor", "none", "edgecolor", "cyan"); %! h = mesh (x1, x2, YintL, ones (10, 10)); %! set (h, "facecolor", "none", "edgecolor", "cyan"); %! hold off %! axis tight %! view (75, 25) %! title ("Gaussian process regression with linear kernel"); %!demo %! ## Projection over basis function with linear kernel %! pp = [2, 2, 0.3, 1]; %! n = 10; %! rand ("seed", 145); %! X = 2 * rand (n, 1) - 1; %! randn ("seed", 45); %! Y = polyval (pp, X) + 0.3 * randn (n, 1); %! %! ## Powers %! px = [sqrt(abs(X)), X, X.^2, X.^3]; %! %! ## Points for interpolation/extrapolation %! Xfit = linspace (-1, 1, 100)'; %! pxi = [sqrt(abs(Xfit)), Xfit, Xfit.^2, Xfit.^3]; %! %! ## Define a prior covariance assuming that the sqrt component is not present %! Sp = 100 * eye (size (px, 2) + 1); %! Sp(2,2) = 1; # We don't believe the sqrt(abs(X)) is present %! %! ## Fit regression model %! [Yfit, Yint, Ysd] = regress_gp (px, Y, pxi, Sp); %! %! ## Plot fitted data %! plot (X, Y, "xk;Data;", Xfit, Yfit, "r-;Estimation;", ... %! Xfit, polyval (pp, Xfit), "g-;True;"); %! axis tight %! axis manual %! hold on %! plot (Xfit, Yint(:,1), "m-;Upper bound;", Xfit, Yint(:,2), "b-;Lower bound;"); %! hold off %! title ("Linear kernel over basis function with prior covariance"); %!demo %! ## Projection over basis function with linear kernel %! pp = [2, 2, 0.3, 1]; %! n = 10; %! rand ("seed", 145); %! X = 2 * rand (n, 1) - 1; %! randn ("seed", 45); %! Y = polyval (pp, X) + 0.3 * randn (n, 1); %! %! ## Powers %! px = [sqrt(abs(X)), X, X.^2, X.^3]; %! %! ## Points for interpolation/extrapolation %! Xfit = linspace (-1, 1, 100)'; %! pxi = [sqrt(abs(Xfit)), Xfit, Xfit.^2, Xfit.^3]; %! %! ## Fit regression model without any assumption on prior covariance %! [Yfit, Yint, Ysd] = regress_gp (px, Y, pxi); %! %! ## Plot fitted data %! plot (X, Y, "xk;Data;", Xfit, Yfit, "r-;Estimation;", ... %! Xfit, polyval (pp, Xfit), "g-;True;"); %! axis tight %! axis manual %! hold on %! plot (Xfit, Yint(:,1), "m-;Upper bound;", Xfit, Yint(:,2), "b-;Lower bound;"); %! hold off %! title ("Linear kernel over basis function without prior covariance"); %!demo %! ## Projection over basis function with rbf kernel %! pp = [2, 2, 0.3, 1]; %! n = 10; %! rand ("seed", 145); %! X = 2 * rand (n, 1) - 1; %! randn ("seed", 45); %! Y = polyval (pp, X) + 0.3 * randn (n, 1); %! %! ## Powers %! px = [sqrt(abs(X)), X, X.^2, X.^3]; %! %! ## Points for interpolation/extrapolation %! Xfit = linspace (-1, 1, 100)'; %! pxi = [sqrt(abs(Xfit)), Xfit, Xfit.^2, Xfit.^3]; %! %! ## Fit regression model with RBF kernel (standard parameters) %! [Yfit, Yint, Ysd] = regress_gp (px, Y, pxi, "rbf"); %! %! ## Plot fitted data %! plot (X, Y, "xk;Data;", Xfit, Yfit, "r-;Estimation;", ... %! Xfit, polyval (pp, Xfit), "g-;True;"); %! axis tight %! axis manual %! hold on %! plot (Xfit, Yint(:,1), "m-;Upper bound;", Xfit, Yint(:,2), "b-;Lower bound;"); %! hold off %! title ("RBF kernel over basis function with standard parameters"); %! text (-0.5, 4, "theta = 5\n g = 0.01"); %!demo %! ## Projection over basis function with rbf kernel %! pp = [2, 2, 0.3, 1]; %! n = 10; %! rand ("seed", 145); %! X = 2 * rand (n, 1) - 1; %! randn ("seed", 45); %! Y = polyval (pp, X) + 0.3 * randn (n, 1); %! %! ## Powers %! px = [sqrt(abs(X)), X, X.^2, X.^3]; %! %! ## Points for interpolation/extrapolation %! Xfit = linspace (-1, 1, 100)'; %! pxi = [sqrt(abs(Xfit)), Xfit, Xfit.^2, Xfit.^3]; %! %! ## Fit regression model with RBF kernel with different parameters %! [Yfit, Yint, Ysd] = regress_gp (px, Y, pxi, "rbf", 10, 0.01); %! %! ## Plot fitted data %! plot (X, Y, "xk;Data;", Xfit, Yfit, "r-;Estimation;", ... %! Xfit, polyval (pp, Xfit), "g-;True;"); %! axis tight %! axis manual %! hold on %! plot (Xfit, Yint(:,1), "m-;Upper bound;", Xfit, Yint(:,2), "b-;Lower bound;"); %! hold off %! title ("GP regression with RBF kernel and non default parameters"); %! text (-0.5, 4, "theta = 10\n g = 0.01"); %! %! ## Fit regression model with RBF kernel with different parameters %! [Yfit, Yint, Ysd] = regress_gp (px, Y, pxi, "rbf", 50, 0.01); %! %! ## Plot fitted data %! figure %! plot (X, Y, "xk;Data;", Xfit, Yfit, "r-;Estimation;", ... %! Xfit, polyval (pp, Xfit), "g-;True;"); %! axis tight %! axis manual %! hold on %! plot (Xfit, Yint(:,1), "m-;Upper bound;", Xfit, Yint(:,2), "b-;Lower bound;"); %! hold off %! title ("GP regression with RBF kernel and non default parameters"); %! text (-0.5, 4, "theta = 50\n g = 0.01"); %! %! ## Fit regression model with RBF kernel with different parameters %! [Yfit, Yint, Ysd] = regress_gp (px, Y, pxi, "rbf", 50, 0.001); %! %! ## Plot fitted data %! figure %! plot (X, Y, "xk;Data;", Xfit, Yfit, "r-;Estimation;", ... %! Xfit, polyval (pp, Xfit), "g-;True;"); %! axis tight %! axis manual %! hold on %! plot (Xfit, Yint(:,1), "m-;Upper bound;", Xfit, Yint(:,2), "b-;Lower bound;"); %! hold off %! title ("GP regression with RBF kernel and non default parameters"); %! text (-0.5, 4, "theta = 50\n g = 0.001"); %! %! ## Fit regression model with RBF kernel with different parameters %! [Yfit, Yint, Ysd] = regress_gp (px, Y, pxi, "rbf", 50, 0.05); %! %! ## Plot fitted data %! figure %! plot (X, Y, "xk;Data;", Xfit, Yfit, "r-;Estimation;", ... %! Xfit, polyval (pp, Xfit), "g-;True;"); %! axis tight %! axis manual %! hold on %! plot (Xfit, Yint(:,1), "m-;Upper bound;", Xfit, Yint(:,2), "b-;Lower bound;"); %! hold off %! title ("GP regression with RBF kernel and non default parameters"); %! text (-0.5, 4, "theta = 50\n g = 0.05"); %!demo %! ## RBF fitting on noiseless 1D Data %! x = [0:2*pi/7:2*pi]'; %! y = 5 * sin (x); %! %! ## Predictive grid of 500 equally spaced locations %! xi = [-0.5:(2*pi+1)/499:2*pi+0.5]'; %! %! ## Fit regression model with RBF kernel %! [Yfit, Yint, Ysd] = regress_gp (x, y, xi, "rbf"); %! %! ## Plot fitted data %! r = mvnrnd (Yfit, diag (Ysd)', 50); %! plot (xi, r', "c-"); %! hold on %! plot (xi, Yfit, "r-;Estimation;", xi, Yint, "b-;Confidence interval;"); %! plot (x, y, ".k;Predictor points;", "markersize", 20) %! plot (xi, 5 * sin (xi), "-y;True Function;"); %! xlim ([-0.5,2*pi+0.5]); %! ylim ([-10,10]); %! hold off %! title ("GP regression with RBF kernel on noiseless 1D data"); %! text (0, -7, "theta = 5\n g = 0.01"); %!demo %! ## RBF fitting on noisy 1D Data %! x = [0:2*pi/7:2*pi]'; %! x = [x; x]; %! y = 5 * sin (x) + randn (size (x)); %! %! ## Predictive grid of 500 equally spaced locations %! xi = [-0.5:(2*pi+1)/499:2*pi+0.5]'; %! %! ## Fit regression model with RBF kernel %! [Yfit, Yint, Ysd] = regress_gp (x, y, xi, "rbf"); %! %! ## Plot fitted data %! r = mvnrnd (Yfit, diag (Ysd)', 50); %! plot (xi, r', "c-"); %! hold on %! plot (xi, Yfit, "r-;Estimation;", xi, Yint, "b-;Confidence interval;"); %! plot (x, y, ".k;Predictor points;", "markersize", 20) %! plot (xi, 5 * sin (xi), "-y;True Function;"); %! xlim ([-0.5,2*pi+0.5]); %! ylim ([-10,10]); %! hold off %! title ("GP regression with RBF kernel on noisy 1D data"); %! text (0, -7, "theta = 5\n g = 0.01"); ## Test input validation %!error regress_gp (ones (20, 2)) %!error regress_gp (ones (20, 2), ones (20, 1)) %!error ... %! regress_gp (ones (20, 2, 3), ones (20, 1), ones (20, 2)) %!error ... %! regress_gp (ones (20, 2), ones (20, 2), ones (20, 2)) %!error ... %! regress_gp (ones (20, 2), ones (15, 1), ones (20, 2)) %!error ... %! regress_gp (ones (20, 2), ones (20, 1), ones (20, 3)) %!error ... %! regress_gp (ones (20, 2), ones (20, 1), ones (10, 2), {[3]}) %!error ... %! regress_gp (ones (20, 2), ones (20, 1), ones (10, 2), "kernel") %!error ... %! regress_gp (ones (20, 2), ones (20, 1), ones (10, 2), "rbf", ones (4)) %!error ... %! regress_gp (ones (20, 2), ones (20, 1), ones (10, 2), "linear", 1) %!error ... %! regress_gp (ones (20, 2), ones (20, 1), ones (10, 2), "rbf", "value") %!error ... %! regress_gp (ones (20, 2), ones (20, 1), ones (10, 2), "rbf", {5}) %!error ... %! regress_gp (ones (20, 2), ones (20, 1), ones (10, 2), ones (3), 5) %!error ... %! regress_gp (ones (20, 2), ones (20, 1), ones (10, 2), "linear", 5) %!error ... %! regress_gp (ones (20, 2), ones (20, 1), ones (10, 2), "rbf", 5, {5}) %!error ... %! regress_gp (ones (20, 2), ones (20, 1), ones (10, 2), "rbf", 5, ones (2)) %!error ... %! regress_gp (ones (20, 2), ones (20, 1), ones (10, 2), 5, 0.01, [1, 1]) %!error ... %! regress_gp (ones (20, 2), ones (20, 1), ones (10, 2), 5, 0.01, "f") %!error ... %! regress_gp (ones (20, 2), ones (20, 1), ones (10, 2), 5, 0.01, "f") %!error ... %! regress_gp (ones (20, 2), ones (20, 1), ones (10, 2), "rbf", 5, 0.01, "f") %!error ... %! regress_gp (ones (20, 2), ones (20, 1), ones (10, 2), "rbf", 5, 0.01, [1, 1]) %!error ... %! regress_gp (ones (20, 2), ones (20, 1), ones (10, 2), "linear", 1) statistics-release-1.6.3/inst/regression_ftest.m000066400000000000000000000214331456127120000220520ustar00rootroot00000000000000## Copyright (C) 1995-2017 Kurt Hornik ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {[@var{h}, @var{pval}, @var{stats}] =} regression_ftest (@var{y}, @var{x}, @var{fm}) ## @deftypefnx {statistics} {[@dots{}] =} regression_ftest (@var{y}, @var{x}, @var{fm}, @var{rm}) ## @deftypefnx {statistics} {[@dots{}] =} regression_ftest (@var{y}, @var{x}, @var{fm}, @var{rm}, @var{Name}, @var{Value}) ## @deftypefnx {statistics} {[@dots{}] =} regression_ftest (@var{y}, @var{x}, @var{fm}, [], @var{Name}, @var{Value}) ## ## F-test for General Linear Regression Analysis ## ## Perform a general linear regression F test for the null hypothesis that the ## full model of the form @qcode{y = b_0 + b_1 * x_1 + b_2 * x_2 + @dots{} + ## b_n * x_n + e}, where n is the number of variables in @var{x}, does not ## perform better than a reduced model, such as @qcode{y = b'_0 + b'_1 * x_1 + ## b'_2 * x_2 + @dots{} + b'_k * x_k + e}, where k < n and it corresponds to the ## first k variables in @var{x}. Explanatory (dependent) variable @var{y} and ## response (independent) variables @var{x} must not contain any missing values ## (NaNs). ## ## The full model, @var{fm}, must be a vector of length equal to the columns of ## @var{x}, in which case the constant term b_0 is assumed 0, or equal to ## the columns of @var{x} plus one, in which case the first element is the ## constant b_0. ## ## The reduced model, @var{rm}, must include the constant term and a subset of ## the variables (columns) in @var{x}. If @var{rm} is not given, then a constant ## term b'_0 is assumed equal to the constant term, b_0, of the full model or 0, ## if the full model, @var{fm}, does not have a constant term. @var{rm} must be ## a vector or a scalar if only a constant term is passed into the function. ## ## Name-Value pair arguments can be used to set statistical significance. ## @qcode{"alpha"} can be used to specify the significance level of the test ## (the default value is 0.05). If you want pass optional Name-Value pair ## without a reduced model, make sure that the latter is passed as an empty ## variable. ## ## If @var{h} is 1 the null hypothesis is rejected, meaning that the full model ## explains the variance better than the restricted model. If @var{h} is 0, it ## can be assumed that the full model does NOT explain the variance any better ## than the restricted model. ## ## The p-value (1 minus the CDF of this distribution at @var{f}) is returned ## in @var{pval}. ## ## Under the null, the test statistic @var{f} follows an F distribution with ## 'df1' and 'df2' degrees of freedom, which are returned as fields in the ## @var{stats} structure along with the test's F-statistic, 'fstat' ## ## @seealso{regression_ttest, regress, regress_gp} ## @end deftypefn function [h, pval, stats] = regression_ftest (y, x, fm, rm, varargin) ## Check for valid input if (nargin < 3) print_usage (); endif ## Check for finite real numbers in Y, X if (! all (isfinite (y)) || ! isreal (y)) error ("regression_ftest: Y must contain finite real numbers."); endif if (! all (isfinite (x(:))) || ! isreal (x)) error ("regression_ftest: X must contain finite real numbers."); endif ## Set default arguments alpha = 0.05; ## Check additional options i = 1; while (i <= length (varargin)) switch lower (varargin{i}) case "alpha" i = i + 1; alpha = varargin{i}; ## Check for valid alpha if (! isscalar (alpha) || ! isnumeric (alpha) || ... alpha <= 0 || alpha >= 1) error ("regression_ftest: invalid value for alpha."); endif otherwise error ("regression_ftest: invalid Name argument."); endswitch i = i + 1; endwhile ## Get size of response (independent) variables [s, v] = size (x); ## Add a constant term of 1s in X x = [ones(s, 1), x]; ## Check the size of explanatory (dependent) variable if (! (isvector (y) && (length (y) == s))) error ("regression_ftest: Y must be a vector of length 'rows (X)'."); endif y = reshape (y, s, 1); ## Check the full model if (! (isvector (fm) && (length (fm) == v || length (fm) == v + 1))) error (strcat (["regression_ftest: full model, FM, must be a vector"], ... [" of length equal to 'rows (X)' or 'rows (X) + 1'."])); endif ## Make it row vector and add a constant = 0 if necessary fm_len = length (fm); fm = reshape (fm, 1, fm_len); if (fm_len == v) fm = [0, fm]; fm_len += 1; endif ## Check the reduced model if (nargin - length (varargin) == 4) if (isempty (rm)) rm = [fm(1), zeros(1, fm_len - 1)]; rm_len = 1; else if (! isvector (rm) || ! isnumeric (rm)) error (strcat (["regression_ftest: reduced model, RM, must be a"], ... [" numeric vector or a scalar."])); endif rm_len = length (rm); if (rm_len >= fm_len - 1) error (strcat (["regression_ftest: reduced model, RM, must have"], ... [" smaller length than the full model, FM."])); endif rm = reshape (rm, 1, rm_len); rm = [rm, zeros(1, fm_len - rm_len)]; endif else rm = [fm(1), zeros(1, fm_len - 1)]; rm_len = 1; endif ## Calculate the fitted response for full and reduced models y_fm = sum (x .* fm, 2); y_rm = sum (x .* rm, 2); ## Calculate Sum of Squares Error for full and reduced models SSE_fm = sumsq (y - y_fm); SSE_rm = sumsq (y - y_rm); ## Calculate the necessary statistics stats.df1 = fm_len - rm_len; stats.df2 = s - v; stats.fstat = ((SSE_rm - SSE_fm) / stats.df1) / (SSE_fm / stats.df2); pval = 1 - fcdf (stats.fstat, stats.df1, stats.df2); ## Determine the test outcome ## MATLAB returns this a double instead of a logical array h = double (pval < alpha); endfunction ## Test input validation %!error regression_ftest (); %!error ... %! regression_ftest ([1 2 3]', [2 3 4; 3 4 5]'); %!error ... %! regression_ftest ([1 2 NaN]', [2 3 4; 3 4 5]', [1 0.5]); %!error ... %! regression_ftest ([1 2 Inf]', [2 3 4; 3 4 5]', [1 0.5]); %!error ... %! regression_ftest ([1 2 3+i]', [2 3 4; 3 4 5]', [1 0.5]); %!error ... %! regression_ftest ([1 2 3]', [2 3 NaN; 3 4 5]', [1 0.5]); %!error ... %! regression_ftest ([1 2 3]', [2 3 Inf; 3 4 5]', [1 0.5]); %!error ... %! regression_ftest ([1 2 3]', [2 3 4; 3 4 3+i]', [1 0.5]); %!error ... %! regression_ftest ([1 2 3]', [2 3 4; 3 4 5]', [1 0.5], [], "alpha", 0); %!error ... %! regression_ftest ([1 2 3]', [2 3 4; 3 4 5]', [1 0.5], [], "alpha", 1.2); %!error ... %! regression_ftest ([1 2 3]', [2 3 4; 3 4 5]', [1 0.5], [], "alpha", [.02 .1]); %!error ... %! regression_ftest ([1 2 3]', [2 3 4; 3 4 5]', [1 0.5], [], "alpha", "a"); %!error ... %! regression_ftest ([1 2 3]', [2 3 4; 3 4 5]', [1 0.5], [], "some", 0.05); %!error ... %! regression_ftest ([1 2 3]', [2 3; 3 4]', [1 0.5]); %!error ... %! regression_ftest ([1 2; 3 4]', [2 3; 3 4]', [1 0.5]); %!error ... %! regression_ftest ([1 2 3]', [2 3 4; 3 4 5]', [1 0.5], ones (2)); %!error ... %! regression_ftest ([1 2 3]', [2 3 4; 3 4 5]', [1 0.5], "alpha"); %!error ... %! regression_ftest ([1 2 3]', [2 3 4; 3 4 5]', [1 0.5], [1 2]); ## Test results statistics-release-1.6.3/inst/regression_ttest.m000066400000000000000000000163031456127120000220700ustar00rootroot00000000000000## Copyright (C) 1995-2017 Kurt Hornik ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{h} =} regression_ttest (@var{y}, @var{x}) ## @deftypefnx {statistics} {[@var{h}, @var{pval}] =} regression_ttest (@var{y}, @var{x}) ## @deftypefnx {statistics} {[@var{h}, @var{pval}, @var{ci}] =} regression_ttest (@var{y}, @var{x}) ## @deftypefnx {statistics} {[@var{h}, @var{pval}, @var{ci}, @var{stats}] =} regression_ttest (@var{y}, @var{x}) ## @deftypefnx {statistics} {[@dots{}] =} regression_ttest (@var{y}, @var{x}, @var{Name}, @var{Value}) ## ## Perform a linear regression t-test. ## ## @code{@var{h} = regression_ttest (@var{y}, @var{x})} tests the null ## hypothesis that the slope @math{beta1} of a simple linear regression equals ## 0. The result is @var{h} = 0 if the null hypothesis cannot be rejected at ## the 5% significance level, or @var{h} = 1 if the null hypothesis can be ## rejected at the 5% level. @var{y} and @var{x} must be vectors of equal ## length with finite real numbers. ## ## The p-value of the test is returned in @var{pval}. A @math{100(1-alpha)%} ## confidence interval for @math{beta1} is returned in @var{ci}. @var{stats} is ## a structure containing the value of the test statistic (@qcode{tstat}), ## the degrees of freedom (@qcode{df}), the slope coefficient (@qcode{beta1}), ## and the intercept (@qcode{beta0}). Under the null, the test statistic ## @var{stats}.@qcode{tstat} follows a @math{T}-distribution with ## @var{stats}.@qcode{df} degrees of freedom. ## ## @code{[@dots{}] = regression_ttest (@dots{}, @var{name}, @var{value})} ## specifies one or more of the following name/value pairs: ## ## @multitable @columnfractions 0.05 0.2 0.75 ## @headitem @tab Name @tab Value ## @item @tab @qcode{"alpha"} @tab the significance level. Default is 0.05. ## ## @item @tab @qcode{"tail"} @tab a string specifying the alternative hypothesis ## @end multitable ## @multitable @columnfractions 0.1 0.25 0.65 ## @item @tab @qcode{"both"} @tab @math{beta1} is not 0 (two-tailed, default) ## @item @tab @qcode{"left"} @tab @math{beta1} is less than 0 (left-tailed) ## @item @tab @qcode{"right"} @tab @math{beta1} is greater than 0 (right-tailed) ## @end multitable ## ## @seealso{regression_ftest, regress, regress_gp} ## @end deftypefn function [h, pval, ci, stats] = regression_ttest (y, x, varargin) ## Check for valid input if (nargin < 2) print_usage (); endif ## Check for finite real numbers in Y, X if (! all (isfinite (y)) || ! isreal (y)) error ("regression_ttest: Y must contain finite real numbers."); endif if (! all (isfinite (x(:))) || ! isreal (x)) error ("regression_ttest: X must contain finite real numbers."); endif # Get number of observations n = length (y); ## Check Y and X have the same number of observations if (! isvector (y) || ! isvector (x) || length (x) != n) error ("regression_ttest: Y and X must be vectors of equal length."); endif ## Set default arguments alpha = 0.05; tail = "both"; ## Check additional options i = 1; while (i <= length (varargin)) switch lower (varargin{i}) case "alpha" i = i + 1; alpha = varargin{i}; ## Check for valid alpha if (! isscalar (alpha) || ! isnumeric (alpha) || ... alpha <= 0 || alpha >= 1) error ("regression_ttest: invalid value for alpha."); endif case "tail" i = i + 1; tail = varargin{i}; if (! any (strcmpi (tail, {"both", "left", "right"}))) error ("regression_ttest: invalid value for tail."); endif otherwise error ("regression_ttest: invalid Name argument."); endswitch i = i + 1; endwhile y_bar = mean (y); x_bar = mean (x); stats.beta1 = cov (x, y) / var (x); stats.beta0 = y_bar - stats.beta1 * x_bar; y_hat = stats.beta0 + stats.beta1 * x_bar; SSE = sum ((y - y_hat) .^ 2); stats.df = n - 2; SE = sqrt (SSE / stats.df); term = SE / sqrt (sum ((x - x_bar) .^ 2)); stats.tstat = stats.beta1 / term; ## Based on the "tail" argument determine the P-value, the critical values, ## and the confidence interval. switch lower (tail) case "both" pval = 2 * (1 - tcdf (abs (stats.tstat), stats.df)); tcrit = - tinv (alpha / 2, stats.df); ci = [stats.beta1 - tcrit * term; stats.beta1 + tcrit * term]; case "left" pval = tcdf (stats.tstat, stats.df); tcrit = - tinv (alpha, stats.df); ci = [-inf; stats.beta1 + tcrit * term]; case "right" pval = 1 - tcdf (stats.tstat, stats.df); tcrit = - tinv (alpha, stats.df); ci = [stats.beta1 - tcrit * term; inf]; endswitch ## Determine the test outcome h = double (pval < alpha); h(isnan (pval)) = NaN; endfunction ## Test input validation %!error regression_ttest (); %!error regression_ttest (1); %!error ... %! regression_ttest ([1 2 NaN]', [2 3 4]'); %!error ... %! regression_ttest ([1 2 Inf]', [2 3 4]'); %!error ... %! regression_ttest ([1 2 3+i]', [2 3 4]'); %!error ... %! regression_ttest ([1 2 3]', [2 3 NaN]'); %!error ... %! regression_ttest ([1 2 3]', [2 3 Inf]'); %!error ... %! regression_ttest ([1 2 3]', [3 4 3+i]'); %!error ... %! regression_ttest ([1 2 3]', [3 4 4 5]'); %!error ... %! regression_ttest ([1 2 3]', [2 3 4]', "alpha", 0); %!error ... %! regression_ttest ([1 2 3]', [2 3 4]', "alpha", 1.2); %!error ... %! regression_ttest ([1 2 3]', [2 3 4]', "alpha", [.02 .1]); %!error ... %! regression_ttest ([1 2 3]', [2 3 4]', "alpha", "a"); %!error ... %! regression_ttest ([1 2 3]', [2 3 4]', "some", 0.05); %!error ... %! regression_ttest ([1 2 3]', [2 3 4]', "tail", "val"); %!error ... %! regression_ttest ([1 2 3]', [2 3 4]', "alpha", 0.01, "tail", "val"); statistics-release-1.6.3/inst/ridge.m000066400000000000000000000156701456127120000175650ustar00rootroot00000000000000## Copyright (C) 2023 Mohammed Azmat Khan ## ## This file is part of the statistics package for GNU Octave. ## ## Octave is free software; you can redistribute it and/or modify it ## under the terms of the GNU General Public License as published by ## the Free Software Foundation; either version 3 of the License, or ## (at your option) any later version. ## ## Octave is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with Octave; see the file COPYING. If not, ## see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{b} =} ridge (@var{y}, @var{X}, @var{k}) ## @deftypefnx {statistics} {@var{b} =} ridge (@var{y}, @var{X}, @var{k}, @var{scaled}) ## ## Ridge regression. ## ## @code{@var{b} = ridge (@var{y}, @var{X}, @var{k})} returns the vector of ## coefficient estimates by applying ridge regression from the predictor matrix ## @var{X} to the response vector @var{y}. Each value of @var{b} is the ## coefficient for the respective ridge parameter given @var{k}. By default, ## @var{b} is calculated after centering and scaling the predictors to have a ## zero mean and standard deviation 1. ## ## @code{@var{b} = ridge (@var{y}, @var{X}, @var{k}, @var{scaled})} performs the ## regression with the specified scaling of the coefficient estimates @var{b}. ## When @qcode{@var{scaled} = 0}, the function restores the coefficients to the ## scale of the original data thus is more useful for making predictions. When ## @qcode{@var{scaled} = 1}, the coefficient estimates correspond to the scaled ## centered data. ## ## @itemize ## @item ## @code{y} must be an @math{Nx1} numeric vector with the response data. ## @item ## @code{X} must be an @math{Nxp} numeric matrix with the predictor data. ## @item ## @code{k} must be a numeric vectir with the ridge parameters. ## @item ## @code{scaled} must be a numeric scalar indicating whether the coefficient ## estimates in @var{b} are restored to the scale of the original data. By ## default, @qcode{@var{scaled} = 1}. ## @end itemize ## ## Further information about Ridge regression can be found at ## @url{https://en.wikipedia.org/wiki/Ridge_regression} ## ## @seealso{lasso, stepwisefit, regress} ## @end deftypefn function b = ridge (y, X, k, scaled) ## Check input arguments if (nargin < 3) error ("ridge: function called with too few input arguments."); endif if (! isvector (y) || columns (y) != 1 || isempty (y)) error ("ridge: Y must be a numeric column vector."); endif if (! ismatrix (X) || isempty (X)) error ("ridge: X must be a numeric matrix."); endif if (rows (y) != rows (X)) error ("ridge: Y and X must contain the same number of rows."); endif ## Parse 4th input argument if (nargin < 4 || isempty (scaled)) unscale = false; elseif (scaled == 1) unscale = false; elseif (scaled == 0) unscale = true; else error ("ridge: wrong value for SCALED argument."); endif ## Force y to a column vector y = y(:); ## Rremove any missing values notnans = ! logical (sum (isnan ([y, X]), 2)); y = y(notnans); X = X(notnans,:); ## Scale and center X to zero mean and StD = 1 m = mean (X); stdx = std (X, 0, 1); z = (X - m) ./ stdx; ## Add pseudo observations Z_pseudo = [z; (sqrt(k(1)) .* eye (columns(X)))]; Y_pseudo = [y; zeros(columns(X), 1)]; ## Compute coefficients b = Z_pseudo \ Y_pseudo; nk = numel (k); ## Compute the coefficient estimates for additional ridge parameters. if (nk >= 2) ## Adding a multiple of the identity matrix to the last p rows. ## b is set to 0 for the current ridge parameter value b(end,nk) = 0; for i=2:nk Z_pseudo(end-columns(X)+1:end, :) = sqrt (k(i)) .* eye (columns (X)); b(:,i) = Z_pseudo \ Y_pseudo; endfor endif ## Changing back to the scale if (unscale) b = b ./ repmat (stdx', 1, nk); b = [mean(y)-m*b; b]; endif endfunction %!demo %! ## Perform ridge regression for a range of ridge parameters and observe %! ## how the coefficient estimates change based on the acetylene dataset. %! %! load acetylene %! %! X = [x1, x2, x3]; %! %! x1x2 = x1 .* x2; %! x1x3 = x1 .* x3; %! x2x3 = x2 .* x3; %! %! D = [x1, x2, x3, x1x2, x1x3, x2x3]; %! %! k = 0:1e-5:5e-3; %! %! b = ridge (y, D, k); %! %! figure %! plot (k, b, "LineWidth", 2) %! ylim ([-100, 100]) %! grid on %! xlabel ("Ridge Parameter") %! ylabel ("Standardized Coefficient") %! title ("Ridge Trace") %! legend ("x1", "x2", "x3", "x1x2", "x1x3", "x2x3") %! %!demo %! %! load carbig %! X = [Acceleration Weight Displacement Horsepower]; %! y = MPG; %! %! n = length(y); %! %! rand("seed",1); % For reproducibility %! %! c = cvpartition(n,'HoldOut',0.3); %! idxTrain = training(c,1); %! idxTest = ~idxTrain; %! %! idxTrain = training(c,1); %! idxTest = ~idxTrain; %! %! k = 5; %! b = ridge(y(idxTrain),X(idxTrain,:),k,0); %! %! % Predict MPG values for the test data using the model. %! yhat = b(1) + X(idxTest,:)*b(2:end); %! scatter(y(idxTest),yhat) %! %! hold on %! plot(y(idxTest),y(idxTest),"r") %! xlabel('Actual MPG') %! ylabel('Predicted MPG') %! hold off %! ## Test output %!test %! b = ridge ([1 2 3 4]', [1 2 3 4; 2 3 4 5]', 1); %! assert (b, [0.5533; 0.5533], 1e-4); %!test %! b = ridge ([1 2 3 4]', [1 2 3 4; 2 3 4 5]', 2); %! assert (b, [0.4841; 0.4841], 1e-4); %!test %! load acetylene %! x = [x1, x2, x3]; %! b = ridge (y, x, 0); %! assert (b,[10.2273;1.97128;-0.601818],1e-4); %!test %! load acetylene %! x = [x1, x2, x3]; %! b = ridge (y, x, 0.0005); %! assert (b,[10.2233;1.9712;-0.6056],1e-4); %!test %! load acetylene %! x = [x1, x2, x3]; %! b = ridge (y, x, 0.001); %! assert (b,[10.2194;1.9711;-0.6094],1e-4); %!test %! load acetylene %! x = [x1, x2, x3]; %! b = ridge (y, x, 0.002); %! assert (b,[10.2116;1.9709;-0.6169],1e-4); %!test %! load acetylene %! x = [x1, x2, x3]; %! b = ridge (y, x, 0.005); %! assert (b,[10.1882;1.9704;-0.6393],1e-4); %!test %! load acetylene %! x = [x1, x2, x3]; %! b = ridge (y, x, 0.01); %! assert (b,[10.1497;1.9695;-0.6761],1e-4); ## Test input validation %!error ridge (1) %!error ridge (1, 2) %!error ridge (ones (3), ones (3), 2) %!error ridge ([1, 2], ones (2), 2) %!error ridge ([], ones (3), 2) %!error ridge (ones (5,1), [], 2) %!error ... %! ridge ([1; 2; 3; 4; 5], ones (3), 3) %!error ... %! ridge ([1; 2; 3], ones (3), 3, 2) %!error ... %! ridge ([1; 2; 3], ones (3), 3, "some") statistics-release-1.6.3/inst/rmmissing.m000066400000000000000000000162721456127120000205020ustar00rootroot00000000000000## Copyright (C) 1995-2023 The Octave Project Developers ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{R} =} rmmissing (@var{A}) ## @deftypefnx {statistics} {@var{R} =} rmmissing (@var{A}, @var{dim}) ## @deftypefnx {statistics} {@var{R} =} rmmissing (@dots{}, @var{Name}, @var{Value}) ## @deftypefnx {statistics} {[@var{R} @var{TF}] =} rmmissing (@dots{}) ## ## Remove missing or incomplete data from an array. ## ## Given an input vector or matrix (2-D array) @var{A}, remove missing data ## from a vector or missing rows or columns from a matrix. @var{A} ## can be a numeric array, char array, or an array of cell strings. ## @var{R} returns the array after removal of missing data. ## ## The values which represent missing data depend on the data type of @var{A}: ## ## @itemize ## @item ## @qcode{NaN}: @code{single}, @code{double}. ## ## @item ## @qcode{' '} (white space): @code{char}. ## ## @item ## @qcode{@{''@}}: string cells. ## @end itemize ## ## Choose to remove rows (default) or columns by setting optional input ## @var{dim}: ## ## @itemize ## @item ## @qcode{1}: rows. ## ## @item ## @qcode{2}: columns. ## @end itemize ## ## Note: data types with no default 'missing' value will always result in ## @code{R == A} and a TF output of @code{false(size(@var{A}))}. ## ## Additional optional parameters are set by @var{Name}-@var{Value} pairs. ## These are: ## ## @itemize ## @item ## @qcode{MinNumMissing}: minimum number of missing values to remove an entry, ## row or column, defined as a positive integer number. E.g.: if ## @qcode{MinNumMissing} is set to @code{2}, remove the row of a numeric matrix ## only if it includes 2 or more NaN. ## @end itemize ## ## Optional return value @var{TF} is a logical array where @code{true} values ## represent removed entries, rows or columns from the original data @var{A}. ## ## @end deftypefn ## ## @seealso{fillmissing, ismissing, standardizeMissing} function [R, TF] = rmmissing (A, varargin) if ((nargin < 1) || (nargin > 4)) print_usage (); endif if ndims(A) > 2 error ("rmmissing: input dimension cannot exceed 2"); endif optDimensionI = 2; # default dimension: rows optMinNumMissingI = 1; ## parse options if (nargin > 1) if (isnumeric (varargin{1})) ## option "dim" switch (varargin{1}) case 1 optDimensionI = 2; case 2 optDimensionI = 1; otherwise error ("rmmissing: 'dim' must be either 1 or 2"); endswitch pair_index = 2; else [r, c] = size (A); ## first non singleton dimension, but only two dimensions considered if (r == 1 && c != 1) optDimensionI = 1; endif pair_index = 1; endif ## parse name-value parameters while (pair_index <= (nargin - 1)) switch (lower (varargin{pair_index})) ## minimum number of missing values to remove entries; ## it must be a positive integer number case "minnummissing" if (! isnumeric (varargin{pair_index + 1}) || ! isscalar (varargin{pair_index + 1}) || floor (varargin{pair_index + 1}) != varargin{pair_index + 1} || varargin{pair_index + 1} < 1) error (["rmmissing: 'MinNumMissing' requires a positive integer"... " number as value"]); endif optMinNumMissingI = varargin{pair_index + 1}; otherwise error ("rmmissing: unknown parameter name '%s'", ... varargin{pair_index}); endswitch pair_index += 2; endwhile endif ## main logic TF = ismissing (A); if (isvector (A)) R = A(TF == 0); elseif (iscellstr(A) || ismatrix (A)) ## matrix: ismissing returns an array, so it must be converted to a row or ## column vector according to the "dim" of choice if (optMinNumMissingI > 1) TF = sum (TF, optDimensionI); TF(TF < optMinNumMissingI) = 0; TF = logical (TF); else TF = any (TF, optDimensionI); endif if (optDimensionI == 2) ## remove the rows R = A((TF == 0), :); else ## remove the columns R = A(:, (TF == 0)); endif else error ("rmmissing: unsupported data"); endif endfunction %!assert (rmmissing ([1,NaN,3]), [1,3]) %!assert (rmmissing ('abcd f'), 'abcdf') %!assert (rmmissing ({'xxx','','xyz'}), {'xxx','xyz'}) %!assert (rmmissing ({'xxx','';'xyz','yyy'}), {'xyz','yyy'}) %!assert (rmmissing ({'xxx','';'xyz','yyy'}, 2), {'xxx';'xyz'}) %!assert (rmmissing ([1,2;NaN,2]), [1,2]) %!assert (rmmissing ([1,2;NaN,2], 2), [2,2]') %!assert (rmmissing ([1,2;NaN,4;NaN,NaN],"MinNumMissing", 2), [1,2;NaN,4]) ## Test second output %!test %! x = [1:6]; %! x([2,4]) = NaN; %! [~, idx] = rmmissing (x); %! assert (idx, logical ([0, 1, 0, 1, 0, 0])); %! assert (class(idx), 'logical'); %! x = reshape (x, [2, 3]); %! [~, idx] = rmmissing (x); %! assert (idx, logical ([0; 1])); %! assert (class(idx), 'logical'); %! [~, idx] = rmmissing (x, 2); %! assert (idx, logical ([1, 1, 0])); %! assert (class(idx), 'logical'); %! [~, idx] = rmmissing (x, 1, "MinNumMissing", 2); %! assert (idx, logical ([0; 1])); %! assert (class(idx), 'logical'); %! [~, idx] = rmmissing (x, 2, "MinNumMissing", 2); %! assert (idx, logical ([0, 0, 0])); %! assert (class(idx), 'logical'); ## Test data type handling %!assert (rmmissing (single ([1 2 NaN; 3 4 5])), single ([3 4 5])) %!assert (rmmissing (logical (ones (3))), logical (ones (3))) %!assert (rmmissing (int32 (ones (3))), int32 (ones (3))) %!assert (rmmissing (uint32 (ones (3))), uint32 (ones (3))) %!assert (rmmissing ({1, 2, 3}), {1, 2, 3}) %!assert (rmmissing ([struct, struct, struct]), [struct, struct, struct]) ## Test empty input handling %!assert (rmmissing ([]), []) %!assert (rmmissing (ones (1,0)), ones (1,0)) %!assert (rmmissing (ones (1,0), 1), ones (1,0)) %!assert (rmmissing (ones (1,0), 2), ones (1,0)) %!assert (rmmissing (ones (0,1)), ones (0,1)) %!assert (rmmissing (ones (0,1), 1), ones (0,1)) %!assert (rmmissing (ones (0,1), 2), ones (0,1)) %!error rmmissing (ones (0,1,2)) ## Test input validation %!error rmmissing () %!error rmmissing (ones(2,2,2)) %!error rmmissing ([1 2; 3 4], 5) %!error rmmissing ([1 2; 3 4], "XXX", 1) %!error <'MinNumMissing'> rmmissing ([1 2; 3 4], 2, "MinNumMissing", -2) %!error <'MinNumMissing'> rmmissing ([1 2; 3 4], "MinNumMissing", 3.8) %!error <'MinNumMissing'> rmmissing ([1 2; 3 4], "MinNumMissing", [1 2 3]) %!error <'MinNumMissing'> rmmissing ([1 2; 3 4], "MinNumMissing", 'xxx') statistics-release-1.6.3/inst/runstest.m000066400000000000000000000300551456127120000203540ustar00rootroot00000000000000## Copyright (C) 2013 Nir Krakauer ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{h} =} runstest (@var{x}) ## @deftypefnx {statistics} {@var{h} =} runstest (@var{x}, @var{v}) ## @deftypefnx {statistics} {@var{h} =} runstest (@var{x}, @qcode{"ud"}) ## @deftypefnx {statistics} {@var{h} =} runstest (@dots{}, @var{Name}, @var{Value}) ## @deftypefnx {statistics} {[@var{h}, @var{pval}, @var{stats}] =} runstest (@dots{}) ## ## Run test for randomness in the vector @var{x}. ## ## @code{@var{h} = runstest (@var{x})} calculates the number of runs of ## consecutive values above or below the mean of @var{x} and tests the null ## hypothesis that the values in the data vector @var{x} come in random order. ## @var{h} is 1 if the test rejects the null hypothesis at the 5% significance ## level, or 0 otherwise. ## ## @code{@var{h} = runstest (@var{x}, @var{v})} tests the null hypothesis based ## on the number of runs of consecutive values above or below the specified ## reference value @var{v}. Values exactly equal to @var{v} are omitted. ## ## @code{@var{h} = runstest (@var{x}, @qcode{"ud"})} calculates the number of ## runs up or down and tests the null hypothesis that the values in the data ## vector @var{x} follow a trend. Too few runs indicate a trend, while too ## many runs indicate an oscillation. Values exactly equal to the preceding ## value are omitted. ## ## @code{@var{h} = runstest (@dots{}, @var{Name}, @var{Value})} specifies ## additional options to the above tests by one or more @var{Name}-@var{Value} ## pair arguments. ## ## @multitable @columnfractions 0.15 0.05 0.8 ## @headitem Name @tab @tab Value ## @item @qcode{"alpha"} @tab @tab the significance level. Default is 0.05. ## ## @item @qcode{"method"} @tab @tab a string specifying the method used to ## compute the p-value of the test. It can be either @qcode{"exact"} to use an ## exact algorithm, or @qcode{"approximate"} to use a normal approximation. The ## default is @qcode{"exact"} for runs above/below, and for runs up/down when ## the length of x is less than or equal to 50. When testing for runs up/down ## and the length of @var{x} is greater than 50, then the default is ## @qcode{"approximate"}, and the @qcode{"exact"} method is not available. ## ## @item @qcode{"tail"} @tab @tab a string specifying the alternative hypothesis ## @end multitable ## @multitable @columnfractions 0.2 0.15 0.05 0.5 ## @item @tab @qcode{"both"} @tab @tab two-tailed (default) ## @item @tab @qcode{"left"} @tab @tab left-tailed ## @item @tab @qcode{"right"} @tab @tab right-tailed ## @end multitable ## ## @seealso{signrank, signtest} ## @end deftypefn function [h, pval, stats] = runstest (x, v, varargin) ## Check arguments if (nargin < 1) print_usage; endif ## Check X being a vector of scalar values if (! isvector (x) || ! isnumeric (x)) error ("runstest: X must be a vector a scalar values."); else ## Remove missing values (NaNs) x(isnan (x)) = []; endif ## Check second argument being either a scalar reference number or "ud" string if (nargin > 1) if (isempty (v)) v = mean (x); endif if (isnumeric (v) && isscalar (v)) x = sign (x - v); rm = x == 0; if (sum (rm) > 0) warning ("runstest: %d elements equal to V were omitted.", sum (rm)); endif x(rm) = []; N = numel(x); UD = false; elseif (strcmpi (v, "ud")) x = diff (x); rm = x == 0; if (sum (rm) > 0) warning ("runstest: %d repeated elements were omitted.", sum (rm)); endif x(rm) = []; N = numel(x) + 1; UD = true else error ("runstest: V must be either a scalar number or 'ud' char string."); endif v = v; else v = mean (x); x = sign (x - v); rm = x == 0; if (sum (rm) > 0) warning ("runstest: %d elements equal to 'mean(X)' were omitted.", ... sum (rm)); endif x(rm) = []; N = numel(x); UD = false; endif ## Get number of runs n_up = sum (x==1); n_dn = numel (x) - n_up; ## Add defaults alpha = 0.05; if (N < 50 || ! UD) method = "exact"; else method = "approximate"; endif tail = "both"; ## Parse optional arguments and validate parameters while (numel (varargin) > 1) switch (lower (varargin{1})) case "alpha" alpha = varargin{2}; if (! isscalar (alpha) || ! isnumeric (alpha) || alpha <= 0 || alpha >= 1) error ("runstest: invalid value for alpha."); endif case "method" method = varargin{2}; if (! any (strcmpi (method, {"exact", "approximate"}))) error ("runstest: invalid value for method."); endif if (strcmpi (method, "exact") && N > 50) warning ("runstest: exact method is not available for N > 50."); method = "approximate"; endif case "tail" tail = varargin{2}; if (! any (strcmpi (tail, {"both", "left", "right"}))) error ("runstest: invalid value for tail."); endif otherwise error ("runstest: invalid optional argument."); endswitch varargin([1:2]) = []; endwhile ## Do the calculations here if (N > 0) R_num = sum (x([1:end-1]) != x([2:end])) + 1; ##R_num = sum ((x(1:(end-1)) .* x(2:end)) < 0) + 1; #number of runs ## Special case if (N == 1) z = NaN; ## Compute with z statistic else ## Handle up/down or above/below if (UD) R_bar = (2 * N - 1) / 3; R_std = sqrt ((16 * N - 29) / 90); else R_bar = 1 + 2 * n_up * n_dn / N; R_std = sqrt (2 * n_up * n_dn * (2 * n_up * n_dn - N) / ... (N ^ 2 * (N - 1))); end ## Handle tail if (strcmpi (tail, "both")) tc = -0.5 * sign (R_num - R_bar); elseif (strcmpi (tail, "left")) tc = 0.5; else tc = -0.5; endif ## Compute z value if (R_std > 0) z = (R_num + tc - R_bar) / R_std; else z = Inf * sign (R_num + tc - R_bar); endif endif ## Exact method if (strcmpi (method, "exact")) if (UD) R_max = N - 1; ## Get precalculated results from rundist.mat file temp = load ("rundist.mat"); runD = temp.rundist; M = runD{N}; p = M / sum (M); p = p([1:R_max]); else R_max = 2 * min ([n_up, n_dn]) + 1; if (n_up == 0 || n_dn == 0) p = 1; else R_vec = [1:R_max]; p = zeros (size (R_vec)); t = mod (R_vec, 2) == 0; ## Compute even if (any (t)) k = R_vec(t) / 2; p(t) = 2 * exp (logBinoCoeff (n_up - 1, k - 1) + ... logBinoCoeff (n_dn - 1, k - 1) - ... logBinoCoeff (N, n_dn)); endif ## Compute odd if (any (! t)) k = floor (R_vec(! t) / 2); logdenom = logBinoCoeff (N, n_dn); p(! t) = exp (logBinoCoeff (n_up - 1, k - 1) + ... logBinoCoeff (n_dn - 1, k) - logdenom) + ... exp (logBinoCoeff (n_up - 1, k) + ... logBinoCoeff (n_dn - 1, k - 1) - logdenom); endif endif endif if (isempty (p)) p_ex = 1; else p_ex = p(R_num); end p_lo = sum (p([1:R_num-1])); p_hi = sum (p([R_num+1:end])); else ## Compute with z statistic p_ex = 0; p_lo = normcdf (z); p_hi = normcdf (-z); end ## Assume a constant vector in data else R_num = NaN; p_ex = 1; p_lo = 0; p_hi = 0; z = NaN; endif ## Compute tail probability if (strcmpi (tail, "both")) pval = min([1, 2*(p_ex + min ([p_lo, p_hi]))]); elseif (strcmpi (tail, "left")) pval = p_ex + p_lo; else pval = p_ex + p_hi; endif ## Retyrn decision of test h = double (pval <= alpha); if (nargout > 2) stats.nruns = R_num; stats.n1 = n_up; stats.n0 = n_dn; stats.z = z; endif endfunction ## Compute the log of the binomial coefficient function logBC = logBinoCoeff(N,n) logBC = gammaln (N + 1) - gammaln (n + 1) - gammaln (N - n + 1); endfunction %!test %! ## NIST beam deflection data %! ## http://www.itl.nist.gov/div898/handbook/eda/section4/eda425.htm %! data = [-213, -564, -35, -15, 141, 115, -420, -360, 203, -338, -431, ... %! 194, -220, -513, 154, -125, -559, 92, -21, -579, -52, 99, -543, ... %! -175, 162, -457, -346, 204, -300, -474, 164, -107, -572, -8, 83, ... %! -541, -224, 180, -420, -374, 201, -236, -531, 83, 27, -564, -112, ... %! 131, -507, -254, 199, -311, -495, 143, -46, -579, -90, 136, ... %! -472, -338, 202, -287, -477, 169, -124, -568, 17, 48, -568, -135, ... %! 162, -430, -422, 172, -74, -577, -13, 92, -534, -243, 194, -355, ... %! -465, 156, -81, -578, -64, 139, -449, -384, 193, -198, -538, 110, ... %! -44, -577, -6, 66, -552, -164, 161, -460, -344, 205, -281, -504, ... %! 134, -28, -576, -118, 156, -437, -381, 200, -220, -540, 83, 11, ... %! -568, -160, 172, -414, -408, 188, -125, -572, -32, 139, -492, ... %! -321, 205, -262, -504, 142, -83, -574, 0, 48, -571, -106, 137, ... %! -501, -266, 190, -391, -406, 194, -186, -553, 83, -13, -577, -49, ... %! 103, -515, -280, 201, 300, -506, 131, -45, -578, -80, 138, -462, ... %! -361, 201, -211, -554, 32, 74, -533, -235, 187, -372, -442, 182, ... %! -147, -566, 25, 68, -535, -244, 194, -351, -463, 174, -125, -570, ... %! 15, 72, -550, -190, 172, -424, -385, 198, -218, -536, 96]; %! [h, p, stats] = runstest (data, median (data)); %! expected_h = 1; %! expected_p = 0.008562; %! expected_z = 2.6229; %! assert (h, expected_h); %! assert (p, expected_p, 1E-6); %! assert (stats.z, expected_z, 1E-4); %!shared x %! x = [45, -60, 1.225, 55.4, -9 27]; %!test %! [h, p, stats] = runstest (x); %! assert (h, 0); %! assert (p, 0.6, 1e-14); %! assert (stats.nruns, 5); %! assert (stats.n1, 3); %! assert (stats.n0, 3); %! assert (stats.z, 0.456435464587638, 1e-14); %!test %! [h, p, stats] = runstest (x, [], "method", "approximate"); %! assert (h, 0); %! assert (p, 0.6481, 1e-4); %! assert (stats.z, 0.456435464587638, 1e-14); %!test %! [h, p, stats] = runstest (x, [], "tail", "left"); %! assert (h, 0); %! assert (p, 0.9, 1e-14); %! assert (stats.z, 1.369306393762915, 1e-14); %!error runstest (ones (2,20)) %!error runstest (["asdasda"]) %!error ... %! runstest ([2 3 4 3 2 3 4], "updown") %!error ... %! runstest ([2 3 4 3 2 3 4], [], "alpha", 0) %!error ... %! runstest ([2 3 4 3 2 3 4], [], "alpha", [0.02 0.2]) %!error ... %! runstest ([2 3 4 3 2 3 4], [], "alpha", 1.2) %!error ... %! runstest ([2 3 4 3 2 3 4], [], "alpha", -0.05) %!error ... %! runstest ([2 3 4 3 2 3 4], [], "method", "some") %!error ... %! runstest ([2 3 4 3 2 3 4], [], "tail", "some") %!error ... %! runstest ([2 3 4 3 2 3 4], [], "option", "some") statistics-release-1.6.3/inst/sampsizepwr.m000066400000000000000000001225231456127120000210530ustar00rootroot00000000000000## Copyright (C) 2022 Andrew Penn ## Copyright (C) 2022 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{n} =} sampsizepwr (@var{testtype}, @var{params}, @var{p1}) ## @deftypefnx {statistics} {@var{n} =} sampsizepwr (@var{testtype}, @var{params}, @var{p1}, @var{power}) ## @deftypefnx {statistics} {@var{power} =} sampsizepwr (@var{testtype}, @var{params}, @var{p1}, [], @var{n}) ## @deftypefnx {statistics} {@var{p1} =} sampsizepwr (@var{testtype}, @var{params}, [], @var{power}, @var{n}) ## @deftypefnx {statistics} {[@var{n1}, @var{n2}] =} sampsizepwr (@qcode{"t2"}, @var{params}, @var{p1}, @var{power}) ## @deftypefnx {statistics} {[@dots{}] =} sampsizepwr (@var{testtype}, @var{params}, @var{p1}, @var{power}, @var{n}, @var{name}, @var{value}) ## ## Sample size and power calculation for hypothesis test. ## ## @code{sampsizepwr} computes the sample size, power, or alternative parameter ## value for a hypothesis test, given the other two values. For example, you can ## compute the sample size required to obtain a particular power for a ## hypothesis test, given the parameter value of the alternative hypothesis. ## ## @code{@var{n} = sampsizepwr (@var{testtype}, @var{params}, @var{p1})} returns ## the sample size N required for a two-sided test of the specified type to have ## a power (probability of rejecting the null hypothesis when the alternative is ## true) of 0.90 when the significance level (probability of rejecting the null ## hypothesis when the null hypothesis is true) is 0.05. @var{params} specifies ## the parameter values under the null hypothesis. P1 specifies the value of ## the single parameter being tested under the alternative hypothesis. For the ## two-sample t-test, N is the value of the equal sample size for both samples, ## @var{params} specifies the parameter values of the first sample under the ## null and alternative hypotheses, and P1 specifies the value of the single ## parameter from the other sample under the alternative hypothesis. ## ## The following TESTTYPE values are available: ## ## @multitable @columnfractions 0.05 0.1 0.85 ## @item @tab "z" @tab one-sample z-test for normally distributed data with ## known standard deviation. @var{params} is a two-element vector [MU0 SIGMA0] ## of the mean and standard deviation, respectively, under the null hypothesis. ## P1 is the value of the mean under the alternative hypothesis. ## @item @tab "t" @tab one-sample t-test or paired t-test for normally ## distributed data with unknown standard deviation. @var{params} is a ## two-element vector [MU0 SIGMA0] of the mean and standard deviation, ## respectively, under the null hypothesis. P1 is the value of the mean under ## the alternative hypothesis. ## @item @tab "t2" @tab two-sample pooled t-test (test for equal means) for ## normally distributed data with equal unknown standard deviations. ## @var{params} is a two-element vector [MU0 SIGMA0] of the mean and standard ## deviation of the first sample under the null and alternative hypotheses. P1 ## is the the mean of the second sample under the alternative hypothesis. ## @item @tab "var" @tab chi-square test of variance for normally distributed ## data. @var{params} is the variance under the null hypothesis. P1 is the ## variance under the alternative hypothesis. ## @item @tab "p" @tab test of the P parameter (success probability) for a ## binomial distribution. @var{params} is the value of P under the null ## hypothesis. P1 is the value of P under the alternative hypothesis. ## @item @tab "r" @tab test of the correlation coefficient parameter for ## significance. @var{params} is the value of r under the null hypothesis. ## P1 is the value of r under the alternative hypothesis. ## @end multitable ## ## The "p" test for the binomial distribution is a discrete test for which ## increasing the sample size does not always increase the power. For N values ## larger than 200, there may be values smaller than the returned N value that ## also produce the desired power. ## ## @code{@var{n} = sampsizepwr (@var{testtype}, @var{params}, @var{p1}, ## @var{power})} returns the sample size N such that the power is @var{power} ## for the parameter value P1. For the two-sample t-test, N is the equal sample ## size of both samples. ## ## @code{[@var{n1}, @var{n2}] = sampsizepwr ("t2", @var{params}, @var{p1}, ## @var{power})} returns the sample sizes @var{n1} and @var{n2} for the two ## samples. These values are the same unless the "ratio" parameter, ## @code{@var{ratio} = @var{n2} / @var{n2}}, is set to a value other than ## the default (See the name/value pair definition of ratio below). ## ## @code{@var{power} = sampsizepwr (@var{testtype}, @var{params}, @var{p1}, [], ## @var{n})} returns the power achieved for a sample size of @var{n} when the ## true parameter value is @var{p1}. For the two-sample t-test, @var{n} is the ## smaller one of the two sample sizes. ## ## @code{@var{p1} = sampsizepwr (@var{testtype}, @var{params}, [], @var{power}, ## @var{n})} returns the parameter value detectable with the specified sample ## size @var{n} and power @var{power}. For the two-sample t-test, @var{n} is ## the smaller one of the two sample sizes. When computing @var{p1} for the "p" ## test, if no alternative can be rejected for a given @var{params}, @var{n} and ## @var{power} value, the function displays a warning message and returns NaN. ## ## @code{[@dots{}] = sampsizepwr (@dots{}, @var{n}, @var{name}, @var{value})} ## specifies one or more of the following @var{name} / @var{value} pairs: ## ## @multitable @columnfractions 0.05 0.15 0.8 ## @item @tab "alpha" @tab significance level of the test (default is 0.05) ## @item @tab "tail" @tab the type of test which can be: ## @end multitable ## ## @multitable @columnfractions 0.1 0.20 0.7 ## @item @tab "both" @tab two-sided test for an alternative @var{p1} not equal ## to @var{params} ## ## @item @tab "right" @tab one-sided test for an alternative @var{p1} larger ## than @var{params} ## ## @item @tab "left" @tab one-sided test for an alternative @var{p1} smaller ## than @var{params} ## @end multitable ## ## @multitable @columnfractions 0.05 0.15 0.8 ## @item @tab "ratio" @tab desired ratio @var{n2} / @var{n2} of the larger ## sample size @var{n2} to the smaller sample size @var{n1}. Used only for the ## two-sample t-test. The value of @code{@var{ratio}} is greater than or equal ## to 1 (default is 1). ## @end multitable ## ## @code{sampsizepwr} computes the sample size, power, or alternative hypothesis ## value given values for the other two. Specify one of these as [] to compute ## it. The remaining parameters (and ALPHA, RATIO) can