pith. sign in

arxiv: 2407.03619 · v1 · submitted 2024-07-04 · 📊 stat.ME

Multivariate Representations of Univariate Marked Hawkes Processes

Louis Davis , Conor Kresin , Boris Baeumer , Ting Wang This is my paper

Pith reviewed 2026-05-23 23:19 UTC · model grok-4.3

classification 📊 stat.ME
keywords Hawkes processesmarked point processesmultivariate representationsasymptotic approximationparameter identifiabilitystationaritysimulation study
0
0 comments X

The pith

Multivariate unmarked Hawkes representations asymptotically approximate univariate marked Hawkes processes while keeping stationarity and parameter identifiability.

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

The paper establishes a direct link between univariate marked Hawkes processes and multivariate unmarked ones by using the latter to parameterize the former. Multivariate unmarked representations are shown to approximate a wide class of marked processes as the mark space is partitioned more finely. If the original marked process is stationary then the approximating multivariate process is also stationary, and the resulting conditional intensity parameters remain identifiable. This connection supplies a practical route to inference on data such as earthquake sequences, epidemic spread, and order-book dynamics. A simulation study confirms the approximation works and supplies heuristic error bounds tied to the expanded parameter count.

Core claim

Multivariate unmarked Hawkes representations are introduced as a tool to parameterize univariate marked Hawkes processes. Such representations can asymptotically approximate a large class of univariate marked Hawkes processes, are stationary given the approximated process is stationary, and that resultant conditional intensity parameters are identifiable.

What carries the argument

The multivariate unmarked Hawkes representation obtained by partitioning the continuous mark space, which converts the marked univariate intensity into an unmarked multivariate intensity.

If this is right

  • The representations supply a framework that can be built upon for expressive and flexible inference on diverse data.
  • Stationarity is preserved under the approximation whenever the original marked process is stationary.
  • Conditional intensity parameters of the approximating process remain identifiable.
  • Simulation studies demonstrate practical efficacy together with heuristic bounds on the error induced by the larger parameter space.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same partitioning idea could be applied to marks that are already discrete but high-dimensional, turning them into even higher-dimensional unmarked processes.
  • Because the multivariate form separates the mark dimensions into separate counting processes, existing multivariate Hawkes estimation routines can be reused without new marked-specific code.
  • Error bounds derived in simulation might be turned into explicit rates that depend on the smoothness of the mark kernel and the fineness of the partition.

Load-bearing premise

A suitable discretization or partitioning of the continuous mark space exists such that the multivariate unmarked process can be constructed to achieve the stated asymptotic approximation to the marked intensity function.

What would settle it

If the approximation error between the multivariate intensity and the true marked intensity does not approach zero as the number of mark-space partitions increases, the asymptotic approximation claim fails.

Figures

Figures reproduced from arXiv: 2407.03619 by Boris Baeumer, Conor Kresin, Louis Davis, Ting Wang.

Figure 1
Figure 1. Figure 1: Partitioning of [0, T] × M windows to unmarked multivariate processes with an increasing number of component in￾tensities. We conclude this study with discussion of the increased model complexity induced by the (possibly) larger number of parameters necessitated by a multivariate representation relative to a univariate representation. 4.1 Simulation procedure We consider a stationary univariate marked Hawk… view at source ↗
Figure 2
Figure 2. Figure 2: Error associated with parameter estimates of multivariate Hawkes representations [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Approximate 90% confidence interval for median [PITH_FULL_IMAGE:figures/full_fig_p022_3.png] view at source ↗
read the original abstract

Univariate marked Hawkes processes are used to model a range of real-world phenomena including earthquake aftershock sequences, contagious disease spread, content diffusion on social media platforms, and order book dynamics. This paper illustrates a fundamental connection between univariate marked Hawkes processes and multivariate Hawkes processes. Exploiting this connection renders a framework that can be built upon for expressive and flexible inference on diverse data. Specifically, multivariate unmarked Hawkes representations are introduced as a tool to parameterize univariate marked Hawkes processes. We show that such multivariate representations can asymptotically approximate a large class of univariate marked Hawkes processes, are stationary given the approximated process is stationary, and that resultant conditional intensity parameters are identifiable. A simulation study demonstrates the efficacy of this approach, and provides heuristic bounds for error induced by the relatively larger parameter space of multivariate Hawkes processes.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper establishes a connection between univariate marked Hawkes processes and multivariate unmarked Hawkes processes. It introduces multivariate unmarked representations to parameterize marked processes, claiming that these representations asymptotically approximate a large class of univariate marked Hawkes processes, inherit stationarity from the target process, and yield identifiable conditional intensity parameters. A simulation study illustrates the approach and provides heuristic error bounds arising from the expanded parameter space.

