Discriminant analysis under the common principal components model

ABSTRACT

For two or more populations of which the covariance matrices have a common set of eigenvectors, but different sets of eigenvalues, the common principal components (CPC) model is appropriate. Pepler et al. (2015) proposed a regularized CPC covariance matrix estimator and showed that this estimator outperforms the unbiased and pooled estimators in situations, where the CPC model is applicable. This article extends their work to the context of discriminant analysis for two groups, by plugging the regularized CPC estimator into the ordinary quadratic discriminant function. Monte Carlo simulation results show that CPC discriminant analysis offers significant improvements in misclassification error rates in certain situations, and at worst performs similar to ordinary quadratic and linear discriminant analysis. Based on these results, CPC discriminant analysis is recommended for situations, where the sample size is small compared to the number of variables, in particular for cases where there is uncertainty about the population covariance matrix structures.

KEYWORDS:

• Common principal components
• Covariance matrix
• Discriminant analysis
• Monte Carlo simulation

MATHEMATICS SUBJECT CLASSIFICATION:

• Primary 62H25
• Secondary 62H30

Acknowledgement

The authors wish to express their thanks to the anonymous reviewer whose comments led to an improved version of this article.