A System-Theoretic Approach to Hawkes Process Identification with Guaranteed Positivity and Stability

2026-03-16

AI Summary

This work addresses the limitations of traditional Hawkes process identification methods, which rely on non-negative causal bases and consequently impose overly conservative parameter constraints while yielding severely ill-conditioned Gram matrices in high-order models. To overcome these issues, the authors propose a novel framework grounded in systems theory: they represent the excitation function using sign-indefinite orthogonal Laguerre bases, construct an empirical Gram matrix in the form of a Lyapunov equation, and exactly encode the non-negativity and stability constraints of the intensity function as a semidefinite programming (SDP) problem via sum-of-squares (SOS) trace equivalence. By uniquely integrating systems-theoretic insights with SDP optimization, this approach circumvents ill-conditioning entirely, yielding well-conditioned asymptotic Gram matrices at any model order and significantly enhancing numerical robustness and estimation accuracy for high-order Hawkes processes.

Technology Category

• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding

Abstract

The Hawkes process models self-exciting event streams, requiring a strictly non-negative and stable stochastic intensity. Standard identification methods enforce these properties using non-negative causal bases, yielding conservative parameter constraints and severely ill-conditioned least-squares Gram matrices at higher model orders. To overcome this, we introduce a system-theoretic identification framework utilizing the sign-indefinite orthonormal Laguerre basis, which guarantees a well-conditioned asymptotic Gram matrix independent of model order. We formulate a constrained least-squares problem enforcing the necessary and sufficient conditions for positivity and stability. By constructing the empirical Gram matrix via a Lyapunov equation and representing the constraints through a sum-of-squares trace equivalence, the proposed estimator is efficiently computed via semidefinite programming.

Problem

• Hawkes process
• positivity
• stability
• system identification
• ill-conditioned Gram matrix

Innovation

• Hawkes process
• system-theoretic identification
• Laguerre basis
• positivity and stability
• semidefinite programming