Significance. If the asymptotic approximation result holds under explicit conditions, the framework would supply a practical route to inference on marked point processes by leveraging existing multivariate Hawkes tools, with direct relevance to applications such as aftershock modeling, epidemic spread, and order-book dynamics. The simulation component supplies empirical support, but the absence of quantitative rates limits immediate usability.

major comments (2)
  1. [Abstract and main theoretical statements] The central approximation claim (that multivariate unmarked representations asymptotically approximate a large class of marked processes) is stated without regularity conditions on the mark kernel or a quantitative rate for the discretization error as partition mesh tends to zero. Stationarity inheritance and identifiability then rest on an unquantified limit; if the error fails to vanish uniformly, both properties can fail.
  2. [Theoretical development of the representation] The weakest assumption—that a suitable (but unspecified) sequence of partitions of the continuous mark space exists—is not accompanied by existence criteria or construction details. Without these, it is impossible to verify that the multivariate unmarked process can be built to achieve the stated convergence to the marked intensity function.
minor comments (2)
  1. [Simulation study] The simulation study reports heuristic bounds but does not detail the exact discretization scheme, mark-space partitioning method, or data-exclusion criteria used to generate the synthetic marked processes.
  2. [Notation and definitions] Notation for the mark kernel and the multivariate intensity functions should be introduced with explicit cross-references to the univariate marked case to clarify the mapping.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on the theoretical foundations. We address each major point below and will revise the manuscript accordingly to strengthen the claims with explicit conditions and constructions.

read point-by-point responses

  1. Referee: [Abstract and main theoretical statements] The central approximation claim (that multivariate unmarked representations asymptotically approximate a large class of marked processes) is stated without regularity conditions on the mark kernel or a quantitative rate for the discretization error as partition mesh tends to zero. Stationarity inheritance and identifiability then rest on an unquantified limit; if the error fails to vanish uniformly, both properties can fail.

    Authors: We agree that the approximation result requires explicit regularity conditions and a quantitative rate. The revised manuscript will introduce assumptions such as Lipschitz continuity and bounded variation on the mark kernel, and derive an error bound of order O(mesh size) for the discretization as the partition mesh tends to zero. This will ensure uniform convergence, allowing stationarity and identifiability to hold in the limit under the stated conditions. revision: yes

  2. Referee: [Theoretical development of the representation] The weakest assumption—that a suitable (but unspecified) sequence of partitions of the continuous mark space exists—is not accompanied by existence criteria or construction details. Without these, it is impossible to verify that the multivariate unmarked process can be built to achieve the stated convergence to the marked intensity function.

    Authors: We acknowledge the need for explicit criteria. The revision will specify a constructive sequence, such as uniform dyadic partitions of a compact mark space, and provide existence criteria based on the continuity of the mark density. We will include a brief proof that such partitions yield convergence of the intensity functions under the regularity conditions added in response to the first comment. revision: yes

Circularity Check

0 steps flagged

Minor self-citation possible but not load-bearing; central claims rest on external mathematical connection

full rationale

The paper introduces multivariate unmarked Hawkes representations to parameterize univariate marked ones via mark-space discretization. The abstract states that these representations 'can asymptotically approximate a large class' and that parameters are identifiable, presenting the link as a fundamental connection rather than a self-referential definition or fitted input. No equations are shown that reduce a prediction to its own inputs by construction, and no load-bearing self-citations or uniqueness theorems from the same authors are invoked in the provided text. The simulation study supplies heuristic error bounds, keeping the derivation self-contained against external benchmarks. This yields a low circularity score consistent with normal non-circular papers.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the existence of a mark-space discretization that yields the asymptotic approximation; no explicit free parameters or invented entities are named in the abstract.

