concorREG -
X and Y are 2 data matrices, n x p and n x q, of p variables and q variables (centered).
The row vector py contains the ky numbers of variables of the ky subsets of Y.
sum(py)=q. Y is the concatenated matrix of matrices Yj, j=1,...,ky.
r is the wanted number of solutions [0
u (p x r) is an orthonormed system associated to :
V (q x r) the orthonormed global axes of Y, and also to :
each of the ky blocks vj of v (q x r), which contains r orthonormed partial
axes relative to Yj, j=1,...,ky.
cri (ky x r) contains the explained variances :
rho2(Yjvj(:,k),Pu(:,k)) var(Yjvj(:,k)) for the solution k, k=1, ...r.
sum(cri,2) contains r successive optimized criteria, with these
orthogonality constraints between solutions : uj'*uj = Ir, j=1, ...kx.
The total explained variance by the first solution is maximal.
Each solution associates one explanatory component to ky explained components Yjvj.
Y*V(:,k) is a mean component of partial explained components Yjvj(:,k).
The columns of P*u are the explanatory components. The matrix X has been
standardized for obtaining the matrix P.
For a set of r solutions, the matrix (Pu)'YV is diagonal : on average,
the explanatory component of one solution is only linked with the components
explained by this explanatory, and is not linked with the explained
components of the other solutions.
The matrices (Pu)'Yjvj are triangular : the explanatory component of one solution
is not linked with each of the partial components explained in the following
solutions.
The definition of the explanatory components depends on the partition vector
py from the second solution.
diag((P*u)'*y*V/n)'.^2 = sum(cri,1)
Hanafi & Lafosse (2001) in Revue de Statistique Appliquee vol.49, n.1.
hanafi@enitiaa-nantes.fr
Roger.Lafosse@lsp.ups-tlse.fr
Authors