## The Annals of Applied Statistics

- Ann. Appl. Stat.
- Volume 8, Number 3 (2014), 1825-1852.

### Nonstationary ETAS models for nonstandard earthquakes

Takao Kumazawa and Yosihiko Ogata

**Full-text: Open access**

#### Abstract

The conditional intensity function of a point process is a useful tool for generating probability forecasts of earthquakes. The epidemic-type aftershock sequence (ETAS) model is defined by a conditional intensity function, and the corresponding point process is equivalent to a branching process, assuming that an earthquake generates a cluster of offspring earthquakes (triggered earthquakes or so-called aftershocks). Further, the size of the first-generation cluster depends on the magnitude of the triggering (parent) earthquake. The ETAS model provides a good fit to standard earthquake occurrences. However, there are nonstandard earthquake series that appear under transient stress changes caused by aseismic forces such as volcanic magma or fluid intrusions. These events trigger transient nonstandard earthquake swarms, and they are poorly fitted by the stationary ETAS model. In this study, we examine nonstationary extensions of the ETAS model that cover nonstandard cases. These models allow the parameters to be time-dependent and can be estimated by the empirical Bayes method. The best model is selected among the competing models to provide the inversion solutions of nonstationary changes. To address issues of the uniqueness and robustness of the inversion procedure, this method is demonstrated on an inland swarm activity induced by the 2011 Tohoku-Oki, Japan earthquake of magnitude 9.0.

#### Article information

**Source**

Ann. Appl. Stat., Volume 8, Number 3 (2014), 1825-1852.

**Dates**

First available in Project Euclid: 23 October 2014

**Permanent link to this document**

https://projecteuclid.org/euclid.aoas/1414091236

**Digital Object Identifier**

doi:10.1214/14-AOAS759

**Mathematical Reviews number (MathSciNet)**

MR3271355

**Zentralblatt MATH identifier**

1304.86011

**Keywords**

Akaike Bayesian Information Criterion change point two-stage ETAS model time-dependent parameters induced seismic activity

#### Citation

Kumazawa, Takao; Ogata, Yosihiko. Nonstationary ETAS models for nonstandard earthquakes. Ann. Appl. Stat. 8 (2014), no. 3, 1825--1852. doi:10.1214/14-AOAS759. https://projecteuclid.org/euclid.aoas/1414091236

