International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP) (12th : 2015)

Sensor network optimization using Bayesian networks, decision graphs, and value of information Malings, Carl; Pozzi, Matteo Jul 31, 2015

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12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Sensor Network Optimization using Bayesian Networks, DecisionGraphs, and Value of InformationCarl MalingsGraduate Student, Dept. of Civil and Environmental Engineering, Carnegie MellonUniversity, Pittsburgh, USAMatteo PozziAssistant Professor, Dept. of Civil and Environmental Engineering, Carnegie MellonUniversity, Pittsburgh, USAABSTRACT: Bayesian Networks (BNs) and decision graphs provide a useful framework for modelingthe uncertain behavior of civil engineering infrastructures subjected to various risks, as well as the po-tential outcomes of risk mitigation actions undertaken by managing agents. These graphs can also guideoptimal sensing and inspection of infrastructure by maximizing the value of information of sensing ef-forts. This paper presents a general framework for modeling infrastructure systems using BNs and forevaluating sensor placement metrics within this model. An example application of the use of the valueof information metric in guiding optimal sensing in a system of infrastructure assets in the San FranciscoBay area subjected to seismic risk is then presented. A parametric study also investigates the sensitivityof the value of information metric to various parameters of the BN system model.1. INTRODUCTIONEffective management of infrastructure requires in-formation about the status of the infrastructure sys-tem so that managers may make informed decisionsto minimize potential losses, in terms of lost rev-enues or potential harm to the public. In orderto cost-effectively collect this information, an op-timization strategy should be used, where the ben-efits of additional information should be weighedagainst their costs. This paper presents such a strat-egy, in which a probabilistic model of an infrastruc-ture system is used to optimize the value of infor-mation of a sensor network monitoring the system.Analyzing the benefits of a sensor network be-fore it is put in place is a form of pre-posterioranalysis. Such analysis requires a model of thesensed system, such that potential sensor measure-ments may be predicted and their consequences as-sessed. We propose a Bayesian Network modelfor this purpose. Bayesian Networks (BNs) area type of probabilistic graphical model (PGM).PGMs represent physical systems using randomvariables with a joint probabilistic distribution. Re-lationships between variables are encoded graphi-cally. The reader is referred to Koller and Friedman(2009) for further background on PGMs. BNs haveapplications in the modeling of infrastructure sys-tems; Bensi et al. (2014) present a BN model of atransportation system subjected to seismic risk. Bymodeling outputs of sensors as observable variableswithin a BN, pre-posterior analysis can also be per-formed. In an infrastructure system BN, similari-ties between components are encoded in the model,allowing observations of one sensor to provide in-formation about multiple components.Within this model for pre-posterior analysis,the usefulness of potential measurements mustbe quantified, such that optimization can be per-formed. A common metric for this purpose is theconditional entropy metric, which quantifies uncer-tainty in one set of random variables conditioned onobservations of another set. The reader is referredto Cover and Thomas (2006) for further informa-tion on conditional entropy. If the goal of a sensor112th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015network is characterized as reducing uncertainty inthe system, this metric can be used to quantify thisgoal and perform sensor network optimization, asdiscussed by Krause (2008).In the case of infrastructure management, how-ever, uncertainty reduction is not necessarily thegoal of sensing. Infrastructure managers use in-formation from sensors to guide decision-makingabout which actions to take to reduce long-termmanagement costs for the system. Potential ac-tions might include closing an unsafe componentto prevent injury to the public or leaving the com-ponent in operation until it fails. Each action comeswith certain costs or losses, such as lost revenues orpotential injuries. To assess the benefits of infor-mation in such decision-making problems, we usethe value of information metric, which quantifiesthe reduction in expected loss in a decision-makingproblem due to the availability of the information.An introduction to value of information is presentedin this