QDA classification of high-dimensional data with rare and weak signals

This paper addresses the two-class classification problem for data with rare and weak signals, under the modern high-dimension setup p >> n. Considering the two-component mixture of Gaussian features with different random mean vector of rare and weak signals but common covariance matrix (homoscedastic Gaussian), Fan (AS 41:2537-2571, 2013) investigated the optimality of linear discriminant analysis (LDA) and proposed an efficient variable selection and classification procedure. We extend their work by incorporating the more general scenario that the two components have different random covariance matrices with difference of rare and weak signals, in order to assess the effect of difference in covariance matrix on classification. Under this model, we investigated the behaviour of quadratic discriminant analysis (QDA) classifier. In theoretical aspect, we derived the successful and unsuccessful classification regions of QDA. For data of rare signals, variable selection will mostly improve the performance of statistical procedures. Thus in implementation aspect, we proposed a variable selection procedure for QDA based on the Higher Criticism Thresholding (HCT) that was proved efficient for LDA. In addition, we conducted extensive simulation studies to demonstrate the successful and unsuccessful classification regions of QDA and evaluate the effectiveness of the proposed HCT thresholded QDA.