naginterfaces.library.rand.int_​poisson_​varmean

naginterfaces.library.rand.int_poisson_varmean(vlamda, statecomm)

int_poisson_varmean generates a vector of pseudorandom integers, each from a discrete Poisson distribution with differing parameter.

For full information please refer to the NAG Library document for g05tk

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/g05/g05tkf.html

Parameters

vlamdafloat, array-like, shape $\left(m\right)$

The means, ${\lambda }_{\text{j}}$, for $\text{j}=1,2,\dots ,m$, of the Poisson distributions.

statecommdict, RNG communication object, modified in place

RNG communication structure.

This argument must have been initialized by a prior call to init_repeat() or init_nonrepeat().

Returns

xint, ndarray, shape $\left(m\right)$

The $m$ pseudorandom numbers from the specified Poisson distributions.

Raises

NagValueError

(errno $1$)

On entry, $m=⟨value⟩$.

Constraint: $m\ge 1$.

(errno $2$)

On entry, at least one element of $vlamda$ is less than zero.

(errno $2$)

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

(errno $3$)

On entry, $statecomm$[‘state’] vector has been corrupted or not initialized.

Notes

int_poisson_varmean generates $m$ integers $x_j$, each from a discrete Poisson distribution with mean ${\lambda }_j$, where the probability of $x_j=I$ is

$$P(x_j=I)=\frac{{\lambda }_j^I\times e^{-{\lambda }_j}}{I!}\text{,}I=0,1,\dots \text{,}$$

where

$${\lambda }_j\ge 0\text{,}j=1,2,\dots ,m\text{.}$$

The methods used by this function have low set up times and are designed for efficient use when the value of the parameter $\lambda$ changes during the simulation. For large samples from a distribution with fixed $\lambda$ using int_poisson() to set up and use a reference vector may be more efficient.

When $\lambda <7.5$ the product of uniforms method is used, see for example Dagpunar (1988). For larger values of $\lambda$ an envelope rejection method is used with a target distribution:

$$\begin{array}{c}\begin{array}{cc}f\left(x\right)=\frac{1}{3}& \text{if}\left|x\right|\le 1\text{,}\\ & \\ f\left(x\right)=\frac{1}{3}{\left|x\right|}^{-3}& \text{otherwise.}\end{array}\end{array}$$

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 $\lambda \ge 87$ 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 init_repeat() (for a repeatable sequence if computed sequentially) or init_nonrepeat() (for a non-repeatable sequence) must be called prior to the first call to int_poisson_varmean.

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