be scalars or arrays of ## the same size. ## ## @seealso{vartest, ttest, ttest2, ztest, binocdf} ## @end deftypefn function [out, N2] = sampsizepwr (TestType, params, p1, power, n, varargin) ## Check for valid number of input arguments narginchk (3, Inf); ## Add defaults for 3 or 4 input arguments if (nargin == 3) power = 0.90; n = []; elseif (nargin == 4) n = []; endif ## Add defaults for extra arguments alpha = 0.05; tail = "both"; ratio = 1; ## Check for valid test type and corresponding size of parameters t_types = {"z", "t", "t2", "var", "p", "r"}; nparams = [2, 2, 2, 1, 1, 1]; if (isempty (TestType) || ! ischar (TestType) || size (TestType, 1) != 1) error ("sampsizepwr: test type must be a non-empty string."); endif if (sum (strcmpi (TestType, t_types)) != 1) error ("sampsizepwr: invalid test type."); endif if (! isnumeric (params)) error ("sampsizepwr: parameters must be numeric."); endif if (length (params) != nparams(strcmpi (TestType, t_types))) error ("sampsizepwr: invalid size of parameters for this test type."); endif ## Check for correct number of output arguments if ((nargout > 1) && ! strcmpi (TestType, "t2")) error ("sampsizepwr: wrong number of output arguments for this test type."); endif Lbound = [-Inf, -Inf, -Inf, 0, 0, -1]; Ubound = [ Inf, Inf, Inf, Inf, 1, 1]; Lbound = Lbound(strcmpi (TestType, t_types)); Ubound = Ubound(strcmpi (TestType, t_types)); ## Check for invalid parameters specific to each test type switch (lower (TestType)) case "z" if (params(2) <= 0) error ("sampsizepwr: negative or zero variance."); endif PowerFunction = @PowerFunction_N; case "t" if (params(2) <= 0) error ("sampsizepwr: negative or zero variance."); endif PowerFunction = @PowerFunction_T; case "t2" if (params(2) <= 0) error ("sampsizepwr: negative or zero variance."); endif PowerFunction = @PowerFunction_T2; case "var" if (params(1) <= 0) error ("sampsizepwr: negative or zero variance."); endif PowerFunction = @PowerFunction_V; case "p" if (params(1) <= 0 || params(1) >= 1) error ("sampsizepwr: out of range probability."); endif PowerFunction = @PowerFunction_P; case "r" if (params(1) <= -1 || params(1) >= 1) error ("sampsizepwr: out of range regression coefficient."); elseif (params(1) == 0) error ("sampsizepwr: regression coefficient must not be 0."); endif PowerFunction = @PowerFunction_R; endswitch ## Get and validate optional parameters numarg = numel (varargin); argpos = 1; while (numarg) argname = varargin{argpos}; switch (lower (argname)) case "alpha" alpha = varargin{argpos + 1}; if (! isnumeric (alpha) || any (alpha(:) <= 0) || any( alpha(:) >= 1)) error ("sampsizepwr: invalid value for 'alpha' parameter."); endif case "tail" tail = varargin{argpos + 1}; if (! ischar (tail) || (size (tail, 1) != 1)) error ("sampsizepwr: 'tail' parameter must be a non-empty string."); endif if (sum (strcmpi (tail, {"left","both","right"})) != 1) error ("sampsizepwr: invalid value for 'tail' parameter."); endif case "ratio" ratio = varargin{argpos + 1}; if (! isnumeric (ratio) || any (ratio(:) < 1)) error ("sampsizepwr: invalid value for 'ratio' parameter."); endif endswitch numarg -= 2; argpos += 2; endwhile ## Check that only one of either p1, power, or n are missing if (isempty(p1) + isempty(power) + isempty(n) != 1) error ("sampsizepwr: only one of either p1, power, or n must be missing."); endif ## Check for valid P1 if (! isempty (p1)) if (! isnumeric (p1)) error ("sampsizepwr: alternative hypothesis parameter must be numeric."); elseif ((! strcmpi (tail, "right") && any (p1(:) <= Lbound)) || ... (! strcmpi (tail, "left") && any (p1(:) >= Ubound))) error ("sampsizepwr: alternative hypothesis parameter out of range."); endif endif ## Check for valid POWER if (! isempty (power) && ... (! isnumeric (power) || any (power(:) <= 0) || any (power(:) >= 1))) error ("sampsizepwr: invalid value for POWER."); endif if (! isempty (power) && any (power(:) <= alpha(:))) error ("sampsizepwr: Cannot compute N or P1 unless POWER > 'alpha'."); endif ## Expand non-empty P1/POWER/N so they are all the same size if (isempty (p1)) [err, power, n, alpha, ratio] = common_size (power, n, alpha, ratio); outclass = getclass (power, n, alpha, ratio); elseif (isempty (power)) [err, p1, n, alpha, ratio] = common_size (p1, n, alpha, ratio); outclass = getclass (p1, n, alpha, ratio); else # n is empty [err, p1, power, alpha, ratio] = common_size (p1, power, alpha, ratio); outclass = getclass (power, p1, alpha, ratio); endif if (err > 0) error ("sampsizepwr: input arguments size mismatch."); end ## Check for valid options when computing N if (isempty (n)) if (any (p1(:) == params(1))) error ("sampsizepwr: Same value for null and alternative hypothesis."); elseif (strcmpi (tail, "left") && any (p1(:) >= params(1))) error ("sampsizepwr: Invalid P1 for testing left tail."); elseif (strcmpi (tail, "right") && any(p1(:) <= params(1))) error ("sampsizepwr: Invalid P1 for testing right tail."); endif endif ## Allocate output of proper size and class out = zeros (size (alpha), outclass); ## Compute whichever one of P1/POWER/N that is now empty if (isempty (p1)) ## Compute effect size given power and sample size switch (lower (TestType)) case "z" ## z (normal) test out(:) = findP1z (params(1), params(2), power, n, alpha, tail); case "t" ## t-test out(:) = findP1t (params(1), params(2), power, n, alpha, tail); case "t2" ## two-sample t-test out(:) = findP1t2 (params(1), params(2), power, n, alpha, tail, ratio); case "var" ## chi-square (variance) test out(:) = findP1v (params(1), power, n, alpha, tail); case "p" ## binomial (p) test out(:) = findP1p (params(1), power, n, alpha, tail); case "r" ## regression coefficient (r) test out(:) = findP1r (params(1), power, n, alpha, tail); endswitch elseif (isempty (power)) ## Compute power given effect size and sample size switch (lower (TestType)) case {"z", "t"} out(:) = PowerFunction (params(1), p1, params(2), alpha, tail, n); case "t2" out(:) = PowerFunction (params(1), p1, params(2), alpha, tail, n, ratio); case {"var", "p", "r"} out(:) = PowerFunction (params(1), p1, alpha, tail, n); endswitch else ## Compute sample size given power and effect size switch (lower (TestType)) case {"z", "t"} ## Calculate one-sided Z value directly out(:) = z1testN (params(1), p1, params(2), power, alpha, tail); ## Iterate upward from there for the other cases if (strcmpi (TestType, "t") || strcmp (tail, "both")) if (strcmpi (TestType, "t")) out = max (out, 2); endif ## Count upward until we get the value we need elem = 1:numel (alpha); while (! isempty (elem)) actualpower = PowerFunction_T (params(1), p1(elem), params(2), ... alpha(elem), tail, out(elem)); elem = elem(actualpower < power(elem)); out(elem) = out(elem) + 1; endwhile endif case "t2" ## Initialize second output argument N2 = zeros (size (alpha), outclass); ## Caculate one-sided two-sample t-test iteratively [out(:), N2(:)] = t1testN (params(1), p1, params(2), power, ... alpha, tail, ratio); case "var" ## Use a binary search method out(:) = searchbinaryN (PowerFunction, [1, 100], params(1), ... p1, power, alpha, tail); case "p" ## Use a binary search method out(:) = searchbinaryN (PowerFunction, [0, 100], params(1), ... p1, power, alpha, tail); ## Adjust for discrete distribution t = out <= 200; if (any (t(:))) ## Try values from 1 up to N (out) and pick the smallest value out(t) = adjdiscreteN (out(t), PowerFunction, params(1), ... p1(t), alpha(t), tail, power(t)); endif if (any (! t(:))) warning ("sampsizepwr: approximate N."); endif case "r" ## Calculate sample size using Student's t distribution out(:) = r1testN (params(1), p1, power, alpha, tail); endswitch endif endfunction ## Define class for output function out = getclass (varargin) if (class (varargin{1}) == "single" || class (varargin{2}) == "single" || ... class (varargin{3}) == "single" || class (varargin{4}) == "single") out = "single"; else out = "double"; endif endfunction ## Sample size calculation for the one-sided Z test function N = z1testN (mu0, mu1, sig, desiredpower, alpha, tail) ## Compute the one-sided normal value directly if (strcmp (tail, "both")) alpha = alpha ./ 2; endif z1 = -norminv (alpha); z2 = norminv (1 - desiredpower); mudiff = abs (mu0 - mu1) / sig; N = ceil (((z1 - z2) ./ mudiff) .^ 2); endfunction ## Sample size calculation for R test function N = rtestN (r0, r1, desiredpower, alpha, tail) ## Compute only for 2-tailed test if (strcmp (tail, "both")) alpha = alpha ./ 2; else error ("sampsizepwr: only 2-tailed testing for regression coefficient."); endif ## Get quantiles of the standard normal deviates for alpha and power Za = norminv (alpha); Zb = norminv (1 - desiredpower); ## Compute difference in regression coefficients rdiff = abs (r0 - r1) C = 0.5 * log ((1 + rdiff) / (1 - rdiff)); ## Compute sample size N = ((Za + Zb) / C) .^ 2 + 3; endfunction ## Find alternative hypothesis parameter value P1 for Z test function mu1 = findP1z (mu0, sig, desiredpower, N, alpha, tail) if (strcmp (tail, "both")) alpha = alpha ./ 2; end sig = sig ./ sqrt (N); ## Get quantiles of the normal or t distribution if (strcmp (tail, "left")) z1 = norminv (alpha); z2 = norminv (desiredpower); else # upper or two-tailed test z1 = norminv (1 - alpha); z2 = norminv (1 - desiredpower); endif mu1 = mu0 + sig .* (z1 - z2); ## For 2-sided test, refine by taking the other tail into account if (strcmp (tail, "both")) elem = 1:numel (alpha); desiredbeta = 1 - desiredpower; betahi = desiredbeta; betalo = zeros (size (desiredbeta)); while (true) ## Compute probability of being below the lower critical value under H1 betalo(elem) = normcdf (-z1(elem) + (mu0 - mu1(elem)) ./ sig(elem)); ## See if the upper and lower probabilities are close enough elem = elem(abs ((betahi(elem) - betalo(elem)) - desiredbeta(elem)) > ... 1e-6 * desiredbeta(elem)); if (isempty (elem)) break endif ## Find a new mu1 by adjusting beta to take lower tail into account betahi(elem) = desiredbeta(elem) + betalo(elem); mu1(elem) = mu0 + sig(elem) .* (z1(elem) - norminv (betahi(elem))); endwhile endif endfunction ## Find alternative hypothesis parameter value P1 for t-test function mu1 = findP1t (mu0, sig, desiredpower, N, alpha, tail) if (strcmp (tail, "both")) a2 = alpha ./ 2; else a2 = alpha; endif ## Get quantiles of the normal or t distribution if (strcmp (tail, "left")) z1 = norminv(alpha); z2 = norminv(desiredpower); else # upper or two-tailed test z1 = norminv(1-a2); z2 = norminv(1-desiredpower); endif mu1 = mu0 + sig .* (z1-z2) ./ sqrt (N); ## Refine using fzero for j=1:numel (mu1) if (mu1(j) > mu0) F0 = @(mu1arg) PowerFunction_T (mu0, max (mu0, mu1arg), sig, alpha(j), ... tail, N(j)) - desiredpower(j); else F0 = @(mu1arg) desiredpower(j) - PowerFunction_T (mu0, min (mu0, ... mu1arg), sig, alpha(j), tail, N(j)); endif mu1(j) = fzero (F0, mu1(j)); endfor endfunction ## Sample size calculation for the one-sided two-sample t-test function [N1, N2] = t1testN (mu0, mu1, sig, desiredpower, alpha, tail, ratio) if (strcmp (tail, "both")) alpha = alpha ./ 2; endif ## Compute the initial value of N, approximated by normal distribution z1 = -norminv (alpha); z2 = norminv (1 - desiredpower); n_0 = ceil ((z1 - z2) .^2 .* (sig ./ abs ((mu0 - mu1))) .^ 2 * 2); ## n need to be > 1, otherwise the degree of freedom of t < 0 n_0(n_0 <= 1) = 2; N = ones (size (n_0)); ## iteratively update the sample size if (strcmp (tail, "both")) for j = 1:numel (n_0) F = @(n) nctcdf (tinv (alpha(j), n + ratio(j) .* n - 2), ... n + ratio(j) .* n - 2, abs (mu1(j) - mu0) ./ ... (sig .* sqrt (1 ./ n + 1 ./ (ratio(j) .* n)))) + ... (1 - nctcdf (- tinv (alpha(j), n + ratio(j) .* n - 2), ... n + ratio(j) .* n - 2, abs (mu1(j) - mu0) ./ ... (sig .* sqrt (1 ./ n + 1 ./ (ratio(j) .* n)))))- ... desiredpower(j); N(j) = localfzero (F, n_0(j), ratio); endfor else for j = 1:numel (n_0) F = @(n) (1 - nctcdf (- tinv (alpha(j), n + ratio(j) .* n - 2), ... n + ratio(j) .* n - 2, abs (mu1(j) - mu0) ./ (sig .* ... sqrt (1 ./ n + 1 ./ (ratio(j) .* n))))) - desiredpower(j); N(j) = localfzero (F, n_0(j), ratio); endfor endif N1 = ceil (N); N2 = ceil (ratio .* N); endfunction ## Find alternative hypothesis parameter value P1 for two-sample t-test function mu1 = findP1t2 (mu0, sig, desiredpower, N, alpha, tail, ratio) if (strcmp (tail, "both")) a2 = alpha ./ 2; else a2 = alpha; endif ## Get quantiles of the normal or t distribution if (strcmp (tail, "left")) t1 = tinv (alpha, N + ratio .* N - 2); t2 = tinv (desiredpower, N + ratio .* N - 2); else # upper or two-tailed test t1 = tinv (1 - a2, N + ratio .* N - 2); # upper tail under H0 t2 = tinv (1 - desiredpower, N + ratio .* N - 2); # lower tail under H1 endif mu1 = mu0 + sig .* (t1 - t2) .* sqrt (1 ./ N + 1 ./ (ratio .* N)); ## Refine using fzero for j = 1:numel (mu1) if (mu1(j) > mu0) F0 = @(mu1arg) PowerFunction_T2 (mu0, max (mu0, mu1arg), sig, ... alpha(j), tail, N(j), ratio(j)) - desiredpower(j); else F0 = @(mu1arg) desiredpower(j) - PowerFunction_T2 (mu0, min (mu0, ... mu1arg), sig, alpha(j), tail, N(j), ratio(j)); endif mu1(j) = fzero (F0, mu1(j)); endfor endfunction ## Find alternative hypothesis parameter value P1 for variance test function p1 = findP1v (p0, desiredpower, N, alpha, tail) ## F and Finv are the cdf and inverse cdf F = @(x,n,p1) chi2cdf (x .* (n - 1) ./ p1, n - 1); # cdf for s^2 Finv = @(p,n,p1) p1 .* chi2inv (p, n - 1) ./ (n - 1); # inverse if (strcmp (tail, "both")) alpha = alpha ./ 2; endif desiredbeta = 1 - desiredpower; ## Calculate critical values and p1 for one-sided test if (! strcmp (tail, "left")) critU = Finv (1 - alpha, N, p0); p1 = 1 ./ Finv (desiredbeta, N, 1 ./ critU); endif if (! strcmp (tail, "right")) critL = Finv (alpha, N, p0); endif if (strcmp (tail, "left")) p1 = 1 ./ Finv (desiredpower, N, 1 ./ critL); endif if (strcmp (tail, "both")) ## For 2-sided test, we have the upper tail probability under H1. ## Refine by taking the other tail into account. elem = 1:numel (alpha); betahi = desiredbeta; betalo = zeros (size (desiredbeta)); while (true) ## Compute probability of being in the lower tail under H1 betalo(elem) = F (critL(elem), N(elem), p1(elem)); ## See if the upper and lower probabilities are close enough obsbeta = betahi(elem) - betalo(elem); elem = elem(abs (obsbeta - desiredbeta(elem)) > 1e-6 * desiredbeta(elem)); if (isempty (elem)) break endif ## Find a new mu1 by adjusting beta to take lower tail into account betahi(elem) = desiredbeta(elem) + betalo(elem); p1(elem) = 1 ./ Finv (betahi(elem), N(elem), 1 ./ critU(elem)); endwhile endif endfunction ## Find alternative hypothesis parameter value P1 for p test function p1 = findP1p (p0, desiredpower, N, alpha, tail) ## Get critical values [critL, critU] = getcritP (p0, N, alpha, tail); ## Use a normal approximation to find P1 values sigma = sqrt (p0 .* (1 - p0) ./ N); p1 = findP1z (p0, sigma, desiredpower, N, alpha, tail); ## Problem if we have no critical region left if (strcmp (tail, "both")) t = (critL == 0 & critU == N); elseif (strcmp (tail, "right")) t = (critU == N); else t = (critL == 0); endif if (any (t)) warning ("sampsizepwr: No Valid Parameter"); p1(t) = NaN; endif ## Force in bounds t = p1 <= 0; if (any (t(:))) p1(t) = p0 / 2; end t = p1 >= 1; if (any (t(:))) p1(t) = 1 - p0 / 2; end ## Refine using fzero for j=1:numel(p1) if (! isnan (p1(j))); if (p1(j) > p0) F0 = @(p1arg) PowerFunction_P (p0, max (p0, min (1, p1arg)), ... alpha(j), tail, N(j), critL(j), critU(j)) - desiredpower(j); else F0 = @(p1arg) desiredpower(j) - PowerFunction_P (p0, max (0, ... min (p0, p1arg)), alpha(j), tail, N(j), critL(j), critU(j)); endif p1(j) = fzero (F0, p1(j)); endif endfor endfunction ## Find alternative hypothesis parameter value P1 for r test function p1 = findP1r (p0, desiredpower, N, alpha, tail) ## Compute only for 2-tailed test if (! strcmp (tail, "both")) error ("sampsizepwr: only 2-tailed testing for regression coefficient."); endif ## Set initial search boundaries for p1 p1_lo = eps; p1_hi = 1 - eps; ## Compute initial sample size N0 according to P0, POWER and ALPHA N0 = rtestN (p0, 0, desiredpower, alpha, tail); ## Find P0 for N0 == N while (N != N0) if (N0 < N) p1_hi = p0; p1 = (p0 + p1_lo) / 2; p0 = p1; N0 = rtestN (p1, 0, desiredpower, alpha, tail); else p1_lo = p0; p1 = (p0 + p1_hi) / 2; p0 = p1; N0 = rtestN (p1, 0, desiredpower, alpha, tail); endif endwhile endfunction ## Get upper and lower critical values for binomial (p) test. function [critL, critU] = getcritP (p0, N, alpha, tail) ## For two-sided tests, this function tries to compute critical values ## favorable for p0<.5. It does this by allocating alpha/2 to the lower ## tail where the probabilities come in larger chunks, then using any ## left-over alpha, probably more than alpha/2, for the upper tail. ## Get part of alpha available for lower tail if (strcmp (tail, "both")) Alo = alpha ./ 2; elseif (strcmp (tail, "left")) Alo = alpha; else Alo = 0; endif ## Calculate critical values critU = N; critL = zeros(size(N)); if (! strcmp (tail, "right")) critL = binoinv (Alo, N, p0); Alo = binocdf (critL, N, p0); t = (critL < N) & (Alo <= alpha / 2); critL(t) = critL(t) + 1; Alo(! t) = Alo(! t) - binopdf (critL(! t), N(! t), p0); endif if (! strcmp (tail, "left")) Aup = max(0, alpha - Alo); critU = binoinv(1 - Aup, N, p0); endif endfunction ## Sample size calculation via binary search function N = searchbinaryN (F, lohi, p0, p1, desiredpower, alpha, tail) ## Find uper and lower bounds nlo = repmat(lohi(1),size(alpha)); nhi = repmat(lohi(2),size(alpha)); obspower = F(p0,p1,alpha,tail,nhi); ## Iterate on n until we achieve the desired power elem = 1:numel (alpha); while (! isempty (elem)) elem = elem(obspower(elem) < desiredpower(elem)); nhi(elem) = nhi(elem) * 2; obspower(elem) = F (p0, p1(elem), alpha(elem), tail, nhi(elem)); endwhile ## Binary search between these bounds for required sample size elem = find(nhi > nlo+1); while (! isempty (elem)) n = floor ((nhi(elem) + nlo(elem)) / 2); obspower = F (p0, p1(elem), alpha(elem), tail, n); toohigh = (obspower > desiredpower(elem)); nhi(elem(toohigh)) = n(toohigh); nlo(elem(! toohigh)) = n(! toohigh); elem = elem(nhi(elem) > nlo(elem) + 1); endwhile N = nhi; endfunction ## Adjust sample size to take discreteness into account function N = adjdiscreteN (N, PowerFunction, p0, p1, alpha, tail, power) for j=1:numel(N) allN = 1:N(j); obspower = PowerFunction (p0, p1(j), alpha(j), tail, allN); N(j) = allN(find (obspower >= power(j), 1, "first")); endfor endfunction ## Normal power calculation function power = PowerFunction_N (mu0, mu1, sig, alpha, tail, n) S = sig ./ sqrt (n); if (strcmp (tail, "both")) critL = norminv (alpha / 2, mu0, S); critU = mu0 + (mu0 - critL); power = normcdf (critL, mu1, S) + normcdf (-critU, -mu1, S); elseif (strcmp (tail, "right")) crit = mu0 + (mu0 - norminv (alpha, mu0, S)); power = normcdf (-crit, -mu1, S); else crit = norminv (alpha, mu0, S); power = normcdf (crit, mu1, S); endif endfunction ## T power calculation function power = PowerFunction_T (mu0, mu1, sig, alpha, tail, n) S = sig ./ sqrt (n); ncp = (mu1 - mu0) ./ S; if (strcmp (tail, "both")) critL = tinv (alpha / 2, n - 1); critU = -critL; power = nctcdf (critL, n - 1, ncp) + nctcdf (-critU, n - 1, -ncp); elseif (strcmp (tail, "right")) crit = tinv (1 - alpha, n - 1); power = nctcdf (-crit, n - 1, -ncp); else crit = tinv (alpha, n - 1); power = nctcdf (crit, n - 1, ncp); endif endfunction ## Two-sample T power calculation function power = PowerFunction_T2 (mu0, mu1, sig, alpha, tail, n, ratio) ncp = (mu1 - mu0) ./ (sig .* sqrt (1 ./ n + 1 ./ (ratio .* n))); if (strcmp (tail, "both")) critL = tinv (alpha / 2, n + ratio .* n - 2); critU = -critL; power = nctcdf (critL, n + ratio .* n - 2, ncp) + ... nctcdf (-critU, n + ratio .* n - 2, -ncp); elseif (strcmp (tail, "right")) crit = tinv (1 - alpha, n + ratio .* n - 2); power = nctcdf (-crit, n + ratio .* n - 2, -ncp); else crit = tinv (alpha, n + ratio .* n - 2); power = nctcdf (crit, n + ratio .* n - 2, ncp); endif endfunction ## Chi-square power calculation function power = PowerFunction_V (v0, v1, alpha, tail, n) if (strcmp (tail, "both")) critU = v0 .* chi2inv (1 - alpha / 2, n - 1); critL = v0 .* chi2inv (alpha / 2, n - 1); power = chi2cdf (critL ./ v1, n - 1) + chi2cdf (critU ./ v1, n - 1); elseif (strcmp (tail, "right")) crit = v0 .* chi2inv (1 - alpha, n - 1); power = chi2cdf (crit ./ v1, n - 1); else crit = v0 .* chi2inv (alpha, n - 1); power = chi2cdf (crit ./ v1, n - 1); endif endfunction ## Binomial power calculation function [power, critL, critU] = PowerFunction_P (p0, p1, alpha, ... tail, n, critL, critU) if (nargin < 6) [critL, critU] = getcritP (p0, n, alpha, tail); endif if (strcmp (tail, "both")) power = binocdf (critL - 1, n, p1) + 1 - binocdf (critU, n, p1); elseif (strcmp (tail, "right")) power = 1 - binocdf (critU , n, p1); else power = binocdf (critL - 1, n, p1); endif endfunction ## Regression power calculation function power = PowerFunction_R (r0, r1, alpha, tail, n) ## Compute only for 2-tailed test if (! strcmp (tail, "both")) error ("sampsizepwr: only 2-tailed testing for regression coefficient."); endif ## Set initial search boundaries for power dp_lo = eps; dp_hi = 1 - eps; power = 0.5; ## Compute initial sample size N0 according to P0, POWER and ALPHA N0 = rtestN (r0, r1, power, alpha, tail); ## Find POWER for N0 == N while (N != N0) if (N0 < N) dp_hi = power; power = (power + dp_lo) / 2; N0 = rtestN (r0, r1, power, alpha, tail); else dp_lo = power; power = (power + pd_hi) / 2; N0 = rtestN (r0, r1, power, alpha, tail); endif endwhile endfunction ## Local zero function for "t2" test function N = localfzero (F, N0, ratio) ## Set minN according to ratio if (ratio >= 2) minN = 1; else minN = 2; endif ## Return minN if function gives a value above zero if (F(minN) > 0) N = minN; return; endif ## Make sure that fzero does not try values below minN if (N0 == minN) N0 = N0 + 1; endif ## Find solution if (F(N0) > 0) N = fzero (F, [minN, N0], optimset ('TolX',1e-6)); # N0 is an upper bound else N = fzero (F, N0, optimset ('TolX',1e-6)); # N0 is a starting value endif endfunction ## Demos %!demo %! ## Compute the mean closest to 100 that can be determined to be %! ## significantly different from 100 using a t-test with a sample size %! ## of 60 and a power of 0.8. %! mu1 = sampsizepwr ("t", [100, 10], [], 0.8, 60); %! disp (mu1); %!demo %! ## Compute the sample sizes required to distinguish mu0 = 100 from %! ## mu1 = 110 by a two-sample t-test with a ratio of the larger and the %! ## smaller sample sizes of 1.5 and a power of 0.6. %! [N1,N2] = sampsizepwr ("t2", [100, 10], 110, 0.6, [], "ratio", 1.5) %!demo %! ## Compute the sample size N required to distinguish p=.26 from p=.2 %! ## with a binomial test. The result is approximate, so make a plot to %! ## see if any smaller N values also have the required power of 0.6. %! Napprox = sampsizepwr ("p", 0.2, 0.26, 0.6); %! nn = 1:250; %! pwr = sampsizepwr ("p", 0.2, 0.26, [], nn); %! Nexact = min (nn(pwr >= 0.6)); %! plot(nn,pwr,'b-', [Napprox Nexact],pwr([Napprox Nexact]),'ro'); %! grid on %!demo %! ## The company must test 52 bottles to detect the difference between a mean %! ## volume of 100 mL and 102 mL with a power of 0.80. Generate a power curve %! ## to visualize how the sample size affects the power of the test. %! %! nout = sampsizepwr('t',[100 5],102,0.80); %! nn = 1:100; %! pwrout = sampsizepwr('t',[100 5],102,[],nn); %! %! figure; %! plot (nn, pwrout, "b-", nout, 0.8, "ro") %! title ("Power versus Sample Size") %! xlabel ("Sample Size") %! ylabel ("Power") ## Input validation %!error ... %! out = sampsizepwr ([], [100, 10], [], 0.8, 60); %!error ... %! out = sampsizepwr (3, [100, 10], [], 0.8, 60); %!error ... %! out = sampsizepwr ({"t", "t2"}, [100, 10], [], 0.8, 60); %!error ... %! out = sampsizepwr ("reg", [100, 10], [], 0.8, 60); %!error ... %! out = sampsizepwr ("t", ["a", "e"], [], 0.8, 60); %!error ... %! out = sampsizepwr ("z", 100, [], 0.8, 60); %!error ... %! out = sampsizepwr ("t", 100, [], 0.8, 60); %!error ... %! out = sampsizepwr ("t2", 60, [], 0.8, 60); %!error ... %! out = sampsizepwr ("var", [100, 10], [], 0.8, 60); %!error ... %! out = sampsizepwr ("p", [100, 10], [], 0.8, 60); %!error ... %! out = sampsizepwr ("r", [100, 10], [], 0.8, 60); %!error ... %! [out, N1] = sampsizepwr ("z", [100, 10], [], 0.8, 60); %!error ... %! [out, N1] = sampsizepwr ("t", [100, 10], [], 0.8, 60); %!error ... %! [out, N1] = sampsizepwr ("var", 2, [], 0.8, 60); %!error ... %! [out, N1] = sampsizepwr ("p", 0.1, [], 0.8, 60); %!error ... %! [out, N1] = sampsizepwr ("r", 0.5, [], 0.8, 60); %!error ... %! out = sampsizepwr ("z", [100, 0], [], 0.8, 60); %!error ... %! out = sampsizepwr ("z", [100, -5], [], 0.8, 60); %!error ... %! out = sampsizepwr ("t", [100, 0], [], 0.8, 60); %!error ... %! out = sampsizepwr ("t", [100, -5], [], 0.8, 60); %!error ... %! [out, N1] = sampsizepwr ("t2", [100, 0], [], 0.8, 60); %!error ... %! [out, N1] = sampsizepwr ("t2", [100, -5], [], 0.8, 60); %!error ... %! out = sampsizepwr ("var", 0, [], 0.8, 60); %!error ... %! out = sampsizepwr ("var", -5, [], 0.8, 60); %!error ... %! out = sampsizepwr ("p", 0, [], 0.8, 60); %!error ... %! out = sampsizepwr ("p", 1.2, [], 0.8, 60); %!error ... %! out = sampsizepwr ("r", -1.5, [], 0.8, 60); %!error ... %! out = sampsizepwr ("r", -1, [], 0.8, 60); %!error ... %! out = sampsizepwr ("r", 1.2, [], 0.8, 60); %!error ... %! out = sampsizepwr ("r", 0, [], 0.8, 60); %!error ... %! out = sampsizepwr ("r", 0.2, [], 0.8, 60, "alpha", -0.2); %!error ... %! out = sampsizepwr ("r", 0.2, [], 0.8, 60, "alpha", 0); %!error ... %! out = sampsizepwr ("r", 0.2, [], 0.8, 60, "alpha", 1.5); %!error ... %! out = sampsizepwr ("r", 0.2, [], 0.8, 60, "alpha", "zero"); %!error ... %! out = sampsizepwr ("r", 0.2, [], 0.8, 60, "tail", 1.5); %!error ... %! out = sampsizepwr ("r", 0.2, [], 0.8, 60, "tail", {"both", "left"}); %!error ... %! out = sampsizepwr ("r", 0.2, [], 0.8, 60, "tail", "other"); %!error ... %! out = sampsizepwr ("r", 0.2, [], 0.8, 60, "ratio", "some"); %!error ... %! out = sampsizepwr ("r", 0.2, [], 0.8, 60, "ratio", 0.5); %!error ... %! out = sampsizepwr ("r", 0.2, [], 0.8, 60, "ratio", [2, 1.3, 0.3]); %!error ... %! out = sampsizepwr ("z", [100, 5], [], [], 60); %!error ... %! out = sampsizepwr ("z", [100, 5], 110, [], []); %!error ... %! out = sampsizepwr ("z", [100, 5], [], 0.8, []); %!error ... %! out = sampsizepwr ("z", [100, 5], 110, 0.8, 60); %!error ... %! out = sampsizepwr ("z", [100, 5], "mu", [], 60); %!error ... %! out = sampsizepwr ("var", 5, -1, [], 60); %!error ... %! out = sampsizepwr ("p", 0.8, 1.2, [], 60, "tail", "right"); %!error ... %! out = sampsizepwr ("r", 0.8, 1.2, [], 60); %!error ... %! out = sampsizepwr ("r", 0.8, -1.2, [], 60); %!error ... %! out = sampsizepwr ("z", [100, 5], 110, 1.2); %!error ... %! out = sampsizepwr ("z", [100, 5], 110, 0); %!error ... %! out = sampsizepwr ("z", [100, 5], 110, 0.05, [], "alpha", 0.1); %!error ... %! out = sampsizepwr ("z", [100, 5], [], [0.8, 0.7], [60, 80, 100]); %!error ... %! out = sampsizepwr ("t", [100, 5], 100, 0.8, []); %!error ... %! out = sampsizepwr ("t", [100, 5], 110, 0.8, [], "tail", "left"); %!error ... %! out = sampsizepwr ("t", [100, 5], 90, 0.8, [], "tail", "right"); ## Warning test %!warning ... %! Napprox = sampsizepwr ("p", 0.2, 0.26, 0.6); %!warning ... %! Napprox = sampsizepwr ("p", 0.30, 0.36, 0.8); ## Results validation %!test %! mu1 = sampsizepwr ("t", [100, 10], [], 0.8, 60); %! assert (mu1, 103.67704316, 1e-8); %!test %! [N1,N2] = sampsizepwr ("t2", [100, 10], 110, 0.6, [], "ratio", 1.5); %! assert (N1, 9); %! assert (N2, 14); %!test %! nn = 1:250; %! pwr = sampsizepwr ("p", 0.2, 0.26, [], nn); %! pwr_out = [0, 0.0676, 0.0176, 0.0566, 0.0181, 0.0431, 0.0802, 0.0322]; %! assert (pwr([1:8]), pwr_out, 1e-4 * ones (1,8)); %! pwr_out = [0.59275, 0.6073, 0.62166, 0.6358, 0.6497, 0.6087, 0.6229, 0.6369]; %! assert (pwr([243:end]), pwr_out, 1e-4 * ones (1,8)); %!test %! nout = sampsizepwr ("t", [100, 5], 102, 0.80); %! assert (nout, 52); %!test %! power = sampsizepwr ("t", [20, 5], 25, [], 5, "Tail", "right"); %! assert (power, 0.5797373588621888, 1e-14); %!test %! nout = sampsizepwr ("t", [20, 5], 25, 0.99, [], "Tail", "right"); %! assert (nout, 18); %!test %! p1out = sampsizepwr ("t", [20, 5], [], 0.95, 10, "Tail", "right"); %! assert (p1out, 25.65317979360237, 1e-14); %!test %! pwr = sampsizepwr ("t2", [1.4, 0.2], 1.7, [], 5, "Ratio", 2); %! assert (pwr, 0.716504004686586, 1e-14); %!test %! n = sampsizepwr ("t2", [1.4, 0.2], 1.7, 0.9, []); %! assert (n, 11); %!test %! [n1, n2] = sampsizepwr ("t2", [1.4, 0.2], 1.7, 0.9, [], "Ratio", 2); %! assert ([n1, n2], [8, 16]); statistics-release-1.6.3/inst/shadow9/000077500000000000000000000000001456127120000176625ustar00rootroot00000000000000statistics-release-1.6.3/inst/shadow9/mad.m000066400000000000000000000317171456127120000206120ustar00rootroot00000000000000## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{m} =} mad (@var{x}) ## @deftypefnx {statistics} {@var{m} =} mad (@var{x}, @var{flag}) ## @deftypefnx {statistics} {@var{m} =} mad (@var{x}, @var{flag}, @qcode{"all"}) ## @deftypefnx {statistics} {@var{m} =} mad (@var{x}, @var{flag}, @var{dim}) ## @deftypefnx {statistics} {@var{m} =} mad (@var{x}, @var{flag}, @var{vecdim}) ## ## Compute the mean or median absolute deviation (MAD). ## ## @code{mad (@var{x})} returns the mean absolute deviation of the values in ## @var{x}. @code{mad} treats NaNs as missing values and removes them. ## ## @itemize ## @item ## If @var{x} is a vector, then @code{mad} returns the mean or median absolute ## deviation of the values in @var{X}. ## @item ## If @var{x} is a matrix, then @code{mad} returns the mean or median absolute ## deviation of each column of @var{X}. ## @item ## If @var{x} is an multidimensional array, then @code{mad (@var{x})} operates ## along the first non-singleton dimension of @var{x}. ## @end itemize ## ## @code{mad (@var{x}, @var{flag})} specifies whether to compute the mean ## absolute deviation (@qcode{flag = 0}, the default) or the median absolute ## deviation (@qcode{flag = 1}). Passing an empty variable, defaults to 0. ## ## @code{mad (@var{x}, @var{flag}, @qcode{"all"})} returns the MAD of all the ## elements in @var{x}. ## ## The optional variable @var{dim} forces @code{mad} to operate over the ## specified dimension, which must be a positive integer-valued number. ## Specifying any singleton dimension in @var{x}, including any dimension ## exceeding @code{ndims (@var{x})}, will result in a MAD equal to ## @code{zeros (size (@var{x}))}, while non-finite elements are returned as NaNs. ## ## @code{mad (@var{x}, @var{flag}, @var{vecdim})} returns the MAD over the ## dimensions specified in the vector @var{vecdim}. For example, if @var{x} ## is a 2-by-3-by-4 array, then @code{mad (@var{x}, [1 2])} returns a ## 1-by-1-by-4 array. Each element of the output array is the median of the ## elements on the corresponding page of @var{x}. If @var{vecdim} indexes all ## dimensions of @var{x}, then it is equivalent to @code{mad (@var{x}, "all")}. ## Any dimension in @var{vecdim} greater than @code{ndims (@var{x})} is ignored. ## ## @seealso{median, mean, mode} ## @end deftypefn function m = mad (x, flag=0, varargin) if (nargin < 1 || nargin > 3) print_usage (); endif if (! (isnumeric (x))) error ("mad: X must be numeric."); endif ## Set initial conditions all_flag = false; perm_flag = false; vecdim_flag = false; dim = []; nvarg = numel (varargin); varg_chars = cellfun ("ischar", varargin); szx = sz_out = size (x); ndx = ndims (x); if (isempty (flag)) flag = 0; endif if (! (flag == 0 || flag == 1)) error ("mad: FLAG must be either 0 or 1."); endif ## Process optional char argument. if (any (varg_chars)) for argin = varargin(varg_chars) switch (tolower (argin{:})) case "all" all_flag = true; otherwise print_usage (); endswitch endfor varargin(varg_chars) = []; nvarg = numel (varargin); endif ## Process special cases for in/out size if (nvarg > 0) ## dim or vecdim provided dim = varargin{1}; vecdim_flag = ! isscalar (dim); if (! (isvector (dim) && dim > 0) || any (rem (dim, 1))) error ("mad: DIM must be a positive integer scalar or vector."); endif ## Adjust sz_out, account for possible dim > ndx by appending singletons sz_out(ndx + 1 : max (dim)) = 1; sz_out(dim(dim <= ndx)) = 1; szx(ndx + 1 : max (dim)) = 1; if (vecdim_flag) ## vecdim - try to simplify first dim = sort (dim); if (! all (diff (dim))) error ("mad: VECDIM must contain non-repeating positive integers."); endif ## dims > ndims(x) and dims only one element long don't affect mad sing_dim_x = find (szx != 1); dim(dim > ndx | szx(dim) == 1) = []; if (isempty (dim)) ## No dims left to process, return input as output m = zeros (sz_out); return; elseif (numel (dim) == numel (sing_dim_x) && unique ([dim, sing_dim_x]) == dim) ## If DIMs cover all nonsingleton ndims(x) it's equivalent to "all" ## (check lengths first to reduce unique overhead if not covered) all_flag = true; endif endif else ## Dim not provided. Determine scalar dimension. if (all_flag) ## Special case 'all': Recast input as dim1 vector, process as normal. x = x(:); szx = [length(x), 1]; dim = 1; sz_out = [1, 1]; elseif (isrow (x)) ## Special case row vector: Avoid setting dim to 1. dim = 2; sz_out = [1, 1]; elseif (ndx == 2 && szx == [0, 0]) ## Special case []: Do not apply sz_out(dim)=1 change. dim = 1; sz_out = [1, 1]; else ## General case: Set dim to first non-singleton, contract sz_out along dim (dim = find (szx != 1, 1)) || (dim = 1); sz_out(dim) = 1; endif endif if (isempty (x)) ## Empty input - output NaN m = NaN (sz_out); return; endif if (all (isnan (x)) || all (isinf (x))) ## NaN or Inf input - output NaN m = NaN (sz_out); return; endif if (szx(dim) == 1) ## Operation along singleton dimension - nothing to do m = zeros (sz_out); m(! isfinite (x)) = NaN; return; endif ## Permute dim to simplify all operations along dim1. At func. end ipermute. if (numel (dim) > 1 || (dim != 1 && ! isvector (x))) perm = 1 : ndx; if (! vecdim_flag) ## Move dim to dim 1 perm([1, dim]) = [dim, 1]; x = permute (x, perm); szx([1, dim]) = szx([dim, 1]); dim = 1; else ## Move vecdims to front perm(dim) = []; perm = [dim, perm]; x = permute (x, perm); ## Reshape all vecdims into dim1 num_dim = prod (szx(dim)); szx(dim) = []; szx = [num_dim, ones(1, numel(dim)-1), szx]; x = reshape (x, szx); dim = 1; endif perm_flag = true; endif if (isvector (x)) if (flag) # Compute median absolute deviation ## Checks above ensure either dim1 or dim2 vector x = sort (x, dim); x = x(! isnan (x)); n = length (x); k = floor ((n + 1) / 2); if (mod (n, 2)) ## odd c = x(k); v = sort (abs (x - c)); m = v(k); else ## even c = (x(k) + x(k + 1)) / 2; v = sort (abs (x - c)); m = (v(k) + v(k + 1)) / 2; endif m(sum (isinf (x)) > 0) = Inf; else # Compute mean absolute deviation x = x(! isnan (x)); n = length (x); m = sum (abs (x - (sum (x) / n))) / n; m(sum (isinf (x)) > 0) = Inf; endif else if (flag) # Compute median absolute deviation m = median (abs (x - median(x, dim, "omitnan")), dim, "omitnan"); m(sum (isinf (x), 2) > 0) = Inf; else # Compute mean absolute deviation idx = isnan (x); n = sum (! idx, dim); x1 = x; x1(idx) = 0; m1 = sum (x1, dim) ./ n; x2 = abs (x - m1); x2(idx) = 0; m = sum (x2, dim) ./ n; m(sum (isinf (x), 2) > 0) = Inf; endif endif if (perm_flag) ## Inverse permute back to correct dimensions m = ipermute (m, perm); endif endfunction %!assert (mad (1), 0) %!assert (mad (1,1), 0) %!assert (mad (1,0,3), 0) %!assert (mad (1,1,3), 0) %!assert (mad (1,[],5), 0) %!assert (mad ([1,2,3]), 2/3) %!assert (mad ([1,2,3],[]), 2/3) %!assert (mad ([1,2,3],0), 2/3) %!assert (mad ([1,2,3],1), 1) %!assert (mad ([1,2,3],0,2), 2/3) %!assert (mad ([1,2,3],[],2), 2/3) %!assert (mad ([1,2,3],1,2), 1) %!assert (mad ([1,2,3],0,1), zeros (1,3)) %!assert (mad ([1,2,3],1,1), zeros (1,3)) %!assert (mad ([1,2,3]',0,2), zeros (3,1)) %!assert (mad ([1,2,3]',1,2), zeros (3,1)) %!assert (mad ([1,2,3]',0,1), 2/3) %!assert (mad ([1,2,3]',1,1), 1) %!assert (mad ([1,2,3]',1,1), 1) ## Test vector or matrix input with scalar DIM %!test %! A = [57, 59, 60, 100, 59, 58, 57, 58, 300, 61, 62, 60, 62, 58, 57]; %! AA = [A;2*A;3*A]; %! m0 = [38.000, 39.333, 40.000, 66.667, 39.333, 38.667, 38.000, 38.667, ... %! 200.000, 40.667, 41.333, 40.000, 41.333, 38.667, 38.000]; %! m1 = [32.569;65.138; 97.707]; %! %! assert (mad (AA), m0, 1e-3); %! assert (mad (AA,1), A); %! assert (mad (AA,1,1), A); %! assert (mad (AA,0,2), m1, 1e-3); %! assert (mad (AA,1,2), [2;4;6]); %! assert (mad (A,0,1), zeros (size (A))); %! assert (mad (A,1,1), zeros (size (A))); ## Test n-dimensional input and optional arguments "all", VECDIM %!test %! x = repmat ([2 2.1 2.2 2 NaN; 3 1 2 NaN 5; 1 1.1 1.4 5 3], [1, 1, 4]); %! m0 = repmat ([0.6667, 0.4667, 0.3111, 1.5, 1], [1, 1, 4]); %! m1 = repmat ([1, 0.1, 0.2, 1.5, 1], [1, 1, 4]); %! assert (mad (x), m0, 1e-4); %! assert (mad (x, 1), m1, 1e-14); %! assert (mad (x, [], [1, 2]), 1.0036 * ones(1,1,4), 1e-4) %! assert (mad (x, 1, [1, 2]), 0.9 * ones(1,1,4), 1e-14) %! assert (mad (x, 0, [1, 3]), m0(1,:,1), 1e-4) %! assert (mad (x, 1, [1, 3]), m1(1,:,1), 1e-14) %! assert (mad (x, 0, [2, 3]), [0.075; 1.25; 1.36], 1e-14) %! assert (mad (x, 1, [2, 3]), [0.05; 1; 0.4], 1e-14) %! assert (mad (x, 0, [1, 2, 3]) == mad (x, 0, "All")) %! assert (mad (x, 1, [1, 2, 3]) == mad (x, 1, "All")) ## Test dimension indexing with vecdim in n-dimensional arrays %!test %! x = repmat ([1:20;6:25], [5 2 6 3]); %! assert (size (mad (x, [], [3 2])), [10 1 1 3]); %! assert (size (mad (x, 0, [1 2])), [1 1 6 3]); %! assert (size (mad (x, 1, [1 2 4])), [1 1 6]); %! assert (size (mad (x, [], [1 4 3])), [1 40]); %! assert (size (mad (x, 1, [1 2 3 4])), [1 1]); ## Test exceeding dimensions %!assert (mad (ones (2,2), 0, 3), zeros (2,2)) %!assert (mad (ones (2,2,2), 1, 99), zeros (2,2,2)) %!assert (mad (magic (3), 1, 3), zeros (3)) %!assert (mad (magic (3), 1, [1 3]), [1, 4, 1]) %!assert (mad (magic (3), 1, [1 99]), [1, 4, 1]) ## Test empty, NaN, Inf inputs %!assert (mad ([]), NaN) %!assert (mad ([], 1), NaN) %!assert (mad (NaN), NaN) %!assert (mad (NaN, 1), NaN) %!assert (mad (Inf), NaN) %!assert (mad (Inf, 1), NaN) %!assert (mad (-Inf), NaN) %!assert (mad (-Inf, 1), NaN) %!assert (mad ([-Inf Inf]), NaN) %!assert (mad ([-Inf Inf], 1), NaN) %!assert (mad ([3 Inf]), Inf) %!assert (mad ([3 4 Inf]), Inf) %!assert (mad ([3 4 Inf], 1), Inf) %!assert (mad ([Inf 3 4]), Inf) %!assert (mad ([Inf 3 4], 1), Inf) %!assert (mad ([Inf 3 Inf]), Inf) %!assert (mad ([Inf 3 Inf], 1), Inf) %!assert (mad ([1 2; 3 Inf]), [1 Inf]) %!assert (mad ([1 2; 3 Inf], 1), [1 Inf]) %!assert (mad ([]), NaN) %!assert (mad (ones(1,0)), NaN) %!assert (mad (ones(0,1)), NaN) %!assert (mad ([], 0, 1), NaN(1,0)) %!assert (mad ([], 0, 2), NaN(0,1)) %!assert (mad ([], 0, 3), NaN(0,0)) %!assert (mad (ones(1,0), 0, 1), NaN(1,0)) %!assert (mad (ones(1,0), 0, 2), NaN(1,1)) %!assert (mad (ones(1,0), 0, 3), NaN(1,0)) %!assert (mad (ones(0,1), 0, 1), NaN(1,1)) %!assert (mad (ones(0,1), 0, 2), NaN(0,1)) %!assert (mad (ones(0,1), 0, 3), NaN(0,1)) %!assert (mad (ones(0,1,0,1), 0, 1), NaN(1,1,0)) %!assert (mad (ones(0,1,0,1), 0, 2), NaN(0,1,0)) %!assert (mad (ones(0,1,0,1), 0, 3), NaN(0,1,1)) %!assert (mad (ones(0,1,0,1), 0, 4), NaN(0,1,0)) ## Test complex inputs (should sort by abs(a)) %!assert (mad([1 3 3i 2 1i]), 1.5297, 1e-4) %!assert (mad([1 3 3i 2 1i], 1), 1) %!assert (mad([1 2 4i; 3 2i 4]), [1, 1.4142, 2.8284], 1e-4) %!assert (mad([1 2 4i; 3 2i 4], 1), [1, 1.4142, 2.8284], 1e-4) %!assert (mad([1 2 4i; 3 2i 4], 1, 2), [1; 1]) %!assert (mad([1 2 4i; 3 2i 4], 0, 2), [1.9493; 1.8084], 1e-4) ## Test input case insensitivity %!assert (mad ([1 2 3], 0, "aLL"), 2/3) %!assert (mad ([1 2 3], 1, "aLL"), 1) ## Test input validation %!error mad () %!error mad (1, 2, 3, 4) %!error mad ("text") %!error mad ({2 3 4}) %!error mad (1, "all", 3) %!error mad (1, "b") %!error mad (1, 1, "foo") %!error mad (1, [] ,ones (2,2)) %!error mad (1, [], 1.5) %!error mad (1, [], 0) %!error mad ([1 2 3], [], [-1 1]) %!error mad(1, [], [1 2 2]) statistics-release-1.6.3/inst/shadow9/mean.m000066400000000000000000000517101456127120000207640ustar00rootroot00000000000000## Copyright (C) 2022 Andreas Bertsatos ## Copyright (C) 2022 Kai Torben Ohlhus ## Copyright (C) 2023 Nicholas Jankowski ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{m} =} mean (@var{x}) ## @deftypefnx {statistics} {@var{m} =} mean (@var{x}, "all") ## @deftypefnx {statistics} {@var{m} =} mean (@var{x}, @var{dim}) ## @deftypefnx {statistics} {@var{m} =} mean (@var{x}, @var{vecdim}) ## @deftypefnx {statistics} {@var{m} =} mean (@dots{}, @var{outtype}) ## @deftypefnx {statistics} {@var{m} =} mean (@dots{}, @var{nanflag}) ## ## Compute the mean of the elements of @var{x}. ## ## @itemize ## @item ## If @var{x} is a vector, then @code{mean(@var{x})} returns the ## mean of the elements in @var{x} defined as ## @tex ## $$ {\rm mean}(x) = \bar{x} = {1\over N} \sum_{i=1}^N x_i $$ ## ## @end tex ## @ifnottex ## ## @example ## mean (@var{x}) = SUM_i @var{x}(i) / N ## @end example ## ## @end ifnottex ## @noindent ## where @math{N} is the length of the @var{x} vector. ## ## @item ## If @var{x} is a matrix, then @code{mean(@var{x})} returns a row vector ## with the mean of each columns in @var{x}. ## ## @item ## If @var{x} is a multidimensional array, then @code{mean(@var{x})} ## operates along the first nonsingleton dimension of @var{x}. ## @end itemize ## ## @code{mean (@var{x}, @var{dim})} returns the mean along the operating ## dimension @var{dim} of @var{x}. For @var{dim} greater than ## @code{ndims (@var{x})}, then @var{m} = @var{x}. ## ## @code{mean (@var{x}, @var{vecdim})} returns the mean over the ## dimensions specified in the vector @var{vecdim}. For example, if @var{x} ## is a 2-by-3-by-4 array, then @code{mean (@var{x}, [1 2])} returns a ## 1-by-1-by-4 array. Each element of the output array is the mean of the ## elements on the corresponding page of @var{x}. If @var{vecdim} indexes all ## dimensions of @var{x}, then it is equivalent to @code{mean (@var{x}, "all")}. ## Any dimension in @var{vecdim} greater than @code{ndims (@var{x})} is ignored. ## ## @code{mean (@var{x}, "all")} returns the mean of all the elements in @var{x}. ## The optional flag "all" cannot be used together with @var{dim} or ## @var{vecdim} input arguments. ## ## @code{mean (@dots{}, @var{outtype})} returns the mean with a specified data ## type, using any of the input arguments in the previous syntaxes. ## @var{outtype} can take the following values: ## @itemize ## @item "default" ## Output is of type double, unless the input is single in which case the output ## is of type single. ## ## @item "double" ## Output is of type double. ## ## @item "native". ## Output is of the same type as the input (@code{class (@var{x})}), unless the ## input is logical in which case the output is of type double or a character ## array in which case an error is produced. ## @end itemize ## ## @code{mean (@dots{}, @var{nanflag})} specifies whether to exclude NaN values ## from the calculation, using any of the input argument combinations in ## previous syntaxes. By default, NaN values are included in the calculation ## (@var{nanflag} has the value "includenan"). To exclude NaN values, set the ## value of @var{nanflag} to "omitnan". ## ## @seealso{trimmean, median, mad, mode} ## @end deftypefn function m = mean (x, varargin) if (nargin < 1 || nargin > 4) print_usage (); endif ## Set initial conditions all_flag = false; omitnan = false; out_flag = false; nvarg = numel (varargin); varg_chars = cellfun ("ischar", varargin); outtype = "default"; szx = size (x); ndx = ndims (x); if (nvarg > 1 && ! varg_chars(2:end)) ## Only first varargin can be numeric print_usage (); endif ## Process any other char arguments. if (any (varg_chars)) for argin = varargin(varg_chars) switch (lower (argin{:})) case "all" all_flag = true; case "omitnan" omitnan = true; case "includenan" omitnan = false; case "default" if (out_flag) error ("mean: only one OUTTYPE can be specified.") endif if (isa (x, "single")) outtype = "single"; else outtype = "double"; endif out_flag = true; case "native" outtype = class (x); if (out_flag) error ("mean: only one OUTTYPE can be specified.") elseif (strcmp (outtype, "logical")) outtype = "double"; elseif (strcmp (outtype, "char")) error ("mean: OUTTYPE 'native' cannot be used with char type inputs."); endif out_flag = true; case "double" if (out_flag) error ("mean: only one OUTTYPE can be specified.") endif outtype = "double"; out_flag = true; otherwise print_usage (); endswitch endfor varargin(varg_chars) = []; nvarg = numel (varargin); endif if (strcmp (outtype, "default")) if (isa (x, "single")) outtype = "single"; else outtype = "double"; endif endif if ((nvarg > 1) || ((nvarg == 1) && ! (isnumeric (varargin{1})))) ## After trimming char inputs can only be one varargin left, must be numeric print_usage (); endif if (! (isnumeric (x) || islogical (x) || ischar (x))) error ("mean: X must be either a numeric, boolean, or character array."); endif ## Process special cases for input/output sizes if (nvarg == 0) ## Single numeric input argument, no dimensions given. if (all_flag) x = x(:); if (omitnan) x = x(! isnan (x)); endif if (any (isa (x, {"int64", "uint64"}))) m = int64_mean (x, 1, numel (x), outtype); else m = sum (x, "double") ./ numel (x); endif else ## Find the first non-singleton dimension. (dim = find (szx != 1, 1)) || (dim = 1); n = szx(dim); if (omitnan) idx = isnan (x); n = sum (! idx, dim); x(idx) = 0; endif if (any (isa (x, {"int64", "uint64"}))) m = int64_mean (x, dim, n, outtype); else m = sum (x, dim, "double") ./ n; endif endif else ## Two numeric input arguments, dimensions given. Note scalar is vector! vecdim = varargin{1}; if (isempty (vecdim) || ! (isvector (vecdim) && all (vecdim > 0) && all (rem (vecdim, 1)==0))) error ("mean: DIM must be a positive integer scalar or vector."); endif if (ndx == 2 && isempty (x) && szx == [0,0]) ## FIXME: This special case handling could be removed once sum ## compatibly handles all sizes of empty inputs. sz_out = szx; sz_out (vecdim(vecdim <= ndx)) = 1; m = NaN (sz_out); else if (isscalar (vecdim)) if (vecdim > ndx) m = x; else n = szx(vecdim); if (omitnan) nanx = isnan (x); n = sum (! nanx, vecdim); x(nanx) = 0; endif if (any (isa (x, {"int64", "uint64"}))) m = int64_mean (x, vecdim, n, outtype); else m = sum (x, vecdim, "double") ./ n; endif endif else vecdim = sort (vecdim); if (! all (diff (vecdim))) error ("mean: VECDIM must contain non-repeating positive integers."); endif ## Ignore dimensions in VECDIM larger than actual array vecdim(find (vecdim > ndims (x))) = []; if (isempty (vecdim)) m = x; else ## Calculate permutation vector remdims = 1 : ndx; # All dimensions remdims(vecdim) = []; # Delete dimensions specified by vecdim nremd = numel (remdims); ## If all dimensions are given, it is equivalent to 'all' flag if (nremd == 0) x = x(:); if (omitnan) x = x(! isnan (x)); endif if (any (isa (x, {"int64", "uint64"}))) m = int64_mean (x, 1, numel (x), outtype); else m = sum (x, "double") ./ numel (x); endif else ## Permute to push vecdims to back perm = [remdims, vecdim]; x = permute (x, perm); ## Reshape to squash all vecdims in final dimension sznew = [szx(remdims), prod(szx(vecdim))]; x = reshape (x, sznew); ## Calculate mean on final dimension dim = nremd + 1; if (omitnan) nanx = isnan (x); x(nanx) = 0; n = sum (! nanx, dim); else n = sznew(dim); endif if (any (isa (x, {"int64", "uint64"}))) m = int64_mean (x, dim, n, outtype); else m = sum (x, dim, "double") ./ n; endif ## Inverse permute back to correct dimensions m = ipermute (m, perm); endif endif endif endif endif ## Convert output if necessary if (! strcmp (class (m), outtype)) if (! islogical (x)) m = feval (outtype, m); endif endif endfunction function m = int64_mean (x, dim, n, outtype) ## Avoid int overflow in large ints. Smaller ints processed as double ## avoids overflow. Large int64 values as double can have floating pt error. ## Use integer math and remainder correction to avoid this. if (any (abs (x(:)) >= flintmax / n)) rmdr = double (rem (x, n)) / n; rmdr_hilo = logical (int8 (rmdr)); # Integer rounding direction indicator ## Native int summation to prevent double precision error, ## then add back in lost round-up/down remainders. m = sum (x/n, dim, "native"); ## rmdr.*!rmdr_hilo = remainders that were rounded down in abs val ## signs retained, can be summed and added back. ## rmdr.*rmdr_hilo = remainders that were rounded up in abs val. ## need to add back difference between 1 and rmdr, retaining sign. rmdr = sum (rmdr .* !rmdr_hilo, dim) - ... sum ((1 - abs (rmdr)) .* rmdr_hilo .