WWW.THESIS.XLIBX.INFO
FREE ELECTRONIC LIBRARY - Thesis, documentation, books
 
<< HOME
CONTACTS



Pages:     | 1 |   ...   | 9 | 10 || 12 | 13 |   ...   | 20 |

«EMPIRICAL GROUND-MOTION MODELS FOR PROBABILISTIC SEISMIC HAZARD ANALYSIS: A GRAPHICAL MODEL PERSPECTIVE Kumulative Dissertation zur Erlangung des ...»

-- [ Page 11 ] --

Equation (4.5) can be interpreted as updating the prior distribution of the parameters using data (via the likelihood function), resulting in the posterior distribution. It is possible to estimate the posterior distribution for a whole batch of data points. Alternatively, one can also update the model sequentially, using one data point at a time, in which case the posterior distribution given the first data point becomes the prior distribution for the second and so on. The latter procedure can be used to easily update a Bayesian model once new data becomes available.

To solve our regression example eq. (4.3), we need to specify our prior information/beliefs of the parameters wH, wI and  into a probability distribution. In conjunction with the likelihood €r(Y {wH ; wI ; }) we can estimate the posterior distribution using eq. (4.5).

function

4.3 Graphical Models

In the previous section we have illustrated the basic ideas behind Bayesian inference, where we €r(¢ D). In simare interested in the conditional distribution of the parameters given the data, ple cases this distribution may be solved analytically, but in many realistic situations the posterior distribution is high-dimensional, complex, and unavailable in closed form. This is primarily due to analytical complications in computing the marginal likelihood. Therefore, one needs to resort to approximate inference methods, such as Markov Chain Monte Carlo (MCMC) sampling of the posterior distribution (e.g. Gilks et al., 1996). MCMC constructs a Markov Chain with the stationary/limiting distribution being the posterior. Here, we will use MCMC to obtain the parameters of our GMM. In particular, we make use of the program OpenBUGS (http://www.openbugs.info/) (Lunn et al., 2009), where BUGS stands for ‘Bayesian inference using Gibbs sampling’.

Gibbs sampling (Geman and Geman, 1984) is one particular MCMC algorithm that exploits conditional independence assumptions between parameters and (un)observables of a data generatGraphical Models

–  –  –

Figure 4.1: Graphical model for a simple linear model of the form y(x; w) = wH + wI x +  ing system such as the one described in eq.

(2). An easy way to encode conditional (in)dependence statements are graphical models, in particular directed acyclic graphs (DAGs; e.g. Spiegelhalter, 1998), which we describe below. For a more detailed introduction to graphical models, see e.g.

Jordan (2004), Koller et al. (2007) or Koller and Friedman (2009).

Each quantity (observable or parameter) of a model corresponds to a node in the graphical model, and arcs between the nodes show direct dependences. The graphical model for our simple regression example [eq. (4.3)] is depicted in Figure 4.1. The graph is a DAG since each edge (connection) is an arrow, and it is acyclic because there is no direct path following the arrows from one node back to itself.

In a graphical model, non-random variables or co-variates are denoted by a rectangular node.

In our example, the data points Xi are considered known and are thus represented by a rectangle.

Elliptical or circular nodes represent either output of mappings (here also shaded) or stochastic quantities. In Fig. 4.1 and eq. (4.4), i is a function of wH ; wI and Xi, i = wH + wI ∗ Xi : (4.7) The functional dependence is denoted by a thick arrow in Figure 4.1. Thin arrows represent stochastic dependences. For example, in our case Yi coincides with a stochastic node associated with a Normal distribution with mean i and standard deviation . From a Bayesian perspective the nodes for wH, wI and  also represent stochastic quantities, which need to be assigned a prior distribution. If we assume a normal prior distribution for each of these parameters, the means and standard deviations of these Gaussians are represented by the nodes in the top line of Figure 4.1.

These nodes are rectangles, because a fixed prior distribution is assumed. Repetitive structures I (in our case the loop over the data points from i = to i = N) are represented by rectangular structures, so-called ‘plates’.

The advantage of a graphical model (DAG) is that it keeps details about distributions and deterministic functions hidden, but communicates the essence (i.e. the direct (in)dependences of Model Setup

Figure 4.2: Earthquakes used in the study.

variables) of a model. This is especially useful for complex models, where we otherwise would have to resort to a large set of equations.

