Statistical properties of a filtered Poisson process with additive random noise: Distributions, correlations and moment estimation

TL;DR

This paper extends filtered Poisson processes by adding noise, deriving statistical properties, and evaluating methods to estimate parameters and distinguish noise types through simulations and analysis.

Contribution

It introduces a comprehensive analysis of additive and dynamical noise effects on filtered Poisson processes, including derivations and comparison of statistical measures.

Findings

Probability density and moments estimate parameters accurately.

Auto-correlation and spectral density identify noise types.

Threshold crossing counts can differentiate noise types.

Abstract

Filtered Poisson processes are often used as reference models for intermittent fluc- tuations in physical systems. Such a process is here extended by adding a noise term, either as a purely additive term to the process or as a dynamical term in a stochastic differential equation. The lowest order moments, probability density function, auto-correlation function and power spectral density are derived and used to identify and compare the effects of the two different noise terms. Monte-Carlo studies of synthetic time series are used to investigate the accuracy of model pa- rameter estimation and to identify methods for distinguishing the noise types. It is shown that the probability density function and the three lowest order moments provide accurate estimations of the parameters, but are unable to separate the noise types. The auto-correlation function and the power spectral density also provide methods for estimating the model parameters, as well as being capable of identifying the noise type. The number of times the signal crosses a prescribed threshold level in the positive direction also promises to be able to differentiate the noise type.