discriminant analysis

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A procedure for the determination of the group to which an individual belongs, based on the characteristics of that individual. Suppose we have measurements on p characteristics for each of a sample of individuals. We know that each individual belongs to one of g groups, but we do not know which. Discriminant analysis attempts to maximize the probability of correct allocation. It differs from cluster analysis in that we have an initial data set, the training set, whose group allocations are known.

For the case g=2, suppose that the p×1 vectors of sample means are 1 and 2. Let n1 and n2 be the numbers of members of the training set falling in the two groups and let S1 and S2 be the variance–covariance matrices for the two parts of the training set. If we define the matrix S by , a future individual with measurement vector x will be assigned to group 1 if and only if , where a′ is the transpose of the vector a given by . The function a′X is called Fisher's linear discriminant function. The terms ‘discriminant analysis’ and ‘discriminant function’ were coined by Sir Ronald Fisher in his articles in 1936 and 1938 that introduced the procedure.

Subjects: Probability and Statistics.

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