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// Scilab ( http://www.scilab.org/ ) - This file is part of Scilab
// Copyright (C) 2000 - INRIA - Carlos Klimann
//
// This file must be used under the terms of the CeCILL.
// This source file is licensed as described in the file COPYING, which
// you should have received as part of this distribution. The terms
// are also available at
// http://www.cecill.info/licences/Licence_CeCILL_V2-en.txt
//
function [lambda,facpr,comprinc]=pca(x)
//
//This function performs several computations known as
//"principal component analysis".
//
//The idea behind this method is to represent in an
//approximative manner a cluster of n individuals in a
//smaller dimensional subspace. In order to do that it
//projects the cluster onto a subspace. The choice of the
//k-dimensional projection subspace is made in such a way
//that the distances in the projection have a minimal
//deformation: we are looking for a k-dimensional subspace
//such that the squares of the distances in the projection
//is as big as possible (in fact in a projection,
//distances can only stretch). In other words, inertia of
//the projection onto the k dimensional subspace must be
//maximal.
//
//x is a nxp (n individuals, p variables) real matrix.
//
//lambda is a px2 numerical matrix. In the first column we
//find the eigenvalues of V, where V is the correlation
//pxp matrix and in the second column are the ratios of
//the corresponding eigenvalue over the sum of
//eigenvalues.
//
//facpr are the principal factors: eigenvectors of V.
//Each column is an eigenvector element of the dual of
//R^p. Is an orthogonal matrix.
//
//comprinc are the principal components. Each column
//(c_i=Xu_i) of this nxn matrix is the M-orthogonal
//projection of individuals onto principal axis. Each one
//of this columns is a linear combination of the variables
//x1, ...,xp with maximum variance under condition
//u'_iM^(-1)u_i=1.
//
//Verification: comprinc*facpr=x
//
//References: Saporta, Gilbert, Probabilites, Analyse des
//Donnees et Statistique, Editions Technip, Paris, 1990.
//
//author: carlos klimann
//
//date: 2002-02-05
//commentary fixed 2003-19-24 ??
// update disable graphics output Allan CORNET 2008
// request user
// splitted in printcomp (matlab compatibility) and show_pca Serge Steer
// 2008
if x==[] then
lambda=%nan;
facpr=%nan;
comprinc=%nan;
return;
end
[facpr,comprinc,lambda]=princomp(wcenter(x,1))
lambda(:,2)=lambda(:,1)/sum(lambda(:,1))
endfunction
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