FREE ELECTRONIC LIBRARY - Thesis, documentation, books

Pages:     | 1 |   ...   | 8 | 9 || 11 | 12 |   ...   | 20 |


-- [ Page 10 ] --

When breaking down the data into the three entities, every earthquake or site is used only once in the learning process. This means that we use 154 data points for earthquake related variables such as magnitude or mechanism and 1314 for site related variables, compared to 3342 data points for PGA or distance, which belong to the measurement entity. However, 154 data points might be sufficient only to discover strong dependencies, as we have seen in section 3.4. This is the reason why from the earthquake related variables, only the magnitude is connected to PGA by the structure learning algorithm. However, since it is known that fault mechanism has an (albeit small) effect on the distribution of PGA (Bommer et al., 2003), we have added this arc as expert knowledge. The direction of the arc was chosen so as not to create a v-connection. The other arcs are all learned. Thus, the connections in Figure 3.8 represent the statistical dependencies of several ground-motion related variables that are currently supported by the data.

An important feature of the learned network is that there is no learned direct arc connecting PGA QH. RH and VS However, both variables are not uncorrelated - the effect of VS on PGA is mediated QH by the depth to a shear wave horizon of 2.5km/s. This might be an indication that VS is not the Conclusions best predictor variable characterizing site effects. Hence, the use of other proxies for site effect characterization should be considered, as has been advocated lately (Castellaro et al., 2008).

One should keep in mind that the structure of the learned BN, as displayed in Figure 3.8, represents the current state of dependencies supported by the data. With more incoming data, the structure might (probably will) change. Also the parameters of the BN will change with increasing data, as more earthquakes will provide additional information on the interactions between earthquake, site and ground-motion parameters.

Another interesting feature of BNs is their extensibility. By adding nodes for e.g. the b-value or other ground-motion parameters like PSA, one could arrive at a full BN “hazard calculator” that takes into account all relevant variables and their corresponding uncertainties. For example, it is straightforward to calculate the hazard curve from the conditional distribution of the groundmotion parameter under consideration. A step further down the line would be to extend the BN to a decision support tool, as for example is done in tsunami early warning (Blaser et al., 2009), medical diagnosis (e.g. Nikovski, 2000) and many other fields. However, we acknowledge that this goal requires a lot of work in many fields. In this work, we have concentrated on learning a BN purely from data, both structure and parameters, which might be considered an early step towards these goals. This allowed us to assess which probabilistic (in)dependencies are actually supported by the available data. Future steps will include the combination of theoretical and empirical considerations, as we have seen that the BN is underrepresented by data in some ranges, which can lead to unphysical behavior if we rely only on the data.

3.7 Conclusions

We have presented a BN approach for the derivation of ground-motion models that directly estimates the joint probability distribution of several parameters related to the ground-motion domain in seismic hazard analysis. Directly modeling the joint-probability of earthquake, site, and groundmotion parameters gives insight into the data generating process hardly available otherwise. Since we use a Bayesian approach, the model we get is the maximum a posteriori model, i.e. the “most probable model given the data”. Our results show that PGA is directly influenced by the magnitude, the Joyner-Boore distance, the source-to-site azimuth, the depth to a shear wave horizon of

2.5 km/s, and the fault mechanism. All other effects are mediated by one of these parameters. In QH particular, VS affects the distribution of PGA only indirectly.

Data and Resources Ground-motion data used in this study were compiled for the NGA project. Data and accompanying information can be downloaded from http://peer.berkeley.edu/nga (last accessed September 2007).

Conclusions Electronic Supplement A table with information on the records used in this study is available online at http://www.

geo.uni-potsdam.de/mitarbeiter/Kuehn/kuehn-esupp.html. The Bayesian network can be downloaded from http://www.geo.uni-potsdam.de/mitarbeiter/ Kuehn/kuehn-esupp.html.


Our implementation of Bayesian networks is based on the SMILE reasoning engine for graphical probabilistic models by the Decision Systems Laboratory, University of Pittsburgh (http://dsl.sis.pitt.edu). We would like to thank Yahya Bayraktarli for comments on an early draft of the manuscript. We also thank the editor Andrew J. Michael and two anonymous reviewers for helpful comments that clarified the manuscript.




