A Bayesian nonparametric model for bounded directional data on the positive orthant of the unit sphere

Abstract

Directional data appears in several branches of research. In some cases, those directional variables are only defined in subsets of the K-dimensional unit sphere. For example, in some applications, angles as measured responses are limited on the positive orthant. Analysis on subsets of the K-dimensional unit sphere is challenging and nowadays there are not many proposals that discuss this topic. Thus, from a methodological point of view, it is important to have probability distributions defined on bounded subsets of the K-dimensional unit sphere. Specifically, in this paper, we introduce a nonparametric Bayesian model to describe directional variables restricted to the first orthant. This model is based on a Dirichlet process mixture model with multivariate projected Gamma densities as kernel distributions. We show how to carry out inference for the proposed model based on a slice sampling scheme. The proposed methodology is illustrated using simulated data sets as well as a real data set.

Keywords:

• Multivariate projected gamma distribution
• Dirichlet process mixture
• circular data
• spherical data

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The work of the first author was supported by CONACYT, Mexico. The second author was partially supported from CONACYT, through Sistema Nacional de Investigadores, Mexico. The support received from the Department of Mathematics of the Metropolitan Autonomous University, Iztapalapa Unit is also gratefully acknowledged. Finally, the authors are grateful to the anonymous reviewers for their detailed and insightful comments.