Constrained Optimization for a Subset of the Gaussian Parsimonious Clustering Models

The expectation-maximization (EM) algorithm is an iterative method for finding maximum likelihood estimates when data are incomplete or are treated as being incomplete. The EM algorithm and its variants are commonly used for parameter estimation in applications of mixture models for clustering and classification. This despite the fact that even the Gaussian mixture model likelihood surface contains many local maxima and is singularity riddled. Previous work has focused on circumventing this problem by constraining the smallest eigenvalue of the component covariance matrices. In this paper, we consider constraining the smallest eigenvalue, the largest eigenvalue, and both the smallest and largest within the family setting. Specifically, a subset of the GPCM family is considered for model-based clustering, where we use a re-parameterized version of the famous eigenvalue decomposition of the component covariance matrices. Our approach is illustrated using various experiments with simulated and real data

Finite mixtures of matrix-variate Poisson-log normal distributions for three-way count data

Three-way data structures, characterized by three entities, the units, the variables and the occasions, are frequent in biological studies. In RNA sequencing, three-way data structures are obtained when high-throughput transcriptome sequencing data are collected for n genes across p conditions at r occasions. Matrix-variate distributions offer a natural way to model three-way data and mixtures of matrix-variate distributions can be used to cluster three-way data. Clustering of gene expression data is carried out as means to discovering gene co-expression networks. In this work, a mixture of matrix-variate Poisson-log normal distributions is proposed for clustering read counts from RNA sequencing. By considering the matrix-variate structure, full information on the conditions and occasions of the RNA sequencing dataset is simultaneously considered, and the number of covariance parameters to be estimated is reduced. A Markov chain Monte Carlo expectation-maximization algorithm is used for parameter estimation and information criteria are used for model selection. The models are applied to both real and simulated data, giving favourable clustering results