Linear prediction of point process times and marks

TL;DR

This paper develops a linear prediction framework for marked temporal point processes, deriving integral equations and proposing recursive solutions, with applications demonstrated through simulations of Hawkes processes.

Contribution

It extends linear prediction theory to marked point processes, deriving Wiener-Hopf equations and introducing two efficient recursive algorithms for different process models.

Findings

Recursive methods are computationally efficient.

Numerical schemes effectively estimate Hawkes process intensity.

The approach generalizes classical prediction to complex stochastic processes.

Abstract

In this paper, we are interested in linear prediction of a particular kind of stochastic process, namely a marked temporal point process. The observations are event times recorded on the real line, with marks attached to each event. We show that in this case, linear prediction extends straightforwardly from the theory of prediction for stationary stochastic processes. Following classical lines, we derive a Wiener-Hopf-type integral equation to characterise the linear predictor, extending the "model independent origin" of the Hawkes process (Jaisson, 2015) as a corollary. We propose two recursive methods to solve the linear prediction problem and show that these are computationally efficient in known cases. The first solves the Wiener-Hopf equation via a set of differential equations. It is particularly well-adapted to autoregressive processes. In the second method, we develop an innovations algorithm tailored for moving-average processes. A small simulation study on two typical examples shows the application of numerical schemes for estimation of a Hawkes process intensity.