The DAG representation also facilitates analysis of probability models, since it encodes conditional independence statements and allows a factorization of the joint probability distribution. It can be shown (Lauritzen et al., 1990) that for any particular €r(¢ Y YX) ∝ €r(¢;Y YX) = €r(V parents [V ] YX); (4.8) V ∈Y ∪¢ where parents[v] specifies the parent set of the nodes from which an arrow points to V (if the parent set is empty the conditional reduces to a marginal). Hence, to specify the full joint probability distribution it is sufficient to define the local “parent-to-child” distributions along the arcs of YX).

the DAG, P (v parents [v] Gibbs sampling is a technique that makes efficient use of these properties by passing information around the DAG in accordance to the independences holding and depending on the local distributions defined.





4.4 Model Setup In this study, our intention is to learn a Bayesian GMM. The central formula in this context is

Bayes rule, eq. (4.5), which we repeat here:

–  –  –

4.4.1 Dataset The dataset we use for constructing the Bayesian GMM is the one compiled by Allen and Wald (2009), which is a global dataset. It contains records from earthquakes in three different tectonic source types: shallow active tectonics, subduction zone and continental interiors. Here, we use only earthquakes from shallow active tectonic regimes.

The dataset of Allen and Wald (2009) comprises 10,163 records from 238 earthquakes from shallow active tectonic regimes. For details on the records compilation, we refer to Allen and Wald (2009). In this work, we use only records up to a rupture distance of 400 km from earthquakes with a moment magnitude greater than 5., which reduces the dataset to 9,872 records from 228 earthquakes.

In the dataset, information on peak ground velocity (PGV), PGA and PSA at 0.3s, 1s and 3s is available. The dataset was originally compiled to reconstruct ground-shaking from recenthistorical earthquakes (Allen et al., 2009) using USGS ShakeMap methodology (Wald et al., 1999). Consequently, the target variables are taken to be the the larger horizontal component.

We use only those records for which all five target quantities (PGV, PGA and PSA at 0.3s, 1s and 3s) are available, which leaves us with 7,957 records from 159 earthquakes, recorded at 2,889 unique stations. The epicenters of the earthquakes are depicted in Figure 4.2.

As predictor variables, we consider moment magnitude MW, rupture distance RRUP, the avQH, erage shear wave velocity in the upper 30 m, VS and the focal mechanism F M. The focal mechanism has three states, normal, strike slip, and reverse. Shear wave velocity values were estimated from topographic gradient using the approach of Wald and Allen (2007), with the minimum QH QH and maximum VS values of 210 m/s and 963.9 m/s, respectively. We group VS into three site categories according to Table 4.1.

For some earthquakes, there is no information on the focal mechanism. Similarly, for some QH stations the value of VS is missing. Nevertheless, the corresponding records can be retained in the analysis, as the uncertainty of the unknown values is taken care of during the analysis. For more details, see section 4.4.2.

In Figure 4.3, we show the magnitude-distance distribution of the used records. Scatter plots between the predictor variables and PGA are depicted in Figure 4.4. We provide Tables with detailed information on the used earthquakes and records in the electronic supplement.

Model Setup 7.5

–  –  –

6.5 6.0 5.5

–  –  –

Almost all published GMMs make the assumption that P GA, P GV and the response spectrum are log-normally distributed, which we follow here. Thus, we introduce a new vector of random variables, Z, which is the log-transformed target vector Y,

–  –  –

We then assume that Z is distributed according to a multivariate normal distribution with a vector ¦.

of means , that are functions of the predictor variables, and a covariance matrix In the development of a GMM, one has to take into account the correlation of records from the same earthquake. This is usually done by invoking an appropriate regression technique that allows for separation of the total variability into between-event and within-event standard deviation, such as a one-step or two-step regression (Joyner and Boore, 1993, 1994) or a random effects algorithm (Abrahamson and Youngs, 1992). Another source of variability is between-station variability, which takes into account that ground motions recorded at the same station are not independent.

However, this variability is only rarely taken into account, e.g. in the work of Chen and Tsai (2002).

In the following, we use the notation introduced by Al Atik et al. (2010) to discern the different components of ground motion variability. Between-event variability is denoted by , while  stands for within-event variability. The respective covariance matrices are denoted by upper case ¨.

letters, i.e. T and We develop our GMM as a multilevel/hierarchical model (Gelman and Hill, 2007), which can be seen as conceptually similar to a two-step regression but with the ability of easily adding extra complexity. The multiple levels allow to take into account grouped data. Hence, one level corresponds to all earthquakes, one level corresponds to all stations, and the intersection of these two levels represents the record of one earthquake recorded at one station.

