Bayesian Analysis of Semiparametric Linear-Circular Models

In many environmental and agricultural studies, data are collected on both linear and circular random variables, with possible dependence between the variables. Classically, the analysis of such data has been carried out in a classical regression framework. We propose a Bayesian hierarchical framework to handle all forms of uncertainty arising in a linear-circular data set. One novelty of our multivariate linear-circular model is that, marginally, the circular component is assumed to be a mixture model with an unknown number of von Mises (or circular normal) distributions. We use the Dirichlet process to introduce variability in the model dimensionality, and develop a simple Gibbs sampling algorithm for simulating the mixture components. Although we illustrate our methodology on von Mises mixtures, it is widely applicable. We thus avoid complicated reversible-jump Markov chain Monte Carlo methods, which are considered ideal for analyzing mixtures of unknown number of distributions. We illustrate our methodologies with simulated and real data sets. Using pseudo-Bayes factors, we also compare different models associated with both fixed and variable numbers of von Mises distributions. Our findings suggest that models associated with varying numbers of mixture components perform at least as well as those with known numbers of mixture components. We tentatively argue that model averaging associated with variable number of mixture components improves the model's predictive power, which compensates for the lack of knowledge of the actual number of mixture components.