A modified linear discriminant analysis for high-dimensional data

Abstract

We deal with the problem of classifying a new observation vector into one of two known multivariate normal populations. Linear discriminant analysis (LDA) is now widely available. However, for high-dimensional data classification problem, due to the small number of samples and the large number of variables, classical LDA has poor performance corresponding to the singularity and instability of the sample covariance matrix. Recently, Xu et al. suggested modified linear discriminant analysis (MLDA). This method is based on the shrink type estimator of the covariance matrix derived by Ledoit and Wolf. This estimator was proposed under the asymptotic framework ${\rm A}_0:n=O(p)$ and $p=O(n)$ when $p\to\infty$. In this paper, we propose a shrink type estimator under more flexible high-dimensional framework. Using this estimator, we define the new MLDA. Through the numerical simulation, the expected correct classification rate of our MLDA is larger than the ones of other discrimination methods when $p>n$. In addition, we consider the limiting value of the expected probability of misclassification (EPMC) under some assumptions.