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

T0 review · 1 major / 2 minor · reviewed 2026-05-21 · grok-4.3

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Pith's one-line read Hawkes process identification can guarantee positivity and stable intensities by solving a constrained least-squares problem via semidefinite programming on a Laguerre basis.

desk verdict The paper's main move is switching to a sign-indefinite Laguerre basis plus Lyapunov Gram matrix and SOS-encoded SDP to get better-conditioned Hawkes fits with claimed positivity and stability guarantees. read the letter →

arxiv 2603.14942 v3 pith:WOLAZ6G2 submitted 2026-03-16 eess.SY cs.SYstat.ME

Xinhui Rong , Girish N. Nair This is my paper
classification eess.SYcs.SYstat.ME
keywords HawkesprocesssystemidentificationpositivityconstraintsstabilityLaguerrebasissemidefiniteprogrammingleast-squaresestimation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper introduces a system-theoretic framework for identifying Hawkes process parameters from event data. It replaces conventional non-negative bases with a sign-indefinite orthonormal Laguerre basis to produce a well-conditioned Gram matrix at any model order. The method formulates identification as a constrained least-squares problem whose constraints enforce the exact necessary and sufficient conditions for the intensity to remain non-negative and stable. These constraints are recast through a sum-of-squares trace equivalence and solved efficiently as a semidefinite program after building the empirical Gram matrix from a Lyapunov equation.

What carries the argument

sign-indefinite orthonormal Laguerre basis together with sum-of-squares trace equivalence to enforce positivity and stability constraints

What would settle it

An observed intensity trajectory generated from the SDP solution that becomes negative or violates stability for some realized history of events would show the constraints are not sufficient.

Watch

Extended reading notes

Core claim

By employing the sign-indefinite orthonormal Laguerre basis, the empirical Gram matrix is constructed via a Lyapunov equation to remain well-conditioned independent of model order. Positivity and stability of the Hawkes intensity are enforced through necessary and sufficient conditions represented via sum-of-squares trace equivalence, allowing the estimator to be computed as a semidefinite program.

Load-bearing premise

The sign-indefinite orthonormal Laguerre basis together with the sum-of-squares trace equivalence exactly captures the necessary and sufficient conditions for positivity and stability of the Hawkes intensity.

Editorial extensions

If this is right

  • The estimator is efficiently computed via semidefinite programming.
  • The asymptotic Gram matrix remains well-conditioned independent of model order.
  • Positivity and stability hold by construction rather than through conservative basis restrictions.
  • Standard non-negative causal bases produce both conservative constraints and severely ill-conditioned matrices at higher orders.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Higher-order models become numerically practical without the conditioning blow-up seen in conventional bases.
  • The Lyapunov-based Gram construction may transfer to identification of other linear state-space representations of point processes.
  • Real-data experiments could test whether the exact positivity guarantee improves out-of-sample intensity prediction compared with post-hoc clipping methods.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

1 major / 2 minor

Summary. The manuscript proposes a system-theoretic framework for identifying Hawkes processes that guarantees a non-negative intensity function and stability of the process. It replaces non-negative causal bases with the sign-indefinite orthonormal Laguerre basis to obtain a well-conditioned asymptotic Gram matrix independent of model order, constructs the empirical Gram matrix from a Lyapunov equation, and encodes the necessary and sufficient positivity and stability constraints via a sum-of-squares trace equivalence that is solved by semidefinite programming.

Significance. If the sum-of-squares trace equivalence is shown to be necessary and sufficient for positivity and stability in the chosen basis, the approach would permit higher-order models with exact enforcement of the required properties and improved numerical conditioning, constituting a useful advance for practical Hawkes identification in applications such as neuroscience and finance.

major comments (1)
  1. [Abstract / constrained least-squares formulation] Abstract and the paragraph describing the constrained least-squares formulation: the claim that the sum-of-squares trace equivalence exactly represents the necessary and sufficient conditions for non-negativity of the kernel and stability (spectral radius <1) must be established rigorously. Because the Laguerre functions are sign-indefinite, the mapping from coefficient vector to admissible kernels is not automatic; any gap between the SOS representation and the true set would mean the SDP either admits invalid kernels or excludes valid ones, undermining the central guarantee.
minor comments (2)
  1. [Gram matrix construction] Clarify the precise statement of the Lyapunov equation used to build the empirical Gram matrix and confirm that it remains well-conditioned for the chosen basis at the orders reported in the numerical examples.
  2. [Numerical implementation] Add a brief remark on how the SDP is initialized and whether warm-starting from an unconstrained least-squares solution is employed in practice.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback. The major comment correctly identifies the need for a fully rigorous proof of the sum-of-squares trace equivalence. We address this point below and will revise the manuscript to include an expanded, self-contained proof.

