UCSC-SOE-23-07: Mixture Modeling for Temporal Point Processes with Memory

Xiaotian Zheng, Athanasios Kottas and Bruno Sanso
05/16/2023 09:27 AM
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 and model checking.
We investigate point process model properties analytically, and illustrate the methodology
with both synthetic and real data examples.