Canonical discriminant analysis is equivalent to canonical correlation
analysis between the quantitative variables and a set of dummy variables
coded from the class variable. In the following notation the dummy
variables will be denoted by y and the quantitative variables by x.
The total sample covariance matrix for the x and y variables is
When c is the number of groups, nt is the number of observations in
group t, and St is the sample covariance matrix for the x variables
in group t, the within-class pooled covariance matrix for the x
The canonical correlations, , are the square roots of the
eigenvalues, , of the following matrix. The corresponding
eigenvectors are vi.
Let V be the matrix with the eigenvectors vi that correspond to
nonzero eigenvalues as columns. The raw canonical coefficients are
calculated as follows
The pooled within-class standardized canonical coefficients are
R = Sp-1/2V
And the total sample standardized canonical coefficients are
P = diag(Sp)1/2R
Let Xc be the matrix with the centered x variables as columns. The
canonical scores may be calculated by any of the following
T = diag(Sxx)1/2R
For the Multivariate tests based on E-1H
E = (n-1)(Syy - SyxSxx-1Sxy)
where n is the total number of observations.
H = (n-1)SyxSxx-1Sxy
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.