Flood Damage and Influencing Factors:A Bayesian Network Perspective

Kristin Vogel 1

kvog@geo.uni-potsdam.de

Carsten Riggelsen 1

riggelsen@geo.uni-potsdam.de

Bruno Merz 2

bmerz@gfz-potsdam.de

Heidi Kreibich 2

kreib@gfz-potsdam.de

Frank Scherbaum 1

fs@geo.uni-potsdam.de

1 Institute of Earth and Environmental Science, University of Potsdam, Germany2 GFZ – GeoForschungszentrum Potsdam, Germany

Abstract

Classical approaches for flood risk assessment relate flood damage for a certain class ofobjects to the inundation depth, while other characteristics of the flooding situation andthe flooded object are widely ignored. Observations on several discrete and continuousvariables collected after the 2002 and 2005/2006 floods in the Elbe and Danube catch-ments in Germany offer a unique data mining opportunity in terms of learning a BayesianNetwork. We take an entirely data-driven stance opting not to discretize continuous vari-ables in advance; rather, we cast the problem in Bayesian framework, and consider themaximum aposteriori of the joint distribution of the triple, network structure, parametersand discretization, as the outcome of the analysis. Moreover, motivated by the work ofMerz et al. (2010), who point out the need of an improved flood damage assessment, were-define the discretization of the target variable, flood loss, once the network has beenlearned. Its domain is split into a large number of intervals and the associated parametersare estimated using a Gaussian kernel density estimator. Although the prediction of therelative flood loss is comparable to state-of-the-art methods, our approach benefits fromcapturing the joint distribution of all factors influencing flood loss.

1 Introduction

Graphical models have in recent years success-fully been employed in earth sciences, givingrise to a wide range of applications, includ-ing Tsunami Early Warning, e.g. (Blaser et al.,2011), Probabilistic Seismic Hazard Analysis,e.g. (Kuehn et al., 2011), and Automatic de-tection and classification of seismic signals, e.g.

(Riggelsen et al., 2007). In this paper we em-bark on another problem: flood damage assess-ment of residential buildings. Typically, thedamage to flooded objects is estimated by stage-damage functions which relate the relative orabsolute damage for a certain class of objectsto the water stage or inundation depth (Merz etal., 2010). Other characteristics of the flooding

situation and of the flooded object are rarelytaken into account, although it is clear thatflood damage is influenced by a variety of fac-tors such as inundation duration, contaminationof flood water, or quality of external responsein a flood situation. The single and joint ef-fects of these parameters on the degree of dam-age are largely unknown and widely neglectedin damage assessments. Moreover, the intrin-sic uncertainties associated with these factorsare largely ignored. Bayesian Networks (BN)pose an interesting formalism for capturing theinterdependencies and the intrinsic uncertaintyinvolved in flood risk assessment.

The paper is organized as follows: After giv-ing a description of the data set in Section 2and a brief introduction into learning BNs inSection 3, we show in Section 4, how we auto-matically discretize continuous variables, basedon the observed data set, using a maximum aposteriori (MAP) score to search simultaneouslyfor the best network structure, parameters anddiscretization. In Section 4.1 we point out, howwe employ the learned BN to estimate the floodloss, using a very fine discretization of the tar-get variable to allow a precise approximationof its continuous conditional distribution func-tions. Finally, the results are shown in Section5 and we conclude in Section 6.

2 Variable Definitions and Dataset

We take a data mining perspective and aim forlearning a BN from observational data. Theobservations are collected after the 2002 and2005/2006 floods in the Elbe and Danube catch-ments in Germany. Results of computer-aidedtelephone interviews with 1135 flood affectedhouseholds yield i.i.d. data, d = ∪k{x(k)}. Top-ics relate to various flood parameters (e.g. con-tamination, water depth), building and house-hold characteristics, precautionary measures,and flood damage to buildings and contents.The raw data were supplemented by estimatesof return periods, building values, loss ratio, i.e.the relation between the building damage andthe building value, and indicators for flow ve-locity, contamination, flood warning, emergency

Xi Predictors Scale and rangeflood parameters

wst Water depth C: 248 cm below ground to 670 cmabove ground

d Inundation duration C: 1 to 1440 hv Flow velocity indicator O: 0=still to 3=high velocitycon Contamination indicator O: 0=no contamination to

