A multivariate projected Gamma model for directional data

Abstract

Analysis of some real phenomena involve directional variables that by their nature are defined only in certain subsets of the k-dimensional unit sphere, ${\mathbb{S}}^k.$ For example, when working with axial data, the support of the associated directional variables turns out to be the interval $(0,\pi ].$ Thus, from a methodological point of view it is important to have probability distributions defined in bounded subsets of ${\mathbb{S}}^k.$ Specifically, in order to describe directional variables restricted to the first orthant, in this paper we introduce the Multivariate Projected Gamma model (MPG). This model is flexible enough and treats observations as projections onto the unit sphere of unobserved responses from a multivariate distribution which is generated as a product of k independent univariate Gamma distributions. Inference about the parameters of the model is based on samples from the corresponding joint posterior density, which is obtained using a Gibbs sampling after the introduction of suitable latent variables. The proposed methodology is illustrated using simulated data sets as well as a real data set.

Keywords:

• Circular data
• Gibbs sampler
• Radial projection
• Spherical data

Mathematics Subject Classification:

• 62H11 and 62F15

Acknowledgments

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