NAG CL Interface
g01erc (prob_vonmises)
1
Purpose
g01erc returns the probability associated with the lower tail of the von Mises distribution between and
.
2
Specification
| double |
g01erc (double t,
double vk,
NagError *fail) |
|
The function may be called by the names: g01erc, nag_stat_prob_vonmises or nag_prob_von_mises.
3
Description
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).
4
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
5
Arguments
-
1:
– double
Input
-
On entry: , the observed von Mises statistic measured in radians.
-
2:
– double
Input
-
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).
6
Error 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: .
7
Accuracy
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 .
8
Parallelism and Performance
g01erc is not threaded in any implementation.
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.
10
Example
This example inputs four values from the von Mises distribution along with the values of the parameter . The probabilities are computed and printed.
10.1
Program Text
10.2
Program Data
10.3
Program Results