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arxiv: 2404.01478 · v1 · submitted 2024-04-01 · 📊 stat.AP

A Multidimensional Fractional Hawkes Process for Multiple Earthquake Mainshock Aftershock Sequences

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

Pith reviewed 2026-05-24 02:24 UTC · model grok-4.3

classification 📊 stat.AP
keywords fractional Hawkes processearthquake point processmagnitude dependent triggeringETAS modelmainshock aftershock sequencesMittag-Leffler kernelmultidimensional Hawkes process
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The pith

A multidimensional fractional Hawkes process that makes magnitude triggering depend on history outperforms the ETAS model on two earthquake catalogs with multiple mainshock-aftershock sequences.

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

The paper challenges the common assumption that earthquake magnitudes are independent of past events and instead builds a model where the size of each quake influences the distribution of future quakes through mutual excitation between magnitude classes. It splits the magnitude range into fixed intervals, treats each interval as a separate point process, and links them with a Mittag-Leffler kernel inside a multidimensional Hawkes framework. When fitted to catalogs from Japan and the Middle America Trench, the new model records better information criteria, cleaner residuals, and stronger retrospective forecasts than the standard ETAS model. A sympathetic reader would care because the approach lets the model reflect real patterns in which large events alter the likelihood of large aftershocks in ways that independent-magnitude models cannot capture.

Core claim

By discretising magnitudes into disjoint intervals and modelling the intervals as mutually exciting subprocesses driven by Mittag-Leffler kernels, the multidimensional fractional Hawkes process incorporates history dependence into the magnitude distribution and produces superior fits, diagnostics, and predictions relative to the ETAS model on data sets that contain successive mainshock-aftershock sequences.

What carries the argument

multidimensional fractional Hawkes process whose subprocesses correspond to magnitude intervals and whose excitation kernel is the Mittag-Leffler density

If this is right

  • The model returns lower information criteria values than ETAS for both the Japan and Middle America Trench catalogs.
  • Residual analysis shows improved agreement with the observed point patterns.
  • Retrospective prediction scores are higher than those obtained with ETAS.
  • The fitted parameters recover magnitude-triggering characteristics already reported in the seismological literature that cannot be recovered from an ETAS fit.

Where Pith is reading between the lines

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

  • The same discretised-mark construction could be tried on other marked point processes where the mark distribution is suspected to depend on history.
  • If the inferred cross-magnitude excitation rates differ systematically between tectonic regions, the model could serve as a diagnostic for regional differences in aftershock productivity.
  • Replacing the fixed magnitude bins with a continuous mark-dependent kernel would test whether the performance gain survives removal of the discretisation step.

Load-bearing premise

Discretising the magnitude range into a fixed set of disjoint intervals and adopting the Mittag-Leffler density as the kernel function is sufficient to capture history-dependent magnitude triggering without material bias.

What would settle it

On a new catalog containing multiple mainshock-aftershock sequences, the multidimensional fractional Hawkes process would need to fail to improve on the ETAS model across information criteria, residual plots, and prediction scores.

Figures

Figures reproduced from arXiv: 2404.01478 by Boris Baeumer, Louis Davis, Ting Wang.

Figure 1
Figure 1. Figure 1: Occurrence maps and magnitude versus time plots for the data sets (A) & (C) [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Mean removed transformed time residuals (MRTT) against the cumulative num [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Mean removed transformed time (MRTT) are the black curves against cumulative number of events (CNE). The solid grey line is the theoretical mean, the dashed and dotted grey lines are the 95% and 99% confidence intervals respectively. (A,C,E,G) correspond to the Middle America Trench and (B,D,F,H) correspond to the Japan data set. For the following SP1 and SP2 stand for subprocess 1 and 2 respectively. (A) … view at source ↗
read the original abstract

