svdcp {BMconcor}

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

a p times q matrix

H

is a row vector which contains the numbers qh, h=1,...,ky, of the partition of x with ky column blocks : sum(qh)=q

r

The number of wanted successive solutions

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]^prime. 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]).

Value

A list with following components:

u

a p times r matrix of kx row blocks uk (pk x r); uk'*uk = 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

s

a kx times ky times r array; with r fixed, each matrix contains kxky values (u_h'*x_{kh}*v_k)^2, the partial (squared) singular values relative to xkh.

Author(s)

Lafosse, R.

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)