Hidden truncation hyperbolic distributions, finite mixtures thereof, and their application for clustering

Abstract

A hidden truncation hyperbolic (HTH) distribution is introduced and finite mixtures thereof are applied for clustering. A stochastic representation of the HTH distribution is given and a density is derived. A hierarchical representation is described, which aids in parameter estimation. Finite mixtures of HTH distributions are presented and their identifiability is proved. The convexity of the HTH distribution is discussed, which is important in clustering applications, and some theoretical results in this direction are presented. The relationship between the HTH distribution and other skewed distributions in the literature is discussed. Illustrations are providedboth of the HTH distribution and application of finite mixtures thereof for clustering.

Mixtures of Hidden Truncation Hyperbolic Factor Analyzers

“…is the squared Mahalanobis distance between x and µ, h p (· | µ, Ω, λ, ω, ω) is the density of a p-dimensional symmetric hyperbolic random variable and H q (· | µ, Σ, λ, ω, ω) is the corresponding q-dimensional CDF. Note that Λ is a p × r skewness matrix, where 1 ≤ r ≤ p. Refer to Murray et al (2017a) for extensive details on the HTH distribution.…”

“…The reader is directed to the supplementary material in Murray et al (2017a) for details on a method for estimating this expectation via a series expansion.…”

“…can easily be estimated using the moments of the truncated symmetric hyperbolic distribution defined in Murray et al (2017a).…”

“…Murray et al (2017a) introduce the hidden truncation hyperbolic (HTH) distribution. The HTH approach is based on a truncated hyperbolic random variable, and Murray et al (2017a) demonstrate the effectiveness of a mixture of HTH distributions for clustering. The HTH distribution contains certain formulations of skew-t and skew-normal distributions as limiting cases.…”

Multivariate Cluster Weighted Models Using Skewed Distributions

“…Additionally, many distributions that parameterize both skewness and tail weight have also been proposed. These include, but are not limited to, work where mixture components follow a skew-t distribution (Lin 2010, Vrbik & McNicholas 2012, 2014, Lee & McLachlan 2014, a normal inverse Gaussian distribution (Karlis & Santourian 2009), a variance-gamma (McNicholas et al 2017), a generalized hyperbolic , a hidden truncation hyperbolic distribution (Murray et al 2017(Murray et al , 2020, or a skewed power exponential distribution (Dang et al 2019). All of these allow for the modelling of skewed data, which when modelled by a Gaussian distribution has a tendency to over fit the true number of components.…”