svdcp function

SVD for a Column Partitioned matrix x

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

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

Returns

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.

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]).

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)

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)