[Paper Review] Hawkes Process Kernel Structure Parametric Search with Renormalization Factors

Rafael Lima, Jaesik Choi|arXiv (Cornell University)|May 24, 2018

Point processes and geometric inequalities14 references3 citations

TL;DR

This paper introduces Renormalization Factors (RFs) to stabilize Maximum Likelihood Estimation (MLE) in Hawkes Processes by constraining self-triggering kernel parameters to stable, stationary configurations. The method improves log-likelihood performance across four parametric kernels—Exponential, Power-Law, Rayleigh, and Tsallis Q-Exponential—especially in ill-conditioned or short-sequence scenarios, with RF-MLE consistently outperforming standard MLE in empirical evaluations.

ABSTRACT

Hawkes Processes are a type of point process for modeling self-excitation, i.e., when the occurrence of an event makes future events more likely to occur. The corresponding self-triggering function of this type of process may be inferred through an Unconstrained Optimization-based method for maximization of its corresponding Loglikelihood function. Unfortunately, the non-convexity of this procedure, along with the ill-conditioning of the initialization of the self- triggering function parameters, may lead to a consequent instability of this method. Here, we introduce Renormalization Factors, over four types of parametric kernels, as a solution to this instability. These factors are derived for each of the self-triggering function parameters, and also for more than one parameter considered jointly. Experimental results show that the Maximum Likelihood Estimation method shows improved performance with Renormalization Factors over sets of sequences of several different lengths.

Motivation & Objective

• To address the instability of unconstrained MLE in Hawkes Processes due to non-convexity and ill-conditioned initialization.
• To derive closed-form Renormalization Factors that constrain self-triggering kernel parameters to stable, stationary configurations.
• To extend stabilization to joint parameter spaces and multiple kernel families, including Exponential, Power-Law, Rayleigh, and Tsallis Q-Exponential.
• To validate the method’s efficacy across diverse sequence lengths and kernel misassignment scenarios.
• To enable robust parameter estimation near unstable parameter regions without sacrificing model fit.

Proposed method

• Proposes Renormalization Factors (RFs) as a regularization mechanism to enforce stationarity in Hawkes Process self-triggering kernel parameters.
• Derives closed-form RF expressions for each individual parameter and for joint parameter sets across four kernel families.
• Applies RFs during MLE optimization to constrain parameter space to stable configurations, preventing explosion to infinite event rates.
• Uses a perturbation-based approach with parameter ε to control the degree of renormalization, enabling exploration near instability boundaries.
• Employs standard MLE with RF constraints (RF-MLE) and compares performance against unconstrained MLE on synthetic sequences.
• Validates the method using synthetic sequences of varying lengths (T=1000, 5000, 10000, 30000) and kernel misassignment scenarios.

Experimental results

Research questions

• RQ1 Can Renormalization Factors stabilize MLE in Hawkes Processes under non-convex optimization and ill-conditioned initialization?
• RQ2 How do RFs improve log-likelihood performance across different parametric kernel families?
• RQ3 To what extent does RF-MLE outperform standard MLE in short-sequence and misassigned kernel scenarios?
• RQ4 Can joint parameter renormalization improve stability and convergence beyond individual parameter constraints?
• RQ5 Does the method maintain robustness when parameters are near unstable configurations?

Key findings

• RF-MLE significantly improves log-likelihood across all tested sequence lengths (T=1000 to T=30000), with consistent gains over standard MLE.
• For T=1000, RF-MLE with ε=0.1 improved log-likelihood in 90% of sequences across all kernel types, with up to 15% improvement in some cases.
• At T=30000, RF-MLE achieved a log-likelihood of -27509.08 (EXP kernel) compared to -27977.34 for standard MLE, a relative improvement of ~1.7%.
• Even under kernel misassignment (e.g., fitting EXP on PWL-generated sequences), RF-MLE showed improvement in 70-80% of cases at T=10000.
• The Rayleigh kernel showed the most consistent improvement, with RF-MLE achieving identical or better log-likelihood than MLE across all sequence types and lengths.
• The method effectively stabilizes MLE in near-unstable parameter regions, preventing explosion to infinite event rates while maintaining model fit.