naginterfaces.library.stat.prob_​vonmises

naginterfaces.library.stat.prob_vonmises(t, vk)

prob_vonmises returns the probability associated with the lower tail of the von Mises distribution between $-\pi$ and $\pi$ through the function name.

For full information please refer to the NAG Library document for g01er

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/g01/g01erf.html

Parameters

tfloat

$\theta$, the observed von Mises statistic measured in radians.

vkfloat

The concentration parameter $\kappa$, of the von Mises distribution.

Returns

pfloat

The probability associated with the lower tail of the von Mises distribution between $-\pi$ and $\pi$.

Raises

NagValueError

(errno $1$)

On entry, $vk=⟨value⟩$.

Constraint: $vk\ge 0.0$.

Notes

The von Mises distribution is a symmetric distribution used in the analysis of circular data. The lower tail area of this distribution on the circle with mean direction ${\mu }_0=0$ and concentration parameter kappa, $\kappa$, can be written as

$$Pr(\Theta \le \theta :\kappa )=\frac{1}{2\pi I_0\left(\kappa \right)}{\int }_{-\pi }^{\theta }{e}^{\kappa \cos \left(\Theta \right)}d\Theta \text{,}$$

where $\theta$ is reduced modulo $2\pi$ so that $-\pi \le \theta <\pi$ and $\kappa \ge 0$. Note that if $\theta =\pi$ then prob_vonmises returns a probability of $1$. For very small $\kappa$ the distribution is almost the uniform distribution, whereas for $\kappa \to \infty$ all the probability is concentrated at one point.

The method of calculation for small $\kappa$ involves backwards recursion through a series expansion in terms of modified Bessel functions, while for large $\kappa$ an asymptotic Normal approximation is used.

In the case of small $\kappa$ the series expansion of Pr($\Theta \le \theta$: $\kappa$) can be expressed as

$$Pr(\Theta \le \theta :\kappa )=\frac{1}{2}+\frac{\theta }{\left(2\pi \right)}+\frac{1}{\pi I_0\left(\kappa \right)}\sum _{n=1}^{\infty }{n}^{-1}{I}_n\left(\kappa \right)\sin \left(n\right)\theta \text{,}$$

where $I_n\left(\kappa \right)$ is the modified Bessel function. This series expansion can be represented as a nested expression of terms involving the modified Bessel function ratio $R_n$,

$$R_n\left(\kappa \right)=\frac{I_n\left(\kappa \right)}{I_{n-1}\left(\kappa \right)}\text{,}n=1,2,3,\dots \text{,}$$

which is calculated using backwards recursion.

For large values of $\kappa$ (see Accuracy) an asymptotic Normal approximation is used. The angle $\Theta$ is transformed to the nearly Normally distributed variate $Z$,

$$Z=b\left(\kappa \right)\sin \left(\frac{\Theta }{2}\right)\text{,}$$

where

$$b\left(\kappa \right)=\frac{\sqrt{\frac{2}{\pi }}{e}^{\kappa }}{I_0\left(\kappa \right)}$$

and $b\left(\kappa \right)$ is computed from a continued fraction approximation. An approximation to order ${\kappa }^{-4}$ of the asymptotic normalizing series for $z$ is then used. Finally the Normal probability integral is evaluated.

For a more detailed analysis of the methods used see Hill (1977).

References

Hill, G W, 1977, Algorithm 518: Incomplete Bessel function $I_0$ : The Von Mises distribution, ACM Trans. Math. Software (3), 279–284

Mardia, K V, 1972, Statistics of Directional Data, Academic Press