nag_prob_von_mises (g01erc) (PDF version)
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NAG C Library Manual

NAG Library Function Document

nag_prob_von_mises (g01erc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_prob_von_mises (g01erc) returns the probability associated with the lower tail of the von Mises distribution between -π and π .

2  Specification

#include <nag.h>
#include <nagg01.h>
double  nag_prob_von_mises (double t, double vk, NagError *fail)

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 μ0=0 and concentration argument kappa, κ, can be written as
PrΘθ:κ=12πI0κ -πθeκcosΘdΘ,
where θ is reduced modulo 2π so that -πθ<π and κ0. Note that if θ=π then nag_prob_von_mises (g01erc) returns a probability of 1. For very small κ the distribution is almost the uniform distribution, whereas for κ all the probability is concentrated at one point.
The method of calculation for small κ involves backwards recursion through a series expansion in terms of modified Bessel functions, while for large κ an asymptotic Normal approximation is used.
In the case of small κ the series expansion of Pr(Θθ: κ) can be expressed as
PrΘθ:κ=12+θ 2π +1πI0κ n=1n-1Inκsinnθ,
where Inκ is the modified Bessel function. This series expansion can be represented as a nested expression of terms involving the modified Bessel function ratio Rn,
Rnκ=Inκ In-1κ ,  n=1,2,3,,
which is calculated using backwards recursion.
For large values of κ (see Section 7) an asymptotic Normal approximation is used. The angle Θ is transformed to the nearly Normally distributed variate Z,
Z=bκsinΘ2,
where
bκ=2π eκ I0κ
and bκ is computed from a continued fraction approximation. An approximation to order κ-4 of the asymptotic normalizing series for z 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 I0: The Von Mises distribution ACM Trans. Math. Software 3 279–284
Mardia K V (1972) Statistics of Directional Data Academic Press

5  Arguments

1:     tdoubleInput
On entry: θ, the observed von Mises statistic measured in radians.
2:     vkdoubleInput
On entry: the concentration parameter κ, of the von Mises distribution.
Constraint: vk0.0.
3:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

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.
NE_REAL
On entry, vk=value.
Constraint: vk0.0.

7  Accuracy

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 vk6.5, the other gives an accuracy of 12 digits and uses the Normal approximation when vk50.0.

8  Further Comments

Using the series expansion for small κ the time taken by nag_prob_von_mises (g01erc) increases linearly with κ; for larger κ, for which the asymptotic Normal approximation is used, the time taken is much less.
If angles outside the region -πθ<π are used care has to be taken in evaluating the probability of being in a region θ1θθ2 if the region contains an odd multiple of π, 2n+1π. The value of Fθ2;κ-Fθ1;κ will be negative and the correct probability should then be obtained by adding one to the value.

9  Example

This example inputs four values from the von Mises distribution along with the values of the argument κ. 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)


nag_prob_von_mises (g01erc) (PDF version)
g01 Chapter Contents
g01 Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012