Bivariate von Mises distribution
In probability theory and statistics, the bivariate von Mises distribution is a probability distribution describing values on a torus. It may be thought of as an analogue on the torus of the bivariate normal distribution. The distribution belongs to the field of directional statistics. The general bivariate von Mises distribution was first proposed by Kanti Mardia in 1975. One of its variants is today used in the field of bioinformatics to formulate a probabilistic model of protein structure in atomic detail, such as backbone-dependent rotamer libraries. Definition. The bivariate von Mises distribution is a probability  distribution defined on the torus, formula_1 in formula_2. The probability density function of the general bivariate von Mises distribution for the angles formula_3 is given by where formula_5 and formula_6 are the means for formula_7 and formula_8, formula_9 and formula_10 their concentration and the matrix formula_11 is related to their correlation. Two commonly used variants of the bivariate von Mises distribution are the sine and cosine variant. The cosine variant of the bivariate von Mises distribution has the probability density function where formula_5 and formula_6 are the means for formula_7 and formula_8, formula_9 and formula_10 their concentration and formula_19 is  related to their correlation. formula_20 is the normalization constant. This distribution with formula_19=0 has been used for kernel density estimates of the distribution of the protein dihedral angles formula_7 and formula_8. The sine variant has the probability density function where the parameters have the same interpretation.