Mixtures of Common Skew-t Factor Analyzers

arXiv.org Machine Learning

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

arXiv.org Machine Learning

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.


Mixtures of Shifted Asymmetric Laplace Distributions

arXiv.org Machine Learning

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.


Multivariate response and parsimony for Gaussian cluster-weighted models

arXiv.org Machine Learning

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 eigen-decomposition 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-\`{a}-vis clustering, over existing methodologies.


Mixtures of Matrix Variate Bilinear Factor Analyzers

arXiv.org Machine Learning

Over the years data has become increasingly higher dimensional, which has prompted an increased need for dimension reduction techniques. This is perhaps especially true for clustering (unsupervised classification) as well as semi-supervised and supervised classification. Although dimension reduction in the area of clustering for multivariate data has been quite thoroughly discussed in the literature, there is relatively little work in the area of three way, or matrix variate, data. Herein, we develop a mixture of matrix variate bilinear factor analyzers (MMVBFA) model for use in clustering high-dimensional matrix variate data. This work can be considered both the first matrix variate bilinear factor analyzers model as well as the first MMVBFA model. Parameter estimation is discussed, and the MMVBFA model is illustrated using simulated and real data.