In many situations, the occurrence of one event increases the likelihood of future events, exhibiting self-triggering behavior, e.g., earthquakes leading to aftershocks, or crime activity in a region leading to further crimes, etc. These systems are usually modelled as Hawkes processes. This presentation focuses on some problems at the interface of generative modeling, optimization, and Spatiotemporal Hawkes processes, with a special emphasis on applications in predictive policing.

A core challenge in applying Hawkes processes to real-world data, such as crime records, is the presence of noisy and missing data. Traditional Maximum Likelihood Estimation (MLE) methods become intractable when dealing with a significant proportion of unreported crimes. To address this, we propose a likelihood-free approach using Wasserstein Generative Adversarial Networks (WGAN) and demonstrate a case study on forecasting crime hotspots in Bogota, Colombia, using only reported crime data. Next, we look at Hawkes networks where activity in one node might trigger further activity across the other nodes. These systems are widely used in predictive policing. Strategic intervention at some nodes (such as enhanced patrolling) can mitigate the spread of events throughout the network. In this context, we explore the problem of optimal intervention strategies under resource constraints to minimize the spread of events in a self-exciting spatial network. Different intervention strategies are compared, and the optimal strategy, formulated as a solution to a mixed integer programming (MILP) problem, outperforms heuristic methods by adapting to clustering and spillover dynamics. Subsequently, we illustrate our methodology using crime data from Los Angeles, CA.

In the last chapter, we investigated shape-constrained non-parametric estimation of triggering kernels in Hawkes processes. While parametric kernels like exponential or power-law are standard, they may not fully capture the true nature of event triggering. Non-parametric methods allow for more flexible kernel shapes, such as monotone decreasing or concave kernels. Our work establishes that computing the NPMLE boils down to solving a convex optimization problem under linear constraints. Then, we describe methodologies to estimate the triggering kernels consistently using regularized NPMLE and illustrate our method using financial market data and earthquake aftershock records.

In addition, we discuss avenues for future research in these areas and general computational challenges in the area of Hawkes processes.