concor: Relative links of several subsets of variables
In concor: Concordance

Description Usage Arguments Details Value References Examples

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

1	concor(x,y,py,r)

`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

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 ∑_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 = ∑ \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.

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}(xu[,k],y_iv_i[,k])^2, the partial measures of link

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.

# 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