End of Discriminant Functions based on Variance-covariance Matrices

Fisher proposes Fisher’s linear discriminant function (LDF) based on maximization of variance ratio. If data satisfies Fisher’s assumption, the same LDF is easily derived by variance-covariance matrix. If variance-covariance matrices of two classes are not same, quadratic discriminant function (QDF) is proposed. These discriminant functions have three problems. First, all discriminant functions except for Revised IP-OLDF can’t discriminate the cases xi on the discriminant hyper plane correctly (the unresolved problem of discriminant analysis). Only Revised IP-OLDF can solve this problem in theoretically. Second, LDF and QDF can’t recognize linear separable data in many cases, and the numbers of misclassifications (NMs) of these functions are usually higher than logistic regression that is used in many users. Third, these are not obtained if the values of some independent variable are constant, because inverse matrix can’t be calculated. JMP prepare the regularized discriminant function if QDF can’t analyze the difficult data. These facts mean that LDF and QDF shouldn’t be used for an important discrimination. On the contrary, Revised IP-OLDF based on Minimum Number of Misclassifications (MNM) criterion resolves these problems. In addition to this, means of error rates of Revised IP-OLDF are less than those of LDF and logistic regression and S-SVM by 100-fold cross validation in many cases.