paper, and a more complete background isprovided by Raiffa and Schlaifer (1961).The decisions of managers, as well as the poten-tial costs of different outcomes, can be included ex-plicitly into a BN model of an infrastructure sys-tem. Such a model is referred to as a decision graphor influence diagram. Pre-posterior analysis canthen be conducted in the decision graph using thevalue of information metric, and an optimal sensornetwork can be designed for the monitoring of themodeled system. In this paper, we present a generalmethod for creating such a model and performingthis optimization. We demonstrate this method onan example infrastructure system in the San Fran-sisco Bay area subjected to seismic risk, as well ason a simple system to perform a parametric analysison the value of information metric.2. PROBLEM FORMULATIONWe now present a BN model of a civil infrastructuresystem, extend this model to a decision graph, anduse this graph to define a metric for optimal sensorplacement for management of the system. Consideran infrastructure system made of n binary compo-nents, which can either be operational or not. Thesystem is subjected to a random risk scenario pa-rameterized by S. Although this scenario is uncer-tain a priori, we assume that, after the occurrenceof the scenario, information about the scenario isavailable to infrastructure managers. The func-tioning of component i is governed by (potentiallymulti-dimensional) variable Wi. From this variable,a binary random variable Xi ∈ {0,1} can be definedas the state variable of component i, i.e., componenti is operational if Xi = 1 and has failed if Xi = 0.Together, the set of variables X = {X1, . . . ,Xn} de-scribes the state of every component of the system.For component i, the manager of this infrastruc-ture system must select an action ai ∈ Ai for themanagement of this component. Depending on thechosen action ai and the state of the component xi,the manager incurs some loss (or reward) Li(xi,ai),a deterministic outcome of the action and state. Fi-nally, the total loss incurred by the manager for theentire system, LG(X ,A), is a function of the jointstate of all system components, as well as all se-lected actions. Here, we assume that this systemloss is the sum of the losses incurred in the man-agement of each component:LG(X ,A) =n∑i=1Li(xi,ai) (1)In selecting which actions to take, an infrastruc-ture system manager may have the ability to ob-serve certain variables, from which he or she mayinfer the states of components within the system.We denote with Z = {Z1, . . . ,Zn} the set of all ob-servations which might be made by the manager,where Zi is the set of observations related to com-ponent i, that is, Zi ⊆Wi. Due to limited time andresources, the manager may not be able to make allpossible observations; therefore, he or she must se-lect a subset Y = {Y1, . . . ,Yn} of these variables tomeasure, where Yi ⊆ Zi, and base their decision onthe specific outcome Y = y of these measurements.Note that Yi may be an empty set if no observationsrelating to component i are selected.A graphical representation of the infrastructuremodel described above is shown in Figure 1. Thesymbols used follow a common convention, de-scribed by Koller and Friedman (2009); circles rep-resent random variables, squares indicate agent ac-tions, and rhombi denote utilities or losses. Ar-212th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015rows or lines indicate relationships between vari-ables. Dashed arrows indicate temporal prece-dence. Shaded circles indicate observed variables.Figure 1: A decision graph for an infrastructure systemwith n components. Variable S describes the risk sce-nario, variable Wi describes component i, Xi describesits state, and observation Yi is made on certain observ-able features of the component Zi ⊆Wi. Action Ai ischosen by the manager of the infrastructure system,who incurs loss Li for component i, and global loss LGfor the whole system.Within the decision graph of Figure 1, we definean optimal sensor network as follows:Y ∗ = argmaxY⊆Z mX(Y ) subject to C(Y )≤ B (2)where Y ∗ is the optimal set of sensed variables,C(Y ) is the cost of measuring Y , B is a fixed bud-get constraint, and mX(Y ) is a metric which quan-tifies how observing Y will improve the managingagent’s ability to effectively and efficiently managethe infrastructure system. We denote this metric asmX(Y ) since X describes the functionality of eachcomponent, and is therefore of primary interest forthe management of the system. Our metric shouldtherefore depend on how well sensor placement Yimproves the manager’s knowledge about X .It is necessary that the optimal sensor placementY ∗ be robust under uncertainty in