We introduce a mixture of generalized hyperbolic distributions as an alternative to the ubiquitous mixture of Gaussian distributions as well as their near relatives of which the mixture of multivariate t and skew-t distributions are predominant. The mathematical development of our mixture of generalized hyperbolic distributions model relies on its relationship with the generalized inverse Gaussian distribution. The latter is reviewed before our mixture models are presented along with details of the aforesaid reliance. Parameter estimation is outlined within the expectation-maximization framework before the clustering performance of our mixture models is illustrated via applications on simulated and real data. In particular, the ability of our models to recover parameters for data from underlying Gaussian and skew-t distributions is demonstrated. Finally, the role of Generalized hyperbolic mixtures within the wider modelbased clustering, classification, and density estimation literature is discussed.

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 generalized 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.

Mixtures of skew-t distributions offer a flexible choice for model-based clustering. A mixture model of this sort can be implemented using a variety of formulations of the skew-t distribution. Herein we develop a mixture of skew-t factor analyzers model for clustering of high-dimensional data using a flexible formulation of the skew-t distribution. Methodological details of our approach, which represents an extension of the mixture of factor analyzers model to a flexible skew-t distribution, are outlined and details of parameter estimation are provided. Clustering results are illustrated and compared to an alternative formulation of the mixture of skew-t factor analyzers model as well as the mixture of factor analyzers model.

An expanded family of mixtures of multivariate power exponential distributions is introduced. While fitting heavy-tails and skewness have received much attention in the model-based clustering literature recently, we investigate the use of a distribution that can deal with both varying tail-weight and peakedness of data. A family of parsimonious models is proposed using an eigen-decomposition of the scale matrix. A generalized expectation-maximization algorithm is presented that combines convex optimization via a minorization-maximization approach and optimization based on accelerated line search algorithms on the Stiefel manifold. Lastly, the utility of this family of models is illustrated using both toy and benchmark data.

We introduce a mixture model whereby each mixture component is itself a mixture of a multivariate Gaussian distribution and a multivariate uniform distribution. Although this model could be used for model-based clustering (model-based unsupervised learning) or model-based classification (model-based semi-supervised learning), we focus on the more general model-based classification framework. In this setting, we fit our mixture models to data where some of the observations have known group memberships and the goal is to predict the memberships of observations with unknown labels. We also present a density estimation example. A generalized expectation-maximization algorithm is used to estimate the parameters and thereby give classifications in this mixture of mixtures model. To simplify the model and the associated parameter estimation, we suggest holding some parameters fixed-this leads to the introduction of more parsimonious models. A simulation study is performed to illustrate how the model allows for bursts of probability and locally higher tails. Two further simulation studies illustrate how the model performs on data simulated from multivariate Gaussian distributions and on data from multivariate t-distributions. This novel approach is also applied to real data and the performance of our approach under the various restrictions is discussed.

A family of parsimonious Gaussian cluster-weighted models is presented. This family concerns a multivariate extension to cluster-weighted modelling that can account for correlations between multivariate responses. Parsimony is attained by constraining parts of an eigendecomposition imposed on the component covariance matrices. A sufficient condition for identifiability is provided and an expectation-maximization algorithm is presented for parameter estimation. Model performance is investigated on both synthetic and classical real data sets and compared with some popular approaches. Finally, accounting for linear dependencies in the presence of a linear regression structure is shown to offer better performance, vis-à-vis clustering, over existing methodologies.

We consider the problem of minimizing an objective function that depends on an orthonormal matrix. This situation is encountered when looking for common principal components, for example, and the Flury method is a popular approach. However, the Flury method is not effective for higher dimensional problems. We obtain several simple majorization-minizmation (MM) algorithms that provide solutions to this problem and are effective in higher dimensions. We then use simulated data to compare them with other approaches in terms of convergence and computational time.

The mixture of factor analyzers model, which has been used successfully for the model-based clustering of high-dimensional data, is extended to generalized hyperbolic mixtures. The development of a mixture of generalized hyperbolic factor analyzers is outlined, drawing upon the relationship with the generalized inverse Gaussian distribution. An alternating expectation-conditional maximization algorithm is used for parameter estimation, and the Bayesian information criterion is used to select the number of factors as well as the number of components. The performance of our generalized hyperbolic factor analyzers model is illustrated on real and simulated data, where it performs favourably compared to its Gaussian analogue and other approaches.