Estimation of Poisson arrival processes under linear models

TL;DR

This paper develops methods for estimating the parameters of a Poisson process with a rate function in a known basis, providing near-optimal guarantees and improved bounds through noise regularization, with practical and empirical insights.

Contribution

It introduces novel estimation guarantees for Poisson processes with basis-constrained rate functions and proposes noise regularization to improve bound dependence.

Findings

Near-optimal estimation bounds achieved

Noise regularization removes dependence on rate function range

Empirical evaluation supports the proposed methods

Abstract

In this paper we consider the problem of estimating the parameters of a Poisson arrival process where the rate function is assumed to lie in the span of a known basis. Our goal is to estimate the basis expansions coefficients given a realization of this process. We establish novel guarantees concerning the accuracy achieved by the maximum likelihood estimate. Our initial result is near-optimal, with the exception of an undesirable dependence on the dynamic range of the rate function. We then show how to remove this dependence through a process of "noise regularization", which results in an improved bound. We conjecture that a similar guarantee should be possible when using a more direct (deterministic) regularization scheme. We conclude with a discussion of practical applications and an empirical examination of the proposed regularization schemes.