NAG Library Function Document

nag_rand_von_mises (g05src)

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 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

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.
NE_INVALID_STATE: On entry, state vector has been corrupted or not initialized.
NE_REAL: On entry, $vk \leq 0.0$ or vk too large: $vk = ⟨value⟩$ .