Most point process models for earthquakes currently in the literature assume the magnitude distribution is i.i.d. potentially hindering the ability of the model to describe the main features of data sets containing multiple earthquake mainshock aftershock sequences in succession. This study presents a novel multidimensional fractional Hawkes process model designed to capture magnitude dependent triggering behaviour by incorporating history dependence into the magnitude distribution. This is done by discretising the magnitude range into disjoint intervals and modelling events with magnitude in these ranges as the subprocesses of a mutually exciting Hawkes process using the Mittag-Leffler density as the kernel function. We demonstrate this model's use by applying it to two data sets, Japan and the Middle America Trench, both containing multiple mainshock aftershock sequences and compare it to the existing ETAS model by using information criteria, residual diagnostics and retrospective prediction performance. We find that for both data sets all metrics indicate that the multidimensional fractional Hawkes process performs favourably against the ETAS model. Furthermore, using the multidimensional fractional Hawkes process we are able to infer characteristics of the data sets that are consistent with results currently in the literature and that cannot be found by using the ETAS model.

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 / 1 minor

Summary. The manuscript proposes a multidimensional fractional Hawkes process for modeling sequences containing multiple mainshock-aftershock earthquake clusters. Magnitudes are discretized into fixed disjoint intervals treated as subprocesses of a mutually exciting Hawkes process; each cross-excitation term uses a Mittag-Leffler density as the triggering kernel to introduce history dependence into the magnitude distribution. The model is fitted to the Japan and Middle America Trench catalogs and is reported to outperform the ETAS model on information criteria, residual diagnostics, and retrospective prediction metrics, while also permitting inference of magnitude-dependent features absent from ETAS.

Significance. If the reported gains survive scrutiny of the discretization step and the fitting procedure is fully reproducible, the construction would supply a concrete route to magnitude-dependent triggering that standard ETAS lacks, potentially improving description of complex seismic catalogs.

major comments (2)
  1. [Abstract (model construction paragraph)] Abstract (paragraph describing model construction): the central claim that the model captures history-dependent magnitude triggering rests on partitioning the continuous magnitude range into a small number of fixed disjoint intervals whose boundaries are chosen once and never varied; nothing demonstrates that the resulting discrete subprocesses recover the same continuous-magnitude dynamics that would appear in an undiscretized formulation, so any performance advantage over ETAS could be an artifact of that arbitrary partition.
  2. [Abstract] Abstract: superior performance is asserted on information criteria, residual diagnostics, and retrospective prediction tasks, yet no derivation of the likelihood, fitting algorithm, optimization details, or uncertainty quantification is supplied, preventing independent verification of the metrics that support the main conclusion.
minor comments (1)
  1. The phrase 'fractional Hawkes process' is used without an explicit definition or reference to the fractional calculus literature; a short clarifying sentence would aid readers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their insightful comments. We address each major comment below and indicate planned revisions.

read point-by-point responses

  1. Referee: [Abstract (model construction paragraph)] Abstract (paragraph describing model construction): the central claim that the model captures history-dependent magnitude triggering rests on partitioning the continuous magnitude range into a small number of fixed disjoint intervals whose boundaries are chosen once and never varied; nothing demonstrates that the resulting discrete subprocesses recover the same continuous-magnitude dynamics that would appear in an undiscretized formulation, so any performance advantage over ETAS could be an artifact of that arbitrary partition.

    Authors: The discretization into fixed intervals is a deliberate modeling choice that enables the multidimensional mutually exciting structure and cross-excitation terms needed to introduce history dependence into the magnitude distribution. Boundaries are chosen using standard seismic magnitude classes for interpretability. We agree that this is an approximation and does not prove equivalence to a hypothetical continuous-magnitude formulation. In revision we will add a sensitivity study across alternative partitions (varying both number and cut-points) and report whether the reported gains over ETAS remain stable. revision: partial

  2. Referee: [Abstract] Abstract: superior performance is asserted on information criteria, residual diagnostics, and retrospective prediction tasks, yet no derivation of the likelihood, fitting algorithm, optimization details, or uncertainty quantification is supplied, preventing independent verification of the metrics that support the main conclusion.

