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princomp

Principal components analysis

Syntax

[facpr,comprinc,lambda,tsquare, explained, mu] = 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.

explained

a column vector of length "number of components". The percentage of variance explained by each principal component.

mu

a row vector of length p. The estimated mean of each variable of x.

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);
x = [1 2 1;2 1 3; 3 2 3]
[facpr, comprinc, lambda, tsquare, explained, mu] = princomp(x, %t);
comprinc * facpr' + ones(3, 1) * mu // == x

See also

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

Bibliography

Saporta, Gilbert, Probabilités, Analyse des Données et Statistique, Editions Technip, Paris, 1990.

History

ВерсияОписание
2024.1.0 princomp now returns the percentage of the variance explained by each principal component and the estimated mean of each variable of x.
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<< pca Multivariate Correl Regress PCA reglin >>

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