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Nonparametric estimation of variable productivity Hawkes processes

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  • Frederic Paik Schoenberg

Abstract

Hawkes models are frequently used to describe point processes that are clustered spatial‐temporally, and have been used in numerous applications including the study of earthquakes, invasive species, and contagious diseases. An extension of the Hawkes model is considered where the productivity is variable. In particular, the case is explored where each point may have its own productivity and a simple analytic formula is derived for the maximum likelihood estimators of these productivities. This estimator is compared with an empirical estimator and ways are explored of stabilizing both estimators by lower truncating, smoothing, and rescaling the estimates. Properties of the estimators are explored in simulations, and the methods are applied to seismological and epidemic datasets to show and quantify substantial variation in productivity.

Suggested Citation

  • Frederic Paik Schoenberg, 2022. "Nonparametric estimation of variable productivity Hawkes processes," Environmetrics, John Wiley & Sons, Ltd., vol. 33(6), September.
    • Handle: RePEc:wly:envmet:v:33:y:2022:i:6:n:e2747
    DOI: 10.1002/env.2747
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    References listed on IDEAS

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    1. Emmanuel Bacry & Iacopo Mastromatteo & Jean-Franc{c}ois Muzy, 2015. "Hawkes processes in finance," Papers 1502.04592, arXiv.org, revised May 2015.
    2. Frederic Paik Schoenberg & Joshua Seth Gordon & Ryan J. Harrigan, 2018. "Analytic computation of nonparametric Marsan–Lengliné estimates for Hawkes point processes," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 30(3), pages 742-757, July.
    3. Kevin Nichols & Frederic Paik Schoenberg, 2014. "Assessing the dependency between the magnitudes of earthquakes and the magnitudes of their aftershocks," Environmetrics, John Wiley & Sons, Ltd., vol. 25(3), pages 143-151, May.
    4. J. Molyneux & J. S. Gordon & F. P. Schoenberg, 2018. "Assessing the predictive accuracy of earthquake strike angle estimates using nonparametric Hawkes processes," Environmetrics, John Wiley & Sons, Ltd., vol. 29(2), March.
    5. Earvin Balderama & Frederic Paik Schoenberg & Erin Murray & Philip W. Rundel, 2012. "Application of Branching Models in the Study of Invasive Species," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 107(498), pages 467-476, June.
    6. Veen, Alejandro & Schoenberg, Frederic P., 2008. "Estimation of SpaceTime Branching Process Models in Seismology Using an EMType Algorithm," Journal of the American Statistical Association, American Statistical Association, vol. 103, pages 614-624, June.
    7. Frederic Paik Schoenberg & Marc Hoffmann & Ryan J. Harrigan, 2019. "A recursive point process model for infectious diseases," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 71(5), pages 1271-1287, October.
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    Cited by:

    1. Francesco Serafini & Finn Lindgren & Mark Naylor, 2023. "Approximation of Bayesian Hawkes process with inlabru," Environmetrics, John Wiley & Sons, Ltd., vol. 34(5), August.

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