NAG Library Routine Document
G01ERF returns the probability associated with the lower tail of the von Mises distribution between and through the function name.
|REAL (KIND=nag_wp) G01ERF
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
is reduced modulo
. Note that if
then G01ERF 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
- 1: T – REAL (KIND=nag_wp)Input
On entry: , the observed von Mises statistic measured in radians.
- 2: VK – REAL (KIND=nag_wp)Input
On entry: the concentration parameter , of the von Mises distribution.
- 3: IFAIL – INTEGERInput/Output
must be set to
. If you are unfamiliar with this parameter you should refer to Section 3.3
in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
is recommended. If the output of error messages is undesirable, then the value
is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is
. When the value is used it is essential to test the value of IFAIL on exit.
unless the routine detects an error or a warning has been flagged (see Section 6
6 Error Indicators and Warnings
If on entry
, explanatory error messages are output on the current error message unit (as defined by X04AAF
Errors or warnings detected by the routine:
|On entry,|| and G01ERF returns .|
G01ERF 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 G01ERF 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 parameter . The probabilities are computed and printed.
9.1 Program Text
Program Text (g01erfe.f90)
9.2 Program Data
Program Data (g01erfe.d)
9.3 Program Results
Program Results (g01erfe.r)