Nonparametric estimation of nonhomogeneous Poisson processes using wavelets

Abstract

Nonhomogeneous Poisson processes (NHPPs) are frequently used in stochastic simulations to model nonstationary point processes. These NHPP models are often constructed by estimating the rate function from one or more observed realizations of the process. Both parametric and nonparametric models have been developed for the NHPP rate function. The current parametric models require prior knowledge of the behavior of the NHPP under study for model selection. The current nonparametric estimators, in general, require the storage of all of the observed data. Other hybrid approaches have also been developed. This paper focuses on the nonparametric estimation of the rate function of a nonhomogeneous Poisson process using wavelets. The advantages of wavelets include the flexibility of a nonparametric estimator enabling one to model the nonstationary rate function of an NHPP without prior knowledge or assumptions about the behavior of the process. Furthermore, this method has some advantages of current nonparametric techniques. Thus, using wavelets we can develop an efficient yet highly flexible NHPP rate function. In this paper, we develop the methodology required for constructing a wavelet estimator for the NHPP rate function. In addition, we present an experimental performance evaluation for this method.