#### References

- Adelfio, G. and Ogata, Y. (2010). Hybrid kernel estimates of space–time earthquake occurrence rates using the epidemic-type aftershock sequence model.
*Ann. Inst. Statist. Math.***62**127–143. - Akaike, H. (1973). Information theory and an extension of the maximum likelihood principle. In
*Second International Symposium on Information Theory*(*Tsahkadsor*, 1971) 267–281. Akadémiai Kiadó, Budapest. - Akaike, H. (1974). A new look at the statistical model identification.
*IEEE Trans. Automat. Control***AC-19**716–723. System identification and time-series analysis. - Akaike, H. (1977). On entropy maximization principle. In
*Applications of Statistics*(P. R. Krishnaian, ed.) 27–41. North-Holland, Amsterdam.Zentralblatt MATH: 0435.62005 - Akaike, H. (1980). Likelihood and the Bayes procedure. In
*Bayesian Statistics*(*Valencia*, 1979) (J. M. Bernardo, M. H. De Groot, D. V. Lindley and A. F. M. Smith, eds.) 143–166. Univ. Press, Valencia, Spain.Zentralblatt MATH: 0495.62085 - Akaike, H. (1985). Prediction and entropy. In
*A Celebration of Statistics*(A. C. Atkinson and E. Fienberg, eds.) 1–24. Springer, New York. - Akaike, H. (1987). Factor analysis and AIC.
*Psychometrika***52**317–332. - Balderama, E., Paik Schoenberg, F., Murray, E. and Rundel, P. W. (2012). Application of branching models in the study of invasive species.
*J. Amer. Statist. Assoc.***107**467–476. - Bansal, A. R. and Ogata, Y. (2013). A non-stationary epidemic type aftershock sequence model for seismicity prior to the December 26, 2004 M9.1 Sumatra-Andaman Islands mega-earthquake.
*J. Geophys. Res.***118**616–629. - Chavez-Demoulina, V. and Mcgillb, J. A. (2012). High-frequency financial data modeling using Hawkes processes.
*J. Bank. Financ.***36**3415–3426. - Daley, D. and Vere-Jones, D. (2003).
*An Introduction to the Theory of Point Processes*, 2nd ed. Springer, New York.Zentralblatt MATH: 1026.60061 - Good, I. J. (1965).
*The Estimation of Probabilities. An Essay on Modern Bayesian Methods*. MIT Press, Cambridge, MA.Zentralblatt MATH: 0168.39603 - Good, I. J. and Gaskins, R. A. (1971). Nonparametric roughness penalties for probability densities.
*Biometrika***58**255–277. - Hainzl, S. and Ogata, Y. (2005). Detecting fluid signals in seismicity data through statistical earthquake modeling.
*J. Geophys. Res.***110**B5, B05S07. - Hassan Zadeh, A. and Sharda, R. (2012). Modeling brand post popularity in online social networks. Social Science Research Network. Available at SSRN 2182711.
- Hawkes, A. G. (1971). Spectra of some self-exciting and mutually exciting point processes.
*Biometrika***58**83–90. - Hawkes, A. G. and Adamopoulos, L. (1973). Cluster models for earthquakes—regional comparisons.
*Bull. Int. Stat. Inst.***45**454–461. - Hawkes, A. G. and Oakes, D. (1974). A cluster process representation of a self-exciting process.
*J. Appl. Probab.***11**493–503. - Herrera, R. and Schipp, B. (2009). Self-exciting extreme value models for stock market crashes. In
*Statistical Inference*,*Econometric Analysis and Matrix Algebra*209–231. Physica-Verlag HD, Heidelberg. - Japan Meteorological Agency (2009). The Iwate-Miyagi Nairiku earthquake in 2008.
*Rep. Coord. Comm. Earthq. Predict***81**101–131. Available at http://cais.gsi.go.jp/YOCHIREN/report/kaihou81/03_04.pdf. - Jordan, T. H., Chen, Y.-T. and Gasparini, P. (2012). Operational earthquake forecasting. State of knowledge and guidelines for utilization.
*Ann. Geophys.***54**315–391. - Kagan, Y. Y. and Knopoff, L. (1987). Statistical short-term earthquake prediction.
*Science***236**1563–1567. - Kendall, D. G. (1949). Stochastic processes and population growth.
*J. Roy. Statist. Soc. Ser. B.***11**230–264. - Kumazawa, T., Ogata, Y. and Toda, S. (2010). Precursory seismic anomalies and transient crustal deformation prior to the 2008 $M_{w}=6.9$ Iwate-Miyagi Nairiku, Japan, earthquake.
*J. Geophys. Res.***115**B10312. - Laplace, P. S. (1774). Memoir on the probability of causes of events. Mémoires de mathématique et de physique, tome sixième (English translation by S. M. Stigler, 1986).
*Statist. Sci.***1**364–378. - Llenos, A. L., Mcguire, J. J. and Ogata, Y. (2009). Modeling seismic swarms triggered by aseismic transients.
