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.
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.
Factor analysis and principal components analysis can be used to model linear relationships between observed variables and linearly map high-dimensional data to a lower-dimensional hidden space. In factor analysis, the observations are modeled as a linear combination ofnormally distributed hidden variables. We describe a nonlinear generalization of factor analysis, called "product analysis", thatmodels the observed variables as a linear combination of products of normally distributed hidden variables. Just as factor analysiscan be viewed as unsupervised linear regression on unobserved, normally distributed hidden variables, product analysis canbe viewed as unsupervised linear regression on products of unobserved, normally distributed hidden variables. The mapping betweenthe data and the hidden space is nonlinear, so we use an approximate variational technique for inference and learning.
Clustering is the process of finding and analyzing underlying group structures in data. In recent years, data as become increasingly higher dimensional and therefore an increased need for dimension reduction techniques for use in clustering. Although such techniques are firmly established in the literature for multivariate data, there is a relative paucity in the area of matrix variate or three way data. Furthermore, these few methods all assume matrix variate normality which is not always sensible if skewness is present. We propose a mixture of bilinear factor analyzers model using four skewed matrix variate distributions, namely the matrix variate skew-t, generalized hyperbolic, variance gamma and normal inverse Gaussian distributions.
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