Mixtures of Multivariate Power Exponential Distributions

Abstract

An expanded family of mixtures of multivariate power exponential distributions is introduced. While fitting heavy-tails and skewness have received much attention in the model-based clustering literature recently, we investigate the use of a distribution that can deal with both varying tail-weight and peakedness of data. A family of parsimonious models is proposed using an eigen-decomposition of the scale matrix. A generalized expectation-maximization algorithm is presented that combines convex optimization via a minorization-maximization approach and optimization based on accelerated line search algorithms on the Stiefel manifold. Lastly, the utility of this family of models is illustrated using both toy and benchmark data.

Model-Based Clustering

“…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. Analogues of all 14 GPCMs, with the option to constrain ν g = ν, are implemented in the teigen package (Andrews et al 2015) for R. While mixtures of t-distributions have been the most popular approach for clustering with heavier tail weight, mixtures of multivariate power exponential (MPE) distributions have emerged as an alternative and are used for clustering by Dang, Browne, and McNicholas (2015). In addition to allowing heavier tails, the MPE distribution also permits lighter tails compared to the Gaussian distribution.…”

“…In addition to allowing heavier tails, the MPE distribution also permits lighter tails compared to the Gaussian distribution. Dang et al (2015) use the parametrization of the MPE distribution given by Gómez, Gómez-Viilegas, and Marin (1998), so that the component density for the mixture of MPEs is given by…”

“…Some well-known distributions arise as special or limiting cases of the MPE distribution, e.g., a double-exponential distribution (β g = 0.5), a Gaussian distribution (β g = 1), or a multivariate uniform distribution (β g → ∞). Dang et al (2015) use analogues of some members of the GPCM family, which, along with the option to constrain β g = β, leads to a family of sixteen mixture models. Contour plots for the bivariate power exponential distribution illustrate some of the flexibility available for different values of β (Figure 2).…”

A matrix variate skew‐t distribution

“…For example, some work has been done using symmetric component densities that parameterize concentration (tail weight), e.g., the t distribution , Andrews & McNicholas 2011, Lin, McNicholas & Hsiu 2014) and the power exponential distribution (Dang, Browne & McNicholas 2015). There has also been work on mixtures for discrete data (e.g., Karlis & Meligkotsidou 2007, Bouguila & ElGuebaly 2009) as well as several examples of mixtures of skewed distributions such as the NIG distribution (Karlis & Santourian 2009, Subedi & McNicholas 2014, the skew-t distribution (Lin 2010, Vrbik & McNicholas 2012, Lee & McLachlan 2014, 2016, the shifted asymmetric Laplace distribution (Morris & McNicholas 2013, Franczak, Browne & McNicholas 2014, the variance-gamma distribution , the generalized hyperbolic distribution , and others (e.g., Elguebaly & Bouguila 2015, Franczak, Tortora, Browne & McNicholas 2015.…”

Estimation of Star-Shaped Distributions

“…The l n,p -elliptically contoured distributions build another big class of star-shaped distributions and are used in [27] to explore to which extent orientation selectivity and contrast gain control can be used to model the statistics of natural images. Mixtures of ecpe distributions are considered for bioinformation data sets in [28]. Texture retrieval using the p-generalized Gaussian densities is studied in [29].…”