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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 matrixV
.- 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 conditionu'_i M^(-1) u_i=1
- lambda
is a
p
column vector. It contains the eigenvalues ofV
, whereV
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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