Computational Details :: SAS/STAT(R) 12.3 User's Guide

Computational Details

General Formulas

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 are denoted by $\mb {y}$ and the quantitative variables by $\mb {x}$ . The total sample covariance matrix for the $\mb {x}$ and $\mb {y}$ variables is

$\mb {S} = \left[\begin{matrix} \mb {S}_{xx} & \mb {S}_{xy} \cr \mb {S}_{yx} & \mb {S}_{yy} \end{matrix}\right]$

When c is the number of groups, is the number of observations in group t, and $\mb {S}_ t$ is the sample covariance matrix for the $\mb {x}$ variables in group t, the within-class pooled covariance matrix for the $\mb {x}$ variables is

$\mb {S}_ p = \frac{1}{\sum n_ t-c} {\sum (n_ t-1)\mb {S}_ t}$

The canonical correlations, $\rho _ i$ , are the square roots of the eigenvalues, $\lambda _ i$ , of the following matrix. The corresponding eigenvectors are $\mb {v}_ i$ .

${\mb {S}_ p}^{-1/2}\mb {S}_{xy}{\mb {S}_{yy}}^{-1}\mb {S}_{yx}{\mb {S}_ p}^{-1/2}$

Let $\mb {V}$ be the matrix with the eigenvectors $\mb {v}_ i$ that correspond to nonzero eigenvalues as columns. The raw canonical coefficients are calculated as follows: