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

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. In addition, Markov Chain Monte Carlo algorithms are presented for Bayesian estimation of both families and a simulation study is used to demonstrate their performance. As a real data example, a meteorological data set is analyzed.