Novel theory and methods for hawkes processes
File(s)
Author(s)
Connell, Andrew Murray
Type
Thesis or dissertation
Abstract
This thesis develops new theory and methodology for Hawkes processes by linking identifiability theory with wavelet-based models for non-stationary event dynamics. The focus is on exponential Hawkes processes with time-varying behaviour and on transforming irregular event data into representations amenable to multiscale analysis.
Chapter 3 establishes a framework for practical identifiability of univariate exponential Hawkes processes. Explicit positive-moment formulae together with tractable representations, bounds, and approximations for the negative intensity moments are obtained, which serve as a basis for identifiability conditions. From these results, an explicit Jacobian together with tractable Fisher information representations, bounds, and approximations are derived. Regions of the parameter space can then be classified as well or poorly determined by considering properties of these matrices. This chapter contributes novel identifiability diagnostics and constraints that stabilise likelihood-based estimation and guide penalty design.
Chapter 4 develops two extensions of trend-locally stationary wavelet (T-LSW) processes: the Absolute Trend-LSW (AT-LSW) and Generalised Trend-LSW (GT-LSW). Building on T-LSW concepts, differencing is employed in both the absolute and general settings. These models can be used to estimate trend and local autocovariance structures. Simulations on the standard Donoho-Johnstone test functions demonstrate improved trend recovery and evolutionary wavelet spectra relative to existing methods. This work motivates Chapter 5, where these ideas are applied to Hawkes processes.
Chapter 5 embeds locally stationary Hawkes processes within the GT-LSW framework. Converting raw event times into a binned regularly indexed series allows for the GT-LSW differencing scheme to be applied. From this, the evolutionary spectrum and background intensity can be recovered, and a non-parametric estimate of the mean density is obtained. Given knowledge of the Hawkes kernel shape, the excitation parameters and the background intensity can then be estimated. Simulations show the effectiveness of the Hawkes GT-LSW pipeline. Finally, this new methodology is applied to real-world earthquake data.
Chapter 3 establishes a framework for practical identifiability of univariate exponential Hawkes processes. Explicit positive-moment formulae together with tractable representations, bounds, and approximations for the negative intensity moments are obtained, which serve as a basis for identifiability conditions. From these results, an explicit Jacobian together with tractable Fisher information representations, bounds, and approximations are derived. Regions of the parameter space can then be classified as well or poorly determined by considering properties of these matrices. This chapter contributes novel identifiability diagnostics and constraints that stabilise likelihood-based estimation and guide penalty design.
Chapter 4 develops two extensions of trend-locally stationary wavelet (T-LSW) processes: the Absolute Trend-LSW (AT-LSW) and Generalised Trend-LSW (GT-LSW). Building on T-LSW concepts, differencing is employed in both the absolute and general settings. These models can be used to estimate trend and local autocovariance structures. Simulations on the standard Donoho-Johnstone test functions demonstrate improved trend recovery and evolutionary wavelet spectra relative to existing methods. This work motivates Chapter 5, where these ideas are applied to Hawkes processes.
Chapter 5 embeds locally stationary Hawkes processes within the GT-LSW framework. Converting raw event times into a binned regularly indexed series allows for the GT-LSW differencing scheme to be applied. From this, the evolutionary spectrum and background intensity can be recovered, and a non-parametric estimate of the mean density is obtained. Given knowledge of the Hawkes kernel shape, the excitation parameters and the background intensity can then be estimated. Simulations show the effectiveness of the Hawkes GT-LSW pipeline. Finally, this new methodology is applied to real-world earthquake data.
Version
Open Access
Date Issued
2025-12-27
Date Awarded
2026-07-01
Copyright Statement
Attribution-NonCommercial-ShareAlike 4.0 International Licence (CC BY NC-SA)
Advisor
Cohen, Ed
McCoy, Emma
Publisher Department
Department of Mathematics
Publisher Institution
Imperial College London
Qualification Level
Doctoral
Qualification Name
Doctor of Philosophy (PhD)