NAG Library Routine Document

G05SRF

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

μ_{0} = 0

and concentration parameter

κ

, can be written as:

f (θ) = \frac{e^{κ \cos θ}}{2 π I_{0} (κ)},

where

θ

is reduced modulo

2 π

so that

- π \leq θ < π

and

κ \geq 0

. 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

n

variates,

θ_{1}, θ_{2}, \dots, θ_{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 routines G05KFF (for a repeatable sequence if computed sequentially) or G05KGF (for a non-repeatable sequence) must be called prior to the first call to G05SRF.

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 Parameters

1: N – INTEGERInput: On entry: $n$ , the number of pseudorandom numbers to be generated.
Constraint: $N \geq 0$ .
2: VK – REAL (KIND=nag_wp)Input: On entry: $κ$ , the concentration parameter of the required von Mises distribution.
Constraint: $0.0 < VK \leq \sqrt{X02ALF} / 2.0$ .
3: STATE( $*$ ) – INTEGER arrayCommunication Array: Note: the actual argument supplied must be the array STATE supplied to the initialization routines G05KFF or G05KGF.
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: X(N) – REAL (KIND=nag_wp) arrayOutput: On exit: the $n$ pseudorandom numbers from the specified von Mises distribution.
5: IFAIL – INTEGERInput/Output: On entry: IFAIL must be set to $0$ , $- 1 or 1$ . 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 $- 1 or 1$ 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 $- 1 or 1$ 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, $N < 0$ .

$IFAIL = 2$

On entry,	$VK \leq 0.0$ ,
or	$VK > \sqrt{X02ALF ()} / 2.0$ .

$IFAIL = 3$

On entry,

STATE vector was not initialized or has been corrupted.

7 Accuracy

Not applicable.

8 Further Comments

For a given number of random variates the generation time increases slightly with increasing

κ

9 Example

This example prints the first five pseudorandom numbers from a von Mises distribution with

κ = 1.0

, generated by a single call to G05SRF, after initialization by G05KFF.

NAG Library Routine DocumentG05SRF

+− Contents

1 Purpose

2 Specification

3 Description