  Generalized evolutionary point processes offer a class of point process
models that allows for either excitation or inhibition based upon the history
of the process. In this regard, we propose modeling which comprises
generalization of the nonlinear Hawkes process. Working within a Bayesian
framework, model fitting is implemented through Markov chain Monte Carlo. This
entails discussion of computation of the likelihood for such point patterns.
Furthermore, for this class of models, we discuss strategies for model
comparison. Using simulation, we illustrate how well we can distinguish these
models from point pattern specifications with conditionally independent event
times, e.g., Poisson processes. Specifically, we demonstrate that these models
can correctly identify true relationships (i.e., excitation or
inhibition/control). Then, we consider a novel extension of the log Gaussian
Cox process that incorporates evolutionary behavior and illustrate that our
model comparison approach prefers the evolutionary log Gaussian Cox process
compared to simpler models. We also examine a real dataset consisting of
violent crime events from the 11th police district in Chicago from the year
2018. This data exhibits strong daily seasonality and changes across the year.
After we account for these data attributes, we find significant but mild
self-excitation, implying that event occurrence increases the intensity of
future events.