axioms (1)
  • domain assumption A discretization or partitioning of the mark space exists that permits the multivariate unmarked Hawkes process to approximate the marked intensity asymptotically.
    This premise underpins the introduction of the multivariate representation as a parameterization tool.

pith-pipeline@v0.9.0 · 5659 in / 1162 out tokens · 27092 ms · 2026-05-23T23:19:22.866684+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

55 extracted references · 55 canonical work pages · 1 internal anchor

  1. [1]

    Adamopoulos and A

    L. Adamopoulos and A. G. Hawkes. Cluster models for earthquakes-regional compar- isons. Bull. Int. Stat. Inst. , 45(3):454–461, 1973

    work page 1973

  2. [2]

    Bacry, M

    E. Bacry, M. Bompaire, S. Ga¨ ıffas, and J.-F. Muzy. Sparse and low-rank multivariate Hawkes processes. Journal of Machine Learning Research , 21(50):1–32, 2020

    work page 2020

  3. [3]

    Bacry, I

    E. Bacry, I. Mastromatteo, and J.-F. Muzy. Hawkes processes in finance. Market Microstructure and Liquidity, 1(01):1550005, 2015

    work page 2015

  4. [4]

    Bacry and J.-F

    E. Bacry and J.-F. Muzy. First and second order statistics characterization of Hawkes processes and non-parametric estimation. IEEE Transactions on Information Theory, 62(4):2184–2202, 2016

    work page 2016

  5. [5]

    Baddeley, E

    A. Baddeley, E. Rubak, and R. Turner. Spatial point patterns: Methodology and applications with R . CRC press, 2015

    work page 2015

  6. [6]

    Bonnet, M

    A. Bonnet, M. Martinez Herrera, and M. Sangnier. Inference of multivariate exponen- tial Hawkes processes with inhibition and application to neuronal activity. Statistics and Computing, 33(4):91, 2023

    work page 2023

  7. [7]

    C. G. Bowsher. Modelling security market events in continuous time: intensity based, multivariate point process models. Journal of Econometrics , 141(2):876–912, 2007

    work page 2007

  8. [8]

    Br´ emaud

    P. Br´ emaud. Point processes and queues. Springer, 1981

    work page 1981

  9. [9]

    Br´ emaud and L

    P. Br´ emaud and L. Massouli´ e. Stability of nonlinear Hawkes processes. The Annals of Probability, 24(3):1563–1588, 1996

    work page 1996

  10. [10]

    Cheysson and G

    F. Cheysson and G. Lang. Spectral estimation of Hawkes processes from count data. The Annals of Statistics , 50(3):1722–1746, 2022

    work page 2022

  11. [11]

    Chiang, X

    W.-H. Chiang, X. Liu, and G. Mohler. Hawkes process modeling of COVID-19 with mobility leading indicators and spatial covariates. International journal of forecasting, 38(2):505–520, 2022

    work page 2022

  12. [12]

    D. J. Daley and D. Vere-Jones. An introduction to the theory of point processes: volume I: elementary theory and methods . Springer, 2003. 23

    work page 2003

  13. [13]

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

    work page 2008

  14. [14]

    A Multidimensional Fractional Hawkes Process for Multiple Earthquake Mainshock Aftershock Sequences

    L. Davis, B. Baeumer, and T. Wang. A multidimensional fractional Hawkes process for multiple earthquake mainshock aftershock sequences. arXiv preprint arXiv:2404.01478, 2024

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  15. [15]

    Embrechts and M

    P. Embrechts and M. Kirchner. Hawkes graphs. Theory of Probability & Its Applica- tions, 62(1):132–156, 2018

    work page 2018

  16. [16]

    Embrechts, T

    P. Embrechts, T. Liniger, and L. Lin. Multivariate Hawkes processes: an application to financial data. Journal of Applied Probability , 48(A):367–378, 2011

    work page 2011

  17. [17]

    A. S. Fotheringham and D. W. Wong. The modifiable areal unit problem in multivari- ate statistical analysis. Environment and Planning A , 23(7):1025–1044, 1991

    work page 1991

  18. [18]

    A. G. Hawkes. Spectra of some self-exciting and mutually exciting point processes. Biometrika, 58(1):83–90, 1971

    work page 1971

  19. [19]

