##
**Comment on “A review of self-exciting spatiotemporal point process and their applications” by Alex Reinhart.**
*(English)*
Zbl 1403.62119

Summary: In my discussion, I would like to comment on our early reactions to A. G. Hawkes’ enlightening paper [Biometrika 58, 83–90 (1971; Zbl 0219.60029)] on the self-exciting model; further, I would like to comment on [A. Reinhart, ibid. 33, No. 3, 299–318 (2018; Zbl 1403.62121)] developments of the extended models with some applications.

### MSC:

62H30 | Classification and discrimination; cluster analysis (statistical aspects) |

60G55 | Point processes (e.g., Poisson, Cox, Hawkes processes) |

62P12 | Applications of statistics to environmental and related topics |

### Keywords:

Akaike Bayesian information criterion (ABIC); causality analysis; conditional intensity function; empirical Bayesian method; epidemic-type aftershock sequence (ETAS) model; maximum-likelihood method; penalized log-likelihood; statistical seismology; study of earthquake predictability### References:

[1] | Adamopoulos, L. (1976). Cluster models for earthquakes: Regional comparisons. Math. Geol.8 463–475. |

[2] | Akaike, H. (1980). Likelihood and the Bayes procedure. In Bayesian Statistics (Valencia, 1979) 143–166. Univ. Press, Valencia. |

[3] | Cox, D. R. and Lewis, P. A. W. (1966). The Statistical Analysis of Series of Events. Methuen & Co., Ltd., London. · Zbl 0148.14005 |

[4] | Fletcher, R. and Powell, M. J. D. (1963/1964). A rapidly convergent descent method for minimization. Comput. J.6 163–168. · Zbl 0132.11603 |

[5] | Good, I. J. and Gaskins, R. A. (1971). Nonparametric roughness penalties for probability densities. Biometrika58 255–277. · Zbl 0221.62012 |

[6] | Hainzl, S. and Ogata, Y. (2005). Detecting fluid signals in seismicity data through statistical earthquake modeling. J. Geophys. Res.110 B05S07. DOI:10.1029/2004JB003247. |

[7] | Han, P., Zhuang, J., Hattori, K. and Ogata, Y. (2016). An interdisciplinary approach for earthquake modelling and forecasting. In 2016 Fall Meeting of the American Geophysical Union (AGU), Moscone Center, San Francisco, CA, 13 December 2016 (Oral). |

[8] | Hawkes, A. G. and Adamopoulos, L. (1973). Cluster models for earthquakes—Regional comparisons. Bull. Int. Stat. Inst.45 454–461. |

[9] | Ogata, Y. (1999). Seismicity analyses through point-process modelling: A review. In Seismicity Patterns, Their Statistical Significance and Physical Meaning (M. Wyss, K. Shimazaki and A. Ito, eds.). Pure and Applied Geophysics155 471–507. Birkhauser, Basel. |

[10] | Jordan, T. H., Chen, Y. T., Gasparini, P., Madariaga, R. and Marzocchi, W. (2011). Operational earthquake forecasting: State of knowledge and guidelines for utilization. Ann. Geophys.54 315–391. |

[11] | Kumazawa, T. and Ogata, Y. (2014). Nonstationary ETAS models for nonstandard earthquakes. Ann. Appl. Stat.8 1825–1852. · Zbl 1304.86011 |

[12] | Kumazawa, T., Ogata, Y., Kimura, K., Maeda, K. and Kobayashi, A. (2016). Background rates of swarm earthquakes that are synchronized with volumetric strain changes. Earth Planet. Sci. Lett.442 51–60. DOI:10.1016/j.epsl.2016.02.049. |

[13] | Lewis, P. A. W. and Shedler, G. S. (1979). Simulation of nonhomogeneous Poisson processes by thinning. Nav. Res. Logist. Q.26 403–413. · Zbl 0497.60003 |

[14] | Nishizawa, O., Lei, X. and Nagao, T. (1994). Hazard function analysis of seismo-electric signals in Greece. In Electromagnetic Phenomena Related to Earthquake Prediction (M. Hayakawa and Y. Fujinawa, eds.) 459–474. Terra Publishing Company, Tokyo. |

