Supplementary Material for Reactive Point Processes: a New Approach to Predicting Power Failures in Underground Electrical Systems

1. Related work. A general linear self-exciting process is identified by the conditional intensity function λ(t) = µ + k 0 t k <t g(t − t k) where µ is the baseline hazard rate. The Hawkes Process (Hawkes, 1971a,b) is a particularly tractable self-exciting point process with an exponential kernel, developed for earthquake modeling. Historically, a lot of effort has gone into modeling earthquakes, e. which consider Epidemic-Type-Aftershock-Sequences (ETAS). The formulation and parametriza-tion of the ETAS model divides the effect of earthquakes into two components, namely background and aftershock events. Background events, corresponding to µ in the expression above, occur independently according to a stationary Poisson process. An earthquake elevates the risk of aftershocks, where the magnitude and temporal properties of this elevated risk are determined by the kernel g. Subsequent works incorporated the spatial distance of past seismic events for modeling the seismic activity at a point of interest (Musmeci and Vere-Jones, 1992; Ogata, 1998) (note that spatial distance elements do not appear to be first order effects for manhole event prediction). In recent applications of SEPP's, Mohler et al. (2011) described a SEPP model that is based on the space-time ETAS model for modeling the rate of crimes. In particular, they drew a parallel between the spread of seismic activity and crime behavior, and proposed a model for residential burglaries that captures the spatial-temporal clustering patterns observed in crime data. Egesdal et al. (2010) studied a similar problem of modeling gang rivalries, where an event involving rival gangs can lead to retaliatory (self-excitatory) acts of violence. The same problem was addressed by Louie, Masaki and Allenby (2010), which extends the model of Egesdal et al. (2010) to incorporate the spatial influence of past events (similar to the space-time ETAS model of Ogata, 1998). Lewis