# 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);

### See also

### Bibliography

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

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