Robustness Properties of k Means and Trimmed k Means

Abstract

The generalized k means method is based on the minimization of the discrepancy between a random variable (or a sample of this random variable) and a set with k points measured through a penalty function Φ. As in the M estimators setting (k = 1), a penalty function, Φ, with unbounded derivative, Ψ, naturally leads to nonrobust generalized k means. However, surprisingly the lack of robustness extends also to the case of bounded Ψ; that is, generalized k means do not inherit the robustness properties of the M estimator from which they came. Attempting to robustify the generalized k means method, the generalized trimmed k means method arises from combining k means idea with a so-called impartial trimming procedure. In this article study generalized k means and generalized trimmed k means performance from the viewpoint of Hampel's robustness criteria; that is, we investigate the influence function, breakdown point, and qualitative robustness, confirming the superiority provided by the trimming. We include the study of two real datasets to make clear the robustness of generalized trimmed k means.