High-Dimensional Clustering with the Contaminated Gaussian Distribution

The contaminated Gaussian distribution represents a simple heavy-tailed elliptical generalization of the Gaussian distribution, differently from the often-considered $t$-distribution, it also allows for automatic detection of outlying or "bad" points in the same way that observations are typically assigned to the groups in the finite mixture model context. Starting from this distribution, we propose the contaminated Gaussian factor analysis model as a method for data reduction and detection of bad points in high-dimensions. A mixture of contaminated Gaussian factor analyzers (MCGFA) model follows therefrom, and extends the recently proposed mixture of contaminated Gaussian distributions to high-dimensional data, i.e., where $p$ (number of dimensions) is large relative to $n$ (sample size). We introduce a family of eight parsimonious models formed by introducing constraints on the covariance structure of the general MCGFA model. We outline a variant of the classical expectation-maximization algorithm for parameter estimation. Various implementation issues are discussed, and the novel model is compared to competing models on both simulated and real data.