6=heavy contaminationrp Return period C: 1 to 848 yrs

warning and emergency measures

wt Early warning lead time C: 0 to 336 hwq Quality of warning O: 1=receiver of warning knew ex-

actly what to do to 6=receiver ofwarning had no idea what to do

ws Indicator of flood warningsource

N: 0=no warning to 4=officialwarning through authorities

wi Indicator of flood warninginformation

O: 0=no helpful information to11=many helpful information

wte Lead time period elapsedwithout using it for emer-gency measures

C: 0 to 335 h

em Emergency measures indi-cator

O: 1=no measures undertaken to17=many measures undertaken

precaution

pre Precautionary measures in-dicator

O: 0=no measures undertaken to38=many, efficient measures un-dertaken

epre Perception of efficiency ofprivate precaution

O: 1=very efficient to 6=not effi-cient at all

fe Flood experience indicator O: 0=no experience to 9=recentflood experience

kh Knowledge of flood hazard N (yes / no)building characteristics

bt Building type N (1=multifamily house, 2= semi-detached house, 3=one-familyhouse)

nfb Number of flats in building C: 1 to 45 flatsfsb Floor space of building C: 45 to 18000 mbq Building quality O: 1=very good to 6=very badbv Building value C: 92244 to 3718677

socio-economic factors

age Age of the interviewed per-son

C: 16 to 95 yrs

hs Household size, i.e. num-ber of persons

C: 1 to 20 people

chi Number of children (< 14years) in household

C: 0 to 6

eld Number of elderly persons(> 65 years) in household

C: 0 to 4

own Ownership structure N (1=tenant; 2=owner of flat;3=owner of building)

inc Monthly net income inclasses

O: 11=below 500 to 16=3000 andmore

socP Socioeconomic status ac-cording to Plapp (2003)

O: 3=very low socioeconomic sta-tus to 13=very high socioeco-nomic status

socS Socioeconomic status ac-cording to Schnell et al(1999)

O: 9=very low socioeconomic sta-tus to 60=very high socioeco-nomic status

flood loss

rloss loss ratio of residentialbuilding

C: 0 = no damage to 1 = totaldamage

Table 1: Description of the candidate predictors(C: continuous, O: ordinal, N: nominal).

measures, precautionary measures, flood expe-rience and socioeconomic variables (Thieken etal., 2005; Elmer et al., 2010). Table 1 lists 28candidate variables allocated to 5 domains andthe predictand rloss, which is the direct dam-age to flooded residential buildings representedas relative value, i.e. fraction of the buildingvalue.

3 Bayesian Network Learning

Formally speaking a BN decomposes a jointprobability distribution/density P (X) intoa product of (local) conditional probabil-ity distributions/densities p(·|·) as P (X) =∏i p(Xi|XPa(i)) according to a directed acyclic

graph (DAG) with vertices Xi and directededges from variables in the parent set XPa(i)

to Xi. In case X is continuous and we donot know p(·|·) in advance, we may approxi-mate the (local) conditional probability distri-butions by first discretizing the continuous vari-ables and rely on contingency tables instead;the challenges that this involves is discussed inSection 4. For such a discrete BN we writeP (X|DAG,θ) =

∏i θXi|XPa(i)

, where θ (the pa-

rameters) are conditional probabilities derivedfrom contingency tables. For the rest of this sec-tion we assume all variables X to be discrete.

BN model selection (learning the joint decom-position as well as the local conditional proba-bilities) is an exercise in traversing the spaceof BNs looking for the one which maximizesa given fitness score. As usual for model se-lection, regularization plays a role in this en-deavour. We use the Bayesian BN MAP score(Riggelsen, 2008) shown to learn BN that arebetter than those derived via the marginal like-lihood score (the BD-score). The BN is selectedas the MAP of the joint posterior (here bothDAG and parameter are being treated as truerandom variables)

P (DAG,Θ|d) ∝ P (d|DAG,Θ)P (Θ, DAG),

where the joint prior is a product, withP (Θ|DAG) defined to be a product Dirichletdistribution and P (DAG) defined to be uni-form over DAGs (we may thus ignore this term

when doing MAP estimation). The simulta-neous joint selection of DAG and parametersyields a DAG which equivalently can be foundby maximizing the structure score

