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princomp

Principal components analysis

Syntax

[facpr,comprinc,lambda,tsquare] = princomp(x,eco)

Arguments

x

is a n-by-p (n individuals, p variables) real matrix.

eco

a boolean, use to allow economy size singular value decomposition.

facpr

A p-by-p matrix. It contains the principal factors: eigenvectors of the correlation matrix V.

comprinc

a n-by-p matrix. It contains the principal components. Each column of this 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'_i M^(-1) u_i=1

lambda

is a p column vector. It contains the eigenvalues of V, where V is the correlation matrix.

tsquare

a n column vector. It contains the Hotelling's T^2 statistic for each data point.

Description

This function performs "principal component analysis" on the n-by-p data matrix x.

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.

To compute principal component analysis with standardized variables may use princomp(wcenter(x,1)) or use the pca function.

Examples

a=rand(100,10,'n');
[facpr,comprinc,lambda,tsquare] = princomp(a);

• wcenter — center and weight
• pca — Computes principal components analysis with standardized variables

Bibliography

Saporta, Gilbert, Probabilites, Analyse des Donnees et Statistique, Editions Technip, Paris, 1990.