Detecting Non-Elliptical Clusters

Abstract

Cluster analysis is used to detect underlying group structure in data. Model-based clustering is a method for performing cluster analysis which involves the fitting of finite mixture models. Regularly these models are fit under the assumption of normality, an assumption that is frequently invalid. To this end, this thesis focuses on the development of novel non-Gaussian mixture models. Three different mixture models are developed using forms of the multivariate skew-t distribution. Given that it is often computationally infeasible to fit mixture models to high-dimensional data, factor analyzers are employed and, in certain cases, the components of the decomposed component-scale matrices are constrained. This leads to the development of flexible, parsimonious models for analyzing high-dimensional data. A novel non-Gaussian distribution is developed and used for cluster analysis via a mixture model. This model is further extended so that it may be applied to high-dimensional data using factor analyzers. For all models and families of models developed in this thesis, the algorithms used for model-fitting and parameter estimation are presented. Real and simulated data sets are used to assess the clustering ability of all models.