S(DAG|d) =∏

i,xPa(i),xi

θn(xi,xPa(i))+α(xi,xPa(i))

xi|xPa(i)

×∏

i,xPa(i)

Γ(∑xi α(xi,xPa(i)))∏

xi Γ(α(xi,xPa(i)))︸ ︷︷ ︸regularization

where the statistics n(·) are the counts of a par-ticular configuration from the data/contingencytable and α(·) are the hyper-parameters ofthe Dirichlet (restricted as to guarantee uni-form DAG scoring equivalence; see (Riggelsen,2008)). The BN parameter estimates requiredfor computing the above score are also the BNparameters, and are given in closed form by

θxi|xPa(i)=n(xi,xPa(i)) + α(xi,xPa(i))

n(xPa(i)) + α(xPa(i)). (1)

The BN is learned using a hill-climber ap-proach in the space of DAGs based on the scoregiven above where arc addition, removal and re-versal are the basic operations. DAG equivalentclasses are simulated using the Repeated Cov-ered Arc Reversal operator (Castelo and Kocka,2003).

4 Automatic Discretization

We want to adhere to an entirely data-drivenapproach for learning BNs and strive for makingnone or at least very weak assumption with re-gard to the functional form of the (local) condi-tional distributions, p(Xi|XPa(i)). A sufficientlyfine discretization of continuous variables, plac-ing “counts” in contingency tables, enables us toapproximate any other (local conditional) dis-tribution, e.g., a Gaussian, but usually with alarger number of parameters.

To transform a continuous variable into a dis-crete one, the number of intervals and theirboundaries have to be chosen carefully. Afine-grained discretization will result in a verysparsely connected BN due to regularization

constraints, ultimately not reflecting the inter-actions we are interested in. On the other hand,too rough a discretization may not provide the“user” with the desired degree of resolution re-quired for proper decision support. Variousdiscretization approaches have been proposedin literature, (Friedman and Goldszmidt, 1996;Monti and Cooper, 1998). Inspired by the lat-ter we present a straightforward extension tothe BN MAP learning score, allowing us to de-termine the “fitness” of a BN and discretizationsimultaneously, conditional on continuous data.

From now on, a superscript c denotes thecontinuous counterpart of a discretized vari-able/configuration, e.g., dc is the original con-tinuous data and d a discretized version thereof.Let Λ define the discretization, that is, the setof interval boundary points for all variables; aconfiguration λ will thus “bin” the original datadc yielding d. Moreover, assume that we haveP (dc|d,Λ); this is the generative model for thecontinuous data given some discretized versionthereof as defined by Λ. Note that this modelis unrelated to the BN; more on this distribu-tion shortly. It follows that the likelihood forobserving dc for a given discretization, networkstructure and parameters (the BN) can be writ-ten as

P (dc|DAG,Θ,Λ) = P (d|DAG,Θ,Λ) P (dc|d,Λ).

Embedded in a Bayesian context, we are nowseeking the MAP of the posterior

P (DAG,Θ,Λ|dc)∝ P (dc|DAG,Θ,Λ) P (DAG,Θ,Λ)

= P (d|DAG,Θ,Λ)︸ ︷︷ ︸1

P (dc|d,Λ)︸ ︷︷ ︸2

× (2)

P (Θ|DAG,Λ)︸ ︷︷ ︸3

P (Λ|DAG)︸ ︷︷ ︸4

P (DAG)︸ ︷︷ ︸5

.

The product of the terms 1, 3 and 5 is equiva-lent to the (joint) BN posterior as introduced inSection 3 (terms 1 and 3 now of course dependon the discretization). Let term 4 be uniform onthe space of all possible discretizations, allowingus to ignore this factor in the MAP estimation.

We define P (Xci |Xi,Λi) according to the fol-

lowing considerations: the discrete Xi is associ-ated with several interval boundaries, such thateach state xi has a lower λxi and upper bound-ary λxi . In between this interval Xc

i is dis-tributed uniformly, outside it is zero; xi thus“picks” the uniform interval in which Xc

i canlie. Effectively we arrive at term 2

P (dc|d,Λ) =∏i

∏xi

(1

λxi − λxi

)n(xi),

which leads to a preference of small intervals,while term 1 and 3 counteract the formation ofa high number of intervals.

