Multivariate and regression models for directional data based on projected P\'olya trees

TL;DR

This paper introduces a Bayesian nonparametric model for directional data by projecting multivariate Pólya trees onto the hypersphere, enabling flexible regression analysis for circular and directional datasets.

Contribution

It develops a novel nonparametric Bayesian framework for directional data using projected Pólya trees, including regression models for directional and linear-directional relationships.

Findings

Models perform well on simulated data.

Effective in real-world directional datasets.

Flexible Bayesian nonparametric approach.

Abstract

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 P\'olya 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 P\'olya tree error and project it to define a linear-directional regression model. We obtain the posterior characterisation of all models and show their performance with simulated and real datasets.