* sign(rmdr), dim); if (any (abs (m(:)) >= flintmax)) if (any (strcmp (outtype, {"int64", "uint64"}))) m += rmdr; else m = double (m) + rmdr; endif else m = double(m) + rmdr; switch (outtype) case "int64" m = int64 (m); case "uint64" m = uint64 (m); endswitch endif else m = double (sum (x, dim, "native")) ./ n; endif endfunction %!test %! x = -10:10; %! y = x'; %! z = [y, y+10]; %! assert (mean (x), 0); %! assert (mean (y), 0); %! assert (mean (z), [0, 10]); %!assert (mean (magic (3), 1), [5, 5, 5]) %!assert (mean (magic (3), 2), [5; 5; 5]) %!assert (mean (logical ([1 0 1 1])), 0.75) %!assert (mean (single ([1 0 1 1])), single (0.75)) %!assert (mean ([1 2], 3), [1 2]) ## Test outtype option %!test %! in = [1 2 3]; %! out = 2; %! assert (mean (in, "default"), mean (in)); %! assert (mean (in, "default"), out); %! assert (mean (in, "double"), out); %! assert (mean (in, "native"), out); %!test %! in = single ([1 2 3]); %! out = 2; %! assert (mean (in, "default"), mean (in)); %! assert (mean (in, "default"), single (out)); %! assert (mean (in, "double"), out); %! assert (mean (in, "native"), single (out)); %!test %! in = logical ([1 0 1]); %! out = 2/3; %! assert (mean (in, "default"), mean (in), eps); %! assert (mean (in, "default"), out, eps); %! assert (mean (in, "double"), out, eps); %! assert (mean (in, "native"), out, eps); %!test %! in = char ("ab"); %! out = 97.5; %! assert (mean (in, "default"), mean (in), eps); %! assert (mean (in, "default"), out, eps); %! assert (mean (in, "double"), out, eps); %!test %! in = uint8 ([1 2 3]); %! out = 2; %! assert (mean (in, "default"), mean (in)); %! assert (mean (in, "default"), out); %! assert (mean (in, "double"), out); %! assert (mean (in, "native"), uint8 (out)); %!test %! in = uint8 ([0 1 2 3]); %! out = 1.5; %! out_u8 = 2; %! assert (mean (in, "default"), mean (in), eps); %! assert (mean (in, "default"), out, eps); %! assert (mean (in, "double"), out, eps); %! assert (mean (in, "native"), uint8 (out_u8)); %! assert (class (mean (in, "native")), "uint8"); %!test # internal sum exceeding intmax %! in = uint8 ([3 141 141 255]); %! out = 135; %! assert (mean (in, "default"), mean (in)); %! assert (mean (in, "default"), out); %! assert (mean (in, "double"), out); %! assert (mean (in, "native"), uint8 (out)); %! assert (class (mean (in, "native")), "uint8"); %!test # fractional answer with internal sum exceeding intmax %! in = uint8 ([1 141 141 255]); %! out = 134.5; %! out_u8 = 135; %! assert (mean (in, "default"), mean (in)); %! assert (mean (in, "default"), out); %! assert (mean (in, "double"), out); %! assert (mean (in, "native"), uint8 (out_u8)); %! assert (class (mean (in, "native")), "uint8"); %!test <54567> # large int64 sum exceeding intmax and double precision limit %! in_same = uint64 ([intmax("uint64") intmax("uint64")-2]); %! out_same = intmax ("uint64")-1; %! in_opp = int64 ([intmin("int64"), intmax("int64")-1]); %! out_opp = -1; %! in_neg = int64 ([intmin("int64") intmin("int64")+2]); %! out_neg = intmin ("int64")+1; %! %! ## both positive %! assert (mean (in_same, "default"), mean (in_same)); %! assert (mean (in_same, "default"), double (out_same)); %! assert (mean (in_same, "double"), double (out_same)); %! assert (mean (in_same, "native"), uint64 (out_same)); %! assert (class (mean (in_same, "native")), "uint64"); %! %! ## opposite signs %! assert (mean (in_opp, "default"), mean (in_opp)); %! assert (mean (in_opp, "default"), double (out_opp)); %! assert (mean (in_opp, "double"), double (out_opp)); %! assert (mean (in_opp, "native"), int64 (out_opp)); %! assert (class (mean (in_opp, "native")), "int64"); %! %! ## both negative %! assert (mean (in_neg, "default"), mean (in_neg)); %! assert (mean (in_neg, "default"), double(out_neg)); %! assert (mean (in_neg, "double"), double(out_neg)); %! assert (mean (in_neg, "native"), int64(out_neg)); %! assert (class (mean (in_neg, "native")), "int64"); ## Additional tests int64 and double precision limits %!test <54567> %! in = [(intmin('int64')+5), (intmax('int64'))-5]; %! assert (mean (in, "native"), int64(-1)); %! assert (class (mean (in, "native")), "int64"); %! assert (mean (double(in)), double(0) ); %! assert (mean (in), double(-0.5) ); %! assert (mean (in, "default"), double(-0.5) ); %! assert (mean (in, "double"), double(-0.5) ); %! assert (mean (in, "all", "native"), int64(-1)); %! assert (mean (in, 2, "native"), int64(-1)); %! assert (mean (in, [1 2], "native"), int64(-1)); %! assert (mean (in, [2 3], "native"), int64(-1)); %! assert (mean ([intmin("int64"), in, intmax("int64")]), double(-0.5)) %! assert (mean ([in; int64([1 3])], 2, "native"), int64([-1; 2])); ## Test input and optional arguments "all", DIM, "omitnan". %!test %! x = [-10:10]; %! y = [x;x+5;x-5]; %! assert (mean (x), 0); %! assert (mean (y, 2), [0, 5, -5]'); %! assert (mean (y, "all"), 0); %! y(2,4) = NaN; %! assert (mean (y', "omitnan"), [0 5.35 -5]); %! z = y + 20; %! assert (mean (z, "all"), NaN); %! assert (mean (z, "all", "includenan"), NaN); %! assert (mean (z, "all", "omitnan"), 20.03225806451613, 4e-14); %! m = [20 NaN 15]; %! assert (mean (z'), m); %! assert (mean (z', "includenan"), m); %! m = [20 25.35 15]; %! assert (mean (z', "omitnan"), m); %! assert (mean (z, 2, "omitnan"), m'); %! assert (mean (z, 2, "native", "omitnan"), m'); %! assert (mean (z, 2, "omitnan", "native"), m'); ## Test boolean input %!test %! assert (mean (true, "all"), 1); %! assert (mean (false), 0); %! assert (mean ([true false true]), 2/3, 4e-14); %! assert (mean ([true false true], 1), [1 0 1]); %! assert (mean ([true false NaN], 1), [1 0 NaN]); %! assert (mean ([true false NaN], 2), NaN); %! assert (mean ([true false NaN], 2, "omitnan"), 0.5); %! assert (mean ([true false NaN], 2, "omitnan", "native"), 0.5); ## Test char inputs %!assert (mean ("abc"), double (98)) %!assert (mean ("ab"), double (97.5), eps) %!assert (mean ("abc", "double"), double (98)) %!assert (mean ("abc", "default"), double (98)) ## Test NaN inputs %!test %! x = magic (4); %! x([2, 9:12]) = NaN; %! assert (mean (x), [NaN 8.5, NaN, 8.5], eps); %! assert (mean (x,1), [NaN 8.5, NaN, 8.5], eps); %! assert (mean (x,2), NaN(4,1), eps); %! assert (mean (x,3), x, eps); %! assert (mean (x, 'omitnan'), [29/3, 8.5, NaN, 8.5], eps); %! assert (mean (x, 1, 'omitnan'), [29/3, 8.5, NaN, 8.5], eps); %! assert (mean (x, 2, 'omitnan'), [31/3; 9.5; 28/3; 19/3], eps); %! assert (mean (x, 3, 'omitnan'), x, eps); ## Test empty inputs %!assert (mean ([]), NaN(1,1)) %!assert (mean (single([])), NaN(1,1,"single")) %!assert (mean ([], 1), NaN(1,0)) %!assert (mean ([], 2), NaN(0,1)) %!assert (mean ([], 3), NaN(0,0)) %!assert (mean (ones(1,0)), NaN(1,1)) %!assert (mean (ones(1,0), 1), NaN(1,0)) %!assert (mean (ones(1,0), 2), NaN(1,1)) %!assert (mean (ones(1,0), 3), NaN(1,0)) %!assert (mean (ones(0,1)), NaN(1,1)) %!assert (mean (ones(0,1), 1), NaN(1,1)) %!assert (mean (ones(0,1), 2), NaN(0,1)) %!assert (mean (ones(0,1), 3), NaN(0,1)) %!assert (mean (ones(0,1,0)), NaN(1,1,0)) %!assert (mean (ones(0,1,0), 1), NaN(1,1,0)) %!assert (mean (ones(0,1,0), 2), NaN(0,1,0)) %!assert (mean (ones(0,1,0), 3), NaN(0,1,1)) %!assert (mean (ones(0,0,1,0)), NaN(1,0,1,0)) %!assert (mean (ones(0,0,1,0), 1), NaN(1,0,1,0)) %!assert (mean (ones(0,0,1,0), 2), NaN(0,1,1,0)) %!assert (mean (ones(0,0,1,0), 3), NaN(0,0,1,0)) ## Test dimension indexing with vecdim in N-dimensional arrays %!test %! x = repmat ([1:20;6:25], [5 2 6 3]); %! assert (size (mean (x, [3 2])), [10 1 1 3]); %! assert (size (mean (x, [1 2])), [1 1 6 3]); %! assert (size (mean (x, [1 2 4])), [1 1 6]); %! assert (size (mean (x, [1 4 3])), [1 40]); %! assert (size (mean (x, [1 2 3 4])), [1 1]); ## Test exceeding dimensions %!assert (mean (ones (2,2), 3), ones (2,2)) %!assert (mean (ones (2,2,2), 99), ones (2,2,2)) %!assert (mean (magic (3), 3), magic (3)) %!assert (mean (magic (3), [1 3]), [5, 5, 5]) %!assert (mean (magic (3), [1 99]), [5, 5, 5]) ## Test results with vecdim in N-dimensional arrays and "omitnan" %!test %! x = repmat ([1:20;6:25], [5 2 6 3]); %! m = repmat ([10.5;15.5], [5 1 1 3]); %! assert (mean (x, [3 2]), m, 4e-14); %! x(2,5,6,3) = NaN; %! m(2,1,1,3) = NaN; %! assert (mean (x, [3 2]), m, 4e-14); %! m(2,1,1,3) = 15.52301255230125; %! assert (mean (x, [3 2], "omitnan"), m, 4e-14); ## Test input case insensitivity %!assert (mean ([1 2 3], "aLL"), 2) %!assert (mean ([1 2 3], "OmitNan"), 2) %!assert (mean ([1 2 3], "DOUBle"), 2) ## Test limits of single precision summation limits on each code path %!assert <*63848> (mean (ones (80e6, 1, "single")), 1, eps) %!assert <*63848> (mean (ones (80e6, 1, "single"), "all"), 1, eps) %!assert <*63848> (mean (ones (80e6, 1, "single"), 1), 1, eps) %!assert <*63848> (mean (ones (80e6, 1, "single"), [1 2]), 1, eps) %!assert <*63848> (mean (ones (80e6, 1, "single"), [1 3]), 1, eps) ## Test limits of double precision summation %!assert <63848> (mean ([flintmax("double"), ones(1, 2^8-1, "double")]), ... %! 35184372088833-1/(2^8), eps(35184372088833)) ## Test input validation %!error mean () %!error mean (1, 2, 3) %!error mean (1, 2, 3, 4) %!error mean (1, "all", 3) %!error mean (1, "b") %!error mean (1, 1, "foo") %!error mean ("abc", "native") %!error mean ({1:5}) %!error mean (1, ones (2,2)) %!error mean (1, 1.5) %!error mean (1, 0) %!error mean (1, []) %!error mean (1, -1) %!error mean (1, -1.5) %!error mean (1, NaN) %!error mean (1, Inf) %!error mean (repmat ([1:20;6:25], [5 2]), -1) %!error mean (repmat ([1:5;5:9], [5 2]), [1 -1]) %!error mean (1, ones(1,0)) %!error mean (1, [2 2]) statistics-release-1.6.3/inst/shadow9/median.m000066400000000000000000000576421456127120000213130ustar00rootroot00000000000000## Copyright (C) 2022-2023 Andreas Bertsatos ## Copyright (C) 2023 Nicholas Jankowski ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{m} =} median (@var{x}) ## @deftypefnx {statistics} {@var{m} =} median (@var{x}, "all") ## @deftypefnx {statistics} {@var{m} =} median (@var{x}, @var{dim}) ## @deftypefnx {statistics} {@var{m} =} median (@var{x}, @var{vecdim}) ## @deftypefnx {statistics} {@var{m} =} median (@dots{}, @var{outtype}) ## @deftypefnx {statistics} {@var{m} =} median (@dots{}, @var{nanflag}) ## ## Compute the median value of the elements of @var{x}. ## ## When the elements of @var{x} are sorted, say ## @code{@var{s} = sort (@var{x})}, the median is defined as ## @tex ## $$ {\rm median} (x) = \cases{s(\lceil N/2\rceil), & $N$ odd; ## \cr (s(N/2)+s(N/2+1))/2, & $N$ even.} $$ ## ## @end tex ## @ifnottex ## ## @example ## @group ## | @var{s}(ceil (N/2)) N odd ## median (@var{x}) = | ## | (@var{s}(N/2) + @var{s}(N/2+1))/2 N even ## @end group ## @end example ## ## @end ifnottex ## @noindent ## where @math{N} is the number of elements of @var{x}. ## ## If @var{x} is an array, then @code{median (@var{x})} operates along the first ## non-singleton dimension of @var{x}. ## ## The optional variable @var{dim} forces @code{median} to operate over the ## specified dimension, which must be a positive integer-valued number. ## Specifying any singleton dimension in @var{x}, including any dimension ## exceeding @code{ndims (@var{x})}, will result in a median equal to @var{x}. ## ## @code{median (@var{x}, @var{vecdim})} returns the median over the ## dimensions specified in the vector @var{vecdim}. For example, if @var{x} ## is a 2-by-3-by-4 array, then @code{median (@var{x}, [1 2])} returns a ## 1-by-1-by-4 array. Each element of the output array is the median of the ## elements on the corresponding page of @var{x}. If @var{vecdim} indexes all ## dimensions of @var{x}, then it is equivalent to ## @code{median (@var{x}, "all")}. Any dimension in @var{vecdim} greater than ## @code{ndims (@var{x})} is ignored. ## ## @code{median (@var{x}, "all")} returns the median of all the elements in ## @var{x}. The optional flag "all" cannot be used together with @var{dim} or ## @var{vecdim} input arguments. ## ## @code{median (@dots{}, @var{outtype})} returns the median with a specified ## data type, using any of the input arguments in the previous syntaxes. ## @var{outtype} can take the following values: ## @itemize ## @item @qcode{"default"} ## Output is of type double, unless the input is single in which case the ## output is of type single. ## ## @item @qcode{"double"} ## Output is of type double. ## ## @item @qcode{"native"} ## Output is of the same type as the input (@code{class (@var{x})}), unless the ## input is logical in which case the output is of type double. ## @end itemize ## ## The optional variable @var{nanflag} specifies whether to include or exclude ## NaN values from the calculation using any of the previously specified input ## argument combinations. The default value for @var{nanflag} is ## @qcode{"includenan"} which keeps NaN values in the calculation. To ## exclude NaN values set the value of @var{nanflag} to @qcode{"omitnan"}. ## The output will still contain NaN values if @var{x} consists of all NaN ## values in the operating dimension. ## ## @seealso{mean, mad, mode} ## @end deftypefn function m = median (x, varargin) if (nargin < 1 || nargin > 4) print_usage (); endif if (! (isnumeric (x) || islogical (x))) error ("median: X must be either numeric or logical."); endif ## Set initial conditions all_flag = false; omitnan = false; perm_flag = false; out_flag = false; vecdim_flag = false; dim = []; nvarg = numel (varargin); varg_chars = cellfun ("ischar", varargin); szx = sz_out = size (x); ndx = ndims (x); outtype = class (x); if (nvarg > 1 && ! varg_chars(2:end)) ## Only first varargin can be numeric print_usage (); endif ## Process any other char arguments. if (any (varg_chars)) for argin = varargin(varg_chars) switch (tolower (argin{:})) case "all" all_flag = true; case "omitnan" omitnan = true; case "includenan" omitnan = false; case "native" if (out_flag) error ("median: only one OUTTYPE can be specified.") endif if (strcmp (outtype, "logical")) outtype = "double"; endif out_flag = true; case "default" if (out_flag) error ("median: only one OUTTYPE can be specified.") endif if (! strcmp (outtype, "single")) outtype = "double"; endif out_flag = true; case "double" if (out_flag) error ("median: only one OUTTYPE can be specified.") endif outtype = "double"; out_flag = true; otherwise print_usage (); endswitch endfor varargin(varg_chars) = []; nvarg = numel (varargin); endif if ((nvarg == 1 && ! isnumeric (varargin{1})) || nvarg > 1) ## After trimming char inputs should only be one numeric varargin left print_usage (); endif ## Process special cases for in/out size if (nvarg > 0) ## dim or vecdim provided if (all_flag) error ("median: 'all' cannot be used with DIM or VECDIM options."); endif dim = varargin{1}; vecdim_flag = ! isscalar (dim); if (! (isvector (dim) && dim > 0) || any (rem (dim, 1))) error ("median: DIM must be a positive integer scalar or vector."); endif ## Adjust sz_out, account for possible dim > ndx by appending singletons sz_out(ndx + 1 : max (dim)) = 1; sz_out(dim(dim <= ndx)) = 1; szx(ndx + 1 : max (dim)) = 1; if (vecdim_flag) ## vecdim - try to simplify first dim = sort (dim); if (! all (diff (dim))) error ("median: VECDIM must contain non-repeating positive integers."); endif ## dims > ndims(x) and dims only one element long don't affect median sing_dim_x = find (szx != 1); dim(dim > ndx | szx(dim) == 1) = []; if (isempty (dim)) ## No dims left to process, return input as output if (! strcmp (class (x), outtype)) m = feval (outtype, x); # convert to outtype else m = x; endif return; elseif (numel (dim) == numel (sing_dim_x) && unique ([dim, sing_dim_x]) == dim) ## If DIMs cover all nonsingleton ndims(x) it's equivalent to "all" ## (check lengths first to reduce unique overhead if not covered) all_flag = true; endif endif else ## Dim not provided. Determine scalar dimension. if (all_flag) ## Special case 'all': Recast input as dim1 vector, process as normal. x = x(:); szx = [length(x), 1]; dim = 1; sz_out = [1 1]; elseif (isrow (x)) ## Special case row vector: Avoid setting dim to 1. dim = 2; sz_out = [1, 1]; elseif (ndx == 2 && szx == [0, 0]) ## Special case []: Do not apply sz_out(dim)=1 change. dim = 1; sz_out = [1, 1]; else ## General case: Set dim to first non-singleton, contract sz_out along dim (dim = find (szx != 1, 1)) || (dim = 1); sz_out(dim) = 1; endif endif if (isempty (x)) ## Empty input - output NaN or class equivalent in pre-determined size switch (outtype) case {"double", "single"} m = NaN (sz_out, outtype); case ("logical") m = false (sz_out); otherwise m = cast (NaN (sz_out), outtype); endswitch return; endif if (szx(dim) == 1) ## Operation along singleton dimension - nothing to do if (! strcmp (class (x), outtype)) m = feval (outtype, x); # convert to outtype else m = x; endif return; endif ## Permute dim to simplify all operations along dim1. At func. end ipermute. if (numel (dim) > 1 || (dim != 1 && ! isvector (x))) perm = 1 : ndx; if (! vecdim_flag) ## Move dim to dim 1 perm([1, dim]) = [dim, 1]; x = permute (x, perm); szx([1, dim]) = szx([dim, 1]); dim = 1; else ## Move vecdims to front perm(dim) = []; perm = [dim, perm]; x = permute (x, perm); ## Reshape all vecdims into dim1 num_dim = prod (szx(dim)); szx(dim) = []; szx = [num_dim, ones(1, numel(dim)-1), szx]; x = reshape (x, szx); dim = 1; endif perm_flag = true; endif ## Find column locations of NaNs nanfree = ! any (isnan (x), dim); if (omitnan && nanfree(:)) ## Don't use omitnan path if no NaNs are present. Prevents any data types ## without a defined NaN from following the omitnan codepath. omitnan = false; endif x = sort (x, dim); # Note: pushes any NaN's to end for omitnan compatibility if (omitnan) ## Ignore any NaN's in data. Each operating vector might have a ## different number of non-NaN data points. if (isvector (x)) ## Checks above ensure either dim1 or dim2 vector x = x(! isnan (x)); n = length (x); k = floor ((n + 1) / 2); if (mod (n, 2)) ## odd m = x(k); else ## even m = (x(k) + x(k + 1)) / 2; endif else ## Each column may have a different n and k. Force index column vector ## for consistent orientation for 2D and nD inputs, then use sub2ind to ## get correct element(s) for each column. n = sum (! isnan (x), 1)(:); k = floor ((n + 1) / 2); m_idx_odd = mod (n, 2) & n; m_idx_even = (! m_idx_odd) & n; m = NaN ([1, szx(2 : end)]); if (ndims (x) > 2) szx = [szx(1), prod(szx(2 : end))]; endif ## Grab kth value, k possibly different for each column if (any (m_idx_odd)) x_idx_odd = sub2ind (szx, k(m_idx_odd), find (m_idx_odd)); m(m_idx_odd) = x(x_idx_odd); endif if (any (m_idx_even)) k_even = k(m_idx_even); x_idx_even = sub2ind (szx, [k_even, k_even+1], ... (find (m_idx_even))(:,[1 1])); m(m_idx_even) = sum (x(x_idx_even), 2) / 2; endif endif else ## No "omitnan". All 'vectors' uniform length. ## All types without a NaN value will use this path. if (all (!nanfree)) m = NaN (sz_out); else if (isvector (x)) n = length (x); k = floor ((n + 1) / 2); m = x(k); if (! mod (n, 2)) ## Even if (any (isa (x, "integer"))) ## avoid int overflow issues m2 = x(k+1); if (sign(m) != sign(m2)) m += m2; m /= 2; else m += (m2 - m) / 2; endif else m += (x(k+1) - m) / 2; endif endif else ## Nonvector, all operations were permuted to be along dim 1 n = szx(1); k = floor ((n + 1) / 2); if (isfloat (x)) m = NaN ([1, szx(2 : end)]); else m = zeros ([1, szx(2 : end)], outtype); endif if (! mod (n, 2)) ## Even if (any (isa(x, "integer"))) ## avoid int overflow issues ## Use flattened index to simplify N-D operations m(1,:) = x(k, :); m2 = x(k+1, :); samesign = prod (sign ([m(1,:); m2]), 1) == 1; m(1,:) = samesign .* m(1,:) + ... (m2 + !samesign .* m(1,:) - samesign .* m(1,:)) / 2; else m(nanfree) = (x(k, nanfree) + x(k+1, nanfree)) / 2; endif else ## Odd. Use flattened index to simplify n-D operations m(nanfree) = x(k, nanfree); endif endif endif endif if (perm_flag) ## Inverse permute back to correct dimensions m = ipermute (m, perm); endif ## Convert output type as requested if (! strcmp (class (m), outtype)) m = feval (outtype, m); endif endfunction %!assert (median (1), 1) %!assert (median ([1,2,3]), 2) %!assert (median ([1,2,3]'), 2) %!assert (median (cat(3,3,1,2)), 2) %!assert (median ([3,1,2]), 2) %!assert (median ([2,4,6,8]), 5) %!assert (median ([8,2,6,4]), 5) %!assert (median (single ([1,2,3])), single (2)) %!assert (median ([1,2], 3), [1,2]) %!test %! x = [1, 2, 3, 4, 5, 6]; %! x2 = x'; %! y = [1, 2, 3, 4, 5, 6, 7]; %! y2 = y'; %! %! assert (median (x) == median (x2) && median (x) == 3.5); %! assert (median (y) == median (y2) && median (y) == 4); %! assert (median ([x2, 2 * x2]), [3.5, 7]); %! assert (median ([y2, 3 * y2]), [4, 12]); ## Test outtype option %!test %! in = [1 2 3]; %! out = 2; %! assert (median (in, "default"), median (in)); %! assert (median (in, "default"), out); %!test %! in = single ([1 2 3]); %! out = 2; %! assert (median (in, "default"), single (median (in))); %! assert (median (in, "default"), single (out)); %! assert (median (in, "double"), double (out)); %! assert (median (in, "native"), single (out)); %!test %! in = uint8 ([1 2 3]); %! out = 2; %! assert (median (in, "default"), double (median (in))); %! assert (median (in, "default"), double (out)); %! assert (median (in, "double"), out); %! assert (median (in, "native"), uint8 (out)); %!test %! in = logical ([1 0 1]); %! out = 1; %! assert (median (in, "default"), double (median (in))); %! assert (median (in, "default"), double (out)); %! assert (median (in, "double"), double (out)); %! assert (median (in, "native"), double (out)); ## Test single input and optional arguments "all", DIM, "omitnan") %!test %! x = repmat ([2 2.1 2.2 2 NaN; 3 1 2 NaN 5; 1 1.1 1.4 5 3], [1, 1, 4]); %! y = repmat ([2 1.1 2 NaN NaN], [1, 1, 4]); %! assert (median (x), y); %! assert (median (x, 1), y); %! y = repmat ([2 1.1 2 3.5 4], [1, 1, 4]); %! assert (median (x, "omitnan"), y); %! assert (median (x, 1, "omitnan"), y); %! y = repmat ([2.05; 2.5; 1.4], [1, 1, 4]); %! assert (median (x, 2, "omitnan"), y); %! y = repmat ([NaN; NaN; 1.4], [1, 1, 4]); %! assert (median (x, 2), y); %! assert (median (x, "all"), NaN); %! assert (median (x, "all", "omitnan"), 2); %!assert (median (cat (3, 3, 1, NaN, 2), "omitnan"), 2) %!assert (median (cat (3, 3, 1, NaN, 2), 3, "omitnan"), 2) ## Test boolean input %!test %! assert (median (true, "all"), logical (1)); %! assert (median (false), logical (0)); %! assert (median ([true false true]), true); %! assert (median ([true false true], 2), true); %! assert (median ([true false true], 1), logical ([1 0 1])); %! assert (median ([true false NaN], 1), [1 0 NaN]); %! assert (median ([true false NaN], 2), NaN); %! assert (median ([true false NaN], 2, "omitnan"), 0.5); %! assert (median ([true false NaN], 2, "omitnan", "native"), double(0.5)); ## Test dimension indexing with vecdim in n-dimensional arrays %!test %! x = repmat ([1:20;6:25], [5 2 6 3]); %! assert (size (median (x, [3 2])), [10 1 1 3]); %! assert (size (median (x, [1 2])), [1 1 6 3]); %! assert (size (median (x, [1 2 4])), [1 1 6]); %! assert (size (median (x, [1 4 3])), [1 40]); %! assert (size (median (x, [1 2 3 4])), [1 1]); ## Test exceeding dimensions %!assert (median (ones (2,2), 3), ones (2,2)) %!assert (median (ones (2,2,2), 99), ones (2,2,2)) %!assert (median (magic (3), 3), magic (3)) %!assert (median (magic (3), [1 3]), [4, 5, 6]) %!assert (median (magic (3), [1 99]), [4, 5, 6]) ## Test results with vecdim in n-dimensional arrays and "omitnan" %!test %! x = repmat ([2 2.1 2.2 2 NaN; 3 1 2 NaN 5; 1 1.1 1.4 5 3], [1, 1, 4]); %! assert (median (x, [3 2]), [NaN NaN 1.4]'); %! assert (median (x, [3 2], "omitnan"), [2.05 2.5 1.4]'); %! assert (median (x, [1 3]), [2 1.1 2 NaN NaN]); %! assert (median (x, [1 3], "omitnan"), [2 1.1 2 3.5 4]); ## Test empty, NaN, Inf inputs %!assert (median (NaN), NaN) %!assert (median (NaN, "omitnan"), NaN) %!assert (median (NaN (2)), [NaN NaN]) %!assert (median (NaN (2), "omitnan"), [NaN NaN]) %!assert (median ([1 NaN 3]), NaN) %!assert (median ([1 NaN 3], 1), [1 NaN 3]) %!assert (median ([1 NaN 3], 2), NaN) %!assert (median ([1 NaN 3]'), NaN) %!assert (median ([1 NaN 3]', 1), NaN) %!assert (median ([1 NaN 3]', 2), [1; NaN; 3]) %!assert (median ([1 NaN 3], "omitnan"), 2) %!assert (median ([1 NaN 3]', "omitnan"), 2) %!assert (median ([1 NaN 3], 1, "omitnan"), [1 NaN 3]) %!assert (median ([1 NaN 3], 2, "omitnan"), 2) %!assert (median ([1 NaN 3]', 1, "omitnan"), 2) %!assert (median ([1 NaN 3]', 2, "omitnan"), [1; NaN; 3]) %!assert (median ([1 2 NaN 3]), NaN) %!assert (median ([1 2 NaN 3], "omitnan"), 2) %!assert (median ([1,2,NaN;4,5,6;NaN,8,9]), [NaN, 5, NaN]) %!assert <*64011> (median ([1,2,NaN;4,5,6;NaN,8,9], "omitnan"), [2.5, 5, 7.5], eps) %!assert (median ([1 2 ; NaN 4]), [NaN 3]) %!assert (median ([1 2 ; NaN 4], "omitnan"), [1 3]) %!assert (median ([1 2 ; NaN 4], 1, "omitnan"), [1 3]) %!assert (median ([1 2 ; NaN 4], 2, "omitnan"), [1.5; 4], eps) %!assert (median ([1 2 ; NaN 4], 3, "omitnan"), [1 2 ; NaN 4]) %!assert (median ([NaN 2 ; NaN 4]), [NaN 3]) %!assert (median ([NaN 2 ; NaN 4], "omitnan"), [NaN 3]) %!assert (median (ones (1, 0, 3)), NaN (1, 1, 3)) %!assert (median (NaN("single")), NaN("single")) %!assert (median (NaN("single"), "omitnan"), NaN("single")) %!assert (median (NaN("single"), "double"), NaN("double")) %!assert (median (single([1 2 ; NaN 4])), single([NaN 3])) %!assert (median (single([1 2 ; NaN 4]), "double"), double([NaN 3])) %!assert (median (single([1 2 ; NaN 4]), "omitnan"), single([1 3])) %!assert (median (single([1 2 ; NaN 4]), "omitnan", "double"), double([1 3])) %!assert (median (single([NaN 2 ; NaN 4]), "double"), double([NaN 3])) %!assert (median (single([NaN 2 ; NaN 4]), "omitnan"), single([NaN 3])) %!assert (median (single([NaN 2 ; NaN 4]), "omitnan", "double"), double([NaN 3])) ## Test omitnan with 2D & 3D inputs to confirm correct sub2ind orientation %!test <*64011> %! x = [magic(3), magic(3)]; %! x([3, 7, 11, 12, 16, 17]) = NaN; %! ynan = [NaN, 5, NaN, NaN, 5, NaN]; %! yomitnan = [5.5, 5, 4.5, 8, 5, 2]; %! assert (median (x), ynan); %! assert (median (x, "omitnan"), yomitnan, eps); %! assert (median (cat (3, x, x)), cat (3, ynan, ynan)); %! assert (median (cat (3, x, x), "omitnan"), cat (3, yomitnan, yomitnan), eps); %!assert (median (Inf), Inf) %!assert (median (-Inf), -Inf) %!assert (median ([-Inf Inf]), NaN) %!assert (median ([3 Inf]), Inf) %!assert (median ([3 4 Inf]), 4) %!assert (median ([Inf 3 4]), 4) %!assert (median ([Inf 3 Inf]), Inf) %!assert (median ([]), NaN) %!assert (median (ones(1,0)), NaN) %!assert (median (ones(0,1)), NaN) %!assert (median ([], 1), NaN(1,0)) %!assert (median ([], 2), NaN(0,1)) %!assert (median ([], 3), NaN(0,0)) %!assert (median (ones(1,0), 1), NaN(1,0)) %!assert (median (ones(1,0), 2), NaN(1,1)) %!assert (median (ones(1,0), 3), NaN(1,0)) %!assert (median (ones(0,1), 1), NaN(1,1)) %!assert (median (ones(0,1), 2), NaN(0,1)) %!assert (median (ones(0,1), 3), NaN(0,1)) %!assert (median (ones(0,1,0,1), 1), NaN(1,1,0)) %!assert (median (ones(0,1,0,1), 2), NaN(0,1,0)) %!assert (median (ones(0,1,0,1), 3), NaN(0,1,1)) %!assert (median (ones(0,1,0,1), 4), NaN(0,1,0)) ## Test complex inputs (should sort by abs(a)) %!assert (median([1 3 3i 2 1i]), 2) %!assert (median([1 2 4i; 3 2i 4]), [2, 1+1i, 2+2i]) ## Test multidimensional arrays %!shared a, b, x, y %! old_state = rand ("state"); %! restore_state = onCleanup (@() rand ("state", old_state)); %! rand ("state", 2); %! a = rand (2,3,4,5); %! b = rand (3,4,6,5); %! x = sort (a, 4); %! y = sort (b, 3); %!assert <*35679> (median (a, 4), x(:, :, :, 3)) %!assert <*35679> (median (b, 3), (y(:, :, 3, :) + y(:, :, 4, :))/2) %!shared ## Clear shared to prevent variable echo for any later test failures ## Test n-dimensional arrays with odd non-NaN data points %!test %! x = ones(15,1,4); %! x([13,15],1,:) = NaN; %! assert (median (x, 1, "omitnan"), ones (1,1,4)) ## Test non-floating point types %!assert (median ([true, false]), true) %!assert (median (logical ([])), false) %!assert (median (uint8 ([1, 3])), uint8 (2)) %!assert (median (uint8 ([])), uint8 (NaN)) %!assert (median (uint8 ([NaN 10])), uint8 (5)) %!assert (median (int8 ([1, 3, 4])), int8 (3)) %!assert (median (int8 ([])), int8 (NaN)) %!assert (median (single ([1, 3, 4])), single (3)) %!assert (median (single ([1, 3, NaN])), single (NaN)) ## Test same sign int overflow when getting mean of even number of values %!assert <54567> (median (uint8 ([253, 255])), uint8 (254)) %!assert <54567> (median (uint8 ([253, 254])), uint8 (254)) %!assert <54567> (median (int8 ([127, 126, 125, 124; 1 3 5 9])), ... %! int8 ([64 65 65 67])) %!assert <54567> (median (int8 ([127, 126, 125, 124; 1 3 5 9]), 2), ... %! int8 ([126; 4])) %!assert <54567> (median (int64 ([intmax("int64"), intmax("int64")-2])), ... %! intmax ("int64") - 1) %!assert <54567> (median ( ... %! int64 ([intmax("int64"), intmax("int64")-2; 1 2]), 2), ... %! int64([intmax("int64") - 1; 2])) %!assert <54567> (median (uint64 ([intmax("uint64"), intmax("uint64")-2])), ... %! intmax ("uint64") - 1) %!assert <54567> (median ( ... %! uint64 ([intmax("uint64"), intmax("uint64")-2; 1 2]), 2), ... %! uint64([intmax("uint64") - 1; 2])) ## Test opposite sign int overflow when getting mean of even number of values %!assert <54567> (median (... %! [intmin('int8') intmin('int8')+5 intmax('int8')-5 intmax('int8')]), ... %! int8(-1)) %!assert <54567> (median ([int8([1 2 3 4]); ... %! intmin('int8') intmin('int8')+5 intmax('int8')-5 intmax('int8')], 2), ... %! int8([3;-1])) %!assert <54567> (median (... %! [intmin('int64') intmin('int64')+5 intmax('int64')-5 intmax('int64')]), ... %! int64(-1)) %!assert <54567> (median ([int64([1 2 3 4]); ... %! intmin('int64') intmin('int64')+5 intmax('int64')-5 intmax('int64')], 2), ... %! int64([3;-1])) ## Test int accuracy loss doing mean of close int64/uint64 values as double %!assert <54567> (median ([intmax("uint64"), intmax("uint64")-2]), ... %! intmax("uint64")-1) %!assert <54567> (median ([intmax("uint64"), intmax("uint64")-2], "default"), ... %! double(intmax("uint64")-1)) %!assert <54567> (median ([intmax("uint64"), intmax("uint64")-2], "double"), ... %! double(intmax("uint64")-1)) %!assert <54567> (median ([intmax("uint64"), intmax("uint64")-2], "native"), ... %! intmax("uint64")-1) ## Test input case insensitivity %!assert (median ([1 2 3], "aLL"), 2) %!assert (median ([1 2 3], "OmitNan"), 2) %!assert (median ([1 2 3], "DOUBle"), 2) ## Test input validation %!error median () %!error median (1, 2, 3) %!error median (1, 2, 3, 4) %!error median (1, "all", 3) %!error median (1, "b") %!error median (1, 1, "foo") %!error <'all' cannot be used with> median (1, 3, "all") %!error <'all' cannot be used with> median (1, [2 3], "all") %!error median ({1:5}) %!error median ("char") %!error median(1, "double", "native") %!error median (1, ones (2,2)) %!error median (1, 1.5) %!error median (1, 0) %!error median ([1 2 3], [-1 1]) %!error median(1, [1 2 2]) statistics-release-1.6.3/inst/shadow9/std.m000066400000000000000000000755151456127120000206470ustar00rootroot00000000000000## Copyright (C) 2022-2023 Andreas Bertsatos ## Copyright (C) 2023 Nicholas Jankowski ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{s} =} std (@var{x}) ## @deftypefnx {statistics} {@var{s} =} std (@var{x}, @var{w}) ## @deftypefnx {statistics} {@var{s} =} std (@var{x}, @var{w}, "all") ## @deftypefnx {statistics} {@var{s} =} std (@var{x}, @var{w}, @var{dim}) ## @deftypefnx {statistics} {@var{s} =} std (@var{x}, @var{w}, @var{vecdim}) ## @deftypefnx {statistics} {@var{s} =} std (@dots{}, @var{nanflag}) ## @deftypefnx {statistics} {[@var{s}, @var{m}] =} std (@dots{}) ## ## Compute the standard deviation of the elements of @var{x}. ## ## @itemize ## @item ## If @var{x} is a vector, then @code{std (@var{x})} returns the standard ## deviation of the elements in @var{x} defined as ## @tex ## $$ {\rm std}(x) = \sqrt{{1\over N-1} \sum_{i=1}^N |x_i - \bar x |^2} $$ ## ## @end tex ## @ifnottex ## ## @example ## std (@var{x}) = sqrt ((1 / (N-1)) * SUM_i (|@var{x}(i) - mean (@var{x})|^2)) ## @end example ## ## @end ifnottex ## @noindent ## where @math{N} is the length of the @var{x} vector. ## ## @item ## If @var{x} is a matrix, then @code{std (@var{x})} returns a row vector with ## the standard deviation of each column in @var{x}. ## ## @item ## If @var{x} is a multi-dimensional array, then @code{std (@var{x})} operates ## along the first non-singleton dimension of @var{x}. ## @end itemize ## ## @code{std (@var{x}, @var{w})} specifies a weighting scheme. When @var{w} = 0 ## (default), the standard deviation is normalized by N-1 (population standard ## deviation), where N is the number of observations. When @var{w} = 1, the ## standard deviation is normalized by the number of observations (sample ## standard deviation). To use the default value you may pass an empty input ## argument [] before entering other options. ## ## @var{w} can also be an array of non-negative numbers. When @var{w} is a ## vector, it must have the same length as the number of elements in the ## operating dimension of @var{x}. If @var{w} is a matrix or n-D array, or the ## operating dimension is supplied as a @var{vecdim} or "all", @var{w} must be ## the same size as @var{x}. NaN values are permitted in @var{w}, will be ## multiplied with the associated values in @var{x}, and can be excluded by the ## @var{nanflag} option. ## ## @code{std (@var{x}, [], @var{dim})} returns the standard deviation along the ## operating dimension @var{dim} of @var{x}. For @var{dim} greater than ## @code{ndims (@var{x})}, then @var{s} is returned as zeros of the same size as ## @var{x} and @var{m} = @var{x}. ## ## @code{std (@var{x}, [], @var{vecdim})} returns the standard deviation over ## the dimensions specified in the vector @var{vecdim}. For example, if @var{x} ## is a 2-by-3-by-4 array, then @code{var (@var{x}, [1 2])} returns a ## 1-by-1-by-4 array. Each element of the output array is the standard ## deviation of the elements on the corresponding page of @var{x}. ## If @var{vecdim} indexes all dimensions of @var{x}, then it is equivalent to ## @code{std (@var{x}, "all")}. Any dimension in @var{vecdim} greater than ## @code{ndims (@var{x})} is ignored. ## ## @code{std (@var{x}, "all")} returns the standard deviation of all the ## elements in @var{x}. The optional flag "all" cannot be used together with ## @var{dim} or @var{vecdim} input arguments. ## ## @code{std (@dots{}, @var{nanflag})} specifies whether to exclude NaN values ## from the calculation using any of the input argument combinations in previous ## syntaxes. The default value for @var{nanflag} is "includenan", and keeps NaN ## values in the calculation. To exclude NaN values, set the value of ## @var{nanflag} to "omitnan". ## ## @code{[@var{s}, @var{m}] = std (@dots{})} also returns the mean of the ## elements of @var{x} used to calculate the standard deviation. If @var{s} is ## the weighted standard deviation, then @var{m} is the weighted mean. ## ## @seealso{var, mean} ## @end deftypefn function [s, m] = std (x, varargin) if (nargin < 1 || nargin > 4) print_usage (); endif ## initialize variables all_flag = false; omitnan = false; nvarg = numel (varargin); varg_chars = cellfun ('ischar', varargin); ## Check all char arguments. if (nvarg == 3 && ! varg_chars(3)) print_usage (); endif if (any (varg_chars)) for i = varargin(varg_chars) switch (lower (i{:})) case "all" all_flag = true; case "omitnan" omitnan = true; case "includenan" omitnan = false; otherwise print_usage (); endswitch endfor varargin(varg_chars) = []; nvarg = numel (varargin); endif # FIXME: when sparse can use broadcast ops, remove sparse checks and hacks sprs_x = issparse (x); w = 0; weighted = false; # true if weight vector/array used vecdim = []; vecempty = true; vecdim_scalar_vector = [false, false]; # [false, false] for empty vecdim szx = size (x); ndx = ndims (x); ## Check numeric arguments if (! (isnumeric (x))) error ("std: X must be a numeric vector or matrix."); endif if (isa (x, "single")) outtype = "single"; else outtype = "double"; endif if (nvarg > 0) if (nvarg > 2 || any (! cellfun ('isnumeric', varargin))) print_usage (); endif ## Process weight input if (any (varargin{1} < 0)) error ("std: weights must not contain any negative values."); endif if (isscalar (varargin{1})) w = varargin{1}; if (! (w == 0 || w == 1) && ! isscalar (x)) error ("std: normalization scalar must be either 0 or 1."); endif elseif (numel (varargin{1}) > 1) weights = varargin{1}; weighted = true; endif if (nvarg > 1) ## Process dimension input vecdim = varargin{2}; if (! (vecempty = isempty (vecdim))) ## Check for empty vecdim, won't change vsv if nonzero size empty vecdim_scalar_vector = [isscalar(vecdim), isvector(vecdim)]; endif if (! (vecdim_scalar_vector(2) && all (vecdim > 0)) ... || any (rem (vecdim, 1))) error ("std: DIM must be a positive integer scalar or vector."); endif if (vecdim_scalar_vector(1) && vecdim > ndx && ! isempty (x)) s = zeros (szx, outtype); sn = ! isfinite (x); s(sn) = NaN; m = x; return; endif if (vecdim_scalar_vector == [0 1] && (! all (diff (sort (vecdim))))) error ("std: VECDIM must contain non-repeating positive integers."); endif endif endif ## Check for conflicting input arguments if (all_flag && ! vecempty) error ("std: 'all' flag cannot be used with DIM or VECDIM options."); endif if (weighted) if (all_flag) if (isvector (weights)) if (numel (weights) != numel (x)) error ("std: weight vector element count does not match X."); endif elseif (! (isequal (size (weights), szx))) error ("std: weight matrix or array does not match X in size."); endif elseif (vecempty) dim = find (szx > 1, 1); if length (dim) == 0 dim = 1; endif if (isvector (weights)) if (numel (weights) != szx(dim)) error (["std: weight vector length does not match operating ", ... "dimension."]); endif elseif (! isequal (size (weights), szx)) error ("std: weight matrix or array does not match X in size."); endif elseif (vecdim_scalar_vector(1)) if (isvector (weights)) if (numel (weights) != szx(vecdim)) error (["std: weight vector length does not match operating ", ... "dimension."]); endif elseif (! isequal (size (weights), szx)) error ("std: weight matrix or array does not match X in size."); endif elseif (vecdim_scalar_vector(2) && ! (isequal (size (weights), szx))) error ("std: weight matrix or array does not match X in size."); endif endif ## Force output for X being empty or scalar if (isempty (x)) if (vecempty && (ndx == 2 || all ((szx) == 0))) s = NaN (outtype); if (nargout > 1) m = NaN (outtype); endif return; endif if (vecdim_scalar_vector(1)) szx(vecdim) = 1; s = NaN (szx, outtype); if (nargout > 1) m = NaN (szx, outtype); endif return; endif endif if (isscalar (x)) if (isfinite (x)) s = zeros (outtype); else s = NaN (outtype); endif if (nargout > 1) m = x; endif return; endif if (nvarg == 0) ## Only numeric input argument, no dimensions or weights. if (all_flag) x = x(:); if (omitnan) x = x(! isnan (x)); endif n = length (x); m = sum (x) ./ n; s = sqrt (sum (abs (x - m) .^ 2) ./ (n - 1 + w)); if (n == 1) s = 0; endif else dim = find (szx > 1, 1); if length (dim) == 0 dim = 1; endif n = szx(dim); if (omitnan) n = sum (! isnan (x), dim); xn = isnan (x); x(xn) = 0; endif m = sum (x, dim) ./ n; dims = ones (1, ndx); dims(dim) = szx(dim); if (sprs_x) m_exp = repmat (m, dims); else m_exp = m .* ones (dims); endif if (omitnan) x(xn) = m_exp(xn); endif s = sqrt (sumsq (x - m_exp, dim) ./ (n - 1 + w)); if (numel (n) == 1) divby0 = n .* ones (size (s)) == 1; else divby0 = n == 1; endif s(divby0) = 0; endif elseif (nvarg == 1) ## Two numeric input arguments, w or weights given. if (all_flag) x = x(:); if (weighted) weights = weights(:); wx = weights .* x; else weights = ones (length (x), 1); wx = x; endif if (omitnan) xn = isnan (wx); wx = wx(! xn); weights = weights(! xn); x = x(! xn); endif n = length (wx); m = sum (wx) ./ sum (weights); if (weighted) s = sqrt (sum (weights .* (abs (x - m) .^ 2)) ./ sum (weights)); else s = sqrt (sum (weights .* (abs (x - m) .^ 2)) ./ (n - 1 + w)); if (n == 1) s = 0; endif endif else dim = find (szx > 1, 1); if length (dim) == 0 dim = 1; endif if (! weighted) weights = ones (szx); wx = x; else if (isvector (weights)) dims = 1:ndx; dims([1, dim]) = [dim, 1]; weights = zeros (szx) + permute (weights(:), dims); endif wx = weights .* x; endif n = size (wx, dim); if (omitnan) xn = isnan (wx); n = sum (! xn, dim); wx(xn) = 0; weights(xn) = 0; endif m = sum (wx, dim) ./ sum (weights, dim); dims = ones (1, ndims (wx)); dims(dim) = size (wx, dim); if (sprs_x) m_exp = repmat (m, dims); else m_exp = m .* ones (dims); endif if (omitnan) x(xn) = m_exp(xn); endif if (weighted) s = sqrt (sum (weights .* ((x - m_exp) .^ 2), dim) ... ./ sum (weights, dim)); else s = sqrt (sumsq (x - m_exp, dim) ./ (n - 1 + w)); if (numel (n) == 1) divby0 = n .* ones (size (s)) == 1; else divby0 = n == 1; endif s(divby0) = 0; endif endif elseif (nvarg == 2) ## Three numeric input arguments, both w or weights and dim or vecdim given. if (vecdim_scalar_vector(1)) if (!weighted) weights = ones (szx); wx = x; else if (isvector (weights)) dims = 1:ndx; dims([1, vecdim]) = [vecdim, 1]; weights = zeros (szx) + permute (weights(:), dims); endif wx = weights .* x; endif n = size (wx, vecdim); if (omitnan) n = sum (! isnan (wx), vecdim); xn = isnan (wx); wx(xn) = 0; weights(xn) = 0; endif m = sum (wx, vecdim) ./ sum (weights, vecdim); dims = ones (1, ndims (wx)); dims(vecdim) = size (wx, vecdim); if (sprs_x) m_exp = repmat (m, dims); else m_exp = m .* ones (dims); endif if (omitnan) x(xn) = m_exp(xn); endif if (weighted) s = sqrt (sum (weights .* ((x - m_exp) .^ 2), vecdim) ... ./ sum (weights, vecdim)); else s = sumsq (x - m_exp, vecdim); sn = isnan (s); s = sqrt (s ./ (n - 1 + w)); if (numel (n) == 1) divby0 = n .* ones (size (s)) == 1; else divby0 = n == 1; endif s(divby0) = 0; s(sn) = NaN; endif else ## Weights and nonscalar vecdim specified ## Ignore exceeding dimensions in VECDIM remdims = 1 : ndx; # all dimensions vecdim(find (vecdim > ndx)) = []; ## Calculate permutation vector remdims(vecdim) = []; # delete dimensions specified by vecdim nremd = numel (remdims); ## If all dimensions are given, it is similar to all flag if (nremd == 0) x = x(:); if (weighted) weights = weights(:); wx = weights .* x; else weights = ones (length (x), 1); wx = x; endif if (omitnan) xn = isnan (wx); wx = wx(! xn); weights = weights(! xn); x = x(! xn); endif n = length (wx); m = sum (wx) ./ sum (weights); if (weighted) s = sqrt (sum (weights .* (abs (x - m) .^ 2)) ./ sum (weights)); else s = sqrt (sum (weights .* (abs (x - m) .^ 2)) ./ (n - 1 + w)); if (n == 1) s = 0; endif endif else ## Apply weights if (weighted) wx = weights .* x; else weights = ones (szx); wx = x; endif ## Permute to bring remaining dims forward perm = [remdims, vecdim]; wx = permute (wx, perm); weights = permute (weights, perm); x = permute (x, perm); ## Reshape to put all vecdims in final dimension szwx = size (wx); sznew = [szwx(1:nremd), prod(szwx(nremd+1:end))]; wx = reshape (wx, sznew); weights = reshape (weights, sznew); x = reshape (x, sznew); ## Calculate var on single, squashed dimension dim = nremd + 1; n = size (wx, dim); if (omitnan) xn = isnan (wx); n = sum (! xn, dim); wx(xn) = 0; weights(xn) = 0; endif m = sum (wx, dim) ./ sum (weights, dim); m_exp = zeros (size (wx)) + m; if (omitnan) x(xn) = m_exp(xn); endif if (weighted) s = sqrt (sum (weights .* ((x - m_exp) .^ 2), dim) ... ./ sum (weights, dim)); else s = sqrt (sumsq (x - m_exp, dim) ./ (n - 1 + w)); if (numel (n) == 1) divby0 = n .* ones (size (s)) == 1; else divby0 = n == 1; endif s(divby0) = 0; endif ## Inverse permute back to correct dimensions s = ipermute (s, perm); if (nargout > 1) m = ipermute (m, perm); endif endif endif endif ## Preserve class type if (nargout < 2) if strcmp (outtype, "single") s = single (s); else s = double (s); endif else if strcmp (outtype, "single") s = single (s); m = single (m); else s = double (s); m = double (m); endif endif endfunction %!assert (std (13), 0) %!assert (std (single (13)), single (0)) %!assert (std ([1,2,3]), 1) %!assert (std ([1,2,3], 1), sqrt (2/3), eps) %!assert (std ([1,2,3], [], 1), [0,0,0]) %!assert (std ([1,2,3], [], 3), [0,0,0]) %!assert (std (5, 99), 0) %!assert (std (5, 99, 1), 0) %!assert (std (5, 99, 2), 0) %!assert (std ([5 3], [99 99], 2), 1) %!assert (std ([1:7], [1:7]), sqrt (3)) %!assert (std ([eye(3)], [1:3]), sqrt ([5/36, 2/9, 1/4]), eps) %!assert (std (ones (2,2,2), [1:2], 3), [(zeros (2,2))]) %!assert (std ([1 2; 3 4], 0, 'all'), std ([1:4])) %!assert (std (reshape ([1:8], 2, 2, 2), 0, [1 3]), sqrt ([17/3 17/3]), eps) %!assert (std ([1 2 3;1 2 3], [], [1 2]), sqrt (0.8), eps) ## Test single input and optional arguments "all", DIM, "omitnan") %!test %! x = [-10:10]; %! y = [x;x+5;x-5]; %! assert (std (x), sqrt (38.5), 1e-14); %! assert (std (y, [], 2), sqrt ([38.5; 38.5; 38.5]), 1e-14); %! assert (std (y, 0, 2), sqrt ([38.5; 38.5; 38.5]), 1e-14); %! assert (std (y, 1, 2), ones (3,1) * sqrt (36.66666666666666), 1e-14); %! assert (std (y, "all"), sqrt (54.19354838709678), 1e-14); %! y(2,4) = NaN; %! assert (std (y, "all"), NaN); %! assert (std (y, "all", "includenan"), NaN); %! assert (std (y, "all", "omitnan"), sqrt (55.01533580116342), 1e-14); %! assert (std (y, 0, 2, "includenan"), sqrt ([38.5; NaN; 38.5]), 1e-14); %! assert (std (y, [], 2), sqrt ([38.5; NaN; 38.5]), 1e-14); %! assert (std (y, [], 2, "omitnan"), ... %! sqrt ([38.5; 37.81842105263158; 38.5]), 1e-14); ## Tests for different weight and omitnan code paths %!assert (std ([4 NaN 6], [1 2 1], "omitnan"), 1, eps) %!assert (std ([4 5 6], [1 NaN 1], "omitnan"), 1, eps) %!assert (std (magic(3), [1 NaN 3], "omitnan"), sqrt(3)*[1 2 1], eps) %!assert (std ([4 NaN 6], [1 2 1], "omitnan", "all"), 1, eps) %!assert (std ([4 NaN 6], [1 2 1], "all", "omitnan"), 1, eps) %!assert (std ([4 5 6], [1 NaN 1], "omitnan", "all"), 1, eps) %!assert (std ([4 NaN 6], [1 2 1], 2, "omitnan"), 1, eps) %!assert (std ([4 5 6], [1 NaN 1], 2, "omitnan"), 1, eps) %!assert (std (magic(3), [1 NaN 3], 1, "omitnan"), sqrt(3)*[1 2 1], eps) %!assert (std (magic(3), [1 NaN 3], 2, "omitnan"), sqrt(3)*[0.5;1;0.5], eps) %!assert (std (4*[4 5; 6 7; 8 9], [1 3], 2, 'omitnan'), sqrt(3)*[1;1;1], eps) %!assert (std ([4 NaN; 6 7; 8 9], [1 1 3], 1, 'omitnan'), [1.6 sqrt(3)/2], eps) %!assert (std (4*[4 NaN; 6 7; 8 9], [1 3], 2, 'omitnan'), sqrt(3)*[0;1;1], eps) %!assert (std (3*reshape(1:18, [3 3 2]), [1 2 3], 1, 'omitnan'), ... %! sqrt(5)*ones(1,3,2), eps) %!assert (std (reshape(1:18, [3 3 2]), [1 2 3], 2, 'omitnan'), ... %! sqrt(5)*ones(3,1,2), eps) %!assert (std (3*reshape(1:18, [3 3 2]), ones (3,3,2), [1 2], 'omitnan'), ... %! sqrt(60)*ones(1,1,2),eps) %!assert (std (3*reshape(1:18, [3 3 2]), ones (3,3,2), [1 4], 'omitnan'), ... %! sqrt(6)*ones(1,3,2),eps) %!assert (std (6*reshape(1:18, [3 3 2]), ones (3,3,2), [1:3], 'omitnan'), ... %! sqrt(969),eps) %!test %! x = reshape(1:18, [3 3 2]); %! x([2, 14]) = NaN; %! w = ones (3,3,2); %! assert (std (16*x, w, [1:3], 'omitnan'), sqrt(6519), eps); %!test %! x = reshape(1:18, [3 3 2]); %! w = ones (3,3,2); %! w([2, 14]) = NaN; %! assert (std (16*x, w, [1:3], 'omitnan'), sqrt(6519), eps); ## Test input case insensitivity %!assert (std ([1 2 3], "aLl"), 1); %!assert (std ([1 2 3], "OmitNan"), 1); %!assert (std ([1 2 3], "IncludeNan"), 1); ## Test dimension indexing with vecdim in n-dimensional arrays %!test %! x = repmat ([1:20;6:25], [5, 2, 6, 3]); %! assert (size (std (x, 0, [3 2])), [10, 1, 1, 3]); %! assert (size (std (x, 1, [1 2])), [1, 1, 6, 3]); %! assert (size (std (x, [], [1 2 4])), [1, 1, 6]); %! assert (size (std (x, 0, [1 4 3])), [1, 40]); %! assert (size (std (x, [], [1 2 3 4])), [1, 1]); ## Test matrix with vecdim, weighted, matrix weights, omitnan %!assert (std (3*magic(3)), sqrt([63 144 63]), eps) %!assert (std (3*magic(3), 'omitnan'), sqrt([63 144 63]), eps) %!assert (std (3*magic(3), 1), sqrt([42 96 42]), eps) %!assert (std (3*magic(3), 1, 'omitnan'), sqrt([42 96 42]), eps) %!assert (std (3*magic(3), ones(1,3), 1), sqrt([42 96 42]), eps) %!assert (std (3*magic(3), ones(1,3), 1, 'omitnan'), sqrt([42 96 42]), eps) %!assert (std (2*magic(3), [1 1 NaN], 1, 'omitnan'), [5 4 1], eps) %!assert (std (3*magic(3), ones(3,3)), sqrt([42 96 42]), eps) %!assert (std (3*magic(3), ones(3,3), 'omitnan'), sqrt([42 96 42]), eps) %!assert (std (3*magic(3), [1 1 1; 1 1 1; 1 NaN 1], 'omitnan'), ... %! sqrt([42 36 42]), eps) %!assert (std (3*magic(3), ones(3,3), 1), sqrt([42 96 42]), eps) %!assert (std (3*magic(3), ones(3,3), 1, 'omitnan'), sqrt([42 96 42]), eps) %!assert (std (3*magic(3), [1 1 1; 1 1 1; 1 NaN 1], 1, 'omitnan'), ... %! sqrt([42 36 42]), eps) %!assert (std (3*magic(3), ones(3,3), [1 4]), sqrt([42 96 42]), eps) %!assert (std (3*magic(3), ones(3,3), [1 4], 'omitnan'), sqrt([42 96 42]), eps) %!assert (std (3*magic(3), [1 1 1; 1 1 1; 1 NaN 1],[1 4],'omitnan'), ... %! sqrt([42 36 42]), eps) ## Test results with vecdim in n-dimensional arrays and "omitnan" %!test %! x = repmat ([1:20;6:25], [5, 2, 6, 3]); %! v = repmat (sqrt (33.38912133891213), [10, 1, 1, 3]); %! assert (std (x, 0, [3, 2]), v, 1e-14); %! v = repmat (sqrt (33.250), [10, 1, 1, 3]); %! assert (std (x, 1, [3, 2]), v, 1e-14); %! x(2,5,6,3) = NaN; %! v(2,1,1,3) = NaN; %! assert (std (x, 1, [3, 2]), v, 1e-14); %! v = repmat (sqrt (33.38912133891213), [10 1 1 3]); %! v(2,1,1,3) = NaN; %! assert (std (x, [], [3, 2]), v, 1e-14); %! v(2,1,1,3) = sqrt (33.40177912169048); %! assert (std (x, [], [3, 2], "omitnan"), v, 1e-14); ## Testing weights vectors & arrays %!assert (std (ones (2,2,2), [1:2], 3), [(zeros (2, 2))]) %!assert (std (magic (3), [1:9], "all"), 2.581988897471611, 1e-14) ## Test exceeding dimensions %!assert (std (ones (2,2), [], 3), zeros (2,2)) %!assert (std (ones (2,2,2), [], 99), zeros (2,2,2)) %!assert (std (magic (3), [], 3), zeros (3,3)) %!assert (std (magic (3), [], 1), sqrt ([7, 16, 7])) %!assert (std (magic (3), [], [1 3]), sqrt ([7, 16, 7])) %!assert (std (magic (3), [], [1 99]), sqrt ([7, 16, 7])) ## Test empty inputs %!assert (std ([]), NaN) %!assert (class (var (single ([]))), "single") %!assert (std ([],[],1), NaN(1,0)) %!assert (std ([],[],2), NaN(0,1)) %!assert (std ([],[],3), []) %!assert (class (var (single ([]), [], 1)), "single") %!assert (std (ones (1,0)), NaN) %!assert (std (ones (1,0), [], 1), NaN(1,0)) %!assert (std (ones (1,0), [], 2), NaN) %!assert (std (ones (1,0), [], 3), NaN(1,0)) %!assert (class (var (ones (1, 0, "single"), [], 1)), "single") %!assert (std (ones (0,1)), NaN) %!assert (std (ones (0,1), [], 1), NaN) %!assert (std (ones (0,1), [], 2), NaN(0,1)) %!assert (std (ones (0,1), [], 3), NaN(0,1)) %!assert (std (ones (1,3,0,2)), NaN(1,1,0,2)) %!assert (std (ones (1,3,0,2), [], 1), NaN(1,3,0,2)) %!assert (std (ones (1,3,0,2), [], 2), NaN(1,1,0,2)) %!assert (std (ones (1,3,0,2), [], 3), NaN(1,3,1,2)) %!assert (std (ones (1,3,0,2), [], 4), NaN(1,3,0)) %!test %! [~, m] = std ([]); %! assert (m, NaN); ## Test optional mean output %!test <*62395> %! [~, m] = std (13); %! assert (m, 13); %! [~, m] = std (single(13)); %! assert (m, single(13)); %! [~, m] = std ([1, 2, 3; 3 2 1], []); %! assert (m, [2 2 2]); %! [~, m] = std ([1, 2, 3; 3 2 1], [], 1); %! assert (m, [2 2 2]); %! [~, m] = std ([1, 2, 3; 3 2 1], [], 2); %! assert (m, [2 2]'); %! [~, m] = std ([1, 2, 3; 3 2 1], [], 3); %! assert (m, [1 2 3; 3 2 1]); ## Test mean output, weighted inputs, vector dims %!test <*62395> %! [~, m] = std (5,99); %! assert (m, 5); %! [~, m] = std ([1:7], [1:7]); %! assert (m, 5); %! [~, m] = std ([eye(3)], [1:3]); %! assert (m, [1/6, 1/3, 0.5], eps); %! [~, m] = std (ones (2,2,2), [1:2], 3); %! assert (m, ones (2,2)); %! [~, m] = std ([1 2; 3 4], 0, 'all'); %! assert (m, 2.5, eps); %! [~, m] = std (reshape ([1:8], 2, 2, 2), 0, [1 3]); %! assert (m, [3.5, 5.5], eps); %!test %! [v, m] = std (4 * eye (2), [1, 3]); %! assert (v, sqrt ([3, 3]), 1e-14); %! assert (m, [1, 3]); ## Test mean output, empty inputs, omitnan %!test <*62395> %! [~, m] = std ([]); %! assert (m, NaN); #%! [~, m] = std ([],[],1); #%! assert (m, NaN(1,0)); #%! [~, m] = std ([],[],2); #%! assert (m, NaN(0,1)); #%! [~, m] = std ([],[],3); #%! assert (m, []); #%! [~, m] = std (ones (1,3,0,2)); #%! assert (m, NaN(1,1,0,2)); ## Test mean output, nD array %!test %! x = repmat ([1:20;6:25], [5, 2, 6, 3]); %! [~, m] = std (x, 0, [3 2]); %! assert (m, mean (x, [3 2])); %! [~, m] = std (x, 0, [1 2]); %! assert (m, mean (x, [1 2])); %! [~, m] = std (x, 0, [1 3 4]); %! assert (m, mean (x, [1 3 4])); %!test %! x = repmat ([1:20;6:25], [5, 2, 6, 3]); %! x(2,5,6,3) = NaN; %! [~, m] = std (x, 0, [3 2], "omitnan"); %! assert (m, mean (x, [3 2], "omitnan")); ## Test Inf and NaN inputs %!test <*63203> %! [v, m] = std (Inf); %! assert (v, NaN); %! assert (m, Inf); %!test <*63203> %! [v, m] = std (NaN); %! assert (v, NaN); %! assert (m, NaN); %!test <*63203> %! [v, m] = std ([1, Inf, 3]); %! assert (v, NaN); %! assert (m, Inf); %!test <*63203> %! [v, m] = std ([1, Inf, 3]'); %! assert (v, NaN); %! assert (m, Inf); %!test <*63203> %! [v, m] = std ([1, NaN, 3]); %! assert (v, NaN); %! assert (m, NaN); %!test <*63203> %! [v, m] = std ([1, NaN, 3]'); %! assert (v, NaN); %! assert (m, NaN); %!test <*63203> %! [v, m] = std ([1, Inf, 3], [], 1); %! assert (v, [0, NaN, 0]); %! assert (m, [1, Inf, 3]); %!test <*63203> %! [v, m] = std ([1, Inf, 3], [], 2); %! assert (v, NaN); %! assert (m, Inf); %!test <*63203> %! [v, m] = std ([1, Inf, 3], [], 3); %! assert (v, [0, NaN, 0]); %! assert (m, [1, Inf, 3]); %!test <*63203> %! [v, m] = std ([1, NaN, 3], [], 1); %! assert (v, [0, NaN, 0]); %! assert (m, [1, NaN, 3]); %!test <*63203> %! [v, m] = std ([1, NaN, 3], [], 2); %! assert (v, NaN); %! assert (m, NaN); %!test <*63203> %! [v, m] = std ([1, NaN, 3], [], 3); %! assert (v, [0, NaN, 0]); %! assert (m, [1, NaN, 3]); %!test <*63203> %! [v, m] = std ([1, 2, 3; 3, Inf, 5]); %! assert (v, sqrt ([2, NaN, 2])); %! assert (m, [2, Inf, 4]); %!test <*63203> %! [v, m] = std ([1, Inf, 3; 3, Inf, 5]); %! assert (v, sqrt ([2, NaN, 2])); %! assert (m, [2, Inf, 4]); %!test <*63203> %! [v, m] = std ([1, 2, 3; 3, NaN, 5]); %! assert (v, sqrt ([2, NaN, 2])); %! assert (m, [2, NaN, 4]); %!test <*63203> %! [v, m] = std ([1, NaN, 3; 3, NaN, 5]); %! assert (v, sqrt ([2, NaN, 2])); %! assert (m, [2, NaN, 4]); %!test <*63203> %! [v, m] = std ([Inf, 2, NaN]); %! assert (v, NaN); %! assert (m, NaN); %!test <*63203> %! [v, m] = std ([Inf, 2, NaN]'); %! assert (v, NaN); %! assert (m, NaN); %!test <*63203> %! [v, m] = std ([NaN, 2, Inf]); %! assert (v, NaN); %! assert (m, NaN); %!test <*63203> %! [v, m] = std ([NaN, 2, Inf]'); %! assert (v, NaN); %! assert (m, NaN); %!test <*63203> %! [v, m] = std ([Inf, 2, NaN], [], 1); %! assert (v, [NaN, 0, NaN]); %! assert (m, [Inf, 2, NaN]); %!test <*63203> %! [v, m] = std ([Inf, 2, NaN], [], 2); %! assert (v, NaN); %! assert (m, NaN); %!test <*63203> %! [v, m] = std ([NaN, 2, Inf], [], 1); %! assert (v, [NaN, 0, NaN]); %! assert (m, [NaN, 2, Inf]); %!test <*63203> %! [v, m] = std ([NaN, 2, Inf], [], 2); %! assert (v, NaN); %! assert (m, NaN); %!test <*63203> %! [v, m] = std ([1, 3, NaN; 3, 5, Inf]); %! assert (v, sqrt ([2, 2, NaN])); %! assert (m, [2, 4, NaN]); %!test <*63203> %! [v, m] = std ([1, 3, Inf; 3, 5, NaN]); %! assert (v, sqrt ([2, 2, NaN])); %! assert (m, [2, 4, NaN]); ## Test sparse/diagonal inputs %!test <*63291> %! [v, m] = std (2 * eye (2)); %! assert (v, sqrt ([2, 2])); %! assert (m, [1, 1]); %!test <*63291> %! [v, m] = std (4 * eye (2), [1, 3]); %! assert (v, sqrt ([3, 3])); %! assert (m, [1, 3]); %!test <*63291> %! [v, m] = std (sparse (2 * eye (2))); %! assert (full (v), sqrt ([2, 2])); %! assert (full (m), [1, 1]); %!test <*63291> %! [v, m] = std (sparse (4 * eye (2)), [1, 3]); %! assert (full (v), sqrt ([3, 3])); %! assert (full (m), [1, 3]); %!test <*63291> %! [v, m] = std (sparse (eye (2))); %! assert (issparse (v)); %! assert (issparse (m)); %!test <*63291> %! [v, m] = std (sparse (eye (2)), [1, 3]); %! assert (issparse (v)); %! assert (issparse (m)); ## Test input validation %!error std () %!error std (1, 2, "omitnan", 3) %!error std (1, 2, 3, 4) %!error std (1, 2, 3, 4, 5) %!error std (1, "foo") %!error std (1, [], "foo") %!error std ([1 2 3], 2) %!error std ([1 2], 2, "all") %!error std ([1 2],0.5, "all") %!error std (1, -1) %!error std (1, [1 -1]) %!error ... %! std ([1 2 3], [1 -1 0]) %!error std ({1:5}) %!error std ("char") %!error std (['A'; 'B']) %!error std (1, [], ones (2,2)) %!error std (1, 0, 1.5) %!error std (1, [], 0) %!error std (1, [], 1.5) %!error std ([1 2 3], [], [-1 1]) %!error ... %! std (repmat ([1:20;6:25], [5 2 6 3]), 0, [1 2 2 2]) %!error ... %! std ([1 2], eye (2)) %!error ... %! std ([1 2 3 4], [1 2; 3 4]) %!error ... %! std ([1 2 3 4], [1 2; 3 4], 1) %!error ... %! std ([1 2 3 4], [1 2; 3 4], [2 3]) %!error ... %! std (ones (2, 2), [1 2], [1 2]) %!error ... %! std ([1 2 3 4; 5 6 7 8], [1 2 1 2 1; 1 2 1 2 1], 1) %!error ... %! std (repmat ([1:20;6:25], [5 2 6 3]), repmat ([1:20;6:25], [5 2 3]), [2 3]) %!error std ([1 2 3; 2 3 4], [1 3 4]) %!error std ([1 2], [1 2 3]) %!error std (1, [1 2]) %!error std ([1 2 3; 2 3 4], [1 3 4], 1) %!error std ([1 2 3; 2 3 4], [1 3], 2) %!error std ([1 2], [1 2], 1) %!error <'all' flag cannot be used with DIM or VECDIM options> ... %! std (1, [], 1, "all") %!error ... %! std ([1 2 3; 2 3 4], [1 3], "all") %!error ... %! std (repmat ([1:20;6:25], [5 2 6 3]), repmat ([1:20;6:25], [5 2 3]), "all") statistics-release-1.6.3/inst/shadow9/var.m000066400000000000000000000743061456127120000206420ustar00rootroot00000000000000## Copyright (C) 2022-2023 Andreas Bertsatos ## Copyright (C) 2023 Nicholas Jankowski ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{v} =} var (@var{x}) ## @deftypefnx {statistics} {@var{v} =} var (@var{x}, @var{w}) ## @deftypefnx {statistics} {@var{v} =} var (@var{x}, @var{w}, "all") ## @deftypefnx {statistics} {@var{v} =} var (@var{x}, @var{w}, @var{dim}) ## @deftypefnx {statistics} {@var{v} =} var (@var{x}, @var{w}, @var{vecdim}) ## @deftypefnx {statistics} {@var{v} =} var (@dots{}, @var{nanflag}) ## @deftypefnx {statistics} {[@var{v}, @var{m}] =} var (@dots{}) ## ## Compute the variance of the elements of @var{x}. ## ## @itemize ## @item ## If @var{x} is a vector, then @code{var(@var{x})} returns the variance of the ## elements in @var{x} defined as ## @tex ## $$ {\rm var}(x) = {1\over N-1} \sum_{i=1}^N |x_i - \bar x |^2 $$ ## ## @end tex ## @ifnottex ## ## @example ## var (@var{x}) = (1 / (N-1)) * SUM_i (|@var{x}(i) - mean (@var{x})|^2) ## @end example ## ## @end ifnottex ## @noindent ## where @math{N} is the length of the @var{x} vector. ## ## @item ## If @var{x} is a matrix, then @code{var (@var{x})} returns a row vector with ## the variance of each column in @var{x}. ## ## @item ## If @var{x} is a multi-dimensional array, then @code{var (@var{x})} operates ## along the first non-singleton dimension of @var{x}. ## @end itemize ## ## @code{var (@var{x}, @var{w})} specifies a weighting scheme. When @var{w} = 0 ## (default), the variance is normalized by N-1 (population variance) where N is ## the number of observations. When @var{w} = 1, the variance is normalized by ## the number of observations (sample variance). To use the default value you ## may pass an empty input argument [] before entering other options. ## ## @var{w} can also be an array of non-negative numbers. When @var{w} is a ## vector, it must have the same length as the number of elements in the ## operating dimension of @var{x}. If @var{w} is a matrix or n-D array, or the ## operating dimension is supplied as a @var{vecdim} or "all", @var{w} must be ## the same size as @var{x}. NaN values are permitted in @var{w}, will be ## multiplied with the associated values in @var{x}, and can be excluded by the ## @var{nanflag} option. ## ## @code{var (@var{x}, [], @var{dim})} returns the variance along the operating ## dimension @var{dim} of @var{x}. For @var{dim} greater than ## @code{ndims (@var{x})} @var{v} is returned as zeros of the same size as ## @var{x} and @var{m} = @var{x}. ## ## @code{var (@var{x}, [], @var{vecdim})} returns the variance over the ## dimensions specified in the vector @var{vecdim}. For example, if @var{x} ## is a 2-by-3-by-4 array, then @code{var (@var{x}, [1 2])} returns a ## 1-by-1-by-4 array. Each element of the output array is the variance of the ## elements on the corresponding page of @var{x}. If @var{vecdim} indexes all ## dimensions of @var{x}, then it is equivalent to @code{var (@var{x}, "all")}. ## Any dimension in @var{vecdim} greater than @code{ndims (@var{x})} is ignored. ## ## @code{var (@var{x}, "all")} returns the variance of all the elements in ## @var{x}. The optional flag "all" cannot be used together with @var{dim} or ## @var{vecdim} input arguments. ## ## @code{var (@dots{}, @var{nanflag})} specifies whether to exclude NaN values ## from the calculation using any of the input argument combinations in previous ## syntaxes. The default value for @var{nanflag} is "includenan", and keeps NaN ## values in the calculation. To exclude NaN values, set the value of ## @var{nanflag} to "omitnan". ## ## @code{[@var{v}, @var{m}] = var (@dots{})} also returns the mean of the ## elements of @var{x} used to calculate the variance. If @var{v} is the ## weighted variance, then @var{m} is the weighted mean. ## ## @seealso{std, mean} ## @end deftypefn function [v, m] = var (x, varargin) if (nargin < 1 || nargin > 4) print_usage (); endif ## initialize variables all_flag = false; omitnan = false; nvarg = numel (varargin); varg_chars = cellfun ('ischar', varargin); ## Check all char arguments. if (nvarg == 3 && ! varg_chars(3)) print_usage (); endif if (any (varg_chars)) for i = varargin(varg_chars) switch (lower (i{:})) case "all" all_flag = true; case "omitnan" omitnan = true; case "includenan" omitnan = false; otherwise print_usage (); endswitch endfor varargin(varg_chars) = []; nvarg = numel (varargin); endif # FIXME: when sparse can use broadcast ops, remove sparse checks and hacks sprs_x = issparse (x); w = 0; weighted = false; # true if weight vector/array used vecdim = []; vecempty = true; vecdim_scalar_vector = [false, false]; # [false, false] for empty vecdim szx = size (x); ndx = ndims (x); ## Check numeric arguments if (! (isnumeric (x))) error ("var: X must be a numeric vector or matrix."); endif if (isa (x, "single")) outtype = "single"; else outtype = "double"; endif if (nvarg > 0) if (nvarg > 2 || any (! cellfun ('isnumeric', varargin))) print_usage (); endif ## Process weight input if (any (varargin{1} < 0)) error ("var: weights must not contain any negative values."); endif if (isscalar (varargin{1})) w = varargin{1}; if (! (w == 0 || w == 1) && ! isscalar (x)) error ("var: normalization scalar must be either 0 or 1."); endif elseif (numel (varargin{1}) > 1) weights = varargin{1}; weighted = true; endif if (nvarg > 1) ## Process dimension input vecdim = varargin{2}; if (! (vecempty = isempty (vecdim))) ## Check for empty vecdim, won't change vsv if nonzero size empty vecdim_scalar_vector = [isscalar(vecdim), isvector(vecdim)]; endif if (! (vecdim_scalar_vector(2) && all (vecdim > 0)) ... || any (rem (vecdim, 1))) error ("var: DIM must be a positive integer scalar or vector."); endif if (vecdim_scalar_vector(1) && vecdim > ndx && ! isempty (x)) ## Scalar dimension larger than ndims(x), variance of any single number ## is zero, except for inf, NaN, and empty values of x. v = zeros (szx, outtype); vn = ! isfinite (x); v(vn) = NaN; m = x; return; endif if (vecdim_scalar_vector == [0 1] && (! all (diff (sort (vecdim))))) error ("var: VECDIM must contain non-repeating positive integers."); endif endif endif ## Check for conflicting input arguments if (all_flag && ! vecempty) error ("var: 'all' flag cannot be used with DIM or VECDIM options."); endif if (weighted) if (all_flag) if (isvector (weights)) if (numel (weights) != numel (x)) error ("var: weight vector element count does not match X."); endif elseif (! (isequal (size (weights), szx))) error ("var: weight matrix or array does not match X in size."); endif elseif (vecempty) dim = find (szx > 1, 1); if length (dim) == 0 dim = 1; endif if (isvector (weights)) if (numel (weights) != szx(dim)) error (["var: weight vector length does not match operating ", ... "dimension."]); endif elseif (! isequal (size (weights), szx)) error ("var: weight matrix or array does not match X in size."); endif elseif (vecdim_scalar_vector(1)) if (isvector (weights)) if (numel (weights) != szx(vecdim)) error (["var: weight vector length does not match operating ", ... "dimension."]); endif elseif (! isequal (size (weights), szx)) error ("var: weight matrix or array does not match X in size."); endif elseif (vecdim_scalar_vector(2) && ! (isequal (size (weights), szx))) error ("var: weight matrix or array does not match X in size."); endif endif ## Force output for X being empty or scalar if (isempty (x)) if (vecempty && (ndx == 2 || all ((szx) == 0))) v = NaN (outtype); if (nargout > 1) m = NaN (outtype); endif return; endif if (vecdim_scalar_vector(1)) szx(vecdim) = 1; v = NaN (szx, outtype); if (nargout > 1) m = NaN (szx, outtype); endif return; endif endif if (isscalar (x)) if (isfinite (x)) v = zeros (outtype); else v = NaN (outtype); endif if (nargout > 1) m = x; endif return; endif if (nvarg == 0) ## Only numeric input argument, no dimensions or weights. if (all_flag) x = x(:); if (omitnan) x = x(! isnan (x)); endif n = length (x); m = sum (x) ./ n; v = sum (abs (x - m) .^ 2) ./ (n - 1 + w); if (n == 1) v = 0; endif else dim = find (szx > 1, 1); if length (dim) == 0 dim = 1; endif n = szx(dim); if (omitnan) n = sum (! isnan (x), dim); xn = isnan (x); x(xn) = 0; endif m = sum (x, dim) ./ n; dims = ones (1, ndx); dims(dim) = szx(dim); if (sprs_x) m_exp = repmat (m, dims); else m_exp = m .* ones (dims); endif if (omitnan) x(xn) = m_exp(xn); endif v = sumsq (x - m_exp, dim) ./ (n - 1 + w); if (numel (n) == 1) divby0 = n .* ones (size (v)) == 1; else divby0 = n == 1; endif v(divby0) = 0; endif elseif (nvarg == 1) ## Two numeric input arguments, w or weights given. if (all_flag) x = x(:); if (weighted) weights = weights(:); wx = weights .* x; else weights = ones (length (x), 1); wx = x; endif if (omitnan) xn = isnan (wx); wx = wx(! xn); weights = weights(! xn); x = x(! xn); endif n = length (wx); m = sum (wx) ./ sum (weights); if (weighted) v = sum (weights .* (abs (x - m) .^ 2)) ./ sum (weights); else v = sum (weights .* (abs (x - m) .^ 2)) ./ (n - 1 + w); if (n == 1) v = 0; endif endif else dim = find (szx > 1, 1); if length (dim) == 0 dim = 1; endif if (! weighted) weights = ones (szx); wx = x; else if (isvector (weights)) dims = 1:ndx; dims([1, dim]) = [dim, 1]; weights = zeros (szx) + permute (weights(:), dims); endif wx = weights .* x; endif n = size (wx, dim); if (omitnan) xn = isnan (wx); n = sum (! xn, dim); wx(xn) = 0; weights(xn) = 0; endif m = sum (wx, dim) ./ sum (weights, dim); dims = ones (1, ndims (wx)); dims(dim) = size (wx, dim); if (sprs_x) m_exp = repmat (m, dims); else m_exp = m .* ones (dims); endif if (omitnan) x(xn) = m_exp(xn); endif if (weighted) v = sum (weights .* ((x - m_exp) .^ 2), dim) ./ sum (weights, dim); else v = sumsq (x - m_exp, dim) ./ (n - 1 + w); if (numel (n) == 1) divby0 = n .* ones (size (v)) == 1; else divby0 = n == 1; endif v(divby0) = 0; endif endif elseif (nvarg == 2) ## Three numeric input arguments, both w or weights and dim or vecdim given. if (vecdim_scalar_vector(1)) if (!weighted) weights = ones (szx); wx = x; else if (isvector (weights)) dims = 1:ndx; dims([1, vecdim]) = [vecdim, 1]; weights = zeros (szx) + permute (weights(:), dims); endif wx = weights .* x; endif n = size (wx, vecdim); if (omitnan) n = sum (! isnan (wx), vecdim); xn = isnan (wx); wx(xn) = 0; weights(xn) = 0; endif m = sum (wx, vecdim) ./ sum (weights, vecdim); dims = ones (1, ndims (wx)); dims(vecdim) = size (wx, vecdim); if (sprs_x) m_exp = repmat (m, dims); else m_exp = m .* ones (dims); endif if (omitnan) x(xn) = m_exp(xn); endif if (weighted) v = sum (weights .* ((x - m_exp) .^ 2), vecdim) ... ./ sum (weights, vecdim); else v = sumsq (x - m_exp, vecdim); vn = isnan (v); v = v ./ (n - 1 + w); if (numel (n) == 1) divby0 = n .* ones (size (v)) == 1; else divby0 = n == 1; endif v(divby0) = 0; v(vn) = NaN; endif else ## Weights and nonscalar vecdim specified ## Ignore exceeding dimensions in VECDIM remdims = 1 : ndx; # all dimensions vecdim(find (vecdim > ndx)) = []; ## Calculate permutation vector remdims(vecdim) = []; # delete dimensions specified by vecdim nremd = numel (remdims); ## If all dimensions are given, it is similar to all flag if (nremd == 0) x = x(:); if (weighted) weights = weights(:); wx = weights .* x; else weights = ones (length (x), 1); wx = x; endif if (omitnan) xn = isnan (wx); wx = wx(! xn); weights = weights(! xn); x = x(! xn); endif n = length (wx); m = sum (wx) ./ sum (weights); if (weighted) v = sum (weights .* (abs (x - m) .^ 2)) ./ sum (weights); else v = sum (weights .* (abs (x - m) .^ 2)) ./ (n - 1 + w); if (n == 1) v = 0; endif endif else ## FIXME: much of the reshaping can be skipped once octave's sum can ## take a vecdim argument. ## Apply weights if (weighted) wx = weights .* x; else weights = ones (szx); wx = x; endif ## Permute to bring remaining dims forward perm = [remdims, vecdim]; wx = permute (wx, perm); weights = permute (weights, perm); x = permute (x, perm); ## Reshape to put all vecdims in final dimension szwx = size (wx); sznew = [szwx(1:nremd), prod(szwx(nremd+1:end))]; wx = reshape (wx, sznew); weights = reshape (weights, sznew); x = reshape (x, sznew); ## Calculate var on single, squashed dimension dim = nremd + 1; n = size (wx, dim); if (omitnan) xn = isnan (wx); n = sum (! xn, dim); wx(xn) = 0; weights(xn) = 0; endif m = sum (wx, dim) ./ sum (weights, dim); m_exp = zeros (sznew) + m; if (omitnan) x(xn) = m_exp(xn); endif if (weighted) v = sum (weights .* ((x - m_exp) .^ 2), dim) ./ sum (weights, dim); else v = sumsq (x - m_exp, dim) ./ (n - 1 + w); if (numel (n) == 1) divby0 = n .* ones (size (v)) == 1; else divby0 = n == 1; endif v(divby0) = 0; endif ## Inverse permute back to correct dimensions v = ipermute (v, perm); if (nargout > 1) m = ipermute (m, perm); endif endif endif endif ## Preserve class type if (nargout < 2) if strcmp (outtype, "single") v = single (v); else v = double (v); endif else if strcmp (outtype, "single") v = single (v); m = single (m); else v = double (v); m = double (m); endif endif endfunction %!assert (var (13), 0) %!assert (var (single (13)), single (0)) %!assert (var ([1,2,3]), 1) %!assert (var ([1,2,3], 1), 2/3, eps) %!assert (var ([1,2,3], [], 1), [0,0,0]) %!assert (var ([1,2,3], [], 3), [0,0,0]) %!assert (var (5, 99), 0) %!assert (var (5, 99, 1), 0) %!assert (var (5, 99, 2), 0) %!assert (var ([5 3], [99 99], 2), 1) %!assert (var ([1:7], [1:7]), 3) %!assert (var ([eye(3)], [1:3]), [5/36, 2/9, 1/4], eps) %!assert (var (ones (2,2,2), [1:2], 3), [(zeros (2,2))]) %!assert (var ([1 2; 3 4], 0, 'all'), var ([1:4])) %!assert (var (reshape ([1:8], 2, 2, 2), 0, [1 3]), [17/3 17/3], eps) %!assert (var ([1 2 3;1 2 3], [], [1 2]), 0.8, eps) ## Test single input and optional arguments "all", DIM, "omitnan") %!test %! x = [-10:10]; %! y = [x;x+5;x-5]; %! assert (var (x), 38.5); %! assert (var (y, [], 2), [38.5; 38.5; 38.5]); %! assert (var (y, 0, 2), [38.5; 38.5; 38.5]); %! assert (var (y, 1, 2), ones (3,1) * 36.66666666666666, 1e-14); %! assert (var (y, "all"), 54.19354838709678, 1e-14); %! y(2,4) = NaN; %! assert (var (y, "all"), NaN); %! assert (var (y, "all", "includenan"), NaN); %! assert (var (y, "all", "omitnan"), 55.01533580116342, 1e-14); %! assert (var (y, 0, 2, "includenan"), [38.5; NaN; 38.5]); %! assert (var (y, [], 2), [38.5; NaN; 38.5]); %! assert (var (y, [], 2, "omitnan"), [38.5; 37.81842105263158; 38.5], 1e-14); ## Tests for different weight and omitnan code paths %!assert (var ([1 NaN 3], [1 2 3], "omitnan"), 0.75, eps) %!assert (var ([1 2 3], [1 NaN 3], "omitnan"), 0.75, eps) %!assert (var (magic(3), [1 NaN 3], "omitnan"), [3 12 3], eps) %!assert (var ([1 NaN 3], [1 2 3], "omitnan", "all"), 0.75, eps) %!assert (var ([1 NaN 3], [1 2 3], "all", "omitnan"), 0.75, eps) %!assert (var ([1 2 3], [1 NaN 3], "omitnan", "all"), 0.75, eps) %!assert (var ([1 NaN 3], [1 2 3], 2, "omitnan"), 0.75, eps) %!assert (var ([1 2 3], [1 NaN 3], 2, "omitnan"), 0.75, eps) %!assert (var (magic(3), [1 NaN 3], 1, "omitnan"), [3 12 3], eps) %!assert (var (magic(3), [1 NaN 3], 2, "omitnan"), [0.75;3;0.75], eps) %!assert (var ([4 4; 4 6; 6 6], [1 3], 2, 'omitnan'), [0;0.75;0], eps) %!assert (var ([4 NaN; 4 6; 6 6], [1 2 3], 1, 'omitnan'), [1 0]) %!assert (var ([4 NaN; 4 6; 6 6], [1 3], 2, 'omitnan'), [0;0.75;0], eps) %!assert (var (3*reshape(1:18, [3 3 2]), [1 2 3], 1, 'omitnan'), ones(1,3,2)*5) %!assert (var (reshape(1:18, [3 3 2]), [1 2 3], 2, 'omitnan'), 5*ones(3,1,2)) %!assert (var (3*reshape(1:18, [3 3 2]), ones (3,3,2), [1 2], 'omitnan'), ... %! 60 * ones(1,1,2)) %!assert (var (3*reshape(1:18, [3 3 2]), ones (3,3,2), [1 4], 'omitnan'), ... %! 6 * ones(1,3,2)) %!assert (var (6*reshape(1:18, [3 3 2]), ones (3,3,2), [1:3], 'omitnan'), 969) %!test %! x = reshape(1:18, [3 3 2]); %! x([2, 14]) = NaN; %! w = ones (3,3,2); %! assert (var (16*x, w, [1:3], 'omitnan'), 6519); %!test %! x = reshape(1:18, [3 3 2]); %! w = ones (3,3,2); %! w([2, 14]) = NaN; %! assert (var (16*x, w, [1:3], 'omitnan'), 6519); ## Test input case insensitivity %!assert (var ([1 2 3], "aLl"), 1); %!assert (var ([1 2 3], "OmitNan"), 1); %!assert (var ([1 2 3], "IncludeNan"), 1); ## Test dimension indexing with vecdim in n-dimensional arrays %!test %! x = repmat ([1:20;6:25], [5, 2, 6, 3]); %! assert (size (var (x, 0, [3 2])), [10, 1, 1, 3]); %! assert (size (var (x, 1, [1 2])), [1, 1, 6, 3]); %! assert (size (var (x, [], [1 2 4])), [1, 1, 6]); %! assert (size (var (x, 0, [1 4 3])), [1, 40]); %! assert (size (var (x, [], [1 2 3 4])), [1, 1]); ## Test matrix with vecdim, weighted, matrix weights, omitnan %!assert (var (3*magic(3)), [63 144 63]) %!assert (var (3*magic(3), 'omitnan'), [63 144 63]) %!assert (var (3*magic(3), 1), [42 96 42]) %!assert (var (3*magic(3), 1, 'omitnan'), [42 96 42]) %!assert (var (3*magic(3), ones(1,3), 1), [42 96 42]) %!assert (var (3*magic(3), ones(1,3), 1, 'omitnan'), [42 96 42]) %!assert (var (2*magic(3), [1 1 NaN], 1, 'omitnan'), [25 16 1]) %!assert (var (3*magic(3), ones(3,3)), [42 96 42]) %!assert (var (3*magic(3), ones(3,3), 'omitnan'), [42 96 42]) %!assert (var (3*magic(3), [1 1 1; 1 1 1; 1 NaN 1], 'omitnan'), [42 36 42]) %!assert (var (3*magic(3), ones(3,3), 1), [42 96 42]) %!assert (var (3*magic(3), ones(3,3), 1, 'omitnan'), [42 96 42]) %!assert (var (3*magic(3), [1 1 1; 1 1 1; 1 NaN 1], 1, 'omitnan'), [42 36 42]) %!assert (var (3*magic(3), ones(3,3), [1 4]), [42 96 42]) %!assert (var (3*magic(3), ones(3,3), [1 4], 'omitnan'), [42 96 42]) %!assert (var (3*magic(3), [1 1 1; 1 1 1; 1 NaN 1],[1 4],'omitnan'), [42 36 42]) ## Test results with vecdim in n-dimensional arrays and "omitnan" %!test %! x = repmat ([1:20;6:25], [5, 2, 6, 3]); %! v = repmat (33.38912133891213, [10, 1, 1, 3]); %! assert (var (x, 0, [3, 2]), v, 1e-14); %! v = repmat (33.250, [10, 1, 1, 3]); %! assert (var (x, 1, [3, 2]), v, 1e-14); %! x(2,5,6,3) = NaN; %! v(2,1,1,3) = NaN; %! assert (var (x, 1, [3, 2]), v, 4e-14); %! v = repmat (33.38912133891213, [10 1 1 3]); %! v(2,1,1,3) = NaN; %! assert (var (x, [], [3, 2]), v, 4e-14); %! v(2,1,1,3) = 33.40177912169048; %! assert (var (x, [], [3, 2], "omitnan"), v, 4e-14); ## Testing weights vector & arrays %!assert (var (ones (2,2,2), [1:2], 3), [(zeros (2, 2))]) %!assert (var (magic (3), [1:9], "all"), 6.666666666666667, 1e-14) ## Test exceeding dimensions %!assert (var (ones (2,2), [], 3), zeros (2,2)) %!assert (var (ones (2,2,2), [], 99), zeros (2,2,2)) %!assert (var (magic (3), [], 3), zeros (3,3)) %!assert (var (magic (3), [], 1), [7, 16, 7]) %!assert (var (magic (3), [], [1 3]), [7, 16, 7]) %!assert (var (magic (3), [], [1 99]), [7, 16, 7]) ## Test empty inputs %!assert (var ([]), NaN) %!assert (class (var (single ([]))), "single") %!assert (var ([],[],1), NaN(1,0)) %!assert (var ([],[],2), NaN(0,1)) %!assert (var ([],[],3), []) %!assert (class (var (single ([]), [], 1)), "single") %!assert (var (ones (1,0)), NaN) %!assert (var (ones (1,0), [], 1), NaN(1,0)) %!assert (var (ones (1,0), [], 2), NaN) %!assert (var (ones (1,0), [], 3), NaN(1,0)) %!assert (class (var (ones (1, 0, "single"), [], 1)), "single") %!assert (var (ones (0,1)), NaN) %!assert (var (ones (0,1), [], 1), NaN) %!assert (var (ones (0,1), [], 2), NaN(0,1)) %!assert (var (ones (0,1), [], 3), NaN(0,1)) %!assert (var (ones (1,3,0,2)), NaN(1,1,0,2)) %!assert (var (ones (1,3,0,2), [], 1), NaN(1,3,0,2)) %!assert (var (ones (1,3,0,2), [], 2), NaN(1,1,0,2)) %!assert (var (ones (1,3,0,2), [], 3), NaN(1,3,1,2)) %!assert (var (ones (1,3,0,2), [], 4), NaN(1,3,0)) %!test %! [~, m] = var ([]); %! assert (m, NaN); ## Test optional mean output %!test <*62395> %! [~, m] = var (13); %! assert (m, 13); %! [~, m] = var (single(13)); %! assert (m, single(13)); %! [~, m] = var ([1, 2, 3; 3 2 1], []); %! assert (m, [2 2 2]); %! [~, m] = var ([1, 2, 3; 3 2 1], [], 1); %! assert (m, [2 2 2]); %! [~, m] = var ([1, 2, 3; 3 2 1], [], 2); %! assert (m, [2 2]'); %! [~, m] = var ([1, 2, 3; 3 2 1], [], 3); %! assert (m, [1 2 3; 3 2 1]); ## Test mean output, weighted inputs, vector dims %!test <*62395> %! [~, m] = var (5,99); %! assert (m, 5); %! [~, m] = var ([1:7], [1:7]); %! assert (m, 5); %! [~, m] = var ([eye(3)], [1:3]); %! assert (m, [1/6, 1/3, 0.5], eps); %! [~, m] = var (ones (2,2,2), [1:2], 3); %! assert (m, ones (2,2)); %! [~, m] = var ([1 2; 3 4], 0, 'all'); %! assert (m, 2.5, eps); %! [~, m] = var (reshape ([1:8], 2, 2, 2), 0, [1 3]); %! assert (m, [3.5, 5.5], eps); %!test %! [v, m] = var (4 * eye (2), [1, 3]); %! assert (v, [3, 3]); %! assert (m, [1, 3]); ## Test mean output, empty inputs, omitnan %!test <*62395> %! [~, m] = var ([]); %! assert (m, NaN); #%! [~, m] = var ([],[],1); #%! assert (m, NaN(1,0)); #%! [~, m] = var ([],[],2); #%! assert (m, NaN(0,1)); #%! [~, m] = var ([],[],3); #%! assert (m, []); #%! [~, m] = var (ones (1,3,0,2)); #%! assert (m, NaN(1,1,0,2)); ## Test mean output, nD array %!test <*62395> %! x = repmat ([1:20;6:25], [5, 2, 6, 3]); %! [~, m] = var (x, 0, [3 2]); %! assert (m, mean (x, [3 2])); %! [~, m] = var (x, 0, [1 2]); %! assert (m, mean (x, [1 2])); %! [~, m] = var (x, 0, [1 3 4]); %! assert (m, mean (x, [1 3 4])); %!test %! x = repmat ([1:20;6:25], [5, 2, 6, 3]); %! x(2,5,6,3) = NaN; %! [~, m] = var (x, 0, [3 2], "omitnan"); %! assert (m, mean (x, [3 2], "omitnan")); ## Test Inf and NaN inputs %!test <*63203> %! [v, m] = var (Inf); %! assert (v, NaN); %! assert (m, Inf); %!test <*63203> %! [v, m] = var (NaN); %! assert (v, NaN); %! assert (m, NaN); %!test <*63203> %! [v, m] = var ([1, Inf, 3]); %! assert (v, NaN); %! assert (m, Inf); %!test <*63203> %! [v, m] = var ([1, Inf, 3]'); %! assert (v, NaN); %! assert (m, Inf); %!test <*63203> %! [v, m] = var ([1, NaN, 3]); %! assert (v, NaN); %! assert (m, NaN); %!test <*63203> %! [v, m] = var ([1, NaN, 3]'); %! assert (v, NaN); %! assert (m, NaN); %!test <*63203> %! [v, m] = var ([1, Inf, 3], [], 1); %! assert (v, [0, NaN, 0]); %! assert (m, [1, Inf, 3]); %!test <*63203> %! [v, m] = var ([1, Inf, 3], [], 2); %! assert (v, NaN); %! assert (m, Inf); %!test <*63203> %! [v, m] = var ([1, Inf, 3], [], 3); %! assert (v, [0, NaN, 0]); %! assert (m, [1, Inf, 3]); %!test <*63203> %! [v, m] = var ([1, NaN, 3], [], 1); %! assert (v, [0, NaN, 0]); %! assert (m, [1, NaN, 3]); %!test <*63203> %! [v, m] = var ([1, NaN, 3], [], 2); %! assert (v, NaN); %! assert (m, NaN); %!test <*63203> %! [v, m] = var ([1, NaN, 3], [], 3); %! assert (v, [0, NaN, 0]); %! assert (m, [1, NaN, 3]); %!test <*63203> %! [v, m] = var ([1, 2, 3; 3, Inf, 5]); %! assert (v, [2, NaN, 2]); %! assert (m, [2, Inf, 4]); %!test <*63203> %! [v, m] = var ([1, Inf, 3; 3, Inf, 5]); %! assert (v, [2, NaN, 2]); %! assert (m, [2, Inf, 4]); %!test <*63203> %! [v, m] = var ([1, 2, 3; 3, NaN, 5]); %! assert (v, [2, NaN, 2]); %! assert (m, [2, NaN, 4]); %!test <*63203> %! [v, m] = var ([1, NaN, 3; 3, NaN, 5]); %! assert (v, [2, NaN, 2]); %! assert (m, [2, NaN, 4]); %!test <*63203> %! [v, m] = var ([Inf, 2, NaN]); %! assert (v, NaN); %! assert (m, NaN); %!test <*63203> %! [v, m] = var ([Inf, 2, NaN]'); %! assert (v, NaN); %! assert (m, NaN); %!test <*63203> %! [v, m] = var ([NaN, 2, Inf]); %! assert (v, NaN); %! assert (m, NaN); %!test <*63203> %! [v, m] = var ([NaN, 2, Inf]'); %! assert (v, NaN); %! assert (m, NaN); %!test <*63203> %! [v, m] = var ([Inf, 2, NaN], [], 1); %! assert (v, [NaN, 0, NaN]); %! assert (m, [Inf, 2, NaN]); %!test <*63203> %! [v, m] = var ([Inf, 2, NaN], [], 2); %! assert (v, NaN); %! assert (m, NaN); %!test <*63203> %! [v, m] = var ([NaN, 2, Inf], [], 1); %! assert (v, [NaN, 0, NaN]); %! assert (m, [NaN, 2, Inf]); %!test <*63203> %! [v, m] = var ([NaN, 2, Inf], [], 2); %! assert (v, NaN); %! assert (m, NaN); %!test <*63203> %! [v, m] = var ([1, 3, NaN; 3, 5, Inf]); %! assert (v, [2, 2, NaN]); %! assert (m, [2, 4, NaN]); %!test <*63203> %! [v, m] = var ([1, 3, Inf; 3, 5, NaN]); %! assert (v, [2, 2, NaN]); %! assert (m, [2, 4, NaN]); ## Test sparse/diagonal inputs %!test <*63291> %! [v, m] = var (2 * eye (2)); %! assert (v, [2, 2]); %! assert (m, [1, 1]); %!test <*63291> %! [v, m] = var (4 * eye (2), [1, 3]); %! assert (v, [3, 3]); %! assert (m, [1, 3]); %!test <*63291> %! [v, m] = var (sparse (2 * eye (2))); %! assert (full (v), [2, 2]); %! assert (full (m), [1, 1]); %!test <*63291> %! [v, m] = var (sparse (4 * eye (2)), [1, 3]); %! assert (full (v), [3, 3]); %! assert (full (m), [1, 3]); %!test<*63291> %! [v, m] = var (sparse (eye (2))); %! assert (issparse (v)); %! assert (issparse (m)); %!test<*63291> %! [v, m] = var (sparse (eye (2)), [1, 3]); %! assert (issparse (v)); %! assert (issparse (m)); ## Test input validation %!error var () %!error var (1, 2, "omitnan", 3) %!error var (1, 2, 3, 4) %!error var (1, 2, 3, 4, 5) %!error var (1, "foo") %!error var (1, [], "foo") %!error var ([1 2 3], 2) %!error var ([1 2], 2, "all") %!error var ([1 2],0.5, "all") %!error var (1, -1) %!error var (1, [1 -1]) %!error ... %! var ([1 2 3], [1 -1 0]) %!error var ({1:5}) %!error var ("char") %!error var (['A'; 'B']) %!error var (1, [], ones (2,2)) %!error var (1, 0, 1.5) %!error var (1, [], 0) %!error var (1, [], 1.5) %!error var ([1 2 3], [], [-1 1]) %!error ... %! var (repmat ([1:20;6:25], [5 2 6 3]), 0, [1 2 2 2]) %!error ... %! var ([1 2], eye (2)) %!error ... %! var ([1 2 3 4], [1 2; 3 4]) %!error ... %! var ([1 2 3 4], [1 2; 3 4], 1) %!error ... %! var ([1 2 3 4], [1 2; 3 4], [2 3]) %!error ... %! var (ones (2, 2), [1 2], [1 2]) %!error ... %! var ([1 2 3 4; 5 6 7 8], [1 2 1 2 1; 1 2 1 2 1], 1) %!error ... %! var (repmat ([1:20;6:25], [5 2 6 3]), repmat ([1:20;6:25], [5 2 3]), [2 3]) %!error var ([1 2 3; 2 3 4], [1 3 4]) %!error var ([1 2], [1 2 3]) %!error var (1, [1 2]) %!error var ([1 2 3; 2 3 4], [1 3 4], 1) %!error var ([1 2 3; 2 3 4], [1 3], 2) %!error var ([1 2], [1 2], 1) %!error <'all' flag cannot be used with DIM or VECDIM options> ... %! var (1, [], 1, "all") %!error ... %! var ([1 2 3; 2 3 4], [1 3], "all") %!error ... %! var (repmat ([1:20;6:25], [5 2 6 3]), repmat ([1:20;6:25], [5 2 3]), "all") statistics-release-1.6.3/inst/sigma_pts.m000066400000000000000000000077451456127120000204650ustar00rootroot00000000000000## Copyright (C) 2017 - Juan Pablo Carbajal ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation; either version 3 of the License, or ## (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program. If not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{pts} =} sigma_pts (@var{n}) ## @deftypefnx {statistics} {@var{pts} =} sigma_pts (@var{n}, @var{m}) ## @deftypefnx {statistics} {@var{pts} =} sigma_pts (@var{n}, @var{m}, @var{K}) ## @deftypefnx {statistics} {@var{pts} =} sigma_pts (@var{n}, @var{m}, @var{K}, @var{l}) ## ## Calculates 2*@var{n}+1 sigma points in @var{n} dimensions. ## ## Sigma points are used in the unscented transfrom to estimate ## the result of applying a given nonlinear transformation to a probability ## distribution that is characterized only in terms of a finite set of statistics. ## ## If only the dimension @var{n} is given the resulting points have zero mean ## and identity covariance matrix. ## If the mean @var{m} or the covaraince matrix @var{K} are given, then the resulting points ## will have those statistics. ## The factor @var{l} scaled the points away from the mean. It is useful to tune ## the accuracy of the unscented transfrom. ## ## There is no unique way of computing sigma points, this function implements the ## algorithm described in section 2.6 "The New Filter" pages 40-41 of ## ## Uhlmann, Jeffrey (1995). "Dynamic Map Building and Localization: New Theoretical Foundations". ## Ph.D. thesis. University of Oxford. ## ## @end deftypefn function pts = sigma_pts (n, m = [], K = [], l = 0) if isempty (K) K = eye (n); endif if isempty (m) m = zeros (1, n); endif if (n ~= length (m)) error ("Dimension and size of mean vector don't match.") endif if any(n ~= size (K)) error ("Dimension and size of covariance matrix don't match.") endif if isdefinite (K) <= 0 error ("Covariance matrix should be positive definite.") endif pts = zeros (2 * n + 1, n); pts(1,:) = m; K = sqrtm ((n + l) * K); pts(2:n+1,:) = bsxfun (@plus, m , K); pts(n+2:end,:) = bsxfun (@minus, m , K); endfunction %!demo %! K = [1 0.5; 0.5 1]; # covaraince matrix %! # calculate and build associated ellipse %! [R,S,~] = svd (K); %! theta = atan2 (R(2,1), R(1,1)); %! v = sqrt (diag (S)); %! v = v .* [cos(theta) sin(theta); -sin(theta) cos(theta)]; %! t = linspace (0, 2*pi, 100).'; %! xe = v(1,1) * cos (t) + v(2,1) * sin (t); %! ye = v(1,2) * cos (t) + v(2,2) * sin (t); %! %! figure(1); clf; hold on %! # Plot ellipse and axes %! line ([0 0; v(:,1).'],[0 0; v(:,2).']) %! plot (xe,ye,'-r'); %! %! col = 'rgb'; %! l = [-1.8 -1 1.5]; %! for li = 1:3 %! p = sigma_pts (2, [], K, l(li)); %! tmp = plot (p(2:end,1), p(2:end,2), ['x' col(li)], ... %! p(1,1), p(1,2), ['o' col(li)]); %! h(li) = tmp(1); %! endfor %! hold off %! axis image %! legend (h, arrayfun (@(x) sprintf ("l:%.2g", x), l, "unif", 0)); %!test %! p = sigma_pts (5); %! assert (mean (p), zeros(1,5), sqrt(eps)); %! assert (cov (p), eye(5), sqrt(eps)); %!test %! m = randn(1, 5); %! p = sigma_pts (5, m); %! assert (mean (p), m, sqrt(eps)); %! assert (cov (p), eye(5), sqrt(eps)); %!test %! x = linspace (0,1,5); %! K = exp (- (x.' - x).^2/ 0.5); %! p = sigma_pts (5, [], K); %! assert (mean (p), zeros(1,5), sqrt(eps)); %! assert (cov (p), K, sqrt(eps)); %!error sigma_pts(2,1); %!error sigma_pts(2,[],1); %!error sigma_pts(2,1,1); %!error sigma_pts(2,[0.5 0.5],[-1 0; 0 0]); statistics-release-1.6.3/inst/signtest.m000066400000000000000000000167151456127120000203340ustar00rootroot00000000000000## Copyright (C) 2014 Tony Richardson ## Copyright (C) 2022 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {[@var{pval}, @var{h}, @var{stats}] =} signtest (@var{x}) ## @deftypefnx {statistics} {[@var{pval}, @var{h}, @var{stats}] =} signtest (@var{x}, @var{m}) ## @deftypefnx {statistics} {[@var{pval}, @var{h}, @var{stats}] =} signtest (@var{x}, @var{y}) ## @deftypefnx {statistics} {[@var{pval}, @var{h}, @var{stats}] =} signtest (@var{x}, @var{y}, @var{Name}, @var{Value}) ## ## Test for median. ## ## Perform a signtest of the null hypothesis that @var{x} is from a distribution ## that has a zero median. @var{x} must be a vector. ## ## If the second argument @var{m} is a scalar, the null hypothesis is that ## X has median m. ## ## If the second argument @var{y} is a vector, the null hypothesis is that ## the distribution of @code{@var{x} - @var{y}} has zero median. ## ## The argument @qcode{"alpha"} can be used to specify the significance level ## of the test (the default value is 0.05). The string ## argument @qcode{"tail"}, can be used to select the desired alternative ## hypotheses. If @qcode{"tail"} is @qcode{"both"} (default) the null is ## tested against the two-sided alternative @code{median (@var{x}) != @var{m}}. ## If @qcode{"tail"} is @qcode{"right"} the one-sided ## alternative @code{median (@var{x}) > @var{m}} is considered. ## Similarly for @qcode{"left"}, the one-sided alternative @code{median ## (@var{x}) < @var{m}} is considered. ## ## When @qcode{"method"} is @qcode{"exact"} the p-value is computed using an ## exact method. When @qcode{"method"} is @qcode{"approximate"} a normal ## approximation is used for the test statistic. When @qcode{"method"} is not ## defined as an optional input argument, then for @code{length (@var{x}) < 100} ## the @qcode{"exact"} method is computed, otherwise the @qcode{"approximate"} ## method is used. ## ## The p-value of the test is returned in @var{pval}. If @var{h} is 0 the ## null hypothesis is accepted, if it is 1 the null hypothesis is rejected. ## @var{stats} is a structure containing the value of the test statistic ## (@var{sign}) and the value of the z statistic (@var{zval}) (only computed ## when the 'method' is 'approximate'. ## ## @end deftypefn function [p, h, stats] = signtest (x, my, varargin) ## Check X being a vector if ! isvector (x) error ("signtest: X must be a vector."); endif ## Add defaults my_default = 0; alpha = 0.05; tail = "both"; ## Matlab compliant default method selection method_present = false; if length (x) < 100 method = "exact"; else method = "approximate"; endif ## When called with a single input argument of second argument is empty if nargin == 1 || isempty (my) my = zeros (size (x)); endif ## If second argument is a scalar convert to vector or check for Y being a ## vector and that X and Y have equal lengths if isscalar (my) my = repmat (my, size (x)); elseif ! isvector (my) error ("signtest: Y must be either a scalar of a vector."); elseif numel (x) != numel (my) error ("signtest: X and Y vectors have different lengths."); endif ## Get optional input arguments i = 1; while (i <= length (varargin)) switch lower (varargin{i}) case "alpha" i = i + 1; alpha = varargin{i}; case "tail" i = i + 1; tail = varargin{i}; case "method" i = i + 1; method = varargin{i}; method_present = true; otherwise error ("signtest: Invalid Name argument."); endswitch i = i + 1; endwhile ## Check values for optional input arguments if (! isnumeric (alpha) || isnan (alpha) || ! isscalar (alpha) ... || alpha <= 0 || alpha >= 1) error ("signtest: alpha must be a numeric scalar in the range (0,1)."); endif if ! isa (tail, 'char') error ("signtest: tail argument must be a string"); elseif sum (strcmpi (tail, {"both", "right", "left"})) < 1 error ("signtest: tail value must be either 'both', right' or 'left'."); endif if ! isa (method, 'char') error("signtest: method argument must be a string"); elseif sum (strcmpi (method, {"exact", "approximate"})) < 1 error ("signtest: method value must be either 'exact' or 'approximate'."); end ## Calculate differences between X and Y vectors: remove equal values of NaNs XY_diff = x(:) - my(:); NO_diff = (XY_diff == 0); XY_diff(NO_diff | isnan (NO_diff)) = []; ## Recalculate remaining length of X vector (after equal or NaNs removal) n = length (XY_diff); ## Check for identical X and Y input arguments if n == 0 p = 1; h = 0; stats.sign = 0; stats.zval = NaN; return; endif ## Re-evaluate method selection if ! method_present && n < 100 method = "exact"; else method = "approximate"; endif ## Get the number of positive and negative elements from X-Y differences pos_n = length (find (XY_diff > 0)); neg_n = n - pos_n; ## Calculate stats according to selected method and tail switch lower(method) case "exact" stats.zval = nan; switch lower(tail) case "both" p = 2 * binocdf (min (neg_n, pos_n), n, 0.5); p = min (1, p); case "left" p = binocdf (pos_n, n, 0.5); case "right" p = binocdf (neg_n, n, 0.5); endswitch stats.zval = NaN; case 'approximate' switch lower(tail) case 'both' z_value = (pos_n - neg_n - sign (pos_n - neg_n)) / sqrt (n); p = 2 * normcdf (- abs (z_value)); case 'left' z_value = (pos_n - neg_n + 1) / sqrt (n); p = normcdf (z_value); case 'right' z_value = (pos_n - neg_n - 1) / sqrt (n); p = normcdf (- z_value); endswitch stats.zval = z_value; endswitch stats.sign = pos_n; h = double (p < alpha); endfunction ## Test suite %!error signtest (); %!error signtest ([]); %!error signtest (ones(1,10), ones(1,8)); %!error signtest (ones(1,10), ones(2,10)); %!error signtest (ones(2,10), 0); %!error signtest (ones(1,10), zeros(1,10), "alpha", 1.4) %!error signtest (ones(1,10), zeros(1,10), "tail", "<") %!error signtest (ones(1,10), zeros(1,10), "method", "some") %!test %! [pval, h, stats] = signtest ([-ones(1, 1000) 1], 0, "tail", "left"); %! assert (pval, 1.091701889420221e-218, 1e-14); %! assert (h, 1); %! assert (stats.zval, -31.5437631079266, 1e-14); %!test %! [pval, h, stats] = signtest ([-2 -1 0 2 1 3 1], 0); %! assert (pval, 0.6875000000000006, 1e-14); %! assert (h, 0); %! assert (stats.zval, NaN); %! assert (stats.sign, 4); %!test %! [pval, h, stats] = signtest ([-2 -1 0 2 1 3 1], 0, "method", "approximate"); %! assert (pval, 0.6830913983096086, 1e-14); %! assert (h, 0); %! assert (stats.zval, 0.4082482904638631, 1e-14); %! assert (stats.sign, 4); statistics-release-1.6.3/inst/silhouette.m000066400000000000000000000151471456127120000206570ustar00rootroot00000000000000## Copyright (C) 2016 Nan Zhou ## Copyright (C) 2021 Stefano Guidoni ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {} silhouette (@var{X}, @var{clust}) ## @deftypefnx {statistics} {[@var{si}, @var{h}] =} silhouette (@var{X}, @var{clust}) ## @deftypefnx {statistics} {[@var{si}, @var{h}] =} silhouette (@dots{}, @var{Metric}, @var{MetricArg}) ## ## Compute the silhouette values of clustered data and show them on a plot. ## ## @var{X} is a n-by-p matrix of n data points in a p-dimensional space. Each ## datapoint is assigned to a cluster using @var{clust}, a vector of n elements, ## one cluster assignment for each data point. ## ## Each silhouette value of @var{si}, a vector of size n, is a measure of the ## likelihood that a data point is accurately classified to the right cluster. ## Defining "a" as the mean distance between a point and the other points from ## its cluster, and "b" as the mean distance between that point and the points ## from other clusters, the silhouette value of the i-th point is: ## ## @tex ## \def\frac#1#2{{\begingroup#1\endgroup\over#2}} ## $$ S_i = \frac{b_i - a_i}{max(a_1,b_i)} $$ ## @end tex ## @ifnottex ## @verbatim ## bi - ai ## Si = ------------ ## max(ai,bi) ## @end verbatim ## @end ifnottex ## ## Each element of @var{si} ranges from -1, minimum likelihood of a correct ## classification, to 1, maximum likelihood. ## ## Optional input value @var{Metric} is the metric used to compute the distances ## between data points. Since @code{silhouette} uses @code{pdist} to compute ## these distances, @var{Metric} is quite similar to the option @var{Metric} of ## pdist and it can be: ## @itemize @bullet ## @item A known distance metric defined as a string: @qcode{Euclidean}, ## @qcode{sqEuclidean} (default), @qcode{cityblock}, @qcode{cosine}, ## @qcode{correlation}, @qcode{Hamming}, @qcode{Jaccard}. ## ## @item A vector as those created by @code{pdist}. In this case @var{X} does ## nothing. ## ## @item A function handle that is passed to @code{pdist} with @var{MetricArg} ## as optional inputs. ## @end itemize ## ## Optional return value @var{h} is a handle to the silhouette plot. ## ## @strong{Reference} ## Peter J. Rousseeuw, Silhouettes: a Graphical Aid to the Interpretation and ## Validation of Cluster Analysis. 1987. doi:10.1016/0377-0427(87)90125-7 ## @end deftypefn ## ## @seealso{dendrogram, evalclusters, kmeans, linkage, pdist} function [si, h] = silhouette (X, clust, metric = "sqeuclidean", varargin) ## check the input parameters if (nargin < 2) print_usage (); endif n = size (clust, 1); ## check size if (! isempty (X)) if (size (X, 1) != n) error ("First dimension of X <%d> doesn't match that of clust <%d>",... size (X, 1), n); endif endif ## check metric if (ischar (metric)) metric = lower (metric); switch (metric) case "sqeuclidean" metric = "squaredeuclidean"; case { "euclidean", "cityblock", "cosine", ... "correlation", "hamming", "jaccard" } ; otherwise error ("silhouette: invalid metric '%s'", metric); endswitch elseif (isnumeric (metric) && isvector (metric)) ## X can be omitted when using this distMatrix = squareform (metric); if (size (distMatrix, 1) != n) error ("First dimension of X <%d> doesn't match that of clust <%d>",... size (distMatrix, 1), n); endif endif ## main si = zeros(n, 1); clusterIDs = unique (clust); # eg [1; 2; 3; 4] m = length (clusterIDs); ## if only one cluster is defined, the silhouette value is not defined if (m == 1) si = NaN * ones (n, 1); return; endif ## distance matrix showing the distance for any two rows of X if (! exist ('distMatrix', 'var')) distMatrix = squareform (pdist (X, metric, varargin{:})); endif ## calculate values of si one by one for iii = 1 : length (si) ## allocate values to clusters groupedValues = {}; for jjj = 1 : m groupedValues{clusterIDs(jjj)} = [distMatrix(iii, ... clust == clusterIDs(jjj))]; endfor ## end allocation ## calculate a(i) ## average distance of iii to all other objects in the same cluster if (length (groupedValues{clust(iii)}) == 1) si(iii) = 1; continue; else a_i = (sum (groupedValues{clust(iii)})) / ... (length (groupedValues{clust(iii)}) - 1); endif ## end a(i) ## calculate b(i) clusterIDs_new = clusterIDs; ## remove the cluster iii in clusterIDs_new(find (clusterIDs_new == clust(iii))) = []; ## average distance of iii to all objects of another cluster a_iii_2others = zeros (length (clusterIDs_new), 1); for jjj = 1 : length (clusterIDs_new) a_iii_2others(jjj) = mean (groupedValues{clusterIDs_new(jjj)}); endfor b_i = min (a_iii_2others); ## end b(i) ## calculate s(i) si(iii) = (b_i - a_i) / (max ([a_i; b_i])); ## end s(i) endfor ## plot ## a poor man silhouette graph vBarsc = zeros (m, 1); vPadding = [0; 0; 0; 0]; Bars = vPadding; for i = 1 : m vBar = si(find (clust == clusterIDs(i))); vBarsc(i) = length (Bars) + (length (vBar) / 2); Bars = [Bars; (sort (vBar, "descend")); vPadding]; endfor figure(); h = barh (Bars, "hist", "facecolor", [0 0.4471 0.7412]); xlabel ("Silhouette Value"); ylabel ("Cluster"); set (gca, "ytick", vBarsc, "yticklabel", clusterIDs); ylim ([0 (length (Bars))]); axis ("ij"); endfunction %!demo %! load fisheriris; %! X = meas(:,3:4); %! cidcs = kmeans (X, 3, "Replicates", 5); %! silhouette (X, cidcs); %! y_labels(cidcs([1 51 101])) = unique (species); %! set (gca, "yticklabel", y_labels); %! title ("Fisher's iris data"); ## Test input validation %!error silhouette (); %!error silhouette ([1 2; 1 1]); %!error silhouette ([1 2; 1 1], [1 2 3]'); %!error silhouette ([1 2; 1 1], [1 2]', "xxx"); statistics-release-1.6.3/inst/slicesample.m000066400000000000000000000220251456127120000207640ustar00rootroot00000000000000## Copyright (C) 1995-2022 The Octave Project Developers ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {[@var{smpl}, @var{neval}] =} slicesample (@var{start}, @var{nsamples}, @var{property}, @var{value}, @dots{}) ## ## Draws @var{nsamples} samples from a target stationary distribution @var{pdf} ## using slice sampling of Radford M. Neal. ## ## Input: ## @itemize ## @item ## @var{start} is a 1 by @var{dim} vector of the starting point of the ## Markov chain. Each column corresponds to a different dimension. ## ## @item ## @var{nsamples} is the number of samples, the length of the Markov chain. ## @end itemize ## ## Next, several property-value pairs can or must be specified, they are: ## ## (Required properties) One of: ## ## @itemize ## @item ## @var{"pdf"}: the value is a function handle of the target stationary ## distribution to be sampled. The function should accept different locations ## in each row and each column corresponds to a different dimension. ## ## or ## ## @item ## @var{logpdf}: the value is a function handle of the log of the target ## stationary distribution to be sampled. The function should accept different ## locations in each row and each column corresponds to a different dimension. ## @end itemize ## ## The following input property/pair values may be needed depending on the ## desired outut: ## ## @itemize ## @item ## "burnin" @var{burnin} the number of points to discard at the beginning, the default ## is 0. ## ## @item ## "thin" @var{thin} omitts @var{m}-1 of every @var{m} points in the generated ## Markov chain. The default is 1. ## ## @item ## "width" @var{width} the maximum Manhattan distance between two samples. ## The default is 10. ## @end itemize ## ## Outputs: ## @itemize ## ## @item ## @var{smpl} is a @var{nsamples} by @var{dim} matrix of random ## values drawn from @var{pdf} where the rows are different random values, the ## columns correspond to the dimensions of @var{pdf}. ## ## @item ## @var{neval} is the number of function evaluations per sample. ## @end itemize ## Example : Sampling from a normal distribution ## ## @example ## @group ## start = 1; ## nsamples = 1e3; ## pdf = @@(x) exp (-.5 * x .