Abstract
In this paper, the Bayesian hypothesis testing basis is proposed for selecting, ranking, and assigning weights to ground motion prediction equations that fits perfectly on the classical definition of a logic tree. The posterior probability of a model being the best model describing the data is calculated, and the definition of Bayes factors is used for selecting and weighting prediction models. Accounting for data correlation is important in model ranking and combination which is missing from the commonly used scoring procedures such as the median likelihood, average log-likelihood, Euclidean distance ranking, and the Bayesian information criterion methods. The proposed method considers data correlation (i.e., within event and between event correlation and correlation between ordinates) by utilizing a multivariate likelihood function. While the proposed procedure is mostly objective and data-driven, the Bayesian updating rule allows for consideration of expert’s judgment by using prior probabilities. The proposed method is applied to subsets of the NGA-West2 dataset, and five selected NGA-West2 models are ranked and weighted in different magnitude and period ranges according to available data.
Similar content being viewed by others
References
Abrahamson N, Wooddell K (2010) Evaluation of evidence for inhibition of very strong ground motions in the Abrahamson and Silva next generation attenuation ground-motion model. Bull Seismol Soc Am 100:2174–2184. https://doi.org/10.1785/0120080278
Abrahamson NA, Silva WJ, Kamai R (2014) Summary of the ASK14 ground motion relation for active crustal regions. Earthquake Spectra 30:1025–1055. https://doi.org/10.1193/070913EQS198M
Al Atik L, Youngs RR (2013) PEER 2013/11 - epistemic uncertainty for NGA-West2 models
Ancheta TD, Darragh RB, Stewart JP, Seyhan E, Silva WJ, Chiou BSJ, Wooddell KE, Graves RW, Kottke AR, Boore DM, Kishida T, Donahue JL (2014) NGA-West2 database. Earthquake Spectra 30:989–1005. https://doi.org/10.1193/070913EQS197M
Arroyo D, Ordaz M, Rueda R (2014) On the selection of ground-motion prediction equations for probabilistic seismic-Hazard analysis. Bull Seismol Soc Am 104:1860–1875. https://doi.org/10.1785/0120130264
Baker JW, Bradley BA (2017) Intensity measure correlations observed in the NGA-West2 database, and dependence of correlations on rupture and site parameters. Earthquake Spectra 33:145–156. https://doi.org/10.1193/060716EQS095M
Berger JO (1990) Robust Bayesian analysis: sensitivity to the prior. J Stat Plan Inference 25:303–328. https://doi.org/10.1016/0378-3758(90)90079-A
Berger JO, Delampady M (1987) Testing precise hypotheses. Stat Sci 2:317–335. https://doi.org/10.1214/ss/1177013238
Bommer JJ, Scherbaum F, Bungum H et al (2005) On the use of logic trees for ground-motion prediction equations in seismic-hazard analysis. Bull Seismol Soc Am 95:377–389. https://doi.org/10.1785/0120040073
Boore DM, Stewart JP, Seyhan E, Atkinson GM (2014) NGA-West 2 equations for predicting PGA , PGV , and 5% -damped PSA for shallow crustal earthquakes. Earthquake Spectra 30:1057–1085. https://doi.org/10.1193/070113EQS184M
Campbell KW, Bozorgnia Y (2014) NGA-West2 ground motion model for the average horizontal components of PGA, PGV, and 5% damped linear acceleration response spectra. Earthquake Spectra 30:1087–1115. https://doi.org/10.1193/062913EQS175M
Castaños H, Lomnitz C (2002) PSHA: is it science? Eng Geol 66:315–317. https://doi.org/10.1016/S0013-7952(02)00039-X
Champion C, Liel A (2012) The effect of near-fault directivity on building seismic collapse risk. Earthq Eng Struct Dyn 41:1391–1409. https://doi.org/10.1002/eqe.1188
Chiou BS-J, Youngs RR (2014) Update of the Chiou and Youngs NGA model for the average horizontal component of peak ground motion and response spectra. Earthquake Spectra 30:1117–1153. https://doi.org/10.1193/072813EQS219M
Chiou B, Darragh R, Gregor N, Silva W (2008) NGA project strong-motion database. Earthquake Spectra 24:23–44. https://doi.org/10.1193/1.2894831