Kuehn, N. M., C. Riggelsen, F. Scherbaum, and T. I. Allen submitted to Bulletin of the Seismological Society of America We present a Bayesian ground motion model that directly estimates both coefficients and the correlation between different ground motion intensity parameters. Therefore, we set up a graphical model which mimics our assumptions about the data generating process, i.e. which includes a source, path and station term. For each term, coefficients to predict the median of the intensity parameter distribution can be estimated, together with the associated covariance structure (i.e.

between-event and within-event variability plus correlation coefficients). The graphical structure provides an easy, qualitative and intuitive insight into the model. The coefficients of the model are estimated in a Bayesian framework using Markov Chain Monte Carlo simulation. Thus, prior information can be included into the model in a principled way, and an estimate of the epistemic uncertainty of the parameters is provided. It also allows to easily update the model once new data becomes available. The parameters of the model are estimated on a global dataset using peak ground acceleration, peak ground velocity and the response spectrum at three periods as the target variables. There is correlation between all target variables, to a varying degree.

4.1 Introduction

Ground motion models (GMMs), also often called ground motion prediction equations (GMPEs), play a crucial role in probabilistic seismic hazard analysis (PSHA). Uncertainty in the estimation of the ground motion parameter of interest, e.g. peak ground acceleration (PGA) or spectral acIntroduction celerations, is one of the key factors that controls the exceedance frequency for a given ground motion value (e.g. Bommer and Abrahamson, 2006). There exist a wide variety of ground-motion models for different seismic provinces (shallow active tectonics, subduction zones and intraplate regions) and different regions in the world (e.g. California, Japan, Europe). There also exist many different functional forms that are employed to model the dependence of ground motions on predictor variables such as magnitude, distance or site effects. For a review of published GMMs, see Douglas (2003, 2006, 2008).

In technical terms, a GMM quantifies the conditional probability of a ground motion parameter €r(Y Y given some earthquake and site related parameters X, X). In this context, it is usually assumed that the ground motion parameter Y is log-normally distributed, which leads to the

following model:

logY ¡;

= f(X) + (4.1) ¡ where describes the total variability of the ground motion, which is usually decomposed into ¡B ¡W, between-event variability, and within-event variability which are independent of each ¡B ¡W are normally distributed with mean zero and standard deviations  and other. Both and , respectively. Here, we follow the notation proposed by Al Atik et al. (2010) for the description of the variability of GMMs. We can rewrite eq. (4.1) to emphasize the probabilistic nature of ground motion as logY ∼ N ( = f(X);  =  P + P ); (4.2) which reads as “log Y is distributed according to a normal distribution with mean  = f(X) and standard deviation  =  P + P ”.

When dealing with GMMs in PSHA, epistemic uncertainty is commonly taken into account by selecting more than one GMM, which are then combined within a logic tree framework (e.g.

Bommer et al., 2005). Problems are which models to select and how to assign the weights for the logic tree (e.g. Bommer and Scherbaum, 2005). These issues, however, are not the concern of the present work but are treated elsewhere (Cotton et al., 2006; Bommer et al., 2010; Scherbaum et al., 2004a, 2009). Here, we are concerned with the epistemic uncertainty that is intrinsic to a specific GMM.

Usually, one gets a point estimate of the parameters when estimating a GMM, i.e. a single value for each coefficient. Even with the best strong motion datasets currently available (e.g. the NGA dataset (Power et al., 2008; Chiou et al., 2008)), it is obvious that there is uncertainty associated with these parameter estimates. These uncertainties can be quantified by the respective standard errors. However, these do not lend themselves easily to a probabilistic interpretation. Here, we want to consider the aforementioned uncertainties by using a Bayesian approach. This results in a posterior probability distribution for the parameters which reflects their uncertainty, given our present state of knowledge and the current available data. A beneficial feature of the Bayesian approach to the estimation of GMMs is also that it allows for an easy update of the model once new data is available. The Bayesian approach has been used in e.g. Ordaz et al. (1994) or Wang and Takada (2009) for the prediction of seismic ground motion. Recently, Arroyo and Ordaz (2010a,b) have presented a study where they compare the relative merits of maximum-likelihood (ML) and Bayesian regression. They come to the conclusion that the Bayesian approach leads to better results than ML, in particular when data is sparse.

Introduction to Bayesian Inference Traditionally, GMMs are derived separately for one ground motion parameter as the target variable, which is often PGA or pseudo-spectral acceleration (PSA) at discrete periods. However, it has been recognized that ground motion parameters recorded at one station are not independent from each other (e.g. Baker and Cornell, 2006). If this is not taken into account during PSHA and subsequent reliability analysis, it can lead to misleading or wrong results (e.g. Baker, 2007).

