Canonical analysis of several sets with another set

Description

Relative proximities of several subsets of variables Yj with another set X. SUCCESSIVE SOLUTIONS

Usage

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concorcano(x,y,py,r)

Arguments

x

is a n x p matrix of p centered variables

y

is a n x q matrix of q centered variables

py

is a row vector which contains the numbers qi, i=1,...,ky, of the ky subsets yi of y : ∑_i q_i=sum(py)=q. py is the partition vector of y

r

is the wanted number of successive solutions

Details

The first solution calculates a standardized canonical component cx[,1] of x associated to ky standardized components cyi[,1] of yi by maximizing ∑_i ρ(cx[,1],cy_i[,1])^2.

The second solution is obtained from the same criterion, with ky orthogonality constraints for having rho(cyi[,1],cyi[,2])=0 (that implies rho(cx[,1],cx[,2])=0). For each of the 1+ky sets, the r canonical components are 2 by 2 zero correlated.

The ky matrices (cx)'*cyi are triangular.

This function uses concor function.

Value

list with following components

cx

is n x r matrix of the r canonical components of x

cy

is n.ky x r matrix. The ky blocks cyi of the rows n*(i-1)+1 : n*i contain the r canonical components relative to Yi

rho2

is a ky x r matrix; each column k contains ky squared canonical correlations ρ(cx[,k],cy_i[,k])^2

References

Hanafi & Lafosse (2001) Generalisation de la regression lineaire simple pour analyser la dependance de K ensembles de variables avec un K+1 eme. Revue de Statistique Appliquee vol.49, n.1

Examples

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x<-matrix(runif(50),10,5);y<-matrix(runif(90),10,9)
x<-scale(x);y<-scale(y)
ca<-concorcano(x,y,c(3,2,4),2)
diag(t(ca$cx)%*%ca$cy[1:10,]/10)^2
ca$rho2[1,]