## Name

princomp — Principal components analysis

## Calling Sequence

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

## Parameters

- 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);

## Bibliography

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