Modeling Event Dynamics by Self-Exciting Processes with Random Memory

TL;DR

The paper addresses the challenge that standard Hawkes processes impose long-lasting or history-wide excitation, which may not capture transient, event-triggered bursts in real data. It introduces an extended Hawkes framework with a latent memory duration $\tau$ and a semi-Markov hot-state $S(t)$, yielding two-component intensities $\lambda_0$ and $\lambda_1$ and enabling a random, finite memory of past events. Parameter estimation is achieved via a Monte Carlo EM algorithm, and the authors develop thinning-based simulation methods to generate synthetic event data that preserve the transient excitation structure. The framework is demonstrated on 2019 Chinese Super League corner-kick data, revealing robust, time-varying baselines and near-doubling of intensity in the hot state, with a mean hot-duration around $2$ minutes; dynamic prediction and simulation show practical utility for real-time match analysis and scenario exploration. Beyond soccer, the approach applies to any recurrent-event setting with transient excitation, offering a flexible tool for inference, prediction, and simulation in domains such as criminology, epidemiology, and reliability analysis.

Abstract

Event history data from sports competitions have recently drawn increasing attention in sports analytics to generate data-driven strategies. Such data often exhibit self-excitation in the event occurrence and dependence within event clusters. The conventional event models based on gap times may struggle to capture those features. In particular, while consecutive events may occur within a short timeframe, the self-excitation effect caused by previous events is often transient and continues for a period of uncertain time. This paper introduces an extended Hawkes process model with random self-excitation duration to formulate the dynamics of event occurrence. We present examples of the proposed model and procedures for estimating the associated model parameters. We employ the collection of the corner kicks in the games of the 2019 regular season of the Chinese Super League to motivate and illustrate the modeling and its usefulness. We also design algorithms for simulating the event process under proposed models. The proposed approach can be adapted with little modification in many other research fields such as Criminology and Infectious Disease.

Figures (4)

• Figure 1: Estimated baseline intensity functions for the home and away teams under four model specifications. Each row corresponds to a different baseline specification (Models (a)--(d)), and each column corresponds to team (home vs. away). Within each panel, solid and dashed black curves represent the estimated baseline intensities under the regular and hot states, respectively. The red curve shows a smoothed baseline intensity derived from a Andersen-Gill model using the Breslow estimator, included for reference.
• Figure 2: Dynamic prediction of corner kick intensity (y-axis) for the home team in the soccer match "Shenzhen vs Shanghai" (Mar 1, 2019). Six representative match times are shown. For each row, the solid black curve displays the estimated intensity based on information observed up to the indicated time (vertical dashed line), and the lighter extension shows the predicted future intensity conditional on the current state of the match. Black crosses represent observed corner kicks, red stars represent goals, and black triangles indicate the end of each half.
• Figure 3: Illustration of the thinning-based simulation algorithm for a time-varying intensity with a latent hot state. Solid and dashed curves indicate the active and inactive intensities, respectively; the solid curve corresponds to the effective intensity $\lambda^\star(t)$ used for thinning at time $t$. Shaded regions denote hot state windows following accepted events. Candidate events generated under the dominating rate are accepted with probability $\lambda^\star(t)/\bar{\lambda}$ (solid points) or rejected otherwise (crosses).
• Figure 4: Cluster size distributions for observed CSL matches and simulated matches. Cluster size is defined as the number of consecutive corner kicks occurring within a pre-specified time threshold. Panels correspond to different threshold durations, with separate curves for the home and away teams. The red curve represents the empirical probability mass function from the observed data, while the grey curves correspond to the 200 simulated seasons.