Estimation of Poisson Arrival Processes Under Linear Models

Abstract

In this paper we consider the problem of estimating the parameters of a Poisson arrival process where the intensity 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 intensity function. We then show how to remove this dependence through a process of "noise regularization", which results in an improved bound under our analysis. 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. * M. Moore and M. Davenport are with the Georgia Institute of Technology in Atlanta, GA, USA. 1 We will typically think of T as simply representing an interval of R, but leave this general to allow for more complex (e.g., spatio-temporal) Poisson processes, to which our analysis also applies.

Determinants of apartment price volatility in Vietnam: a comparison between Hanoi and Ho Chi Minh City

“…Stochastic processes allow us to model uncertain quantities that evolve over time. One of the most fundamental and important of such processes is the Poisson process 1 arXiv:2402.12808v1 [cs.LG] 20 Feb 2024 (Komaki, 2021;Jahani et al, 2021;Moore and Davenport, 2019;Alizadeh and Papp, 2013) as it is appropriate for modelling a large number of completely random phenomena. Therefore, the Poisson process has been widely used in various real-world applications such as taxi routing (Son et al, 2022), queueing systems (Brown et al, 2005), and e-mail arrival counting (Alizadeh et al, 2008;Alizadeh and Papp, 2013).…”

“…While estimation of homogeneous Poisson processes (HPP), where the rate is constant, is trivial, the problem of estimating nonhomogeneous Poisson processes (NHPP), where the rate is a function of time, is considered more challenging (Brown et al, 2005;Alizadeh and Papp, 2013;Avanzi et al, 2021). Moreover, all of the estimation methods in the literature so far have relied on dividing observed data into bins (binning method), very often with equal length (Moore and Davenport, 2019). While it is reasonable to use binning method for statistical estimation with infinite amount of data as it helps to reduce the estimation difficulty by transforming a NHPP into a HPP (see further explanations in section 4), it could create some serious problems when working with finite amount of data (which is usually the case for real-world applications).…”