read point-by-point responses

  1. Referee: [Abstract / constrained least-squares formulation] Abstract and the paragraph describing the constrained least-squares formulation: the claim that the sum-of-squares trace equivalence exactly represents the necessary and sufficient conditions for non-negativity of the kernel and stability (spectral radius <1) must be established rigorously. Because the Laguerre functions are sign-indefinite, the mapping from coefficient vector to admissible kernels is not automatic; any gap between the SOS representation and the true set would mean the SDP either admits invalid kernels or excludes valid ones, undermining the central guarantee.

    Authors: We agree that a rigorous, self-contained demonstration of necessity and sufficiency is essential for the central claim. In the manuscript the equivalence is obtained by combining the completeness of the orthonormal Laguerre basis in L2 with the Lyapunov representation of the Gram matrix; the SOS trace condition then becomes both necessary (by the Riesz representation theorem applied to the quadratic form induced by the kernel) and sufficient (by the positive-semidefinite factorization of the coefficient matrix). The sign-indefiniteness of individual basis functions is immaterial because the constraint is imposed on the quadratic form over the entire function space rather than on pointwise coefficient signs. Nevertheless, the current exposition is concise and may leave the logical steps implicit. In the revision we will insert a dedicated lemma (with full proof) immediately after the problem formulation that (i) shows the admissible set of coefficient vectors is exactly the feasible set of the SDP, (ii) proves that every feasible solution yields a non-negative kernel whose induced operator has spectral radius strictly less than one, and (iii) confirms that every admissible kernel possesses a unique Laguerre coefficient vector satisfying the trace condition. This addition will eliminate any possible gap between the SOS representation and the true constraint set. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; derivation uses independent system-theoretic tools

full rationale

The paper's central estimator is a constrained least-squares problem whose positivity and stability constraints are encoded via sum-of-squares trace equivalence after constructing the empirical Gram matrix from a Lyapunov equation. These steps rely on standard, externally validated tools (orthonormal Laguerre basis, Lyapunov equations, SDP) rather than any self-definitional loop, fitted-input-as-prediction, or load-bearing self-citation. The abstract explicitly separates the construction of the Gram matrix (via Lyapunov) from the constraint representation (via SOS trace equivalence), and the reader's assessment confirms the framework does not reduce results to quantities defined by the same fitted parameters. No quoted reduction of the form 'Eq. X equals input Y by construction' appears. The derivation therefore remains self-contained against external benchmarks in control theory and convex optimization.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The approach rests on standard mathematical properties of orthonormal Laguerre functions and Lyapunov equations for Gram matrix construction; no new free parameters or invented entities are introduced in the abstract.

assumptions (2)
  • standard math Orthonormal Laguerre basis yields well-conditioned asymptotic Gram matrix independent of model order
    Invoked to overcome ill-conditioning of traditional bases
  • domain assumption Sum-of-squares trace equivalence represents necessary and sufficient positivity and stability conditions
    Used to convert constraints into SDP form

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Cite this review

Pith. "Pith review of A System-Theoretic Approach to Hawkes Process Identification with Guaranteed Positivity and Stability." pith.science (2026). https://pith.science/paper/WOLAZ6G2

@misc{pith2026260314942,
  author       = {Pith},
  title        = {Pith review of: A System-Theoretic Approach to Hawkes Process Identification with Guaranteed Positivity and Stability},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WOLAZ6G2}},
  note         = {Machine review of arXiv:2603.14942}
}
read the original 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.

Figures

Figures reproduced from arXiv: 2603.14942 by the authors.

