The function may be called by the names: g01erc, nag_stat_prob_vonmises or nag_prob_von_mises.
3Description
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 g01erc 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 Section 7) 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).
4References
Hill G W (1977) Algorithm 518: Incomplete Bessel function : The Von Mises distribution ACM Trans. Math. Software3 279–284
Mardia K V (1972) Statistics of Directional Data Academic Press
5Arguments
1: – doubleInput
On entry: , the observed von Mises statistic measured in radians.
2: – doubleInput
On entry: the concentration parameter , of the von Mises distribution.
Constraint:
.
3: – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_REAL
On entry, .
Constraint: .
7Accuracy
g01erc uses one of two sets of constants depending on the value of machine precision. One set gives an accuracy of six digits and uses the Normal approximation when , the other gives an accuracy of digits and uses the Normal approximation when .
8Parallelism and Performance
g01erc is not threaded in any implementation.
9Further Comments
Using the series expansion for small the time taken by g01erc increases linearly with ; for larger , for which the asymptotic Normal approximation is used, the time taken is much less.
If angles outside the region are used care has to be taken in evaluating the probability of being in a region if the region contains an odd multiple of , . The value of will be negative and the correct probability should then be obtained by adding one to the value.
10Example
This example inputs four values from the von Mises distribution along with the values of the parameter . The probabilities are computed and printed.