The Asymptotic Approximation of EPMC for Linear Discriminant Rules Using a Moore-Penrose Inverse Matrix in High Dimension

Abstract

We consider the discriminant rule in a high-dimensional setting, i.e., when the number of feature variables p is comparable to or larger than the number of observations N. The discriminant rule must be modified in order to cope with singular sample covariance matrix in high-dimension. One way to do so is by considering the Moor-Penrose inverse matrix. Recently, Srivastava (2006) proposed maximum likelihood ratio rule by using Moor-Penrose inverse matrix of sample covariance matrix. In this article, we consider the linear discriminant rule by using Moor-Penrose inverse matrix of sample covariance matrix (LDRMP). With the discriminant rule, the expected probability of misclassification (EPMC) is commonly used as measure of the classification accuracy. We investigate properties of EPMC for LDRMP in high-dimension as well as the one of the maximum likelihood rule given by Srivastava (2006). From our asymptotic results, we show that the classification accuracy of LDRMP depends on new distance. Additionally, our asymptotic result is verified by using the Monte Carlo simulation.

Keywords:

• Expected probability of misclassification
• High-dimensional approximations
• Linear discriminant rule
• Maximum likelihood classification rule

Mathematics Subject Classification:

• Primary 62H12
• Secondary 62H30

Acknowledgments

This work was supported by Japan Society for the Promotion of Science (JSPS). The authors would like to thank the referee for valuable comments and a careful reading of this article. They would also like to thank Professors Yasunori Fujikoshi and Muni S. Srivastava for their advice and encouragement.