concoreg: Redundancy of sets yj by one set x
In concor: Concordance

Description Usage Arguments Details Value References Examples

Regression of several subsets of variables Yj by another set X. SUCCESSIVE SOLUTIONS

1	concoreg(x,y,py,r)

`x`	is a n x p matrix of p centered explanatory variables
`y`	is a n x q matrix of q centered variables
`py`	is a row vector which contains the numbers q_i, i=1,...,ky, of the ky subsets y_i of y : ∑_i q_i = sum(py) = q. py is the partition vector of y
`r`	is the wanted number of successive solutions

The first solution calculates 1+ky normed vectors: the component cx[,1] in R^n associated to the ky vectors vi[,1]'s of R^{q_i}, by maximizing varexp1=∑_i ρ(cx[,1],y_i*v_i[,1])^2 \mbox{var}(y_i*v_i[,1])), with 1+ky norm constraints. A explanatory component cx[,k] is associated to ky partial explained components yi*vi[,k] and also to a global explained component y*V[,k]. ρ(cx[,k],y*V[,k])^2 \mbox{var}(y*V[,k])= \mbox{varexpk}. The total explained variance by the first solution is maximal.

The second solution is obtained from the same criterion, but after replacing each yi by y_i-y_i*v_i[,1]*v_i[,1]'. And so on for the successive solutions 1,2,...,r . The biggest number of solutions may be r=inf(n,p,q_i), when the matrices x'*yi are supposed with full rank. For a set of r solutions, the matrix (cx)'*y*V 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 (cx)'*y_j*v_j 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.

This function is using concor function

list with following components

`cx`	the n x r matrix of the r explanatory components
`v`	is a q x r matrix of ky row blocks v_i (q_i x r) of axes in Rqi relative to yi; v_i'v_i = \mbox{Id}*
`V`	is a q x r matrix of axes in Rq relative to y; V'V = \mbox{Id}*
`varexp`	is a ky x r matrix; each column k contains ky explained variances ρ(cx[,k],y_iv_i[,k])^2 \mbox{var}(y_iv_i[,k])

Hanafi & Lafosse (2001) Generalisation de la regression lineaire simple pour analyser la dependance de K ensembles de variables avec un K+1 eme. Revue de Statistique Appliquee vol.49, n.1.

Chessel D. & Hanafi M. (1996) Analyses de la Co-inertie de K nuages de points. Revue de Statistique Appliquee vol.44, n.2. (this ACOM analysis of one multiset is obtained by the command : concoreg(Y,Y,py,r))

x<-matrix(runif(50),10,5);y<-matrix(runif(90),10,9)
x<-scale(x);y<-scale(y)
co<-concoreg(x,y,c(3,2,4),2)
((t(co$cx[,1])%*%y[,1:3]%*%co$v[1:3,1])/10)^2;co$varexp[1,1]
t(co$cx)%*%co$cx /10
diag(t(co$cx)%*%y%*%co$V/10)^2
sum(co$varexp[,1]);sum(co$varexp[,2])