Bivariate von Mises distribution

Definition

The bivariate von Mises distribution is a probability distribution defined on the torus, ${\displaystyle S^1\times S^1}$ in ${\displaystyle {\mathbb{R}}^3}$. The probability density function of the general bivariate von Mises distribution for the angles ${\displaystyle \varphi ,\psi \in [0,2\pi ]}$ is given by

${\displaystyle f(\varphi ,\psi )\propto \exp [{\kappa }_1\cos (\varphi -\mu )+{\kappa }_2\cos (\psi -\nu )+(\cos (\varphi -\mu ),\sin (\varphi -\mu ))\mathbf{A}(\cos (\psi -\nu ),\sin (\psi -\nu ){)}^T],}$

where ${\displaystyle \mu }$ and ${\displaystyle \nu }$ are the means for ${\displaystyle \varphi }$ and ${\displaystyle \psi }$, ${\displaystyle {\kappa }_1}$ and ${\displaystyle {\kappa }_2}$ their concentration and the matrix ${\displaystyle \mathbf{A}\in \mathbb{M}(2,2)}$ 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

${\displaystyle f(\varphi ,\psi )=Z_c({\kappa }_1,{\kappa }_2,{\kappa }_3)\exp [{\kappa }_1\cos (\varphi -\mu )+{\kappa }_2\cos (\psi -\nu )-{\kappa }_3\cos (\varphi -\mu -\psi +\nu )],}$

where ${\displaystyle \mu }$ and ${\displaystyle \nu }$ are the means for ${\displaystyle \varphi }$ and ${\displaystyle \psi }$, ${\displaystyle {\kappa }_1}$ and ${\displaystyle {\kappa }_2}$ their concentration and ${\displaystyle {\kappa }_3}$ is related to their correlation. ${\displaystyle Z_c}$ is the normalization constant. This distribution with ${\displaystyle {\kappa }_3}$=0 has been used for kernel density estimates of the distribution of the protein dihedral angles ${\displaystyle \varphi }$ and ${\displaystyle \psi }$.

The sine variant has the probability density function

${\displaystyle f(\varphi ,\psi )=Z_s({\kappa }_1,{\kappa }_2,{\kappa }_3)\exp [{\kappa }_1\cos (\varphi -\mu )+{\kappa }_2\cos (\psi -\nu )+{\kappa }_3\sin (\varphi -\mu )\sin (\psi -\nu )],}$

where the parameters have the same interpretation.