Extending mixtures of multivariate t-factor analyzers

Abstract

Model-based clustering typically involves the development of a family of mixture models and the imposition of these models upon data. The best member of the family is then chosen using some criterion and the associated parameter estimates lead to predicted group memberships, or clusterings. This paper describes the extension of the mixtures of multivariate t-factor analyzers model to include constraints on the degrees of freedom, the factor loadings, and the error variance matrices. The result is a family of six mixture models, including parsimonious models. Parameter estimates for this family of models are derived using an alternating expectation-conditional maximization algorithm and convergence is determined based on Aitken's acceleration. Model selection is carried out using the Bayesian information criterion (BIC) and the integrated completed likelihood (ICL). This novel family of mixture models is then applied to simulated and real data where clustering performance meets or exceeds that of established model-based clustering methods. The simulation studies include a comparison of the BIC and the ICL as model selection techniques for this novel family of models. Application to simulated data with larger dimensionality is also explored.

Model-based clustering of high-dimensional data: A review

“…Among the related works, we may cite the robust versions [2,66] of the MFA models which relies on t-distributions and the GMMDR technique [89] which iteratively looks for the subspace containing the most relevant information for clustering. Let us finally notice that some subspace clustering methods have been modified for performing variable selection through ℓ 1 penalization.…”

teigen: An R Package for Model-Based Clustering and Classification via the Multivariate t Distribution

“…As such, the teigen package will scale variables to have mean 0 and variance 1 by default (see Section 3 for controlling this option). In addition, constraints are also imposed on the degrees of freedom ν g , following Andrews and McNicholas (2011a). Thus, taking all possible combinations of these constraints into consideration would result in a 28-model family (see Table 1), 24 of which were originally developed for the tEIGEN family (Andrews and McNicholas 2012).…”

Model-Based Clustering

“…The mixture of t-distributions model has only G more free parameters than the mixture of Gaussian distributions. Andrews and McNicholas (2011a) consider the option to constrain degrees of freedom to be equal across components, i.e., ν g = ν, which can lead to improved classification performance by effectively allowing the borrowing of information across components to estimate the degrees of freedom. Andrews and McNicholas (2012) introduce a t-analogue of 12 members of the GPCM family of models by imposing the constraints in Table 1 on the component scale matrices Σ g , while also allowing the constraint ν g = ν.…”