*Earth Planet. Sci. Lett.***281**59–69. - Lombardi, A. M., Cocco, M. and Marzocchi, W. (2010). On the increase of background seismicity rate during the 1997–1998 Umbria–Marche, central Italy, sequence: Apparent variation or fluid-driven triggering?
*Bull. Seismol. Soc. Amer.***100**1138–1152. - Lomnitz, C. (1974).
*Global Tectonic and Earthquake Risk*. Elsevier, Amsterdam. - Mohler, G. O., Short, M. B., Brantingham, P. J., Schoenberg, F. P. and Tita, G. E. (2011). Self-exciting point process modeling of crime.
*J. Amer. Statist. Assoc.***106**100–108. - Ogata, Y. (1978). The asymptotic behaviour of maximum likelihood estimators for stationary point processes.
*Ann. Inst. Statist. Math.***30**243–261. - Ogata, Y. (1981). On Lewis’ simulation method for point processes.
*IEEE Trans. Inform. Theory***27**23–31. - Ogata, Y. (1985). Statistical models for earthquake occurrences and residual analysis for point processes. Research Memorandum No. 388 (21 May), The Institute of Statistical Mathematics, Tokyo. Available at http://www.ism.ac.jp/editsec/resmemo-e.html.
- Ogata, Y. (1986). Statistical models for earthquake occurrences and residual analysis for point processes.
*Mathematical Seismology***1**228–281. - Ogata, Y. (1988). Statistical models for earthquake occurrences and residual analysis for point processes.
*J. Amer. Statist. Assoc.***83**9–27. - Ogata, Y. (1989). Statistical model for standard seismicity and detection of anomalies by residual analysis.
*Tectonophysics***169**159–174. - Ogata, Y. (1992). Detection of precursory relative quiescence before great earthquakes through a statistical model.
*J. Geophys. Res.***97**19845–19871. - Ogata, Y. (1998). Space–time point-process models for earthquake occurrences.
*Ann. Inst. Statist. Math.***50**379–402. - Ogata, Y. (1999). Seismicity analysis through point-process modeling: A review.
*Pure Appl. Geophys.***155**471–507. - Ogata, Y. (2001). Exploratory analysis of earthquake clusters by likelihood-based trigger models.
*J. Appl. Probab.***38A**202–212. - Ogata, Y. (2004). Space–time model for regional seismicity and detection of crustal stress changes.
*J. Geophys. Res.***109**B03308. - Ogata, Y. (2005). Detection of anomalous seismicity as a stress change sensor.
*J. Geophys. Res.***110**B05S06. - Ogata, Y. (2006a). Seismicity anomaly scenario prior to the major recurrent earthquakes off the East coast of Miyagi prefecture, northern Japan.
*Tectonophysics***424**291–306. - Ogata, Y. (2006b). Fortran programs statistical analysis of seismicity—Updated version, (SASeis2006).
*Computer Science Monograph*No. 33, The Institute of Statistical Mathematics, Tokyo, Japan. Available at http://www.ism.ac.jp/editsec/csm/index_j.html. - Ogata, Y. (2007). Seismicity and geodetic anomalies in a wide preceding the Niigata-Ken-Chuetsu earthquake of 23 October 2004, central Japan.
*J. Geophys. Res.***112**B10301. - Ogata, Y. (2010). Anomalies of seismic activity and transient crustal deformations preceding the 2005 M7.0 earthquake west of Fukuoka.
*Pure Appl. Geophys.***167**1115–1127. - Ogata, Y. (2011a). Long-term probability forecast of the regional seismicity that was induced by the M9 Tohoku-Oki earthquake.
*Report of the Coordinating Committee for Earthquake Prediction***88**92–99. - Ogata, Y. (2011b). Significant improvements of the space–time ETAS model for forecasting of accurate baseline seismicity.
*Earth Planets Space***63**217–229. - Ogata, Y. (2012). Tohoku earthquake aftershock activity (in Japanese).
*Report of the Coordinating Committee for Earthquake Prediction***88**100–103. - Ogata, Y., Jones, L. M. and Toda, S. (2003). When and where the aftershock activity was depressed: Contrasting decay patterns of the proximate large earthquakes in southern California.
*J. Geophys. Res.***108**B6, 2318. - Ogata, Y. and Katsura, K. (1993). Analysis of temporal and special heterogeneity of magnitude frequency distribution inferred from earthquake catalogues.
*Geophys. J. Int.***113**727–738. - Ogata, Y. and Katsura, K. (2006). Immediate and updated forecasting of aftershock hazard.
*Geophys. Res. Lett.***33**L10305. - Ogata, Y., Katsura, K. and Tanemura, M. (2003). Modelling heterogeneous space–time occurrences of earthquakes and its residual analysis.
*J. Roy. Statist. Soc. Ser. C***52**499–509. - Okutani, T. and Ide, S. (2011). Statistic analysis of swarm activities around the Boso Peninsula, Japan: Slow slip events beneath Tokyo Bay?
