concor function

Relative links of several subsets of variables

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

concor(x, y, py, r)

Arguments

• x: are the n times p and n times q matrices of p and q centered column
• 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

Returns

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

• V: A q times r matrix of axes in Rq relative to y; Vprime*V = 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

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.

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)

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.

Author(s)