The Noise Component in Model-based Cluster Analysis

Abstract

The so-called noise-component has been introduced by Banfield and Raftery (1993) to improve the robustness of cluster analysis based on the normal mixture model. The idea is to add a uniform distribution over the convex hull of the data as an additional mixture component. While this yields good results in many practical applications, there are some problems with the original proposal: 1) As shown by Hennig (2004), the method is not breakdown-robust. 2) The original approach doesn’t define a proper ML estimator, and doesn’t have satisfactory asymptotic properties.

We discuss two alternatives. The first one consists of replacing the uniform distribution by a fixed constant, modelling an improper uniform distribution that doesn’t depend on the data. This can be proven to be more robust, though the choice of the involved tuning constant is tricky. The second alternative is to approximate the ML-estimator of a mixture of normals with a uniform distribution more precisely than it is done by the “convex hull” approach. The approaches are compared by simulations and for a real data example.