An interpretable family of projected normal distributions and a related copula model for Bayesian analysis of hypertoroidal data

TL;DR

This paper develops two interpretable families of projected normal distributions for Bayesian analysis of hypertoroidal data, featuring closed-form marginals, flexible extensions, and intuitive parameters, demonstrated on meteorological data.

Contribution

Introduces two new families of projected normal distributions with clear parameters and marginal properties, suitable for Bayesian analysis of hypertoroidal data.

Findings

Distributions are closed under marginalization.

Flexible extension can model any univariate marginal.

Effective Bayesian estimation via MCMC demonstrated.

Abstract

This paper introduces two families of probability distributions for Bayesian analysis of hypertoroidal data. The first family consists of symmetric distributions derived from the projection of multivariate normal distributions under specific parameter constraints. This family is closed under marginalization and hence any marginal distribution belongs to a lower-dimensional case of the same family. In particular the univariate marginal of the family is the unimodal case of the projected normal distribution on the circle. The second family is a flexible extension of the copula case of the first family, which can accommodate any univariate marginal distributions. Unlike existing models derived via projection, both families have the common advantage that their parameters possess a clear and intuitive interpretation. The use of latent variables simplifies Bayesian estimation using Markov chain Monte Carlo algorithms. The usefulness of the proposed families is demonstrated through the analysis of a meteorological dataset.