What exactly is generalized? Canonical correlations start with Rij, a
symmetric matrix of Pearson correlation coefficients based on linear
relations. This function starts with a more general non-symmetric R*ij
produced by gmcmtx0() as an input. This is a superior measure
of dependence, allowing for nonlinear dependencies. It generalizes Hotelling's
derivation for the nonlinear case.
This function uses data on two sets of column vectors. LHS set [x1, x2 .. xr]
has r=nLHS number of columns
with coefficients alpha, and
the larger RHS set [xr+1, xr+2, .. xp] has nRHS=(p-r) columns and RHS
coefficients beta. Must arrange the sets so that the larger set
in on RHS with coefficients beta estimated first from an eigenvector
of the problem [A* beta = rho^2 beta], where A* is a partitioning of our
generalized matrix of (non-symmetric) correlation coefficients.
canonRho(mtx, nLHS = 2, sgn = 1, verbo = FALSE, ridg = c(0, 0))eigenvalue computing matrix for Generalized canonical correlations
Generalized canonical correlation coefficient
RHS coefficient vector
LHS coefficient vector
Input matrix of generalized correlation coefficients R*
number of columns in the LHS set, default=2
preferred sign of coefficients default=1 for positive, use sgn= -1 if prior knowledge suggests that negative signs of coefficients are more realistic
logical, verbo=FALSE default means do not print results
two regularization constants added before computing matrix inverses of S11 and S22, respectively, with default=c(0,0). Some suggest ridg=c(0.01,0.01) for stable results
Prof. H. D. Vinod, Economics Dept., Fordham University, NY.
Vinod, H. D. 'Matrix Algebra Topics in Statistics and Economics Using R', Chapter 4 in 'Handbook of Statistics: Computational Statistics with R', Vol.32, co-editors: M. B. Rao and C.R. Rao. New York: North Holland, Elsevier Science Publishers, 2014, pp. 143-176.
Vinod, H. D. 'Canonical ridge and econometrics of joint production,' Journal of Econometrics, vol. 4, 147--166.
Vinod, H. D. 'New exogeneity tests and causal paths,' Chapter 2 in 'Handbook of Statistics: Conceptual Econometrics Using R', Vol.32, co-editors: H. D. Vinod and C.R. Rao. New York: North Holland, Elsevier Science Publishers, 2019, pp. 33-64.
See gmcmtx0.
if (FALSE) {
set.seed(99)
mtx2=matrix(sample(1:25),nrow=5)
g1=gmcmtx0(mtx2)
canonRho(g1,verbo=TRUE)
}#'
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