Stepwise Procedures In Discriminant Analysis

THE GENERAL FORM OF A LINEAR DISCRIMINANT FUNCTION

Given that X1, X2... XP follow a multivariate normal distribution, then

Hi - Expected value of X, E1(X), in population one (1)

H2 - Expected value of X, E2(X), in population two (2) given that X is a p-variate random vector.

Discriminant functions perform the main tasks of discrimination and classification, as they provide the rules, upon which decisions are made.

They help us to see the extent to which different populations overlap one another or diverge from one another.

CRITERIA FOR GOOD DISCRIMINANT FUNCTIONS

In order to obtain a discriminant function that helps to assign an unknown observation to a group with a low error rate, we need a criterion of goodness of classification. Below are some criteria for good discriminant functions.

Fisher’s Criterion

Fisher (1936) suggested using a linear combination of the observations and choosing the coefficients so that the ratio of the square of the differences of the means of the linear combination in the two groups to its variance is maximized. He considered the Linear function, Y = X+X, with mean, Y = X+^ in population 1(A1) and Y = X+p2 in population 2(A2). If we assume that the covariance matrices, T = T2 = T, its variance is X+T X.

X is chosen to maximize

^ = (X+m - X+ ^)2- -  -           -           (2.4)

X is used for the separation of population only, we may then multiply X by any desired constant.

This means that X is proportional to Z-1 (g1-g2). For unknown parameters, g1, g2 and Z are estimated by X1, X2 and S, respectively. The assignment procedure is to assign an individual to â–²1 if Y = (g1-g2) +Z-1X is closer to Y1 =( g1-g2) +Z-1 g1 than to Y2 = (g1-g2) +Z-1 g2 (for known parameters). For unknown parameter, the assignment procedure is to assign an individual to

^1(population 1)

if Y= (X1-X2) +S-1X is closer”to Y1_= (X1-X2T + S-1X1 than to Y2 = (X1-X2) + S-1 X2. Or assign an individual to population 2 if Y = (X1-X2) +S-1X is closer to

Y2= (X1-X2) +St1X2 than to Y1.