[15] | Ogata, Y. (1981). On Lewis’ simulation method for point processes. IEEE Trans. Inform. Theory27 23–31. · Zbl 0449.60037 |

[16] | Ogata, Y. (1988). Statistical models for earthquake occurrences and residual analysis for point processes. J. Amer. Statist. Assoc.83 9–27. |

[17] | Ogata, Y. (2004). Space-time model for regional seismicity and detection of crustal stress changes. J. Geophys. Res.109 B03308. |

[18] | Ogata, Y. (2011). Significant improvements of the space-time ETAS model for forecasting of accurate baseline seismicity. Earth Planets Space63 217–229. |

[19] | Ogata, Y. (2013). A prospect of earthquake prediction research. Statist. Sci.28 521–541. · Zbl 1331.86028 |

[20] | Ogata, Y. (2017). Statistics of earthquake activity: Models and methods for earthquake predictability studies. Annu. Rev. Earth Planet. Sci.45 497–527. DOI:10.1146/annurev-earth-063016-015918. |

[21] | Ogata, Y. and Akaike, H. (1982). On linear intensity models for mixed doubly stochastic Poisson and self-exciting point processes. J. Roy. Statist. Soc. Ser. B44 102–107. · Zbl 0496.62074 |

[22] | Ogata, Y., Akaike, H. and Katsura, K. (1982). The application of linear intensity models to the investigation of causal relations between a point process and another stochastic process. Ann. Inst. Statist. Math.34 373–387. · Zbl 0516.62085 |

[23] | Ogata, Y. and Katsura, K. (1986). Point-process models with linearly parametrized intensity for application to earthquake data. J. Appl. Probab.23A 291–310. · Zbl 0583.62100 |

[24] | Ogata, Y., Matsu’ura, R. S. and Katsura, K. (1993). Fast likelihood computation of epidemic type aftershock-sequence model. Geophys. Res. Lett.20 2143–2146. |

[25] | Ogata, Y., Katsura, K., Falcone, G., Nanjo, K. Z. and Zhuang, J. (2013). Comprehensive and topical evaluations of earthquake forecasts in terms of number, time, space, and magnitude. Bull. Seismol. Soc. Am.103 1692–1708. |

[26] | Ozaki, T. (1979). Maximum likelihood estimation of Hawkes’ self-exciting point processes. Ann. Inst. Statist. Math.31 145–155. · Zbl 0447.62081 |

[27] | Parzen, E., Tanabe, K. and Kitagawa, G. (eds.) (1998). Selected Papers of Hirotugu Akaike. Springer, New York. · Zbl 0902.62100 |

[28] | Rubin, I. (1972). Regular point processes and their detection. IEEE Trans. Inform. Theory18 547–557. · Zbl 0246.60042 |

[29] | Utsu, T., Ogata, Y. and Matsu’ura, R. S. (1995). The centenary of the Omori formula for a decay law of aftershock activity. J. Phys. Earth43 1–33. |

[30] | Vere-Jones, D. (1970). Stochastic models for earthquake occurrence. J. Roy. Statist. Soc. Ser. B32 1–62. · Zbl 0201.27702 |

[31] | Vere-Jones, D. and Ozaki, T. (1982). Some examples of statistical inference applied to earthquake data. Ann. Inst. Statist. Math.34 189–207. |

[32] | Wang, T., Zhuang, J., Kato, T. and Bebbington, M. (2013). Assessing the potential improvement in short-term earthquake forecasts from incorporation of GPS data. Geophys. Res. Lett.40 2631–2635. |

[33] | Whittle, P. (1951). Hypothesis Testing in Times Series Analysis. Almqvist & Wiksells Boktryckeri AB, Uppsala. · Zbl 0045.41301 |

[34] | Zhuang, J., Vere-Jones, D., Guan, H., Ogata, Y. and Ma, L. (2005). Preliminary analysis of observations on the ultra-low frequency electric field in a region around Beijing. Pure Appl. Geophys.162 1367–1396. DOI:10.1007/s00024-004-2674-3. |

[35] | Zhuang, J. |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.