On the circular correlation coefficients for bivariate von Mises distributions on a torus

This paper studies circular correlations for the bivariate von Mises sine and cosine distributions. These are two simple and appealing models for bivariate angular data with five parameters each that have interpretations connected to those in the ordinary bivariate normal model. However, the variability and association of the angle pairs cannot be easily deduced from the model parameters unlike the bivariate normal. Thus to compute such summary measures, tools from circular statistics are needed. We derive analytic expressions and study the properties of the Jammalamadaka–Sarma and Fisher–Lee circular correlation coefficients for the von Mises sine and cosine models. Likelihood-based inference of these coefficients from sample data is then presented. The correlation coefficients are illustrated with numerical and visual examples, and the maximum likelihood estimators are assessed on simulated and real data, with comparisons to their non-parametric counterparts. Implementations of these computations for practical use are provided in our R package BAMBI.