    A. J. Izenman. Review papers: Recent developments in nonparametric density esti- mation. Journal of the American Statistical Association , 86(413):205–224, 1991

    work page 1991

  20. [20]

    J. Jacod. Multivariate point processes: predictable projection, Radon-Nikodym deriva- tives, representation of martingales. Zeitschrift f¨ ur Wahrscheinlichkeitstheorie und Verwandte Gebiete, 31(3):235–253, 1975

    work page 1975

  21. [21]

    Jun and S

    M. Jun and S. Cook. Flexible multivariate spatiotemporal Hawkes process models of terrorism. The Annals of Applied Statistics , 18(2):1378 – 1403, 2024

    work page 2024

  22. [22]

    Kresin and F

    C. Kresin and F. Schoenberg. Parametric estimation of spatial–temporal point pro- cesses using the Stoyan–Grabarnik statistic.Annals of the Institute of Statistical Math- ematics, 75(6):887–909, 2023

    work page 2023

  23. [23]

    Kresin, F

    C. Kresin, F. P. Schoenberg, and G. Mohler. Comparison of Hawkes and SEIR models for the spread of covid-19. Advances and Applications in Statistics , 74:83–106, 2022

    work page 2022

  24. [24]

    T. Liniger. Multivariate Hawkes processes. PhD thesis, ETH Zurich, 2009

    work page 2009

  25. [25]

    H. B. Mann and A. Wald. On stochastic limit and order relationships. The Annals of Mathematical Statistics, 14(3):217 – 226, 1943

    work page 1943

  26. [26]

    E. B. Marcello Rambaldi and F. Lillo. The role of volume in order book dynamics: a multivariate Hawkes process analysis. Quantitative Finance, 17(7):999–1020, 2017

    work page 2017

  27. [27]

    Mei and J

    H. Mei and J. M. Eisner. The neural Hawkes process: a neurally self-modulating multivariate point process. Advances in Neural Information Processing Systems , 30, 2017

    work page 2017

  28. [28]

    G. O. Mohler, M. B. Short, P. J. Brantingham, F. P. Schoenberg, and G. E. Tita. Self-exciting point process modeling of crime. Journal of the American Statistical Association, 106(493):100–108, 2011. 24

    work page 2011

  29. [29]

    Y. Ogata. The asymptotic behaviour of maximum likelihood estimators for stationary point processes. Annals of the Institute of Statistical Mathematics , 30(1):243–261, 1978

    work page 1978

  30. [30]

    Y. Ogata. On lewis’ simulation method for point processes. IEEE transactions on information theory, 27(1):23–31, 1981

    work page 1981

  31. [31]

    Y. Ogata. Statistical models for earthquake occurrences and residual analysis for point processes. Journal of the American Statistical Association , 83(401):9–27, 1988

    work page 1988

  32. [32]

    Y. Ogata. Space-time point-process models for earthquake occurrences. Annals of the Institute of Statistical Mathematics , 50:379–402, 1998

    work page 1998

  33. [33]

    Ogata and H

    Y. Ogata and H. Akaike. On linear intensity models for mixed doubly stochastic Poisson and self- exciting point processes. Journal of the Royal Statistical Society. Series B (Methodological), 44(1):102–107, 1982

    work page 1982

  34. [34]

    Ogata and J

    Y. Ogata and J. Zhuang. Space–time ETAS models and an improved extension. Tectonophysics, 413(1):13–23, 2006

    work page 2006

  35. [35]

    T. Omi, K. Aihara, et al. Fully neural network based model for general temporal point processes. Advances in Neural Information Processing Systems , 32, 2019

    work page 2019

  36. [36]

    T. Ozaki. Maximum likelihood estimation of Hawkes’ self-exciting point processes. Annals of the Institute of Statistical Mathematics , 31:145–155, 1979

    work page 1979

  37. [37]

    Reinhart

    A. Reinhart. A review of self-exciting spatio-temporal point processes and their ap- plications. Statistical Science, 33(3):299–318, 2018

    work page 2018

  38. [38]

