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

Xinhui Rong, Girish N. Nair

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.

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

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.
Paper Structure (12 sections, 8 theorems, 22 equations, 2 figures)

This paper contains 12 sections, 8 theorems, 22 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hawkes Processes
  4. Laguerre Bases
  5. Unconstrained LS for ON-HL
  6. The Truncated Hawkes Intensity
  7. The Unconstrained LS Estimator and the Asymptotics
  8. System-Theoretic Computation of the Estimators
  9. Positive and Stable LS for ON-HL
  10. Simulations
  11. Conclusions and Future Work
  12. Appendix: Proofs

Key Result

Lemma 1

Roh06 The orthonormal Laguerre polynomials $u_j(t)$ with $w(t)=\beta e^{-\beta t}\mathbf{1}_{t\geq0}$, satisty a three-term recursion with $\rho_j = \frac{2\beta}{j+1}, \kappa_j=\frac{2j+1}{j+1}, \gamma_j=\frac{j}{j+1}, u_0(t)=\sqrt{2/\beta}$, and $u_j(t)\triangleq0$ for $j<0$. As a property, $\frac{1}{\rho_{j}}=\frac{\gamma_{j+1}}{\rho_{j+1}}$. For an arbitrary dimension $m\geq1$, the three-term

Figures (2)

  • Figure 1: Quantiles of the constrained LS estimators.
  • Figure 2: The squared $L_2$ HIR approximation errors for the constrained LS.

Theorems & Definitions (8)

  • Lemma 1
  • Theorem 1
  • Proposition 1.1
  • Proposition 1.2
  • Theorem 2
  • Proposition 2.1
  • Lemma 2
  • Theorem 3