In Figure 4.5, we show the GMM of this study as a graphical model. The two plates correspond to the two levels, where indices e and s denote the eth earthquake and sth station, respectively.

Figure 4.5 can be thought of as a conceptual model of the data generation.

The concept of the model is as follows:

–  –  –

The central node in Figure 4.5 is the vector of target variables, denoted by Zes, which is the eth earthquake recorded at the sth station. Zes is distributed according to a multivariate normal distri¨.

bution with mean vector Z;es and covariance matrix The mean vector Z;es is the sum of an event term, a station term and a record term, as can be seen in eq. (4.12). The event term E e is common to all records from the same earthquake e and is itself distributed according to a multivariate normal distribution, with mean vector E;e and covariance matrix T [eq. (4.13)]. Correspondingly, the station term S;s is common to all records recorded at the same station k. In principle, one could assume that the station term is also sampled from a multivariate normal distribution. However, a stable estimation of its covariance requires stations with multiple recordings, which are not Model Setup

–  –  –

abundant in our dataset. Hence, we assume that the station term is not a random variable, but a constant (strictly speaking, we assume that the components of the covariance matrix are all zero).

¨, T are the within-event and between-event covariances, respectively.

The means of the event, record and station terms are functions of parameters and the predictor

variables:

–  –  –

where superscript t denotes the tth target variable, while at, bt and ct are the coefficients for the individual functions. These parameters are all assumed to be independent of each other.

¨ The parameters a, b, c, as well as the covariances and T, are displayed as stochastic nodes, since they are treated as random variables. Therefore, they are assigned a prior distribution, whose parameters are represented by the rectangular (i.e. fixed) nodes of Figure 4.5. For a, b, and c, the prior distributions are independent univariate normal distribution with means a, b, c and ¨ standard deviations a, b, c, respectively. For the covariances and T, the prior distributions, €r €r denoted ¨ and T, are uniform distributions over their respective Cholesky decompositions (Weisstein, 2010). For more details on the prior distributions, see section 4.4.3.

We have settled on the following functional forms for f, g and h. These are based on geophysical considerations and generalization error determined by 10-fold cross-validation (e.g. Kuehn et al., 2009a; Hastie et al., 2001).

–  –  –

1. Specify prior distributions on the stress drop, QH and the slope of the geometric attenuation.

2. Generate a synthetic dataset from these parameters using stochastic simulations (Boore, 2003).

3. Regress the function of eqs. (4.17) to (4.18) on the synthetic dataset.

4. Take the estimated coefficients and their standard errors as prior distributions for the parameters.

–  –  –

4.5 Results In the previous section, we have specified the model, the prior distribution and the dataset. Using Bayes’ rule [eq. (4.5)], we can now estimate the posterior distribution of the parameters given €r(¢ €r(¢ D). As described before, the model is too complicated to estimate D) the data, analytically, so we resort to approximate inference, using MCMC sampling to obtain samples from the posterior distribution of each parameter. From the sequence of samples we can compute several summary statistics, such as mean values, standard deviations, quantiles and so on. The histogram of the sampled values serves as an approximation to the posterior probability density function.



Pages:     | 1 |   ...   | 9 | 10 || 12 | 13 |   ...   | 20 |


Similar works:

«White Paper Two Flash Technologies Compared: NOR vs NAND Written by: Arie Tal OCTOBER 02 91-SR-012-04-8L REV. 1.0 Two Flash Technologies Compared: NOR vs. NAND Introduction Two main technologies dominate the non-volatile flash memory market today: NOR and NAND. NOR flash was first introduced by Intel in 1988, revolutionizing a market that was then dominated by EPROM and EEPROM devices. NAND flash architecture was introduced by Toshiba in 1989. Most hardware engineers are not familiar with the...»

«MODEL POLICIES for INTERNAL GOOD GOVERNANCE in VOLUNTARY ORGANIZATIONS VOICE OF THE VOLUNTARY SECTOR Model Policies for Internal Good Governance in Voluntary Organizations Author: Voluntary Action Network India (VANI) December 2011 Copyright © Voluntary Action Network India The content of this book can be reproduced in whole or in parts with due acknowledgment to the publisher. Supported by: ChildFund India Edited and Designed by: Sumitra Srinivasan Published by: Voluntary Action Network India...»

