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

μ_{0} = 0

and concentration argument kappa,

κ

, can be written as

\Pr (Θ \leq θ : κ) = \frac{1}{2 π I_{0} (κ)} \int_{- π}^{θ} e^{κ \cos Θ} d Θ,

where

θ

is reduced modulo

2 π

so that

- π \leq θ < π

and

κ \geq 0

. Note that if

θ = π

then nag_stat_prob_vonmises (g01er) returns a probability of

1

. For very small

κ

the distribution is almost the uniform distribution, whereas for

κ \to \infty

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(

Θ \leq θ

κ

) can be expressed as

\Pr (Θ \leq θ : κ) = \frac{1}{2} + \frac{θ}{(2 π)} + \frac{1}{π I_{0} (κ)} \sum_{n = 1}^{\infty} n^{- 1} I_{n} (κ) \sin n θ,

where

I_{n} (κ)

is the modified Bessel function. This series expansion can be represented as a nested expression of terms involving the modified Bessel function ratio

R_{n}

R_{n} (κ) = \frac{I_{n} (κ)}{I_{n - 1} (κ)}, n = 1, 2, 3, \dots,

which is calculated using backwards recursion.

For large values of

κ

(see Accuracy) an asymptotic Normal approximation is used. The angle

Θ

is transformed to the nearly Normally distributed variate

Z

Z = b (κ) \sin \frac{Θ}{2},

where

b (κ) = \frac{\sqrt{\frac{2}{π}} e^{κ}}{I_{0} (κ)}

and

b (κ)

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).

References

Hill G W (1977) Algorithm 518: Incomplete Bessel function

I_{0}

: The Von Mises distribution ACM Trans. Math. Software 3 279–284

Mardia K V (1972) Statistics of Directional Data Academic Press

Parameters

Compulsory Input Parameters

1: $t$ – double scalar: $θ$ , the observed von Mises statistic measured in radians.
2: $vk$ – double scalar: The concentration parameter $κ$ , of the von Mises distribution.

Constraint: $vk \geq 0.0$ .

Optional Input Parameters

None.

Output Parameters

1: $result$ – double scalar: The result of the function.
2: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

$ifail = 1$

On entry,

vk < 0.0

and nag_stat_prob_vonmises (g01er) returns

0

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

Accuracy

nag_stat_prob_vonmises (g01er) 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

vk \geq 6.5

, the other gives an accuracy of

12

digits and uses the Normal approximation when

vk \geq 50.0

Further Comments

Using the series expansion for small

κ

the time taken by nag_stat_prob_vonmises (g01er) 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

- π \leq θ < π

are used care has to be taken in evaluating the probability of being in a region

θ_{1} \leq θ \leq θ_{2}

if the region contains an odd multiple of

π

(2 n + 1) π

. The value of

F (θ_{2}; κ) - F (θ_{1}; κ)

will be negative and the correct probability should then be obtained by adding one to the value.

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.

Open in the MATLAB editor: g01er_example

function g01er_example


fprintf('g01er example results\n\n');

% Lower tail probabilities for Von Mises distribution
t  = [7; 2.8; 1.0; -1.4];
vk = [0; 2.4; 1.0;  1.3];

fprintf('     t         vk     probability\n');
for j = 1:numel(t)

          [p, ifail] = g01er( ...
                              t(j), vk(j));

  fprintf('%9.4f%10.4f%12.4f\n', t(j), vk(j), p);
end

g01er example results

     t         vk     probability
   7.0000    0.0000      0.6141
   2.8000    2.4000      0.9983
   1.0000    1.0000      0.7944
  -1.4000    1.3000      0.1016

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox