Combinatoric classification of multivariate normal variates

Abstract

Consider c l a s s i f y i n g a n n x 1 o b g e r v a t i o n v e c t o r a s coming from one of two m u l t i v a r i a t e normal d i s t r i b u t i o n s which d i f f e r both i n mean v e c t o r s and c o v a r i a n c e m a t r i c e s . A c l a s s of d i sc r i m i n a t i o n r u l e s based upon n independent u n i v a r i a t e discrim-

Algorithm AS 271: General Optimal Combinatoric Classification

“…Combinatoric classification was introduced in Zeis and Smith (1974) and developed in Dunn andSmith (1980, 1982) with a further application in Dunn (1982). The algorithm on which subroutine MCE is based is described in detail in Dunn and Smith (1980). The a priori probability that an element comes from population i (i = 1, 2) is denoted by a., For the ith (i = 1, 2, ... , n) marginal classification rule, the probability of misclassifying an element from population 1 is denoted by Pi and the probability of misclassifying an element from population 2 is denoted by qi' As in Dunn and Smith (1980), it is assumed that the marginal rules are indexed such that n,~P2 ... «o.. For the joint procedure O"k,A, let the misclassification probabilities for elements from population 1 and population 2 be P, k A and q, k A respectively.…”

“…(For implicit enumeration in integer programming, see Taha (1975).) The implicit enumeration algorithm obtains those index sets which are defined to be terminal feasible solutions by Dunn and Smith (1980). For each terminal feasible solution Cthe matrix PSTARc is computed and searched for the smallest entry.…”