NAME

       math::PCA - Package for Principal Component Analysis

SYNOPSIS

       package require Tcl  ?8.6?

       package require math::linearalgebra  1.0

       ::math::PCA::createPCA data ?args?

       $pca using ?number?|?-minproportion value?

       $pca eigenvectors ?option?

       $pca eigenvalues ?option?

       $pca proportions ?option?

       $pca approximate observation

       $pca approximatOriginal

       $pca scores observation

       $pca distance observation

       $pca qstatistic observation ?option?

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DESCRIPTION

       The  PCA  package  provides  a  means  to  perform principal components analysis in Tcl, using an object-
       oriented  technique  as  facilitated  by  TclOO.  It   actually   defines   a   single   public   method,
       ::math::PCA::createPCA,  which  constructs  an  object  based  on the data that are passed to perform the
       actual analysis.

       The methods of the PCA objects that are created with this command allow  one  to  examine  the  principal
       components,  to  approximate  (new) observations using all or a selected number of components only and to
       examine the properties of the components and the statistics of the approximations.

       The package has been modelled after the PCA example provided by the original linear algebra package by Ed
       Hume.

COMMANDS

       The math::PCA package provides one public command:

       ::math::PCA::createPCA data ?args?
              Create  a  new  object,  based  on  the data that are passed via the data argument.  The principal
              components may be  based  on  either  correlations  or  covariances.   All  observations  will  be
              normalised according to the mean and standard deviation of the original data.

              list data
                     - A list of observations (see the example below).

              list args
                     -  A  list  of  key-value  pairs  defining  the  options.  Currently there is only one key:
                     -covariances. This indicates if covariances are to be used (if the value is 1)  or  instead
                     correlations (value is 0). The default is to use correlations.

       The PCA object that is created has the following methods:

       $pca using ?number?|?-minproportion value?
              Set  the  number  of  components  to  be used in the analysis (the number of retained components).
              Returns the number of components, also if no argument is given.

              int number
                     - The number of components to be retained

              double value
                     - Select the number of components based on the minimum  proportion  of  variation  that  is
                     retained by them. Should be a value between 0 and 1.

       $pca eigenvectors ?option?
              Return the eigenvectors as a list of lists.

              string option
                     -  By default only the retained components are returned.  If all eigenvectors are required,
                     use the option -all.

       $pca eigenvalues ?option?
              Return the eigenvalues as a list of lists.

              string option
                     - By default only the  eigenvalues  of  the  retained  components  are  returned.   If  all
                     eigenvalues are required, use the option -all.

       $pca proportions ?option?
              Return  the proportions for all components, that is, the amount of variations that each components
              can explain.

       $pca approximate observation
              Return an approximation of the observation based on the retained components

              list observation
                     - The values for the observation.

       $pca approximatOriginal
              Return an approximation of the original data, using the retained components. It is  a  convenience
              method that works on the complete set of original data.

       $pca scores observation
              Return the scores per retained component for the given observation.

              list observation
                     - The values for the observation.

       $pca distance observation
              Return  the  distance between the given observation and its approximation. (Note: this distance is
              based on the normalised vectors.)

              list observation
                     - The values for the observation.

       $pca qstatistic observation ?option?
              Return the Q statistic, basically the square of the distance, for the given observation.

              list observation
                     - The values for the observation.

              string option
                     - If the observation is part of the original data, you may want  to  use  the  corrected  Q
                     statistic. This is achieved with the option "-original".

EXAMPLE

       TODO: NIST example

BUGS, IDEAS, FEEDBACK

       This  document,  and  the package it describes, will undoubtedly contain bugs and other problems.  Please
       report such in the category PCA of the Tcllib  Trackers  [http://core.tcl.tk/tcllib/reportlist].   Please
       also report any ideas for enhancements you may have for either package and/or documentation.

       When proposing code changes, please provide unified diffs, i.e the output of diff -u.

       Note  further  that  attachments  are strongly preferred over inlined patches. Attachments can be made by
       going to the Edit form of the ticket immediately after its creation, and then using the left-most  button
       in the secondary navigation bar.

KEYWORDS

       PCA, math, statistics, tcl

CATEGORY

       Mathematics