the risk scenarioS. Since S is not known a priori, we must computeour metric for specific values of s ∈ S; we denotethese scenario-specific metric values with mX |s(Y ).We then compute mX(Y ) by taking the expectedvalue over potential scenarios:mX(Y ) = ES[mX |s(Y )] =∫smX |s(Y )p(s)ds (3)where ES [·] represents the statistical expectationunder the distribution p(S) of S. To evaluate (3),we adopt a Monte Carlo sampling approach:mX(Y )≈1nsns∑i=1mX |si(Y ) (4)where ns is the number of scenarios s1, . . . ,sns sam-pled independently from p(S).3. VALUE OF INFORMATION FOR SEN-SOR PLACEMENTValue of information quantifies the explicit benefit,in terms of a reduction in expected losses, that aninfrastructure manager would see after implement-ing a sensor network to measure Y . For componenti, the expected loss for this component under sce-nario s ∈ S, without any observations of Y , is:ELi,s( /0) = minAiEXi|s [Li(xi,ai)] (5)That is, a manager should choose an action ai ∈ Aiwhich minimizes his or her expected loss under thepossible outcomes of Xi in scenario s. Note that,with binary components, there are only two out-comes for Xi: 0 (a failure) or 1 (no failure).Given an observation y of Y , the managing agentcan update the probability distribution of the stateof Xi based on the conditional distribution p(Xi|y,s).Taking this into account, the expected loss giventhat an observation of Y will be available before anaction is chosen is given in Eq. (6):ELi,s(Y ) = EY |s[minAiEXi|y,s [Li(xi,ai)]](6)where the inner expectation is taken over the condi-tional distribution of Xi given a specific observationof Y = y, and the outer expectation is taken over thedistribution for these observations in scenario S= s.The outer expectation accounts for the fact that theobservation y is not known a priori, but the opti-mal action will depend on this observation (thus,the minimization over Ai is within this expectation).For component i, the value of information of ob-serving Y before choosing an action is the differ-ence between the expected loss for this component312th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015given this information and the expected loss with-out information:VoIi,s(Y ) = ELi,s( /0)−ELi,s(Y ) (7)That is, the value of information ofY is the decreasein expected loss due to the availability of an obser-vation of Y prior to making a decision.For the entire system, since the system lossis equivalent to the sum of individual componentlosses, by Eq. (1), and since the expectation is lin-ear, the value of information of Y for the system is:VoIs(Y ) = ELGs ( /0)−ELGs (Y ) =n∑i=1VoIi,s(Y ) (8)To maximize the net benefit of a sensor network,the difference of the value provided by this sen-sor network, quantified as VoIs(Y ), and the cost ofimplementing this network, C(Y ), should be max-imized. We can use this difference as a metric toassess sensor network Y under scenario s:mX |s(Y ) = VoIs(Y )−C(Y ) (9)where the dependence of the metric on X is implicitin the formulation of the value of information.4. METHODOLOGYIn section 4.1, a general method for computingmX(Y ) in a PGM as outlined in Figure 1 is de-scribed. In section 4.2, details for implement-ing this method in a Gaussian graphical model arebriefly presented. The reader is referred to Malingsand Pozzi (2014) for details. In section 4.3, an algo-rithm is given for efficiently approximating Eq. (2).4.1. General metric evaluationThe evaluation of a generic metric mX(Y ) for someproposed sensor network measuring Y in an infras-tructure system PGM as outlined in section 2 canbe performed by the following steps:1. Begin with the distributions p(S), p(W |s),p(X |w,s), and p(Y |w,s) which parameterizethe PGM.2. Marginalize over W to obtain distributionsp(X |s) and p(Y |s).3. For any observation Y = y, perform inferencewithin the PGM to obtain the updated distribu-tion p(W |y,s).4. Use this distribution to obtain an updated dis-tribution p(X |y,s).5. Define a mapping mX |y,s(Y ) from the proba-bility distributions p(X |s) and p(X |y,s) to themetric to be evaluated. Note that we have as-sumed in section 2 that the value of metricmX |s(Y ) depends on how an observation y ofY allows for updating knowledge about X un-der scenario s, from p(X |s) to p(X |y,s).6. Take the expectation of mX |y,s(Y ) over p(Y |s)to compute mX |s(Y ). This accounts for the factthat observation y is not known a priori.7. Take the expectation of mX |s(Y ) over p(S) tocompute mX(Y ), as in Eq. (3), following theapproach of Eq. (4).Details on marginalization and inference in PGMsare given by Koller and Friedman (2009).Note that for the value of information met-ric, mX |y,s(Y ) = ∑ni=1 {νi,s( /0)−νi,s(y)} − C(Y ),where