Reinhart 2018 - Self-Exciting Spatio-Temporal Point Processes - Overview

Summary

Reinhart (2018, Statistical Science) reviews self-exciting spatio-temporal point processes (Hawkes processes and ETAS-type models): event rates that depend on the history of earlier events, so that events trigger further events. The paper unifies estimation (maximum likelihood, EM with branching structure), stochastic declustering, simulation, inference, Bayesian methods and diagnostics, and surveys earthquake, crime, epidemic and network applications. Key takeaway: the cluster (branching) representation with latent parent indicators $u_i$ drives EM, declustering, simulation and Bayesian sampling alike.

Overview

Research question. How can we model, fit and check point-process data where events cluster because they cause each other (aftershocks, repeat burglaries, contagion), without aggregating to grid cells (the Modifiable Areal Unit Problem)?

Contribution. A synthesis of literature isolated within each application field: a common definition via the conditional intensity, a common likelihood, and a common toolbox.

Key results (reviewed, not new).

• Conditional intensity $\lambda (s,t)=\mu (s)+{\sum }_{i:t_i<t}g(s-s_i,t-t_i)$ (Reinhart 2018 - Hawkes Processes and Conditional Intensity).
• Log-likelihood $\ell (\Theta )={\sum }_i\log \lambda (s_i,t_i)-\iint \lambda$; EM via triggering probabilities $\mathrm{Pr}(u_i=j)=g_{ij}/{\lambda }_i$ (Reinhart 2018 - Likelihood, EM and Bayesian Estimation).
• Stochastic declustering (ZOV 2002, FLP, MISD) (Reinhart 2018 - Stochastic Declustering).
• Simulation by thinning or by the cluster representation; asymptotic vs bootstrap inference; thinning/Voronoi residual diagnostics (Reinhart 2018 - Simulation, Inference and Diagnostics).
• Applications: ETAS aftershocks (Reinhart 2018 - ETAS Earthquake Models and Omori Law), crime, epidemics, networks (Reinhart 2018 - Crime, Epidemic and Network Applications).

Main Content

Motivating PyMC examples

Source of the example list: https://redam94.github.io/pymc-challenges/examples.html.

• E65 - Self-exciting point processes: Hawkes and ETAS aftershock forecasting (USGS ComCat, 2019 Ridgecrest sequence; conditional intensity, branching ratio, closed-form compensator, Omori-Utsu, Gutenberg-Richter b-value).
• E07 - Geo-temporal events: induced earthquakes in Oklahoma (contrast: a log-Gaussian Cox process, which has no event-to-event triggering; see Simpson 2016 - Going Off Grid - Overview).

Source Map

Paper section	Pages	Covered in
1 Introduction (regression/MAUP, LGCP, cluster processes)	1-3	Reinhart 2018 - Hawkes Processes and Conditional Intensity
2.1 Hawkes processes, Poisson-cluster representation	3-4	Reinhart 2018 - Hawkes Processes and Conditional Intensity
2.2 Spatio-temporal form, $m=\iint g$	4-5	Reinhart 2018 - Hawkes Processes and Conditional Intensity
2.3 Marks	5-6	Reinhart 2018 - Hawkes Processes and Conditional Intensity
2.4 Log-likelihood, Janossy density, integral approximations	6-8	Reinhart 2018 - Likelihood, EM and Bayesian Estimation
3.1 Maximum likelihood, EM, boundary effects	9-10	Reinhart 2018 - Likelihood, EM and Bayesian Estimation
3.2 Stochastic declustering (Alg. 1-3, MISD)	11-14	Reinhart 2018 - Stochastic Declustering
3.3 Simulation (Alg. 4-5)	14-16	Reinhart 2018 - Simulation, Inference and Diagnostics
3.4 Asymptotics, bootstrap (Alg. 6), partial likelihood	16-17	Reinhart 2018 - Simulation, Inference and Diagnostics
3.5 Bayesian approaches	17-18	Reinhart 2018 - Likelihood, EM and Bayesian Estimation
3.6 Model selection and diagnostics (thinning, super-thinning, Voronoi residuals, Fig. 3)	18-19	Reinhart 2018 - Simulation, Inference and Diagnostics
4.1 Earthquake / ETAS models	19-22	Reinhart 2018 - ETAS Earthquake Models and Omori Law
4.2 Crime forecasting (Mohler)	22-23	Reinhart 2018 - Crime, Epidemic and Network Applications
4.3 Epidemic forecasting (Meyer; Schoenberg)	23-25	Reinhart 2018 - Crime, Epidemic and Network Applications
4.4 Events on social networks	25-26	Reinhart 2018 - Crime, Epidemic and Network Applications
5 Conclusions (interpretation caveat)	26	Reinhart 2018 - Crime, Epidemic and Network Applications
Figures 1-3	4, 6, 20	Reinhart 2018 - Hawkes Processes and Conditional Intensity, Reinhart 2018 - Simulation, Inference and Diagnostics

Examples

See the worked branching-ratio and EM sketches in Reinhart 2018 - Likelihood, EM and Bayesian Estimation and the Ridgecrest-style ETAS PyMC sketch in Reinhart 2018 - ETAS Earthquake Models and Omori Law.

Connections

• Relevance to marketing measurement: self-excitation is the natural model for bursty user-level or event-level data (organic virality, retargeting cascades, repeat purchases, support-ticket storms). A Hawkes background $\mu$ plus triggering $g$ separates exogenous demand (what media can move) from endogenous word-of-mouth/contagion, which matters when attributing lift. Treat this as an analogy; the paper itself covers seismology, crime, epidemics and networks only.
• Contrasts with the Cox-process route: Simpson 2016 - Going Off Grid - Overview (random but not history-dependent intensity).
• Conditional-intensity likelihood parallels hazard modelling: Survival Analysis.

See Also

Reinhart 2018 - Self-Exciting Spatio-Temporal Point Processes | Hierarchical Models | Prior Predictive Checking