Robust Clustering with Normal Mixture Models: A Pseudo $β$-Likelihood Approach

As in other estimation scenarios, likelihood based estimation in the normal mixture set-up is highly non-robust against model misspecification and presence of outliers (apart from being an ill-posed optimization problem). We propose a robust alternative to the ordinary likelihood approach for this estimation problem which performs simultaneous estimation and data clustering and leads to subsequent anomaly detection. To invoke robustness, we follow, in spirit, the methodology based on the minimization of the density power divergence (or alternatively, the maximization of the $\beta$-likelihood) under suitable constraints. An iteratively reweighted least squares approach has been followed in order to compute our estimators for the component means (or equivalently cluster centers) and component dispersion matrices which leads to simultaneous data clustering. Some exploratory techniques are also suggested for anomaly detection, a problem of great importance in the domain of statistics and machine learning. Existence and consistency of the estimators are established under the aforesaid constraints. We validate our method with simulation studies under different set-ups; it is seen to perform competitively or better compared to the popular existing methods like K-means and TCLUST, especially when the mixture components (i.e., the clusters) share regions with significant overlap or outlying clusters exist with small but non-negligible weights. Two real datasets are also used to illustrate the performance of our method in comparison with others along with an application in image processing. It is observed that our method detects the clusters with lower misclassification rates and successfully points out the outlying (anomalous) observations from these datasets.