📂Probability Distribution

Bivariate von Mises Distribution

Definition ¹

The mean direction $\mu, \nu \in \mathbb{R}$ and concentration $\kappa_{1}, \kappa_{2} > 0$ with respect to a certain matrix $A \in \mathbb{R}^{2 \times 2}$ have a continuous probability distribution $\text{vM}^{2} \left( \mu , \nu , \kappa_{1} , \kappa_{2} \right)$ with a probability density function that is proportional to the following, which is called the Bivariate von Mises Distribution: $$ \exp \left[ \kappa_{1} \cos \left( \theta - \mu \right) + \kappa_{2} \cos \left( \phi - \nu \right) + \begin{bmatrix} \cos \left( \theta - \mu \right) & \sin \left( \theta - \mu \right) \end{bmatrix} A \begin{bmatrix} \cos \left( \phi - \nu \right) \\ \sin \left( \phi - \nu \right) \end{bmatrix} \right] $$ Simplifying it to when $A = \begin{bmatrix} \alpha & 0 \\ 0 & \beta \end{bmatrix}$ yields $$ \exp \left[ \begin{align*} & \kappa_{1} \cos \left( \theta - \mu \right) + \kappa_{2} \cos \left( \phi - \nu \right) \\ &+ \alpha \cos \left( \theta - \mu \right) \cos \left( \phi - \nu \right) \\ &+ \beta \sin \left( \theta - \mu \right) \sin \left( \phi - \nu \right) \end{align*} \right] $$ For practical purposes, the following models, which further reduce the parameters, are famous.

Sine Model

When setting $\alpha = 0$ and $\beta = \lambda$, the bivariate von Mises distribution with the following probability density function is shortly referred to as the Sine Model: $$ f_{s} \left( \theta , \phi \right) := c \left( \kappa_{1} , \kappa_{2} \right) \exp \left[ \begin{align*} & \kappa_{1} \cos \left( \theta - \mu \right) + \kappa_{2} \cos \left( \phi - \nu \right) \\ &+ \lambda \sin \left( \theta - \mu \right) \sin \left( \phi - \nu \right) \end{align*} \right] \qquad , \left( \theta , \phi \right) \in \left[ 0, 2 \pi \right]^{2} $$ Here, $c \left( \kappa_{1} , \kappa_{2} \right)$ is a normalizing constant given as follows. $$ c \left( \kappa_{1} , \kappa_{2} \right) := 4 \pi^{2} \sum_{m=1}^{\infty} \binom{2m}{m} \left( {{ \lambda^{2} } \over { 4 \kappa_{1} \kappa_{2} }} \right)^{m} I_{m} \left( \kappa_{1} \right) I_{m} \left( \kappa_{2} \right) $$

Cosine Model

When setting $\alpha = \beta = - \kappa_{3}$ and satisfying $\min \left\{ \kappa_{1} , \kappa_{2} \right\} \ge \kappa_{3} \ge 0$, the bivariate von Mises distribution with the following probability density function is shortly referred to as the Cosine Model: $$ f_{c} \left( \theta , \phi \right) := c \left( \kappa_{1} , \kappa_{2} , \kappa_{3} \right) \exp \left[ \begin{align*} & \kappa_{1} \cos \left( \theta - \mu \right) + \kappa_{2} \cos \left( \phi - \nu \right) \\ &- \kappa_{3} \cos \left( \theta - \mu - \phi + \nu \right) \end{align*} \right] \qquad , \left( \theta , \phi \right) \in \left[ 0, 2 \pi \right]^{2} $$ Here, $c \left( \kappa_{1} , \kappa_{2} , \kappa_{3} \right)$ is a normalizing constant given as follows. $$ c \left( \kappa_{1} , \kappa_{2} , \kappa_{3} \right) := 4 \pi^{2} \left[ I_{0} \left( \kappa_{1} \right) I_{0} \left( \kappa_{2} \right) I_{0} \left( \kappa_{3} \right) + 2 \sum_{p=1}^{\infty} I_{p} \left( \kappa_{1} \right) I_{p} \left( \kappa_{2} \right) I_{p} \left( \kappa_{3} \right) \right] $$

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• $I_{\nu}$ is a Modified Bessel Function of the First Kind of order $\nu$, and to understand why such complex functions are used, refer to the post on why modified Bessel functions of the first kind appear in directional statistics.

Description ²

If the von Mises Distribution can be considered a normal distribution on a unit circle $S^{1}$, then the bivariate von Mises distribution can be seen as a normal distribution on a torus $S^{1} \times S^{1}$ as shown in the figure above.

You might wonder what the point is of studying donuts, but in reality, similar motifs can be found in applications such as in bioinformatics, where it’s relevant to understand the molecular structure of proteins and the angles at which they are connected.