A Bayesian model for truncated regression for the estimation of empirical ground-motion models
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A Bayesian model for truncated regression for the estimation of empirical ground‑motion models Nicolas Martin Kuehn1 · Tadahiro Kishida2 · Mohammad AlHamaydeh3 · Grigorios Lavrentiadis4 · Yousef Bozorgnia1 Received: 8 April 2020 / Accepted: 25 August 2020 © Springer Nature B.V. 2020
Abstract We present a Bayesian model for the estimation of ground-motion models that allows one to account for truncated data. Truncated data occurs in ground-motion model development because instruments do not record continuously, but only when triggered. The model is formulated as a multi-level model and incorporates event and station terms. The model considers truncation on one variable [e.g., peak ground acceleration (PGA)], and models the joint occurrence of PGA and other ground-motion intensity measures, while conditioning on the truncation for PGA. Initially, we perform numerical experiments on simulated data sets and show that not taking data truncation into account leads to biased models. Regressions using the proposed truncated model can recapture the functions used in the simulation well, and perform comparable to alternative approaches used in the past. Subsequently, we show the impact of the truncated model on observed ground-motion data representing moderate and high trigger levels, 2–4 gal and 10 gal, respectively. Differences to a model that does not take truncation into account occur at larger distances, and are more severe for the high trigger level data. For untruncated regression, the values of the standard deviations are underestimated. Keywords Ground-motion model · Bayesian regression · Truncated data
1 Introduction Empirical ground-motion models (GMMs) are an important part of probabilistic seismic hazard analysis (PSHA). They describe the conditional distribution of a ground-motion intensity measure (such as peak ground acceleration (PGA), or response spectral ordinates (PSA) at different periods), given an earthquake/site scenario. Empirical GMMs are typically derived from a regression analysis on a strong-motion data set, where the model parameters (coefficients and standard deviations) are estimated during the regression. Electronic supplementary material The online version of this article (https://doi.org/10.1007/s1051 8-020-00943-8) contains supplementary material, which is available to authorized users. * Nicolas Martin Kuehn [email protected] Extended author information available on the last page of the article
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Bulletin of Earthquake Engineering
Due to the nature of strong-motion data sets, there are challenges during the regression. There are repeated measurements of the same earthquake at multiple stations or at the same station, which means that the data is multilevel in nature (Gelman and Hill 2006), leading to correlations between different observations. In addition, the predictor variables (such as magnitude or time-averaged shear wave velocity in the upper 30m of soil, VS30 ) are associated with uncertainty (Rhoades 1997; Moss 2011; Kuehn and Abrahamson 2018; Wang
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