# Modeling and Estimation for Self-Exciting Spatio-Temporal Models of Terrorist Activity

## TL;DR

This paper develops a novel Bayesian approach to model terrorism spread by integrating self-excitation into spatio-temporal hierarchical models, enabling more accurate inference and comparison of violence diffusion mechanisms.

## Contribution

It introduces a new modeling framework that combines spatio-temporal diffusion with self-excitation, and demonstrates efficient inference using Laplace approximations.

## Key findings

- Self-excitation significantly improves model fit for terrorism data.
- Choice of process model affects conclusions on violence spread.
- Efficient inference method enables practical application to real data.

## Abstract

Spatio-temporal hierarchical modeling is an extremely attractive way to model the spread of crime or terrorism data over a given region, especially when the observations are counts and must be modeled discretely. The spatio-temporal diffusion is placed, as a matter of convenience, in the process model allowing for straightforward estimation of the diffusion parameters through Bayesian techniques. However, this method of modeling does not allow for the existence of self-excitation, or a temporal data model dependency, that has been shown to exist in criminal and terrorism data. In this manuscript we will use existing theories on how violence spreads to create models that allow for both spatio-temporal diffusion in the process model as well as temporal diffusion, or self-excitation, in the data model. We will further demonstrate how Laplace approximations similar to their use in Integrated Nested Laplace Approximation can be used to quickly and accurately conduct inference of self-exciting spatio-temporal models allowing practitioners a new way of fitting and comparing multiple process models. We will illustrate this approach by fitting a self-exciting spatio-temporal model to terrorism data in Iraq and demonstrate how choice of process model leads to differing conclusions on the existence of self-excitation in the data and differing conclusions on how violence is spreading spatio-temporally.