Proper Loss Functions for Nonlinear Hawkes Processes

Temporal point processes are a statistical framework for modelling the times at which events of interest occur. The Hawkes process is a well-studied instance of this framework that captures self-exciting behaviour, wherein the occurrence of one event increases the likelihood of future events. Such processes have been successfully applied to model phenomena ranging from earthquakes to behaviour in a social network. We propose a framework to design new loss functions to train linear and nonlinear Hawkes processes. This captures standard maximum likelihood as a special case, but allows for other losses that guarantee convex objective functions (for certain types of kernel), and admit simpler optimisation. We illustrate these points with three concrete examples: for linear Hawkes processes, we provide a least-squares style loss potentially admitting closed-form optimisation; for exponential Hawkes processes, we reduce training to a weighted logistic regression; and for sigmoidal Hawkes processes, we propose an asymmetric form of logistic regression.