Hawkes Process Modelling in Temporal Data Analysis

Hawkes processes are a class of stochastic point processes that model the occurrence of events whose likelihood increases in response to past activity. By incorporating a self-exciting mechanism, these models capture clustering and cascade effects in temporal data, offering a principled framework for understanding the dynamics of phenomena as diverse as seismic sequences, neuron firing, financial transactions and traffic incidents. At the core of a Hawkes model lies an intensity function composed of a baseline rate and a sum of triggering kernels, each of which quantifies the influence of a prior event on future activity. Extensions to the basic univariate form have introduced multivariate and spatial variants to account for interactions among multiple event types or across geographical regions, together with non-linear and state-dependent formulations that embody feedback loops and regime shifts. Recent advances have focused on enhancing the fidelity of the model in practice through non-parametric kernel estimation, fractional-order memory kernels and efficient sampling algorithms. These developments have widened the applicability of Hawkes processes to real-world tasks such as forecasting aftershock sequences, modelling high-frequency limit order books, analysing patterns of road accidents and characterising irregularly sampled medical records. Across these applications, Hawkes modelling offers both explanatory insight into event interdependence and practical tools for prediction and risk assessment.

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Recent work on state-dependent Hawkes processes has introduced a coupling between a latent state variable and the excitation mechanism, enabling application to high-frequency financial data. In this approach, the self-excitation kernel evolves according to the state of the order-book, capturing how bid–ask spread or queue imbalance modulate event clustering. In the domain of traffic safety, a non-parametric spatio-temporal Hawkes model has been applied to motorway incident data, distinguishing between primary accidents driven by background traffic patterns and secondary incidents triggered by earlier disruptions. This dual-component model employs kernel smoothing and likelihood estimation to reveal temporal decay scales and spatial influence ranges for self-excitation. On the theoretical front, a fractional Hawkes formulation has been developed in which the triggering kernel follows a Mittag-Leffler law, imparting power-law memory to the intensity function. Analytical characterisation of the expected intensity and event counts under this framework has provided new insight into long-range dependence and facilitated accurate simulation of processes with heavy-tailed inter-event times.

Hawkes Process Modelling in Temporal Data Analysis publication trend

The graph below shows the total number of articles in hawkes process modelling in temporal data analysis across all publications each year (not limited to Nature Index journals).

Technical terms

Point process: A mathematical object representing random occurrences of events in time or space.

Self-exciting process: A process in which past events increase the probability of future events occurring shortly afterwards.

Intensity function: A time-varying function that defines the instantaneous rate of event occurrence in a point process.

Triggering kernel: A function describing how the influence of a past event on the intensity decays over time or space.

References

1. State-dependent Hawkes processes and their application to limit order book modelling. Quantitative Finance (2021).
2. A Non-Parametric Hawkes Process Model of Primary and Secondary Accidents on a UK Smart Motorway. Journal of the Royal Statistical Society Series C (Applied Statistics) (2021).
3. A fractional Hawkes process II: Further characterization of the process. Physica A Statistical Mechanics and its Applications (2023).