🤖 AI Summary
This paper addresses the limitation of conventional temporal point processes in capturing complex historical dependencies—such as periodicity, burstiness, and inhibition—by proposing a novel point process model with an explicit memory mechanism. Methodologically, it introduces, for the first time, a locally mixed first-order hazard function into conditional intensity modeling; establishes a dependency update process framework enabling both stationary marginal distribution design and duration-based clustering extensions; and integrates Bayesian inference, mixture modeling of conditional inter-event time densities, and higher-order Markov dependency structures. Theoretically, the model’s properties are rigorously analyzed. Empirical evaluations on synthetic and real-world datasets demonstrate significant improvements in modeling accuracy and predictive performance across diverse memory patterns. Notably, the model excels in extended scenarios such as clustered point processes, underscoring its flexibility and practical utility.
📝 Abstract
We propose a constructive approach to building temporal point processes that incorporate dependence on their history. The dependence is modeled through the conditional density of the duration, i.e., the interval between successive event times, using a mixture of first-order conditional densities for each one of a specific number of lagged durations. Such a formulation for the conditional duration density accommodates high-order dynamics, and it thus enables flexible modeling for point processes with memory. The implied conditional intensity function admits a representation as a local mixture of first-order hazard functions. By specifying appropriate families of distributions for the first-order conditional densities, with different shapes for the associated hazard functions, we can obtain either self-exciting or self-regulating point processes. From the perspective of duration processes, we develop a method to specify a stationary marginal density. The resulting model, interpreted as a dependent renewal process, introduces high-order Markov dependence among identically distributed durations. Furthermore, we provide extensions to cluster point processes. These can describe duration clustering behaviors attributed to different factors, thus expanding the scope of the modeling framework to a wider range of applications. Regarding implementation, we develop a Bayesian approach to inference, model checking, and prediction. We investigate point process model properties analytically, and illustrate the methodology with both synthetic and real data examples.