Chousianitis K, Del Gaudio V, Pierri P, Tselentis G-A (2018) Regional ground-motion prediction equations for amplitude-, frequency response-, and duration-based parameters for Greece. Earthq Eng Struct Dyn 47:2252–2274. https://doi.org/10.1002/eqe.3067
Cotton F, Scherbaum F, Bommer JJ, Bungum H (2006) Criteria for selecting and adjusting ground-motion models for specific target regions: application to Central Europe and rock sites. J Seismol 10:137–156. https://doi.org/10.1007/s10950-005-9006-7
de Almeida AAD, Assumpção M, Bommer JJ, Drouet S, Riccomini C, Prates CLM (2019) Probabilistic seismic hazard analysis for a nuclear power plant site in Southeast Brazil. J Seismol 23:1–23. https://doi.org/10.1007/s10950-018-9755-8
Delavaud E, Cotton F, Akkar S, Scherbaum F, Danciu L, Beauval C, Drouet S, Douglas J, Basili R, Sandikkaya MA, Segou M, Faccioli E, Theodoulidis N (2012a) Toward a ground-motion logic tree for probabilistic seismic hazard assessment in Europe. J Seismol 16:451–473. https://doi.org/10.1007/s10950-012-9281-z
Delavaud E, Scherbaum F, Kuehn N, Allen T (2012b) Testing the global applicability of ground-motion prediction equations for active shallow crustal regions. Bull Seismol Soc Am 102:707–721. https://doi.org/10.1785/0120110113
Draper D (1995) Assessment and propagation of model uncertainty. J R Stat Soc Ser B 57:45–97
Drouet S, Scherbaum F, Cotton F, Souriau A (2007) Selection and ranking of ground motion models for seismic hazard analysis in the Pyrenees. J Seismol 11:87–100. https://doi.org/10.1007/s10950-006-9039-6
Du W, Wang G (2013) A simple ground-motion prediction model for cumulative absolute velocity and model validation. Earthq Eng Struct Dyn 42:1189–1202. https://doi.org/10.1002/eqe.2266
Eskandarinejad A, Zafarani H, Jahanandish M (2018) Comparison of conventional and Monte Carlo simulation-based probabilistic seismic hazard analyses for shiraz city, southern Iran. J Seismol 22:1629–1643. https://doi.org/10.1007/s10950-018-9790-5
Field CA, Welsh AH (2007) Bootstrapping clustered data. J R Stat Soc Ser B Stat Methodol 69:369–390. https://doi.org/10.1111/j.1467-9868.2007.00593.x
FitzGerald THB, Dolan RJ, Friston KJ (2014) Model averaging, optimal inference, and habit formation. Front Hum Neurosci 8:457. https://doi.org/10.3389/fnhum.2014.00457
Gamse S, Zhou W-H, Tan F, Yuen KV, Oberguggenberger M (2018) Hydrostatic-season-time model updating using Bayesian model class selection. Reliab Eng Syst Saf 169:40–50. https://doi.org/10.1016/j.ress.2017.07.018
Idriss IM (2014) An NGA-West2 empirical model for estimating the horizontal spectral values generated by shallow crustal earthquakes. Earthquake Spectra 30:1155–1177. https://doi.org/10.1193/070613EQS195M
Jeffreys H (1961) The theory of probability, 3rd edn. Oxford University Press, New York
Joyner WB, Boore DM (1993) Methods for regression analysis of strong-motion data. Bull Seismol Soc Am 83:469–487
Kabir G, Tesfamariam S, Sadiq R (2015) Predicting water main failures using Bayesian model averaging and survival modelling approach. Reliab Eng Syst Saf 142:498–514. https://doi.org/10.1016/j.ress.2015.06.011
Kale O, Akkar S (2013) A new procedure for selecting and ranking ground-motion prediction equations (GMPEs): the Euclidean distance-based ranking (EDR) method. Bull Seismol Soc Am 103:1069–1084. https://doi.org/10.1785/0120120134
Kale Ö, Akkar S (2017) A ground-motion logic-tree scheme for regional seismic Hazard studies. Earthquake Spectra 33:837–856. https://doi.org/10.1193/051316EQS080M
Kass RE, Wasserman L (1996) The selection of prior distributions by formal rules. J Am Stat Assoc 91:1343. https://doi.org/10.2307/2291752
Kohrangi M, Vamvatsikos D, Bazzurro P (2018) Pulse-like versus non-pulse-like ground motion records: spectral shape comparisons and record selection strategies. Earthq Eng Struct Dyn 48:46–64. https://doi.org/10.1002/eqe.3122
Krinitzsky EL (1995) Problems with logic trees in earthquake hazard evaluation. Eng Geol 39:1–3. https://doi.org/10.1016/0013-7952(94)00060-F
