concor (version 1.0-0.1)

svdcp: SVD for a Column Partitioned matrix x

Description

SVD for a Column Partitioned matrix x. r global successive solutions

Usage

svdcp(x,H,r)

Arguments

x

is a p x q matrix

H

is a row vector which contains the numbers qi, i=1,...,kx, of the partition of x with kx column blocks xi : $\sum q_i = q.$

r

is the wanted number of successive solutions.

Value

• list with following components
• u
  is a p x r matrix; u'*u = Identity
• v
  is a q x r matrix of kx row blocks vi (qi x r); vi'*vi = Identity
• s2
  is a kx x r matrix; each column k contains kx values $(u[,k]'*x_i*v_i[,k])^2$, the partial (squared) singular values relative to xi

Details

The first solution calculates 1+kx normed vectors: the vector u[,1] of $R^p$ associated to the kx vectors vi[,1]'s of $R^{q_i}$. by maximizing $\sum_i (u[,1]'*x_i*v_i[,1])^2$, with 1+kx norm constraints. A value $(u[,1]'*x_i*v_i[,1])^2$ measures the relative link between $R^p$ and $R^{q_i}$ associated to xi. It corresponds to a partial squared singular value notion, since $\sum_i (u[,1]'*x_i*v_i[,1])^2=s^2$, where s is the usual first singular value of x.

The second solution is obtained from the same criterion, but after replacing each xi by xi-xi*vi[,1]*vi[,1]'. And so on for the successive solutions 1,2,...,r . The biggest number of solutions may be r=inf(p,qi), when the xi's are supposed with full rank; then rmax=min([min(H),p]).

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

x<-matrix(runif(200),10,20)
s<-svdcp(x,c(5,5,10),1)
ss<-svd(x);ss$d[1]^2
sum(s$s2)