Rank-Based Mixture Models for Temporal Point Processes

Temporal point process, an important area in stochastic process, has been extensively studied in both theory and applications. The classical theory on point process focuses on time-based framework, where a conditional intensity function at each given time can fully describe the process. However, such a framework cannot directly capture important overall features/patterns in the process, for example, characterizing a center-outward rank or identifying outliers in a given sample. In this article, we propose a new, data-driven model for regular point process. Our study provides a probabilistic model using two factors: (1) the number of events in the process, and (2) the conditional distribution of these events given the number. The second factor is the key challenge. Based on the equivalent inter-event representation, we propose two frameworks on the inter-event times (IETs) to capture large variability in a given process—One is to model the IETs directly by a Dirichlet mixture, and the other is to model the isometric logratio transformed IETs by a classical Gaussian mixture. Both mixture models can be properly estimated using a Dirichlet process (for the number of components) and Expectation-Maximization algorithm (for parameters in the models). In particular, we thoroughly examine the new models on the commonly used Poisson processes. We finally demonstrate the effectiveness of the new framework using two simulations and one real experimental dataset.