Modeling and simulation of a nonhomogeneous poisson process having cyclic behavior

Abstract

In this paper we develop a unified approach to modeling and simulation of a nonhomogeneous Poisson process whose rate function exhibits cyclic behavior as well as a long-term evolutionary trend. The approach can be applied whether the oscillation frequency of the cyclic behavior is known or unknown. To model such a process, we use an exponential rate function whose exponent includes both a polynomial and a trigonometric component.Maximum likelihood estimates of the unknown continuous parameters of this function are obtained numerically, and the degree of the polynomial component is determined by a likelihood ratio test. If the oscillation frequency is unknown, then an initial estimate of this parameter is obtained via spectral analysis of the observed series of events; initial estimates of the remaining trigonometric (respectively, polynomial) parameters are computed from a standard maximum likelihood (respectively, moment-matching) procedure for an exponential-trigonometric (respectively, exponential-polynomial) rate function. To simulate the fitted process by the method of thinning, we present (a) a procedure for constructing an optimal piecewise linear majorizing rate function; and(b)a "piecewise thinning" simulation procedure based on the inverse transform method for generating events from a piecewise linear rate function. These procedures are applied to the storm-arrival process observed at an off-shore drilling site.

Keywords:

• Nonhomogeneous Poisson process
• exponential-polynomiai-trigonometric rate function
• simulation
• thinning