νi,s( /0) = minAi EXi|s [Li(xi,ai)] and νi,s(y) =minAi EXi|y,s [Li(xi,ai)]. Evaluation of the valueof information metric therefore only involves themarginal distributions p(Xi|y,s) and p(Xi|s), ratherthan the joint distributions p(X |y,s) and p(X |s).The ability to evaluate value of information inthis way is due to the decomposition of system-level loss into the sum of component-level lossesin Eq. (1). Computation of the value of in-formation is therefore relatively efficient, sincep(X1|y,s), . . . ,p(Xn|y,s) only require the computa-tion of n values, i.e., the probability of failurefor each component, to parameterize them whilep(X |y,s) requires 2n−1 values to parameterize.4.2. Gaussian linear modelsFollowing the approach of Ditlevsen and Madsen(1996), we assume that component i is fully char-acterized by the demand (di) and the capacity toresist this demand (ci), i.e., Wi = {ci,di}. Fur-thermore, we assume that under scenario s, thesevariables, via an appropriate transformation, can berepresented by joint Gaussian distributions:w=[cd]∼N([µCµD],[ΣC 00 ΣD])(10)412th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015where d = [d1, . . . ,dn]T and c = [c1, . . . ,cn]T, µDand µC are the corresponding means, ΣD and ΣCthe covariance matrices, and 0 is an n by n matrixof zeros. A multivariate Gaussian distribution is de-noted by N (·, ·), with the mean vector as the firstargument and the covariance matrix as the second.This model assumes capacities are marginally inde-pendent of demands. We also assume that candidateobservation variables Z are noisy measurements ofthe capacity and demand for each component, i.e.z = w+ ε , where ε represents the random errorof these observations, assumed to be a zero meanGaussian random vector independent of w with adiagonal covariance matrix.To encode the proposed sensor network Y , weuse the matrix A. For m observed variables selectedfrom 2n potentially observed variables, A is an m by2n matrix. Each row of A has all entries zero, ex-cept for an entry of one in the position correspond-ing to the selected observable variable. We modelthe observations of sensors as:y= Az= A(w+ ε) (11)The relationship between X and W is given by:x= I(c−d≥ 0) = I([ I −I ]w≥ 0)(12)where I[·] is an indicator function, taking on value1 when its argument is true and 0 otherwise, and Irepresents and n by n identity matrix. Under thismodel, a component fails when the demand placedon it exceeds its capacity.Equations (10), (11), and (12) define the distri-butions for p(W |s), p(Y |w,s), and p(X |w,s) respec-tively, where the latter is a deterministic function ofw. These distributions, together with p(S), whichgives a distribution on the parameters µD, µC, ΣD,and ΣC (which may differ for different scenarios),are all that are needed for the general procedureoutlined in section 4.1. Inference and marginaliza-tion for Gaussian random variables are outlined byKoller and Friedman (2009).4.3. Greedy sensor placementAn exact solution of the optimal sensor placementproblem of Eq. (2) would require the enumerationof all subsets Y of Z, which is computationally pro-hibitive in all but the smallest problems. An al-ternative, approximate solution approach involvesa heuristic known as the greedy algorithm, as dis-cussed by Krause (2008). In this method, singleelements of Z are iteratively added to the set Y ,where these elements most improve the objectiveto be optimized. Pseudo-code for implementationof the greedy algorithm is given in Algorithm 1.Input: Z; mX(·); C(·); BY ← /0 ;while C(Y ) < B, |Z|> 0 doy∗← argmaxy∈Z mX(y∪Y ) ;Y ← Y ∪ y∗ ;Z← Z\y∗ ;foreach z ∈ Z doif C(Y ∪ z) > B thenZ← Z\z ;endendendreturn Y ;Algorithm 1: Pseudo-code for the greedy algo-rithm. mX(·) is evaluated as outlined in sec-tion 4.1. Based on algorithms of Krause (2008).5. EXAMPLE APPLICATION TO SEISMICRISK IN SAN FRANCISCOAs an illustrative application of sensor placementoptimization to a practical infrastructure manage-ment problem, we examine an infrastructure sys-tem consisting of 18 bridges and 9 tunnels in theSan Francisco Bay area subjected to seismic risk. Itshould be noted that this system will merely serveto illustrate the application of the metric and tech-niques discussed above, and is not meant to be apractical recommendation. Seismic risk scenariosare modeled using a homogeneous Poisson processfor earthquake occurrence, based on the model out-lined by Anagnos and Kiremidjian (1988), with em-pirical data for the San Francisco Bay area pre-sented by Field et al. (2009) and USGS (2008). Us-ing this generative model, for the evaluation of (4),ns = 1000 sample seismic scenarios are generated,512th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015with each scenario s consisting of an earthquakewith magnitude M and epicenter location E.Earthquake demands are defined in terms of peakground acceleration, using attenuation equationspresented by Douglas (2011). Ground accelerationsare modeled as lognormal random variables; underthe logarithm transformation, these become Gaus-sian random variables, as in section 4.2. Correla-tions between demands are modeled by a squared-exponential kernel function, as in Bensi et al.