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 g01erf 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 Section 7) 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).

4

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

5

Arguments

1: $t$ – Real (Kind=nag_wp)Input: On entry: $θ$ , the observed von Mises statistic measured in radians.
2: $vk$ – Real (Kind=nag_wp)Input: On entry: the concentration parameter $κ$ , of the von Mises distribution.

Constraint: $vk \geq 0.0$ .
3: $ifail$ – IntegerInput/Output: On entry: ifail must be set to $0$ , $- 1 or 1$ . If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $- 1 or 1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$ . When the value $- 1 or 1$ is used it is essential to test the value of ifail on exit.

On exit: $ifail = 0$ unless the routine detects an error or a warning has been flagged (see Section 6).

6

Error Indicators and Warnings

If on entry

ifail = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail = 1$: On entry, $vk = 〈value〉$ .
Constraint: $vk \geq 0.0$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7

Accuracy

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

vk \geq 6.5

, the other gives an accuracy of

12

digits and uses the Normal approximation when

vk \geq 50.0

8

Parallelism and Performance

g01erf is not threaded in any implementation.

9

Further Comments

Using the series expansion for small

κ

the time taken by g01erf 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.

10

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.

NAG Library Routine Document

g01erf (prob_vonmises)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

6

Error Indicators and Warnings

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

10.2

Program Data

10.3

Program Results