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20.1 Von Mises distribution

20.1.1 Probability density function

If $\mu \in \mathbb{R}$ and $\kappa \in \mathbb{R}^+$, then for $y \in \mathbb{R}$, $$\text{VonMises}(y|\mu,\kappa) = \frac{\exp(\kappa\cos(y-\mu))}{2\pi I_0(\kappa)} \!.$$ In order for this density to properly normalize, $y$ must be restricted to some interval $(c, c + 2\pi)$ of length $2 \pi$, because $$\int_{c}^{c + 2\pi} \text{VonMises}(y|\mu,\kappa) dy = 1.$$ Similarly, if $\mu$ is a parameter, it will typically be restricted to the same range as $y$.

If $\kappa > 0$, a von Mises distribution with its $2 \pi$ interval of support centered around its location $\mu$ will have a single mode at $\mu$; for example, restricting $y$ to $(-\pi,\pi)$ and taking $\mu = 0$ leads to a single local optimum at the mode $\mu$. If the location $\mu$ is not in the center of the support, the density is circularly translated and there will be a second local maximum at the boundary furthest from the mode. Ideally, the parameterization and support will be set up so that the bulk of the probability mass is in a continuous interval around the mean $\mu$.

For $\kappa = 0$, the Von Mises distribution corresponds to the circular uniform distribution with density $1 / (2 \pi)$ (independently of the values of $y$ or $\mu$).

20.1.2 Sampling statement

y ~ von_mises(mu, kappa)

Increment target log probability density with von_mises_lupdf(y | mu, kappa).

20.1.3 Stan functions

R von_mises_lpdf(reals y | reals mu, reals kappa)
The log of the von mises density of y given location mu and scale kappa.

R von_mises_lupdf(reals y | reals mu, reals kappa)
The log of the von mises density of y given location mu and scale kappa dropping constant additive terms.

R von_mises_rng(reals mu, reals kappa)
Generate a Von Mises variate with location mu and scale kappa (i.e. returns values in the interval $[(\mu \mod 2\pi)-\pi,(\mu \mod 2\pi)+\pi]$); may only be used in transformed data and generated quantities blocks. For a description of argument and return types, see section vectorized PRNG functions.

20.1.4 Numerical stability

Evaluating the Von Mises distribution for $\kappa > 100$ is numerically unstable in the current implementation. Nathanael I. Lichti suggested the following workaround on the Stan users group, based on the fact that as $\kappa \rightarrow \infty$, $$\text{VonMises}(y|\mu,\kappa) \rightarrow \text{Normal}(\mu, \sqrt{1 / \kappa}).$$ The workaround is to replace y ~ von_mises(mu,kappa) with

 if (kappa < 100)
   y ~ von_mises(mu, kappa);
 else
   y ~ normal(mu, sqrt(1 / kappa));