Figure 1
Figure 1. Quantiles of the constrained LS estimators. [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. The squared L2 HIR approximation errors for the constrained LS. LS computations. Using an SOS characterisation of non￾negative polynomials, we formulated a trace-based SDP that enforces both positivity and stability of the identified inten￾sity. Simulations confirmed that ON-HL maintains numerical robustness and achieves lower HIR approximation error than the Erlang-based model. In the future, we will extend the ON-… view at source ↗

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Works this paper leans on

30 extracted references · 30 canonical work pages

  1. [1]

    Bialek, R

    W. Bialek, R. R. Van Steveninck, F. Rieke, and D. Warland,Spikes: Exploring the Neural Code. MIT press, 1997

    work page 1997

  2. [2]

    Multivari- ate Hawkes process models of the occurrence of regulatory elements,

    L. Carstensen, A. Sandelin, O. Winther, and N. R. Hansen, “Multivari- ate Hawkes process models of the occurrence of regulatory elements,” BMC Bioinformatics, vol. 11, pp. 456–474, 2010

    work page 2010

  3. [3]

    Hawkes processes in finance,

    E. Bacry, I. Mastromatteo, and J.-F. Muzy, “Hawkes processes in finance,”Mark. Microstruct. Liq., vol. 1, no. 1, p. 1550005, 2015

    work page 2015

  4. [4]

    Event-based vision: A survey,

    G. Gallego, T. Delbruck, G. Orchard, C. Bartolozzi, B. Taba, A. Censi, S. Leutenegger, A. J. Davison, J. Conradt, K. Daniilidis, and D. Scara- muzza, “Event-based vision: A survey,”IEEE Trans. Pattern Anal. Mach. Intell., vol. 44, no. 01, pp. 154–180, 2022

    work page 2022

  5. [5]

    D. Shi, L. Shi, and T. Chen,Event-Based State Estimation: A Stochas- tic Perspective. New York: Springer, 2015

    work page 2015

  6. [6]

    Maximum likelihood estimation of Hawkes’ self-exciting point processes,

    T. Ozaki, “Maximum likelihood estimation of Hawkes’ self-exciting point processes,”Ann. Inst. Stat. Math., vol. 31, pp. 145–155, 1979

    work page 1979

  7. [7]

    Truncated Hawkes point pro- cess modeling: System theory and system identification,

    B. I. Godoy, V . Solo, and S. A. Pasha, “Truncated Hawkes point pro- cess modeling: System theory and system identification,”Automatica, vol. 113, p. 108733, 2020

    work page 2020

  8. [8]

    A nonparametric EM algorithm for multi- scale Hawkes processes,

    E. Lewis and G. Mohler, “A nonparametric EM algorithm for multi- scale Hawkes processes,”J. Nonparam. Stat., vol. 1, pp. 1–20, 2011

    work page 2011

Show all 30 references
  1. [9]

    Estimation of space–time branching process models in seismology using an EM–type algorithm,

    A. Veen and F. P. Schoenberg, “Estimation of space–time branching process models in seismology using an EM–type algorithm,”J. Am. Stat. Assoc., vol. 103, no. 482, pp. 614–624, 2008

    2008

  2. [10]

    Sparse and low- rank multivariate Hawkes processes,

    E. Bacry, M. Bompaire, S. Ga ¨ıffas, and J.-F. Muzy, “Sparse and low- rank multivariate Hawkes processes,”J. Machn. Learn. Res., vol. 21, no. 50, pp. 1–32, 2020

    2020

  3. [11]

    Lasso and probabilistic inequalities for multivariate point processes,

    N. R. Hansen, P. Reynaud-Bouret, and V . Rivoirard, “Lasso and probabilistic inequalities for multivariate point processes,”Bernoulli, vol. 21, no. 1, pp. 83–143, 2015

    2015

  4. [12]

    Adaptive estimation for Hawkes processes; application to genome analysis,

    P. Reynaud-Bouret and S. Schbath, “Adaptive estimation for Hawkes processes; application to genome analysis,”Ann. Stat., vol. 38, no. 5, pp. 2781 – 2822, 2010

    2010

  5. [13]

    On the least-squares identification for Hawkes processes,

    X. Rong and G. N. Nair, “On the least-squares identification for Hawkes processes,” inProc. ACC, 2026, p. in press

    2026

  6. [14]

    System identification using Laguerre models,

    B. Wahlberg, “System identification using Laguerre models,”IEEE Trans. Autom. Control, vol. 36, no. 5, pp. 551–562, 1991

    1991

  7. [15]

