### Mixtures of Skewed Matrix Variate Bilinear Factor Analyzers

Clustering is the process of finding and analyzing underlying group structures in data. In recent years, data as become increasingly higher dimensional and therefore an increased need for dimension reduction techniques for use in clustering. Although such techniques are firmly established in the literature for multivariate data, there is a relative paucity in the area of matrix variate or three way data. Furthermore, these few methods all assume matrix variate normality which is not always sensible if skewness is present. We propose a mixture of bilinear factor analyzers model using four skewed matrix variate distributions, namely the matrix variate skew-t, generalized hyperbolic, variance gamma and normal inverse Gaussian distributions.

### Mixtures of Common Skew-t Factor Analyzers

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.

### A Mixture of Generalized Hyperbolic Factor Analyzers

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.

### Parsimonious Shifted Asymmetric Laplace Mixtures

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.

### Clustering Higher Order Data: Finite Mixtures of Multidimensional Arrays

There have been many examples of clustering multivariate (i.e., two-way) data using finite mixture models (see, e.g., reviews by Fraley and Raftery, 2002; Bouveyron and Brunet-Saumard, 2014; McNicholas, 2016b). More recently, there have been some notable examples of clustering threeway data using finite mixtures of matrix-variate distributions (e.g., Viroli, 2011; Anderlucci et al., 2015; Gallaugher and McNicholas, 2018a). This work on clustering three-way data is timely in the sense that the variety of data that require clustering continues to increase. Furthermore, there is no reason to believe that this need ends with three-way data. An approach for clustering multi-way data is introduced based on a finite mixture of multidimensional arrays. While some might refer to such structures as'tensors', and so write about clustering tensor-variate data, we prefer the nomenclature multidimensional array to avoid confusion with the term'tensor' as used in engineering and physics, e.g., tensor fields.