    Authors: Section 3 derives the likelihood for the multidimensional fractional Hawkes process; Section 4 describes the maximum-likelihood fitting procedure and numerical optimization; uncertainty is obtained from the observed information matrix. To improve verifiability we will expand these sections with explicit likelihood expressions, pseudocode for the optimization routine, and additional implementation details. We will also release the fitting code upon publication. revision: yes

Circularity Check

0 steps flagged

No circularity: model is an explicit extension validated on external data

full rationale

The derivation defines a new process via magnitude discretization into fixed intervals and Mittag-Leffler kernels for cross-excitation, then evaluates it against ETAS using standard IC, residual, and retrospective prediction metrics on two independent catalogs. No equation reduces a claimed result to a fitted input by construction, no uniqueness theorem is imported from self-citation, and no prediction is statistically forced by the fitting procedure itself. The central performance claim therefore rests on observable data comparisons rather than tautology.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

Abstract implies modeling choices but does not enumerate fitted values or background axioms; free parameters and domain assumptions are inferred from the described construction.

free parameters (2)
  • magnitude interval boundaries
    Discretization into disjoint intervals defines the subprocesses; boundaries are a modeling choice that affects all subsequent inference.
  • Mittag-Leffler kernel parameters
    Parameters of the Mittag-Leffler density are required for the excitation kernel and are expected to be estimated from data.
axioms (1)
  • domain assumption Earthquake events partitioned by magnitude interval form a collection of mutually exciting point processes
    Central modeling premise stated when the multidimensional Hawkes structure is introduced.

pith-pipeline@v0.9.0 · 5728 in / 1209 out tokens · 25546 ms · 2026-05-24T02:24:26.632076+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Multivariate Representations of Univariate Marked Hawkes Processes

    stat.ME 2024-07 unverdicted novelty 6.0

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

Reference graph

Works this paper leans on

48 extracted references · 48 canonical work pages · cited by 1 Pith paper · 1 internal anchor

  1. [1]

    Akaike, H. (1974). A new look at the statistical model identification. IEEE transactions on automatic control , 19(6):716--723

    work page 1974

  2. [2]

    Bacry, E., Delattre, S., Hoffmann, M., and Muzy, J.-F. (2013). Some limit theorems for H awkes processes and application to financial statistics. Stochastic Processes and their Applications , 123(7):2475--2499

    work page 2013

  3. [3]

    S., Harte, D

    Bebbington, M. S., Harte, D. S., and Jaum \'e , S. C. (2010). Repeated intermittent earthquake cycles in the S an F rancisco B ay region. Pure and applied geophysics , 167:801--818

    work page 2010

  4. [4]

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

    work page 2007

  5. [5]

    and Stindl, T

    Chen, F. and Stindl, T. (2018). Direct likelihood evaluation for the renewal H awkes process. Journal of Computational and Graphical Statistics , 27(1):119--131

    work page 2018

  6. [6]

    Chen, J., Hawkes, A., and Scalas, E. (2021). A fractional H awkes process. In Nonlocal and Fractional Operators , pages 121--131. Springer

    work page 2021

  7. [7]

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

    work page 2003

  8. [8]

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

    work page 2008

  9. [9]

    Davis, L., Baeumer, B., and Wang, T. (2024). A F ractional M odel for E arthquakes. arXiv preprint arXiv:2403.00142

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  10. [10]

    and Zhu, L

    Gao, X. and Zhu, L. (2018). Functional central limit theorems for stationary H awkes processes and application to infinite-server queues. Queueing Systems , 90:161--206

    work page 2018

  11. [11]

    Garrappa, R. (2015). Numerical evaluation of two and three parameter M ittag- L effler functions. SIAM Journal on Numerical Analysis , 53(3):1350--1369

    work page 2015

  12. [12]

