The periodically-observed time-homogeneous Poisson process, with applications in violence and loss

Hornberger, Zachary T.

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https://hdl.handle.net/2142/129911 Copy

Description

Title

The periodically-observed time-homogeneous Poisson process, with applications in violence and loss

Author(s)

Hornberger, Zachary T.

Issue Date

2025-06-30

Director of Research (if dissertation) or Advisor (if thesis)

• King, Douglas M
• Jacobson, Sheldon H

Doctoral Committee Chair(s)

King, Douglas M

Committee Member(s)

• Sowers, Richard B
• Sreenivas, Ramavarapu S

Department of Study

Industrial&Enterprise Sys Eng

Discipline

Industrial Engineering

Degree Granting Institution

University of Illinois Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

• Poisson Process
• periodic observation
• discretization error
• social contagion

Abstract

When analytical tools can show that arrival data for practical systems are indistinguishable from those of a time-homogeneous Poisson process (THPP), important insights can be drawn from these systems; this is particularly true for phenomena of violence and loss, where events occur with minimal warning or apparent rationale. Practical constraints (e.g., manpower, money, technology) frequently prevent the exact arrival data of a system from being collected, and subsequent analysis must be performed on the available approximations of the arrival data. Although a common approach to modeling in the presence of periodic observation is simply to ignore the error associated with these approximations of arrival data, researchers have noted that this impedes proper analysis of these data. Therefore, a practitioner's ability to effectively identify THPPs in periodically observed practical systems---and therefore draw important insights about these systems---relies on the availability of analytical tools that can properly accommodate periodically observed data. This research introduces the periodically-observed time-homogeneous Poisson process (PTPP) framework, which explicitly accounts for the presence of periodic observation within THPP systems. This framework enables the study of how the act of periodic observation during data collection can influence insights drawn from THPP systems. First, periodic observation is shown to pose challenges to traditional goodness-of-fit techniques for determining whether arrival data originate from a THPP. This research demonstrates that the implementation of the PTPP model for goodness-of-fit testing can overcome these challenges. Second, the effects of periodic observation on the distribution of recorded interarrival times is quantified and studied. Third, the finite behavior of the PTPP system is assessed with respect to both the mixing time of the PTPP model as well as the moments of the recorded interarrival distribution of a PTPP after $n$ events. Fourth, periodic observation is shown to introduce a source of bias into the estimation of the arrival rate parameter, and the magnitude of this bias is computed. Fifth, the limitations of implementing the PTPP model for goodness-of-fit testing are explored as they relate to Type II error (i.e., the inability to distinguish periodically observed non-homogeneous Poisson processes from periodically observed THPPs). In addition to introducing the PTPP framework and studying the phenomenon of periodic observation within the context of THPPs, this research also applies the PTPP to real-world scenarios of violence and loss where data regarding these events are recorded periodically. First, this research studies the emergence of search-and-rescue events in New York wilderness areas and whether they behave similarly to periodically observed THPPs. Second, this research integrates the PTPP into a larger three-phase methodology to evaluate the social contagion hypothesis for US mass killings from 2006-2023. These applications demonstrate that the PTPP framework can contribute valuable insights to the study of phenomena where periodic observation is largely unavoidable.

Graduation Semester

2025-08

Type of Resource

Thesis

Handle URL

https://hdl.handle.net/2142/129911

Copyright and License Information

Copyright 2025 Zachary Hornberger

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