Abstract
Self-exciting spatiotemporal Hawkes processes have found increasing use in the study of large-scale public health threats, ranging from gun violence and earthquakes to wildfires and viral contagion. Whereas many such applications feature locational uncertainty, that is, the exact spatial positions of individual events are unknown, most Hawkes model analyses to date have ignored spatial coarsening present in the data. Three particular 21st century public health crises—urban gun violence, rural wildfires and global viral spread—present qualitatively and quantitatively varying uncertainty regimes that exhibit: (a) different collective magnitudes of spatial coarsening, (b) uniform and mixed magnitude coarsening, (c) differently shaped uncertainty regions and—less orthodox—(d) locational data distributed within the “wrong” effective space. We explicitly model such uncertainties in a Bayesian manner and jointly infer unknown locations together with all parameters of a reasonably flexible Hawkes model, obtaining results that are practically and statistically distinct from those obtained while ignoring spatial coarsening. This work also features two different secondary contributions: first, to facilitate Bayesian inference of locations and background rate parameters, we make a subtle yet crucial change to an established kernel-based rate model, and second, to facilitate the same Bayesian inference at scale, we develop a massively parallel implementation of the model’s log-likelihood gradient with respect to locations and thus avoid its quadratic computational cost in the context of Hamiltonian Monte Carlo. Our examples involve thousands of observations and allow us to demonstrate practicality at moderate scales.
Funding Statement
The first author was supported by NIH Grant K25 AI153816.
The third author was supported by NIH Grants U19 AI135995 and R01 AI153044 and NSF Grant DMS1264153.
Acknowledgments
We gratefully acknowledge support from NVIDIA Corporation with the donation of parallel computing resources used for this research. We also thank Rick Schoenberg and Sudipto Banerjee for their insight and helpful suggestions.
Citation
Andrew J. Holbrook. Xiang Ji. Marc A. Suchard. "Bayesian mitigation of spatial coarsening for a Hawkes model applied to gunfire, wildfire and viral contagion." Ann. Appl. Stat. 16 (1) 573 - 595, March 2022. https://doi.org/10.1214/21-AOAS1517
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