Multivariate and regression models for directional data based on projected Pólya trees

Projected distributions have proved to be useful in the study of circular and directional data. Although any multivariate distribution can be used to produce a projected model, these distributions are typically parametric. In this article we consider a multivariate Pólya tree on R k and project it to the unit hypersphere S k to define a new Bayesian nonparametric model for directional data. We study the properties of the proposed model and in particular, concentrate on the implied conditional distributions of some directions given the others to define a directional–directional regression model. We also define a multivariate linear regression model with Pólya tree errors and project it to define a linear-directional regression model. We obtain the posterior characterisation of all models via their full conditional distributions. Metropolis-Hastings steps are required, where random walk proposal distributions are optimised with a novel adaptation scheme. We show the performance of our models with simulated and real datasets.