Computational Details

The CANDISC Procedure

Computational Details

General Formulas

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

$S = [\matrix{ S_{xx} & S_{xy} \cr S_{yx} & S_{yy} }]$

When c is the number of groups, n_t is the number of observations in group t, and S_t is the sample covariance matrix for the x variables in group t, the within-class pooled covariance matrix for the x variables is

$S_p = {1 \over \sum n_t-c}{\sum (n_t-1)S_t}$

The canonical correlations, $\rho_i$ , are the square roots of the eigenvalues, $\lambda_i$ , of the following matrix. The corresponding eigenvectors are v_i.

S_p^-1/2S_xyS_yy^-1S_yxS_p^-1/2

Let V be the matrix with the eigenvectors v_i that correspond to nonzero eigenvalues as columns. The raw canonical coefficients are calculated as follows

R = S_p^-1/2V

The pooled within-class standardized canonical coefficients are

P = diag(S_p)^1/2R

And the total sample standardized canonical coefficients are

T = diag(S_xx)^1/2R

Let X_c be the matrix with the centered x variables as columns. The canonical scores may be calculated by any of the following

X_c R

X_c diag(S_p)^-1/2P

X_c diag(S_xx)^-1/2T

For the Multivariate tests based on E^-1H

E = (n-1)(S_yy - S_yxS_xx^-1S_xy)

H = (n-1)S_yxS_xx^-1S_xy

where n is the total number of observations.

Chapter Contents
Previous
Next
Top