Hybrid linear classifier for jointly normal data: theory

Abstract

Classifier design for a given classification task needs to take into consideration both the complexity of the classifier and the size of the data set that is available for training the classifier. With limited training data, as often is the situation in computer-aided diagnosis of medical images, a classifier with simple structure (e.g., a linear classifier) is more robust and therefore preferred. We consider the two-class classification problem in which the feature data arise from two multivariate normal distributions. A linear function is used to combine the multi-dimensional feature vector onto a scalar variable. This scalar variable, however, is generally not an ideal decision variable unless the covariance matrices of the two classes are equal. We propose using the likelihood ratio of this scalar variable as a decision variable and, thus, generalizing the traditional classification paradigm to a hybrid two-stage procedure: a linear combination of the feature vector elements to form a scalar variable followed by a nonlinear, nonmonotic transformation that maps the scalar variable onto its likelihood ratio (i.e., the ideal decision variable, given the scalar variable). We show that the traditional Fisher's linear discriminant function is generally not the optimal linear function for the first stage in this two-stage paradigm. We further show that the optimal linear function can be obtained with a numerical optimization procedure using the area under the "proper" ROC curve as the objective function.