concor - Analyzing the contributions of each subset of variables Yj to its link with another set X
X and Y are 2 data matrices, n x p and n x q, of p variables and q variables (centered) measured on the same set of n cases. The row vector py contains the ky numbers of variables of each of the ky subsets of Y. sum(py)=q.
Y is the concatenated matrix of ky matrices Yj, j=1,...,ky.
r is the wanted number of solutions [0
u (p x r) are the orthonormed axes of X. They are associated to :
V (q x r) the orthonormed global axes of Y, and to :
the sub-blocks vj of v (q x r), the orthonormed partial axes of Yj.
The ky x r matrix cri contains the values cov2(Yj vj(:,k), Xu(:,k)), k=1...r,
which for each k are the measures of the relative links of Yj with X.
For each solution k, sumj cov2(Yjvj(:,k),Xu(:,k)) is equal to
cov2(Xu(:,k),YV(:,k)), and
that corresponds to these 2 independently optimized criteria, with respective
norm constraints on the axes vj(:,k) and u(:,k), or on the axes u(:,k) and
V(:,k);
Each column of YV represents a mean component of the respective
partial components Yjvj.
For a set of r solutions, the matrix u'X'YV is diagonal and the matrices
u'X'Yjvj are triangular.
Example of use :
To make some "GPA" : so, by posing the compromise X = Y, "procrustes" rotations
to the compromise X then are : Yj*(vj*u').
Lafosse & Hanafi (1997) in Revue de Statistique Appliquee.
(concor.m corresponds to the svdcp.m function)
hanafi@enitiaa-nantes.fr
Roger.Lafosse@lsp.ups-tlse.fr
Examples
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