^ 2) / (pi ^ .5 * 2 ^ .5); ## [smpl, accept] = slicesample (start, nsamples, "pdf", pdf, "thin", 4); ## histfit (smpl); ## @end group ## @end example ## ## @seealso{rand, mhsample, randsample} ## @end deftypefn function [smpl, neval] = slicesample (start, nsamples, varargin) if (nargin < 4) error ("slicesample: function called with too few input arguments."); endif sizestart = size (start); pdf = []; logpdf = []; width = 10; burnin = 0; thin = 1; for k = 1:2:length (varargin) if (ischar (varargin{k})) switch lower (varargin{k}) case "pdf" if (isa (varargin{k+1}, "function_handle")) pdf = varargin{k+1}; else error ("slicesample: pdf must be a function handle."); endif case "logpdf" if (isa (varargin{k+1}, "function_handle")) pdf = varargin{k+1}; else error ("slicesample: logpdf must be a function handle."); endif case "width" if (numel (varargin{k+1}) == 1 || numel (varargin{k+1}) == sizestart(2)) width = varargin{k+1}(:).'; else error ("slicesample: width must be a scalar or 1 by dim vector."); endif case "burnin" if (varargin{k+1}>=0) burnin = varargin{k+1}; else error ("slicesample: burnin must be greater than or equal to 0."); endif case "thin" if (varargin{k+1}>=1) thin = varargin{k+1}; else error ("slicesample: thin must be greater than or equal to 1."); endif otherwise warning (["slicesample: Ignoring unknown option " varargin{k}]); endswitch else error (["slicesample: " varargin{k} " is not a valid property."]); endif endfor if (! isempty (pdf) && isempty (logpdf)) logpdf = @(x) rloge (pdf (x)); elseif (isempty (pdf) && isempty (logpdf)) error ("slicesample: pdf or logpdf must be input."); endif dim = sizestart(2); smpl = zeros (nsamples, dim); if (all (sizestart == [1 dim])) smpl(1, :) = start; else error ("slicesample: start must be a 1 by dim vector."); endif maxit = 100; neval = 0; fgraterthan = @(x, fxc) logpdf (x) >= fxc; ti = burnin + nsamples * thin; rndexp = rande (ti, 1); crand = rand (ti, dim); prand = rand (ti, dim); xc = smpl(1, :); for i = 1:ti neval++; sliceheight = logpdf (xc) - rndexp(i); c = width .* crand(i, :); lb = xc - c; ub = xc + width - c; #Only for single variable as bounds can not be found with point when dim > 1 if (dim == 1) for k=1:maxit neval++; if (! fgraterthan (lb, sliceheight)) break endif lb -= width; end if (k == maxit) warning ("slicesample: Step out exceeded maximum iterations"); endif for k = 1:maxit neval++; if (! fgraterthan (ub, sliceheight)) break endif ub += width; end if (k == maxit) warning ("slicesample: Step out exceeded maximum iterations"); endif end xp = (ub - lb) .* prand(i, :) + lb; for k=1:maxit neval++; isgt = fgraterthan (xp,sliceheight); if (all (isgt)) break endif lc = ! isgt & xp < xc; uc = ! isgt & xp > xc; lb(lc) = xp(lc); ub(uc) = xp(uc); xp = (ub - lb) .* rand (1, dim) + lb; end if (k == maxit) warning ("slicesample: Step in exceeded maximum iterations"); endif xc = xp; if (i > burnin) indx = (i - burnin) / thin; if rem (indx, 1) == 0 smpl(indx, :) = xc; end end end neval = neval / (nsamples * thin + burnin); endfunction function y = rloge (x) y = -inf (size (x)); xg0 = x > 0; y(xg0) = log (x(xg0)); endfunction %!demo %! ## Define function to sample %! d = 2; %! mu = [-1; 2]; %! rand ("seed", 5) # for reproducibility %! Sigma = rand (d); %! Sigma = (Sigma + Sigma'); %! Sigma += eye (d)*abs (eigs (Sigma, 1, "sa")) * 1.1; %! pdf = @(x)(2*pi)^(-d/2)*det(Sigma)^-.5*exp(-.5*sum((x.'-mu).*(Sigma\(x.'-mu)),1)); %! %! ## Inputs %! start = ones (1,2); %! nsamples = 500; %! K = 500; %! m = 10; %! rande ("seed", 4); rand ("seed", 5) # for reproducibility %! [smpl, accept] = slicesample (start, nsamples, "pdf", pdf, "burnin", K, "thin", m, "width", [20, 30]); %! figure; %! hold on; %! plot (smpl(:,1), smpl(:,2), 'x'); %! [x, y] = meshgrid (linspace (-6,4), linspace(-3,7)); %! z = reshape (pdf ([x(:), y(:)]), size(x)); %! mesh (x, y, z, "facecolor", "None"); %! %! ## Using sample points to find the volume of half a sphere with radius of .5 %! f = @(x) ((.25-(x(:,1)+1).^2-(x(:,2)-2).^2).^.5.*(((x(:,1)+1).^2+(x(:,2)-2).^2)<.25)).'; %! int = mean (f (smpl) ./ pdf (smpl)); %! errest = std (f (smpl) ./ pdf (smpl)) / nsamples^.5; %! trueerr = abs (2/3*pi*.25^(3/2)-int); %! fprintf ("Monte Carlo integral estimate int f(x) dx = %f\n", int); %! fprintf ("Monte Carlo integral error estimate %f\n", errest); %! fprintf ("The actual error %f\n", trueerr); %! mesh (x,y,reshape (f([x(:), y(:)]), size(x)), "facecolor", "None"); %!demo %! ## Integrate truncated normal distribution to find normilization constant %! pdf = @(x) exp (-.5*x.^2)/(pi^.5*2^.5); %! nsamples = 1e3; %! rande ("seed", 4); rand ("seed", 5) # for reproducibility %! [smpl, accept] = slicesample (1, nsamples, "pdf", pdf, "thin", 4); %! f = @(x) exp (-.5 * x .^ 2) .* (x >= -2 & x <= 2); %! x = linspace (-3, 3, 1000); %! area (x, f(x)); %! xlabel ("x"); %! ylabel ("f(x)"); %! int = mean (f (smpl) ./ pdf (smpl)); %! errest = std (f (smpl) ./ pdf (smpl)) / nsamples ^ 0.5; %! trueerr = abs (erf (2 ^ 0.5) * 2 ^ 0.5 * pi ^ 0.5 - int); %! fprintf("Monte Carlo integral estimate int f(x) dx = %f\n", int); %! fprintf("Monte Carlo integral error estimate %f\n", errest); %! fprintf("The actual error %f\n", trueerr); ## Test output %!test %! start = 0.5; %! nsamples = 1e3; %! pdf = @(x) exp (-.5*(x-1).^2)/(2*pi)^.5; %! [smpl, accept] = slicesample (start, nsamples, "pdf", pdf, "thin", 2, "burnin", 0, "width", 5); %! assert (mean (smpl, 1), 1, .1); %! assert (var (smpl, 1), 1, .2); ## Test input validation %!error slicesample (); %!error slicesample (1); %!error slicesample (1, 1); statistics-release-1.6.3/inst/squareform.m000066400000000000000000000103031456127120000206430ustar00rootroot00000000000000## Copyright (C) 2015 Carnë Draug ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation; either version 3 of the License, or ## (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{z} =} squareform (@var{y}) ## @deftypefnx {statistics} {@var{y} =} squareform (@var{z}) ## @deftypefnx {statistics} {@var{z} =} squareform (@var{y}, @qcode{"tovector"}) ## @deftypefnx {statistics} {@var{y} =} squareform (@var{z}, @qcode{"tomatrix"}) ## ## Interchange between distance matrix and distance vector formats. ## ## Converts between an hollow (diagonal filled with zeros), square, and ## symmetric matrix and a vector with of the lower triangular part. ## ## Its target application is the conversion of the vector returned by ## @code{pdist} into a distance matrix. It performs the opposite operation ## if input is a matrix. ## ## If @var{x} is a vector, its number of elements must fit into the ## triangular part of a matrix (main diagonal excluded). In other words, ## @code{numel (@var{x}) = @var{n} * (@var{n} - 1) / 2} for some integer ## @var{n}. The resulting matrix will be @var{n} by @var{n}. ## ## If @var{x} is a distance matrix, it must be square and the diagonal entries ## of @var{x} must all be zeros. @code{squareform} will generate a warning if ## @var{x} is not symmetric. ## ## The second argument is used to specify the output type in case there ## is a single element. It will defaults to @qcode{"tomatrix"} otherwise. ## ## @seealso{pdist} ## @end deftypefn function y = squareform (x, method) if (nargin < 1 || nargin > 2) print_usage (); elseif (! isnumeric (x) || ! ismatrix (x)) error ("squareform: Y or Z must be a numeric matrix or vector."); endif if (nargin == 1) ## This is ambiguous when numel (x) == 1, but that's the whole reason ## why the "method" option exists. if (isvector (x)) method = "tomatrix"; else method = "tovector"; endif endif switch (tolower (method)) case "tovector" if (! issquare (x)) error ("squareform: Z is not a square matrix."); elseif (any (diag (x) != 0)) error ("squareform: Z is not a hollow matrix."); elseif (! issymmetric(x)) warning ("squareform:symmetric", "squareform: Z is not a symmetric matrix"); endif y = vec (tril (x, -1, "pack"), 2); case "tomatrix" ## the dimensions of y are the solution to the quadratic formula for: ## length (x) = (sy - 1) * (sy / 2) sy = (1 + sqrt (1 + 8 * numel (x))) / 2; if (fix (sy) != sy) error ("squareform: the numel of Y cannot form a square matrix."); endif y = zeros (sy, class (x)); y(tril (true (sy), -1)) = x; # fill lower triangular part y += y.'; # and then the upper triangular part otherwise error ("squareform: invalid METHOD '%s'.", method); endswitch endfunction %!shared v, m %! v = 1:6; %! m = [0 1 2 3;1 0 4 5;2 4 0 6;3 5 6 0]; ## make sure that it can go both directions automatically %!assert (squareform (v), m) %!assert (squareform (squareform (v)), v) %!assert (squareform (m), v) ## treat row and column vectors equally %!assert (squareform (v'), m) ## handle 1 element input properly %!assert (squareform (1), [0 1;1 0]) %!assert (squareform (1, "tomatrix"), [0 1; 1 0]) %!assert (squareform (0, "tovector"), zeros (1, 0)) %!warning squareform ([0 1 2; 3 0 4; 5 6 0]); ## confirm that it respects input class %!test %! for c = {@single, @double, @uint8, @uint32, @uint64} %! f = c{1}; %! assert (squareform (f (v)), f (m)) %! assert (squareform (f (m)), f (v)) %! endfor statistics-release-1.6.3/inst/standardizeMissing.m000066400000000000000000000164431456127120000223340ustar00rootroot00000000000000## Copyright (C) 1995-2023 The Octave Project Developers ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{B} =} standardizeMissing (@var{A}, @var{indicator}) ## ## Replace data values specified by @var{indicator} in @var{A} by the ## standard 'missing' data value for that data type. ## ## @var{A} can be a numeric scalar or array, a character vector or array, or ## a cell array of character vectors (a.k.a. string cells). ## ## @var{indicator} can be a scalar or an array containing values to be ## replaced by the 'missing' value for the class of @var{A}, and should have ## a data type matching @var{A}. ## ## 'missing' values are defined as : ## ## @itemize ## @item ## @qcode{NaN}: @code{single}, @code{double} ## ## @item ## @qcode{" "} (white space): @code{char} ## ## @item ## @qcode{@{""@}} (empty string in cell): string cells. ## @end itemize ## ## Compatibility Notes: ## @itemize ## @item ## Octave's implementation of @code{standardizeMissing} ## does not restrict @var{indicator} of type @qcode{char} to row vectors. ## ## @item ## All numerical and logical inputs for @var{A} and @var{indicator} may ## be specified in any combination. The output will be the same class as ## @var{A}, with the @var{indicator} converted to that data type for ## comparison. Only @code{single} and @code{double} have defined 'missing' ## values, so @var{A} of other data types will always output ## @var{B} = @var{A}. ## @end itemize ## ## @end deftypefn ## ## @seealso{fillmissing, ismissing, rmmissing} function A = standardizeMissing (A, indicator) if (nargin != 2) print_usage (); endif input_class = class(A); do_nothing_flag = false; ## Set missing_val. ## Compatibility requirements: ## Numeric/logical: Only double and single have 'missing' values defined, ## so other numeric and logical pass through unchanged. ## Other: only char & cellstr have 'missing' values defined, others produce ## error. ## ## TODO: as implemented in Octave, add table, string, timetable, ## categorical, datetime, and duration to class checks and BISTs ## ## TODO: when 'missing' class is implemented, these switch blocks can all ## be removed and the final assignment updated to call missing instead of ## missing_val if (isnumeric(A) || islogical(A)) switch (input_class) case "double" missing_val = NaN ("double"); case "single" missing_val = NaN ("single"); otherwise do_nothing_flag = true; endswitch else switch (input_class) case "char" missing_val = " "; case "cell" if iscellstr(A) missing_val = {""}; else error ("stardardizeMissing: only cells of strings are supported."); endif otherwise error ("standardizeMissing: unsupported data type %s.", input_class); endswitch endif if (! do_nothing_flag) ## if A is an array of cell strings and indicator just a string, ## convert indicator to a cell string with one element if (iscellstr (A) && ischar (indicator) && ! iscellstr (indicator)) indicator = {indicator}; endif if ((isnumeric (A) && ! (isnumeric (indicator) || islogical (indicator))) || (ischar (A) && ! ischar (indicator)) || (iscellstr (A) && ! (iscellstr (indicator)))) error (strcat (["standardizeMissing: 'indicator' and 'A' must"], ... [" have the same data type."])); endif A(ismember (A, indicator)) = missing_val; endif endfunction ## numeric tests %!assert (standardizeMissing (1, 1), NaN) %!assert (standardizeMissing (1, 0), 1) %!assert (standardizeMissing (eye(2), 1), [NaN 0;0 NaN]) %!assert (standardizeMissing ([1:3;4:6], [2 3; 4 5]), [1, NaN, NaN; NaN, NaN, 6]) %!assert (standardizeMissing (cat (3,1,2,3,4), 3), cat (3,1,2,NaN,4)) ## char and cellstr tests %!assert (standardizeMissing ('foo', 'a'), 'foo') %!assert (standardizeMissing ('foo', 'f'), ' oo') %!assert (standardizeMissing ('foo', 'o'), 'f ') %!assert (standardizeMissing ('foo', 'oo'), 'f ') %!assert (standardizeMissing ({'foo'}, 'f'), {'foo'}) %!assert (standardizeMissing ({'foo'}, {'f'}), {'foo'}) %!assert (standardizeMissing ({'foo'}, 'test'), {'foo'}) %!assert (standardizeMissing ({'foo'}, {'test'}), {'foo'}) %!assert (standardizeMissing ({'foo'}, 'foo'), {''}) %!assert (standardizeMissing ({'foo'}, {'foo'}), {''}) ## char and cellstr array tests %!assert (standardizeMissing (['foo';'bar'], 'oar'), ['f ';'b ']) %!assert (standardizeMissing (['foo';'bar'], ['o';'a';'r']), ['f ';'b ']) %!assert (standardizeMissing (['foo';'bar'], ['o ';'ar']), ['f ';'b ']) %!assert (standardizeMissing ({'foo','bar'}, 'foo'), {'','bar'}) %!assert (standardizeMissing ({'foo','bar'}, 'f'), {'foo','bar'}) %!assert (standardizeMissing ({'foo','bar'}, {'foo', 'a'}), {'','bar'}) %!assert (standardizeMissing ({'foo'}, {'f', 'oo'}), {'foo'}) %!assert (standardizeMissing ({'foo','bar'}, {'foo'}), {'','bar'}) %!assert (standardizeMissing ({'foo','bar'}, {'foo', 'a'}), {'','bar'}) ## numeric type preservation tests %!assert (standardizeMissing (double (1), single (1)), double (NaN)) %!assert (standardizeMissing (single (1), single (1)), single (NaN)) %!assert (standardizeMissing (single (1), double (1)), single (NaN)) %!assert (standardizeMissing (single (1), true), single (NaN)) %!assert (standardizeMissing (double (1), int32(1)), double (NaN)) ## Passttrough tests %!assert (standardizeMissing (true, true), true) %!assert (standardizeMissing (true, 1), true) %!assert (standardizeMissing (int32 (1), int32 (1)), int32 (1)) %!assert (standardizeMissing (int32 (1), 1), int32 (1)) %!assert (standardizeMissing (uint32 (1), uint32 (1)), uint32 (1)) %!assert (standardizeMissing (uint32 (1), 1), uint32 (1)) ## Test input validation %!error standardizeMissing (); %!error standardizeMissing (1); %!error standardizeMissing (1,2,3); %!error standardizeMissing ({'abc', 1}, 1); %!error standardizeMissing (struct ('a','b'), 1); %!error <'indicator' and 'A' must have > standardizeMissing ([1 2 3], {1}); %!error <'indicator' and 'A' must have > standardizeMissing ([1 2 3], 'a'); %!error <'indicator' and 'A' must have > standardizeMissing ([1 2 3], struct ('a', 1)); %!error <'indicator' and 'A' must have > standardizeMissing ('foo', 1); %!error <'indicator' and 'A' must have > standardizeMissing ('foo', {1}); %!error <'indicator' and 'A' must have > standardizeMissing ('foo', {'f'}); %!error <'indicator' and 'A' must have > standardizeMissing ('foo', struct ('a', 1)); %!error <'indicator' and 'A' must have > standardizeMissing ({'foo'}, 1); %!error <'indicator' and 'A' must have > standardizeMissing ({'foo'}, 1); statistics-release-1.6.3/inst/stepwisefit.m000066400000000000000000000137661456127120000210450ustar00rootroot00000000000000## Copyright (C) 2013-2021 Nir Krakauer ## Copyright (C) 2014 Mikael Kurula ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {[@var{X_use}, @var{b}, @var{bint}, @var{r}, @var{rint}, @var{stats}] =} stepwisefit (@var{y}, @var{X}, @var{penter} = 0.05, @var{premove} = 0.1, @var{method} = "corr") ## ## Linear regression with stepwise variable selection. ## ## @subheading Arguments ## ## @itemize @bullet ## @item ## @var{y} is an @var{n} by 1 vector of data to fit. ## @item ## @var{X} is an @var{n} by @var{k} matrix containing the values of @var{k} potential predictors. No constant term should be included (one will always be added to the regression automatically). ## @item ## @var{penter} is the maximum p-value to enter a new variable into the regression (default: 0.05). ## @item ## @var{premove} is the minimum p-value to remove a variable from the regression (default: 0.1). ## @item ## @var{method} sets how predictors are selected at each step, either based on their correlation with the residuals ("corr", default) ## or on the p values of their regression coefficients when they are successively added ("p"). ## @end itemize ## ## @subheading Return values ## ## @itemize @bullet ## @item ## @var{X_use} contains the indices of the predictors included in the final regression model. The predictors are listed in the order they were added, so typically the first ones listed are the most significant. ## @item ## @var{b}, @var{bint}, @var{r}, @var{rint}, @var{stats} are the results of @code{[b, bint, r, rint, stats] = regress(y, [ones(size(y)) X(:, X_use)], penter);} ## @end itemize ## @subheading References ## ## @enumerate ## @item ## N. R. Draper and H. Smith (1966). @cite{Applied Regression Analysis}. Wiley. Chapter 6. ## ## @end enumerate ## @seealso{regress} ## @end deftypefn function [X_use, b, bint, r, rint, stats] = stepwisefit(y, X, penter = 0.05, premove = 0.1, method = "corr") if nargin >= 3 && isempty(penter) penter = 0.05; endif if nargin >= 4 && isempty(premove) premove = 0.1; endif #remove any rows with missing entries notnans = !any (isnan ([y X]) , 2); y = y(notnans); X = X(notnans,:); n = numel(y); #number of data points k = size(X, 2); #number of predictors X_use = []; v = 0; #number of predictor variables in regression model iter = 0; max_iters = 100; #maximum number of interations to do r = y; while 1 iter++; #decide which variable to add to regression, if any added = false; if numel(X_use) < k X_inds = zeros(k, 1, "logical"); X_inds(X_use) = 1; switch lower (method) case {"corr"} [~, i_to_add] = max(abs(corr(X(:, ~X_inds), r))); #try adding the variable with the highest correlation to the residual from current regression i_to_add = (1:k)(~X_inds)(i_to_add); #index within the original predictor set [b_new, bint_new, r_new, rint_new, stats_new] = regress(y, [ones(n, 1) X(:, [X_use i_to_add])], penter); case {"p"} z_vals=zeros(k,1); for j=1:k if ~X_inds(j) [b_j, bint_j, ~,~ ,~] = regress(y, [ones(n, 1) X(:, [X_use j])], penter); z_vals(j) = abs(b_j(end)) / (bint_j(end, 2) - b_j(end)); endif endfor [~, i_to_add] = max(z_vals); #try adding the variable with the largest z-value (smallest partial p-value) [b_new, bint_new, r_new, rint_new, stats_new] = regress(y, [ones(n, 1) X(:, [X_use i_to_add])], penter); otherwise error("stepwisefit: invalid value for method") endswitch z_new = abs(b_new(end)) / (bint_new(end, 2) - b_new(end)); if z_new > 1 #accept new variable added = true; X_use = [X_use i_to_add]; b = b_new; bint = bint_new; r = r_new; rint = rint_new; stats = stats_new; v = v + 1; endif endif #decide which variable to drop from regression, if any dropped = false; if v > 0 t_ratio = tinv(1 - premove/2, n - v - 1) / tinv(1 - penter/2, n - v - 1); #estimate the ratio between the z score corresponding to premove to that corresponding to penter [z_min, i_min] = min(abs(b(2:end)) ./ (bint(2:end, 2) - b(2:end))); if z_min < t_ratio #drop a variable dropped = true; X_use(i_min) = []; [b, bint, r, rint, stats] = regress(y, [ones(n, 1) X(:, X_use)], penter); v = v - 1; endif endif #terminate if no change in the list of regression variables if ~added && ~dropped break endif if iter >= max_iters warning('stepwisefit: maximum iteration count exceeded before convergence') break endif endwhile endfunction %!test %! % Sample data from Draper and Smith (n = 13, k = 4) %! X = [7 1 11 11 7 11 3 1 2 21 1 11 10; ... %! 26 29 56 31 52 55 71 31 54 47 40 66 68; ... %! 6 15 8 8 6 9 17 22 18 4 23 9 8; ... %! 60 52 20 47 33 22 6 44 22 26 34 12 12]'; %! y = [78.5 74.3 104.3 87.6 95.9 109.2 102.7 72.5 93.1 115.9 83.8 113.3 109.4]'; %! [X_use, b, bint, r, rint, stats] = stepwisefit(y, X); %! assert(X_use, [4 1]) %! assert(b, regress(y, [ones(size(y)) X(:, X_use)], 0.05)) %! [X_use, b, bint, r, rint, stats] = stepwisefit(y, X, 0.05, 0.1, "corr"); %! assert(X_use, [4 1]) %! assert(b, regress(y, [ones(size(y)) X(:, X_use)], 0.05)) %! [X_use, b, bint, r, rint, stats] = stepwisefit(y, X, [], [], "p"); %! assert(X_use, [4 1]) %! assert(b, regress(y, [ones(size(y)) X(:, X_use)], 0.05)) statistics-release-1.6.3/inst/tabulate.m000066400000000000000000000127131456127120000202670ustar00rootroot00000000000000## Copyright (C) 2003 Alberto Terruzzi ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {} tabulate (@var{x}) ## @deftypefnx {statistics} {@var{table} =} tabulate (@var{x}) ## ## Calculate a frequency table. ## ## @code{tabulate (x)} displays a frequency table of the data in the vector ## @var{x}. For each unique value in @var{x}, the tabulate function shows the ## number of instances and percentage of that value in @var{x}. ## ## @code{@var{table} = tabulate (@var{x})} returns the frequency table, ## @var{table}, as a numeric matrix when @var{x} is numeric and as a cell array ## otherwise. When an output argument is requested, @code{tabulate} does not ## print the frequency table in the command window. ## ## If @var{x} is numeric, any missing values (@qcode{NaNs}) are ignored. ## ## If all the elements of @var{x} are positive integers, then the frequency ## table includes 0 counts for the integers between 1 and @qcode{max (@var{x})} ## that do not appear in @var{x}. ## ## @seealso{bar, pareto} ## @end deftypefn function table = tabulate (x) ## Check input for being either numeric or a cell array if (! (isnumeric (x) && isvector (x)) && ! (iscellstr (x) && isvector (x)) && ! ischar (x)) error (strcat (["tabulate: X must be either a numeric vector, a"], ... [" vector cell array of strings, or a character matrix."])); endif ## Remove missing values (NaNs) if numeric if (isnumeric (x)) x(isnan (x)) = []; endif ## Handle positive integers separately if (isnumeric (x) && all (x == fix (x)) && all (x > 0)); [count, value] = hist (x, (1:max (x))); posint = true; else [g, gn, gl] = grp2idx (x); [count, value] = hist (g, (1:length (gn))); posint = false; endif ## Calculate percentages percent = 100 * count ./ sum (count); ## Display results is no output argument if (nargout == 0) if (posint) fprintf (" Value Count Percent\n"); fprintf (" %5d %5d %6.2f%%\n", value', count', percent'); else valw = max (cellfun ("length", gn)); valw = max ([5, min([50, valw])]); header = sprintf (" %%%ds %%5s %%6s\n", valw); result = sprintf (" %%%ds %%5d %%6.2f%%%%\n", valw); fprintf (header, "Value", "Count", "Percent"); for i = 1:length (gn) fprintf (result, gn{i}, count(i), percent(i)); endfor endif ## Create output table else if (posint) table = [value', count', percent']; elseif (isnumeric (x)) table = [gl, count', percent']; else table = [gn, num2cell([count', percent'])]; endif endif endfunction %!demo %! ## Generate a frequency table for a vector of data in a cell array %! load patients %! %! ## Display the first seven entries of the Gender variable %! gender = Gender(1:7) %! %! ## Compute the equency table that shows the number and %! ## percentage of Male and Female patients %! tabulate (Gender) %!demo %! ## Create a frequency table for a vector of positive integers %! load patients %! %! ## Display the first seven entries of the Gender variable %! height = Height(1:7) %! %! ## Create a frequency table that shows, in its second and third columns, %! ## the number and percentage of patients with a particular height. %! table = tabulate (Height); %! %! ## Display the first and last seven entries of the frequency table %! first = table(1:7,:) %! %! last = table(end-6:end,:) %!demo %! ## Create a frequency table from a character array %! load carsmall %! %! ## Tabulate the data in the Origin variable, which shows the %! ## country of origin of each car in the data set %! tabulate (Origin) %!demo %! ## Create a frequency table from a numeric vector with NaN values %! load carsmall %! %! ## The carsmall dataset contains measurements of 100 cars %! total_cars = length (MPG) %! ## For six cars, the MPG value is missing %! missingMPG = length (MPG(isnan (MPG))) %! %! ## Create a frequency table using MPG %! tabulate (MPG) %! table = tabulate (MPG); %! %! ## Only 94 cars were used %! valid_cars = sum (table(:,2)) %!test %! load patients %! table = tabulate (Gender); %! assert (table{1,1}, "Male"); %! assert (table{2,1}, "Female"); %! assert (table{1,2}, 47); %! assert (table{2,2}, 53); %!test %! load patients %! table = tabulate (Height); %! assert (table(end-4,:), [68, 15, 15]); %! assert (table(end-3,:), [69, 8, 8]); %! assert (table(end-2,:), [70, 11, 11]); %! assert (table(end-1,:), [71, 10, 10]); %! assert (table(end,:), [72, 4, 4]); %!error tabulate (ones (3)) %!error tabulate ({1, 2, 3, 4}) %!error ... %! tabulate ({"a", "b"; "a", "c"}) statistics-release-1.6.3/inst/tiedrank.m000066400000000000000000000111431456127120000202630ustar00rootroot00000000000000## Copyright (C) 2022 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {[@var{r}, @var{tieadj}]} = tiedrank (@var{x}) ## @deftypefnx {statistics} {[@var{r}, @var{tieadj}]} = tiedrank (@var{x}, @var{tieflag}) ## @deftypefnx {statistics} {[@var{r}, @var{tieadj}]} = tiedrank (@var{x}, @var{tieflag}, @var{bidir}) ## ## @code{[@var{r}, @var{tieadj}] = tiedrank (@var{x})} computes the ranks of the ## values in vector @var{x}. If any values in @var{x} are tied, @code{tiedrank} ## computes their average rank. The return value @var{tieadj} is an adjustment ## for ties required by the nonparametric tests @code{signrank} and ## @code{ranksum}, and for the computation of Spearman's rank correlation. ## ## @code{[@var{r}, @var{tieadj}] = tiedrank (@var{x}, 1)} computes the ranks of ## the values in the vector @var{x}. @var{tieadj} is a vector of three ## adjustments for ties required in the computation of Kendall's tau. ## @code{tiedrank (@var{x}, 0)} is the same as @code{tiedrank (@var{x})}. ## ## @code{[@var{r}, @var{tieadj}] = tiedrank (@var{x}, 0, 1)} computes the ranks ## from each end, so that the smallest and largest values get rank 1, the next ## smallest and largest get rank 2, etc. These ranks are used in the ## Ansari-Bradley test. ## ## @end deftypefn function [r, tieadj] = tiedrank (x, tieflag, bidir) ## Check input arguments and add defauls if (nargin < 1 || nargin > 3) print_usage (); endif if (nargin < 2) tieflag = false; endif if (nargin < 3) bidir = false; endif ## X must be a vector if isvector (x) ## Sort X and leave NaNs at the end of vector [sx, idx] = sort (x(:)); NaNs = sum (isnan (x)); xLen = length (x) - NaNs; ## Count ranks from low end if ! bidir ranks = [1:xLen NaN(1,NaNs)]'; ## Count ranks from both ends else ## For even number of samples if mod(xLen,2)==0 ranks = [(1:xLen/2), (xLen/2:-1:1), NaN(1,NaNs)]'; ## For odd number of samples else ranks = [(1:(xLen+1)/2), ((xLen-1)/2:-1:1), NaN(1,NaNs)]'; endif endif ## Define number of adjustments if ! tieflag tieadj = 0; else tieadj = [0; 0; 0]; endif ## Check precision of X if isa (x, "single") ranks = single (ranks); tieadj = single (tieadj); endif ## Adjust for ties ties = sx(1:xLen-1) >= sx(2:xLen); tieloc = [find(ties); xLen+2]; maxTies = length (tieloc); tiecount = 1; while (tiecount < maxTies) tiestart = tieloc(tiecount); ntied = 2; while(tieloc(tiecount+1) == tieloc(tiecount)+1) tiecount = tiecount + 1; ntied = ntied + 1; endwhile if ! tieflag tieadj = tieadj + ntied * (ntied - 1) * (ntied + 1) / 2; else n2minusn = ntied * (ntied - 1); tieadj = tieadj + [n2minusn/2; n2minusn*(ntied-2); n2minusn*(2*ntied+5)]; endif ## Compute mean of tied ranks ranks(tiestart:tiestart+ntied-1) = ... sum (ranks(tiestart:tiestart+ntied-1)) / ntied; tiecount = tiecount + 1; endwhile ## Reshape ranks including NaN where required. r(idx) = ranks; r = reshape (r, size (x)); else error ("X must be a vector"); endif endfunction ## testing against mileage data and results from Matlab %!test %! mileage = [33.3, 34.5, 37.4; 33.4, 34.8, 36.8; ... %! 32.9, 33.8, 37.6; 32.6, 33.4, 36.6; ... %! 32.5, 33.7, 37.0; 33.0, 33.9, 36.7]; %! [r,tieadj] = tiedrank([10, 20, 30, 40, 50]); %! assert (r, [1, 2, 3, 4, 5]); %! assert (tieadj, 0); %! [r,tieadj] = tiedrank([10, 20, 30, 40, 50]'); %! assert (r, [1; 2; 3; 4; 5]); %!test %! mileage = [33.3, 34.5, 37.4; 33.4, 34.8, 36.8; ... %! 32.9, 33.8, 37.6; 32.6, 33.4, 36.6; ... %! 32.5, 33.7, 37.0; 33.0, 33.9, 36.7]; %! [r,tieadj] = tiedrank([10, 20, 30, 40, 50], 1); %! assert (r, [1, 2, 3, 4, 5]); %! assert (tieadj, [0 0 0]'); statistics-release-1.6.3/inst/trimmean.m000066400000000000000000000271271456127120000203070ustar00rootroot00000000000000## Copyright (C) 2001 Paul Kienzle ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{m} =} trimmean (@var{x}, @var{p}) ## @deftypefnx {statistics} {@var{m} =} trimmean (@var{x}, @var{p}, @var{flag}) ## @deftypefnx {statistics} {@var{m} =} trimmean (@dots{}, @qcode{"all"}) ## @deftypefnx {statistics} {@var{m} =} trimmean (@dots{}, @var{dim}) ## @deftypefnx {statistics} {@var{m} =} trimmean (@dots{}, @var{dim}) ## ## Compute the trimmed mean. ## ## The trimmed mean of @var{x} is defined as the mean of @var{x} excluding the ## highest and lowest @math{k} data values of @var{x}, calculated as ## @qcode{@var{k} = n * (@var{p} / 100) / 2)}, where @var{n} is the sample size. ## ## @code{@var{m} = trimmean (@var{x}, @var{p})} returns the mean of @var{x} ## after removing the outliers in @var{x} defined by @var{p} percent. ## @itemize ## @item If @var{x} is a vector, then @code{trimmean (@var{x}, @var{p})} is the ## mean of all the values of @var{x}, computed after removing the outliers. ## @item If @var{x} is a matrix, then @code{trimmean (@var{x}, @var{p})} is a ## row vector of column means, computed after removing the outliers. ## @item If @var{x} is a multidimensional array, then @code{trimmean} operates ## along the first nonsingleton dimension of @var{x}. ## @end itemize ## ## To specify the operating dimension(s) when @var{x} is a matrix or a ## multidimensional array, use the @var{dim} or @var{vecdim} input argument. ## ## @code{trimmean} treats @qcode{NaN} values in @var{x} as missing values and ## removes them. ## ## @code{@var{m} = trimmean (@var{x}, @var{p}, @var{flag})} specifies how to ## trim when @math{k}, i.e. half the number of outliers, is not an integer. ## @var{flag} can be specified as one of the following values: ## @multitable @columnfractions 0.2 0.05 0.75 ## @headitem Value @tab @tab Description ## @item @qcode{"round"} @tab @tab Round @math{k} to the nearest integer. This ## is the default. ## @item @qcode{"floor"} @tab @tab Round @math{k} down to the next smaller ## integer. ## @item @qcode{"weighted"} @tab @tab If @math{k = i + f}, where @math{i} is an ## integer and @math{f} is a fraction, compute a weighted mean with weight ## @math{(1 – f)} for the @math{(i + 1)}-th and @math{(n – i)}-th values, and ## full weight for the values between them. ## @end multitable ## ## @code{@var{m} = trimmean (@dots{}, @qcode{"all"})} returns the trimmed mean ## of all the values in @var{x} using any of the input argument combinations in ## the previous syntaxes. ## ## @code{@var{m} = trimmean (@dots{}, @var{dim})} returns the trimmed mean along ## the operating dimension @var{dim} specified as a positive integer scalar. If ## not specified, then the default value is the first nonsingleton dimension of ## @var{x}, i.e. whose size does not equal 1. If @var{dim} is greater than ## @qcode{ndims (@var{X})} or if @qcode{size (@var{x}, @var{dim})} is 1, then ## @code{trimmean} returns @var{x}. ## ## @code{@var{m} = trimmean (@dots{}, @var{vecdim})} returns the trimmed mean ## over the dimensions specified in the vector @var{vecdim}. For example, if ## @var{x} is a 2-by-3-by-4 array, then @code{mean (@var{x}, [1 2])} returns a ## 1-by-1-by-4 array. Each element of the output array is the mean of the ## elements on the corresponding page of @var{x}. If @var{vecdim} indexes all ## dimensions of @var{x}, then it is equivalent to @code{mean (@var{x}, "all")}. ## Any dimension in @var{vecdim} greater than @code{ndims (@var{x})} is ignored. ## ## @seealso{mean} ## @end deftypefn function m = trimmean (x, p, varargin) if (nargin < 2 || nargin > 4) print_usage; endif if (p < 0 || p >= 100) error ("trimmean: invalid percent."); endif ## Parse extra arguments if (nargin < 3) flag = []; dim = []; elseif (nargin < 4) if (ischar (varargin{1}) && ! strcmpi (varargin{1}, "all")) flag = varargin{1}; dim = []; elseif (isnumeric (varargin{1}) || strcmpi (varargin{1}, "all")) flag = []; dim = varargin{1}; endif else flag = varargin{1}; dim = varargin{2}; endif ## Get size of X szx = size (x); ndx = numel (szx); ## Handle special case X = [] if (isempty (x)) m = NaN (class (x)); return endif ## Check FLAG if (isempty (flag)) flag = "round"; endif if (! any (strcmpi (flag, {"round", "floor", "weighted"}))) error ("trimmean: invalid FLAG argument."); endif ## Check DIM if (isempty (dim)) (dim = find (szx != 1, 1)) || (dim = 1); endif if (strcmpi (dim, "all")) x = x(:); dim = 1; endif if (! (isvector (dim) && all (dim > 0) && all (rem (dim, 1) == 0))) error ("trimmean: DIM must be a positive integer scalar or vector."); endif vecdim_flag = false; if (numel (dim) > 1) dim = sort (dim); if (! all (diff (dim))) error ("trimmean: VECDIM must contain non-repeating positive integers."); endif vecdim_flag = true; endif ## If DIM is a scalar greater than ndims, return X if (isscalar (dim) && dim > ndx) m = x; return endif ## If DIM is a scalar and size (x, dim) == 1, return X if (isscalar (dim) && size (x, dim) == 1) m = x; return endif ## If DIM is a vector, ignore any value > ndims (x) if (numel (dim) > 1) dim(dim > ndx) = []; endif ## Permute dim to simplify all operations along dim1. At func. end ipermute. if (numel (dim) > 1 || (dim != 1 && ! isvector (x))) perm = 1:ndx; if (! vecdim_flag) ## Move dim to dim 1 perm([1, dim]) = [dim, 1]; x = permute (x, perm); szx([1, dim]) = szx([dim, 1]); dim = 1; else ## Move vecdims to front perm(dim) = []; perm = [dim, perm]; x = permute (x, perm); ## Reshape all vecdims into dim1 num_dim = prod (szx(dim)); szx(dim) = []; szx = [num_dim, ones(1, numel(dim)-1), szx]; x = reshape (x, szx); dim = 1; endif perm_flag = true; else perm_flag = false; endif ## Create output matrix sizem = size (x); sizem(dim) = 1; ## Sort X along 1st dimensions x = sort (x); ## No missing data, all columns have the same length if (! any (isnan (x(:)))) if (isempty (x)) n = 0; else n = size (x, 1); end m = trim (x, n, p, flag, sizem); m = reshape (m, sizem); ## With missing data, each column is computed separately else m = NaN (sizem, class (x)); for j = 1:prod (sizem(2:end)) n = find (! isnan (x(:,j)), 1, "last"); m(j) = trim (x(:,j), n, p, flag, [1, 1]); endfor endif ## Inverse permute back to correct dimensions (if necessary) if (perm_flag) m = ipermute (m, perm); endif endfunction ## Help function for handling different flags function m = trim (x, n, p, flag, sizem) switch (lower (flag)) case "round" k = n * p / 200; k0 = round (k - eps (k)); if (! isempty (n) && n > 0 && k0 < n / 2) m = mean (x((k0+1):(n-k0),:), 1); else m = NaN (sizem, class (x)); endif case "floor" k0 = floor (n * p / 200); if (! isempty (n) && n > 0 && k0 < n / 2) m = mean (x((k0+1):(n-k0),:), 1); else m = NaN (sizem, class (x)); endif case "weighted" k = n * p / 200; k0 = floor (k); fr = 1 + k0 - k; if (! isempty (n) && n > 0 && (k0 < n / 2 || fr > 0)) m = (sum (x((k0+2):(n-k0-1),:),1) + fr * x(k0+1,:) + fr * x(n-k0,:)) ... / (max (0, n - 2 * k0 - 2) + 2 * fr); else m = NaN (sizem, class (x)); endif endswitch endfunction ## Test output %!test %! x = reshape (1:40, [5, 4, 2]); %! x([3, 37]) = -100; %! assert (trimmean (x, 10, "all"), 19.4722, 1e-4); %!test %! x = reshape (1:40, [5, 4, 2]); %! x([3, 37]) = -100; %! out = trimmean (x, 10, [1, 2]); %! assert (out(1,1,1), 10.3889, 1e-4); %! assert (out(1,1,2), 29.6111, 1e-4); %!test %! x = reshape (1:40, [5, 4, 2]); %! x([3, 37]) = -100; %! x([4, 38]) = NaN; %! assert (trimmean (x, 10, "all"), 19.3824, 1e-4); %!test %! x = reshape (1:40, [5, 4, 2]); %! x([3, 37]) = -100; %! out = trimmean (x, 10, 1); %! assert (out(:,:,1), [-17.6, 8, 13, 18]); %! assert (out(:,:,2), [23, 28, 33, 10.6]); %!test %! x = reshape (1:40, [5, 4, 2]); %! x([3, 37]) = -100; %! x([4, 38]) = NaN; %! out = trimmean (x, 10, 1); %! assert (out(:,:,1), [-23, 8, 13, 18]); %! assert (out(:,:,2), [23, 28, 33, 3.75]); %!test %! x = reshape (1:40, [5, 4, 2]); %! x([3, 37]) = -100; %! out = trimmean (x, 10, 2); %! assert (out(:,:,1), [8.5; 9.5; -15.25; 11.5; 12.5]); %! assert (out(:,:,2), [28.5; -4.75; 30.5; 31.5; 32.5]); %!test %! x = reshape (1:40, [5, 4, 2]); %! x([3, 37]) = -100; %! x([4, 38]) = NaN; %! out = trimmean (x, 10, 2); %! assert (out(:,:,1), [8.5; 9.5; -15.25; 14; 12.5]); %! assert (out(:,:,2), [28.5; -4.75; 28; 31.5; 32.5]); %!test %! x = reshape (1:40, [5, 4, 2]); %! x([3, 37]) = -100; %! out = trimmean (x, 10, [1, 2, 3]); %! assert (out, trimmean (x, 10, "all")); ## Test N-D array with NaNs %!test %! x = reshape (1:40, [5, 4, 2]); %! x([3, 37]) = -100; %! x([4, 38]) = NaN; %! out = trimmean (x, 10, [1, 2]); %! assert (out(1,1,1), 10.7647, 1e-4); %! assert (out(1,1,2), 29.1176, 1e-4); %!test %! x = reshape (1:40, [5, 4, 2]); %! x([3, 37]) = -100; %! x([4, 38]) = NaN; %! out = trimmean (x, 10, [1, 3]); %! assert (out, [2.5556, 18, 23, 11.6667], 1e-4); %!test %! x = reshape (1:40, [5, 4, 2]); %! x([3, 37]) = -100; %! x([4, 38]) = NaN; %! out = trimmean (x, 10, [2, 3]); %! assert (out, [18.5; 2.3750; 3.2857; 24; 22.5], 1e-4); %!test %! x = reshape (1:40, [5, 4, 2]); %! x([3, 37]) = -100; %! x([4, 38]) = NaN; %! out = trimmean (x, 10, [1, 2, 3]); %! assert (out, trimmean (x, 10, "all")); %!test %! x = reshape (1:40, [5, 4, 2]); %! x([3, 37]) = -100; %! x([4, 38]) = NaN; %! out = trimmean (x, 10, [2, 3, 5]); %! assert (out, [18.5; 2.3750; 3.2857; 24; 22.5], 1e-4); ## Test special cases %!assert (trimmean (reshape (1:40, [5, 4, 2]), 10, 4), reshape(1:40, [5, 4, 2])) %!assert (trimmean ([], 10), NaN) %!assert (trimmean ([1;2;3;4;5], 10, 2), [1;2;3;4;5]) ## Test input validation %!error trimmean (1) %!error trimmean (1,2,3,4,5) %!error trimmean ([1 2 3 4], -10) %!error trimmean ([1 2 3 4], 100) %!error trimmean ([1 2 3 4], 10, "flag") %!error trimmean ([1 2 3 4], 10, "flag", 1) %!error ... %! trimmean ([1 2 3 4], 10, -1) %!error ... %! trimmean ([1 2 3 4], 10, "floor", -1) %!error ... %! trimmean (reshape (1:40, [5, 4, 2]), 10, [-1, 2]) %!error ... %! trimmean (reshape (1:40, [5, 4, 2]), 10, [1, 2, 2]) statistics-release-1.6.3/inst/ttest.m000066400000000000000000000154061456127120000176330ustar00rootroot00000000000000## Copyright (C) 2014 Tony Richardson ## Copyright (C) 2022 Andrew Penn ## Copyright (C) 2022 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {[@var{h}, @var{pval}, @var{ci}, @var{stats}] =} ttest (@var{x}) ## @deftypefnx {statistics} {[@var{h}, @var{pval}, @var{ci}, @var{stats}] =} ttest (@var{x}, @var{m}) ## @deftypefnx {statistics} {[@var{h}, @var{pval}, @var{ci}, @var{stats}] =} ttest (@var{x}, @var{y}) ## @deftypefnx {statistics} {[@var{h}, @var{pval}, @var{ci}, @var{stats}] =} ttest (@var{x}, @var{m}, @var{Name}, @var{Value}) ## @deftypefnx {statistics} {[@var{h}, @var{pval}, @var{ci}, @var{stats}] =} ttest (@var{x}, @var{y}, @var{Name}, @var{Value}) ## ## Test for mean of a normal sample with unknown variance. ## ## Perform a t-test of the null hypothesis @code{mean (@var{x}) == ## @var{m}} for a sample @var{x} from a normal distribution with unknown ## mean and unknown standard deviation. Under the null, the test statistic ## @var{t} has a Student's t distribution. The default value of ## @var{m} is 0. ## ## If the second argument @var{y} is a vector, a paired-t test of the ## hypothesis @code{mean (@var{x}) = mean (@var{y})} is performed. If @var{x} ## and @var{y} are vectors, they must have the same size and dimensions. ## ## @var{x} (and @var{y}) can also be matrices. For matrices, @qcode{ttest} ## performs separate t-tests along each column, and returns a vector of results. ## @var{x} and @var{y} must have the same number of columns. The Type I error ## rate of the resulting vector of @var{pval} can be controlled by entering ## @var{pval} as input to the function @qcode{multcompare}. ## ## @qcode{ttest} treats NaNs as missing values, and ignores them. ## ## Name-Value pair arguments can be used to set various options. ## @qcode{"alpha"} can be used to specify the significance level ## of the test (the default value is 0.05). @qcode{"tail"}, can be used ## to select the desired alternative hypotheses. If the value is ## @qcode{"both"} (default) the null is tested against the two-sided ## alternative @code{mean (@var{x}) != @var{m}}. ## If it is @qcode{"right"} the one-sided alternative @code{mean (@var{x}) ## > @var{m}} is considered. Similarly for @qcode{"left"}, the one-sided ## alternative @code{mean (@var{x}) < @var{m}} is considered. ## When argument @var{x} is a matrix, @qcode{"dim"} can be used to select ## the dimension over which to perform the test. (The default is the ## first non-singleton dimension). ## ## If @var{h} is 1 the null hypothesis is rejected, meaning that the tested ## sample does not come from a Student's t distribution. If @var{h} is 0, then ## the null hypothesis cannot be rejected and it can be assumed that @var{x} ## follows a Student's t distribution. The p-value of the test is returned in ## @var{pval}. A 100(1-alpha)% confidence interval is returned in @var{ci}. ## ## @var{stats} is a structure containing the value of the test statistic ## (@var{tstat}), the degrees of freedom (@var{df}) and the sample's standard ## deviation (@var{sd}). ## ## @seealso{hotelling_ttest, ttest2, hotelling_ttest2} ## @end deftypefn function [h, p, ci, stats] = ttest (x, my, varargin) ## Set default arguments my_default = 0; alpha = 0.05; tail = "both"; ## Find the first non-singleton dimension of x dim = min (find (size (x) != 1)); if (isempty (dim)) dim = 1; endif if (nargin == 1) my = my_default; endif i = 1; while (i <= length (varargin)) switch lower (varargin{i}) case "alpha" i = i + 1; alpha = varargin{i}; case "tail" i = i + 1; tail = varargin{i}; case "dim" i = i + 1; dim = varargin{i}; otherwise error ("ttest: Invalid Name argument."); endswitch i = i + 1; endwhile if (! isa (tail, "char")) error ("ttest: tail argument must be a string."); endif if (any (and (! isscalar (my), size (x) != size (my)))) error ("ttest: Arrays in paired test must be the same size."); endif ## Set default values if arguments are present but empty if (isempty (my)) my = my_default; endif ## This adjustment allows everything else to remain the ## same for both the one-sample t test and paired tests. x = x - my; if (! isscalar (my)) my = 0; endif ## Calculate the test statistic value (tval) n = sum (!isnan (x), dim); x_bar = mean (x, dim, "omitnan"); stats.tstat = []; stats.df = n - 1; stats.sd = std (x, 0, dim, "omitnan"); x_bar_std = stats.sd ./ sqrt(n); tval = (x_bar) ./ x_bar_std; stats.tstat = tval; ## Based on the "tail" argument determine the P-value, the critical values, ## and the confidence interval. switch lower (tail) case "both" p = 2 * (1 - tcdf (abs (tval), n - 1)); tcrit = - tinv (alpha / 2, n - 1); ci = [x_bar-tcrit.*x_bar_std; x_bar+tcrit.*x_bar_std] + my; case "left" p = tcdf (tval, n - 1); tcrit = - tinv (alpha, n - 1); ci = [-inf*ones(size(x_bar)); my+x_bar+tcrit.*x_bar_std]; case "right" p = 1 - tcdf (tval, n - 1); tcrit = - tinv (alpha, n - 1); ci = [my+x_bar-tcrit.*x_bar_std; inf*ones(size(x_bar))]; otherwise error ("ttest: Invalid value for tail argument."); endswitch ## Reshape the ci array to match MATLAB shaping if (isscalar (x_bar) && dim == 2) ci = ci(:)'; elseif (size (x_bar, 2) < size (x_bar, 1)) ci = reshape (ci(:), length (x_bar), 2); endif ## Determine the test outcome ## MATLAB returns this a double instead of a logical array h = double (p < alpha); endfunction %!test %! x = 8:0.1:12; %! [h, pval, ci] = ttest (x, 10); %! assert (h, 0) %! assert (pval, 1, 10*eps) %! assert (ci, [9.6219 10.3781], 1E-5) %! [h, pval, ci0] = ttest (x, 0); %! assert (h, 1) %! assert (pval, 0) %! assert (ci0, ci, 2e-15) %! [h, pval, ci] = ttest (x, 10, "tail", "right", "dim", 2, "alpha", 0.05); %! assert (h, 0) %! assert (pval, 0.5, 10*eps) %! assert (ci, [9.68498 Inf], 1E-5) %!error ttest ([8:0.1:12], 10, "tail", "invalid"); %!error ttest ([8:0.1:12], 10, "tail", 25); statistics-release-1.6.3/inst/ttest2.m000066400000000000000000000153421456127120000177140ustar00rootroot00000000000000## Copyright (C) 2014 Tony Richardson ## Copyright (C) 2022 Andrew Penn ## Copyright (C) 2022 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {[@var{h}, @var{pval}, @var{ci}, @var{stats}]} = ttest2 (@var{x}, @var{y}) ## @deftypefnx {statistics} {[@var{h}, @var{pval}, @var{ci}, @var{stats}]} = ttest2 (@var{x}, @var{y}, @var{Name}, @var{Value}) ## ## Perform a t-test to compare the means of two groups of data under the null ## hypothesis that the groups are drawn from distributions with the same mean. ## ## @var{x} and @var{y} can be vectors or matrices. For matrices, @qcode{ttest2} ## performs separate t-tests along each column, and returns a vector of results. ## @var{x} and @var{y} must have the same number of columns. The Type I error ## rate of the resulting vector of @var{pval} can be controlled by entering ## @var{pval} as input to the function @qcode{multcompare}. ## ## @qcode{ttest2} treats NaNs as missing values, and ignores them. ## ## For a nested t-test, use @qcode{anova2}. ## ## The argument @qcode{"alpha"} can be used to specify the significance level ## of the test (the default value is 0.05). The string argument @qcode{"tail"}, ## can be used to select the desired alternative hypotheses. If @qcode{"tail"} ## is @qcode{"both"} (default) the null is tested against the two-sided ## alternative @code{mean (@var{x}) != @var{m}}. If @qcode{"tail"} is ## @qcode{"right"} the one-sided alternative @code{mean (@var{x}) > @var{m}} is ## considered. Similarly for @qcode{"left"}, the one-sided alternative ## @code{mean (@var{x}) < @var{m}} is considered. ## ## When @qcode{"vartype"} is @qcode{"equal"} the variances are assumed to be ## equal (this is the default). When @qcode{"vartype"} is @qcode{"unequal"} the ## variances are not assumed equal. ## ## When argument @var{x} and @var{y} are matrices the @qcode{"dim"} argument can ## be used to select the dimension over which to perform the test. ## (The default is the first non-singleton dimension.) ## ## If @var{h} is 0 the null hypothesis is accepted, if it is 1 the null ## hypothesis is rejected. The p-value of the test is returned in @var{pval}. ## A 100(1-alpha)% confidence interval is returned in @var{ci}. @var{stats} ## is a structure containing the value of the test statistic (@var{tstat}), ## the degrees of freedom (@var{df}) and the sample standard deviation ## (@var{sd}). ## ## @seealso{hotelling_ttest2, anova1, hotelling_ttest, ttest} ## @end deftypefn function [h, p, ci, stats] = ttest2 (x, y, varargin) ## Set defaults alpha = 0.05; tail = "both"; vartype = "equal"; ## Find the first non-singleton dimension of x dim = min (find (size (x) != 1)); if (isempty (dim)) dim = 1; endif ## Evaluate optional input arguments i = 1; while ( i <= length(varargin) ) switch lower(varargin{i}) case "alpha" i = i + 1; alpha = varargin{i}; case "tail" i = i + 1; tail = varargin{i}; case "vartype" i = i + 1; vartype = varargin{i}; case "dim" i = i + 1; dim = varargin{i}; otherwise error ("ttest2: Invalid Name argument."); endswitch i = i + 1; endwhile ## Error checking if (! isa (tail, "char")) error ("ttest2: tail argument must be a string."); endif if (size (x, abs (dim - 3)) != size (y, abs (dim - 3))) error ("ttest2: The data in a 2-sample t-test must be commensurate") endif ## Calculate mean, variance and size of each sample m = sum (!isnan (x), dim); n = sum (!isnan (y), dim); x_bar = mean (x, dim, "omitnan") - mean (y, dim, "omitnan"); s1_var = var (x, 0, dim, "omitnan"); s2_var = var (y, 0, dim, "omitnan"); ## Perform test-specific calculations switch lower (vartype) case "equal" stats.tstat = []; stats.df = (m + n - 2); sp_var = ((m - 1) .* s1_var + (n - 1) .* s2_var) ./ stats.df; stats.sd = sqrt (sp_var); x_bar_std = sqrt (sp_var .* (1 ./ m + 1 ./ n)); n_sd = 1; case "unequal" stats.tstat = []; se1 = sqrt (s1_var ./ m); se2 = sqrt (s2_var ./ n); sp_var = s1_var ./ m + s2_var ./ n; stats.df = ((se1 .^ 2 + se2 .^ 2) .^ 2 ./ ... (se1 .^ 4 ./ (m - 1) + se2 .^ 4 ./ (n - 1))); stats.sd = [sqrt(s1_var); sqrt(s2_var)]; x_bar_std = sqrt (sp_var); n_sd = 2; otherwise error ("ttest2: Invalid value for vartype argument."); end stats.tstat = x_bar ./ x_bar_std; ## Based on the "tail" argument determine the P-value, the critical values, ## and the confidence interval. switch lower(tail) case "both" p = 2 * (1 - tcdf (abs (stats.tstat), stats.df)); tcrit = - tinv (alpha / 2, stats.df); ci = [x_bar-tcrit.*x_bar_std; x_bar+tcrit.*x_bar_std]; case "left" p = tcdf (stats.tstat, stats.df); tcrit = - tinv (alpha, stats.df); ci = [-inf*ones(size(x_bar)); x_bar+tcrit.*x_bar_std]; case "right" p = 1 - tcdf (stats.tstat, stats.df); tcrit = - tinv (alpha, stats.df); ci = [x_bar-tcrit.