Discretization and BN learning are nested it-eratively: learn a BN for a given discretization,followed by learning a new discretization, andso on. For a given discretization maximizing(2) with respect to the BN-pair (DAG, Θ) isequivalent to maximizing the MAP BN scorealone (which is the same as learning the DAGvia S(DAG|d, λ) also implying the BN parame-ter estimates) because term 2 is independent ofthe BN. To find the interval boundaries for thediscretization, the variables are discretized iter-atively until (2) stops improving, or until a pre-defined number of iterations has been reached.At each step we select a variable and employ abinary search for the “best” discretization fix-ating the intervals for the other variables.

4.1 A Single Continuous Target

The “optimal” discretization will not necessar-ily result in the required resolution for a par-ticular target variable of interest. In our caserloss is of primary interest, and the discretiza-tion described in the last section leads to a dis-cretization into 5 intervals. To achieve a finerresolution we treat, once we learned the BN,the target variable Xi as discrete with a verylarge number of states, defined by splitting therange of Xc

i into nXi intervals, e.g., nXi = 512.For simplicity we use equidistant intervals ofwidth ∆Xi . The states of Xi are the midpointsof the corresponding intervals. Because of thelarge number of states, the estimator (1) forθXi|XPa(i)

, or θXj |XPa(j)when Xi ∈ XPa(j), is

based on very few observations leading to weakestimates.

To avoid this problem, we use a Gaussian ker-nel density estimator to adopt the parameter es-timation in the following manner. For θXi|XPa(i)

we set

θxi|xPa(i)= ∆XiPXc

i |xPa(i)(xi),

where PXci |xPa(i)

(xi) =

1

n(xPa(i))√

2π h

∑k|x(k)

Pa(i)=xPa(i)

exp

(−(xc

(k)

i − xi)2

2h2

)

is the Gaussian kernel density estimator, with abandwidth, h, according to Silverman’s “rule ofthumb” (Silverman, 1986), over all observationsof Xc

i , for which XPa(i) = xPa(i).

For Xi ∈ XPa(j) we use Bayes theorem torewrite

p(Xj |XPa(j)) =p(Xj ,XPa(j))∑xj p(xj ,XPa(j))

=p(Xi|Xj ,XPa(j)−Xi

) p(Xj ,XPa(j)−Xi)∑

xj p(Xi|xj ,XPa(j)−Xi) p(xj ,XPa(j)−Xi

)

=p(Xi|Xj ,XPa(j)−Xi

) p(Xj |XPa(j)−Xi)∑

xj p(Xi|xj ,XPa(j)−Xi) p(xj |XPa(j)−Xi

),

and we set

θxj |xPa(j)=

∆XiPXci |xj ,xPa(j)−Xi

(xi) θxj |xPa(j)−Xi∑xj ∆XiPXc

i |xj ,xPa(j)−Xi(xi) θxj |xPa(j)−Xi

.

Because of the large number of states for thetarget variable, inference can become time andspace consuming. For relatively small/sparsenetworks this is not a big issue per se and in ourparticular case it has not posed any significantproblem.

Mixtures of truncated Exponentials (MTE)are an alternative to approximate continu-ous distributions (Moral et al., 2001). More-over using MTEs efficient inference is possible(Langseth et al., 2009). Methods for their con-struction are given by e.g. Rumı et al. (2006)and Langseth et al. (2010). However, finding

an optimal MTE-representation for a condi-tional/multivariate distribution from data is notrivial task. Moreover, learning both networkstructure and MTEs simultaneously from datais even more challenging. We leave this task forfuture work.

5 Results

We apply the BN-learning and discretizationmethods which are described in the last sectionsto the data set of Section 2. Continuous andordinal variables are treated in the same man-ner and both discretized. Thus, the numbersof states is reduced for continuous as well as fordiscrete variables. Only the number of states forthe nominal variables (ws, kh, bt, own) remainsas given.

For rloss the majority of the observations isgathered close to the lower domain boundary;taking the logarithm of rloss results in moreequal spread over the domain. To avoid an in-finite domain range, the lower boundary, whichcorresponds to buildings with no damage, wasset to log(5.5 ·10−6), where 5.5 ·10−5 is the min-imal observed loss ratio of damaged buildings.

