Mixtures of Factor Analyzers with Fundamental Skew Symmetric Distributions
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Mixtures of factor analyzers (MFA) provide a powerful tool for modelling high-dimensional datasets. In recent years, several generalizations of MFA have been developed where the normality assumption of the factors and/or of the errors was relaxed to allow for skewness in the data. However, due to the form of the adopted component densities, the distribution of the factors/errors in most of these models are typically limited to modelling skewness concentrated in a single direction. We introduce a more flexible finite mixture of factor analyzers based on the family of scale mixtures of canonical fundamental skew normal (SMCFUSN) distributions. This very general family of skew distributions can capture various types of skewness and asymmetry in the data. In particular, the proposed mixture of SMCFUSN factor analyzers (SMCFUSNFA) model can simultaneously accommodate multiple directions of skewness. As such, it encapsulates many commonly used models as special and/or limiting cases, such as models of some versions of skew normal and skew t-factor analyzers, and skew hyperbolic factor analyzers. For illustration, we focus on the t-distribution member of the family of SMCFUSN distributions, leading to the model based on canonical fundamental skew t-factor analyzers (CFUSTFA). Parameter estimation can be carried out by maximum likelihood via an EM-type algorithm. The usefulness and potential of the proposed model are demonstrated using two real datasets.
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