(2014). Capacities of components are also mod-eled as lognormal random variables using fragilitycurves presented in Hazus (2012). Correlations be-tween capacity variables are assumed to be higherfor components with a similar overall typology.These lognormal variables are collected in the vec-tor w′, such that log(w′) = w is Gaussian.It is assumed that a binary decision must be un-dertaken as to whether or not to close each poten-tially damaged bridge or tunnel in the wake of theevent. The option to close down the componentcomes with a certain cost in terms of lost toll rev-enues and service loss, which is roughly estimatedfor each component considered. If the componentis not closed, no costs would be incurred, but if thecomponent is severely damaged and fails while inuse, a high cost of failure is incurred.Detailed measurements of the capacities of eachcomponent and/or the ground acceleration at eachlocation would potentially be available, at a certaincost. These costs are assumed to consist of an in-stallation cost for sensors as well as ongoing main-tenance costs for the sensor network which are dis-counted to their present value using a discount rateof 5%. Sensor noise is modeled as a multiplicativelognormal error ε ′ with median 1, such that under alogarithm transformation, this noise would be zeromean additive Gaussian noise ε , as in Eq. (11):log(w′ε ′) = log(w′)+ log(ε ′) = w+ ε (13)Further details of the model outlined above are pro-vided in Malings and Pozzi (2014).Figure 2 shows the optimal measurement selec-tions for the management of this infrastructure sys-tem based on the value of information metric ofEq. (9). Capacity measurements for the GoldenGate Bridge and the Caldecott tunnel, as well asdemand measurements at both locations and at theSan Francisco-Oakland Bay Bridge, are indicatedas the optimal measurement set for the managementof this example infrastructure system.Figure 2: Optimal measure selections based on thevalue of information metric. Background image fromwww.maps.google.com.6. PARAMETRIC ANALYSISA parametric study is presented to provide insightinto the sensitivity of the value of information met-ric to some of the parameters of the BN used to rep-resent an infrastructure system. This study is per-formed on an example system with 12 componentssubjected to a single magnitude 7 earthquake sce-nario, as shown in Figure 3. Demands on all com-ponents are computed for this scenario as discussedin section 5. Component capacities are defined us-ing lognormal distributions, with median capacitiesfor each type as listed in Table 1 and coefficient ofvariation of 0.6 for all types. Components of thesame type have correlated capacities, with the cor-relation coefficient listed in Table 1 between anypair of components of the same type. The binarydecision described in section 5 is again used, withclosure cost set at $10 million for all componentsand failure cost set as shown in Table 1. Sensors aremodeled as having multiplicative lognormal noise612th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Table 1: BN model parameters for the system of Fig-ure 3. The parameters to be varied are C f , µ IIC , and ρ .Component Cost of Median CorrelationType Failure Capacity Coefficient[$M] [g]I C f 1 0.1II 100 µ IIC 0.1III 100 1 ρwith median 1 and coefficient of variation 0.2. Allsensors are assigned a cost of $1 million, and a bud-get constraint of $1 million is used, such that only 1sensor will be selected. For this parametric study,Figure 3: Example system for this parametric study.Example measures Y1, Y2, Y3, and Y4 are shown.we track the value of information for four proposedmeasurements, Y1, Y2, Y3, and Y4, each of which areselected as the most optimal measurement for thesystem under a specific setting of the parametersC f , µC(II), and ρ .We begin by studying the effect of varying failurecost parameter C f from 100 to 10 for componentsof type I. At C f = 100, demand measurement Y1 oncomponent 2 has the highest value of information.As C f is lowered, the relative value of informationfor Y2 compared to Y1 grows as C f is decreased.This is due to the lower expected costs for com-ponents of type I compared to components of othertypes as C f decreases. It eventually becomes