# NAG C Library Function Document

## 1Purpose

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

## 2Specification

 #include #include
 double nag_prob_von_mises (double t, double vk, NagError *fail)

## 3Description

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 ${\mu }_{0}=0$ and concentration parameter kappa, $\kappa$, can be written as
 $PrΘ≤θ:κ=12πI0κ ∫-πθeκcos⁡ΘdΘ,$
where $\theta$ is reduced modulo $2\pi$ so that $-\pi \le \theta <\pi$ and $\kappa \ge 0$. Note that if $\theta =\pi$ then nag_prob_von_mises (g01erc) returns a probability of $1$. For very small $\kappa$ the distribution is almost the uniform distribution, whereas for $\kappa \to \infty$ all the probability is concentrated at one point.
The method of calculation for small $\kappa$ involves backwards recursion through a series expansion in terms of modified Bessel functions, while for large $\kappa$ an asymptotic Normal approximation is used.
In the case of small $\kappa$ the series expansion of Pr($\Theta \le \theta$: $\kappa$) can be expressed as
 $PrΘ≤θ:κ=12+θ 2π +1πI0κ ∑n=1∞n-1Inκsin⁡nθ,$
where ${I}_{n}\left(\kappa \right)$ is the modified Bessel function. This series expansion can be represented as a nested expression of terms involving the modified Bessel function ratio ${R}_{n}$,
 $Rnκ=Inκ In-1κ , n=1,2,3,…,$
which is calculated using backwards recursion.
For large values of $\kappa$ (see Section 7) an asymptotic Normal approximation is used. The angle $\Theta$ is transformed to the nearly Normally distributed variate $Z$,
 $Z=bκsin⁡Θ2,$
where
 $bκ=2π eκ I0κ$
and $b\left(\kappa \right)$ is computed from a continued fraction approximation. An approximation to order ${\kappa }^{-4}$ 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).

## 4References

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

## 5Arguments

1:    $\mathbf{t}$doubleInput
On entry: $\theta$, the observed von Mises statistic measured in radians.
2:    $\mathbf{vk}$doubleInput
On entry: the concentration parameter $\kappa$, of the von Mises distribution.
Constraint: ${\mathbf{vk}}\ge 0.0$.
3:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation 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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_REAL
On entry, ${\mathbf{vk}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{vk}}\ge 0.0$.

## 7Accuracy

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 ${\mathbf{vk}}\ge 6.5$, the other gives an accuracy of $12$ digits and uses the Normal approximation when ${\mathbf{vk}}\ge 50.0$.

## 8Parallelism and Performance

nag_prob_von_mises (g01erc) is not threaded in any implementation.

Using the series expansion for small $\kappa$ the time taken by nag_prob_von_mises (g01erc) increases linearly with $\kappa$; for larger $\kappa$, for which the asymptotic Normal approximation is used, the time taken is much less.
If angles outside the region $-\pi \le \theta <\pi$ are used care has to be taken in evaluating the probability of being in a region ${\theta }_{1}\le \theta \le {\theta }_{2}$ if the region contains an odd multiple of $\pi$, $\left(2n+1\right)\pi$. The value of $F\left({\theta }_{2}\text{;}\kappa \right)-F\left({\theta }_{1}\text{;}\kappa \right)$ will be negative and the correct probability should then be obtained by adding one to the value.

## 10Example

This example inputs four values from the von Mises distribution along with the values of the parameter $\kappa$. The probabilities are computed and printed.

### 10.1Program Text

Program Text (g01erce.c)

### 10.2Program Data

Program Data (g01erce.d)

### 10.3Program Results

Program Results (g01erce.r)