Point processes on directed linear network

TL;DR

This paper develops methods for modeling and analyzing point processes on directed linear networks, extending classical temporal models to network settings with applications to neurological data.

Contribution

It introduces a framework for point processes on directed linear networks, including likelihood derivation, simulation algorithms, and model checking, extending well-known models like Hawkes processes.

Findings

Derived explicit likelihood expressions for models

Developed two simulation algorithms for these processes

Applied methods to neurological data analysis

Abstract

In this paper we consider point processes specified on directed linear networks, i.e. linear networks with associated directions. We adapt the so-called conditional intensity function used for specifying point processes on the time line to the setting of directed linear networks. For models specified by such a conditional intensity function, we derive an explicit expression for the likelihood function, specify two simulation algorithms (the inverse method and Ogata's modified thinning algorithm), and consider methods for model checking through the use of residuals. We also extend the results and methods to the case of a marked point process on a directed linear network. Furthermore, we consider specific classes of point process models on directed linear networks (Poisson processes, Hawkes processes, non-linear Hawkes processes, self-correcting processes, and marked Hawkes processes), all adapted from well-known models in the temporal setting. Finally, we apply the results and methods to analyse simulated and neurological data.