g05 Chapter Contents
g05 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_rand_von_mises (g05src)

## 1  Purpose

nag_rand_von_mises (g05src) generates a vector of pseudorandom numbers from a von Mises distribution with concentration parameter $\kappa$.

## 2  Specification

 #include #include
 void nag_rand_von_mises (Integer n, double vk, Integer state[], double x[], NagError *fail)

## 3  Description

The von Mises distribution is a symmetric distribution used in the analysis of circular data. The PDF (probability density function) of this distribution on the circle with mean direction ${\mu }_{0}=0$ and concentration parameter $\kappa$, can be written as:
 $fθ= eκcos⁡θ 2πI0κ ,$
where $\theta$ is reduced modulo $2\pi$ so that $-\pi \le \theta <\pi$ and $\kappa \ge 0$. 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 $n$ variates, ${\theta }_{1},{\theta }_{2},\dots ,{\theta }_{n}$, are generated using an envelope rejection method with a wrapped Cauchy target distribution as proposed by Best and Fisher (1979) and described by Dagpunar (1988).
One of the initialization functions nag_rand_init_repeatable (g05kfc) (for a repeatable sequence if computed sequentially) or nag_rand_init_nonrepeatable (g05kgc) (for a non-repeatable sequence) must be called prior to the first call to nag_rand_von_mises (g05src).

## 4  References

Best D J and Fisher N I (1979) Efficient simulation of the von Mises distribution Appl. Statist. 28 152–157
Dagpunar J (1988) Principles of Random Variate Generation Oxford University Press
Mardia K V (1972) Statistics of Directional Data Academic Press

## 5  Arguments

1:    $\mathbf{n}$IntegerInput
On entry: $n$, the number of pseudorandom numbers to be generated.
Constraint: ${\mathbf{n}}\ge 0$.
2:    $\mathbf{vk}$doubleInput
On entry: $\kappa$, the concentration parameter of the required von Mises distribution.
Constraint: $0.0<{\mathbf{vk}}\le \sqrt{{\mathbf{nag_real_largest_number}}}/2.0$.
3:    $\mathbf{state}\left[\mathit{dim}\right]$IntegerCommunication Array
Note: the dimension, $\mathit{dim}$, of this array is dictated by the requirements of associated functions that must have been previously called. This array MUST be the same array passed as argument state in the previous call to nag_rand_init_repeatable (g05kfc) or nag_rand_init_nonrepeatable (g05kgc).
On entry: contains information on the selected base generator and its current state.
On exit: contains updated information on the state of the generator.
4:    $\mathbf{x}\left[{\mathbf{n}}\right]$doubleOutput
On exit: the $n$ pseudorandom numbers from the specified von Mises distribution.
5:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 2.7 in How to Use the NAG Library and its Documentation).

## 6  Error 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.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
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_INVALID_STATE
On entry, state vector has been corrupted or not initialized.
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, ${\mathbf{vk}}\le 0.0$ or vk too large: ${\mathbf{vk}}=〈\mathit{\text{value}}〉$.

Not applicable.

## 8  Parallelism and Performance

nag_rand_von_mises (g05src) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
Please consult the x06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

For a given number of random variates the generation time increases slightly with increasing $\kappa$.

## 10  Example

This example prints the first five pseudorandom numbers from a von Mises distribution with $\kappa =1.0$, generated by a single call to nag_rand_von_mises (g05src), after initialization by nag_rand_init_repeatable (g05kfc).

### 10.1  Program Text

Program Text (g05srce.c)

None.

### 10.3  Program Results

Program Results (g05srce.r)