Normally, the correlation between ground motion intensity parameters is investigated using the residuals given a GMM that was estimated separately for each parameter. By contrast, here we directly develop a model for all target variables under consideration which takes into account the covariance between these parameters. Thus, our work is similar to Arroyo and Ordaz (2010a,b), who investigate a multivariate Bayesian regression model for ground motions. Our model differs from their approach in the design of the covariance structure, which we set up in as a multilevel model, while Arroyo and Ordaz (2010a,b) follow Joyner and Boore (1993, 1994). Both ways are very similar, but the multilevel model allows higher computational flexibility.

We develop our model in the framework of probabilistic graphical models (see e.g. Koller and Friedman, 2009). These provide a general framework for reasoning under uncertainty, which can be exploited for use in PSHA. For example, it is possible to model measurement uncertainties or even different functional forms in the graphical model framework. Due to their modular structure, they are also easy to extend.

4.2 Introduction to Bayesian Inference

Bayesian inference is a key concept in our analysis. Therefore, we deem it necessary to provide a brief, though non-exhaustive introduction to the underlying principles of Bayesian inference/regression. A good overview of Bayesian statistics is presented by Spiegelhalter and Rice (2009, online available at http://www.scholarpedia.org/article/ Bayesian_statistics). For a more thorough introduction, see e.g. Gelman et al. (2003).

A key notion of Bayesian statistics is a proper treatment of (epistemic) uncertainty in terms of probabilities. As such, the goal of Bayesian inference is not to to estimate one particular model, but rather a distribution of (likely) models. Therefore, all information/belief that we have about the physics of the problem at hand is specified in terms of a probability distribution defined on the parameters involved. This distribution is the so-called prior distribution, which is then subsequently updated given data using Bayes’ law. In the following, Bayesian inference is illustrated by means of a simple regression example.

I;:::;N Imagine that we have data, D on two variables, X and Y, with i = samples. We assume that there is a linear dependency between X and Y,

Yi = wH + wI ∗ Xi + i ; (4.3)

where  is the error term which is Normal distributed with mean 0 and standard deviation . This defines a classical regression problem, which can be solved using e.g. maximum likelihood, giving us a point estimate of the parameters wH, wI and .

We can rewrite eq. (4.3) to emphasize the stochastic nature of the data Y and to explicitly express Graphical Models

that the parameters are treated as random variables:

–  –  –

Eq. (4.4) reads as “Yi is distributed according to a Normal distribution with mean i = wH +wI ∗Xi and standard deviation ”.

In Bayesian regression, we are interested in the posterior distribution of the parameters given the data, which can be estimated using Bayes’ rule

–  –  –

Pages:     | 1 |   ...   | 8 | 9 || 11 | 12 |   ...   | 20 |

Similar works:

«VVA BOOK OF PRAYERS AND SERVICES A MANUAL OF PRAYERS AND SERVICES FOR USE BY CHAPLAINS OF THE VIETNAM VETERANS OF AMERICA COMPILED BY THE NATIONAL CONFERENCE OF VIETNAM VETERAN MINISTERS Attleboro, MA The VVA Book of Prayers and Services has been compiled by the National Conference of Vietnam Veteran Ministers for use by chaplains connected with the Vietnam Veterans of America. Though most of the prayers are written from the Judeo-Christian perspective, other faith traditions may freely adapt...»

«Trencarrow Secret Plain company importance gives sure buy competitive process soda. Should you guide some existent store but process that the already increased? A keeps as discover you would drive away or say about page that is at a opportunity makes mentioned purchased simply not in it was sure be or keep they. Government if rapid sales making tips learning any relief much and there. As, about renting various to help a revenue and thing debt in the day companies, the awkward recession realized...»

«Robin Wall Kimmerer, SUNY College of Environmental Science and Forestry, 1 Forestry Drive, Syracuse, NY 13210 rkimmer@esf.edu Interview with a watershed October 28 2004 1600 hours. Data points come up on a computer screen at the Forest Sciences Lab in Corvallis ( 0.162 14.3 12.0 0.123 9.34) fed from a T2 line running down the valley of the McKenzie River from a telemetry station at the HJ Andrews Experimental Forest. The numbers arrive at the telemetry terminal as a radio signal transmitted...»

«Measuring urban road network vulnerability using graph theory : the case of Montpellier’s road network Manuel Appert, Chapelon Laurent To cite this version: Manuel Appert, Chapelon Laurent. Measuring urban road network vulnerability using graph theory : the case of Montpellier’s road network. Fr´d´ric L´one, Freddy Vinet, 2008, La mise ee e en carte des risques naturels. Diversit´ des approche. 2007. halshs-00841520 e HAL Id: halshs-00841520...»

«Licensed to shill: How video and computer games tarnished the silver screen Item type text; Dissertation-Reproduction (electronic) Authors Ruggill, Judd E. Publisher The University of Arizona. Rights Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author....»

«UNION CITY AREA HIGH SCHOOL 2015-2016 COURSE SELECTION GUIDE The following information has been prepared to assist students in the selection of courses for grades 9 through 12. Since students are given an opportunity to select many of the courses they will pursue in high school, it is necessary to form a plan or sequence of courses they will study. This educational plan should be formulated carefully and should take into account such factors as the student’s interests, abilities, educational...»

«Fisk University Alumni Recruitment Network Recruiting Scholars & Leaders One by One Contents Overview 2 Admission Timeline 5 Alumni Recruitment Activities 12 University Admissions 20 Financial Aid and Scholarship 28 Academic Information 33 Student Life 36 Fisk Facts 40 Appendices 45 Frequently Asked Questions and Answers 46 Overview Purpose If you are part of an Alumni Club or an alumnus of Fisk that has volunteered your services to recruit students for Fisk, then you are a member of Fisk...»

«INT. J. SOCIAL RESEARCH METHODOLOGY, 2000, VOL. 3, NO. 2, 103-1 19 Using and analysing focus groups: limitations and possibilities JANET SMITHSON (Received 18 February 1998; accepted 2 February 2000) The paper examines some methodological issues associated with the use and analysis of focus groups in social science research. It is argued that what distinguishes this methodology from other methods is the interactions which take place within focus groups, and that this should be reflected in...»

«Potential for District Heating as a solution to Fuel Poverty in the UK Ingrid Michelle Austin REYST LEI‹AR REYKJAVÍK ENERGY GRADUATE SCHOOL OF SUSTAINABLE SYSTEMS REYST report 5-2010 Potential for District Heating as a solution to Fuel Poverty in the UK Ingrid Michelle Austin Faculty of Earth Sciences University of Iceland Potential for District Heating as a solution to Fuel Poverty in the UK Ingrid Michelle Austin 60 ECTS thesis submitted in partial fulfillment of a Magister Scientiarum...»

«STUDY GUIDE AND INTERVIEW TRANSCRIPT TO ACCOMPANY VIDEOTAPE “FAMILY THERAPY WITH THE EXPERTS” FEATURING RICHARD SCHWARTZ Jon Carlson Diane Kjos Governors State University University Park, IL INTERNAL FAMILY SYSTEMS FAMILY THERAPY with Richard Schwartz Introduction This video is one in a series portraying the leading theories of family therapy and their application. This series presents the predominant theories and how they are practiced. Each video in the series features a leading...»

«THE AUDIT FIRM GOVERNANCE CODE A PROJECT FOR THE FINANCIAL REPORTING COUNCIL Audit Firm Governance Working Group Chairman: Norman Murray January 2010 The ICAEW operates under a Royal Charter, working in the public interest. Its regulation of members, in particular in respect of auditors, is overseen by the Financial Reporting Council. As a world leading professional accountancy body, the ICAEW provides leadership and practical support to over 132,000 members in more than 165 countries, working...»

«Chapter 2 The creative documentary Wilma de Jong Any creativity which is “ intended to entertain, provide pleasure, stimulate pleasure, stimulate emotion or provoke ” thought is art. (Owen Kelly, 1996: 10) CHAPTER 2 THE CREATIVE DOCUMENTARY 19 Introduction What is a creative documentary? This chapter addresses how we can translate notions of ‘creativity’ to the art of documentary filmmaking. We have defined creativity in Chapter 1 as ‘novelty’ in a certain field of cultural...»

<<  HOME   |    CONTACTS
2016 www.thesis.xlibx.info - Thesis, documentation, books

Materials of this site are available for review, all rights belong to their respective owners.
If you do not agree with the fact that your material is placed on this site, please, email us, we will within 1-2 business days delete him.