Franczak, Brian C., McNicholas, Paul D., Browne, Ryan P., Murray, Paula M.

A family of parsimonious shifted asymmetric Laplace mixture models is introduced. We extend the mixture of factor analyzers model to the shifted asymmetric Laplace distribution. Imposing constraints on the constitute parts of the resulting decomposed component scale matrices leads to a family of parsimonious models. An explicit two-stage parameter estimation procedure is described, and the Bayesian information criterion and the integrated completed likelihood are compared for model selection. This novel family of models is applied to real data, where it is compared to its Gaussian analogue within clustering and classification paradigms.

McNicholas, Sharon M., McNicholas, Paul D., Ashlock, Daniel A.

The expectation-maximization (EM) algorithm is almost ubiquitous for parameter estimation in model-based clustering problems; however, it can become stuck at local maxima, due to its single path, monotonic nature. Rather than using an EM algorithm, an evolutionary algorithm (EA) is developed. This EA facilitates a different search of the fitness landscape, i.e., the likelihood surface, utilizing both crossover and mutation. Furthermore, this EA represents an efficient approach to "hard" model-based clustering and so it can be viewed as a sort of generalization of the k-means algorithm, which is itself equivalent to a classification EM algorithm for a Gaussian mixture model with spherical component covariances. The EA is illustrated on several data sets, and its performance is compared to k-means clustering as well as model-based clustering with an EM algorithm.

Franczak, Brian C., Browne, Ryan P., McNicholas, Paul D.

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.

Murray, Paula M., McNicholas, Paul D., Browne, Ryan P.

A mixture of common skew-t factor analyzers model is introduced for model-based clustering of high-dimensional data. By assuming common component factor loadings, this model allows clustering to be performed in the presence of a large number of mixture components or when the number of dimensions is too large to be well-modelled by the mixtures of factor analyzers model or a variant thereof. Furthermore, assuming that the component densities follow a skew-t distribution allows robust clustering of skewed data. The alternating expectation-conditional maximization algorithm is employed for parameter estimation. We demonstrate excellent clustering performance when our model is applied to real and simulated data.This paper marks the first time that skewed common factors have been used.

Tortora, Cristina, McNicholas, Paul D., Browne, Ryan P.

Model-based clustering imposes a finite mixture modelling structure on data for clustering. Finite mixture models assume that the population is a convex combination of a finite number of densities, the distribution within each population is a basic assumption of each particular model. Among all distributions that have been tried, the generalized hyperbolic distribution has the advantage that is a generalization of several other methods, such as the Gaussian distribution, the skew t-distribution, etc. With specific parameters, it can represent either a symmetric or a skewed distribution. While its inherent flexibility is an advantage in many ways, it means the estimation of more parameters than its special and limiting cases. The aim of this work is to propose a mixture of generalized hyperbolic factor analyzers to introduce parsimony and extend the method to high dimensional data. This work can be seen as an extension of the mixture of factor analyzers model to generalized hyperbolic mixtures. The performance of our generalized hyperbolic factor analyzers is illustrated on real data, where it performs favourably compared to its Gaussian analogue.