NAG Library Function Document

nag_rngs_von_mises (g05lpc)

The von Mises distribution is a symmetric distribution used in the analysis of circular data. The probability density function of this distribution on the circle with mean direction

μ_{0} = 0

and concentration parameter kappa,

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_rngs_init_repeatable (g05kbc) (for a repeatable sequence if computed sequentially) or nag_rngs_init_nonrepeatable (g05kcc) (for a non-repeatable sequence) must be called prior to the first call to nag_rngs_von_mises (g05lpc).

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: vk – doubleInput: On entry: $κ$ , the concentration parameter of the required von Mises distribution.
Constraint: $vk > 0.0$ .
2: n – IntegerInput: On entry: $n$ , the number of pseudorandom numbers to be generated.
Constraint: $n \geq 0$ .
3: x[n] – doubleOutput: On exit: the $n$ pseudorandom numbers from the specified von Mises distribution.
4: igen – IntegerInput: On entry: must contain the identification number for the generator to be used to return a pseudorandom number and should remain unchanged following initialization by a prior call to nag_rngs_init_repeatable (g05kbc) or nag_rngs_init_nonrepeatable (g05kcc).
5: iseed[ $4$ ] – IntegerCommunication Array: On entry: contains values which define the current state of the selected generator.
On exit: contains updated values defining the new state of the selected generator.
6: 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_REAL: On entry, $vk = 〈value〉$ .
Constraint: $vk > 0.0$ .