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

TL;DR

This paper introduces a novel system-theoretic method for identifying Hawkes processes that guarantees positivity and stability, overcoming limitations of traditional approaches at higher model orders.

Contribution

It proposes a new identification framework using orthonormal Laguerre basis and semidefinite programming to ensure well-conditioned estimation and enforce key properties.

Findings

The method guarantees a well-conditioned Gram matrix regardless of model order.

It enforces positivity and stability through a constrained least-squares formulation.

The approach is computationally efficient via semidefinite programming.

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.