Bivariate von Mises distribution

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 ${\displaystyle \phi ,\psi \in [0,2\pi ]}$ is given by^[1]

${\displaystyle f(\phi ,\psi )\propto \exp[\kappa _{1}\cos(\phi -\mu )+\kappa _{2}\cos(\psi -\nu )+(\cos(\phi -\mu ),\sin(\phi -\mu ))\mathbf {A} (\cos(\psi -\nu ),\sin(\psi -\nu ))^{T}],}$

where ${\displaystyle \mu }$ and ${\displaystyle \nu }$ are the means for ${\displaystyle \phi }$ 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^[3] has the probability density function

${\displaystyle f(\phi ,\psi )=Z_{c}(\kappa _{1},\kappa _{2},\kappa _{3})\ \exp[\kappa _{1}\cos(\phi -\mu )+\kappa _{2}\cos(\psi -\nu )-\kappa _{3}\cos(\phi -\mu -\psi +\nu )],}$

where ${\displaystyle \mu }$ and ${\displaystyle \nu }$ are the means for ${\displaystyle \phi }$ 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 ${\displaystyle \phi }$ and ${\displaystyle \psi }$.^[4]

The sine variant has the probability density function^[5]

${\displaystyle f(\phi ,\psi )=Z_{s}(\kappa _{1},\kappa _{2},\kappa _{3})\ \exp[\kappa _{1}\cos(\phi -\mu )+\kappa _{2}\cos(\psi -\nu )+\kappa _{3}\sin(\phi -\mu )\sin(\psi -\nu )],}$

where the parameters have the same interpretation.

• Von Mises distribution, a similar distribution on the one-dimensional unit circle
• Kent distribution, a related distribution on the two-dimensional unit sphere
• von Mises–Fisher distribution
• Directional statistics