@article{HermkesKuehnRiggelsen2014,
  author    = {Hermkes, Marcel and K{\"u}hn, Nicolas M. and Riggelsen, Carsten},
  title     = {Simultaneous quantification of epistemic and aleatory uncertainty in GMPEs using Gaussian process regression},
  series = {Bulletin of earthquake engineering : official publication of the European Association for Earthquake Engineering},
  volume    = {12},
  journal   = {Bulletin of earthquake engineering : official publication of the European Association for Earthquake Engineering},
  number    = {1},
  publisher = {Springer},
  address   = {Dordrecht},
  issn      = {1570-761X},
  doi       = {10.1007/s10518-013-9507-7},
  pages     = {449 -- 466},
  year      = {2014},
  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.},
  language  = {en}
}
@phdthesis{Kuehn2010,
  author    = {K{\"u}hn, Nicolas M.},
  title     = {Empirical ground-motion models for probabilistic seismic hazard analysis : a graphical model perspective},
  address   = {Potsdam},
  pages     = {125 S.},
  year      = {2010},
  language  = {en}
}
@article{KuehnRiggelsenScherbaum2011,
  author    = {K{\"u}hn, Nicolas M. and Riggelsen, Carsten and Scherbaum, Frank},
  title     = {Modeling the joint probability of earthquake, site, and ground-motion parameters using bayesian networks},
  series = {Bulletin of the Seismological Society of America},
  volume    = {101},
  journal   = {Bulletin of the Seismological Society of America},
  number    = {1},
  publisher = {Seismological Society of America},
  address   = {El Cerrito},
  issn      = {0037-1106},
  doi       = {10.1785/0120100080},
  pages     = {235 -- 249},
  year      = {2011},
  abstract  = {Bayesian networks are a powerful and increasingly popular tool for reasoning under uncertainty, offering intuitive insight into (probabilistic) data-generating processes. They have been successfully applied to many different fields, including bioinformatics. In this paper, Bayesian networks are used to model the joint-probability distribution of selected earthquake, site, and ground-motion parameters. This provides a probabilistic representation of the independencies and dependencies between these variables. In particular, contrary to classical regression, Bayesian networks do not distinguish between target and predictors, treating each variable as random variable. The capability of Bayesian networks to model the ground-motion domain in probabilistic seismic hazard analysis is shown for a generic situation. A Bayesian network is learned based on a subset of the Next Generation Attenuation (NGA) dataset, using 3342 records from 154 earthquakes. Because no prior assumptions about dependencies between particular parameters are made, the learned network displays the most probable model given the data. The learned network shows that the ground-motion parameter (horizontal peak ground acceleration, PGA) is directly connected only to the moment magnitude, Joyner-Boore distance, fault mechanism, source-to-site azimuth, and depth to a shear-wave horizon of 2: 5 km/s (Z2.5). In particular, the effect of V-S30 is mediated by Z2.5. Comparisons of the PGA distributions based on the Bayesian networks with the NGA model of Boore and Atkinson (2008) show a reasonable agreement in ranges of good data coverage.},
  language  = {en}
}
@article{KuehnScherbaumRiggelsen2009,
  author    = {K{\"u}hn, Nicolas M. and Scherbaum, Frank and Riggelsen, Carsten},
  title     = {Deriving empirical ground-motion models : balancing data constraints and physical assumptions to optimize prediction capability},
  issn      = {0037-1106},
  doi       = {10.1785/0120080136},
  year      = {2009},
  abstract  = {Empirical ground-motion models used in seismic hazard analysis are commonly derived by regression of observed ground motions against a chosen set of predictor variables. Commonly, the model building process is based on residual analysis and/or expert knowledge and/or opinion, while the quality of the model is assessed by the goodness-of-fit to the data. Such an approach, however, bears no immediate relation to the predictive power of the model and with increasing complexity of the models is increasingly susceptible to the danger of overfitting. Here, a different, primarily data-driven method for the development of ground-motion models is proposed that makes use of the notion of generalization error to counteract the problem of overfitting. Generalization error directly estimates the average prediction error on data not used for the model generation and, thus, is a good criterion to assess the predictive capabilities of a model. The approach taken here makes only few a priori assumptions. At first, peak ground acceleration and response spectrum values are modeled by flexible, nonphysical functions (polynomials) of the predictor variables. The inclusion of a particular predictor and the order of the polynomials are based on minimizing generalization error. The approach is illustrated for the next generation of ground-motion attenuation dataset. The resulting model is rather complex, comprising 48 parameters, but has considerably lower generalization error than functional forms commonly used in ground-motion models. The model parameters have no physical meaning, but a visual interpretation is possible and can reveal relevant characteristics of the data, for example, the Moho bounce in the distance scaling. In a second step, the regression model is approximated by an equivalent stochastic model, making it physically interpretable. The resulting resolvable stochastic model parameters are comparable to published models for western North America. In general, for large datasets generalization error minimization provides a viable method for the development of empirical ground-motion models.},
  language  = {en}
}
@article{ScherbaumKuehn2011,
  author    = {Scherbaum, Frank and K{\"u}hn, Nicolas M.},
  title     = {Logic tree branch weights and probabilities summing up to one is not enough},
  series = {Earthquake spectra : the professional journal of the Earthquake Engineering Research Institute},
  volume    = {27},
  journal   = {Earthquake spectra : the professional journal of the Earthquake Engineering Research Institute},
  number    = {4},
  publisher = {Earthquake Engineering Research Institute},
  address   = {Oakland},
  issn      = {8755-2930},
  doi       = {10.1193/1.3652744},
  pages     = {1237 -- 1251},
  year      = {2011},
  abstract  = {Logic trees have become the most popular tool for the quantification of epistemic uncertainties in probabilistic seismic hazard assessment (PSHA). In a logic-tree framework, epistemic uncertainty is expressed in a set of branch weights, by which an expert or an expert group assigns degree-of-belief values to the applicability of the corresponding branch models. Despite the popularity of logic-trees, however, one finds surprisingly few clear commitments to what logic-tree branch weights are assumed to be (even by hazard analysts designing logic trees). In the present paper we argue that it is important for hazard analysts to accept the probabilistic framework from the beginning for assigning logic-tree branch weights. In other words, to accept that logic-tree branch weights are probabilities in the axiomatic sense, independent of one's preference for the philosophical interpretation of probabilities. We demonstrate that interpreting logic-tree branch weights merely as a numerical measure of "model quality," which are then subsequently normalized to sum up to unity, will with increasing number of models inevitably lead to an apparent insensitivity of hazard curves on the logic-tree branch weights, which may even be mistaken for robustness of the results. Finally, we argue that assigning logic-tree branch weights in a sequential fashion may improve their logical consistency.},
  language  = {en}
}