concorCANO - Canonical analysis between one subset X and several subsets Yj
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
P is a matrix with n rows which is obtained from X by standardization.
Py is a matrix with n rows which contains the standardized matrices Pyj
respectively associated to Yj. The number of the columns of Pyj is ry(j).
The r columns of P*u are the r canonical variables relative to X.
v contains ky blocks vj, each of them with r columns. v1=v(1:ry(1),:),
v2=v(ry(1)+1:ry(1)+ry(2),:), ...
Pyj*vj are the r canonical variables of Yj, respectively associated with the
r canonical variables of X.
For each set, the canonical variables are standardized and 2 by 2 zero
correlated.
rho2 (ky x r) contains the canonical coefficients rho2(Yjvj(:,k),Pu(:,k))
associated to the solution k, k=1, ...r.
The total sum of squared correlations of the the first solution is maximal.
Py*V contains (for each k) weighted means of the Pyj*vj
(the weights, are for each fixed k, rho(Yjvj(:,k),Pu(:,k)).
For the set of r solutions, the matrix (P*u)'Py*V is diagonal.
The matrices (Pu)'Pyjvj are triangular.
diag((P*u)'*Py*V/n)'.^2 = sum(rho2,1)
Hanafi & Lafosse (2001) in Revue de Statistique Appliquee vol.49, n.1.
hanafi@enitiaa-nantes.fr
Roger.Lafosse@lsp.ups-tlse.fr
Authors