Simulation methods for poisson processes in nonstationary systems

Abstract

The nonhomogeneous Poisson process is a widely used model for a series of events (stochastic point process) in which the "rate" or "intensity" of occurrence of points varies, usually with time. The process has the characteristic properties that the number of points in any finite set of nonoverlapping intervals are mutually independent random variables, and that the number of points in any of these intervals has a Poisson distribution. In this paper we first discuss several general methods for simulation of the one-dimensional nonhomogeneous Poisson process; these include time-scale transformation of a homogeneous (rate one) Poisson process via the inverse of the integrated rate function, generation of the individual intervals between points, and generation of a Poisson number of order statistics from a fixed density function. We then state a particular and very efficient method for simulation of nonhomogeneous Poisson processes with log-linear rate function. The method is based on an identity relating the nonhomogeneous Poisson process to the gap statistics from a random number of exponential random variables with suitably chosen parameters. This method can also be used, at the cost of programming complexity and some memory, as the basis for a very efficient technique for simulation of nonhomogeneous Poisson processes with more complicated rate functions such as a log-quadratic rate function. Finally, we describe a simple and relatively efficient new method for simulation of one-dimensional and two-dimensional nonhomogeneous Poisson processes. The method is applicable for any given rate function and is based on controlled deletion of points in a Poisson process with a rate function that dominates the given rate function. In its simplest implementation, the method obviates the need for numerical integration of the rate function, ~or ordering of points, and for generation of Poisson variates. The thinning method is also applicable to the generation of individual intervals between points r as is required in many programs for discrete-event simulations.