Kulkarni RB, Youngs RR, Coppersmith K (1984) Assessment of confidence intervals for results of seismic hazard analysis. In: 8th world conference on earthquake engineering, San Francisco, pp 263–270
Ling Y, Mahadevan S (2013) Quantitative model validation techniques: new insights. Reliab Eng Syst Saf 111:217–231. https://doi.org/10.1016/j.ress.2012.11.011
Liu D, Wang S, Zhang C, Tomovic M (2018) Bayesian model averaging based reliability analysis method for monotonic degradation dataset based on inverse Gaussian process and gamma process. Reliab Eng Syst Saf 180:25–38. https://doi.org/10.1016/j.ress.2018.06.019
Mak S, Clements RA, Schorlemmer D (2017a) Empirical evaluation of hierarchical ground-motion models: score uncertainty and model weighting. Bull Seismol Soc Am 107:949–965. https://doi.org/10.1785/0120160232
Mak S, Cotton F, Schorlemmer D (2017b) Measuring the performance of ground-motion models: the importance of being independent. Seismol Res Lett 88:1212–1217
Mak S, Cotton F, Gerstenberger M, Schorlemmer D (2018) An evaluation of the applicability of NGA-West2 ground-motion models for Japan and New Zealand. Bull Seismol Soc Am 108:836–856. https://doi.org/10.1785/0120170146
Marzocchi W, Jordan TH (2014) Testing for ontological errors in probabilistic forecasting models of natural systems. Proc Natl Acad Sci 111:11973–11978. https://doi.org/10.1073/pnas.1410183111
Marzocchi W, Taroni M, Selva J (2015) Accounting for epistemic uncertainty in PSHA: logic tree and ensemble modeling. Bull Seismol Soc Am 105:2151–2159. https://doi.org/10.1785/0120140131
McGuire RK (2004) Seismic hazard and risk analysis. Earthquake Engineering Research Institute, Oakland
Mullins J, Mahadevan S (2014) Variable-fidelity model selection for stochastic simulation. Reliab Eng Syst Saf 131:40–52. https://doi.org/10.1016/j.ress.2014.06.011
Nannapaneni S, Mahadevan S (2016) Reliability analysis under epistemic uncertainty. Reliab Eng Syst Saf 155:9–20. https://doi.org/10.1016/j.ress.2016.06.005
Nguyen KTP, Fouladirad M, Grall A (2018) Model selection for degradation modeling and prognosis with health monitoring data. Reliab Eng Syst Saf 169:105–116. https://doi.org/10.1016/j.ress.2017.08.004
Park I, Grandhi RV (2014) A Bayesian statistical method for quantifying model form uncertainty and two model combination methods. Reliab Eng Syst Saf 129:46–56. https://doi.org/10.1016/j.ress.2014.04.023
Park I, Amarchinta HK, Grandhi RV (2010) A Bayesian approach for quantification of model uncertainty. Reliab Eng Syst Saf 95:777–785. https://doi.org/10.1016/j.ress.2010.02.015
Passarelli L, Maccaferri F, Rivalta E, Dahm T, Abebe Boku E (2013) A probabilistic approach for the classification of earthquakes as ‘triggered’ or ‘not triggered’. J Seismol 17:165–187. https://doi.org/10.1007/s10950-012-9289-4
Petersen MD, Moschetti MP, Powers PM, Mueller CS, Haller KM, Frankel AD, Zeng Y, Rezaeian S, Harmsen SC, Boyd OS, Field N, Chen R, Rukstales KS, Luco N, Wheeler RL, Williams RA, Olsen AH (2015) The 2014 United States National Seismic Hazard Model. Earthquake Spectra 31:S1–S30. https://doi.org/10.1193/120814EQS210M
Raftery AE, Madigan D, Hoeting JA (1997) Bayesian model averaging for linear regression models. J Am Stat Assoc 92:179–191. https://doi.org/10.1080/01621459.1997.10473615
Raftery AE, Gneiting T, Balabdaoui F, Polakowski M (2005) Using Bayesian model averaging to calibrate forecast ensembles. Mon Weather Rev 133:1155–1174. https://doi.org/10.1175/MWR2906.1
Rebba R, Mahadevan S (2006) Validation of models with multivariate output. Reliab Eng Syst Saf 91:861–871. https://doi.org/10.1016/j.ress.2005.09.004
Rebba R, Mahadevan S, Huang S (2006) Validation and error estimation of computational models. Reliab Eng Syst Saf 91:1390–1397. https://doi.org/10.1016/j.ress.2005.11.035
Roselli P, Marzocchi W, Faenza L (2016) Toward a new probabilistic framework to score and merge ground-motion prediction equations: the case of the Italian region. Bull Seismol Soc Am 106:720–733. https://doi.org/10.1785/0120150057
