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_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: $n$ – IntegerInput: On entry: $n$ , the number of pseudorandom numbers to be generated.

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

Constraint: $0.0 < vk \leq \sqrt{nag_real_largest_number} / 2.0$ .
3: $state [\dim]$ – IntegerCommunication Array: Note: the dimension, $\dim$ , 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: $x [n]$ – doubleOutput: On exit: the $n$ pseudorandom numbers from the specified von Mises distribution.
5: $fail$ – 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.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_INT: On entry, $n = 〈value〉$ .
Constraint: $n \geq 0$ .
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.

An unexpected error has been triggered by this function. Please contact NAG.
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, $vk \leq 0.0$ or vk too large: $vk = 〈value〉$ .

7 Accuracy

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.