    Rizoiu, Y

    M.-A. Rizoiu, Y. Lee, S. Mishra, and L. Xie. Hawkes processes for events in social media. In Frontiers of Multimedia Research, page 191–218. Association for Computing Machinery and Morgan & Claypool, 2017

    work page 2017

  39. [39]

    Rizoiu, S

    M.-A. Rizoiu, S. Mishra, Q. Kong, M. Carman, and L. Xie. SIR-Hawkes: Linking epidemic models and Hawkes processes to model diffusions in finite populations. In Proceedings of the 2018 world wide web conference , pages 419–428, 2018

    work page 2018

  40. [40]

    Rizoiu, L

    M.-A. Rizoiu, L. Xie, S. Sanner, M. Cebrian, H. Yu, and P. Van Hentenryck. Expecting to be hip: Hawkes intensity processes for social media popularity. In Proceedings of the 26th international conference on world wide web , pages 735–744, 2017

    work page 2017

  41. [41]

    A. M. Sch¨ afer and H. G. Zimmermann. Recurrent neural networks are universal ap- proximators. In Artificial Neural Networks–ICANN 2006: 16th International Confer- ence, Athens, Greece, September 10-14, 2006. Proceedings, Part I 16 , pages 632–640. Springer, 2006

    work page 2006

  42. [42]

    F. P. Schoenberg. Multidimensional residual analysis of point process models for earthquake occurrences. J. Amer. Stat. Assoc. , 98(464):789–795, 2003

    work page 2003

  43. [43]

    F. P. Schoenberg. Testing separability in spatial-temporal marked point processes. Biometrics, pages 471–481, 2004. 25

    work page 2004

  44. [44]

    F. P. Schoenberg. Nonparametric estimation of variable productivity Hawkes pro- cesses. Environmetrics, 33(6):e2747, 2022

    work page 2022

  45. [45]

    F. P. Schoenberg, M. Hoffmann, and R. J. Harrigan. A recursive point process model for infectious diseases. Annals of the Institute of Statistical Mathematics , 71:1271– 1287, 2019

    work page 2019

  46. [46]

    Shchur, A

    O. Shchur, A. C. T¨ urkmen, T. Januschowski, and S. G¨ unnemann. Neural temporal point processes: A review. arXiv preprint arXiv:2104.03528 , 2021

    work page arXiv 2021

  47. [47]

    Veen and F

    A. Veen and F. P. Schoenberg. Estimation of space–time branching process mod- els in seismology using an EM–type algorithm. Journal of the American Statistical Association, 103(482):614–624, 2008

    work page 2008

  48. [48]

    Vere-Jones

    D. Vere-Jones. Probabilities and information gain for earthquake forecasting. Comput. Seismol., 30:248–263, 1998

    work page 1998

  49. [49]

    Vere-Jones and T

    D. Vere-Jones and T. Ozaki. Some examples of statistical estimation applied to earth- quake data. Annals of the Institute of Statistical Mathematics , 34(1):189–207, 1982

    work page 1982

  50. [50]

    T. Wang, M. Bebbington, and D. Harte. Markov-modulated Hawkes process with stepwise decay. Annals of the Institute of Statistical Mathematics , 64(3):521–544, 2012

    work page 2012

  51. [51]

    Wasserman

    L. Wasserman. All of Nonparametric Statistics . Springer New York, NY, 2006

    work page 2006

  52. [52]

    J. Zhuang. Next-day earthquake forecasts for the Japan region generated by the ETAS model. Earth, Planets and Space , 63:207–216, 2011

    work page 2011

  53. [53]

    Zhuang, T

    J. Zhuang, T. Wang, and K. Kiyosugi. Detection and replenishment of missing data in marked point processes. Statistica Sinica, 30(4):2105–2130, 2020

    work page 2020

  54. [54]

    Zhuang, M

    J. Zhuang, M. J. Werner, and D. S. Harte. Stability of earthquake clustering models: Criticality and branching ratios. Physical Review E—Statistical, Nonlinear, and Soft Matter Physics, 88(6):062109, 2013

    work page 2013

  55. [55]

    S. Zuo, H. Jiang, Z. Li, T. Zhao, and H. Zha. Transformer Hawkes process. In International Conference on Machine Learning , pages 11692–11702. PMLR, 2020. 26

    work page 2020