concor (version 1.0-0.1)

concor: Relative links of several subsets of variables

Description

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

Usage

concor(x,y,py,r)

Arguments

x,y

are n x p and n x q matrices of p and q centered columns

py

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

r

is the wanted number of successive solutions

Value

• list with following components
• u
  is a p x r matrix of axes in Rp relative to x; u'*u = Identity
• v
  is a q x r matrix of ky row blocks vi (qi x r) of axes in Rqi relative to yi; vi'*vi = Identity
• V
  is a q x r matrix of axes in Rq relative to y; V'*V = Identity
• cov2
  is a ky x 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

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 $\sum_i \mbox{cov}(x*u[,k],y_i*v_i[,k])^2$, 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]. $\mbox{cov}(x*u[,k],y*V[,k])^2 = \sum \mbox{cov}(x*u[,k],y_i*v_i[,k])^2$. 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

Lafosse R. & Hanafi M.(1997) Concordance d'un tableau avec K tableaux: Definition de K+1 uples synthetiques. Revue de Statistique Appliquee vol.45,n.4.

Examples

# To make some "GPA" : so, by posing the compromise X = Y,
# "procrustes" rotations to the "compromise X" then are :
# Yj*(vj*u').

x<-matrix(runif(50),10,5);y<-matrix(runif(90),10,9)
x<-scale(x);y<-scale(y)
co<-concor(x,y,c(3,2,4),2)
((t(x%*%co$u[,1])%*%y[,1:3]%*%co$v[1:3,1])/10)^2;co$cov2[1,1] 
t(x%*%co$u)%*%y%*%co$V