concorgm {BMconcor}

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)

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)

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.

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

Author(s)

Lafosse, R.

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,]