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.[1] [2] One of its variants is today used in the field of bioinformatics to formulate a probabilistic model of protein structure in atomic detail, [3] [4] such as backbone-dependent rotamer libraries.
The bivariate von Mises distribution is a probability distribution defined on the torus,
S1 x S1
R3
\phi,\psi\in[0,2\pi]
f(\phi,\psi)\propto\exp[\kappa1\cos(\phi-\mu)+\kappa2\cos(\psi-\nu)+(\cos(\phi-\mu),\sin(\phi-\mu))A(\cos(\psi-\nu),\sin(\psi-\nu))T],
where
\mu
\nu
\phi
\psi
\kappa1
\kappa2
A\inM(2,2)
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
f(\phi,\psi)=Zc(\kappa1,\kappa2,\kappa3) \exp[\kappa1\cos(\phi-\mu)+\kappa2\cos(\psi-\nu)-\kappa3\cos(\phi-\mu-\psi+\nu)],
where
\mu
\nu
\phi
\psi
\kappa1
\kappa2
\kappa3
Zc
\kappa3
\phi
\psi
The sine variant has the probability density function[5]
f(\phi,\psi)=Zs(\kappa1,\kappa2,\kappa3) \exp[\kappa1\cos(\phi-\mu)+\kappa2\cos(\psi-\nu)+\kappa3\sin(\phi-\mu)\sin(\psi-\nu)],
where the parameters have the same interpretation.