There are missing values in the data, likelymissing (completely) at random, M(C)AR. Forconvenience, we for now simply replace them bysampling from the observed values of the cor-responding variable. Principled iterative meth-ods like Expectation Maximization (EM) are in-tractable for our purpose, since learning bothdiscretization and the BN means collecting suf-ficient statistics via inference (in the E-stepof EM) disproportionally often. In the futurethe Markov Blanket Predictor (Riggelsen, 2006)will be employed, which is a fast one-pass ap-proximation to EM, by restricting the attentionto predictors of the Markov Blanket of a vari-able with a missing observation.

Figure 1 (left) shows the BN learned for allvariables listed in Table 1, starting from aninitial “Tree Augmented Naive Bayes” networkthat was obtained using the method describedin (Vogel et al., 2012) with rloss as class vari-able. Some of the variables (wte and fe) have

−12 −10 −8 −6 −4 −2 0

0.0

0.1

0.2

0.3

0.4

cond. probability of rloss

log rloss

good precaution and warning

bad precaution and warning

Figure 1: left: Bayesian network learned over the discretized data; Numbers in the nodes give thenumber of discrete states learned.right: Conditional Probabilities of rloss for specific flood events using a coarse automaticallylearned discretization (shaded histograms) and interval refinement according to Section 4.1 (con-tinuous lines).

zero in/out-degree as the data apparently didnot support any interactions with other vari-ables. Variables that belong to the same sub-domain, are in most cases linked via a shortpath. Even though all domains are included inthe network, there is a clear distinction of thedomains visible. Building characteristics andsocio-economic factors have only indirect im-pact on the relative building loss, while floodparameters and precautions are closely relatedto the target variable. This gives us an ideaabout the importance of the variables for thecalculation of the relative building loss.

After the network was learned, rloss is se-lected as target variable and the number ofstates redefined viz. Section 4.1. Thus, weget an almost continuous approximation of theconditional probability function of rloss. Fig-ure 1 (right) illustrates the effect of the in-terval refinement. It shows the conditionaldistribution of rloss for the fine discretiza-tion in contrast to the coarse one for a floodevent with water depth between 9 cm and100 cm, a return period between 1 and 99years and different precaution and warning lev-els (good: 1≤wq≤2, 13≤pre≤38, 1≤epre≤5;bad: 3≤wq≤6, 0≤pre≤2, epre=6).

We compare the performance of the BN interms of the rloss-prediction to flood damage

assessment approaches currently used in Ger-many, namely to the stage-damage-function ap-proach and to FLEMOps+r (Elmer et al., 2010).For the stage-damage function approach, a rootfunction is fitted to the damage data of certainobject classes using least squares, i.e., the rel-ative damage is a function of the water depthonly. FLEMOps+r has been developed usingthe same data set and it has been shown toprovide superior results compared to other ap-proaches currently used in Germany. FLE-MOps+r calculates the building loss ratio forprivate households using five classes of inunda-tion depth, three intervals of flood frequency,three individual building types, two classes ofbuilding quality, three classes of contaminationand three classes of private precaution. Inessence, the data set is stratified into 27 sub-samples and the average loss ratio is used asdamage estimator (Elmer et al., 2010).

Additionally we compare the learned BN tothe Naive Bayes (NB) and Tree AugmentedNaive Bayes (TAN) learned from the same dataset. These models are set up as restricted BNswith one single target variable in mind. We re-fer to (Vogel et al., 2012) for a correspondingdescription of an automatic discretization andinterval refinement according to Section 4.1. Ofcourse the independence restrictions imposed by

sd−f FL BN NB TAN

0.08

0.12

0.16

RMSE

sd−f FL BN NB TAN

0.4

0.5

0.6

0.7

0.8

correlation coefficiant

Figure 2: Comparison of flood damage esti-mation models (sd-f: stage damage function;FL: FLEMOps+r−model developed from samedata set; BN: Bayesian Network; NB: NaiveBayes; TAN: Tree Augmented Naive Bayes).

these models do not (unlikely) obey “reality”and can not be used to gain insight into the“workings” of the underlying system. However,in terms of predictions for a single target, theyhave shown to often outperform BNs.1

To be able to compare to FLEMOps+r, wefollow a evaluation policy commonly used inthe field of hydrology: 100 bootstrap samples,each with 100 households, are drawn from thedata set. We complete the results of the stage-damage-function and the FLEMOps+r modelwith the rloss-predictions we get from theBN, NB and TAN by using the expectation ofthe conditional rloss-distribution as predictedvalue. The predictions are quantified by theroot mean squared error (RMSE) and the Pear-son correlation coefficient.

