Description Usage Arguments Details Value References Examples
SVD for a Column Partitioned matrix x. r global successive solutions
1 | svdcp(x,H,r)
|
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. |
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]).
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 |
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.
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