svdcp: SVD for a Column Partitioned matrix x

In concor: Concordance

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 : ∑ q_i = q.
r	is the wanted number of 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 ∑_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 ∑_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]).

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

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)

concor documentation built on May 2, 2019, 7:25 a.m.