Robust Linear Discriminant Analysis for Multiple Groups: Influence and Classification Efficiencies

Abstract

Linear discriminant analysis for multiple groups is typically carried out using Fisher's method. This method relies on the sample averages and covariance matrices computed from the different groups constituting the training sample. Since sample averages and covariance matrices are not robust, it is proposed to use robust estimators of location and covariance instead, yielding a robust version of Fisher's method. In this paper expressions are derived for the influence that an observation in the training set has on the error rate of the Fisher method for multiple linear discriminant analysis. These influence functions on the error rate turn out to be unbounded for the classical rule, but bounded when using a robust approach. Using these influence functions, we compute relative classification efficiencies of the robust procedures with respect to the classical method. It is shown that, by using an appropriate robust estimator, the loss in classification efficiency at the normal model remains limited. These findings are confirmed by finite sample simulations.

Keywords: Classification efficiency, Discriminant analysis, Error rate, Fisher rule, Influence function, Multiple groups, Robustness