# princomp

Principal components analysis

## Calling Sequence

    [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);

## See Also

- wcenter
- pca

## Bibliography

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