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.TH "math::PCA" n 1\&.1 tcllib "Principal Components Analysis"
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.SH NAME
math::PCA \- Package for Principal Component Analysis
.SH SYNOPSIS
package require \fBTcl ?8\&.6 9?\fR
.sp
package require \fBmath::linearalgebra 1\&.1\fR
.sp
\fB::math::PCA::createPCA\fR \fIdata\fR ?args?
.sp
\fB$pca using\fR ?number?|?-minproportion value?
.sp
\fB$pca eigenvectors\fR ?option?
.sp
\fB$pca eigenvalues\fR ?option?
.sp
\fB$pca proportions\fR ?option?
.sp
\fB$pca approximate\fR \fIobservation\fR
.sp
\fB$pca approximatOriginal\fR
.sp
\fB$pca scores\fR \fIobservation\fR
.sp
\fB$pca distance\fR \fIobservation\fR
.sp
\fB$pca qstatistic\fR \fIobservation\fR ?option?
.sp
.BE
.SH DESCRIPTION
.PP
The PCA package provides a means to perform principal components analysis
in Tcl, using an object-oriented technique as facilitated by TclOO\&. It
actually defines a single public method, \fI::math::PCA::createPCA\fR,
which constructs an object based on the data that are passed to perform
the actual analysis\&.
.PP
The methods of the PCA objects that are created with this command allow one
to examine the principal components, to approximate (new) observations
using all or a selected number of components only and to examine the
properties of the components and the statistics of the approximations\&.
.PP
The package has been modelled after the PCA example provided by the
original linear algebra package by Ed Hume\&.
.SH COMMANDS
The \fImath::PCA\fR package provides one public command:
.TP
\fB::math::PCA::createPCA\fR \fIdata\fR ?args?
Create a new object, based on the data that are passed via the \fIdata\fR argument\&.
The principal components may be based on either correlations or covariances\&.
All observations will be normalised according to the mean and standard deviation
of the original data\&.
.RS
.TP
list \fIdata\fR
- A list of observations (see the example below)\&.
.TP
list \fIargs\fR
- A list of key-value pairs defining the options\&. Currently there is
only one key: \fI-covariances\fR\&. This indicates if covariances are to be used
(if the value is 1) or instead correlations (value is 0)\&. The default is to use
correlations\&.
.RE
.PP
The PCA object that is created has the following methods:
.TP
\fB$pca using\fR ?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\&.
.RS
.TP
int \fInumber\fR
- The number of components to be retained
.TP
double \fIvalue\fR
- 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\&.
.RE
.TP
\fB$pca eigenvectors\fR ?option?
Return the eigenvectors as a list of lists\&.
.RS
.TP
string \fIoption\fR
- By default only the \fIretained\fR components are returned\&.
If all eigenvectors are required, use the option \fI-all\fR\&.
.RE
.TP
\fB$pca eigenvalues\fR ?option?
Return the eigenvalues as a list of lists\&.
.RS
.TP
string \fIoption\fR
- By default only the eigenvalues of the \fIretained\fR components are returned\&.
If all eigenvalues are required, use the option \fI-all\fR\&.
.RE
.TP
\fB$pca proportions\fR ?option?
Return the proportions for all components, that is, the amount of variations that each
components can explain\&.
.TP
\fB$pca approximate\fR \fIobservation\fR
Return an approximation of the observation based on the retained components
.RS
.TP
list \fIobservation\fR
- The values for the observation\&.
.RE
.TP
\fB$pca approximatOriginal\fR
Return an approximation of the original data, using the retained components\&. It is
a convenience method that works on the complete set of original data\&.
.TP
\fB$pca scores\fR \fIobservation\fR
Return the scores per retained component for the given observation\&.
.RS
.TP
list \fIobservation\fR
- The values for the observation\&.
.RE
.TP
\fB$pca distance\fR \fIobservation\fR
Return the distance between the given observation and its approximation\&. (Note:
this distance is based on the normalised vectors\&.)
.RS
.TP
list \fIobservation\fR
- The values for the observation\&.
.RE
.TP
\fB$pca qstatistic\fR \fIobservation\fR ?option?
Return the Q statistic, basically the square of the distance, for the given observation\&.
.RS
.TP
list \fIobservation\fR
- The values for the observation\&.
.TP
string \fIoption\fR
- 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"\&.
.RE
.PP
.SH EXAMPLE
TODO: NIST example
.SH "BUGS, IDEAS, FEEDBACK"
This document, and the package it describes, will undoubtedly contain
bugs and other problems\&.
Please report such in the category \fIPCA\fR of the
\fITcllib Trackers\fR [http://core\&.tcl\&.tk/tcllib/reportlist]\&.
Please also report any ideas for enhancements you may have for either
package and/or documentation\&.
.PP
When proposing code changes, please provide \fIunified diffs\fR,
i\&.e the output of \fBdiff -u\fR\&.
.PP
Note further that \fIattachments\fR are strongly preferred over
inlined patches\&. Attachments can be made by going to the \fBEdit\fR
form of the ticket immediately after its creation, and then using the
left-most button in the secondary navigation bar\&.
.SH KEYWORDS
PCA, math, statistics, tcl
.SH CATEGORY
Mathematics