*x_bar_std; inf*ones(size(x_bar))]; otherwise error ("ttest2: Invalid value for tail argument."); endswitch ## Reshape the ci array to match MATLAB shaping if (isscalar (x_bar) && dim == 2) ci = ci(:)'; stats.sd = stats.sd(:)'; elseif (size (x_bar, 2) < size (x_bar, 1)) ci = reshape (ci(:), length (x_bar), 2); stats.sd = reshape (stats.sd(:), length (x_bar), n_sd); endif ## Determine the test outcome ## MATLAB returns this a double instead of a logical array h = double (p < alpha); endfunction %!test %! a = 1:5; %! b = 6:10; %! b(5) = NaN; %! [h,p,ci,stats] = ttest2 (a,b); %! assert (h, 1); %! assert (p, 0.002535996080258229, 1e-14); %! assert (ci, [-6.822014919225481, -2.17798508077452], 1e-14); %! assert (stats.tstat, -4.582575694955839, 1e-14); %! assert (stats.df, 7); %! assert (stats.sd, 1.4638501094228, 1e-13); %!error ttest2 ([8:0.1:12], [8:0.1:12], "tail", "invalid"); %!error ttest2 ([8:0.1:12], [8:0.1:12], "tail", 25); statistics-release-1.6.3/inst/vartest.m000066400000000000000000000201201456127120000201450ustar00rootroot00000000000000## Copyright (C) 2014 Tony Richardson ## Copyright (C) 2022 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{h} =} vartest (@var{x}, @var{v}) ## @deftypefnx {statistics} {@var{h} =} vartest (@var{x}, @var{v}, @var{name}, @var{value}) ## @deftypefnx {statistics} {[@var{h}, @var{pval}] =} vartest (@dots{}) ## @deftypefnx {statistics} {[@var{h}, @var{pval}, @var{ci}] =} vartest (@dots{}) ## @deftypefnx {statistics} {[@var{h}, @var{pval}, @var{ci}, @var{stats}] =} vartest (@dots{}) ## ## One-sample test of variance. ## ## @code{@var{h} = vartest (@var{x}, @var{v})} performs a chi-square test of the ## hypothesis that the data in the vector @var{x} come from a normal ## distribution with variance @var{v}, against the alternative that @var{x} ## comes from a normal distribution with a different variance. The result is ## @var{h} = 0 if the null hypothesis ("variance is V") cannot be rejected at ## the 5% significance level, or @var{h} = 1 if the null hypothesis can be ## rejected at the 5% level. ## ## @var{x} may also be a matrix or an N-D array. For matrices, @code{vartest} ## performs separate tests along each column of @var{x}, and returns a vector of ## results. For N-D arrays, @code{vartest} works along the first non-singleton ## dimension of @var{x}. @var{v} must be a scalar. ## ## @code{vartest} treats NaNs as missing values, and ignores them. ## ## @code{[@var{h}, @var{pval}] = vartest (@dots{})} returns the p-value. That ## is the probability of observing the given result, or one more extreme, by ## chance if the null hypothesisis true. ## ## @code{[@var{h}, @var{pval}, @var{ci}] = vartest (@dots{})} returns a ## 100 * (1 - @var{alpha})% confidence interval for the true variance. ## ## @code{[@var{h}, @var{pval}, @var{ci}, @var{stats}] = vartest (@dots{})} ## returns a structure with the following fields: ## ## @multitable @columnfractions 0.05 0.2 0.75 ## @item @tab @qcode{chisqstat} @tab the value of the test statistic ## @item @tab @qcode{df} @tab the degrees of freedom of the test ## @end multitable ## ## @code{[@dots{}] = vartest (@dots{}, @var{name}, @var{value}), @dots{}} ## specifies one or more of the following name/value pairs: ## ## @multitable @columnfractions 0.05 0.2 0.75 ## @headitem @tab Name @tab Value ## @item @tab @qcode{"alpha"} @tab the significance level. Default is 0.05. ## ## @item @tab @qcode{"dim"} @tab dimension to work along a matrix or an N-D ## array. ## ## @item @tab @qcode{"tail"} @tab a string specifying the alternative hypothesis ## @end multitable ## @multitable @columnfractions 0.1 0.15 0.75 ## @item @tab @qcode{"both"} @tab variance is not @var{v} (two-tailed, default) ## @item @tab @qcode{"left"} @tab variance is less than @var{v} (left-tailed) ## @item @tab @qcode{"right"} @tab variance is greater than @var{v} ## (right-tailed) ## @end multitable ## ## @seealso{ttest, ztest, kstest} ## @end deftypefn function [h, pval, ci, stats] = vartest (x, v, varargin) ## Validate input arguments if (nargin < 2) error ("vartest: too few input arguments."); endif if (! isscalar (v) || ! isnumeric(v) || ! isreal(v) || v < 0) error ("vartest: invalid value for variance."); endif ## Add defaults alpha = 0.05; tail = "both"; dim = []; if (nargin > 2 && mod (numel (varargin(:)), 2) == 0) for idx = 3:2:nargin name = varargin{idx-2}; value = varargin{idx-1}; switch (lower (name)) case "alpha" alpha = value; if (! isscalar (alpha) || ! isnumeric (alpha) || ... alpha <= 0 || alpha >= 1) error ("vartest: invalid value for alpha."); endif case "tail" tail = value; if (! any (strcmpi (tail, {"both", "left", "right"}))) error ("vartest: invalid value for tail."); endif case "dim" dim = value; if (! isscalar (dim) || ! ismember (dim, 1:ndims (x))) error ("vartest: invalid value for operating dimension."); endif otherwise error ("vartest: invalid name for optional arguments."); endswitch endfor elseif (nargin > 2 && mod (numel (varargin(:)), 2) != 0) error ("vartest: optional arguments must be in name/value pairs."); endif ## Figure out which dimension mean will work along if (isempty (dim)) dim = find (size (x) != 1, 1); endif ## Replace all NaNs with zeros is_nan = isnan (x); x_dims = ndims (x); x(is_nan) = 0; ## Find sample size for each group (if more than one) if (any (is_nan(:))) sz = sum (! is_nan, dim); else sz = size (x, dim); endif ## Find degrees of freedom for each group (if more than one) df = max (sz - 1, 0); ## Calculate mean for each group (if more than one) x_mean = sum (x, dim) ./ max (1, sz); ## Center data if (isscalar (x_mean)) x_centered = x - x_mean; else rep = ones (1, x_dims); rep(dim) = size (x, dim); x_centered = x - repmat (x_mean, rep); endif ## Replace all NaNs with zeros x_centered(is_nan) = 0; ## Calculate chi-square statistic sumsq = sum (abs (x_centered) .^ 2, dim); if (v > 0) chisqstat = sumsq ./ v; else chisqstat = Inf (size (sumsq)); chisqstat(sumsq == 0) = NaN; endif ## Calculate p-value for the test and confidence intervals (if requested) if (strcmpi (tail, "both")) pval = chi2cdf (chisqstat, df); pval = 2 * min (pval, 1 - pval); if (nargout > 2) ci = cat (dim, sumsq ./ chi2inv (1 - alpha / 2, df), ... sumsq ./ chi2inv (alpha / 2, df)); endif elseif (strcmpi (tail, "right")) pval = chi2cdf (chisqstat, df); if (nargout > 2) ci = cat (dim, sumsq ./ chi2inv (1 - alpha, df), Inf (size (pval))); endif elseif (strcmpi (tail, "left")) pval = chi2cdf (chisqstat, df); if (nargout > 2) ci = cat (dim, zeros (size (pval)), sumsq ./ chi2inv (alpha, df)); endif endif ## Determine the test outcome h = double (pval < alpha); h(isnan (pval)) = NaN; ## Create stats output structure (if requested) if (nargout > 3) stats = struct ("chisqstat", chisqstat, "df", df); endif endfunction ## Test input validation %!error vartest (); %!error vartest ([1, 2, 3, 4], -0.5); %!error ... %! vartest ([1, 2, 3, 4], 1, "alpha", 0); %!error ... %! vartest ([1, 2, 3, 4], 1, "alpha", 1.2); %!error ... %! vartest ([1, 2, 3, 4], 1, "alpha", "val"); %!error ... %! vartest ([1, 2, 3, 4], 1, "tail", "val"); %!error ... %! vartest ([1, 2, 3, 4], 1, "alpha", 0.01, "tail", "val"); %!error ... %! vartest ([1, 2, 3, 4], 1, "dim", 3); %!error ... %! vartest ([1, 2, 3, 4], 1, "alpha", 0.01, "tail", "both", "dim", 3); %!error ... %! vartest ([1, 2, 3, 4], 1, "alpha", 0.01, "tail", "both", "badoption", 3); %!error ... %! vartest ([1, 2, 3, 4], 1, "alpha", 0.01, "tail"); ## Test results %!test %! load carsmall %! [h, pval, ci] = vartest (MPG, 7^2); %! assert (h, 1); %! assert (pval, 0.04335086742174443, 1e-14); %! assert (ci, [49.397; 88.039], 1e-3); statistics-release-1.6.3/inst/vartest2.m000066400000000000000000000222751456127120000202440ustar00rootroot00000000000000## Copyright (C) 2014 Tony Richardson ## Copyright (C) 2022 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{h} =} vartest2 (@var{x}, @var{y}) ## @deftypefnx {statistics} {@var{h} =} vartest2 (@var{x}, @var{y}, @var{name}, @var{value}) ## @deftypefnx {statistics} {[@var{h}, @var{pval}] =} vartest2 (@dots{}) ## @deftypefnx {statistics} {[@var{h}, @var{pval}, @var{ci}] =} vartest2 (@dots{}) ## @deftypefnx {statistics} {[@var{h}, @var{pval}, @var{ci}, @var{stats}] =} vartest2 (@dots{}) ## ## Two-sample F test for equal variances. ## ## @code{@var{h} = vartest2 (@var{x}, @var{y})} performs an F test of the ## hypothesis that the independent data in vectors @var{x} and @var{y} come from ## normal distributions with equal variance, against the alternative that they ## come from normal distributions with different variances. The result is ## @var{h} = 0 if the null hypothesis ("variance are equal") cannot be rejected ## at the 5% significance level, or @var{h} = 1 if the null hypothesis can be ## rejected at the 5% level. ## ## @var{x} and @var{y} may also be matrices or N-D arrays. For matrices, ## @code{vartest2} performs separate tests along each column and returns a ## vector of results. For N-D arrays, @code{vartest2} works along the first ## non-singleton dimension and @var{x} and @var{y} must have the same size along ## all the remaining dimensions. ## ## @code{vartest} treats NaNs as missing values, and ignores them. ## ## @code{[@var{h}, @var{pval}] = vartest (@dots{})} returns the p-value. That ## is the probability of observing the given result, or one more extreme, by ## chance if the null hypothesisis true. ## ## @code{[@var{h}, @var{pval}, @var{ci}] = vartest (@dots{})} returns a ## @math{100 * (1 - @var{alpha})%} confidence interval for the true ratio ## var(X)/var(Y). ## ## @code{[@var{h}, @var{pval}, @var{ci}, @var{stats}] = vartest (@dots{})} ## returns a structure with the following fields: ## ## @multitable @columnfractions 0.05 0.2 0.75 ## @item @tab @qcode{fstat} @tab the value of the test statistic ## @item @tab @qcode{df1} @tab the numerator degrees of freedom of the test ## @item @tab @qcode{df2} @tab the denominator degrees of freedom of the test ## @end multitable ## ## @code{[@dots{}] = vartest (@dots{}, @var{name}, @var{value}), @dots{}} ## specifies one or more of the following name/value pairs: ## ## @multitable @columnfractions 0.05 0.2 0.75 ## @headitem @tab Name @tab Value ## @item @tab @qcode{"alpha"} @tab the significance level. Default is 0.05. ## ## @item @tab @qcode{"dim"} @tab dimension to work along a matrix or an N-D ## array. ## ## @item @tab @qcode{"tail"} @tab a string specifying the alternative hypothesis ## @end multitable ## @multitable @columnfractions 0.1 0.15 0.75 ## @item @tab @qcode{"both"} @tab variance is not @var{v} (two-tailed, default) ## @item @tab @qcode{"left"} @tab variance is less than @var{v} (left-tailed) ## @item @tab @qcode{"right"} @tab variance is greater than @var{v} ## (right-tailed) ## @end multitable ## ## @seealso{ttest2, kstest2, bartlett_test, levene_test} ## @end deftypefn function [h, pval, ci, stats] = vartest2 (x, y, varargin) ## Validate input arguments if (nargin < 2) error ("vartest2: too few input arguments."); endif if (isscalar (x) || isscalar(y)) error ("vartest2: X and Y must be vectors or matrices or N-D arrays."); endif ## If X and Y are vectors make them the same orientation if (isvector (x) && isvector (y)) if (size (x, 1) == 1) y = y(:)'; else y = y(:); endif endif ## Add defaults alpha = 0.05; tail = "both"; dim = []; if (nargin > 2 && mod (numel (varargin(:)), 2) == 0) for idx = 3:2:nargin name = varargin{idx-2}; value = varargin{idx-1}; switch (lower (name)) case "alpha" alpha = value; if (! isscalar (alpha) || ! isnumeric (alpha) || ... alpha <= 0 || alpha >= 1) error ("vartest2: invalid value for alpha."); endif case "tail" tail = value; if (! any (strcmpi (tail, {"both", "left", "right"}))) error ("vartest2: invalid value for tail."); endif case "dim" dim = value; if (! isscalar (dim) || ! ismember (dim, 1:ndims (x))) error ("vartest2: invalid value for operating dimension."); endif otherwise error ("vartest2: invalid name for optional arguments."); endswitch endfor elseif (nargin > 2 && mod (numel (varargin(:)), 2) != 0) error ("vartest2: optional arguments must be in name/value pairs."); endif ## Figure out which dimension mean will work along if (isempty (dim)) dim = find (size (x) != 1, 1); endif ## Check that all non-working dimensions of X and Y are of equal size x_size = size (x); y_size = size (y); x_size(dim) = 1; y_size(dim) = 1; if (! isequal (x_size, y_size)) error ("vartestt2: input size mismatch."); endif ## Compute statistics for each sample [df1, x_var] = getstats(x,dim); [df2, y_var] = getstats(y,dim); ## Compute F statistic F = NaN (size (x_var)); t1 = (y_var > 0); F(t1) = x_var(t1) ./ y_var(t1); t2 = (x_var > 0) & ! t1; F(t2) = Inf; ## Calculate p-value for the test and confidence intervals (if requested) if (strcmpi (tail, "both")) pval = 2 * min (fcdf (F, df1, df2), 1 - fcdf (F, df1, df2)); if (nargout > 2) ci = cat (dim, F .* finv (alpha / 2, df2, df1), ... F ./ finv (alpha / 2, df1, df2)); endif elseif (strcmpi (tail, "right")) pval = 1 - fcdf (F, df1, df2); if (nargout > 2) ci = cat (dim, F .* finv (alpha, df2, df1), Inf (size (F))); endif elseif (strcmpi (tail, "left")) pval = fcdf (F, df1, df2); if (nargout > 2) ci = cat (dim, zeros (size (F)), F ./ finv (alpha, df1, df2)); endif endif ## Determine the test outcome h = double (pval < alpha); h(isnan (pval)) = NaN; ## Create stats output structure (if requested) if (nargout > 3) stats = struct ("fstat", F, "df1", df1, "df2", df2); endif endfunction ## Compute statistics for one sample function [df, data_var] = getstats (data, dim) ## Calculate sample size and df by ignoring NaNs is_nan = isnan (data); n_data = sum (! is_nan, dim); df = max (n_data - 1, 0); ## Calculate mean data(is_nan) = 0; m_data = sum (data, dim) ./ max (1, n_data); ## Calculate variance if (isscalar (m_data)) c_data = data - m_data; else rep = ones (1, ndims (data)); rep(dim) = size (data, dim); c_data = data - repmat (m_data, rep); end c_data(is_nan) = 0; data_var = sum (abs (c_data) .^ 2,dim); t = (df > 0); data_var(t) = data_var(t) ./ df(t); data_var(! t) = NaN; ## Make df a scalar if possible if (numel (df) > 1 && all (df(:) == df(1))) df = df(1); end endfunction ## Test input validation %!error vartest2 (); %!error vartest2 (ones (20,1)); %!error ... %! vartest2 (rand (20,1), 5); %!error ... %! vartest2 (rand (20,1), rand (25,1)*2, "alpha", 0); %!error ... %! vartest2 (rand (20,1), rand (25,1)*2, "alpha", 1.2); %!error ... %! vartest2 (rand (20,1), rand (25,1)*2, "alpha", "some"); %!error ... %! vartest2 (rand (20,1), rand (25,1)*2, "alpha", [0.05, 0.001]); %!error ... %! vartest2 (rand (20,1), rand (25,1)*2, "tail", [0.05, 0.001]); %!error ... %! vartest2 (rand (20,1), rand (25,1)*2, "tail", "some"); %!error ... %! vartest2 (rand (20,1), rand (25,1)*2, "dim", 3); %!error ... %! vartest2 (rand (20,1), rand (25,1)*2, "alpha", 0.001, "dim", 3); %!error ... %! vartest2 (rand (20,1), rand (25,1)*2, "some", 3); %!error ... %! vartest2 (rand (20,1), rand (25,1)*2, "some"); ## Test results %!test %! load carsmall %! [h, pval, ci, stat] = vartest2 (MPG(Model_Year==82), MPG(Model_Year==76)); %! assert (h, 0); %! assert (pval, 0.6288022362718455, 1e-13); %! assert (ci, [0.4139; 1.7193], 1e-4); %! assert (stat.fstat, 0.8384, 1e-4); %! assert (stat.df1, 30); %! assert (stat.df2, 33); statistics-release-1.6.3/inst/vartestn.m000066400000000000000000000404461456127120000203400ustar00rootroot00000000000000## Copyright (C) 2022 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {} vartestn (@var{x}) ## @deftypefnx {statistics} {} vartestn (@var{x}, @var{group}) ## @deftypefnx {statistics} {} vartestn (@dots{}, @var{name}, @var{value}) ## @deftypefnx {statistics} {@var{p} =} vartestn (@dots{}) ## @deftypefnx {statistics} {[@var{p}, @var{stats}] =} vartestn (@dots{}) ## @deftypefnx {statistics} {[@var{p}, @var{stats}] =} vartestn (@dots{}, @var{name}, @var{value}) ## ## Test for equal variances across multiple groups. ## ## @code{@var{h} = vartestn (@var{x})} performs Bartlett's test for equal ## variances for the columns of the matrix @var{x}. This is a test of the null ## hypothesis that the columns of @var{x} come from normal distributions with ## the same variance, against the alternative that they come from normal ## distributions with different variances. The result is displayed in a summary ## table of statistics as well as a box plot of the groups. ## ## @code{vartestn (@var{x}, @var{group})} requires a vector @var{x}, and a ## @var{group} argument that is a categorical variable, vector, string array, or ## cell array of strings with one row for each element of @var{x}. Values of ## @var{x} corresponding to the same value of @var{group} are placed in the same ## group. ## ## @code{vartestn} treats NaNs as missing values, and ignores them. ## ## @code{@var{p} = vartestn (@dots{})} returns the probability of observing the ## given result, or one more extreme, by chance under the null hypothesis that ## all groups have equal variances. Small values of @var{p} cast doubt on the ## validity of the null hypothesis. ## ## @code{[@var{p}, @var{stats}] = vartestn (@dots{})} returns a structure with ## the following fields: ## ## @multitable @columnfractions 0.05 0.2 0.75 ## @item @tab @qcode{chistat} @tab -- the value of the test statistic ## @item @tab @qcode{df} @tab -- the degrees of freedom of the test ## @end multitable ## ## ## @code{[@var{p}, @var{stats}] = vartestn (@dots{}, @var{name}, @var{value})} ## specifies one or more of the following @var{name}/@var{value} pairs: ## ## @multitable @columnfractions 0.20 0.8 ## @item @qcode{"display"} @tab @qcode{"on"} to display a boxplot and table, or ## @qcode{"off"} to omit these displays. Default @qcode{"on"}. ## ## @item @qcode{"testtype"} @tab One of the following strings to control the ## type of test to perform ## @end multitable ## ## @multitable @columnfractions 0.03 0.25 0.72 ## @item @tab @qcode{"Bartlett"} @tab Bartlett's test (default). ## ## @item @tab @qcode{"LeveneQuadratic"} @tab Levene's test computed by ## performing anova on the squared deviations of the data values from their ## group means. ## ## @item @tab @qcode{"LeveneAbsolute"} @tab Levene's test computed by performing ## anova on the absolute deviations of the data values from their group means. ## ## @item @tab @qcode{"BrownForsythe"} @tab Brown-Forsythe test computed by ## performing anova on the absolute deviations of the data values from the group ## medians. ## ## @item @tab @qcode{"OBrien"} @tab O'Brien's modification of Levene's test with ## @math{W=0.5}. ## @end multitable ## ## The classical Bartlett's test is sensitive to the assumption that the ## distribution in each group is normal. The other test types are more robust ## to non-normal distributions, especially ones prone to outliers. For these ## tests, the STATS output structure has a field named @qcode{fstat} containing ## the test statistic, and @qcode{df1} and @qcode{df2} containing its numerator ## and denominator degrees of freedom. ## ## @seealso{vartest, vartest2, anova1, bartlett_test, levene_test} ## @end deftypefn function [p, stats] = vartestn (x, group, varargin) ## Validate input arguments if (nargin < 1) error ("vartestn: too few input arguments."); endif if (isscalar (x)) error ("vartestn: X must be a vector or a matrix."); endif if (nargin < 2) group = []; endif if (nargin > 1 && any (strcmpi (group, {"display", "testtype"}))) varargin = [{group} varargin]; group = []; endif if (isvector (x) && (nargin < 2 || isempty (group ))) error ("vartestn: if X is a vector then a group vector is required."); endif ## Add defaults plotdata = true; testtype = "Bartlett"; if (numel (varargin(:)) > 0 && mod (numel (varargin(:)), 2) == 0) for idx = 1:2:numel (varargin(:)) name = varargin{idx}; value = varargin{idx+1}; switch (lower (name)) case "display" plotdata = value; if (! any (strcmpi (plotdata, {"on", "off"}))) error ("vartestn: invalid value for display."); endif if (strcmpi (plotdata, "on")) plotdata = true; else plotdata = false; endif case "testtype" testtype = value; if (! any (strcmpi (testtype, {"Bartlett", "LeveneAbsolute", ... "LeveneQuadratic", "BrownForsythe", "OBrien"}))) error ("vartestn: invalid value for testtype."); endif otherwise error ("vartestn: invalid name for optional arguments."); endswitch endfor elseif (numel (varargin(:)) > 0 && mod (numel (varargin(:)), 2) != 0) error ("vartestn: optional arguments must be in name/value pairs."); endif ## Convert group to cell array from character array, make it a column if (! isempty (group) && ischar (group)) group = cellstr (group); endif if (size (group, 1) == 1) group = group'; endif ## If x is a matrix, convert it to column vector and create a ## corresponging column vector for groups if (length (x) < prod (size (x))) [n, m] = size (x); x = x(:); gi = reshape (repmat ((1:m), n, 1), n*m, 1); if (length (group) == 0) ## no group names are provided group = gi; elseif (size (group, 1) == m) ## group names exist and match columns group = group(gi,:); else error ("vartestn: columns in X and GROUP length do not match."); endif endif ## Check that x and group are the same size if (! all (numel (x) == numel (group))) error ("vartestn: GROUP must be a vector with the same number of rows as x."); endif ## Identify NaN values (if any) and remove them from X along with ## their corresponding values from group vector nonan = ! isnan (x); x = x(nonan); group = group(nonan, :); ## Convert group to indices and separate names [group_id, group_names] = grp2idx (group); group_id = group_id(:); ## Compute group summary statistics [group_mean, group_ster, group_size] = grpstats (x, group_id, ... {"mean", "sem", "numel"}); ## Compute group degreed of freedom and variances group_DF = group_size - 1; groupVAR = group_size .* group_ster .^ 2; sum_DF = sum (group_DF); ## Caculate pooled variance if (sum_DF > 0) pooledVAR = sum (group_DF .* groupVAR) / sum_DF; else pooledVAR = NaN; end ## Get number of groups k = length (group_DF); ## Test for equal variance according to specified testtype switch (lower (testtype)) case "bartlett" ## Calculate degrees of freedom Bdf = max(0, sum (group_DF > 0) - 1); ## Get valid groups msgroups = group_DF > 0; ## For valid groups if (Bdf > 0 && sum_DF > 0) B = log (pooledVAR) * sum (group_DF) - ... sum (group_DF(msgroups) .* log (groupVAR(msgroups))); C = 1 + (sum (1 ./ group_DF(msgroups)) - 1 / sum (group_DF)) / (3 * Bdf); F = B / C; else F = NaN; endif ## Compute p-value p = 1 - chi2cdf (F, Bdf); testname = "Bartlett's statistic "; if (nargout > 1) stats = struct("chisqstat", F, "df", Bdf); endif case {"leveneabsolute", "levenequadratic"} ## Remove single-sample groups ssgroups = find (group_size < 2); msgroups = ! ismember (group_id, ssgroups); ## Center each group with mean x_center = x(msgroups) - group_mean(group_id(msgroups)); ## Get number of valid groups (group size > 1) n_groups = length (group_size) - length (ssgroups); ## Perform one-way anova and extract results from the anova table if (n_groups > 1) if (strcmpi (testtype, "LeveneAbsolute")) [p, atab] = anova1 (abs (x_center), group_id(msgroups), "off"); testname = "Levene's statistic (absolute) "; else [p, atab] = anova1 (x_center .^ 2, group_id(msgroups), "off"); testname = "Levene's statistic (quadratic) "; endif ## Get F statistic and both degrees of freedom F = atab{2,5}; Bdf = [atab{2,3}, atab{3,3}]; else p = NaN; F = NaN; Bdf = [0, (length (x_center) - n_groups)]; endif if (nargout > 1) stats = struct("fstat", F, "df", Bdf); endif case "brownforsythe" ## Remove single-sample groups ssgroups = find (group_size < 2); msgroups = ! ismember (group_id, ssgroups); ## Calculate group medians group_md = grpstats (x, group_id, "median"); ## Center each group with median xcbf = x(msgroups) - group_md(group_id(msgroups)); ## Get number of valid groups (group size > 1) n_groups = length(group_size) - length(ssgroups); ## Perform one-way anova and extract results from the anova table if (n_groups > 1) [p, atab] = anova1 (abs (xcbf), group_id(msgroups), "off"); ## Get F statistic and both degrees of freedom F = atab{2,5}; Bdf = [atab{2,3}, atab{3,3}]; else p = NaN; F = NaN; Bdf = [0, (length (xcbf) - n_groups)]; end testname = "Brown-Forsythe statistic "; if (nargout > 1) stats = struct("fstat", F, "df", Bdf); endif case "obrien" ## Remove single-sample groups ssgroups = find (group_size < 2); msgroups = ! ismember (group_id, ssgroups); ## Center each group with mean x_center = x(msgroups) - group_mean(group_id(msgroups)); ## Calculate OBrien Z_ij xcs = x_center.^2; W = 0.5; xcw = ((W + group_size(group_id(msgroups)) - 2) .* ... group_size(group_id(msgroups)) .* xcs - W .* ... (group_size(group_id(msgroups)) - 1) .* ... groupVAR(group_id(msgroups))) ./ ... ((group_size(group_id(msgroups)) - 1) .* ... (group_size(group_id(msgroups)) - 2)); ## Get number of valid groups (group size > 1) n_groups = length(group_size) - length(ssgroups); ## Perform one-way anova and extract results from the anova table if (n_groups > 1) [p, atab] = anova1 (xcw, group_id(msgroups), "off"); ## Get F statistic and both degrees of freedom F = atab{2,5}; Bdf = [atab{2,3}, atab{3,3}]; else p = NaN; F = NaN; Bdf = [0, length(xcw)-n_groups]; end testname = "OBrien statistic "; if (nargout > 1) stats = struct("fstat", F, "df", Bdf); endif endswitch ## Print Group Summary Table (unless opted out) if (nargout == 0 || plotdata) groupSTD = sqrt (groupVAR); printf ("\n Group Summary Table\n\n"); printf ("Group Count Mean Std Dev\n"); printf ("------------------------------------------------------------\n"); for i = 1:k printf ("%-20s %10i %9.4f %1.6f\n", ... group_names{i}, group_size(i), group_mean(i), groupSTD(i)); endfor printf ("Pooled Groups %10i %9.4f %1.6f\n", ... sum (group_size), mean (group_mean), mean (groupSTD)); printf ("Pooled valid Groups %10i %9.4f %1.6f\n\n", ... sum (group_size(group_id(msgroups))), ... mean (group_mean(group_id(msgroups))), ... mean (groupSTD(group_id(msgroups)))); printf ("%s %7.5f\n", testname, F); if (numel (Bdf) == 1) printf ("Degrees of Freedom %10i\n", Bdf); else printf ("Degrees of Freedom %10i, %3i\n", Bdf(1), Bdf(2)); endif printf ("p-value %1.6f\n\n", p); endif ## Plot data using BOXPLOT (unless opted out) if (plotdata) boxplot (x, group_id, "Notch", "on", "Labels", group_names); endif endfunction %!demo %! ## Test the null hypothesis that the variances are equal across the five %! ## columns of data in the students’ exam grades matrix, grades. %! %! load examgrades %! vartestn (grades) %!demo %! ## Test the null hypothesis that the variances in miles per gallon (MPG) are %! ## equal across different model years. %! %! load carsmall %! vartestn (MPG, Model_Year) %!demo %! ## Use Levene’s test to test the null hypothesis that the variances in miles %! ## per gallon (MPG) are equal across different model years. %! %! load carsmall %! p = vartestn (MPG, Model_Year, "TestType", "LeveneAbsolute") %!demo %! ## Test the null hypothesis that the variances are equal across the five %! ## columns of data in the students’ exam grades matrix, grades, using the %! ## Brown-Forsythe test. Suppress the display of the summary table of %! ## statistics and the box plot. %! %! load examgrades %! [p, stats] = vartestn (grades, "TestType", "BrownForsythe", "Display", "off") ## Test input validation %!error vartestn (); %!error vartestn (1); %!error ... %! vartestn ([1, 2, 3, 4, 5, 6, 7]); %!error ... %! vartestn ([1, 2, 3, 4, 5, 6, 7], []); %!error ... %! vartestn ([1, 2, 3, 4, 5, 6, 7], "TestType", "LeveneAbsolute"); %!error ... %! vartestn ([1, 2, 3, 4, 5, 6, 7], [], "TestType", "LeveneAbsolute"); %!error ... %! vartestn ([1, 2, 3, 4, 5, 6, 7], [1, 1, 1, 2, 2, 2, 2], "Display", "some"); %!error ... %! vartestn (ones (50,3), "Display", "some"); %!error ... %! vartestn (ones (50,3), "Display", "off", "testtype", "some"); %!error ... %! vartestn (ones (50,3), [], "som"); %!error ... %! vartestn (ones (50,3), [], "some", "some"); %!error ... %! vartestn (ones (50,3), [1, 2], "Display", "off"); ## Test results %!test %! load examgrades %! [p, stat] = vartestn (grades, "Display", "off"); %! assert (p, 7.908647337018238e-08, 1e-14); %! assert (stat.chisqstat, 38.7332, 1e-4); %! assert (stat.df, 4); %!test %! load examgrades %! [p, stat] = vartestn (grades, "Display", "off", "TestType", "LeveneAbsolute"); %! assert (p, 9.523239714592791e-07, 1e-14); %! assert (stat.fstat, 8.5953, 1e-4); %! assert (stat.df, [4, 595]); %!test %! load examgrades %! [p, stat] = vartestn (grades, "Display", "off", "TestType", "LeveneQuadratic"); %! assert (p, 7.219514351897161e-07, 1e-14); %! assert (stat.fstat, 8.7503, 1e-4); %! assert (stat.df, [4, 595]); %!test %! load examgrades %! [p, stat] = vartestn (grades, "Display", "off", "TestType", "BrownForsythe"); %! assert (p, 1.312093241723211e-06, 1e-14); %! assert (stat.fstat, 8.4160, 1e-4); %! assert (stat.df, [4, 595]); %!test %! load examgrades %! [p, stat] = vartestn (grades, "Display", "off", "TestType", "OBrien"); %! assert (p, 8.235660885480556e-07, 1e-14); %! assert (stat.fstat, 8.6766, 1e-4); %! assert (stat.df, [4, 595]); statistics-release-1.6.3/inst/violin.m000066400000000000000000000267631456127120000200000ustar00rootroot00000000000000## Copyright (C) 2016 - Juan Pablo Carbajal ## Copyright (C) 2022-2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This progrm is free software; you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation; either version 3 of the License, or ## (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program. If not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {} violin (@var{x}) ## @deftypefnx {statistics} {@var{h} =} violin (@var{x}) ## @deftypefnx {statistics} {@var{h} =} violin (@dots{}, @var{property}, @var{value}, @dots{}) ## @deftypefnx {statistics} {@var{h} =} violin (@var{hax}, @dots{}) ## @deftypefnx {statistics} {@var{h} =} violin (@dots{}, @code{"horizontal"}) ## ## Produce a Violin plot of the data @var{x}. ## ## The input data @var{x} can be a N-by-m array containg N observations of m ## variables. It can also be a cell with m elements, for the case in which the ## variables are not uniformly sampled. ## ## The following @var{property} can be set using @var{property}/@var{value} pairs ## (default values in parenthesis). ## The value of the property can be a scalar indicating that it applies ## to all the variables in the data. ## It can also be a cell/array, indicating the property for each variable. ## In this case it should have m columns (as many as variables). ## ## @table @asis ## ## @item Color ## (@asis{"y"}) Indicates the filling color of the violins. ## ## @item Nbins ## (50) Internally, the function calls @command{hist} to compute the histogram ## of the data. This property indicates how many bins to use. ## See @command{help hist} for more details. ## ## @item SmoothFactor ## (4) The fuction performs simple kernel density estimation and automatically ## finds the bandwith of the kernel function that best approximates the ## histogram using optimization (@command{sqp}). ## The result is in general very noisy. To smooth the result the bandwidth is ## multiplied by the value of this property. The higher the value the smoother ## the violings, but values too high might remove features from the data ## distribution. ## ## @item Bandwidth ## (NA) If this property is given a value other than NA, it sets the bandwith of ## the kernel function. No optimization is peformed and the property ## @asis{SmoothFactor} is ignored. ## ## @item Width ## (0.5) Sets the maximum width of the violins. Violins are centered at integer ## axis values. The distance between two violin middle axis is 1. Setting a ## value higher thna 1 in this property will cause the violins to overlap. ## @end table ## ## If the string @asis{"Horizontal"} is among the input arguments, the violin ## plot is rendered along the x axis with the variables in the y axis. ## ## The returned structure @var{h} has handles to the plot elements, allowing ## customization of the visualization using set/get functions. ## ## Example: ## ## @example ## title ("Grade 3 heights"); ## axis ([0,3]); ## set (gca, "xtick", 1:2, "xticklabel", @{"girls"; "boys"@}); ## h = violin (@{randn(100,1)*5+140, randn(130,1)*8+135@}, "Nbins", 10); ## set (h.violin, "linewidth", 2) ## @end example ## ## @seealso{boxplot, hist} ## @end deftypefn function h = violin (ax, varargin) old_hold = ishold (); # First argument is not an axis if (~ishandle (ax) || ~isscalar (ax)) x = ax; ax = gca (); else x = varargin{1}; varargin(1) = []; endif ###################### ## Parse parameters ## parser = inputParser (); parser.CaseSensitive = false; parser.FunctionName = 'violin'; parser.addParamValue ('Nbins', 50); parser.addParamValue ('SmoothFactor', 4); parser.addParamValue ('Bandwidth', NA); parser.addParamValue ('Width', 0.5); parser.addParamValue ('Color', "y"); parser.addSwitch ('Horizontal'); parser.parse (varargin{:}); res = parser.Results; c = res.Color; # Color of violins if (ischar (c)) c = c(:); endif nb = res.Nbins; # Number of bins in histogram sf = res.SmoothFactor; # Smoothing factor for kernel estimation r0 = res.Bandwidth; # User value for KDE bandwith to prevent optimization is_horiz = res.Horizontal; # Whether the plot must be rotated width = res.Width; # Width of the violins clear parser res ###################### ## Make everything a cell for code simplicity if (~iscell (x)) [N Nc] = size (x); x = mat2cell (x, N, ones (1, Nc)); else Nc = numel (x); endif try [nb, c, sf, r0, width] = to_cell (nb, c, sf, r0, width, Nc); catch err if strcmp (err.identifier, "to_cell:element_idx") n = str2num (err.message); txt = {"Nbins", "Color", "SmoothFactor", "Bandwidth", "Width"}; error ("Octave:invalid-input-arg", ... ["options should be scalars or call/array with as many values as" ... " numbers of variables in the data (wrong size of %s)."], txt{n}); else rethrow (lasterror()) endif end ## Build violins [px py mx] = cellfun (@(y,n,s,r)build_polygon(y, n, s, r), ... x, nb, sf, r0, "unif", 0); Nc = 1:numel (px); Ncc = mat2cell (Nc, 1, ones (1, Nc(end))); ## get hold state old_hold = ishold (); ## Draw plain violins tmp = cellfun (@(x,y,n,u, w)patch(ax, (w * x + n)(:), y(:) ,u'), ... px, py, Ncc, c, width); h.violin = tmp; hold on ## Overlay mean value tmp = cellfun (@(z,y)plot(ax, z, y,'.k', "markersize", 6), Ncc, mx); h.mean = tmp; ## Overlay median Mx = cellfun (@median, x, "unif", 0); tmp = cellfun (@(z,y)plot(ax, z, y, 'ok'), Ncc, Mx); h.median = tmp; ## Overlay 1nd and 3th quartiles LUBU = cellfun (@(x,y)abs(quantile(x,[0.25 0.75])-y), x, Mx, "unif", 0); tmp = cellfun (@(x,y,z)errorbar(ax, x, y, z(1),z(2)), Ncc, Mx, LUBU)(:); ## Flatten errorbar output handles tmp2 = allchild (tmp); if (~iscell (tmp2)) tmp2 = mat2cell (tmp2, ones(length (tmp2), 1), 1); endif tmp = mat2cell (tmp, ones (length (tmp), 1), 1); tmp = cellfun (@vertcat, tmp, tmp2, "unif", 0); h.quartile = cell2mat (tmp); hold off ## Rotate the plot if it is horizontal if (is_horiz) structfun (@swap_axes, h); set (ax, "ytick", Nc); else set (ax, "xtick", Nc); endif if (nargout < 1); clear h; endif ## restore hold state if (old_hold) hold on endif endfunction function y = stdnormal_pdf (x) y = (2 * pi)^(- 1/2) * exp (- x .^ 2 / 2); endfunction function k = kde(x,r) k = mean (stdnormal_pdf (x / r)) / r; k /= max (k); endfunction function [px py mx] = build_polygon (x, nb, sf, r) N = size (x, 1); mx = mean (x); sx = std (x); X = (x - mx ) / sx; [count bin] = hist (X, nb); count /= max (count); Y = X - bin; if isna (r) r0 = 1.06 * N^(1/5); r = sqp (r0, @(r)sumsq (kde(Y,r) - count), [], [], 1e-3, 1e2); else sf = 1; endif sig = sf * r; ## Create violin polygon ## smooth tails: extend to 1.83 sigmas, i.e. ~99% of data. xx = linspace (0, 1.83 * sig, 5); bin = [bin(1)-fliplr(xx) bin bin(end)+xx]; py = [bin; fliplr(bin)].' * sx + mx; v = kde (X-bin, sig).'; px = [v -flipud(v)]; endfunction function tf = swap_axes (h) tmp = mat2cell (h(:), ones (length (h),1), 1); tmpy = cellfun(@(x)get(x, "ydata"), tmp, "unif", 0); tmpx = cellfun(@(x)get(x, "xdata"), tmp, "unif", 0); cellfun (@(h,x,y)set (h, "xdata", y, "ydata", x), tmp, tmpx, tmpy); tf = true; endfunction function varargout = to_cell (varargin) m = varargin{end}; varargin(end) = []; for i = 1:numel(varargin) x = varargin{i}; if (isscalar (x)) x = repmat (x, m, 1); endif if (iscell (x)) if (numel(x) ~= m) # no dimension equals m error ("to_cell:element_idx", "%d\n",i); endif varargout{i} = x; continue endif sz = size (x); d = find (sz == m); if (isempty (d)) # no dimension equals m error ("to_cell:element_idx", "%d\n",i); elseif (length (d) == 2) ## both dims are m, choose 1st elseif (d == 1) # 2nd dimension is m --> transpose x = x.'; sz = fliplr (sz); endif varargout{i} = mat2cell (x, sz(1), ones (m,1)); endfor endfunction %!demo %! clf %! x = zeros (9e2, 10); %! for i=1:10 %! x(:,i) = (0.1 * randn (3e2, 3) * (randn (3,1) + 1) + 2 * randn (1,3))(:); %! endfor %! h = violin (x, "color", "c"); %! axis tight %! set (h.violin, "linewidth", 2); %! set (gca, "xgrid", "on"); %! xlabel ("Variables") %! ylabel ("Values") %!demo %! clf %! data = {randn(100,1)*5+140, randn(130,1)*8+135}; %! subplot (1,2,1) %! title ("Grade 3 heights - vertical"); %! set (gca, "xtick", 1:2, "xticklabel", {"girls"; "boys"}); %! violin (data, "Nbins", 10); %! axis tight %! %! subplot(1,2,2) %! title ("Grade 3 heights - horizontal"); %! set (gca, "ytick", 1:2, "yticklabel", {"girls"; "boys"}); %! violin (data, "horizontal", "Nbins", 10); %! axis tight %!demo %! clf %! data = exprnd (0.1, 500,4); %! violin (data, "nbins", {5,10,50,100}); %! axis ([0 5 0 max(data(:))]) %!demo %! clf %! data = exprnd (0.1, 500,4); %! violin (data, "color", jet(4)); %! axis ([0 5 0 max(data(:))]) %!demo %! clf %! data = repmat(exprnd (0.1, 500,1), 1, 4); %! violin (data, "width", linspace (0.1,0.5,4)); %! axis ([0 5 0 max(data(:))]) %!demo %! clf %! data = repmat(exprnd (0.1, 500,1), 1, 4); %! violin (data, "nbins", [5,10,50,100], "smoothfactor", [4 4 8 10]); %! axis ([0 5 0 max(data(:))]) ## Test plotting %!test %! hf = figure ("visible", "off"); %! unwind_protect %! data = exprnd (0.1, 500,4); %! violin (data, "color", jet(4)); %! axis ([0 5 0 max(data(:))]) %! unwind_protect_cleanup %! close (hf); %! end_unwind_protect %!test %! hf = figure ("visible", "off"); %! unwind_protect %! data = {randn(100,1)*5+140, randn(130,1)*8+135}; %! subplot (1,2,1) %! title ("Grade 3 heights - vertical"); %! set (gca, "xtick", 1:2, "xticklabel", {"girls"; "boys"}); %! violin (data, "Nbins", 10); %! axis tight %! unwind_protect_cleanup %! close (hf); %! end_unwind_protect %!test %! hf = figure ("visible", "off"); %! unwind_protect %! data = {randn(100,1)*5+140, randn(130,1)*8+135}; %! subplot (1,2,1) %! title ("Grade 3 heights - vertical"); %! set (gca, "xtick", 1:2, "xticklabel", {"girls"; "boys"}); %! violin (data, "Nbins", 10); %! axis tight %! subplot(1,2,2) %! title ("Grade 3 heights - horizontal"); %! set (gca, "ytick", 1:2, "yticklabel", {"girls"; "boys"}); %! violin (data, "horizontal", "Nbins", 10); %! axis tight %! unwind_protect_cleanup %! close (hf); %! end_unwind_protect %!test %! hf = figure ("visible", "off"); %! unwind_protect %! data = repmat(exprnd (0.1, 500,1), 1, 4); %! violin (data, "nbins", [5,10,50,100], "smoothfactor", [4 4 8 10]); %! axis ([0 5 0 max(data(:))]) %! unwind_protect_cleanup %! close (hf); %! end_unwind_protect %!test %! hf = figure ("visible", "off"); %! unwind_protect %! data = repmat(exprnd (0.1, 500,1), 1, 4); %! violin (data, "width", linspace (0.1,0.5,4)); %! axis ([0 5 0 max(data(:))]) %! unwind_protect_cleanup %! close (hf); %! end_unwind_protect statistics-release-1.6.3/inst/wblplot.m000066400000000000000000000271741456127120000201600ustar00rootroot00000000000000## Copyright (C) 2014 Bj{\"o}rn Vennberg ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it ## under the terms of the GNU General Public License as published by ## the Free Software Foundation; either version 3 of the License, or ## (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program. If not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {} wblplot (@var{data}, @dots{}) ## @deftypefnx {statistics} {@var{handle} =} wblplot (@var{data}, @dots{}) ## @deftypefnx {statistics} {[@var{handle}, @var{param}] =} wblplot (@var{data}) ## @deftypefnx {statistics} {[@var{handle}, @var{param}] =} wblplot (@var{data}, @var{censor}) ## @deftypefnx {statistics} {[@var{handle}, @var{param}] =} wblplot (@var{data}, @var{censor}, @var{freq}) ## @deftypefnx {statistics} {[@var{handle}, @var{param}] =} wblplot (@var{data}, @var{censor}, @var{freq}, @var{confint}) ## @deftypefnx {statistics} {[@var{handle}, @var{param}] =} wblplot (@var{data}, @var{censor}, @var{freq}, @var{confint}, @var{fancygrid}) ## @deftypefnx {statistics} {[@var{handle}, @var{param}] =} wblplot (@var{data}, @var{censor}, @var{freq}, @var{confint}, @var{fancygrid}, @var{showlegend}) ## ## Plot a column vector @var{data} on a Weibull probability plot using rank ## regression. ## ## @var{censor}: optional parameter is a column vector of same size as ## @var{data} with 1 for right censored data and 0 for exact observation. ## Pass [] when no censor data are available. ## ## @var{freq}: optional vector same size as @var{data} with the number of ## occurences for corresponding data. ## Pass [] when no frequency data are available. ## ## @var{confint}: optional confidence limits for ploting upper and lower ## confidence bands using beta binomial confidence bounds. If a single ## value is given this will be used such as LOW = a and HIGH = 1 - a. ## Pass [] if confidence bounds is not requested. ## ## @var{fancygrid}: optional parameter which if set to anything but 1 will turn ## off the fancy gridlines. ## ## @var{showlegend}: optional parameter that when set to zero(0) turns off the ## legend. ## ## If one output argument is given, a @var{handle} for the data marker and ## plotlines is returned, which can be used for further modification of line and ## marker style. ## ## If a second output argument is specified, a @var{param} vector with scale, ## shape and correlation factor is returned. ## ## @seealso{normplot, wblpdf} ## @end deftypefn function [handle, param] = wblplot (data, censor = [], freq = [], ... confint = [], fancygrid = 1, showlegend = 1) [mm, nn] = size (data); if (mm > 1 && nn > 1) error ("wblplot: can only handle a single data vector") elseif (mm == 1 && nn > 1) data = data(:); mm = nn; endif if (any (data <= 0)) error ("wblplot: data vector must be positive and non zero") endif if (isempty (freq)) freq = ones (mm, 1); N = mm; else [mmf nnf] = size (freq); if ((mmf == mm && nnf == 1) || (mmf == 1 && nnf == mm)) freq = freq(:); N = sum (freq); ## Total number of samples if (any (freq <= 0)) error ("wblplot: frequency vector must be positive non zero integers") endif else error ("wblplot: frequency must be vector of same length as data") endif endif if (isempty (censor)) censor = zeros(mm,1); else [mmc, nnc] = size(censor); if ((mmc == mm && nnc == 1) || (mmc == 1 && nnc == mm)) censor = censor(:); else error ("wblplot: censor must be a vector of same length as data") endif ## Make sure censored data is sorted corectly so that no censored samples ## are processed before failures if they have the same time. if (any (censor > 0)) ind = find (censor > 0); ind2 = find (data(1:end-1) == data(2:end)); if ((! isempty (ind)) && (! isempty (ind2))) if (any (ind == ind2)) tmp = censor(ind2); censor(ind2) = censor(ind2 + 1); censor(ind2+1) = tmp; tmp = freq(ind2); freq(ind2) = freq(ind2 + 1); freq(ind2 + 1) = tmp; endif endif endif endif ## Determine the order number wbdat = zeros (length (find (censor == 0)), 3); Op = 0; Oi = 0; c = N; nf = 0; for k = 1 : mm if (censor(k, 1) == 0) nf = nf + 1; wbdat(nf, 1) = data(k, 1); for s = 1 : freq(k, 1); Oi = Op + ((N + 1) - Op) / (1 + c); Op = Oi; c = c - 1; endfor wbdat(nf, 3) = Oi; else c = c - freq(k, 1); endif endfor ## Compute median rank a = wbdat(:, 3) ./ (N - wbdat(:, 3) + 1); f = finv(0.5, 2 * (N - wbdat(:, 3) + 1), 2 * wbdat(:, 3)); wbdat(:, 2) = a ./ (f+a); datx = log (wbdat(:,1)); daty = log (log (1 ./ (1 - wbdat(:,2)))); ## Rank regression poly = polyfit (datx, daty, 1); ## Shape factor beta_rry = poly(1); ## Scale factor eta_rry = exp (-(poly(2) / beta_rry)); ## Determine min-max values of view port aa = ceil (log10 (max (wbdat(:,1)))); bb = log10 (max (wbdat(:,1))); if ((aa - bb) < 0.2) aa = ceil (log10 (max (wbdat(:,1)))) + 1; endif xmax = 10 ^ aa; if ((log10 (min (wbdat(:,1))) - floor (log10 (min (wbdat(:,1))))) < 0.2) xmin = 10 ^ (floor (log10 (min (wbdat(:,1)))) - 1); else xmin = 10 ^ floor (log10 (min (wbdat(:,1)))); endif if (min (wbdat(:,2)) > 0.20) ymin = log (log (1 / (1 - 0.1))); elseif (min (wbdat(:,2)) > 0.02) ymin = log (log (1 / (1 - 0.01))); elseif (min (wbdat(:,2)) > 0.002) ymin = log (log (1 / (1 - 0.001))); else ymin = log (log (1 / (1 - 0.0001))); endif ymax= log (log (1 / (1 - 0.999))); x = [0;0]; y = [0;0]; label = char('0.10', '1.00', '10.00', '99.00'); prob = [0.001 0.01 0.1 0.99]; tick = log (log (1 ./ (1 - prob))); xbf = [xmin; xmax]; ybf = polyval (poly, log (xbf)); newplot(); x(1, 1) = xmin; x(2, 1) = xmax; if (fancygrid == 1) for k = 1 : 4 ## Y major grids x(1, 1) = xmin; x(2, 1) = xmax*10; y(1, 1) = log (log (1 / (1 - 10 ^ (-k)))); y(2, 1) = y(1, 1); ymajorgrid(k) = line (x, y, 'LineStyle', '-', 'Marker', 'none', ... 'Color', [1 0.75 0.75], 'LineWidth', 0.1); endfor ## Y Minor grids 2 - 9 x(1, 1) = xmin; x(2, 1) = xmax * 10; for m = 1 : 4 for k = 1 : 8 y(1, 1) = log (log (1 / (1 - ((k + 1) / (10 ^ m))))); y(2, 1) = y(1, 1); yminorgrid(k) = line (x, y, 'LineStyle', '-', 'Marker', 'none', ... 'Color', [0.75 1 0.75], 'LineWidth', 0.1); endfor endfor ## X-axis grid y(1, 1) = ymin; y(2, 1) = ymax; for m = log10 (xmin) : log10 (xmax) x(1, 1) = 10 ^ m; x(2, 1) = x(1, 1); y(1, 1) = ymin; y(2, 1) = ymax; xmajorgrid(k) = line (x, y, 'LineStyle', '-', 'Marker', 'none', ... 'Color', [1 0.75 0.75]); for k = 1 : 8 ## X Minor grids - 2 - 9 x(1, 1) = (k + 1) * (10 ^ m); x(2, 1) = (k + 1) * (10 ^ m); xminorgrid(k) = line (x, y, 'LineStyle', '-', 'Marker', 'none', ... 'Color', [0.75 1 0.75], 'LineWidth', 0.1); endfor endfor endif set (gca, 'XScale', 'log'); set (gca, 'YTick', tick, 'YTickLabel', label); xlabel ('Data', 'FontSize', 12); ylabel ('Unreliability, F(t)=1-R(t)', 'FontSize', 12); title ('Weibull Probability Plot', 'FontSize', 12); set (gcf, 'Color', [0.9, 0.9, 0.9]); set (gcf, 'name', 'WblPlot'); hold on h = plot (wbdat(:,1), daty, 'o'); set (h, 'markerfacecolor', [0, 0, 1]); set (h, 'markersize', 8); h2 = line (xbf, ybf, 'LineStyle', '-', 'Marker', 'none', ... 'Color', [0.25 0.25 1], 'LineWidth', 1); ## If requested plot beta binomial confidens bounds if (! isempty (confint)) cb_high = []; cb_low = []; if (length (confint) == 1) if (confint > 0.5) cb_high = confint; cb_low = 1 - confint; else cb_high = 1 - confint; cb_low = confint; endif else cb_high = confint(2); cb_low = confint(1); endif conf = zeros (N + 4, 3); betainv = 1 / beta_rry; N2 = [1:N]'; N2 = [0.3; 0.7; N2; N2(end) + 0.5; N2(end) + 0.8]; ## Extend the ends a bit ypos = medianranks (0.5, N, N2); conf(:, 1) = eta_rry * log (1 ./ (1 - ypos)) .^ betainv; conf(:, 2) = medianranks (cb_low, N, N2); conf(:, 3) = medianranks (cb_high, N, N2); confy = log (log (1 ./ (1 - conf(:,2:3)))); confu = [conf(:,1) confy]; if (conf(1,1) > xmin) ## It looks better to extend the lines. p1 = polyfit (log (conf(1:2,1)), confy(1:2,1), 1); y1 = polyval (p1, log (xmin)); p2 = polyfit (log (conf(1:2,1)), confy(1:2,2), 1); y2 = polyval (p2, log (xmin)); confu = [xmin y1 y2; confu]; endif if (conf(end,1) < xmax) p3 = polyfit (log (conf(end-1:end,1)), confy(end-1:end,1), 1); y3 = polyval (p3, log (xmax)); p4 = polyfit (log (conf(end-1:end,1)), confy(end-1:end,2), 1); y4 = polyval (p4, log (xmax)); confu = [confu; xmax, y3, y4]; endif h3 = plot (confu(:,1), confu(:,2:3), 'LineStyle', '-' ,'Marker', 'none', ... 'Color', [1 0.25 0.25], 'LineWidth', 1); endif ## Correlation coefficient rsq = corr (datx, daty); if (showlegend == 1) s1 = sprintf (' RRY\n \\beta=%.3f \n \\eta=%.2f \n \\rho=%.4f', ... beta_rry, eta_rry, rsq); if (! isempty (confint)) s2 = sprintf ('CB_H=%.2f', cb_high); s3 = sprintf ('CB_L=%.2f', cb_low); legend ([h; h2; h3], "Data", s1, s2, s3, "location", "northeastoutside"); else legend ([h; h2], "Data", s1, "location", "northeastoutside"); endif legend ("boxoff"); endif axis ([xmin, xmax, ymin, (log (log (1 / (1 - 0.99))))]); hold off if (nargout >= 2) param = [eta_rry, beta_rry, rsq]; if (! isempty (confint)) handle = [h; h2; h3]; else handle = [h; h2]; endif endif if (nargout == 1) if (! isempty (confint)) handle = [h; h2; h3]; else handle = [h; h2]; endif endif endfunction function ret = medianranks (alpha, n, ii) a = ii ./ (n - ii + 1); f = finv (alpha, 2 * (n - ii + 1), 2 * ii); ret = a ./ (f + a); endfunction %!demo %! x = [16 34 53 75 93 120]; %! wblplot (x); %!demo %! x = [2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67]'; %! c = [0 1 0 1 0 1 1 1 0 0 1 0 1 0 1 1 0 1 1]'; %! [h, p] = wblplot (x, c); %! p %!demo %! x = [16, 34, 53, 75, 93, 120, 150, 191, 240 ,339]; %! [h, p] = wblplot (x, [], [], 0.05); %! p %! ## Benchmark Reliasoft eta = 146.2545 beta 1.1973 rho = 0.9999 %!demo %! x = [46 64 83 105 123 150 150]; %! c = [0 0 0 0 0 0 1]; %! f = [1 1 1 1 1 1 4]; %! wblplot (x, c, f, 0.05); %!demo %! x = [46 64 83 105 123 150 150]; %! c = [0 0 0 0 0 0 1]; %! f = [1 1 1 1 1 1 4]; %! ## Subtract 30.92 from x to simulate a 3 parameter wbl with gamma = 30.92 %! wblplot (x - 30.92, c, f, 0.05); ## Test plotting %!test %! hf = figure ("visible", "off"); %! unwind_protect %! x = [16, 34, 53, 75, 93, 120, 150, 191, 240 ,339]; %! [h, p] = wblplot (x, [], [], 0.05); %! assert (numel (h), 4) %! assert (p(1), 146.2545, 1E-4) %! assert (p(2), 1.1973, 1E-4) %! assert (p(3), 0.9999, 5E-5) %! unwind_protect_cleanup %! close (hf); %! end_unwind_protect statistics-release-1.6.3/inst/x2fx.m000066400000000000000000000207171456127120000173600ustar00rootroot00000000000000## Copyright (C) 2022 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . ## -*- texinfo -*- ## @deftypefn {statistics} {[@var{d}, @var{model}, @var{termstart}, @var{termend}] =} x2fx (@var{x}) ## @deftypefnx {statistics} {[@var{d}, @var{model}, @var{termstart}, @var{termend}] =} x2fx (@var{x}, @var{model}) ## @deftypefnx {statistics} {[@var{d}, @var{model}, @var{termstart}, @var{termend}] =} x2fx (@var{x}, @var{model}, @var{categ}) ## @deftypefnx {statistics} {[@var{d}, @var{model}, @var{termstart}, @var{termend}] =} x2fx (@var{x}, @var{model}, @var{categ}, @var{catlevels}) ## ## Convert predictors to design matrix. ## ## @code{@var{d} = x2fx (@var{x}, @var{model})} converts a matrix of predictors ## @var{x} to a design matrix @var{d} for regression analysis. Distinct ## predictor variables should appear in different columns of @var{x}. ## ## The optional input @var{model} controls the regression model. By default, ## @code{x2fx} returns the design matrix for a linear additive model with a ## constant term. @var{model} can be any one of the following strings: ## ## @multitable @columnfractions 0.05 0.2 0.75 ## @item @tab "linear" @tab Constant and linear terms (the default) ## @item @tab "interaction" @tab Constant, linear, and interaction terms ## @item @tab "quadratic" @tab Constant, linear, interaction, and squared terms ## @item @tab "purequadratic" @tab Constant, linear, and squared terms ## @end multitable ## ## If @var{x} has n columns, the order of the columns of @var{d} for a full ## quadratic model is: ## ## @itemize ## @item ## The constant term. ## @item ## The linear terms (the columns of X, in order 1,2,...,n). ## @item ## The interaction terms (pairwise products of columns of @var{x}, in order ## (1,2), (1,3), ..., (1,n), (2,3), ..., (n-1,n). ## @item ## The squared terms (in the order 1,2,...,n). ## @end itemize ## ## Other models use a subset of these terms, in the same order. ## ## Alternatively, MODEL can be a matrix specifying polynomial terms of arbitrary ## order. In this case, MODEL should have one column for each column in X and ## one r for each term in the model. The entries in any r of MODEL are powers ## for the corresponding columns of @var{x}. For example, if @var{x} has ## columns X1, X2, and X3, then a row [0 1 2] in @var{model} would specify the ## term (X1.