A Scalable EM Algorithm for Hawkes Processes

Halpin and De Boeck (2013) considered the time series analysis of bivariate event data in the context of dyadic interaction. They proposed the use of point processes (e.g., Daley and Vere-Jones 2003), and in particular Hawkes processes (Hawkes 1971; Hawkes and Oakes 1974), as way to capture the temporal dependence between the actions of two individuals. Here an action is treated as an occurrence, which is a discrete event that is viewed as having negligible duration relative to the period of observation. Occurrences may be contrasted with events that are viewed as extended in time (e.g., states, regimes). Examples of occurrences during the course of a conversation include specific types of statements (e.g., criticism, questions, lies) or nonverbal behaviors (e.g., laughter, facial expressions, gestures). Point processes are especially well suited to cases where human interaction is mediated by technology (e.g., text-messaging, emailing, chatting, tweeting), because such interactions are naturally parsed as series of time-stamped events. We can also view interaction more broadly, including, say, a student’s interactions with an intelligent tutor, or a gamer’s interactions with a virtual agent. The fundamental idea is to represent an interaction as a series of discrete, instantaneous actions. The theory of point processes then provides a general statistical framework for modelling the timing of those actions—how their probability changes in continuous time, how this depends on previous actions, and how the actions of two or more people may be coordinated in time. The approach to estimation taken by Halpin and De Boeck (2013) was based on the so-called branching structure representation of the Hawkes process, which they showed to be amenable to the EM algorithm (see also Veen and Schoenberg 2008).