Covariance estimation for limited training samples

Abstract

In Gaussian maximum likelihood classification, the mean vector and covariance matrix are usually estimated from training samples. When the training sample size is small compared to dimensionality, the sample estimates, especially the covariance matrix becomes highly variable and consequently, the classifier performs poorly. In particular, if the number of training samples is less than dimensionality, the sample covariance estimate becomes singular so the quadratic classifier cannot be applied. Unfortunately, the problem of limited training samples is prevalent in remote sensing applications. While the recent progress in sensor technology has increased the number of spectral features making possible more classes to be identified, the training data remain expensive and difficult to acquire. In this work, the problem of small training set size on the classification performance is addressed by introducing a covariance estimation method for limited training samples. The proposed approach can be viewed as an intermediate method between linear and quadratic classifiers by selecting an appropriate mixture of covariance matrices. The mixture of covariance matrices is formulated under an empirical Bayesian setting which is advantageous when the training sample size reflects the prior of each class. The experimental results show that the proposed method improves the classification performance when training sample sizes are limited.