Robust linear discriminant procedures using projection pursuit methods.
Zen-Yi Chen · Deep Blue (University of Michigan) · 1989
Two projection indices are proposed for the construction of robust 2-sample linear discriminant functions using projection pursuit methods. The first robust projection index robustifies the classical Fisher ratio of between-class variation to within-class variation. The second is the total (weighted) error rate, and here the estimators of the cutoff points involved in their calculations are robustified. Based on these projection indices, robust linear discriminant functions are constructed using a numerical projection pursuit optimization algorithm. In addition, various cutoff points, in forming robust linear discriminant procedures, are implemented, and Monte Carlo studies are conducted in a well-designed setting. The results show that projection pursuit discriminant functions, derived from robustified indices, perform well under various distributional situations with regard to their empirical error rates. At the same time, the use of a rank cutoff, an adaptive cutoff, or a robustified cutoff enhances the robustness of associated discriminant procedures. In general, the discriminant procedures constructed from the second projection index are more robust than those constructed from the first index in terms of error rates. From the demonstration of our Monte Carlo study, we have introduced a feasible procedure for the construction of robust linear discriminant functions using projection pursuit methods in optimizing a robustified projection index. From a theoretical point of view, this study first defines projection pursuit measures (estimates) of our two proposed indices, as well as their associated discriminant coefficient vectors. Under some restrictions of interest, we derive the qualitative robustness, the breakdown points, and the influence functions for the projection pursuit measures (estimates) of our projection indices in any given projection axis. The linear discriminant functions, constructed by the projection pursuit estimates of associated discriminant coefficient vectors, are expected to be robust in the same sense as the robustness of the corresponding projection indices. Lachenbruch (1982) pointed out: "A formal definition of robustness of discriminant function in the sense of Huber (1981) is not presently available." Based on our theoretical derivations of robustified projection indices in the sense of Huber (1981) and Hampel et al. (1986), we have provided a stepping-stone in achieving this goal.