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NAG Library Function Document

nag_rand_compd_poisson (g05tkc)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

▸▿ 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1

Purpose

nag_rand_compd_poisson (g05tkc) generates a vector of pseudorandom integers, each from a discrete Poisson distribution with differing parameter.

2

Specification

#include <nag.h>

#include <nagg05.h>

void	nag_rand_compd_poisson (Integer m, const double vlamda[], Integer state[], Integer x[], NagError *fail)

3

Description

nag_rand_compd_poisson (g05tkc) generates

m

integers

x_{j}

, each from a discrete Poisson distribution with mean

λ_{j}

, where the probability of

x_{j} = I

P (x_{j} = I) = \frac{λ_{j}^{I} \times e^{- λ_{j}}}{I!}, I = 0, 1, \dots,

where

λ_{j} \geq 0, j = 1, 2, \dots, m .

The methods used by this function have low set up times and are designed for efficient use when the value of the parameter

λ

changes during the simulation. For large samples from a distribution with fixed

λ

using nag_rand_poisson (g05tjc) to set up and use a reference vector may be more efficient.

When

λ < 7.5

the product of uniforms method is used, see for example Dagpunar (1988). For larger values of

λ

an envelope rejection method is used with a target distribution:

\begin{matrix} f (x) = \frac{1}{3} & if ​ |x| \leq 1, \\ f (x) = \frac{1}{3} {|x|}^{- 3} & otherwise. \end{matrix}

This distribution is generated using a ratio of uniforms method. A similar approach has also been suggested by Ahrens and Dieter (1989). The basic method is combined with quick acceptance and rejection tests given by Maclaren (1990). For values of

λ \geq 87

Stirling's approximation is used in the computation of the Poisson distribution function, otherwise tables of factorials are used as suggested by Maclaren (1990).

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_compd_poisson (g05tkc).

4

References

Ahrens J H and Dieter U (1989) A convenient sampling method with bounded computation times for Poisson distributions Amer. J. Math. Management Sci. 1–13

Dagpunar J (1988) Principles of Random Variate Generation Oxford University Press

Maclaren N M (1990) A Poisson random number generator Personal Communication

5

Arguments

1: $m$ – IntegerInput: On entry: $m$ , the number of Poisson distributions for which pseudorandom variates are required.

Constraint: $m \geq 1$ .
2: $vlamda [m]$ – const doubleInput: On entry: the means, $λ_{j}$ , for $j = 1, 2, \dots, m$ , of the Poisson distributions.

Constraint: $0.0 \leq vlamda [j - 1] \leq nag_max_integer / 2.0$ , for $j = 1, 2, \dots, m$ .
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: $x [m]$ – IntegerOutput: On exit: the $m$ pseudorandom numbers from the specified Poisson distributions.
5: $fail$ – NagError *Input/Output: The NAG error argument (see Section 3.7 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, $m = 〈value〉$ .
Constraint: $m \geq 1$ .
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.
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_ARRAY: On entry, at least one element of vlamda is less than zero.

On entry, at least one element of vlamda is too large.

7

Accuracy

Not applicable.

8

Parallelism and Performance

nag_rand_compd_poisson (g05tkc) 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.

9

Further Comments

None.

10

Example

This example prints ten pseudorandom integers from five Poisson distributions with means

λ_{1} = 0.5

λ_{2} = 5

λ_{3} = 10

λ_{4} = 500

and

λ_{5} = 1000

. These are generated by ten calls to nag_rand_compd_poisson (g05tkc), after initialization by nag_rand_init_repeatable (g05kfc).

10.1

Program Text

Program Text (g05tkce.c)

10.2

Program Data

None.

10.3

Program Results

Program Results (g05tkce.r)

nag_rand_compd_poisson (g05tkc) (PDF version)

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NAG Library Manual

NAG Library Function Document

nag_rand_compd_poisson (g05tkc)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results