Self-Exciting Process

A self-exciting process is any point process whose conditional intensity $\lambda (t)$ depends on, and increases in response to, the process's own past event history. The Hawkes process is the canonical, linear self-exciting process, its intensity is baseline plus an additive sum of decaying kernel contributions from past events. More general (nonlinear) self-exciting processes replace this additive structure with a nonlinear function of the past, $\lambda (t)=g\left(\mu +{\sum }_{t_i<t}\varphi (t-t_i)\right)$ for some nonlinear link function $g$, relaxing the requirement that excitation contributions simply add up linearly. Self-excitation is contrasted with self-correction (or self-inhibition), where past events instead suppress the near-future intensity, an important and distinct sub-family used to model regularly-spaced or inhibited event sequences (e.g. neuronal spike trains with a refractory period).

• In the overwhelming majority of applied statistics and machine learning literature, 'self-exciting process' and 'Hawkes process' are used essentially interchangeably, the distinction drawn in this dictionary, general property vs. specific linear formalization, matters mainly when discussing nonlinear extensions or self-correcting counterparts.
• Self-excitation is a genuinely different clustering mechanism from the Cox process's over-dispersion: a Cox process clusters because of a shared, unobserved external random environment, while a self-exciting process clusters because events causally trigger further events, a distinction with real scientific consequences for how the two models should be interpreted.
• Beyond seismology and finance, self-exciting process models have become a standard tool in computational social science for modeling social media cascades, retweet chains, and information diffusion, wherever one event plausibly triggers a wave of closely-following related events.