BMconcor (version 2.0.0)

concorgm: Analyzing a set of partial links between Xi and Yj

Description

Analyzing a set of partial links between Xi and Yj, SUCCESSIVE SOLUTIONS

Usage

concorgm(x, px, y, py, r)

Value

A list with following components:

u

a p times r matrix of axes in Rp relative to x; u^prime*u = Identity

v

a q times r matrix of ky row blocks vi (qi x r) of axes in Rqi relative to yi; vi^prime*vi = Identity

cov2

a ky times r matrix; each column k contains ky squared covariances \(\mbox{cov}((x)(u[,k]),(y_i)(v_i[,k]))^2\), the partial measures of link

Arguments

x

are the n times p and n times q matrices of p and q centered column

px

A row vector which contains the numbers pi, i=1,...,kx, of the kx subsets xi of x : sum(pi)=sum(px)=p. px is the partition vector of x

y

See x

py

The partition vector of y. A row vector containing the numbers qi for i = 1,...,ky of the ky subsets yi of y : sum(qi)=sum(py)=q.

r

The number of wanted successive solutions rmax <= min(min(px),min(py),n)

Author

Lafosse, R.

Details

The first solution calculates 1+kx normed vectors: the vector u[:,1] of Rp associated to the ky vectors vi[:,1]'s of Rqi, by maximizing sum(cov((x)(u[,k]),(y_i)(v_i[,k]))^2), with 1+ky norm constraints on the axes. A component (x)(u[,k]) is associated to ky partial components (yi)(vi)[,k] and to a global component y*V[,k]. cov((x)(u[,k]),(y)(V[,k]))^2 = sum(cov((x)(u[,k]),(y_i)(v_i[,k]))^2)(y)(V[,k]) is a global component of the components (yi)(vi[,k]). The second solution is obtained from the same criterion, but after replacing each yi by \(y_i-(y_i)(v_i[,1])(v_i[,1]')\). And so on for the successive solutions 1,2,...,r. The biggest number of solutions may be r=inf(n, p, qi), when the (x')(yi')(s) are supposed with full rank; then rmax=min(c(min(py),n,p)). For a set of r solutions, the matrix u'X'YV is diagonal and the matrices u'X'Yjvj are triangular (good partition of the link by the solutions). concor.m is the svdcp.m function applied to the matrix x'y.

References

Kissita, Cazes, Hanafi & Lafosse (2004) Deux methodes d'analyse factorielle du lien entre deux tableaux de variables partitionn?es. Revue de Statistique Appliqu?e, Vol 52, n. 3, 73-92.

Examples

x <- matrix(runif(50),10,5);y <- matrix(runif(90),10,9)
x <- scale(x);y <- scale(y)
cg <- concorgm(x,c(2,3),y,c(3,2,4),2)
cg$cov2[1,1,]