Location and scale mixtures of Gaussians with flexible tail behaviour: Properties, inference and application to multivariate clustering

Abstract

International audienceThe family of location and scale mixtures of Gaussians has the ability to generate a number of flexible distributional forms. The family nests as particular cases several important asymmetric distributions like the Generalized Hyperbolic distribution. The Generalized Hyperbolic distribution in turn nests many other well known distributions such as the Normal Inverse Gaussian. In a multivariate setting, an extension of the standard location and scale mixture concept is proposed into a so called multiple scaled framework which has the advantage of allowing different tail and skewness behaviours in each dimension with arbitrary correlation between dimensions. Estimation of the parameters is provided via an EM algorithm and extended to cover the case of mixtures of such multiple scaled distributions for application to clustering. Assessments on simulated and real data confirm the gain in degrees of freedom and flexibility in modelling data of varying tail behaviour and directional shape

Inverse regression approach to robust nonlinear high-to-low dimensional mapping

“…As another future work, our derivations would be similar for other members in the scale mixture of Gaussians family or among other elliptical distributions. It would be interesting to study in a regression context, the use of distributions with even more flexible tails such as multiple scale Student distributions [15] or various skew-t [23,26,28] or Normal Inverse Gaussian distributions and more generally non-elliptically contoured distributions [32,41]. At last, another interesting direction of research would be to further complement the model with sparsity inducing penalties in particular for situations where interpreting the influential covariates is important.…”

Comparison and Classification of Flexible Distributions for Multivariate Skew and Heavy-Tailed Data

“…In order to provide a wider variety of distributional forms, Wraith and Forbes [41] proposed an extension of their multiple scale mixtures to multiple location-scale mixtures of multinormal distributions. This extension allows different tail and skewness behavior in each dimension of the variable space with arbitrary correlation between dimensions.…”

Supervised and Unsupervised Classification Using Mixture Models

“…First, in case of discrete data, multinomial models can be used, see for instance Bouguila et al (2003), Celeux and Govaert (1991) or Goldstein and Dillon (1978). Second, to tackle the case of heavytailed or asymmetric data, several extensions of Gaussian models have been recently introduced: Skew normal distribution (Vilca et al, 2014), t-distributions (Andrews andMcNicholas, 2012 or Forbes andWraith, 2014), asymmetric Laplace distribution (Franczak et al, 2014) and skew t-distributions (Lee andMcLachlan, 2013 or Wraith andForbes, 2015). Finally, an extension of the Gaussian mixture model to non quantitative data has been proposed by Bouveyron et.…”