Linear discriminant analysis and transvariation

In this paper, a two group linear discriminant function (LDF) is derived by minimizing Gini’s transvariation probability. This solution is a special case of the projection pursuit method and improves the performance of Fisher’s LDF when the conditions for its optimality do not hold. Two groups are said to transvariate with respect to a variable Y (here a linear combination of the observed variables) if there exists at least one pair of units, belonging to different groups, in which the difference in sign between the Y values is opposite to that of the corresponding group mean values. As any difference satisfying this condition is called “a transvariation”, transvariation probability is defined as the ratio of the number of observed transvariations to its maximum possible value. When transvariation probability is used to measure group separation a linear discriminant function may be obtained as the linear combination along which transvariation probability is minimum. The performances of the proposed method are tested through a wide simulation study and on a real data set.