Pattern recognition based on scale invariant discriminant functions

Abstract

Some probability models for classifying individuals as belonging to one of two or more populations using scale invariant discriminant functions are considered. The investigation is motivated by practical situations where the observed data on an individual are in the form of ratios of some basic measurements or measurements scaled by an unknown nonnegative number. The probability models are obtained by considering a p-vector random variable X with a known distribution and deriving the distribution of the random vector Y = [G(X)]^-1 X, where G(X) is a nonnegative measure of size such that G(λ X) = λG(X) for λ > 0. Explicit expressions are obtained for the densities of what are called angular Gaussian; compositional Gaussian, type 1; and compositional Gaussian, type 2 distributions.