# pca

Computes principal components analysis with standardized variables

## Calling Sequence

    [lambda,facpr,comprinc] = pca(x)

## Arguments

x

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

  Note that `pca` center and normalize the columns of `x` to produce principal components analysis with standardized variables.

lambda

  is a p x 2 numerical matrix. In the first column we find the eigenvalues of V, where V is the correlation p x p 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`.

comprinc

  are the principal components. Each column (c_i=Xu_i) of this n x n 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`

## Description

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.

Warning, the graphical part of the old version of `pca` has been removed. It can now be performed using the show_pca function.

## Examples

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

## See Also

- show_pca
- princomp

## Bibliography

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