The Effect of Unequal Variance-Covariance Matrices on Fisher's Linear Discriminant Function

Abstract

This paper undertakes an investigation of the effect of unequal variance-covariance matrices on Fisher's linear discriminant function when used for discrimination or risk estimation. The behavior of this function is compared with the optimal quadratic form when the parameters of the two populations are assumed to be known. The problem is reduced to canonical form, and formulas are developed both for the correlation coefficient of the linear and quadratic functions, and for the probabilities of misclassification resulting from use of each of these functions. Numerical values of these measures are obtained for a number of cases in which one variance-covariance matrix is a multiple d of the other. These calculations are carried out for d equal to 0.1, 0.2, 0.5, 1, 2, 5, and 10; for T2, the squared linear distance between the two populations, equal to 0, 1, 2, 4, and 8; for 7r, the relative frequency of population 1, equal to 2, 3, and 6; and for p, the number of variables, equal to 1, 2, 6, and 10.