NAG Library Function Document
nag_prob_von_mises (g01erc) returns the probability associated with the lower tail of the von Mises distribution between and .
||nag_prob_von_mises (double t,
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 argument kappa,
, can be written as
is reduced modulo
. Note that if
then nag_prob_von_mises (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
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
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)
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
t – doubleInput
On entry: , the observed von Mises statistic measured in radians.
vk – doubleInput
On entry: the concentration parameter , of the von Mises distribution.
fail – NagError *Input/Output
The NAG error argument (see Section 3.6
in the Essential Introduction).
6 Error Indicators and Warnings
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
On entry, .
nag_prob_von_mises (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 .
Using the series expansion for small the time taken by nag_prob_von_mises (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.
This example inputs four values from the von Mises distribution along with the values of the argument . The probabilities are computed and printed.
9.1 Program Text
Program Text (g01erce.c)
9.2 Program Data
Program Data (g01erce.d)
9.3 Program Results
Program Results (g01erce.r)