Constrained Optimization for a Subset of the Gaussian Parsimonious Clustering Models

The expectation-maximization (EM) algorithm is an iterative method for finding maximum likelihood estimates when data are incomplete or are treated as being incomplete. The EM algorithm and its variants are commonly used for parameter estimation in applications of mixture models for clustering and classification. This despite the fact that even the Gaussian mixture model likelihood surface contains many local maxima and is singularity riddled. Previous work has focused on circumventing this problem by constraining the smallest eigenvalue of the component covariance matrices. In this paper, we consider constraining the smallest eigenvalue, the largest eigenvalue, and both the smallest and largest within the family setting. Specifically, a subset of the GPCM family is considered for model-based clustering, where we use a re-parameterized version of the famous eigenvalue decomposition of the component covariance matrices. Our approach is illustrated using various experiments with simulated and real data

Mixtures of Shifted Asymmetric Laplace Distributions

A mixture of shifted asymmetric Laplace distributions is introduced and used for clustering and classification. A variant of the EM algorithm is developed for parameter estimation by exploiting the relationship with the general inverse Gaussian distribution. This approach is mathematically elegant and relatively computationally straightforward. Our novel mixture modelling approach is demonstrated on both simulated and real data to illustrate clustering and classification applications. In these analyses, our mixture of shifted asymmetric Laplace distributions performs favourably when compared to the popular Gaussian approach. This work, which marks an important step in the non-Gaussian model-based clustering and classification direction, concludes with discussion as well as suggestions for future work