Growth-Invariant Discriminant Functions and Generalized Distances

Abstract

It is commonly assumed that the method of discriminants cannot be used when the groups to be discriminated are not homogeneous. An important special case is that of groups of organisms in which growth does not cease abruptly at maturity, and where it is not possible to determine the age of an individual to be classified. Provided however that a linear relation can be assumed for the law of individual growth (or other nuisance factor) it is possible to compute discriminant functions and canonical vectors which are invariant with respect to individual growth, while retaining the property of maximising the ratio of between-group to within-group variance. The same technique permits the resolution of the generalized distance (Mahalanobis'' D2) into 2 additive components, which are respectively independent of and wholly determined by the growth factor, as expressed by the identity (xA - xB)[image]W-1(xA - xB) = (xA - XB)[image]W-1 [W - g(g[image]W-1g[image]] W-1(xA - xB) + (xA - xB)[image]W-1g(g[image]W-1g)-1g[image]W-1 (xA - xB). The practical procedure, for the case of 2 nuisance factors, is illustrated by a numerical example.