concorPLS - Regression PLS of subsets Yj by one set X.
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 qj 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
The columns of Cx (n x r) are the r explanatory components.
They are associated to :
V (q x r) the orthonormed global axes of Y, and also to :
kx matrices vj (qj xr), blocks of v (q x r), normed partial axes of
the respective Yj, j=1,...,ky.
cri (ky x r) contains the explained variances :
for the solution column k, ky values rho2(Yjvj(:,k),Cx(:,k)) var(Yjvj(:,k)).
u (p x r) is not very useful : only u(,1) contains the scores of the variables
of X (Cx(:,1) is the standardized component X*u(:,1) ).
Each solution associates one explanatory component to ky explained components Yjvj.
A column of Y*V is a mean component of partial explained components Yjvj.
For a set of r solutions, the matrix Cx'*Y*V is
triangular : on average, the explanatory component of one solution
is not linked with the mean explanation given by the following solutions.
The definition of the explanatory components does not depend on the partition
vector py.
The orthogonality constraints for successive solutions are :
V'*V = Ir and Cx'*Cx =Ir.
diag(Cx'*y*V/n)'.^2 = sum(cri,1) = sumj diag(Cx'*Yj*vj/n)'.^2
Another PLS in Vivien & Sabatier (2001) Revue de Stat appli. vol.49 n.1
myrtille.vivien@univ-montp1.fr
Authors