    Garrappa, R. (2022). The M ittag- L effler function. https://au.mathworks.com/matlabcentral/fileexchange/48154-the-mittag-leffler-function

    work page 2022

  13. [13]

    and Richter, C

    Gutenberg, B. and Richter, C. F. (1944). Frequency of earthquakes in California *. Bulletin of the Seismological Society of America , 34(4):185--188

    work page 1944

  14. [14]

    A., Scalas, E., Chen, J., and Hawkes, A

    Habyarimana, C., Aduda, J. A., Scalas, E., Chen, J., and Hawkes, A. G. (2022). A fractional H awkes process II : F urther characterization of the process

    work page 2022

  15. [15]

    Harte, D. (2010). Pt P rocess: An R P ackage for M odelling M arked P oint P rocess I ndexed by T ime. Journal of Statistical Software , 35

    work page 2010

  16. [16]

    Harte, D. (2013). Bias in fitting the ETAS model: a case study based on N ew Z ealand seismicity. Geophysical Journal International , 192(1):390--412

    work page 2013

  17. [17]

    Harte, D. (2019). Evaluation of earthquake stochastic models based on their real-time forecasts: a case study of K aikoura 2016. Geophysical Journal International , 217(3):1894--1914

    work page 2019

  18. [18]

    J., Mathai, A

    Haubold, H. J., Mathai, A. M., and Saxena, R. K. (2009). Mittag-leffler functions and their applications

    work page 2009

  19. [19]

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

    work page 1971

  20. [20]

    Hawkes, A. G. (1973). Cluster models for earthquakes-regional comparisons. Bull. Int. Stat. Inst. , 45(3):454--461

    work page 1973

  21. [21]

    and Sornette, D

    Helmstetter, A. and Sornette, D. (2003). Foreshocks explained by cascades of triggered seismicity. Journal of Geophysical Research: Solid Earth , 108(B10)

    work page 2003

  22. [22]

    Kolev, A. A. and Ross, G. J. (2019). Inference for ETAS models with non- P oissonian mainshock arrival times. Statistics and Computing , 29(5):915--931

    work page 2019

  23. [23]

    and Ogata, Y

    Kumazawa, T. and Ogata, Y. (2014). Nonstationary ETAS models for nonstandard earthquakes. The Annals of Applied Statistics , 8(3):1825--1852

    work page 2014

  24. [24]

    Lewis, P. W. and Shedler, G. S. (1979). Simulation of nonhomogeneous P oisson processes by thinning. Naval research logistics quarterly , 26(3):403--413

    work page 1979

  25. [25]

    Liu, J., Vere-Jones, D., Ma, L., Shi, Y.-L., and Zhuang, J.-C. (1998). The principle of coupled stress release model and its application. Acta Seismologica Sinica , 11:273--281

    work page 1998

  26. [26]

    and Nonnenmacher, T

    Metzler, R. and Nonnenmacher, T. F. (2003). Fractional relaxation processes and fractional rheological models for the description of a class of viscoelastic materials. International Journal of Plasticity , 19(7):941--959

    work page 2003

  27. [27]

    and Schoenberg, F

    Nichols, K. and Schoenberg, F. P. (2014). Assessing the dependency between the magnitudes of earthquakes and the magnitudes of their aftershocks: MAGNITUDE DEPENDENCE . Environmetrics , 25(3):143--151

    work page 2014

  28. [28]

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

    work page 1978

  29. [29]

    Ogata, Y. (1981). On L ewis' simulation method for point processes. IEEE transactions on information theory , 27(1):23--31

    work page 1981

  30. [30]

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

    work page 1988

  31. [31]