^0).*(X2.^1).*(X3.^2). A row of all zeros in @var{model} specifies ## a constant term, which you can omit. ## ## @code{@var{d} = x2fx (@var{x}, @var{model}, @var{categ})} treats columns with ## numbers listed in the vector @var{categ} as categorical variables. Terms ## involving categorical variables produce dummy variable columns in @var{d}. ## Dummy variables are computed under the assumption that possible categorical ## levels are completely enumerated by the unique values that appear in the ## corresponding column of @var{x}. ## ## @code{@var{d} = x2fx (@var{x}, @var{model}, @var{categ}, @var{catlevels})} ## accepts a vector @var{catlevels} the same length as @var{categ}, specifying ## the number of levels in each categorical variable. In this case, values in ## the corresponding column of @var{x} must be integers in the range from 1 to ## the specified number of levels. Not all of the levels need to appear in ## @var{x}. ## ## @end deftypefn function [D, model, termstart, termend] = x2fx (x, model, categ, catlevels) ## Get matrix size [m,n] = size(x); ## Get data class if (isa (x, "single")) data_class = 'single'; else data_class = 'double'; endif ## Check for input arguments if (nargin < 2 || isempty (model)) model = 'linear'; endif if (nargin < 3) categ = []; endif if (nargin < 4) catlevels = []; endif ## Convert models parsed as strings to numerical matrix if (ischar (model)) if (strcmpi (model, "linear") || strcmpi (model, "additive")) interactions = false; quadratic = false; elseif (strcmpi (model, "interactions")) interactions = true; quadratic = false; elseif (strcmpi (model, "quadratic")) interactions = true; quadratic = true; elseif (strcmpi (model, "purequadratic")) interactions = false; quadratic = true; else D = feval (model, x); termstart = []; termend = []; return endif I = eye(n); ## Construct interactions part if (interactions && n > 1) [r, c] = find (tril (ones (n) ,-1)); nt = length(r); intpart = zeros(nt,n); intpart(sub2ind (size (intpart),(1:nt)', r)) = 1; intpart(sub2ind (size (intpart),(1:nt)', c)) = 1; else intpart = zeros(0,n); endif ## Construct quadratic part if (quadratic) quadpart = 2 * I; quadpart(categ,:) = []; else quadpart = zeros(0,n); endif model = [zeros(1,n); I]; model = [model; intpart; quadpart]; endif ## Process each categorical variable catmember = ismember (1:n, categ); var_DF = ones(1,n); if (isempty (catlevels)) ## Get values of each categorical variable and replace them with integers for idx=1:length(categ) categ_idx = categ(idx); [Y, I, J] = unique (x(:,categ_idx)); var_DF(categ_idx) = length (Y) - 1; x(:,categ_idx) = J; endfor else ## Ensure all categorical variables take valid values var_DF(categ) = catlevels - 1; for idx = 1:length (categ) categ_idx = categ(idx); if (any (! ismember (x(:,categ_idx), 1:catlevels(idx)))) error("x2fx: wrong value %f in category %d.", ... catlevels(idx), categ_idx); endif endfor endif ## Get size of model martix [r, c] = size (model); ## Check for equal number of columns between x and model if (c != n) error("x2fx: wrong number of columns between x and model"); endif ## Check model category column for values greater than 1 if (any (model(:,categ) > 1, 1)) error("x2fx: wrong values in model's category column"); endif ## Allocate space for the dummy variables for all terms termdf = prod (max (1, (model > 0) .* repmat (var_DF, r, 1)), 2); termend = cumsum (termdf); termstart = termend - termdf + 1; D = zeros (m, termend(end), data_class); allrows = (1:m)'; for idx = 1:r cols = termstart(idx):termend(idx); pwrs = model(idx,:); t = pwrs > 0; C = 1; if (any (t)) if (any (pwrs(! catmember))) pwrs_cat = pwrs .* !catmember; C = ones (size (x, 1), 1); collist = find (pwrs_cat > 0); for idx = 1:length (collist) categ_idx = collist(idx); C = C .* x(:,categ_idx) .^ pwrs_cat(categ_idx); endfor endif if (any (pwrs(catmember) > 0)) Z = zeros (m, termdf(idx)); collist = find (pwrs > 0 & catmember); xcol = x(:,collist(1)); keep = (xcol <= var_DF(collist(1))); colnum = xcol; cumdf = 1; for idx=2:length(collist) cumdf = cumdf * var_DF(collist(idx-1)); xcol = x(:,collist(idx)); keep = keep & (xcol <= var_DF(collist(idx))); colnum = colnum + cumdf * (xcol - 1); endfor if (length (C) > 1) C = C(keep); endif Z(sub2ind(size(Z),allrows(keep),colnum(keep))) = C; C = Z; endif endif D(:,cols) = C; endfor endfunction %!test %! X = [1, 10; 2, 20; 3, 10; 4, 20; 5, 15; 6, 15]; %! D = x2fx(X,'quadratic'); %! assert (D(1,:) , [1, 1, 10, 10, 1, 100]); %! assert (D(2,:) , [1, 2, 20, 40, 4, 400]); %!test %! X = [1, 10; 2, 20; 3, 10; 4, 20; 5, 15; 6, 15]; %! model = [0, 0; 1, 0; 0, 1; 1, 1; 2, 0]; %! D = x2fx(X,model); %! assert (D(1,:) , [1, 1, 10, 10, 1]); %! assert (D(2,:) , [1, 2, 20, 40, 4]); %! assert (D(4,:) , [1, 4, 20, 80, 16]); %!error x2fx ([1, 10; 2, 20; 3, 10], [0; 1]); %!error x2fx ([1, 10, 15; 2, 20, 40; 3, 10, 25], [0, 0; 1, 0; 0, 1; 1, 1; 2, 0]); %!error x2fx ([1, 10; 2, 20; 3, 10], "whatever"); statistics-release-1.6.3/inst/ztest.m000066400000000000000000000167021456127120000176410ustar00rootroot00000000000000## Copyright (C) 2014 Tony Richardson ## Copyright (C) 2022 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{h} =} ztest (@var{x}, @var{m}, @var{sigma}) ## @deftypefnx {statistics} {@var{h} =} ztest (@var{x}, @var{m}, @var{sigma}, @var{Name}, @var{Value}) ## @deftypefnx {statistics} {[@var{h}, @var{pval}] =} ztest (@dots{}) ## @deftypefnx {statistics} {[@var{h}, @var{pval}, @var{ci}] =} ztest (@dots{}) ## @deftypefnx {statistics} {[@var{h}, @var{pval}, @var{ci}, @var{zvalue}] =} ztest (@dots{}) ## ## One-sample Z-test. ## ## @code{@var{h} = ztest (@var{x}, @var{v})} performs a Z-test of the hypothesis ## that the data in the vector @var{x} come from a normal distribution with mean ## @var{m}, against the alternative that @var{x} comes from a normal ## distribution with a different mean @var{m}. The result is @var{h} = 0 if the ## null hypothesis ("mean is M") cannot be rejected at the 5% significance ## level, or @var{h} = 1 if the null hypothesis can be rejected at the 5% level. ## ## @var{x} may also be a matrix or an N-D array. For matrices, @code{ztest} ## performs separate tests along each column of @var{x}, and returns a vector of ## results. For N-D arrays, @code{ztest} works along the first non-singleton ## dimension of @var{x}. @var{m} and @var{sigma} must be a scalars. ## ## @code{ztest} treats NaNs as missing values, and ignores them. ## ## @code{[@var{h}, @var{pval}] = ztest (@dots{})} returns the p-value. That ## is the probability of observing the given result, or one more extreme, by ## chance if the null hypothesisis true. ## ## @code{[@var{h}, @var{pval}, @var{ci}] = ztest (@dots{})} returns a ## 100 * (1 - @var{alpha})% confidence interval for the true mean. ## ## @code{[@var{h}, @var{pval}, @var{ci}, @var{zvalue}] = ztest (@dots{})} ## returns the value of the test statistic. ## ## @code{[@dots{}] = ztest (@dots{}, @var{Name}, @var{Value}, @dots{})} ## specifies one or more of the following @var{Name}/@var{Value} pairs: ## ## @multitable @columnfractions 0.05 0.2 0.75 ## @headitem @tab @var{Name} @tab @var{Value} ## @item @tab "alpha" @tab the significance level. Default is 0.05. ## ## @item @tab "dim" @tab dimension to work along a matrix or an N-D array. ## ## @item @tab "tail" @tab a string specifying the alternative hypothesis: ## @end multitable ## @multitable @columnfractions 0.1 0.15 0.75 ## @item @tab "both" @tab "mean is not @var{m}" (two-tailed, default) ## @item @tab "left" @tab "mean is less than @var{m}" (left-tailed) ## @item @tab "right" @tab "mean is greater than @var{m}" (right-tailed) ## @end multitable ## ## @seealso{ttest, vartest, signtest, kstest} ## @end deftypefn function [h, pval, ci, zvalue] = ztest (x, m, sigma, varargin) ## Validate input arguments if (nargin < 3) error ("ztest: too few input arguments."); endif if (! isscalar (m) || ! isnumeric(m) || ! isreal(m)) error ("ztest: invalid value for mean."); endif if (! isscalar (sigma) || ! isnumeric(sigma) || ! isreal(sigma) || sigma < 0) error ("ztest: invalid value for standard deviation."); endif ## Add defaults alpha = 0.05; tail = "both"; dim = []; if (nargin > 3) for idx = 4:2:nargin name = varargin{idx-3}; value = varargin{idx-2}; switch (lower (name)) case "alpha" alpha = value; if (! isscalar (alpha) || ! isnumeric (alpha) || ... alpha <= 0 || alpha >= 1) error ("ztest: invalid VALUE for alpha."); endif case "tail" tail = value; if (! any (strcmpi (tail, {"both", "left", "right"}))) error ("ztest: invalid VALUE for tail."); endif case "dim" dim = value; if (! isscalar (dim) || ! ismember (dim, 1:ndims (x))) error ("ztest: invalid VALUE for operating dimension."); endif otherwise error ("ztest: invalid NAME for optional arguments."); endswitch endfor endif ## Figure out which dimension mean will work along if (isempty (dim)) dim = find (size (x) != 1, 1); endif ## Replace all NaNs with zeros is_nan = isnan (x); ## Find sample size for each group (if more than one) if (any (is_nan(:))) sz = sum (! is_nan, dim); else sz = size (x, dim); endif ## Calculate mean, strandard error and z-value for each group x_mean = sum (x(! is_nan), dim) ./ max (1, sz); stderr = sigma ./ sqrt (sz); zvalue = (x_mean - m) ./ stderr; ## Calculate p-value for the test and confidence intervals (if requested) if (strcmpi (tail, "both")) pval = 2 * normcdf (- abs (zvalue), 0, 1); if (nargout > 2) crit = norminv (1 - alpha / 2, 0, 1) .* stderr; ci = cat (dim, x_mean - crit, x_mean + crit); endif elseif (strcmpi (tail, "right")) p = normcdf (- zvalue,0,1); if (nargout > 2) crit = norminv (1 - alpha, 0, 1) .* stderr; ci = cat (dim, x_mean - crit, Inf (size (p))); endif elseif (strcmpi (tail, "left")) p = normcdf (zvalue, 0, 1); if (nargout > 2) crit = norminv (1 - alpha, 0, 1) .* stderr; ci = cat (dim, - Inf (size (p)), x_mean + crit); endif endif ## Determine the test outcome h = double (pval < alpha); h(isnan (pval)) = NaN; endfunction ## Test input validation %!error ztest (); %!error ... %! ztest ([1, 2, 3, 4], 2, -0.5); %!error ... %! ztest ([1, 2, 3, 4], 1, 2, "alpha", 0); %!error ... %! ztest ([1, 2, 3, 4], 1, 2, "alpha", 1.2); %!error ... %! ztest ([1, 2, 3, 4], 1, 2, "alpha", "val"); %!error ... %! ztest ([1, 2, 3, 4], 1, 2, "tail", "val"); %!error ... %! ztest ([1, 2, 3, 4], 1, 2, "alpha", 0.01, "tail", "val"); %!error ... %! ztest ([1, 2, 3, 4], 1, 2, "dim", 3); %!error ... %! ztest ([1, 2, 3, 4], 1, 2, "alpha", 0.01, "tail", "both", "dim", 3); %!error ... %! ztest ([1, 2, 3, 4], 1, 2, "alpha", 0.01, "tail", "both", "badoption", 3); ## Test results %!test %! load carsmall %! [h, pval, ci] = ztest (MPG, mean (MPG, "omitnan"), std (MPG, "omitnan")); %! assert (h, 0); %! assert (pval, 1, 1e-14); %! assert (ci, [22.094; 25.343], 1e-3); %!test %! load carsmall %! [h, pval, ci] = ztest (MPG, 26, 8); %! assert (h, 1); %! assert (pval, 0.00568359158544743, 1e-14); %! assert (ci, [22.101; 25.335], 1e-3); %!test %! load carsmall %! [h, pval, ci] = ztest (MPG, 26, 4); %! assert (h, 1); %! assert (pval, 3.184168011941316e-08, 1e-14); %! assert (ci, [22.909; 24.527], 1e-3); statistics-release-1.6.3/inst/ztest2.m000066400000000000000000000132151456127120000177170ustar00rootroot00000000000000## Copyright (C) 1996-2017 Kurt Hornik ## Copyright (C) 2023 Andreas Bertsatos ## ## This file is part of the statistics package for GNU Octave. ## ## This program is free software: you can redistribute it and/or ## modify it under the terms of the GNU General Public License as ## published by the Free Software Foundation, either version 3 of the ## License, or (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {statistics} {@var{h} =} ztest2 (@var{x1}, @var{n1}, @var{x2}, @var{n2}) ## @deftypefnx {statistics} {@var{h} =} ztest2 (@var{x1}, @var{n1}, @var{x2}, @var{n2}, @var{Name}, @var{Value}) ## @deftypefnx {statistics} {[@var{h}, @var{pval}] =} ztest2 (@dots{}) ## @deftypefnx {statistics} {[@var{h}, @var{pval}, @var{zvalue}] =} ztest2 (@dots{}) ## ## Two proportions Z-test. ## ## If @var{x1} and @var{n1} are the counts of successes and trials in one ## sample, and @var{x2} and @var{n2} those in a second one, test the null ## hypothesis that the success probabilities @math{p1} and @math{p2} are the ## same. The result is @var{h} = 0 if the null hypothesis cannot be rejected at ## the 5% significance level, or @var{h} = 1 if the null hypothesis can be ## rejected at the 5% level. ## ## Under the null, the test statistic @var{zvalue} approximately follows a ## standard normal distribution. ## ## The size of @var{h}, @var{pval}, and @var{zvalue} is the common size of ## @var{x}, @var{n1}, @var{x2}, and @var{n2}, which must be scalars or of common ## size. A scalar input functions as a constant matrix of the same size as the ## other inputs. ## ## @code{[@var{h}, @var{pval}] = ztest2 (@dots{})} returns the p-value. That ## is the probability of observing the given result, or one more extreme, by ## chance if the null hypothesisis true. ## ## @code{[@var{h}, @var{pval}, @var{zvalue}] = ztest2 (@dots{})} returns the ## value of the test statistic. ## ## @code{[@dots{}] = ztest2 (@dots{}, @var{Name}, @var{Value}, @dots{})} ## specifies one or more of the following @var{Name}/@var{Value} pairs: ## ## @multitable @columnfractions 0.05 0.2 0.75 ## @headitem @tab @var{Name} @tab @var{Value} ## @item @tab @qcode{"alpha"} @tab the significance level. Default is 0.05. ## ## @item @tab @qcode{"tail"} @tab a string specifying the alternative hypothesis ## @end multitable ## @multitable @columnfractions 0.1 0.25 0.65 ## @item @tab @qcode{"both"} @tab @math{p1} is not @math{p2} ## (two-tailed, default) ## @item @tab @qcode{"left"} @tab @math{p1} is less than @math{p2} ## (left-tailed) ## @item @tab @qcode{"right"} @tab @math{p1} is greater than @math{p2} ## (right-tailed) ## @end multitable ## ## @seealso{chi2test, fishertest} ## @end deftypefn function [h, pval, zvalue] = ztest2 (x1, n1, x2, n2, varargin) if (nargin < 4) print_usage (); endif if (! isscalar (x1) || ! isscalar (n1) || ! isscalar (x2) || ! isscalar (n2)) [retval, x1, n1, x2, n2] = common_size (x1, n1, x2, n2); if (retval > 0) error ("ztest2: X1, N1, X2, and N2 must be of common size or scalars."); endif endif if (iscomplex (x1) || iscomplex (n1) || iscomplex (x2) || iscomplex(n2)) error ("ztest2: X1, N1, X2, and N2 must not be complex."); endif ## Add defaults and parse optional arguments alpha = 0.05; tail = "both"; if (nargin > 4) params = numel (varargin); if ((params / 2) != fix (params / 2)) error ("ztest2: optional arguments must be in NAME-VALUE pairs.") endif for idx = 1:2:params name = varargin{idx}; value = varargin{idx+1}; switch (lower (name)) case "alpha" alpha = value; if (! isscalar (alpha) || ! isnumeric (alpha) || ... alpha <= 0 || alpha >= 1) error ("ztest2: invalid VALUE for alpha."); endif case "tail" tail = value; if (! any (strcmpi (tail, {"both", "left", "right"}))) error ("ztest2: invalid VALUE for tail."); endif otherwise error ("ztest2: invalid NAME for optional arguments."); endswitch endfor endif p1 = x1 ./ n1; p2 = x2 ./ n2; pc = (x1 + x2) ./ (n1 + n2); zvalue = (p1 - p2) ./ sqrt (pc .* (1 - pc) .* (1 ./ n1 + 1 ./ n2)); cdf = normcdf (zvalue); if (strcmpi (tail, "both")) pval = 2 * min (cdf, 1 - cdf); elseif (strcmpi (tail, "right")) pval = 1 - cdf; elseif (strcmpi (tail, "left")) pval = cdf; endif ## Determine the test outcome h = double (pval < alpha); h(isnan (pval)) = NaN; endfunction ## Test input validation %!error ztest2 (); %!error ztest2 (1); %!error ztest2 (1, 2); %!error ztest2 (1, 2, 3); %!error ... %! ztest2 (1, 2, 3, 4, "alpha") %!error ... %! ztest2 (1, 2, 3, 4, "alpha", 0); %!error ... %! ztest2 (1, 2, 3, 4, "alpha", 1.2); %!error ... %! ztest2 (1, 2, 3, 4, "alpha", "val"); %!error ... %! ztest2 (1, 2, 3, 4, "tail", "val"); %!error ... %! ztest2 (1, 2, 3, 4, "alpha", 0.01, "tail", "val"); %!error ... %! ztest2 (1, 2, 3, 4, "alpha", 0.01, "tail", "both", "badoption", 3); statistics-release-1.6.3/io.github.gnu_octave.statistics.metainfo.xml000066400000000000000000000022271456127120000260670ustar00rootroot00000000000000 io.github.gnu_octave.statistics org.octave.Octave FSFAP GPL-3.0-or-later Statistics The Statistics package for GNU Octave Octave Statistics

The Statistics package for GNU Octave is a collection of functions for statistical analysis.

https://github.com/gnu-octave/statistics/issues https://gnu-octave.github.io/statistics Octave Community octave-maintainers@gnu.org statistics-release-1.6.3/src/000077500000000000000000000000001456127120000161165ustar00rootroot00000000000000statistics-release-1.6.3/src/Makefile000066400000000000000000000003611456127120000175560ustar00rootroot00000000000000# Makefile for compiling required oct files all: $(MKOCTFILE) libsvmread.cc $(MKOCTFILE) libsvmwrite.cc $(MKOCTFILE) svmpredict.cc svm.cpp svm_model_octave.cc $(MKOCTFILE) svmtrain.cc svm.cpp svm_model_octave.cc statistics-release-1.6.3/src/libsvmread.cc000066400000000000000000000131631456127120000205610ustar00rootroot00000000000000/* Copyright (C) 2022 Andreas Bertsatos Adapted from MATLAB libsvmwrite.c file from the LIBSVM 3.25 (2021) library by Chih-Chung Chang and Chih-Jen Lin. This file is part of the statistics package for GNU Octave. This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 3 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with this program; if not, see . */ #include #include #include #include #include #include #include #include #include #include #include #include #include #ifndef max #define max(x,y) (((x)>(y))?(x):(y)) #endif #ifndef min #define min(x,y) (((x)<(y))?(x):(y)) #endif using namespace std; static char *line; static int max_line_len; static char* readline(FILE *input) { int len; if(fgets(line,max_line_len,input) == NULL) { return NULL; } while(strrchr(line,'\n') == NULL) { max_line_len *= 2; line = (char *) realloc(line, max_line_len); len = (int) strlen(line); if(fgets(line+len,max_line_len-len,input) == NULL) { break; } } return line; } // read the file in libsvm format void read(string filename, ColumnVector &label_vec, SparseMatrix &instance_mat) { int max_index, min_index, inst_max_index; size_t elements, k, i, l=0; FILE *fp = fopen(filename.c_str(),"r"); char *endptr; octave_idx_type *ir, *jc; double *labels, *samples; if(fp == NULL) { printf("can't open input file %s\n",filename.c_str()); return; } max_line_len = 1024; line = (char *) malloc(max_line_len*sizeof(char)); max_index = 0; min_index = 1; // our index starts from 1 elements = 0; while(readline(fp) != NULL) { char *idx, *val; // features int index = 0; inst_max_index = -1; strtok(line," \t"); while (1) { idx = strtok(NULL,":"); val = strtok(NULL," \t"); if(val == NULL) break; errno = 0; index = (int) strtol(idx,&endptr,10); if(endptr == idx || errno != 0 || *endptr != '\0' || index <= inst_max_index) { printf("libsvmread: wrong input format at line %d.\n", (int)l+1); return; } else inst_max_index = index; min_index = min(min_index, index); elements++; } max_index = max(max_index, inst_max_index); l++; } rewind(fp); // y label_vec = ColumnVector(l, 1); // x^T if (min_index <= 0) { octave_idx_type r = max_index-min_index+1; octave_idx_type c = l; octave_idx_type val = elements; instance_mat = SparseMatrix(r, c, val); } else { octave_idx_type r = max_index-min_index+1; octave_idx_type c = l; octave_idx_type val = elements; instance_mat = SparseMatrix(r, c, val); } labels = (double*)label_vec.data(); samples = (double*)instance_mat.data(); ir = (octave_idx_type*)instance_mat.ridx(); jc = (octave_idx_type*)instance_mat.cidx(); k=0; for(i=0;i start from 0 ir[k] = strtol(idx,&endptr,10) - min_index; errno = 0; samples[k] = strtod(val,&endptr); if (endptr == val || errno != 0 || (*endptr != '\0' && !isspace(*endptr))) { printf("libsvmread: wrong input format at line %d.\n", (int)i+1); return; } ++k; } } jc[l] = k; fclose(fp); free(line); // transpose instance sparse matrix in row format instance_mat.transpose(); } DEFUN_DLD (libsvmread, args, nargout, "-*- texinfo -*- \n\n\ @deftypefn {statistics} {[@var{labels}, @var{data}] =} libsvmread (@var{filename})\n\ \n\ \n\ This function reads the labels and the corresponding instance_matrix from a \ LIBSVM data file and stores them in @var{labels} and @var{data} respectively. \ These can then be used as inputs to @code{svmtrain} or @code{svmpredict} \ function. \ \n\ \n\ @end deftypefn") { if(args.length() != 1 || nargout != 2) { error ("libsvmread: wrong number of input or output arguments."); } if(!args(0).is_string()) { error ("libsvmread: filename must be a string."); } string filename = args(0).string_value(); octave_value_list retval(nargout); ColumnVector label_vec; SparseMatrix instance_mat; read(filename, label_vec, instance_mat); retval(0) = label_vec; retval(1) = instance_mat.transpose(); return retval; } /* %!error [L, D] = libsvmread (24); %!error ... %! D = libsvmread ("filename"); %!test %! [L, D] = libsvmread (file_in_loadpath ("heart_scale.dat")); %! assert (size (L), [270, 1]); %! assert (size (D), [270, 13]); %!test %! [L, D] = libsvmread (file_in_loadpath ("heart_scale.dat")); %! assert (issparse (L), false); %! assert (issparse (D), true); */ statistics-release-1.6.3/src/libsvmwrite.cc000066400000000000000000000111111456127120000207670ustar00rootroot00000000000000/* Copyright (C) 2022 Andreas Bertsatos Adapted from MATLAB libsvmwrite.c file from the LIBSVM 3.25 (2021) library by Chih-Chung Chang and Chih-Jen Lin. This file is part of the statistics package for GNU Octave. This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 3 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with this program; if not, see . */ #include #include #include #include #include #include #include #include #include #include #include #include using namespace std; void write(string filename, ColumnVector label_vec, SparseMatrix instance_mat) { // open file FILE *fp = fopen(filename.c_str(),"w"); if (fp == NULL) { error ("libsvmwrite: error opening file for write."); } else { // check for equal number of instances and labels int im_rows = (int)instance_mat.rows(); int lv_rows = (int)label_vec.rows(); if(im_rows != lv_rows) { // close file fclose (fp); remove (filename.c_str()); error ("libsvmwrite: length of label vector does not match instances."); } // transpose instance sparse matrix in column format SparseMatrix instance_mat_col = instance_mat.transpose(); octave_idx_type *ir, *jc, k, low, high; size_t i, l, label_vector_row_num; double *samples, *labels; // each column is one instance labels = (double*)label_vec.data(); samples = (double*)instance_mat_col.data(); ir = (octave_idx_type*)instance_mat_col.ridx(); jc = (octave_idx_type*)instance_mat_col.cidx(); for(int i = 0; i < lv_rows; i++) { fprintf(fp, "%.17g", labels[i]); low = jc[i], high = jc[i+1]; for(k=low;k 0) { error ("libsvmwrite: wrong number of output arguments."); return octave_value_list(); } // Transform the input Matrix to libsvm format if(args.length() == 3) { if(!args(1).is_double_type() || !args(2).is_double_type()) { error ("libsvmwrite: label vector and instance matrix must be double."); } if(!args(0).is_string()) { error ("libsvmwrite: filename must be a string."); } string filename = args(0).string_value(); if(args(2).issparse()) { ColumnVector label_vec = args(1).column_vector_value(); SparseMatrix instance_mat = args(2).sparse_matrix_value(); write(filename, label_vec, instance_mat); } else { error ("libsvmwrite: instance_matrix must be sparse."); } } else { error ("libsvmwrite: wrong number of input arguments."); } return octave_value_list(); } /* %!shared L, D %! [L, D] = libsvmread (file_in_loadpath ("heart_scale.dat")); %!error libsvmwrite ("", L, D); %!error ... %! libsvmwrite (tempname (), [L;L], D); %!error ... %! OUT = libsvmwrite (tempname (), L, D); %!error ... %! libsvmwrite (tempname (), single (L), D); %!error libsvmwrite (13412, L, D); %!error ... %! libsvmwrite (tempname (), L, full (D)); %!error ... %! libsvmwrite (tempname (), L, D, D); */ statistics-release-1.6.3/src/svm.cpp000066400000000000000000002007551456127120000174400ustar00rootroot00000000000000/* Copyright (C) 2021 Chih-Chung Chang and Chih-Jen Lin This file is part of the statistics package for GNU Octave. Permission granted by Chih-Jen Lin to the package maintainer to include this ile and double license under GPLv3 by means of personal communication. This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 3 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with this program; if not, see . */ #include #include #include #include #include #include #include #include #include #include "svm.h" int libsvm_version = LIBSVM_VERSION; typedef float Qfloat; typedef signed char schar; #ifndef min template static inline T min(T x,T y) { return (x static inline T max(T x,T y) { return (x>y)?x:y; } #endif template static inline void swap(T& x, T& y) { T t=x; x=y; y=t; } template static inline void clone(T*& dst, S* src, int n) { dst = new T[n]; memcpy((void *)dst,(void *)src,sizeof(T)*n); } static inline double powi(double base, int times) { double tmp = base, ret = 1.0; for(int t=times; t>0; t/=2) { if(t%2==1) ret*=tmp; tmp = tmp * tmp; } return ret; } #define INF HUGE_VAL #define TAU 1e-12 #define Malloc(type,n) (type *)malloc((n)*sizeof(type)) static void print_string_stdout(const char *s) { fputs(s,stdout); fflush(stdout); } static void (*svm_print_string) (const char *) = &print_string_stdout; #if 1 static void info(const char *fmt,...) { char buf[BUFSIZ]; va_list ap; va_start(ap,fmt); vsnprintf(buf,sizeof(buf),fmt,ap); va_end(ap); (*svm_print_string)(buf); } #else static void info(const char *fmt,...) {} #endif // // Kernel Cache // // l is the number of total data items // size is the cache size limit in bytes // class Cache { public: Cache(int l,long int size); ~Cache(); // request data [0,len) // return some position p where [p,len) need to be filled // (p >= len if nothing needs to be filled) int get_data(const int index, Qfloat **data, int len); void swap_index(int i, int j); private: int l; long int size; struct head_t { head_t *prev, *next; // a circular list Qfloat *data; int len; // data[0,len) is cached in this entry }; head_t *head; head_t lru_head; void lru_delete(head_t *h); void lru_insert(head_t *h); }; Cache::Cache(int l_,long int size_):l(l_),size(size_) { head = (head_t *)calloc(l,sizeof(head_t)); // initialized to 0 size /= sizeof(Qfloat); size -= l * sizeof(head_t) / sizeof(Qfloat); size = max(size, 2 * (long int) l); // cache must be large enough for two columns lru_head.next = lru_head.prev = &lru_head; } Cache::~Cache() { for(head_t *h = lru_head.next; h != &lru_head; h=h->next) free(h->data); free(head); } void Cache::lru_delete(head_t *h) { // delete from current location h->prev->next = h->next; h->next->prev = h->prev; } void Cache::lru_insert(head_t *h) { // insert to last position h->next = &lru_head; h->prev = lru_head.prev; h->prev->next = h; h->next->prev = h; } int Cache::get_data(const int index, Qfloat **data, int len) { head_t *h = &head[index]; if(h->len) lru_delete(h); int more = len - h->len; if(more > 0) { // free old space while(size < more) { head_t *old = lru_head.next; lru_delete(old); free(old->data); size += old->len; old->data = 0; old->len = 0; } // allocate new space h->data = (Qfloat *)realloc(h->data,sizeof(Qfloat)*len); size -= more; swap(h->len,len); } lru_insert(h); *data = h->data; return len; } void Cache::swap_index(int i, int j) { if(i==j) return; if(head[i].len) lru_delete(&head[i]); if(head[j].len) lru_delete(&head[j]); swap(head[i].data,head[j].data); swap(head[i].len,head[j].len); if(head[i].len) lru_insert(&head[i]); if(head[j].len) lru_insert(&head[j]); if(i>j) swap(i,j); for(head_t *h = lru_head.next; h!=&lru_head; h=h->next) { if(h->len > i) { if(h->len > j) swap(h->data[i],h->data[j]); else { // give up lru_delete(h); free(h->data); size += h->len; h->data = 0; h->len = 0; } } } } // // Kernel evaluation // // the static method k_function is for doing single kernel evaluation // the constructor of Kernel prepares to calculate the l*l kernel matrix // the member function get_Q is for getting one column from the Q Matrix // class QMatrix { public: virtual Qfloat *get_Q(int column, int len) const = 0; virtual double *get_QD() const = 0; virtual void swap_index(int i, int j) const = 0; virtual ~QMatrix() {} }; class Kernel: public QMatrix { public: Kernel(int l, svm_node * const * x, const svm_parameter& param); virtual ~Kernel(); static double k_function(const svm_node *x, const svm_node *y, const svm_parameter& param); virtual Qfloat *get_Q(int column, int len) const = 0; virtual double *get_QD() const = 0; virtual void swap_index(int i, int j) const // no so const... { swap(x[i],x[j]); if(x_square) swap(x_square[i],x_square[j]); } protected: double (Kernel::*kernel_function)(int i, int j) const; private: const svm_node **x; double *x_square; // svm_parameter const int kernel_type; const int degree; const double gamma; const double coef0; static double dot(const svm_node *px, const svm_node *py); double kernel_linear(int i, int j) const { return dot(x[i],x[j]); } double kernel_poly(int i, int j) const { return powi(gamma*dot(x[i],x[j])+coef0,degree); } double kernel_rbf(int i, int j) const { return exp(-gamma*(x_square[i]+x_square[j]-2*dot(x[i],x[j]))); } double kernel_sigmoid(int i, int j) const { return tanh(gamma*dot(x[i],x[j])+coef0); } double kernel_precomputed(int i, int j) const { return x[i][(int)(x[j][0].value)].value; } }; Kernel::Kernel(int l, svm_node * const * x_, const svm_parameter& param) :kernel_type(param.kernel_type), degree(param.degree), gamma(param.gamma), coef0(param.coef0) { switch(kernel_type) { case LINEAR: kernel_function = &Kernel::kernel_linear; break; case POLY: kernel_function = &Kernel::kernel_poly; break; case RBF: kernel_function = &Kernel::kernel_rbf; break; case SIGMOID: kernel_function = &Kernel::kernel_sigmoid; break; case PRECOMPUTED: kernel_function = &Kernel::kernel_precomputed; break; } clone(x,x_,l); if(kernel_type == RBF) { x_square = new double[l]; for(int i=0;iindex != -1 && py->index != -1) { if(px->index == py->index) { sum += px->value * py->value; ++px; ++py; } else { if(px->index > py->index) ++py; else ++px; } } return sum; } double Kernel::k_function(const svm_node *x, const svm_node *y, const svm_parameter& param) { switch(param.kernel_type) { case LINEAR: return dot(x,y); case POLY: return powi(param.gamma*dot(x,y)+param.coef0,param.degree); case RBF: { double sum = 0; while(x->index != -1 && y->index !=-1) { if(x->index == y->index) { double d = x->value - y->value; sum += d*d; ++x; ++y; } else { if(x->index > y->index) { sum += y->value * y->value; ++y; } else { sum += x->value * x->value; ++x; } } } while(x->index != -1) { sum += x->value * x->value; ++x; } while(y->index != -1) { sum += y->value * y->value; ++y; } return exp(-param.gamma*sum); } case SIGMOID: return tanh(param.gamma*dot(x,y)+param.coef0); case PRECOMPUTED: //x: test (validation), y: SV return x[(int)(y->value)].value; default: return 0; // Unreachable } } // An SMO algorithm in Fan et al., JMLR 6(2005), p. 1889--1918 // Solves: // // min 0.5(\alpha^T Q \alpha) + p^T \alpha // // y^T \alpha = \delta // y_i = +1 or -1 // 0 <= alpha_i <= Cp for y_i = 1 // 0 <= alpha_i <= Cn for y_i = -1 // // Given: // // Q, p, y, Cp, Cn, and an initial feasible point \alpha // l is the size of vectors and matrices // eps is the stopping tolerance // // solution will be put in \alpha, objective value will be put in obj // class Solver { public: Solver() {}; virtual ~Solver() {}; struct SolutionInfo { double obj; double rho; double upper_bound_p; double upper_bound_n; double r; // for Solver_NU }; void Solve(int l, const QMatrix& Q, const double *p_, const schar *y_, double *alpha_, double Cp, double Cn, double eps, SolutionInfo* si, int shrinking); protected: int active_size; schar *y; double *G; // gradient of objective function enum { LOWER_BOUND, UPPER_BOUND, FREE }; char *alpha_status; // LOWER_BOUND, UPPER_BOUND, FREE double *alpha; const QMatrix *Q; const double *QD; double eps; double Cp,Cn; double *p; int *active_set; double *G_bar; // gradient, if we treat free variables as 0 int l; bool unshrink; // XXX double get_C(int i) { return (y[i] > 0)? Cp : Cn; } void update_alpha_status(int i) { if(alpha[i] >= get_C(i)) alpha_status[i] = UPPER_BOUND; else if(alpha[i] <= 0) alpha_status[i] = LOWER_BOUND; else alpha_status[i] = FREE; } bool is_upper_bound(int i) { return alpha_status[i] == UPPER_BOUND; } bool is_lower_bound(int i) { return alpha_status[i] == LOWER_BOUND; } bool is_free(int i) { return alpha_status[i] == FREE; } void swap_index(int i, int j); void reconstruct_gradient(); virtual int select_working_set(int &i, int &j); virtual double calculate_rho(); virtual void do_shrinking(); private: bool be_shrunk(int i, double Gmax1, double Gmax2); }; void Solver::swap_index(int i, int j) { Q->swap_index(i,j); swap(y[i],y[j]); swap(G[i],G[j]); swap(alpha_status[i],alpha_status[j]); swap(alpha[i],alpha[j]); swap(p[i],p[j]); swap(active_set[i],active_set[j]); swap(G_bar[i],G_bar[j]); } void Solver::reconstruct_gradient() { // reconstruct inactive elements of G from G_bar and free variables if(active_size == l) return; int i,j; int nr_free = 0; for(j=active_size;j 2*active_size*(l-active_size)) { for(i=active_size;iget_Q(i,active_size); for(j=0;jget_Q(i,l); double alpha_i = alpha[i]; for(j=active_size;jl = l; this->Q = &Q; QD=Q.get_QD(); clone(p, p_,l); clone(y, y_,l); clone(alpha,alpha_,l); this->Cp = Cp; this->Cn = Cn; this->eps = eps; unshrink = false; // initialize alpha_status { alpha_status = new char[l]; for(int i=0;iINT_MAX/100 ? INT_MAX : 100*l); int counter = min(l,1000)+1; while(iter < max_iter) { // show progress and do shrinking if(--counter == 0) { counter = min(l,1000); if(shrinking) do_shrinking(); info("."); } int i,j; if(select_working_set(i,j)!=0) { // reconstruct the whole gradient reconstruct_gradient(); // reset active set size and check active_size = l; info("*"); if(select_working_set(i,j)!=0) break; else counter = 1; // do shrinking next iteration } ++iter; // update alpha[i] and alpha[j], handle bounds carefully const Qfloat *Q_i = Q.get_Q(i,active_size); const Qfloat *Q_j = Q.get_Q(j,active_size); double C_i = get_C(i); double C_j = get_C(j); double old_alpha_i = alpha[i]; double old_alpha_j = alpha[j]; if(y[i]!=y[j]) { double quad_coef = QD[i]+QD[j]+2*Q_i[j]; if (quad_coef <= 0) quad_coef = TAU; double delta = (-G[i]-G[j])/quad_coef; double diff = alpha[i] - alpha[j]; alpha[i] += delta; alpha[j] += delta; if(diff > 0) { if(alpha[j] < 0) { alpha[j] = 0; alpha[i] = diff; } } else { if(alpha[i] < 0) { alpha[i] = 0; alpha[j] = -diff; } } if(diff > C_i - C_j) { if(alpha[i] > C_i) { alpha[i] = C_i; alpha[j] = C_i - diff; } } else { if(alpha[j] > C_j) { alpha[j] = C_j; alpha[i] = C_j + diff; } } } else { double quad_coef = QD[i]+QD[j]-2*Q_i[j]; if (quad_coef <= 0) quad_coef = TAU; double delta = (G[i]-G[j])/quad_coef; double sum = alpha[i] + alpha[j]; alpha[i] -= delta; alpha[j] += delta; if(sum > C_i) { if(alpha[i] > C_i) { alpha[i] = C_i; alpha[j] = sum - C_i; } } else { if(alpha[j] < 0) { alpha[j] = 0; alpha[i] = sum; } } if(sum > C_j) { if(alpha[j] > C_j) { alpha[j] = C_j; alpha[i] = sum - C_j; } } else { if(alpha[i] < 0) { alpha[i] = 0; alpha[j] = sum; } } } // update G double delta_alpha_i = alpha[i] - old_alpha_i; double delta_alpha_j = alpha[j] - old_alpha_j; for(int k=0;k= max_iter) { if(active_size < l) { // reconstruct the whole gradient to calculate objective value reconstruct_gradient(); active_size = l; info("*"); } fprintf(stderr,"\nWARNING: reaching max number of iterations\n"); } // calculate rho si->rho = calculate_rho(); // calculate objective value { double v = 0; int i; for(i=0;iobj = v/2; } // put back the solution { for(int i=0;iupper_bound_p = Cp; si->upper_bound_n = Cn; info("\noptimization finished, #iter = %d\n",iter); delete[] p; delete[] y; delete[] alpha; delete[] alpha_status; delete[] active_set; delete[] G; delete[] G_bar; } // return 1 if already optimal, return 0 otherwise int Solver::select_working_set(int &out_i, int &out_j) { // return i,j such that // i: maximizes -y_i * grad(f)_i, i in I_up(\alpha) // j: minimizes the decrease of obj value // (if quadratic coefficeint <= 0, replace it with tau) // -y_j*grad(f)_j < -y_i*grad(f)_i, j in I_low(\alpha) double Gmax = -INF; double Gmax2 = -INF; int Gmax_idx = -1; int Gmin_idx = -1; double obj_diff_min = INF; for(int t=0;t= Gmax) { Gmax = -G[t]; Gmax_idx = t; } } else { if(!is_lower_bound(t)) if(G[t] >= Gmax) { Gmax = G[t]; Gmax_idx = t; } } int i = Gmax_idx; const Qfloat *Q_i = NULL; if(i != -1) // NULL Q_i not accessed: Gmax=-INF if i=-1 Q_i = Q->get_Q(i,active_size); for(int j=0;j= Gmax2) Gmax2 = G[j]; if (grad_diff > 0) { double obj_diff; double quad_coef = QD[i]+QD[j]-2.0*y[i]*Q_i[j]; if (quad_coef > 0) obj_diff = -(grad_diff*grad_diff)/quad_coef; else obj_diff = -(grad_diff*grad_diff)/TAU; if (obj_diff <= obj_diff_min) { Gmin_idx=j; obj_diff_min = obj_diff; } } } } else { if (!is_upper_bound(j)) { double grad_diff= Gmax-G[j]; if (-G[j] >= Gmax2) Gmax2 = -G[j]; if (grad_diff > 0) { double obj_diff; double quad_coef = QD[i]+QD[j]+2.0*y[i]*Q_i[j]; if (quad_coef > 0) obj_diff = -(grad_diff*grad_diff)/quad_coef; else obj_diff = -(grad_diff*grad_diff)/TAU; if (obj_diff <= obj_diff_min) { Gmin_idx=j; obj_diff_min = obj_diff; } } } } } if(Gmax+Gmax2 < eps || Gmin_idx == -1) return 1; out_i = Gmax_idx; out_j = Gmin_idx; return 0; } bool Solver::be_shrunk(int i, double Gmax1, double Gmax2) { if(is_upper_bound(i)) { if(y[i]==+1) return(-G[i] > Gmax1); else return(-G[i] > Gmax2); } else if(is_lower_bound(i)) { if(y[i]==+1) return(G[i] > Gmax2); else return(G[i] > Gmax1); } else return(false); } void Solver::do_shrinking() { int i; double Gmax1 = -INF; // max { -y_i * grad(f)_i | i in I_up(\alpha) } double Gmax2 = -INF; // max { y_i * grad(f)_i | i in I_low(\alpha) } // find maximal violating pair first for(i=0;i= Gmax1) Gmax1 = -G[i]; } if(!is_lower_bound(i)) { if(G[i] >= Gmax2) Gmax2 = G[i]; } } else { if(!is_upper_bound(i)) { if(-G[i] >= Gmax2) Gmax2 = -G[i]; } if(!is_lower_bound(i)) { if(G[i] >= Gmax1) Gmax1 = G[i]; } } } if(unshrink == false && Gmax1 + Gmax2 <= eps*10) { unshrink = true; reconstruct_gradient(); active_size = l; info("*"); } for(i=0;i i) { if (!be_shrunk(active_size, Gmax1, Gmax2)) { swap_index(i,active_size); break; } active_size--; } } } double Solver::calculate_rho() { double r; int nr_free = 0; double ub = INF, lb = -INF, sum_free = 0; for(int i=0;i0) r = sum_free/nr_free; else r = (ub+lb)/2; return r; } // // Solver for nu-svm classification and regression // // additional constraint: e^T \alpha = constant // class Solver_NU: public Solver { public: Solver_NU() {} void Solve(int l, const QMatrix& Q, const double *p, const schar *y, double *alpha, double Cp, double Cn, double eps, SolutionInfo* si, int shrinking) { this->si = si; Solver::Solve(l,Q,p,y,alpha,Cp,Cn,eps,si,shrinking); } private: SolutionInfo *si; int select_working_set(int &i, int &j); double calculate_rho(); bool be_shrunk(int i, double Gmax1, double Gmax2, double Gmax3, double Gmax4); void do_shrinking(); }; // return 1 if already optimal, return 0 otherwise int Solver_NU::select_working_set(int &out_i, int &out_j) { // return i,j such that y_i = y_j and // i: maximizes -y_i * grad(f)_i, i in I_up(\alpha) // j: minimizes the decrease of obj value // (if quadratic coefficeint <= 0, replace it with tau) // -y_j*grad(f)_j < -y_i*grad(f)_i, j in I_low(\alpha) double Gmaxp = -INF; double Gmaxp2 = -INF; int Gmaxp_idx = -1; double Gmaxn = -INF; double Gmaxn2 = -INF; int Gmaxn_idx = -1; int Gmin_idx = -1; double obj_diff_min = INF; for(int t=0;t= Gmaxp) { Gmaxp = -G[t]; Gmaxp_idx = t; } } else { if(!is_lower_bound(t)) if(G[t] >= Gmaxn) { Gmaxn = G[t]; Gmaxn_idx = t; } } int ip = Gmaxp_idx; int in = Gmaxn_idx; const Qfloat *Q_ip = NULL; const Qfloat *Q_in = NULL; if(ip != -1) // NULL Q_ip not accessed: Gmaxp=-INF if ip=-1 Q_ip = Q->get_Q(ip,active_size); if(in != -1) Q_in = Q->get_Q(in,active_size); for(int j=0;j= Gmaxp2) Gmaxp2 = G[j]; if (grad_diff > 0) { double obj_diff; double quad_coef = QD[ip]+QD[j]-2*Q_ip[j]; if (quad_coef > 0) obj_diff = -(grad_diff*grad_diff)/quad_coef; else obj_diff = -(grad_diff*grad_diff)/TAU; if (obj_diff <= obj_diff_min) { Gmin_idx=j; obj_diff_min = obj_diff; } } } } else { if (!is_upper_bound(j)) { double grad_diff=Gmaxn-G[j]; if (-G[j] >= Gmaxn2) Gmaxn2 = -G[j]; if (grad_diff > 0) { double obj_diff; double quad_coef = QD[in]+QD[j]-2*Q_in[j]; if (quad_coef > 0) obj_diff = -(grad_diff*grad_diff)/quad_coef; else obj_diff = -(grad_diff*grad_diff)/TAU; if (obj_diff <= obj_diff_min) { Gmin_idx=j; obj_diff_min = obj_diff; } } } } } if(max(Gmaxp+Gmaxp2,Gmaxn+Gmaxn2) < eps || Gmin_idx == -1) return 1; if (y[Gmin_idx] == +1) out_i = Gmaxp_idx; else out_i = Gmaxn_idx; out_j = Gmin_idx; return 0; } bool Solver_NU::be_shrunk(int i, double Gmax1, double Gmax2, double Gmax3, double Gmax4) { if(is_upper_bound(i)) { if(y[i]==+1) return(-G[i] > Gmax1); else return(-G[i] > Gmax4); } else if(is_lower_bound(i)) { if(y[i]==+1) return(G[i] > Gmax2); else return(G[i] > Gmax3); } else return(false); } void Solver_NU::do_shrinking() { double Gmax1 = -INF; // max { -y_i * grad(f)_i | y_i = +1, i in I_up(\alpha) } double Gmax2 = -INF; // max { y_i * grad(f)_i | y_i = +1, i in I_low(\alpha) } double Gmax3 = -INF; // max { -y_i * grad(f)_i | y_i = -1, i in I_up(\alpha) } double Gmax4 = -INF; // max { y_i * grad(f)_i | y_i = -1, i in I_low(\alpha) } // find maximal violating pair first int i; for(i=0;i Gmax1) Gmax1 = -G[i]; } else if(-G[i] > Gmax4) Gmax4 = -G[i]; } if(!is_lower_bound(i)) { if(y[i]==+1) { if(G[i] > Gmax2) Gmax2 = G[i]; } else if(G[i] > Gmax3) Gmax3 = G[i]; } } if(unshrink == false && max(Gmax1+Gmax2,Gmax3+Gmax4) <= eps*10) { unshrink = true; reconstruct_gradient(); active_size = l; } for(i=0;i i) { if (!be_shrunk(active_size, Gmax1, Gmax2, Gmax3, Gmax4)) { swap_index(i,active_size); break; } active_size--; } } } double Solver_NU::calculate_rho() { int nr_free1 = 0,nr_free2 = 0; double ub1 = INF, ub2 = INF; double lb1 = -INF, lb2 = -INF; double sum_free1 = 0, sum_free2 = 0; for(int i=0;i 0) r1 = sum_free1/nr_free1; else r1 = (ub1+lb1)/2; if(nr_free2 > 0) r2 = sum_free2/nr_free2; else r2 = (ub2+lb2)/2; si->r = (r1+r2)/2; return (r1-r2)/2; } // // Q matrices for various formulations // class SVC_Q: public Kernel { public: SVC_Q(const svm_problem& prob, const svm_parameter& param, const schar *y_) :Kernel(prob.l, prob.x, param) { clone(y,y_,prob.l); cache = new Cache(prob.l,(long int)(param.cache_size*(1<<20))); QD = new double[prob.l]; for(int i=0;i*kernel_function)(i,i); } Qfloat *get_Q(int i, int len) const { Qfloat *data; int start, j; if((start = cache->get_data(i,&data,len)) < len) { for(j=start;j*kernel_function)(i,j)); } return data; } double *get_QD() const { return QD; } void swap_index(int i, int j) const { cache->swap_index(i,j); Kernel::swap_index(i,j); swap(y[i],y[j]); swap(QD[i],QD[j]); } ~SVC_Q() { delete[] y; delete cache; delete[] QD; } private: schar *y; Cache *cache; double *QD; }; class ONE_CLASS_Q: public Kernel { public: ONE_CLASS_Q(const svm_problem& prob, const svm_parameter& param) :Kernel(prob.l, prob.x, param) { cache = new Cache(prob.l,(long int)(param.cache_size*(1<<20))); QD = new double[prob.l]; for(int i=0;i*kernel_function)(i,i); } Qfloat *get_Q(int i, int len) const { Qfloat *data; int start, j; if((start = cache->get_data(i,&data,len)) < len) { for(j=start;j*kernel_function)(i,j); } return data; } double *get_QD() const { return QD; } void swap_index(int i, int j) const { cache->swap_index(i,j); Kernel::swap_index(i,j); swap(QD[i],QD[j]); } ~ONE_CLASS_Q() { delete cache; delete[] QD; } private: Cache *cache; double *QD; }; class SVR_Q: public Kernel { public: SVR_Q(const svm_problem& prob, const svm_parameter& param) :Kernel(prob.l, prob.x, param) { l = prob.l; cache = new Cache(l,(long int)(param.cache_size*(1<<20))); QD = new double[2*l]; sign = new schar[2*l]; index = new int[2*l]; for(int k=0;k*kernel_function)(k,k); QD[k+l] = QD[k]; } buffer[0] = new Qfloat[2*l]; buffer[1] = new Qfloat[2*l]; next_buffer = 0; } void swap_index(int i, int j) const { swap(sign[i],sign[j]); swap(index[i],index[j]); swap(QD[i],QD[j]); } Qfloat *get_Q(int i, int len) const { Qfloat *data; int j, real_i = index[i]; if(cache->get_data(real_i,&data,l) < l) { for(j=0;j*kernel_function)(real_i,j); } // reorder and copy Qfloat *buf = buffer[next_buffer]; next_buffer = 1 - next_buffer; schar si = sign[i]; for(j=0;jl; double *minus_ones = new double[l]; schar *y = new schar[l]; int i; for(i=0;iy[i] > 0) y[i] = +1; else y[i] = -1; } Solver s; s.Solve(l, SVC_Q(*prob,*param,y), minus_ones, y, alpha, Cp, Cn, param->eps, si, param->shrinking); double sum_alpha=0; for(i=0;il)); for(i=0;il; double nu = param->nu; schar *y = new schar[l]; for(i=0;iy[i]>0) y[i] = +1; else y[i] = -1; double sum_pos = nu*l/2; double sum_neg = nu*l/2; for(i=0;ieps, si, param->shrinking); double r = si->r; info("C = %f\n",1/r); for(i=0;irho /= r; si->obj /= (r*r); si->upper_bound_p = 1/r; si->upper_bound_n = 1/r; delete[] y; delete[] zeros; } static void solve_one_class( const svm_problem *prob, const svm_parameter *param, double *alpha, Solver::SolutionInfo* si) { int l = prob->l; double *zeros = new double[l]; schar *ones = new schar[l]; int i; int n = (int)(param->nu*prob->l); // # of alpha's at upper bound for(i=0;il) alpha[n] = param->nu * prob->l - n; for(i=n+1;ieps, si, param->shrinking); delete[] zeros; delete[] ones; } static void solve_epsilon_svr( const svm_problem *prob, const svm_parameter *param, double *alpha, Solver::SolutionInfo* si) { int l = prob->l; double *alpha2 = new double[2*l]; double *linear_term = new double[2*l]; schar *y = new schar[2*l]; int i; for(i=0;ip - prob->y[i]; y[i] = 1; alpha2[i+l] = 0; linear_term[i+l] = param->p + prob->y[i]; y[i+l] = -1; } Solver s; s.Solve(2*l, SVR_Q(*prob,*param), linear_term, y, alpha2, param->C, param->C, param->eps, si, param->shrinking); double sum_alpha = 0; for(i=0;iC*l)); delete[] alpha2; delete[] linear_term; delete[] y; } static void solve_nu_svr( const svm_problem *prob, const svm_parameter *param, double *alpha, Solver::SolutionInfo* si) { int l = prob->l; double C = param->C; double *alpha2 = new double[2*l]; double *linear_term = new double[2*l]; schar *y = new schar[2*l]; int i; double sum = C * param->nu * l / 2; for(i=0;iy[i]; y[i] = 1; linear_term[i+l] = prob->y[i]; y[i+l] = -1; } Solver_NU s; s.Solve(2*l, SVR_Q(*prob,*param), linear_term, y, alpha2, C, C, param->eps, si, param->shrinking); info("epsilon = %f\n",-si->r); for(i=0;il); Solver::SolutionInfo si; switch(param->svm_type) { case C_SVC: solve_c_svc(prob,param,alpha,&si,Cp,Cn); break; case NU_SVC: solve_nu_svc(prob,param,alpha,&si); break; case ONE_CLASS: solve_one_class(prob,param,alpha,&si); break; case EPSILON_SVR: solve_epsilon_svr(prob,param,alpha,&si); break; case NU_SVR: solve_nu_svr(prob,param,alpha,&si); break; } info("obj = %f, rho = %f\n",si.obj,si.rho); // output SVs int nSV = 0; int nBSV = 0; for(int i=0;il;i++) { if(fabs(alpha[i]) > 0) { ++nSV; if(prob->y[i] > 0) { if(fabs(alpha[i]) >= si.upper_bound_p) ++nBSV; } else { if(fabs(alpha[i]) >= si.upper_bound_n) ++nBSV; } } } info("nSV = %d, nBSV = %d\n",nSV,nBSV); decision_function f; f.alpha = alpha; f.rho = si.rho; return f; } // Platt's binary SVM Probablistic Output: an improvement from Lin et al. static void sigmoid_train( int l, const double *dec_values, const double *labels, double& A, double& B) { double prior1=0, prior0 = 0; int i; for (i=0;i 0) prior1+=1; else prior0+=1; int max_iter=100; // Maximal number of iterations double min_step=1e-10; // Minimal step taken in line search double sigma=1e-12; // For numerically strict PD of Hessian double eps=1e-5; double hiTarget=(prior1+1.0)/(prior1+2.0); double loTarget=1/(prior0+2.0); double *t=Malloc(double,l); double fApB,p,q,h11,h22,h21,g1,g2,det,dA,dB,gd,stepsize; double newA,newB,newf,d1,d2; int iter; // Initial Point and Initial Fun Value A=0.0; B=log((prior0+1.0)/(prior1+1.0)); double fval = 0.0; for (i=0;i0) t[i]=hiTarget; else t[i]=loTarget; fApB = dec_values[i]*A+B; if (fApB>=0) fval += t[i]*fApB + log(1+exp(-fApB)); else fval += (t[i] - 1)*fApB +log(1+exp(fApB)); } for (iter=0;iter= 0) { p=exp(-fApB)/(1.0+exp(-fApB)); q=1.0/(1.0+exp(-fApB)); } else { p=1.0/(1.0+exp(fApB)); q=exp(fApB)/(1.0+exp(fApB)); } d2=p*q; h11+=dec_values[i]*dec_values[i]*d2; h22+=d2; h21+=dec_values[i]*d2; d1=t[i]-p; g1+=dec_values[i]*d1; g2+=d1; } // Stopping Criteria if (fabs(g1)= min_step) { newA = A + stepsize * dA; newB = B + stepsize * dB; // New function value newf = 0.0; for (i=0;i= 0) newf += t[i]*fApB + log(1+exp(-fApB)); else newf += (t[i] - 1)*fApB +log(1+exp(fApB)); } // Check sufficient decrease if (newf=max_iter) info("Reaching maximal iterations in two-class probability estimates\n"); free(t); } static double sigmoid_predict(double decision_value, double A, double B) { double fApB = decision_value*A+B; // 1-p used later; avoid catastrophic cancellation if (fApB >= 0) return exp(-fApB)/(1.0+exp(-fApB)); else return 1.0/(1+exp(fApB)) ; } // Method 2 from the multiclass_prob paper by Wu, Lin, and Weng static void multiclass_probability(int k, double **r, double *p) { int t,j; int iter = 0, max_iter=max(100,k); double **Q=Malloc(double *,k); double *Qp=Malloc(double,k); double pQp, eps=0.005/k; for (t=0;tmax_error) max_error=error; } if (max_error=max_iter) info("Exceeds max_iter in multiclass_prob\n"); for(t=0;tl); double *dec_values = Malloc(double,prob->l); // random shuffle for(i=0;il;i++) perm[i]=i; for(i=0;il;i++) { int j = i+rand()%(prob->l-i); swap(perm[i],perm[j]); } for(i=0;il/nr_fold; int end = (i+1)*prob->l/nr_fold; int j,k; struct svm_problem subprob; subprob.l = prob->l-(end-begin); subprob.x = Malloc(struct svm_node*,subprob.l); subprob.y = Malloc(double,subprob.l); k=0; for(j=0;jx[perm[j]]; subprob.y[k] = prob->y[perm[j]]; ++k; } for(j=end;jl;j++) { subprob.x[k] = prob->x[perm[j]]; subprob.y[k] = prob->y[perm[j]]; ++k; } int p_count=0,n_count=0; for(j=0;j0) p_count++; else n_count++; if(p_count==0 && n_count==0) for(j=begin;j 0 && n_count == 0) for(j=begin;j 0) for(j=begin;jx[perm[j]],&(dec_values[perm[j]])); // ensure +1 -1 order; reason not using CV subroutine dec_values[perm[j]] *= submodel->label[0]; } svm_free_and_destroy_model(&submodel); svm_destroy_param(&subparam); } free(subprob.x); free(subprob.y); } sigmoid_train(prob->l,dec_values,prob->y,probA,probB); free(dec_values); free(perm); } // Return parameter of a Laplace distribution static double svm_svr_probability( const svm_problem *prob, const svm_parameter *param) { int i; int nr_fold = 5; double *ymv = Malloc(double,prob->l); double mae = 0; svm_parameter newparam = *param; newparam.probability = 0; svm_cross_validation(prob,&newparam,nr_fold,ymv); for(i=0;il;i++) { ymv[i]=prob->y[i]-ymv[i]; mae += fabs(ymv[i]); } mae /= prob->l; double std=sqrt(2*mae*mae); int count=0; mae=0; for(i=0;il;i++) if (fabs(ymv[i]) > 5*std) count=count+1; else mae+=fabs(ymv[i]); mae /= (prob->l-count); info("Prob. model for test data: target value = predicted value + z,\nz: Laplace distribution e^(-|z|/sigma)/(2sigma),sigma= %g\n",mae); free(ymv); return mae; } // label: label name, start: begin of each class, count: #data of classes, perm: indices to the original data // perm, length l, must be allocated before calling this subroutine static void svm_group_classes(const svm_problem *prob, int *nr_class_ret, int **label_ret, int **start_ret, int **count_ret, int *perm) { int l = prob->l; int max_nr_class = 16; int nr_class = 0; int *label = Malloc(int,max_nr_class); int *count = Malloc(int,max_nr_class); int *data_label = Malloc(int,l); int i; for(i=0;iy[i]; int j; for(j=0;jparam = *param; model->free_sv = 0; // XXX if(param->svm_type == ONE_CLASS || param->svm_type == EPSILON_SVR || param->svm_type == NU_SVR) { // regression or one-class-svm model->nr_class = 2; model->label = NULL; model->nSV = NULL; model->probA = NULL; model->probB = NULL; model->sv_coef = Malloc(double *,1); if(param->probability && (param->svm_type == EPSILON_SVR || param->svm_type == NU_SVR)) { model->probA = Malloc(double,1); model->probA[0] = svm_svr_probability(prob,param); } decision_function f = svm_train_one(prob,param,0,0); model->rho = Malloc(double,1); model->rho[0] = f.rho; int nSV = 0; int i; for(i=0;il;i++) if(fabs(f.alpha[i]) > 0) ++nSV; model->l = nSV; model->SV = Malloc(svm_node *,nSV); model->sv_coef[0] = Malloc(double,nSV); model->sv_indices = Malloc(int,nSV); int j = 0; for(i=0;il;i++) if(fabs(f.alpha[i]) > 0) { model->SV[j] = prob->x[i]; model->sv_coef[0][j] = f.alpha[i]; model->sv_indices[j] = i+1; ++j; } free(f.alpha); } else { // classification int l = prob->l; int nr_class; int *label = NULL; int *start = NULL; int *count = NULL; int *perm = Malloc(int,l); // group training data of the same class svm_group_classes(prob,&nr_class,&label,&start,&count,perm); if(nr_class == 1) info("WARNING: training data in only one class. See README for details.