Self-exciting point processes

Alex Reinhart – Updated June 25, 2018 notebooks · refsmmat.com

See also spatiotemporal point processes and Mutually exciting point processes.

Also known as Hawkes processes.

A review, focusing on temporal processes modeling social media events, is Rizoiu, M.-A., Lee, Y., Mishra, S., & Xie, L. (2017). A tutorial on Hawkes proceses for events in social media. https://arxiv.org/abs/1708.06401

Properties

ETAS

The basic approach comes from Epidemic-Type Aftershock Models, where earthquakes are caused by some constant background process and then induce further aftershocks when they arrive. There’s a whole series of papers by Ogata; some highlights from the field:

Simulation

Thinning is a common technique, but there are better ways:

Epidemic/endemic models

The ETAS has been adapted to epidemiology (see also epidemic models):

Stochastic declustering

A self-exciting point process can be interpreted as a Poisson cluster process, as mentioned above. It could be interesting to decluster it, meaning to remove the events which were “excited” by another, and leave only the background events which occurred spontaneously. (In the earthquake literature, this means removing the aftershocks and keeping only the main shocks.) Stochastic declustering is this procedure.

Evaluation

How do we evaluate predictions made by a self-exciting point process model?

Renewal Hawkes processes

What if the underlying background process is not Poisson but some other general renewal process? This is much more flexible, e.g. allowing Weibull interevent arrival times and very different clustering behaviors of the background process, but computationally challenging.