genpca {GeoXp}R Documentation

Generalized Principal Component Analysis (PCA)

Description

The function `genpca' computes a generalized Principal Component Analysis (PCA). It calculates the principal components, the coordinates of the variables and in these principals components axes and the inertia of these principal components.

Usage

genpca(data, w=rep(1/nrow(data),length=nrow(data)), 
m=diag(ncol(data)), center=NULL, reduc=TRUE)

Arguments

data matrix $n times p$
w vector of size n of weight (by default : $weight=t(1/n,...,1/n)$)
m matrix $p times p$ (by default : metric=Identity matrix)
center boolean. if TRUE, centered PCA (by default : center=TRUE)
reduc boolean. if TRUE, reduced PCA (by default : reduce=TRUE)

Details

Let

W=diag(w)

x=data=(x_1',...,x_n')'

with

x_i=(x_i^1,...,x_i^p)


Let

1_n=(1,...,1)'

with n rows and :

1_p=(1,...,1)'

with p rows. Normalization of weight :

w_i=frac{w_i}{sum_iw_i}

Vector of means :

bar{x}=(bar{x^1},...,bar{x^p})'

with:

bar{x^j}=sum_iw_ix_i^j


If center=True,

x_c=x-1_nbar{x}'

Standart deviation :

(σ^j)^2=sum_iw_i(x_i^j)^2-(bar{x^j})^2

Σ=diag((σ^1)^2,...,(σ^p)^2)'

If reduc=True :

x_{cr}=x_c times Σ^{-1/2}

Variance-Covariance matrix:

C=x_{cr}'Wx_{cr}

Cholesky decomposition : M=LL' where M=m
Let

C_l=LCL'

Let U and D as :

C_lU=UD

with D=diag(λ_1,...,λ_p)
Let

V=L'U


Then :
Coordinates of individuals in the principals components basis :

CC=x_{cr}V

Coordinates of variables in principals components :

VC=CVD^{-1/2}

Inertia :

I=D1_p

Value

Returns `inertia' vector of size $p$ with percent of inertia of each component (corresponding to $I$), `casecoord' matrix $n times p$ (corresponding to matrix $CC$), `varcoord' matrix $p times p$ (corresponding to matrix $VC$).

Author(s)

Thomas-Agnan C., Aragon Y., Ruiz-Gazen A., Laurent T., Robidou L.

References

Aragon Yves, Perrin Olivier, Ruiz-Gazen Anne, Thomas-Agnan Christine (2008), ``Statistique et Econométrie pour données géoréférencées : modèles et études de cas''

Caussinus H., Fekri M., Hakam S., Ruiz-Gazen A. (2003) , ``A monitoring display of Multivariate Outliers'', Computational Statistics and Data Analysis, vol. 44, 1-2, 237-252.

See Also

clustermap,pcamap


[Package GeoXp version 1.3 Index]