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

    work page 1998

  32. [32]

    and Zhuang, J

    Ogata, Y. and Zhuang, J. (2006). Space–time ETAS models and an improved extension. Tectonophysics , 413(1):13--23. Critical Point Theory and Space-Time Pattern Formation in Precursory Seismicity

    work page 2006

  33. [33]

    and Karlis, D

    Orfanogiannaki, K. and Karlis, D. (2018). Multivariate P oisson hidden M arkov models with a case study of modelling seismicity. Australian & New Zealand Journal of Statistics , 60(3):301--322

    work page 2018

  34. [34]

    L., Vazquez-Caamal, M

    Ram \' rez-Herrera, M.-T., Corona, N., Cerny, J., Castillo-Aja, R., Melgar, D., Lagos, M., Goguitchaichvili, A., Machain, M. L., Vazquez-Caamal, M. L., Ortu \ n o, M., et al. (2020). Sand deposits reveal great earthquakes and tsunamis at M exican P acific C oast. Scientific Reports , 10(1):11452

    work page 2020

  35. [35]

    Schwarz, G. (1978). Estimating the dimension of a model. The annals of statistics , pages 461--464

    work page 1978

  36. [36]

    Shi, Y.-L., Liu, J., Vere-Jones, D., Zhuang, J.-C., and Ma, L. (1998). Application of mechanical and statistical models to the study of seismicity of synthetic earthquakes and the prediction of natural ones. Acta Seismologica Sinica , 11:421--430

    work page 1998

  37. [37]

    and Chen, F

    Stindl, T. and Chen, F. (2018). Likelihood based inference for the multivariate renewal H awkes process. Computational Statistics & Data Analysis , 123:131--145

    work page 2018

  38. [38]

    and Chen, F

    Stindl, T. and Chen, F. (2023). EM algorithm for the estimation of the RETAS model. Journal of Computational and Graphical Statistics , pages 1--11

    work page 2023

  39. [39]

    Can `` M ega Q uakes" really happen? L ike a magnitude 10 or larger? https://www.usgs.gov/faqs/can-megaquakes-really-happen-magnitude-10-or-larger

    USGS (2023). Can `` M ega Q uakes" really happen? L ike a magnitude 10 or larger? https://www.usgs.gov/faqs/can-megaquakes-really-happen-magnitude-10-or-larger

    work page 2023

  40. [40]

    Utsu, T. (1961). A statistical study on the occurrence of aftershocks. Geophys. Mag. , 30:521--605

    work page 1961

  41. [41]

    Utsu, T. (1971). Aftershocks and earthquake statistics (2): further investigation of aftershocks and other earthquake sequences based on a new classification of earthquake sequences. Journal of the Faculty of Science, Hokkaido University. Series 7, Geophysics , 3(4):197--266

    work page 1971

  42. [42]

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

    work page 1998

  43. [43]

    Wakita, K. (2013). Geology and tectonics of J apanese islands: a review--the key to understanding the geology of A sia. Journal of Asian Earth Sciences , 72:75--87

    work page 2013

  44. [44]

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

    work page 2012

  45. [45]

    Wang, T., Zhuang, J., Obara, K., and Tsuruoka, H. (2017). Hidden M arkov modelling of sparse time series from non-volcanic tremor observations. Journal of the Royal Statistical Society Series C: Applied Statistics , 66(4):691--715

    work page 2017

  46. [46]

    Wheatley, S., Filimonov, V., and Sornette, D. (2016). The H awkes process with renewal immigration & its estimation with an EM algorithm. Computational Statistics & Data Analysis , 94:120--135

    work page 2016

  47. [47]

    Y., Liu, A., Chen, J., and Hawkes, A

    Yang, S. Y., Liu, A., Chen, J., and Hawkes, A. (2018). Applications of a multivariate hawkes process to joint modeling of sentiment and market return events. Quantitative finance , 18(2):295--310

    work page 2018

  48. [48]

    Zhang, Y., Fan, J., Marzocchi, W., Shapira, A., Hofstetter, R., Havlin, S., and Ashkenazy, Y. (2020). Scaling laws in earthquake memory for interevent times and distances. Physical Review Research , 2(1):013264

    work page 2020