G01 Chapter Contents
G01 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentG01ERF

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

G01ERF returns the probability associated with the lower tail of the von Mises distribution between $-\pi$ and $\pi$ through the function name.

## 2  Specification

 FUNCTION G01ERF ( T, VK, IFAIL)
 REAL (KIND=nag_wp) G01ERF
 INTEGER IFAIL REAL (KIND=nag_wp) T, VK

## 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 ${\mu }_{0}=0$ 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 G01ERF 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).

## 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  Parameters

1:     T – REAL (KIND=nag_wp)Input
On entry: $\theta$, the observed von Mises statistic measured in radians.
2:     VK – REAL (KIND=nag_wp)Input
On entry: the concentration parameter $\kappa$, of the von Mises distribution.
Constraint: ${\mathbf{VK}}\ge 0.0$.
3:     IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to $0$, $-1\text{​ or ​}1$. 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 $-1\text{​ or ​}1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is $0$. When the value $-\mathbf{1}\text{​ or ​}\mathbf{1}$ is used it is essential to test the value of IFAIL on exit.
On exit: ${\mathbf{IFAIL}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6  Error Indicators and Warnings

If on entry ${\mathbf{IFAIL}}={\mathbf{0}}$ or $-{\mathbf{1}}$, explanatory error messages are output on the current error message unit (as defined by X04AAF).
Errors or warnings detected by the routine:
${\mathbf{IFAIL}}=1$
 On entry, ${\mathbf{VK}}<0.0$ and G01ERF returns $0$.

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

Using the series expansion for small $\kappa$ the time taken by G01ERF 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.

## 9  Example

This example inputs four values from the von Mises distribution along with the values of the parameter $\kappa$. 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)