We study the problem of simulating a class of nonstationary spatio-temporal Poisson processes. The Poisson intensity function is non-stationary and piecewise linear in both the time dimension and the spatial location dimensions. We propose an exact simulation algorithm based on the inversion method. This simulation algorithm adopts three advantages. First, the entire procedure involves only closed-form computation with no need for numerical integration or numerical inversion of any function. Each step in the algorithm only requires exact arithmetic operations. Second, the proposed algorithm is sample efficient, especially compared to the thinning method when the maximum intensity value is much larger than the minimum intensity value. Third, the algorithm generates arrivals sequentially, one at a time in ascending order, so that they can be conveniently fed into real-time or online decision-making tools.

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 varialbes, and that the number of points in any of these intervals has a Poisson distribution. This paper first discusses several general methods for simulation of the one-dimensional nonhomogeneous Poisson process. Then a particular and very efficient method for simulation of nonhomogeneous Poisson processes is stated 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. Finally, a simple and relatively efficient new method for simulation of one-dimensional and two-dimensional nonhomogeneous Poisson processes is described. 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.

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