moreworthwhile to directly monitor a component withFigure 4: Parametric study on C f , with µ IIC = 1 andρ = 0.1.higher expected costs. The two sharp bends in eachcurve of Figure 4 are due to the choice of optimalmanagement action for components 1 and 2 chang-ing as the relative costs of these actions change.Figure 5: Parametric study on µ IIC , with C f = 10 andρ = 0.1. Probability of failure for component 4 is plot-ted on the horizontal axis.Next, the median capacity of type II components,µ IIC , is varied from 1 to 0.1, increasing the probabil-ity of failure for these components. The probabil-ity of failure of component 4, where measure Y2 istaken, is used as the axis of Figure 5. As this prob-ability of failure increases, the value of measuringY2 declines, as this component is more likely to fail,and therefore the optimal action, to close the com-ponent, is no longer in question. It soon becomes712th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015more valuable to take measureY3 instead, as the op-timal management action for the component wherethis measure is taken is still unclear.Figure 6: Parametric study on ρ , with C f = 10 andµ IIC = 0.1.Finally, the correlation coefficient for compo-nents of type III, ρ , is varied from 0.1 to 0.9, asshown in Figure 6. As this coefficient increases, thevalue of information for measuring Y4, the capacityof a component of type III, also increases, since in-formation about the capacities of many componentsin the system will be gained from this single mea-surement. Note that VoIs(Y3) remains constant.7. CONCLUSIONSThis paper presents a general framework for mod-eling systems of infrastructure using BNs and de-cision graphs. Within such a model, methods forcomputing sensor placement metrics in general,and the value of information metric in particular,are presented. These methods are then demon-strated using an example system of infrastructureassets in the San Francisco Bay area subjected toseismic risk. A parametric analysis is also con-ducted to demonstrate the sensitivity of the value ofinformation metric to various BN parameters. Theability to directly trade off value of information andsensor system cost using the metric demonstratedabove can allow for senor network selections whichmaximize the net benefit of the information pro-vided by these networks in terms of minimizing ex-pected losses for decision-making problems in in-frastructure management.8. REFERENCESAnagnos, T. and Kiremidjian, A. (1988). “A reviewof earthquake occurrence models for seismic hazardanalysis.” Probabilistic Engineering Mechanics, 3(1),3–11.Bensi, M., Der Kiureghian, A., and Straub, D. (2014).“Framework for post-earthquake risk assessment anddecision making for infrastructure systems.” ASCE-ASME Journal of Risk and Uncertainty in Engineer-ing Systems, Part A: Civil Engineering, 9, 1–17.Cover, T. and Thomas, J. (2006). Elements of Informa-tion Theory. John Wiley & Sons, Inc., Hoboken, NewJersey, USA, 2 edition.Ditlevsen, O. and Madsen, H. O. (1996). Structural Reli-ability Methods. John Wiley & Sons Ltd, Chichester,UK, 1 edition.Douglas, J. (2011). “Ground-motion prediction equa-tions, 1964-2010.” PEER PEER Report 2011/102, Pa-cific Earthquake Engineering Research Center, Col-lege of Engineering, University of California, Berke-ley, Berkeley, California, USA (April).Field, E. H., Dawson, T. E., Felzer, K. R., Frankel, A. D.,Gupta, V., Jordan, T. H., Parsons, T., Petersen, M. D.,Stein, R. S., Weldon II, R. J., and Wills, C. J. (2009).“Uniform california earthquake rupture forecast, ver-sion 2 (UCERF 2).” Bulletin of the Seismological So-ciety of America, 99(4), 2053–2107.Hazus (2012). “Hazus-MH 2.1 technical manual: Earth-quake model – multi-hazard loss estimation method-ology.” Report no., Department of Homeland Secu-rity, Federal Emergency Management Agency, Wash-ington, DC, USA.Koller, D. and Friedman, N. (2009). ProbabilisticGraphical Models: Principles and Techniques. Mas-sachusetts Institute of Technology, Cambridge, Mas-sachusetts, USA.Krause, A. (2008). “Optimizing sensing: Theory and ap-plications.” Ph.D. thesis, Carnegie Mellon University,Pittsburgh, PA 15213 (December).Malings, C. and Pozzi, M. (2014). “Conditional en-tropy and value of information based metrics for sen-sor placement in infrastructure systems.” Proceedingsof the 6th World Conference on Strucutral Controland Monitoring, Barcelona, Spain (July).Raiffa, H. and Schlaifer, R. (1961). Applied statisti-cal decision theory. Harvard University Press, Cam-bridge, Mass., USA.USGS (2008). “Uniform california earthquake ruptureforecast.” Report No. 3027, United States GeologicalSurvey (April).8

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