\n"); svm_node **x = Malloc(svm_node *,l); int i; for(i=0;ix[perm[i]]; // calculate weighted C double *weighted_C = Malloc(double, nr_class); for(i=0;iC; for(i=0;inr_weight;i++) { int j; for(j=0;jweight_label[i] == label[j]) break; if(j == nr_class) fprintf(stderr,"WARNING: class label %d specified in weight is not found\n", param->weight_label[i]); else weighted_C[j] *= param->weight[i]; } // train k*(k-1)/2 models bool *nonzero = Malloc(bool,l); for(i=0;iprobability) { probA=Malloc(double,nr_class*(nr_class-1)/2); probB=Malloc(double,nr_class*(nr_class-1)/2); } int p = 0; for(i=0;iprobability) svm_binary_svc_probability(&sub_prob,param,weighted_C[i],weighted_C[j],probA[p],probB[p]); f[p] = svm_train_one(&sub_prob,param,weighted_C[i],weighted_C[j]); for(k=0;k 0) nonzero[si+k] = true; for(k=0;k 0) nonzero[sj+k] = true; free(sub_prob.x); free(sub_prob.y); ++p; } // build output model->nr_class = nr_class; model->label = Malloc(int,nr_class); for(i=0;ilabel[i] = label[i]; model->rho = Malloc(double,nr_class*(nr_class-1)/2); for(i=0;irho[i] = f[i].rho; if(param->probability) { model->probA = Malloc(double,nr_class*(nr_class-1)/2); model->probB = Malloc(double,nr_class*(nr_class-1)/2); for(i=0;iprobA[i] = probA[i]; model->probB[i] = probB[i]; } } else { model->probA=NULL; model->probB=NULL; } int total_sv = 0; int *nz_count = Malloc(int,nr_class); model->nSV = Malloc(int,nr_class); for(i=0;inSV[i] = nSV; nz_count[i] = nSV; } info("Total nSV = %d\n",total_sv); model->l = total_sv; model->SV = Malloc(svm_node *,total_sv); model->sv_indices = Malloc(int,total_sv); p = 0; for(i=0;iSV[p] = x[i]; model->sv_indices[p++] = perm[i] + 1; } int *nz_start = Malloc(int,nr_class); nz_start[0] = 0; for(i=1;isv_coef = Malloc(double *,nr_class-1); for(i=0;isv_coef[i] = Malloc(double,total_sv); p = 0; for(i=0;isv_coef[j-1][q++] = f[p].alpha[k]; q = nz_start[j]; for(k=0;ksv_coef[i][q++] = f[p].alpha[ci+k]; ++p; } free(label); free(probA); free(probB); free(count); free(perm); free(start); free(x); free(weighted_C); free(nonzero); for(i=0;il; int *perm = Malloc(int,l); int nr_class; if (nr_fold > l) { nr_fold = l; fprintf(stderr,"WARNING: # folds > # data. Will use # folds = # data instead (i.e., leave-one-out cross validation)\n"); } fold_start = Malloc(int,nr_fold+1); // stratified cv may not give leave-one-out rate // Each class to l folds -> some folds may have zero elements if((param->svm_type == C_SVC || param->svm_type == NU_SVC) && nr_fold < l) { int *start = NULL; int *label = NULL; int *count = NULL; svm_group_classes(prob,&nr_class,&label,&start,&count,perm); // random shuffle and then data grouped by fold using the array perm int *fold_count = Malloc(int,nr_fold); int c; int *index = Malloc(int,l); for(i=0;ix[perm[j]]; subprob.y[k] = prob->y[perm[j]]; ++k; } for(j=end;jx[perm[j]]; subprob.y[k] = prob->y[perm[j]]; ++k; } struct svm_model *submodel = svm_train(&subprob,param); if(param->probability && (param->svm_type == C_SVC || param->svm_type == NU_SVC)) { double *prob_estimates=Malloc(double,svm_get_nr_class(submodel)); for(j=begin;jx[perm[j]],prob_estimates); free(prob_estimates); } else for(j=begin;jx[perm[j]]); svm_free_and_destroy_model(&submodel); free(subprob.x); free(subprob.y); } free(fold_start); free(perm); } int svm_get_svm_type(const svm_model *model) { return model->param.svm_type; } int svm_get_nr_class(const svm_model *model) { return model->nr_class; } void svm_get_labels(const svm_model *model, int* label) { if (model->label != NULL) for(int i=0;inr_class;i++) label[i] = model->label[i]; } void svm_get_sv_indices(const svm_model *model, int* indices) { if (model->sv_indices != NULL) for(int i=0;il;i++) indices[i] = model->sv_indices[i]; } int svm_get_nr_sv(const svm_model *model) { return model->l; } double svm_get_svr_probability(const svm_model *model) { if ((model->param.svm_type == EPSILON_SVR || model->param.svm_type == NU_SVR) && model->probA!=NULL) return model->probA[0]; else { fprintf(stderr,"Model doesn't contain information for SVR probability inference\n"); return 0; } } double svm_predict_values(const svm_model *model, const svm_node *x, double* dec_values) { int i; if(model->param.svm_type == ONE_CLASS || model->param.svm_type == EPSILON_SVR || model->param.svm_type == NU_SVR) { double *sv_coef = model->sv_coef[0]; double sum = 0; for(i=0;il;i++) sum += sv_coef[i] * Kernel::k_function(x,model->SV[i],model->param); sum -= model->rho[0]; *dec_values = sum; if(model->param.svm_type == ONE_CLASS) return (sum>0)?1:-1; else return sum; } else { int nr_class = model->nr_class; int l = model->l; double *kvalue = Malloc(double,l); for(i=0;iSV[i],model->param); int *start = Malloc(int,nr_class); start[0] = 0; for(i=1;inSV[i-1]; int *vote = Malloc(int,nr_class); for(i=0;inSV[i]; int cj = model->nSV[j]; int k; double *coef1 = model->sv_coef[j-1]; double *coef2 = model->sv_coef[i]; for(k=0;krho[p]; dec_values[p] = sum; if(dec_values[p] > 0) ++vote[i]; else ++vote[j]; p++; } int vote_max_idx = 0; for(i=1;i vote[vote_max_idx]) vote_max_idx = i; free(kvalue); free(start); free(vote); return model->label[vote_max_idx]; } } double svm_predict(const svm_model *model, const svm_node *x) { int nr_class = model->nr_class; double *dec_values; if(model->param.svm_type == ONE_CLASS || model->param.svm_type == EPSILON_SVR || model->param.svm_type == NU_SVR) dec_values = Malloc(double, 1); else dec_values = Malloc(double, nr_class*(nr_class-1)/2); double pred_result = svm_predict_values(model, x, dec_values); free(dec_values); return pred_result; } double svm_predict_probability( const svm_model *model, const svm_node *x, double *prob_estimates) { if ((model->param.svm_type == C_SVC || model->param.svm_type == NU_SVC) && model->probA!=NULL && model->probB!=NULL) { int i; int nr_class = model->nr_class; double *dec_values = Malloc(double, nr_class*(nr_class-1)/2); svm_predict_values(model, x, dec_values); double min_prob=1e-7; double **pairwise_prob=Malloc(double *,nr_class); for(i=0;iprobA[k],model->probB[k]),min_prob),1-min_prob); pairwise_prob[j][i]=1-pairwise_prob[i][j]; k++; } if (nr_class == 2) { prob_estimates[0] = pairwise_prob[0][1]; prob_estimates[1] = pairwise_prob[1][0]; } else multiclass_probability(nr_class,pairwise_prob,prob_estimates); int prob_max_idx = 0; for(i=1;i prob_estimates[prob_max_idx]) prob_max_idx = i; for(i=0;ilabel[prob_max_idx]; } else return svm_predict(model, x); } static const char *svm_type_table[] = { "c_svc","nu_svc","one_class","epsilon_svr","nu_svr",NULL }; static const char *kernel_type_table[]= { "linear","polynomial","rbf","sigmoid","precomputed",NULL }; int svm_save_model(const char *model_file_name, const svm_model *model) { FILE *fp = fopen(model_file_name,"w"); if(fp==NULL) return -1; char *old_locale = setlocale(LC_ALL, NULL); if (old_locale) { old_locale = strdup(old_locale); } setlocale(LC_ALL, "C"); const svm_parameter& param = model->param; fprintf(fp,"svm_type %s\n", svm_type_table[param.svm_type]); fprintf(fp,"kernel_type %s\n", kernel_type_table[param.kernel_type]); if(param.kernel_type == POLY) fprintf(fp,"degree %d\n", param.degree); if(param.kernel_type == POLY || param.kernel_type == RBF || param.kernel_type == SIGMOID) fprintf(fp,"gamma %.17g\n", param.gamma); if(param.kernel_type == POLY || param.kernel_type == SIGMOID) fprintf(fp,"coef0 %.17g\n", param.coef0); int nr_class = model->nr_class; int l = model->l; fprintf(fp, "nr_class %d\n", nr_class); fprintf(fp, "total_sv %d\n",l); { fprintf(fp, "rho"); for(int i=0;irho[i]); fprintf(fp, "\n"); } if(model->label) { fprintf(fp, "label"); for(int i=0;ilabel[i]); fprintf(fp, "\n"); } if(model->probA) // regression has probA only { fprintf(fp, "probA"); for(int i=0;iprobA[i]); fprintf(fp, "\n"); } if(model->probB) { fprintf(fp, "probB"); for(int i=0;iprobB[i]); fprintf(fp, "\n"); } if(model->nSV) { fprintf(fp, "nr_sv"); for(int i=0;inSV[i]); fprintf(fp, "\n"); } fprintf(fp, "SV\n"); const double * const *sv_coef = model->sv_coef; const svm_node * const *SV = model->SV; for(int i=0;ivalue)); else while(p->index != -1) { fprintf(fp,"%d:%.8g ",p->index,p->value); p++; } fprintf(fp, "\n"); } setlocale(LC_ALL, old_locale); free(old_locale); if (ferror(fp) != 0 || fclose(fp) != 0) return -1; else return 0; } static char *line = NULL; static int max_line_len; static char* readline(FILE *input) { int len; if(fgets(line,max_line_len,input) == NULL) return NULL; while(strrchr(line,'\n') == NULL) { max_line_len *= 2; line = (char *) realloc(line,max_line_len); len = (int) strlen(line); if(fgets(line+len,max_line_len-len,input) == NULL) break; } return line; } // // FSCANF helps to handle fscanf failures. // Its do-while block avoids the ambiguity when // if (...) // FSCANF(); // is used // #define FSCANF(_stream, _format, _var) do{ if (fscanf(_stream, _format, _var) != 1) return false; }while(0) bool read_model_header(FILE *fp, svm_model* model) { svm_parameter& param = model->param; // parameters for training only won't be assigned, but arrays are assigned as NULL for safety param.nr_weight = 0; param.weight_label = NULL; param.weight = NULL; char cmd[81]; while(1) { FSCANF(fp,"%80s",cmd); if(strcmp(cmd,"svm_type")==0) { FSCANF(fp,"%80s",cmd); int i; for(i=0;svm_type_table[i];i++) { if(strcmp(svm_type_table[i],cmd)==0) { param.svm_type=i; break; } } if(svm_type_table[i] == NULL) { fprintf(stderr,"unknown svm type.\n"); return false; } } else if(strcmp(cmd,"kernel_type")==0) { FSCANF(fp,"%80s",cmd); int i; for(i=0;kernel_type_table[i];i++) { if(strcmp(kernel_type_table[i],cmd)==0) { param.kernel_type=i; break; } } if(kernel_type_table[i] == NULL) { fprintf(stderr,"unknown kernel function.\n"); return false; } } else if(strcmp(cmd,"degree")==0) FSCANF(fp,"%d",¶m.degree); else if(strcmp(cmd,"gamma")==0) FSCANF(fp,"%lf",¶m.gamma); else if(strcmp(cmd,"coef0")==0) FSCANF(fp,"%lf",¶m.coef0); else if(strcmp(cmd,"nr_class")==0) FSCANF(fp,"%d",&model->nr_class); else if(strcmp(cmd,"total_sv")==0) FSCANF(fp,"%d",&model->l); else if(strcmp(cmd,"rho")==0) { int n = model->nr_class * (model->nr_class-1)/2; model->rho = Malloc(double,n); for(int i=0;irho[i]); } else if(strcmp(cmd,"label")==0) { int n = model->nr_class; model->label = Malloc(int,n); for(int i=0;ilabel[i]); } else if(strcmp(cmd,"probA")==0) { int n = model->nr_class * (model->nr_class-1)/2; model->probA = Malloc(double,n); for(int i=0;iprobA[i]); } else if(strcmp(cmd,"probB")==0) { int n = model->nr_class * (model->nr_class-1)/2; model->probB = Malloc(double,n); for(int i=0;iprobB[i]); } else if(strcmp(cmd,"nr_sv")==0) { int n = model->nr_class; model->nSV = Malloc(int,n); for(int i=0;inSV[i]); } else if(strcmp(cmd,"SV")==0) { while(1) { int c = getc(fp); if(c==EOF || c=='\n') break; } break; } else { fprintf(stderr,"unknown text in model file: [%s]\n",cmd); return false; } } return true; } svm_model *svm_load_model(const char *model_file_name) { FILE *fp = fopen(model_file_name,"rb"); if(fp==NULL) return NULL; char *old_locale = setlocale(LC_ALL, NULL); if (old_locale) { old_locale = strdup(old_locale); } setlocale(LC_ALL, "C"); // read parameters svm_model *model = Malloc(svm_model,1); model->rho = NULL; model->probA = NULL; model->probB = NULL; model->sv_indices = NULL; model->label = NULL; model->nSV = NULL; // read header if (!read_model_header(fp, model)) { fprintf(stderr, "ERROR: fscanf failed to read model\n"); setlocale(LC_ALL, old_locale); free(old_locale); free(model->rho); free(model->label); free(model->nSV); free(model); return NULL; } // read sv_coef and SV int elements = 0; long pos = ftell(fp); max_line_len = 1024; line = Malloc(char,max_line_len); char *p,*endptr,*idx,*val; while(readline(fp)!=NULL) { p = strtok(line,":"); while(1) { p = strtok(NULL,":"); if(p == NULL) break; ++elements; } } elements += model->l; fseek(fp,pos,SEEK_SET); int m = model->nr_class - 1; int l = model->l; model->sv_coef = Malloc(double *,m); int i; for(i=0;isv_coef[i] = Malloc(double,l); model->SV = Malloc(svm_node*,l); svm_node *x_space = NULL; if(l>0) x_space = Malloc(svm_node,elements); int j=0; for(i=0;iSV[i] = &x_space[j]; p = strtok(line, " \t"); model->sv_coef[0][i] = strtod(p,&endptr); for(int k=1;ksv_coef[k][i] = strtod(p,&endptr); } while(1) { idx = strtok(NULL, ":"); val = strtok(NULL, " \t"); if(val == NULL) break; x_space[j].index = (int) strtol(idx,&endptr,10); x_space[j].value = strtod(val,&endptr); ++j; } x_space[j++].index = -1; } free(line); setlocale(LC_ALL, old_locale); free(old_locale); if (ferror(fp) != 0 || fclose(fp) != 0) return NULL; model->free_sv = 1; // XXX return model; } void svm_free_model_content(svm_model* model_ptr) { if(model_ptr->free_sv && model_ptr->l > 0 && model_ptr->SV != NULL) free((void *)(model_ptr->SV[0])); if(model_ptr->sv_coef) { for(int i=0;inr_class-1;i++) free(model_ptr->sv_coef[i]); } free(model_ptr->SV); model_ptr->SV = NULL; free(model_ptr->sv_coef); model_ptr->sv_coef = NULL; free(model_ptr->rho); model_ptr->rho = NULL; free(model_ptr->label); model_ptr->label= NULL; free(model_ptr->probA); model_ptr->probA = NULL; free(model_ptr->probB); model_ptr->probB= NULL; free(model_ptr->sv_indices); model_ptr->sv_indices = NULL; free(model_ptr->nSV); model_ptr->nSV = NULL; } void svm_free_and_destroy_model(svm_model** model_ptr_ptr) { if(model_ptr_ptr != NULL && *model_ptr_ptr != NULL) { svm_free_model_content(*model_ptr_ptr); free(*model_ptr_ptr); *model_ptr_ptr = NULL; } } void svm_destroy_param(svm_parameter* param) { free(param->weight_label); free(param->weight); } const char *svm_check_parameter(const svm_problem *prob, const svm_parameter *param) { // svm_type int svm_type = param->svm_type; if(svm_type != C_SVC && svm_type != NU_SVC && svm_type != ONE_CLASS && svm_type != EPSILON_SVR && svm_type != NU_SVR) return "unknown svm type"; // kernel_type, degree int kernel_type = param->kernel_type; if(kernel_type != LINEAR && kernel_type != POLY && kernel_type != RBF && kernel_type != SIGMOID && kernel_type != PRECOMPUTED) return "unknown kernel type"; if((kernel_type == POLY || kernel_type == RBF || kernel_type == SIGMOID) && param->gamma < 0) return "gamma < 0"; if(kernel_type == POLY && param->degree < 0) return "degree of polynomial kernel < 0"; // cache_size,eps,C,nu,p,shrinking if(param->cache_size <= 0) return "cache_size <= 0"; if(param->eps <= 0) return "eps <= 0"; if(svm_type == C_SVC || svm_type == EPSILON_SVR || svm_type == NU_SVR) if(param->C <= 0) return "C <= 0"; if(svm_type == NU_SVC || svm_type == ONE_CLASS || svm_type == NU_SVR) if(param->nu <= 0 || param->nu > 1) return "nu <= 0 or nu > 1"; if(svm_type == EPSILON_SVR) if(param->p < 0) return "p < 0"; if(param->shrinking != 0 && param->shrinking != 1) return "shrinking != 0 and shrinking != 1"; if(param->probability != 0 && param->probability != 1) return "probability != 0 and probability != 1"; if(param->probability == 1 && svm_type == ONE_CLASS) return "one-class SVM probability output not supported yet"; // check whether nu-svc is feasible if(svm_type == NU_SVC) { int l = prob->l; int max_nr_class = 16; int nr_class = 0; int *label = Malloc(int,max_nr_class); int *count = Malloc(int,max_nr_class); int i; for(i=0;iy[i]; int j; for(j=0;jnu*(n1+n2)/2 > min(n1,n2)) { free(label); free(count); return "specified nu is infeasible"; } } } free(label); free(count); } return NULL; } int svm_check_probability_model(const svm_model *model) { return ((model->param.svm_type == C_SVC || model->param.svm_type == NU_SVC) && model->probA!=NULL && model->probB!=NULL) || ((model->param.svm_type == EPSILON_SVR || model->param.svm_type == NU_SVR) && model->probA!=NULL); } void svm_set_print_string_function(void (*print_func)(const char *)) { if(print_func == NULL) svm_print_string = &print_string_stdout; else svm_print_string = print_func; } statistics-release-1.6.3/src/svm.h000066400000000000000000000102451456127120000170760ustar00rootroot00000000000000/* Copyright (C) 2021 Chih-Chung Chang and Chih-Jen Lin This file is part of the statistics package for GNU Octave. Permission granted by Chih-Jen Lin to the package maintainer to include this ile and double license under GPLv3 by means of personal communication. This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 3 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with this program; if not, see . */ #ifndef _LIBSVM_H #define _LIBSVM_H #define LIBSVM_VERSION 325 #ifdef __cplusplus extern "C" { #endif extern int libsvm_version; struct svm_node { int index; double value; }; struct svm_problem { int l; double *y; struct svm_node **x; }; enum { C_SVC, NU_SVC, ONE_CLASS, EPSILON_SVR, NU_SVR }; /* svm_type */ enum { LINEAR, POLY, RBF, SIGMOID, PRECOMPUTED }; /* kernel_type */ struct svm_parameter { int svm_type; int kernel_type; int degree; /* for poly */ double gamma; /* for poly/rbf/sigmoid */ double coef0; /* for poly/sigmoid */ /* these are for training only */ double cache_size; /* in MB */ double eps; /* stopping criteria */ double C; /* for C_SVC, EPSILON_SVR and NU_SVR */ int nr_weight; /* for C_SVC */ int *weight_label; /* for C_SVC */ double* weight; /* for C_SVC */ double nu; /* for NU_SVC, ONE_CLASS, and NU_SVR */ double p; /* for EPSILON_SVR */ int shrinking; /* use the shrinking heuristics */ int probability; /* do probability estimates */ }; // // svm_model // struct svm_model { struct svm_parameter param; /* parameter */ int nr_class; /* number of classes, = 2 in regression/one class svm */ int l; /* total #SV */ struct svm_node **SV; /* SVs (SV[l]) */ double **sv_coef; /* coefficients for SVs in decision functions (sv_coef[k-1][l]) */ double *rho; /* constants in decision functions (rho[k*(k-1)/2]) */ double *probA; /* pariwise probability information */ double *probB; int *sv_indices; /* sv_indices[0,...,nSV-1] are values in [1,...,num_traning_data] to indicate SVs in the training set */ /* for classification only */ int *label; /* label of each class (label[k]) */ int *nSV; /* number of SVs for each class (nSV[k]) */ /* nSV[0] + nSV[1] + ... + nSV[k-1] = l */ /* XXX */ int free_sv; /* 1 if svm_model is created by svm_load_model*/ /* 0 if svm_model is created by svm_train */ }; struct svm_model *svm_train(const struct svm_problem *prob, const struct svm_parameter *param); void svm_cross_validation(const struct svm_problem *prob, const struct svm_parameter *param, int nr_fold, double *target); int svm_save_model(const char *model_file_name, const struct svm_model *model); struct svm_model *svm_load_model(const char *model_file_name); int svm_get_svm_type(const struct svm_model *model); int svm_get_nr_class(const struct svm_model *model); void svm_get_labels(const struct svm_model *model, int *label); void svm_get_sv_indices(const struct svm_model *model, int *sv_indices); int svm_get_nr_sv(const struct svm_model *model); double svm_get_svr_probability(const struct svm_model *model); double svm_predict_values(const struct svm_model *model, const struct svm_node *x, double* dec_values); double svm_predict(const struct svm_model *model, const struct svm_node *x); double svm_predict_probability(const struct svm_model *model, const struct svm_node *x, double* prob_estimates); void svm_free_model_content(struct svm_model *model_ptr); void svm_free_and_destroy_model(struct svm_model **model_ptr_ptr); void svm_destroy_param(struct svm_parameter *param); const char *svm_check_parameter(const struct svm_problem *prob, const struct svm_parameter *param); int svm_check_probability_model(const struct svm_model *model); void svm_set_print_string_function(void (*print_func)(const char *)); #ifdef __cplusplus } #endif #endif /* _LIBSVM_H */ statistics-release-1.6.3/src/svm_model_octave.cc000066400000000000000000000214061456127120000217560ustar00rootroot00000000000000/* Copyright (C) 2022 Andreas Bertsatos Based on the Octave LIBSVM wrapper created by Alan Meeson (2014) based on an earlier version of the LIBSVM (3.18) library for MATLAB. Current implementation is based on LIBSVM 3.25 (2021) by Chih-Chung Chang and Chih-Jen Lin. This file is part of the statistics package for GNU Octave. This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 3 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with this program; if not, see . */ #include #include #include #include #include #include "svm.h" #define NUM_OF_RETURN_FIELD 11 #define Malloc(type,n) (type *)malloc((n)*sizeof(type)) static const char *field_names[] = { "Parameters", "nr_class", "totalSV", "rho", "Label", "sv_indices", "ProbA", "ProbB", "nSV", "sv_coef", "SVs" }; const char *model_to_octave_structure(octave_value_list &plhs, int num_of_feature, struct svm_model *model) { int i, j, n; double *ptr; octave_scalar_map osm_model; // Parameters ColumnVector cm_parameters (5); cm_parameters(0) = model->param.svm_type; cm_parameters(1) = model->param.kernel_type; cm_parameters(2) = model->param.degree; cm_parameters(3) = model->param.gamma; cm_parameters(4) = model->param.coef0; osm_model.assign("Parameters", cm_parameters); // nr_class osm_model.assign("nr_class", octave_value(model->nr_class)); // total SV osm_model.assign("totalSV", octave_value(model->l)); // rho n = model->nr_class*(model->nr_class-1)/2; ColumnVector cm_rho(n); for(i = 0; i < n; i++) { cm_rho(i) = model->rho[i]; } osm_model.assign("rho", cm_rho); // Label if(model->label) { ColumnVector cm_label(model->nr_class); for(i = 0; i < model->nr_class; i++) { cm_label(i) = model->label[i]; } osm_model.assign("Label", cm_label); } else { osm_model.assign("Label", ColumnVector(0)); } // sv_indices if(model->sv_indices) { ColumnVector cm_sv_indices(model->l); for(i = 0; i < model->l; i++) { cm_sv_indices(i) = model->sv_indices[i]; } osm_model.assign("sv_indices", cm_sv_indices); } else { osm_model.assign("sv_indices", ColumnVector(0)); } // probA if(model->probA != NULL) { ColumnVector cm_proba(n); for(i = 0; i < n; i++) { cm_proba(i) = model->probA[i]; } osm_model.assign("ProbA", cm_proba); } else { osm_model.assign("ProbA", ColumnVector(0)); } // probB if(model->probB != NULL) { ColumnVector cm_probb(n); for(i = 0; i < n; i++) { cm_probb(i) = model->probB[i]; } osm_model.assign("ProbB", cm_probb); } else { osm_model.assign("ProbB", ColumnVector(0)); } // nSV if(model->nSV) { ColumnVector cm_nsv(model->nr_class); for(i = 0; i < model->nr_class; i++) { cm_nsv(i) = model->nSV[i]; } osm_model.assign("nSV", cm_nsv); } else { osm_model.assign("nSV", ColumnVector(0)); } // sv_coef Matrix m_sv_coef(model->l, model->nr_class-1); for (i = 0; i < model->nr_class-1; i++) { for(j = 0; j < model->l; j++) { m_sv_coef(j,i) = model->sv_coef[i][j]; } } osm_model.assign("sv_coef", m_sv_coef); // SVs { int ir_index, nonzero_element; octave_idx_type *ir, *jc; //mxArray *pprhs[1], *pplhs[1]; if(model->param.kernel_type == PRECOMPUTED) { nonzero_element = model->l; num_of_feature = 1; } else { nonzero_element = 0; for(i = 0; i < model->l; i++) { j = 0; while(model->SV[i][j].index != -1) { nonzero_element++; j++; } } } // SV in column, easier accessing SparseMatrix sm_rhs = SparseMatrix((octave_idx_type)num_of_feature, (octave_idx_type)model->l, (octave_idx_type)nonzero_element); ir = sm_rhs.ridx(); jc = sm_rhs.cidx(); ptr = (double*) sm_rhs.data(); jc[0] = ir_index = 0; for(i = 0; i < model->l; i++) { if(model->param.kernel_type == PRECOMPUTED) { // make a (1 x model->l) matrix ir[ir_index] = 0; ptr[ir_index] = model->SV[i][0].value; ir_index++; jc[i+1] = jc[i] + 1; } else { int x_index = 0; while (model->SV[i][x_index].index != -1) { ir[ir_index] = model->SV[i][x_index].index - 1; ptr[ir_index] = model->SV[i][x_index].value; ir_index++, x_index++; } jc[i+1] = jc[i] + x_index; } } // transpose back to SV in row sm_rhs = sm_rhs.transpose(); osm_model.assign("SVs", sm_rhs); } /* return */ plhs(0) = osm_model; return NULL; } struct svm_model *octave_matrix_to_model(octave_scalar_map &octave_model, const char **msg) { int i, j, n, num_of_fields; double *ptr; int id = 0; struct svm_node *x_space; struct svm_model *model; model = Malloc(struct svm_model, 1); model->rho = NULL; model->probA = NULL; model->probB = NULL; model->label = NULL; model->sv_indices = NULL; model->nSV = NULL; model->free_sv = 1; // XXX //Parameters ColumnVector cm_parameters = octave_model.getfield("Parameters").column_vector_value(); model->param.svm_type = (int)cm_parameters(0); model->param.kernel_type = (int)cm_parameters(1); model->param.degree = (int)cm_parameters(2); model->param.gamma = cm_parameters(3); model->param.coef0 = cm_parameters(4); //nr_class model->nr_class = (int)octave_model.getfield("nr_class").int_value(); //total SV model->l = (int)octave_model.getfield("totalSV").int_value(); //rho n = model->nr_class * (model->nr_class-1)/2; model->rho = (double*) malloc(n*sizeof(double)); ColumnVector cm_rho = octave_model.getfield("rho").column_vector_value(); for(i = 0; i < n; i++) { model->rho[i] = cm_rho(i); } //label if (!octave_model.getfield("Label").isempty()) { model->label = (int*) malloc(model->nr_class*sizeof(int)); ColumnVector cm_label = octave_model.getfield("Label").column_vector_value(); for(i = 0; i < model->nr_class; i++) { model->label[i] = (int)cm_label(i); } } //sv_indices if (!octave_model.getfield("sv_indices").isempty()) { model->sv_indices = (int*) malloc(model->l*sizeof(int)); ColumnVector cv_svi = octave_model.getfield("sv_indices").column_vector_value(); for(i = 0; i < model->l; i++) { model->sv_indices[i] = (int)cv_svi(i); } } // probA if(!octave_model.getfield("ProbA").isempty()) { model->probA = (double*) malloc(n*sizeof(double)); ColumnVector cv_proba = octave_model.getfield("ProbA").column_vector_value(); for(i = 0; i < n; i++) { model->probA[i] = cv_proba(i); } } // probB if(!octave_model.getfield("ProbB").isempty()) { model->probB = (double*) malloc(n*sizeof(double)); ColumnVector cv_probb = octave_model.getfield("ProbB").column_vector_value(); for(i = 0; i < n; i++) { model->probB[i] = cv_probb(i); } } // nSV if(!octave_model.getfield("nSV").isempty()) { model->nSV = (int*) malloc(model->nr_class*sizeof(int)); ColumnVector cv_nsv = octave_model.getfield("nSV").column_vector_value(); for(i = 0; i < model->nr_class; i++) { model->nSV[i] = (int)cv_nsv(i); } } // sv_coef Matrix m_sv_coef = octave_model.getfield("sv_coef").matrix_value(); ptr = (double*) m_sv_coef.data(); model->sv_coef = (double**) malloc((model->nr_class-1)*sizeof(double)); for(i = 0; i < model->nr_class - 1; i++ ) { model->sv_coef[i] = (double*) malloc((model->l)*sizeof(double)); } for(i = 0; i < model->nr_class - 1; i++) { for(j = 0; j < model->l; j++) { model->sv_coef[i][j] = ptr[i*(model->l)+j];//m_sv_coef(i,j); } } // SV { int sr, sc, elements; int num_samples; octave_idx_type *ir, *jc; // transpose SV SparseMatrix sm_sv = octave_model.getfield("SVs").sparse_matrix_value(); sm_sv = sm_sv.transpose(); sr = (int)sm_sv.cols(); sc = (int)sm_sv.rows(); ptr = (double*)sm_sv.data(); ir = sm_sv.ridx(); jc = sm_sv.cidx(); num_samples = (int)sm_sv.nzmax(); elements = num_samples + sr; model->SV = (struct svm_node **) malloc(sr * sizeof(struct svm_node *)); x_space = (struct svm_node *)malloc(elements * sizeof(struct svm_node)); // SV is in column for(i = 0; i < sr; i++) { int low = (int)jc[i], high = (int)jc[i+1]; int x_index = 0; model->SV[i] = &x_space[low+i]; for(j = low; j < high; j++) { model->SV[i][x_index].index = (int)ir[j] + 1; model->SV[i][x_index].value = ptr[j]; x_index++; } model->SV[i][x_index].index = -1; } id++; } return model; } statistics-release-1.6.3/src/svm_model_octave.h000066400000000000000000000023771456127120000216260ustar00rootroot00000000000000/* Copyright (C) 2022 Andreas Bertsatos Based on the Octave LIBSVM wrapper created by Alan Meeson (2014) based on an earlier version of the LIBSVM (3.18) library for MATLAB. Current implementation is based on LIBSVM 3.25 (2021) by Chih-Chung Chang and Chih-Jen Lin. This file is part of the statistics package for GNU Octave. This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 3 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with this program; if not, see . */ #include #include #include "svm.h" const char *model_to_octave_structure(octave_value_list &plhs, int num_of_feature, struct svm_model *model); struct svm_model *octave_matrix_to_model(octave_scalar_map &octave_struct, const char **error_message); statistics-release-1.6.3/src/svmpredict.cc000066400000000000000000000327041456127120000206130ustar00rootroot00000000000000/* Copyright (C) 2022 Andreas Bertsatos Based on the Octave LIBSVM wrapper adapted by Alan Meeson (2014) based on an earlier version of the LIBSVM (3.18) library for MATLAB. Current implementation is based on LIBSVM 3.25 (2021) by Chih-Chung Chang and Chih-Jen Lin. This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 3 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with this program; if not, see . */ #include #include #include #include #include #include #include #include #include "svm.h" #include "svm_model_octave.h" #define CMD_LEN 2048 int print_null(const char *s,...) {return 0;} int (*info)(const char *fmt,...) = &printf; void read_sparse_instance(const SparseMatrix &args, int index, struct svm_node *x) { int i, j, low, high; octave_idx_type *ir, *jc; double *samples; ir = (octave_idx_type*)args.ridx(); jc = (octave_idx_type*)args.cidx(); samples = (double*)args.data(); // each column is one instance j = 0; low = (int)jc[index], high = (int)jc[index+1]; for(i=low;iparam.kernel_type == PRECOMPUTED) { // precomputed kernel requires dense matrix, so we make one t_data = args(1).matrix_value(); } else { //If it's a sparse matrix with a non PRECOMPUTED kernel, transpose it pplhs = args(1).sparse_matrix_value().transpose(); } } else { t_data = args(1).matrix_value(); } ptr_instance = (double*)t_data.data(); if(predict_probability) { if(svm_type==NU_SVR || svm_type==EPSILON_SVR) { info("Prob. model for test data: target value = predicted value + z,\nz: Laplace distribution e^(-|z|/sigma)/(2sigma),sigma=%g\n", svm_get_svr_probability(model)); } else { prob_estimates = (double*) malloc(nr_class*sizeof(double)); } } ColumnVector cv_predictions(testing_instance_number); tplhs(0) = cv_predictions; if(predict_probability) { // prob estimates are in plhs[2] if(svm_type == C_SVC || svm_type == NU_SVC) { Matrix m_pe(testing_instance_number, nr_class); tplhs(2) = m_pe; } else { Matrix m_pe(0,0); tplhs(2) = m_pe; } } else { // decision values are in plhs[2] if(svm_type == ONE_CLASS || svm_type == EPSILON_SVR || svm_type == NU_SVR || nr_class == 1) { Matrix m_pe(testing_instance_number, 1); tplhs(2) = m_pe; } else { Matrix m_pe(testing_instance_number, nr_class*(nr_class-1)/2); tplhs(2) = m_pe; } } ptr_predict_label = (double*)tplhs(0).column_vector_value().data(); ptr_prob_estimates = (double*)tplhs(2).matrix_value().data(); ptr_dec_values = (double*)tplhs(2).matrix_value().data(); x = (struct svm_node*)malloc((feature_number+1)*sizeof(struct svm_node)); for(instance_index=0;instance_indexparam.kernel_type != PRECOMPUTED) { read_sparse_instance(pplhs, instance_index, x); } else { for(i = 0; i < feature_number; i++) { x[i].index = i+1; x[i].value = ptr_instance[testing_instance_number*i+instance_index]; } x[feature_number].index = -1; } if(predict_probability) { if(svm_type == C_SVC || svm_type == NU_SVC) { predict_label = svm_predict_probability(model, x, prob_estimates); ptr_predict_label[instance_index] = predict_label; for(i = 0; i < nr_class; i++) { ptr_prob_estimates[instance_index + i * testing_instance_number] = prob_estimates[i]; } } else { predict_label = svm_predict(model,x); ptr_predict_label[instance_index] = predict_label; } } else { if(svm_type == ONE_CLASS || svm_type == EPSILON_SVR || svm_type == NU_SVR) { double res; predict_label = svm_predict_values(model, x, &res); ptr_dec_values[instance_index] = res; } else { double *dec_values = (double *) malloc(sizeof(double) * nr_class*(nr_class-1)/2); predict_label = svm_predict_values(model, x, dec_values); if(nr_class == 1) { ptr_dec_values[instance_index] = 1; } else { for(i = 0; i < (nr_class * (nr_class - 1)) / 2; i++) { ptr_dec_values[instance_index + i * testing_instance_number] = dec_values[i]; } } free(dec_values); } ptr_predict_label[instance_index] = predict_label; } if(predict_label == target_label) { ++correct; } error += (predict_label-target_label)*(predict_label-target_label); sump += predict_label; sumt += target_label; sumpp += predict_label*predict_label; sumtt += target_label*target_label; sumpt += predict_label*target_label; ++total; } if(svm_type==NU_SVR || svm_type==EPSILON_SVR) { info("Mean squared error = %g (regression)\n",error/total); info("Squared correlation coefficient = %g (regression)\n", ((total*sumpt-sump*sumt)*(total*sumpt-sump*sumt))/ ((total*sumpp-sump*sump)*(total*sumtt-sumt*sumt))); } else { info("Accuracy = %g%% (%d/%d) (classification)\n", (double)correct/total*100,correct,total); } // return accuracy, mean squared error, squared correlation coefficient ColumnVector cv_acc(3); ptr = (double*)cv_acc.data(); ptr[0] = (double)correct/total*100; ptr[1] = error/total; ptr[2] = ((total*sumpt-sump*sumt)*(total*sumpt-sump*sumt))/ ((total*sumpp-sump*sump)*(total*sumtt-sumt*sumt)); tplhs(1) = cv_acc; free(x); if(prob_estimates != NULL) { free(prob_estimates); } switch(nlhs) { case 3: plhs(2) = tplhs(2); plhs(1) = tplhs(1); case 1: case 0: plhs(0) = tplhs(0); } } DEFUN_DLD (svmpredict, args, nargout, "-*- texinfo -*- \n\n\ @deftypefn {statistics} {@var{predicted_label} =} svmpredict (@var{labels}, @var{data}, @var{model})\n\ @deftypefnx {statistics} {@var{predicted_label} =} svmpredict (@var{labels}, @var{data}, @var{model}, ""libsvm_options"")\n\ @deftypefnx {statistics} {[@var{predicted_label}, @var{accuracy}, @var{decision_values}] =} svmpredict (@var{labels}, @var{data}, @var{model}, ""libsvm_options"")\n\ @deftypefnx {statistics} {[@var{predicted_label}, @var{accuracy}, @var{prob_estimates}] =} svmpredict (@var{labels}, @var{data}, @var{model}, ""libsvm_options"")\n\ \n\ \n\ This function predicts new labels from a testing instance matrtix based on an \ SVM @var{model} created with @code{svmtrain}. \ \n\ \n\ @itemize \n\ @item @var{labels} : An m by 1 vector of prediction labels. If labels \ of test data are unknown, simply use any random values. (type must be double) \ \n\ \n\ @item @var{data} : An m by n matrix of m testing instances with n features. \ It can be dense or sparse. (type must be double) \ \n\ \n\ @item @var{model} : The output of @code{svmtrain} function. \ \n\ \n\ @item @code{libsvm_options} : A string of testing options in the same format \ as that of LIBSVM. \ \n\ \n\ @end itemize \ \n\ \n\ @code{libsvm_options} :\n\ \n\ @itemize \n\ @item @code{-b} : probability_estimates; whether to predict probability \ estimates. For one-class SVM only 0 is supported.\n\ \n\ @end itemize \ \n\ @multitable @columnfractions 0.1 0.1 0.8 \n\ @item @tab 0 @tab return decision values. (default) \n\ \n\ @item @tab 1 @tab return probability estimates. \ \n\ @end multitable \ \n\ \n\ @itemize \n\ @item @code{-q} : quiet mode. (no outputs) \ \n\ @end itemize \ \n\ \n\ The @code{svmpredict} function has three outputs. The first one, \ @var{predicted_label}, is a vector of predicted labels. The second output, \ @var{accuracy}, is a vector including accuracy (for classification), mean \ squared error, and squared correlation coefficient (for regression). The \ third is a matrix containing decision values or probability estimates \ (if @code{-b 1}' is specified). If @math{k} is the number of classes in \ training data, for decision values, each row includes results of predicting \ @math{k(k-1)/2} binary-class SVMs. For classification, @math{k = 1} is a \ special case. Decision value +1 is returned for each testing instance, \ instead of an empty vector. For probabilities, each row contains @math{k} \ values indicating the probability that the testing instance is in each class. \ Note that the order of classes here is the same as @code{Label} field in the \ @var{model} structure. \ \n\ @end deftypefn") { int nlhs = nargout; int nrhs = args.length(); octave_value_list plhs(nlhs); int prob_estimate_flag = 0; struct svm_model *model; info = &print_null; if(nlhs == 2 || nlhs > 3) { error ("svmpredict: wrong number of output arguments."); } if(nrhs > 4 || nrhs < 3) { error ("svmpredict: wrong number of input arguments."); } if(!args(0).is_double_type() || !args(1).is_double_type()) { error ("svmpredict: label vector and instance matrix must be double."); } if(args(2).isstruct()) { const char *error_msg; // parse options if(nrhs==4) { int i, argc = 1; char cmd[CMD_LEN], *argv[CMD_LEN/2]; // put options in argv[] strncpy(cmd, args(3).string_value().c_str(), CMD_LEN); if((argv[argc] = strtok(cmd, " ")) != NULL) { while((argv[++argc] = strtok(NULL, " ")) != NULL); } for(i=1;i=argc) && argv[i-1][1] != 'q') { fake_answer(nlhs, plhs); return plhs; } switch(argv[i-1][1]) { case 'b': prob_estimate_flag = atoi(argv[i]); break; case 'q': i--; info = &print_null; break; default: printf("svmpredict: unknown option: -%c\n", argv[i-1][1]); fake_answer(nlhs, plhs); return plhs; } } } octave_scalar_map osm_model = args(2).scalar_map_value(); model = octave_matrix_to_model(osm_model, &error_msg); if (model == NULL) { printf("svmpredict: can't read model: %s\n", error_msg); fake_answer(nlhs, plhs); return plhs; } if(prob_estimate_flag) { if(svm_check_probability_model(model)==0) { svm_free_and_destroy_model(&model); error ("svmpredict: model does not support probabiliy estimates.\n"); } } else { if(svm_check_probability_model(model)!=0) info("Model supports probability estimates, but disabled in prediction.\n"); } predict(nlhs, plhs, args, model, prob_estimate_flag); // destroy model svm_free_and_destroy_model(&model); } else { error ("svmpredict: model should be a struct array."); } return plhs; } /* %!test %! [L, D] = libsvmread (file_in_loadpath ("heart_scale.dat")); %! model = svmtrain (L, D, '-c 1 -g 0.07'); %! [predict_label, accuracy, dec_values] = svmpredict (L, D, model); %! assert (size (predict_label), size (dec_values)); %! assert (accuracy, [86.666, 0.533, 0.533]', [1e-3, 1e-3, 1e-3]'); %! assert (dec_values(1), 1.225836001973273, 1e-14); %! assert (dec_values(2), -0.3212992933043805, 1e-14); %! assert (predict_label(1), 1); %!shared L, D, model %! [L, D] = libsvmread (file_in_loadpath ("heart_scale.dat")); %! model = svmtrain (L, D, '-c 1 -g 0.07'); %!error ... %! [p, a] = svmpredict (L, D, model); %!error p = svmpredict (L, D); %!error ... %! p = svmpredict (single (L), D, model); %!error p = svmpredict (L, D, 123); */ statistics-release-1.6.3/src/svmtrain.cc000066400000000000000000000364231456127120000203000ustar00rootroot00000000000000/* Copyright (C) 2022 Andreas Bertsatos Based on the Octave LIBSVM wrapper adapted by Alan Meeson (2014) based on an earlier version of the LIBSVM (3.18) library for MATLAB. Current implementation is based on LIBSVM 3.25 (2021) by Chih-Chung Chang and Chih-Jen Lin. This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 3 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with this program; if not, see . */ #include #include #include #include #include #include #include #include #include "svm.h" #include "svm_model_octave.h" #define CMD_LEN 2048 #define Malloc(type,n) (type *)malloc((n)*sizeof(type)) void print_null(const char *s) {} //void print_string_octave(const char *s) {printf(s);} // svm arguments struct svm_parameter param; // set by parse_command_line struct svm_problem prob; // set by read_problem struct svm_model *model; struct svm_node *x_space; int cross_validation; int nr_fold; double do_cross_validation() { int i; int total_correct = 0; double total_error = 0; double sumv = 0, sumy = 0, sumvv = 0, sumyy = 0, sumvy = 0; double *target = Malloc(double,prob.l); double retval = 0.0; svm_cross_validation(&prob,¶m,nr_fold,target); if(param.svm_type == EPSILON_SVR || param.svm_type == NU_SVR) { for(i=0;i 2) { // put options in argv[] strncpy(cmd, args(2).string_value().c_str(), CMD_LEN); //mxGetString(args[2], cmd, mxGetN(args[2]) + 1); if((argv[argc] = strtok(cmd, " ")) != NULL) while((argv[++argc] = strtok(NULL, " ")) != NULL) ; } // parse options for(i=1;i=argc && argv[i-1][1] != 'q') // since option -q has no parameter return 1; switch(argv[i-1][1]) { case 's': param.svm_type = atoi(argv[i]); break; case 't': param.kernel_type = atoi(argv[i]); break; case 'd': param.degree = atoi(argv[i]); break; case 'g': param.gamma = atof(argv[i]); break; case 'r': param.coef0 = atof(argv[i]); break; case 'n': param.nu = atof(argv[i]); break; case 'm': param.cache_size = atof(argv[i]); break; case 'c': param.C = atof(argv[i]); break; case 'e': param.eps = atof(argv[i]); break; case 'p': param.p = atof(argv[i]); break; case 'h': param.shrinking = atoi(argv[i]); break; case 'b': param.probability = atoi(argv[i]); break; case 'q': print_func = &print_null; i--; break; case 'v': cross_validation = 1; nr_fold = atoi(argv[i]); if(nr_fold < 2) { printf("n-fold cross validation: n must >= 2\n"); return 1; } break; case 'w': ++param.nr_weight; param.weight_label = (int *)realloc(param.weight_label,sizeof(int)*param.nr_weight); param.weight = (double *)realloc(param.weight,sizeof(double)*param.nr_weight); param.weight_label[param.nr_weight-1] = atoi(&argv[i-1][2]); param.weight[param.nr_weight-1] = atof(argv[i]); break; default: printf("svmtrain: unknown option -%c\n", argv[i-1][1]); return 1; } } svm_set_print_string_function(print_func); return 0; } // read in a problem (in svmlight format) int read_problem_dense(ColumnVector &label_vec, Matrix &instance_mat) { int i, j, k; int elements, max_index, sc, label_vector_row_num; double *samples, *labels; prob.x = NULL; prob.y = NULL; x_space = NULL; labels = (double*)label_vec.data();//mxGetPr(label_vec); samples = (double*)instance_mat.data(); sc = (int)instance_mat.cols(); elements = 0; // the number of instance prob.l = (int)instance_mat.rows(); label_vector_row_num = (int)label_vec.rows(); if(label_vector_row_num!=prob.l) { printf("svmtrain: length of label vector does not match # of instances.\n"); return -1; } if(param.kernel_type == PRECOMPUTED) elements = prob.l * (sc + 1); else { for(i = 0; i < prob.l; i++) { for(k = 0; k < sc; k++) if(samples[k * prob.l + i] != 0) elements++; // count the '-1' element elements++; } } prob.y = Malloc(double,prob.l); prob.x = Malloc(struct svm_node *,prob.l); x_space = Malloc(struct svm_node, elements); max_index = sc; j = 0; for(i = 0; i < prob.l; i++) { prob.x[i] = &x_space[j]; prob.y[i] = labels[i]; for(k = 0; k < sc; k++) { if(param.kernel_type == PRECOMPUTED || samples[k * prob.l + i] != 0) { x_space[j].index = k + 1; x_space[j].value = samples[k * prob.l + i]; j++; } } x_space[j++].index = -1; } if(param.gamma == 0 && max_index > 0) param.gamma = 1.0/max_index; if(param.kernel_type == PRECOMPUTED) for(i=0;i max_index) { printf("svmtrain: wrong input format: sample_serial_number out of range\n"); return -1; } } return 0; } int read_problem_sparse(ColumnVector &label_vec, SparseMatrix &instance_mat) { int i, j, k, low, high; octave_idx_type *ir, *jc; int elements, max_index, num_samples, label_vector_row_num; double *samples, *labels; // transposed instance sparse matrix SparseMatrix instance_mat_col = instance_mat.transpose(); prob.x = NULL; prob.y = NULL; x_space = NULL; // each column is one instance labels = (double*)label_vec.data(); samples = (double*)instance_mat_col.data(); ir = (octave_idx_type*)instance_mat_col.ridx(); jc = (octave_idx_type*)instance_mat_col.cidx(); num_samples = (int)instance_mat_col.nzmax(); // the number of instance prob.l = (int)instance_mat_col.cols(); label_vector_row_num = (int)label_vec.rows(); if(label_vector_row_num!=prob.l) { printf("svmtrain: length of label vector does not match # of instances.\n"); return -1; } elements = num_samples + prob.l; max_index = (int)instance_mat_col.rows(); prob.y = Malloc(double,prob.l); prob.x = Malloc(struct svm_node *,prob.l); x_space = Malloc(struct svm_node, elements); j = 0; for(i=0;i 0) { param.gamma = 1.0/max_index; } return 0; } static void fake_answer(int nlhs, octave_value_list &plhs) { int i; for(i=0;i 1) { error ("svmtrain: wrong number of output arguments."); } // Transform the input Matrix to libsvm format if(nrhs > 1 && nrhs < 4) { int err; if(!args(0).is_double_type() || !args(1).is_double_type()) { error ("svmtrain: label vector and instance matrix must be double."); } if(parse_command_line(nrhs, args, NULL)) { svm_destroy_param(¶m); error ("svmtrain: wrong values in parameter string."); } if(args(1).issparse()) { if(param.kernel_type == PRECOMPUTED) { // precomputed kernel requires dense matrix, so we make one ColumnVector cv_lab = args(0).column_vector_value(); Matrix m_dat = args(1).matrix_value(); err = read_problem_dense(cv_lab, m_dat); } else { ColumnVector cv_lab = args(0).column_vector_value(); SparseMatrix m_dat = args(1).sparse_matrix_value(); err = read_problem_sparse(cv_lab, m_dat); } } else { ColumnVector cv_lab = args(0).column_vector_value(); Matrix m_dat = args(1).matrix_value(); err = read_problem_dense(cv_lab, m_dat); } // svmtrain's original code error_msg = svm_check_parameter(&prob, ¶m); if(err || error_msg) { if (error_msg != NULL) { printf("svmtrain: %s\n", error_msg); } svm_destroy_param(¶m); free(prob.y); free(prob.x); free(x_space); fake_answer(nlhs, plhs); return plhs; } if(cross_validation) { double ptr = do_cross_validation(); plhs(0) = octave_value(ptr); } else { int nr_feat = (int)args(1).matrix_value().cols(); const char *error_msg; model = svm_train(&prob, ¶m); error_msg = model_to_octave_structure(plhs, nr_feat, model); if(error_msg) { printf("svmtrain: can't convert libsvm model to matrix structure: %s\n", error_msg); } svm_free_and_destroy_model(&model); } svm_destroy_param(¶m); free(prob.y); free(prob.x); free(x_space); return plhs; } else { error ("svmtrain: wrong number of input arguments."); } } /* %!test %! [L, D] = libsvmread (file_in_loadpath ("heart_scale.dat")); %! model = svmtrain(L, D, '-c 1 -g 0.07'); %! [predict_label, accuracy, dec_values] = svmpredict(L, D, model); %! assert (isstruct (model), true); %! assert (isfield (model, "Parameters"), true); %! assert (model.totalSV, 130); %! assert (model.nr_class, 2); %! assert (size (model.Label), [2, 1]); %!shared L, D %! [L, D] = libsvmread (file_in_loadpath ("heart_scale.dat")); %!error [L, D] = svmtrain (L, D); %!error ... %! model = svmtrain (single (L), D); %!error ... %! model = svmtrain (L, D, "", ""); */