concorcano function

Canonical analysis of several sets with another set

Relative proximities of several subsets of variables Yj with another set X. SUCCESSIVE SOLUTIONS

concorcano(x, y, py, r)

Arguments

• x: are the n times p and n times q matrices of p and q centered column
• y: See x
• py: The partition vector of y. A row vector containing the numbers qi for i = 1,...,ky of the ky subsets yi of y : sum(qi)=sum(py)=q.
• r: The number of wanted successive solutions

Returns

A list with following components: - cx: a n times r matrix of the r canonical components of x

• cy: a n.ky times r matrix. The ky blocks cyi of the rows n*(i-1)+1 : n*i contain the r canonical components relative to Yi

• rho2: a ky times r matrix; each column k contains ky squared canonical correlations $\rho(cx[,k],cy_i[,k])^2$

Details

The first solution calculates a standardized canonical component cx[,1] of x associated to ky standardized components cyi[,1] of yi by maximizing $\sum_i \rho(cx[,1],cy_i[,1])^2$. The second solution is obtained from the same criterion, with ky orthogonality constraints for having rho(cyi[,1],cyi[,2])=0 (that implies rho(cx[,1],cx[,2])=0). For each of the 1+ky sets, the r canonical components are 2 by 2 zero correlated. The ky matrices (cx)'*cyi are triangular. This function uses concor function.

Examples

x <- matrix(runif(50),10,5);y <- matrix(runif(90),10,9)
x <- scale(x);y <- scale(y)
ca <- concorcano(x,y,c(3,2,4),2)

References

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

Author(s)