Abstract
This paper presents a Bayesian non-parametric method based on Gaussian Process (GP) regression to derive ground-motion models for peak-ground parameters and response spectral ordinates. Due to its non-parametric nature there is no need to specify any fixed functional form as in parametric regression models. A GP defines a distribution over functions, which implicitly expresses the uncertainty over the underlying data generating process. An advantage of GP regression is that it is possible to capture the whole uncertainty involved in ground-motion modeling, both in terms of aleatory variability as well as epistemic uncertainty associated with the underlying functional form and data coverage. The distribution over functions is updated in a Bayesian way by computing the posterior distribution of the GP after observing ground-motion data, which in turn can be used to make predictions. The proposed GP regression models is evaluated on a subset of the RESORCE data base for the SIGMA project. The experiments show that GP models have a better generalization error than a simple parametric regression model. A visual assessment of different scenarios demonstrates that the inferred GP models are physically plausible.
Similar content being viewed by others
Notes
For application of the GP models these constituents are necessary. The specific values can be found in the electronic supplementary.
References
Abrahamson NA, Youngs RR (1992) A stable algorithm for regression analysis using the random effects model. Bull Seismol Soc Am 82(1):505–510
Abrahamson N, Atkinson G, Boore D, Bozorgnia Y, Campbell K, Chiou B, Silva W, Idriss IM, Youngs R (2008) Comparisons of the NGA ground-motion relations. Earthq Spectra 24(1):45–66
Akkar S, Bommer JJ (2010) Empirical equations for the prediction of PGA, PGV, and spectral accelerations in Europe, the Mediterranean Region, and the Middle East. Seismol Res Lett 81(2):195–206
Akkar S, Sandıkkaya MA, Şenyurt M, Azari AS, Ay BÖ, Traversa P, Douglas J, Cotton F, Luzi L, Hernandez B, Godey S (2013) Reference database for seismic ground-motion in Europe (RESORCE). Bull Earthq Eng. doi:10.1007/s10518-013-9506-8
Al-Atik L, Abrahamson N, Bommer JJ, Scherbaum F, Cotton F, Kuehn N (2010) The variability of ground-motion prediction models and its components. Seismol Res Lett 81(5):794–801
Anderson JG, Lei Y (1994) Nonparametric description of peak acceleration as a function of magnitude, distance, and site in Guerrero, Mexico. Bull Seismol Soc Am 84(4):1003–1017
Arroyo D, Ordaz M (2010a) Multivariate Bayesian regression analysis applied to ground-motion prediction equations, part 1: theory and synthetic example. Bull Seismol Soc Am 100(4):1551–1567
Arroyo D, Ordaz M (2010b) Multivariate Bayesian regression analysis applied to ground-motion prediction equations, part 2: numerical example with actual data. Bull Seismol Soc Am 100(4):1568–1577
Arroyo D, Ordaz M (2011) On the forecasting of ground-motion parameters for probabilistic seismic hazard analysis. Earthq Spectra 27(1):1
Bommer JJ (2012) Challenges of building logic trees for probabilistic seismic hazard analysis. Earthq Spectra 28(4):1723–1735
Bommer JJ, Abrahamson NA (2006) Why do modern probabilistic seismic-hazard analyses often lead to increased hazard estimates? Bull Seismol Soc Am 96(6):1967–1977
Bommer JJ, Scherbaum F (2008) The use and misuse of logic trees in probabilistic seismic hazard analysis. Earthq Spectra 24(4):997
Bommer JJ, Scherbaum F, Bungum H, Cotton F, Sabetta F, Abrahamson NA (2005) On the use of logic trees for ground-motion prediction equations in seismic-hazard analysis. Bull Seismol Soc Am 95(2):377–389
Bommer JJ, Stafford PJ, Alarcon JE, Akkar S (2007) The influence of magnitude range on empirical ground-motion prediction. Bull Seismol Soc Am 97(6):2152–2170
Bora SS, Scherbaum F, Stafford PJ, Kuehn N (2013) Fourier spectral- and duration models for the generation of response spectra adjustable to different source-, propagation-, and site conditions. Bull Earthq Eng. doi:10.1007/s10518-013-9482-z
Derras B, Bard P-Y, Cotton F, Bekkouche A (2012) Adapting the neural network approach to PGA prediction: an example based on the KiK-net data. Bull Seismol Soc Am 102(4):1446–1461
Derras B, Cotton F, Bard P-Y (2013) Towards fully data driven ground-motion prediction models for Europe. Bull Earthq Eng. doi: 10.1007/s10518-013-9481-0
Douglas J, Jousset P (2011) Modeling the difference in ground-motion magnitude-scaling in small and large earthquakes. Seismol Res Lett 82(4):504–508
Douglas J, Akkar S, Ameri G, Bard P-Y, Bindi D, Bommer JJ, Bora SS, Cotton F, Derras B, Hermkes M, Kuehn NM, Luzi L, Massa M, Pacor F, Riggelsen C, Sandikkaya MA, Scherbaum F, Stafford PJ, Traversa P (2013) Comparisons among the five ground-motion models developed using RESORCE for the prediction of response spectral accelerations due to earthquakes in Europe and the Middle East. Bull Earthq Eng. doi:10.1007/s10518-013-9522-8
Goovaerts P (1997) Geostatistics for natural resources evaluation. Oxford University Press, USA
Hermkes M, Kuehn NM, Riggelsen C (2012) Learning task relatedness via Dirichlet process priors for linear regression models. In: 20th European symposium on artificial neural networks, computational intelligence and machine learning, pp 293–298
Joyner WB, Boore DM (1993) Methods for regression analysis of strong-motion data. Bull Seismol Soc Am 83(2):469–487
Joyner WB, Boore DM (1994) Erratum to ’Methods for regression analysis of strong-motion data’. Bull Seismol Soc Am 84(3):955–956
Kuehn NM, Scherbaum F, Riggelsen C (2009) Deriving empirical ground-motion models: balancing data constraints and physical assumptions to optimize prediction capability. Bull Seismol Soc Am 99(4):2335–2347
Kulkarni RB, Youngs RR, Coppersmith KJ (1984) Assessment of confidence intervals for results of seismic hazard analysis. In: Proceedings of 8th world conference on earthquake engineering, pp 263–270
Musson RMW (2012) PSHA validated by quasi observational means. Seismol Res Lett 83(1):130–134
Neal RM (1996) Bayesian learning for neural networks. Springer, Secaucus, NJ
Power M, Chiou B, Abrahamson N, Bozorgnia Y, Shantz T, Roblee C (2008) An overview of the NGA project. Earthq Spectra 24:3
Quiñonero-Candela J, Rasmussen CE (2005) A unifying view of sparse approximate gaussian process regression. J Mach Learn Res 6: 1939–1960
Rasmussen CE, Williams CKI (2006) Gaussian processes for machine learning. MIT Press, Cambridge, MA
Scherbaum F, Kuehn NM (2011) Logic tree branch weights and probabilities: summing up to one is not enough. Earthq Spectra 27(4):1237–1251
Scherbaum F, Kuehn NM, Ohrnberger M, Koehler A (2010) Exploring the proximity of ground-motion models using high-dimensional visualization techniques. Earthqu Spectra 26(4):1117–1138
Snelson E, Ghahramani Z (2007) Local and global sparse Gaussian process approximations. InL Proceedings of artificial intelligence and statistics (AISTATS)
Stein ML (1999) Interpolation of spatial data: some theory for kriging (Springer Series in Statistics), 1st edn. Springer, Berlin
Tezcan J, Cheng Q (2012) Support vector regression for estimating earthquake response spectra. Bull Earthq Eng 10:1205–1219
Tezcan J, Piolatto A (2012) A probabilistic nonparametric model for the vertical-to-horizontal spectral ratio. J Earthq Eng 10:142–157
Toro GR (2006) The effects of ground-motion uncertainty on seismic hazard results: examples and approximate results. In: Annual meeting of the Seismological Society of America, San Francisco, CA, USA
Williams CKI, Rasmussen CE (1996) Gaussian processes for regression. In: Advances in neural information processing systems, vol 8, pp 514–520. MIT press
Acknowledgments
We thank Frank Scherbaum for comments on an early draft of the paper. We thank Adrian Rodriguez-Marek and an anonymous reviewer for comments which helped to clarify the manuscript. This work was (partly) funded by the German Research Foundation, grant DFG RI 2037/2-1.
Author information
Authors and Affiliations
Corresponding author
Electronic supplementary material
Below is the link to the electronic supplementary material.
Rights and permissions
About this article
Cite this article
Hermkes, M., Kuehn, N.M. & Riggelsen, C. Simultaneous quantification of epistemic and aleatory uncertainty in GMPEs using Gaussian process regression. Bull Earthquake Eng 12, 449–466 (2014). https://doi.org/10.1007/s10518-013-9507-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10518-013-9507-7