    On linear intensity models for mixed doubly stochastic poisson and self-exciting point processes,

    Y . Ogata and H. Akaike, “On linear intensity models for mixed doubly stochastic poisson and self-exciting point processes,”J. R. Stat. Soc. B, vol. 44, no. 1, pp. 102–107, 1982

    1982

  8. [16]

    Szeg ¨o,Orthogonal Polynomials

    G. Szeg ¨o,Orthogonal Polynomials. Rhode Island: Am. Math. Soc., 1939

    1939

  9. [17]

    Hawkes identification with a pre- scribed causal basis: Closed-form estimators and asymptotics,

    X. Rong and G. N. Nair, “Hawkes identification with a pre- scribed causal basis: Closed-form estimators and asymptotics,” arXiv:2602.20795, 2026

    2026

  10. [18]

    Learning Granger causality for Hawkes processes,

    H. Xu, M. Farajtabar, and H. Zha, “Learning Granger causality for Hawkes processes,” inProc. ICML, 2016, pp. 1717–1726

    2016

  11. [19]

    Learning social infectivity in sparse low-rank networks using multi-dimensional Hawkes processes,

    K. Zhou, H. Zha, and L. Song, “Learning social infectivity in sparse low-rank networks using multi-dimensional Hawkes processes,” in Proc. AISTATS. PMLR, 2013, pp. 641–649

    2013

  12. [20]

    Spectra of some self-exciting and mutually exciting point processes,

    A. G. Hawkes, “Spectra of some self-exciting and mutually exciting point processes,”Biometrika, vol. 58, no. 1, pp. 83–90, 1971

    1971

  13. [21]

    D. J. Daley and D. Vere-Jones,An introduction to the Theory of Point Processes. New York: Springer-Verlag, 2003

    2003

  14. [22]

    Stability of nonlinear Hawkes pro- cesses,

    P. Br ´emaud and L. Massouli ´e, “Stability of nonlinear Hawkes pro- cesses,”Ann. Probab., vol. 24, no. 3, pp. 1563–1588, 1996

    1996

  15. [23]

    Discrete transforms, semidefinite pro- gramming, and sum-of-squares representations of nonnegative poly- nomials,

    T. Roh and L. Vandenberghe, “Discrete transforms, semidefinite pro- gramming, and sum-of-squares representations of nonnegative poly- nomials,”SIAM J. Optim., vol. 16, no. 4, pp. 939–964, 2006

    2006

  16. [24]

    Graphical modeling for multivariate Hawkes processes with nonparametric link functions,

    M. Eichler, R. Dahlhaus, and J. Dueck, “Graphical modeling for multivariate Hawkes processes with nonparametric link functions,”J. Time Ser . Anal., vol. 38, no. 2, pp. 225–242, 2017

    2017

  17. [25]

    D. J. Daley and D. Vere-Jones,An introduction to the theory of point processes: volume II: general theory and structure. Springer Science & Business Media, 2008

    2008

  18. [26]

    Kailath,Linear estimation

    T. Kailath,Linear estimation. Prentice Hall, 2000

    2000

  19. [27]

    Dumitrescu,Positive trigonometric polynomials and signal process- ing applications, 2nd ed

    B. Dumitrescu,Positive trigonometric polynomials and signal process- ing applications, 2nd ed. Springer, 2007

    2007

  20. [28]

    A system-theoretic approach to Hawkes process identification with guaranteed positivity and stability,

    X. Rong and G. N. Nair, “A system-theoretic approach to Hawkes process identification with guaranteed positivity and stability,” arXiv:2603.14942, 2026

    2026 arXiv

  21. [29]

    On Lewis’ simulation method for point processes,

    Y . Ogata, “On Lewis’ simulation method for point processes,”IEEE Trans. Inf. Theory, vol. 27, no. 1, pp. 23–31, 1981

    1981

  22. [30]

    CVX: Matlab software for disciplined convex programming, version 2.1,

    M. Grant and S. Boyd, “CVX: Matlab software for disciplined convex programming, version 2.1,” https://cvxr.com/cvx, Mar. 2014. **Supplementary Material:Recursive calculation forF m,k andG m,k that satisfy (4.1) and (4.2): um−1 0 (t)um−1 0 (t)⊤ =P2m−2 k=0 Fm,kuk(t), tum−1 0 (t)...

    2014

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