Bivariate von Mises densities for angular data with applications to protein bioinformatics

Mardia, K.V., Taylor, C.C. & Subramaniam, G.K.

Motivated by a fundamental problem in bioinformatics, we examine two natural bivariate von Mises distributions referred to as the Sine model and Cosine model. These distributions have five parameters as in the bivariate normal distribution and, for concentrated data, these tend to a bivariate normal distribution. These are analyzed and their main properties derived. Specifically, conditions on the parameters are established which result in bimodal behaviour for the joint density and the marginal distribution, and proofs are given. An interesting situation can occur in which the joint density is bimodal, but the marginal distributions are unimodal. By considering contour plots of the densities and trigonometric moments we make comparisons of the two models by an assessment of how the parameters can capture correlation, and it is seen that the Cosine model is preferred in this respect. Efficient simulation methods are described for both the Sine and Cosine model. The Cosine model, and mixtures thereof, are fitted to real data using the EM algorithm, with applications in conformational angles in protein structure.

Keywords: Bivariate angular data, Bivariate circular mixture, Directional statistics, Distribution on Torus, Myoglobin, Protein conformational angles, Ramachandran diagrams.