Robust improper maximum likelihood: tuning, computation, and a comparison with other methods for robust Gaussian clustering

The two main topics of this paper are the introduction of the "optimally tuned improper maximum likelihood estimator" (OTRIMLE) for robust clustering based on the multivariate Gaussian model for clusters, and a comprehensive simulation study comparing the OTRIMLE to Maximum Likelihood in Gaussian mixtures with and without noise component, mixtures of t-distributions, and the TCLUST approach for trimmed clustering. The OTRIMLE uses an improper constant density for modelling outliers and noise. This can be chosen optimally so that the non-noise part of the data looks as close to a Gaussian mixture as possible. Some deviation from Gaussianity can be traded in for lowering the estimated noise proportion. Covariance matrix constraints and computation of the OTRIMLE are also treated. In the simulation study, all methods are confronted with setups in which their model assumptions are not exactly fulfilled, and in order to evaluate the experiments in a standardized way by misclassification rates, a new model-based definition of "true clusters" is introduced that deviates from the usual identification of mixture components with clusters. In the study, every method turns out to be superior for one or more setups, but the OTRIMLE achieves the most satisfactory overall performance. The methods are also applied to two real datasets, one without and one with known "true" clusters

An adequacy approach for deciding the number of clusters for OTRIMLE robust Gaussian mixture-based clustering

We introduce a new approach to deciding the number of clusters. The approach is applied to Optimally Tuned Robust Improper Maximum Likelihood Estimation (OTRIMLE; Coretto & Hennig, Journal of the American Statistical Association111, 1648-1659) of a Gaussian mixture model allowing for observations to be classified as 'noise', but it can be applied to other clustering methods as well. The quality of a clustering is assessed by a statistic Q that measures how close the within-cluster distributions are to elliptical unimodal distributions that have the only mode in the mean. This non-parametric measure allows for non-Gaussian clusters as long as they have a good quality according to Q. The simplicity of a model is assessed by a measure S that prefers a smaller number of clusters unless additional clusters can reduce the estimated noise proportion substantially. The simplest model is then chosen that is adequate for the data in the sense that its observed value of Q is not significantly larger than what is expected for data truly generated from the fitted model, as can be assessed by parametric bootstrap. The approach is compared with model-based clustering using the Bayesian information criterion (BIC) and the integrated complete likelihood (ICL) in a simulation study and on real two data sets