Multivariate Hawkes Processes for Incomplete Biased Data.

Multivariate Hawkes processes have been widely used in many applications such as crime detection and disaster rescue forecast to model events that exhibit self-exciting properties. One of the biggest challenges is that data collected from real world is usually incomplete, and even biased. The training of a machine learning model using such data can introduce biased predictions. For example, event hotspot predictions using biased data can make the visibility of minority groups (e.g., communities of racial minorities) more apparent. While there have been some explorations in developing Hawkes processes for event data, none of those methods deals with incomplete biased data where events of certain markers (e.g., events reported from racial minorities) may be missing or heavily underrepresented. In this paper, we propose a novel Multivariate Hawkes model to tackle the incomplete biased data challenge. First, we assume that there is possibility that events can be missing between any two observed events and we define a n ovel likelihood function integrating missing window probabilities. A Markov Chain Monte Carlo (MCMC) sampling framework is used to generate virtual event data probabilistically in missing windows. Second, we propose to incorporate event marker features such as geographic information to regularize the infectivity kernel matrix between markers. In such a way, the MCMC sampler is encouraged to generate more virtual events with markers that are biased. Both observed and virtual events will contribute to the model estimation through maximizing the log-likelihood. We carry on experiments over several real-world datasets, and our model improves prediction accuracy in comparison with the state-of-arts.