*Earth Planets Space***63**419–426. - Omi, T., Ogata, Y., Hirata, Y. and Aihara, K. (2013). Forecasting large aftershocks within one day after the main shock.
*Sci. Rep.***3**2218. - Peng, R. D., Schoenberg, F. P. and Woods, J. A. (2005). A space–time conditional intensity model for evaluating a wildfire hazard index.
*J. Amer. Statist. Assoc.***100**26–35. - Reasenberg, P. (1985). Second-order moment of central California seismicity, 1969–1982.
*J. Geophys. Res.***90**B7, 5479–5495. - Schoenberg, F. P., Peng, R. and Woods, J. (2003). On the distribution of wild fire sizes.
*Environmetrics***14**583–592. - Terakawa, T., Hashimoto, C. and Matsu’ura, M. (2013). Changes in seismic activity following the 2011 Tohoku-Oki earthquake: Effects of pore fluid pressure.
*Earth Planet. Sci. Lett.***365**17–24. - Terakawa, T. and Matsu’uara, M. (2010). The 3-d tectonic stress fields in and around Japan inverted from centroid moment tensor data of seismic events.
*Tectonics***29**(TC6008). - Terakawa, T., Miller, S. and Deichmann, N. (2012). High fluid pressure and triggered earthquakes in the enhanced geothermal system in Basel, Switzerland.
*J. Geophys. Res.***117**B07305, 15 pp. - Toda, S., Lian, L. and Ross, S. (2011). Using the 2011 M${}={}$9.0 Tohoku earthquake to test the Coulomb stress triggering hypothesis and to calculate faults brought closer to failure.
*Earth Planets Space***63**725–730. - Toda, S., Stein, R. S. and Jian, L. (2011). Widespread seismicity excitation throughout central Japan following the 2011 M${}={}$9.0 Tohoku earthquake, and its interpretation in terms of Coulomb stress transfer.
*Geophys. Res. Lett.***38**L00G03. - Tsuruoka, H. (1996). Development of seismicity analysis software on workstation (in Japanese).
*Tech. Res. Rep.***2**34–42. Earthq. Res. Inst., Univ. of Tokyo, Tokyo. - Utsu, T. (1961). Statistical study on the occurrence of aftershocks.
*Geophys. Mag.***30**521–605. - Utsu, T. (1962). On the nature of three Alaskan aftershock sequences of 1957 and 1958.
*Bull. Seismol. Soc. Amer.***52**279–297. - Utsu, T. (1969). Aftershocks and earthquake statistics (I)—Some parameters which characterize an aftershock sequence and their interrelations.
*J. Fac. Sci. Hokkaido Univ.*,*Ser. VII***3**129–195. - Utsu, T. (1970). Aftershocks and earthquake statistics (II)—Further investigation of aftershocks and other earthquake sequences based on a new classification of earthquake sequences.
*J. Fac. Sci. Hokkaido Univ.*,*Ser. VII***3**197–266. - Utsu, T. (1971). Aftershocks and earthquake statistics (III)—Analyses of the distribution of earthquakes in magnitude, time, and space with special consideration to clustering characteristics of earthquake occurrence (1).
*J. Fac. Sci. Hokkaido Univ.*,*Ser. VII***3**379–441. - Utsu, T. (1972). Aftershocks and earthquake statistics (IV)—Analyses of the distribution of earthquakes in magnitude, time, and space with special consideration to clustering characteristics of earthquake occurrence (2).
*J. Fac. Sci. Hokkaido Univ.*,*Ser. VII***4**1–42. - Utsu, T., Ogata, Y. and Matsu’ura, R. S. (1995). The centenary of the Omori formula for a decay law of aftershock activity.
*J. Seismol. Soc. Japan***7**233–240. - Utsu, T. and Seki, A. (1955). A relation between the area of after-shock region and the energy of main shock (in Japanese).
*Zisin*(2)**7**233–240. - Vere-Jones, D. (1970). Stochastic models for earthquake occurrence.
*J. Roy. Statist. Soc. Ser. B***32**1–62. - Vere-Jones, D. and Davies, R. B. (1966). A statistical study of earthquakes in the main seismic area of New Zealand. Part II: Time series analyses.
*N. Z. J. Geol. Geophys.***9**251–284. - Zhuang, J. and Ogata, Y. (2006). Properties of the probability distribution associated with the largest earthquake in a cluster and their implications to foreshocks.
*Phys. Rev. E***73**046134. - Zhuang, J., Ogata, Y. and Vere-Jones, D. (2002). Stochastic declustering of space–time earthquake occurrences.
*J. Amer. Statist. Assoc.***97**369–380. - Zhuang, J., Ogata, Y. and Vere-Jones, D. (2004). Analyzing earthquake clustering features by using stochastic reconstruction.
*J. Geophys. Res.***109**B5, B05301.

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