naginterfaces.library.stat.prob_​vonmises

naginterfaces.library.stat.prob_vonmises(t, vk)[source]

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

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

https://www.nag.com/numeric/nl/nagdoc_28.4/flhtml/g01/g01erf.html

Parameters
tfloat

, the observed von Mises statistic measured in radians.

vkfloat

The concentration parameter , of the von Mises distribution.

Returns
pfloat

The probability associated with the lower tail of the von Mises distribution between and .

Raises
NagValueError
(errno )

On entry, .

Constraint: .

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 and concentration parameter kappa, , can be written as

where is reduced modulo so that and . Note that if then prob_vonmises returns a probability of . For very small the distribution is almost the uniform distribution, whereas for all the probability is concentrated at one point.

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

In the case of small the series expansion of Pr(: ) can be expressed as

where is the modified Bessel function. This series expansion can be represented as a nested expression of terms involving the modified Bessel function ratio ,

which is calculated using backwards recursion.

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

where

and is computed from a continued fraction approximation. An approximation to order of the asymptotic normalizing series for 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 : The Von Mises distribution, ACM Trans. Math. Software (3), 279–284

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