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Likelihood-Free Estimation for Spatiotemporal Hawkes processes with missing data and application to predictive policing

Pramit Das, Moulinath Banerjee, Yuekai Sun

TL;DR

The paper tackles parameter estimation for spatiotemporal Hawkes processes in predictive policing when crime data are incomplete due to non-reporting. It proposes a likelihood-free, Wasserstein GAN framework with exact, interpretable Hawkes generators to recover the underlying parameters $\theta=(\mu,\alpha,\beta,\sigma^2)$ from distorted observations. Applied to a Bogotá-based crime simulation, the method improves parameter recovery and hotspot forecasts despite substantial missingness, demonstrated through interarrival-based goodness-of-fit tests and hotspot overlap metrics. This work advances accountability in predictive policing by enabling robust inference under missing data and discusses identifiability and future extensions to richer spatiotemporal generators and unknown missingness mechanisms.

Abstract

With the growing use of AI technology, many police departments use forecasting software to predict probable crime hotspots and allocate patrolling resources effectively for crime prevention. The clustered nature of crime data makes self-exciting Hawkes processes a popular modeling choice. However, one significant challenge in fitting such models is the inherent missingness in crime data due to non-reporting, which can bias the estimated parameters of the predictive model, leading to inaccurate downstream hotspot forecasts, often resulting in over or under-policing in various communities, especially the vulnerable ones. Our work introduces a Wasserstein Generative Adversarial Networks (WGAN) driven likelihood-free approach to account for unreported crimes in Spatiotemporal Hawkes models. We demonstrate through empirical analysis how this methodology improves the accuracy of parametric estimation in the presence of data missingness, leading to more reliable and efficient policing strategies.

Likelihood-Free Estimation for Spatiotemporal Hawkes processes with missing data and application to predictive policing

TL;DR

The paper tackles parameter estimation for spatiotemporal Hawkes processes in predictive policing when crime data are incomplete due to non-reporting. It proposes a likelihood-free, Wasserstein GAN framework with exact, interpretable Hawkes generators to recover the underlying parameters from distorted observations. Applied to a Bogotá-based crime simulation, the method improves parameter recovery and hotspot forecasts despite substantial missingness, demonstrated through interarrival-based goodness-of-fit tests and hotspot overlap metrics. This work advances accountability in predictive policing by enabling robust inference under missing data and discusses identifiability and future extensions to richer spatiotemporal generators and unknown missingness mechanisms.

Abstract

With the growing use of AI technology, many police departments use forecasting software to predict probable crime hotspots and allocate patrolling resources effectively for crime prevention. The clustered nature of crime data makes self-exciting Hawkes processes a popular modeling choice. However, one significant challenge in fitting such models is the inherent missingness in crime data due to non-reporting, which can bias the estimated parameters of the predictive model, leading to inaccurate downstream hotspot forecasts, often resulting in over or under-policing in various communities, especially the vulnerable ones. Our work introduces a Wasserstein Generative Adversarial Networks (WGAN) driven likelihood-free approach to account for unreported crimes in Spatiotemporal Hawkes models. We demonstrate through empirical analysis how this methodology improves the accuracy of parametric estimation in the presence of data missingness, leading to more reliable and efficient policing strategies.

Paper Structure

This paper contains 15 sections, 36 equations, 5 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction to Hawkes Processes and Predictive Policing
  2. Likelihood-Free Learning for Stochastic Processes
  3. Generative Adversarial Networks (GAN) and Wasserstein GAN (WGAN)
  4. Recurrent Neural Network and Long Short-Term Memory Networks
  5. Generative models for Temporal point processes
  6. Proposed Wasserstein GAN-based method with exact interpretable model generators
  7. WGAN-based parametric estimation in Hawkes Process with missingness
  8. Experiment using different initialization points
  9. Goodness of Fit tests for Hawkes data with missingness
  10. WGAN-based learning for Bogota data accounting for missingness
  11. Discussion
  12. Supplementary Materials
  13. Branching structure and EM algorithm for inference in Hawkes process
  14. Some comments on Identifiability of the Hawkes process parameters
  15. Sequential deep learning models - Recurrent Neural nets and Long Short term memory network

Figures (5)

  • Figure 1: Bogota map and crime statistics: figure from NiljanaAkpinarpredpolicing_2021
  • Figure 2: Bogota district-wise comparison of expected crimes
  • Figure 3: Comparison of true vs. estimated crime hotspots for the Bogota crime data
  • Figure 4: Architecture of RNN from https://colah.github.io/posts/2015-08-Understanding-LSTMs/
  • Figure 5: Structure of LSTM from https://colah.github.io/posts/2015-08-Understanding-LSTMs/