svdbip: SVD for one bipartitioned matrix x
In concor: Concordance

Description Usage Arguments Details Value References Examples

SVD for bipartitioned matrix x. r successive Solutions

1	svdbip(x,K,H,r)

`x`	is a p x q matrix
`K`	is a row vector which contains the numbers pk, k=1,...,kx, of the partition of x with kx row blocks : `sum(pk)=p`
`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`	is the wanted number of successive solutions

The first solution calculates kx+ky normed vectors: kx vectors uk[:,1] of R^{p_k} associated to ky vectors vh[:,1]'s of R^{q_h}, by maximizing ∑_k ∑_h (u_k[:,1]'*x_{kh}*v_h[:,1])^2, with kx+ky norm constraints. A value (u_k[,1]'*x_{kh}*v_h[,1])^2 measures the relative link between R^{p_k} and R^{q_h} associated to the block xkh.

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

When K=p (or H=q, with t(x)), svdcp function is better. When H=q and K=p, it is the usual svd (with squared singular values).

Convergence of algorithm may be not global. So the below proposed initialisation of the algorithm may be not very suitable for some data sets. Several different random initialisations with normed vectors might be considered and the best result then choosen.

list with following components

`u`	is a p x r matrix of kx row blocks uk (pk x r); uk'*uk = Identity.
`v`	is a q x r matrix of ky row blocks vh (qh x r); vh'*vh = Identity
`s2`	is a kx x ky x 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.

Kissita G., Cazes P., Hanafi M. & Lafosse (2004) Deux methodes d'analyse factorielle du lien entre deux tableaux de variables partitiones. Revue de Statistique Appliquee.

x<-matrix(runif(200),10,20)
s<-svdbip(x,c(3,4,3),c(5,15),3)
zu<-cbind(x[1:3,1:5]%*%s$v[1:5,1],x[1:3,6:20]%*%s$v[6:20,1])
czu<-svd(zu);
czu$u[,1]%*%s$u[1:3,2:3]
czu$u[,1] # is a compromise between the vectors xj*vj[,1],
# orthogonal to the partial vectors uk[,k] relative to the
# following solutions (k>1); (in a same way, the singular
# vectors ui and vj of an usual SVD of x verifies ui'*(x*vj)=0,
#when i is not equal to j)