Fitting a Filtered Poisson Process

Abstract

A technique has been developed for generating an approximate characterization of a class of non-Gaussian stochastic processes. The class of processes considered are basically those non-Gaussian stationary processes for which the fourth moments of the univariate distributions are greater than the fourth moments of corresponding Gaussian approximations to the processes. In addition, the processes are assumed to be one-dimensional with continuous sample functions and the corresponding autocovariance functions are assumed to go to zero within finite time intervals. The characterization is accomplished by fitting a filtered Poisson process to the non-Gaussian process by matching the autocovariance function and an arbitrary number of the lower order univariate moments of the two processes. In matching the autocovariance function, a problem mathematically equivalent to the “factorization” of the power spectrum of a stationary process is solved numerically using an iterative procedure. The main advantage of using the filtered Poisson process is that linear operations on these processes yield filtered Poisson processes. This then simplifies the determination of probabilistic information of the output of a stochastic process passed through linear filters. Such considerations might be required, for example, in reliability problems involving random vibrations of linear mechanical systems.