Computational Details :: SAS/STAT(R) 13.1 User's Guide

Computational Details

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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 are denoted 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, $n_ t$ 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 that contains the eigenvectors $\mb {v}_ i$ that correspond to nonzero eigenvalues as columns. The raw canonical coefficients are calculated as follows:

$\mb {R} = {\mb {S}_ p}^{-1/2}\mb {V}$

The pooled within-class standardized canonical coefficients are

$\mb {P} = \mr {diag}(\mb {S}_ p)^{1/2}\mb {R}$

The total sample standardized canonical coefficients are

$\mb {T} = \mr {diag}(\mb {S}_{xx})^{1/2}\mb {R}$

Let $\mb {X}_ c$ be the matrix that contains the centered $\mb {x}$ variables as columns. The canonical scores can be calculated by any of the following:

$\mb {X}_ c \, \mb {R}$

$\mb {X}_ c \, \mr {diag}(\mb {S}_ p)^{-1/2}\mb {P}$

$\mb {X}_ c \, \mr {diag}(\mb {S}_{xx})^{-1/2}\mb {T}$

For the multivariate tests based on $\mb {E}^{-1}\mb {H}$ ,

$\mb {E} = (n-1)(\mb {S}_{yy} - \mb {S}_{yx}\mb {S}_{xx}^{-1}\mb {S}_{xy})$

$\mb {H} = (n-1)\mb {S}_{yx}\mb {S}_{xx}^{-1}\mb {S}_{xy}$

where n is the total number of observations.

The CANDISC Procedure

Computational Details

General Formulas