«1736A LSR 267 July 13, 2004 Dear Reader: Enclosed are two Records of Decision (ROD) for the Upper Siuslaw Late-Successional Reserve Restoration Plan. One ROD addresses watershed restoration actions and the other ROD addresses upland thinning actions. In both RODs, I adopt the actions of Alternative D. I am issuing two separate RODs for the plan for several reasons. First, these two classes of actions will require different implementation processes; second, these two classes of actions will...»

«In presenting this dissertation/thesis as a partial fulfillment of the requirements for an advanced degree from Emory University, I agree that the Library of the University shall make it available for inspection and circulation in accordance with its regulations governing materials of this type. I agree that permission to copy from, or to publish, this thesis/dissertation may be granted by the professor under whose direction it was written when such copying or publication is solely for...»

«Experiment 11 – Carbohydrates Carbohydrates are a class of natural compounds that contain either an aldehyde or a ketone group and many hydroxyl groups – they are often called polyhydroxy aldehydes or ketones. A monosaccharide consists of a single carbohydrate molecule, containing between 3 and 7 carbons. Examples of monosaccharides are glucose and fructose. A disaccharide consists of two monosaccharides that are linked together. Sucrose and lactose are disaccharides. A polysaccharide...»

«SUSTAINABLE FUTURE, REQUISITE HOLISM, AND SOCIAL RESPONSIBILITY (Against the current abuse of free market society) Stane Božičnik, Timi Ećimović and Matjaž Mulej With co-authors ANSTED University, British Virgin Islands, and ANSTED Service Centre, Penang, Malaysia School of Environmental Sciences In Cooperation with SEM Institute for Climate Change, Korte, Slovenia And IRDO Institute for development of Social Responsibility, Maribor, Slovenia ANSTED UNIVERSITY, BVI OF UK SCHOOL OF...»

«Field Test Edition                                                 Spring 2010    Poems to Play with (in Class) Sample Unit of Study for Grades 3-5 Office of Curriculum, Standards and Academic Engagement   Department of English Language Arts Field Test Edition                                                 Spring 2010    NYC Department of Education Department of English Language Arts Unit...»

«Administration and Staff  Director Richard P. Appelbaum 1998­1999 Administrative and Technical Staff Jan Holtzclaw, Personnel/Payroll Jan Jacobson, Purchasing/Accounting John Lin, Systems Administrator  Tim Schmidt, Office Manager Jerrel Sorensen, Accounts Manager 1998­1999 Research Development Specialist Barbara Herr Harthorn 1998­1999 Advisory Committee Michael Jochim, (Chair), Anthropology Richard Appelbaum, ex­officio, ISBER Director ...»

«Issue 55 / e-Post / January 2011 Editor Welcome to the first edition of e-Post of 2011 and happy new year! In this issue, we offer you our selection of outstanding issues from 2010 on which we expect further news this year. For advice about any of the topics covered, please contact Martin Brewer, David Mills, Gillie Scoular or your usual Mills & Reeve contact. If you would like to receive information about key developments more quickly, please sign up to an email alert or RSS feed from our blog...»

«TCS-103117 Pettman (D):156x234mm 11/02/2009 15:21 Page 1 Love in the Time of Tamagotchi Dominic Pettman Abstract There is a popular conception among many Zeitgeist watchers, especially in places like the US, Western Europe and Australia, of the urbanized East as existing somehow further into the future. As William Gibson once stated: ‘The future is here; it just isn’t equally distributed yet.’ This kind of cultural fetishism extends to not only technolust, but the practices that new...»

«MRI Tissue Classification with Neighborhood Statistics: A Nonparametric, Entropy-Minimizing Approach Tolga Tasdizen1, Suyash P. Awate1, Ross T. Whitaker1, and Norman L. Foster2 School of Computing, University of Utah Department of Neurology, University of Michigan Abstract. We introduce a novel approach for magnetic resonance image (MRI) brain tissue classification by learning image neighborhood statistics from noisy input data using nonparametric density estimation. The method models...»

«Report of the ASA Workgroup on Master’s Degrees 06 November 2012 Submitted by John Bailer (baileraj@MiamiOH.edu) – Chair of the ASA Workgroup on Master’s Degrees Roger Hoerl (roger.hoerl@gmail.com) David Madigan (david.madigan@columbia.edu) Jill Montaquila (jillmontaquila@westat.com) Tommy Wright (tommy.wright@census.gov) ASA support: Ron Wasserstein (ron@amstat.org) – Staff liaison to the ASA Workgroup on Master’s Degrees Keith Crank – Initial liaison to Workgroup Table of Contents...»





 
<<  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.