Saito T, Beck JL (2010) Bayesian model selection for ARX models and its application to structural health monitoring. Earthq Eng Struct Dyn 39:1737–1759. https://doi.org/10.1002/eqe.1006
Salahshoor H, Lyubushin A, Shabani E, Kazemian J (2018) Comparison of Bayesian estimates of peak ground acceleration (Amax) with PSHA in Iran. J Seismol 22:1515–1527. https://doi.org/10.1007/s10950-018-9782-5
Scherbaum F, Kuehn NM (2011) Logic tree branch weights and probabilities: summing up to one is not enough. Earthquake Spectra 27:1237–1251. https://doi.org/10.1193/1.3652744
Scherbaum F, Cotton F, Smit P (2004) On the use of response spectral-reference data for the selection and ranking of ground-motion models for seismic-Hazard analysis in regions of moderate seismicity: the case of rock motion. Bull Seismol Soc Am 94:2164–2185. https://doi.org/10.1785/0120030147
Scherbaum F, Bommer JJ, Bungum H et al (2005) Composite ground-motion models and logic trees: methodology, sensitivities, and uncertainties. Bull Seismol Soc Am 95:1575–1593. https://doi.org/10.1785/0120040229
Scherbaum F, Delavaud E, Riggelsen C (2009) Model selection in seismic Hazard analysis: an information-theoretic perspective. Bull Seismol Soc Am 99:3234–3247
Secanell R, Martin C, Viallet E, Senfaute G (2018) A Bayesian methodology to update the probabilistic seismic hazard assessment. Bull Earthq Eng 16:2513–2527. https://doi.org/10.1007/s10518-017-0137-3
Shahi SK, Baker JW (2014) An efficient algorithm to identify strong-velocity pulses in multicomponent ground motions. Bull Seismol Soc Am 104:2456–2466. https://doi.org/10.1785/0120130191
Shahidzadeh MS, Yazdani A (2017) A Bayesian updating applied to earthquake ground-motion prediction equations for Iran. J Earthq Eng 21:290–324. https://doi.org/10.1080/13632469.2016.1158754
Stafford PJ (2019) Continuous integration of data into ground-motion models using Bayesian updating. J Seismol 23:39–57. https://doi.org/10.1007/s10950-018-9792-3
Wang M, Takada T (2009) A Bayesian framework for prediction of seismic ground motion. Bull Seismol Soc Am 99:2348–2364. https://doi.org/10.1785/0120080017
Waseem M, Lateef A, Ahmad I, Khan S, Ahmed W (2019) Seismic hazard assessment of Afghanistan. J Seismol 23:217–242. https://doi.org/10.1007/s10950-018-9802-5
Wasserman L (2000) Bayesian model selection and model averaging. J Math Psychol 44:92–107. https://doi.org/10.1006/jmps.1999.1278
Yang D, Zhou J (2015) A stochastic model and synthesis for near-fault impulsive ground motions. Earthq Eng Struct Dyn 44:243–264. https://doi.org/10.1002/eqe.2468
Yazdani A, Kowsari M (2013) Earthquake ground-motion prediction equations for northern Iran. Nat Hazards 69:1877–1894. https://doi.org/10.1007/s11069-013-0778-8
Yazdani A, Nicknam A, Eftekhari SN, Yousefi Dadras E, (2016) Sensitivity of Near‐Fault PSHA results to input variables based on Information Theory. Bull Seismol Soc Am 106(4):1858–1866. https://doi.org/10.1785/0120160006
Zhang R, Mahadevan S (2003) Bayesian methodology for reliability model acceptance. Reliab Eng Syst Saf 80:95–103. https://doi.org/10.1016/S0951-8320(02)00269-7
Zio E, Apostolakis GE (1996) Two methods for the structured assessment of model uncertainty by experts in performance assessments of radioactive waste repositories. Reliab Eng Syst Saf 54:225–241. https://doi.org/10.1016/S0951-8320(96)00078-6
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Highlights
• A full Bayesian approach is presented for selecting, ranking, and weighting.
• Bayesian model averaging is used to combine multiple GMPEs.
• Bayes factors are utilized for ranking and weighting ground motion models.
• Multivariate likelihood function is used for consideration of correlation.
• The expert’s judgment and data-driven procedures can be merged by the Bayesian rule.
Rights and permissions
About this article
Cite this article
Shahidzadeh, M., Yazdani, A. & Eftekhari, S.N. Multivariate Bayesian hypothesis testing for ground motion model selection. J Seismol 24, 511–529 (2020). https://doi.org/10.1007/s10950-020-09924-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10950-020-09924-5