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 functions nag_rand_init_repeat (g05kf) (for a repeatable sequence if computed sequentially) or nag_rand_init_nonrepeat (g05kg) (for a non-repeatable sequence) must be called prior to the first call to nag_rand_dist_vonmises (g05sr).

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

Parameters

Compulsory Input Parameters

1: $n$ – int64int32nag_int scalar: $n$ , the number of pseudorandom numbers to be generated.

Constraint: $n \geq 0$ .
2: $vk$ – double scalar: $κ$ , the concentration parameter of the required von Mises distribution.

Constraint: $0.0 < vk \leq \sqrt{x02al} / 2.0$ .
3: $state (:)$ – int64int32nag_int array: Note: the actual argument supplied must be the array state supplied to the initialization routines nag_rand_init_repeat (g05kf) or nag_rand_init_nonrepeat (g05kg).

Contains information on the selected base generator and its current state.

Optional Input Parameters

None.

Output Parameters

1: $state (:)$ – int64int32nag_int array: Contains updated information on the state of the generator.
2: $x (n)$ – double array: The $n$ pseudorandom numbers from the specified von Mises distribution.
3: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

$ifail = 1$: Constraint: $n \geq 0$ .

$ifail = 2$: On entry, $vk \leq 0.0$ or vk too large:

$ifail = 3$: On entry, state vector has been corrupted or not initialized.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

Accuracy

Not applicable.

Further Comments

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

κ

Example

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

κ = 1.0

, generated by a single call to nag_rand_dist_vonmises (g05sr), after initialization by nag_rand_init_repeat (g05kf).

Open in the MATLAB editor: g05sr_example

function g05sr_example


fprintf('g05sr example results\n\n');

% Initialize the base generator to a repeatable sequence
seed  = [int64(1762543)];
genid = int64(1);
subid = int64(1);
[state, ifail] = g05kf( ...
                        genid, subid, seed);

% Number of variates
n = int64(5);

% Parameters
vk = 1;

% Generate variates from von Mises distribution
[state, x, ifail] = g05sr( ...
                           n, vk, state);

disp('Variates');
disp(x);

g05sr example results

Variates
    1.2947
   -1.9542
   -0.6464
   -1.4172
    1.2536

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox