genpca: 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 $n x p$

vector of size n of weight (by default : $weight=t(1/n,...,1/n)$)

matrix $p x p$ (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)

Value

Returns `inertia' vector of size p with percent of inertia of each component (corresponding to I), `casecoord' matrix $n x p$ (corresponding to matrix CC), `varcoord' matrix $p x n$ (corresponding to matrix VC0).

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, $$x_c=x-1_n\bar{x}'$$ Standart deviation : $$(\sigma^j)^2=\sum_iw_i(x_i^j)^2-(\bar{x^j})^2$$ $$\Sigma=diag((\sigma^1)^2,...,(\sigma^p)^2)'$$ If reduc=True : $$x_{cr}=x_c \times \Sigma^{-1/2}$$ 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 $D=diag(lambda_1,...,lambda_p)$ 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 : $$I=D1_p$$

References

Thibault Laurent, Anne Ruiz-Gazen, Christine Thomas-Agnan (2012), GeoXp: An R Package for Exploratory Spatial Data Analysis. Journal of Statistical Software, 47(2), 1-23.