Probabilistic model for two dependent circular variables

Motivated by problems in molecular biology and molecular physics, we propose a five-parameter torus analogue of the bivariate normal distribution for modelling the distribution of two circular random variables.The conditional distributions of the proposed distribution are von Mises. The marginal distributions are symmetric around their means and are either unimodal or bimodal. The type of shape depends on the configuration of parameters, and we derive the conditions that ensure a specific shape. The utility of the proposed distribution is illustrated by the modelling of angular variables in a short linear peptide.