It is important to stress that no separate test-sets are used: the bootstrap policy describeduses parts of the training data for performanceevaluation. This is not legitimate per se, andmay influence the results considerably (opti-mistically). However, since the number of freeparameters in the stage-damage function andthe FLEMOps+r model are relatively small andthe BN MAP criterion accounts for model com-plexity (it regularizes) these models will notover-fit and consequently performance testingon the training data will not yield overly opti-mistic results. Moreover, the BN MAP score is

1In fact, discriminative models are even more likelyto perform well, e.g., logistic regression.

not a fitness measure of predictive performanceof any target variable in particular, but rather,provides the predictive performance “overall”for all variables jointly. For the NB and TANnetworks the number of free parameters is quitelarge and over-fitting might be a problem mean-ing that they on separate test-set may performless well.

Figure 2 shows the performance measures forthe 100 bootstrap samples in boxplots. It in-dicates that in terms of predicting rloss theBN performs well compared to the FLEMOps+r(the “best” method currently in use). TheNaive Bayes and especially the Tree AugmentedNaive Bayes show an improvement in the rlossprediction. However, it is important to notethat the BN model in fact provides us with thejoint distribution (the “correct” (in)dependencerelationships) and it is therefore somewhat un-fair to compare directly with approaches tryingto improve upon the predictive performance ofa single target variable only.

6 Conclusion

A BN has been learned from real-life data de-scribing flood related observations on 29 vari-ables, the majority continuous and some dis-crete. In general continuous variables pose achallenge, and often discretization is performedas a pre-processing step prior to BN model se-lection. We have extended the BN MAP modelselection metric to score not only BNs but si-multaneously take the proper discretization intoaccount, providing an entirely data-driven ap-proach to learn from a Bayesian maximum apos-teriori (MAP) perspective. From a data miningpoint of view, the BN indeed does reveal andconfirm non-trivial interactions. The learnednetwork captures the (in)dependencies reveal-ing connectivity between flood loss and warn-ing, emergency measures and socio-economicfactors, which are widely neglected in flood riskassessment. Additionally, from a predictionpoint of view where the performance of a par-ticular target is of interest, kernel estimationimproves upon the BN “multivariate” view byincreasing the degree of resolution (number of

states) required for proper decision support (of-ten derived/dependent on a single target vari-able). Compared to existing primarily deter-ministic flood damage estimation procedures,the BN shows a comparable performance withthe added benefit of capturing and reasoningunder uncertainty.

Acknowledgements

This work is supported by the Potsdam Research Cluster

for Georisk Analysis, Environmental Change and Sus-

tainability, PROGRESS, a joint project of university and

external organizations in the region of Potsdam-Berlin

(Germany).

References

Lilian Blaser, Matthias Ohrnberger, CarstenRiggelsen, Andrey Babeyko, and FrankScherbaum. 2011. Bayesian networks fortsunami early warning. Geophysical JournalInternational, 185(3):1431–1443.

Robert Castelo and Tonas Kocka. 2003. Oninclusion-driven learning of Bayesian networks.The Journal of Machine Learning Research,4:527–574.

Florian Elmer, Annegret H. Thieken, Ina Pech, andHeidi Kreibich. 2010. Influence of Flood Fre-quency on Residential Building Losses. Natu-ral Hazards and Earth System Sciences, 10:2145–2159.

Nir Friedman and Moises Goldszmidt. 1996. Dis-cretizing Continuous Attributes While Learn-ing Bayesian Networks. In Proceedings of theThirteenth International Conference on MachineLearning, pages 157–165.

Nicolas M. Kuehn, Carsten Riggelsen, and FrankScherbaum. 2011. Modeling the Joint Probabilityof Earthquake, Site, and Ground-Motion Param-eters Using Bayesian Networks. Bulletin of theSeismological Society of America, 101(1):235–249.

Helge Langseth, Thomas D. Nielsen, Rafael Rumı,and Antonio Salmeron. 2009. Inference in hy-brid Bayesian networks. Reliability Engineering& System Safety, 94(10):1499–1509.

Helge Langseth, Thomas D Nielsen, Rafael Rum,and Antonio Salmeron. 2010. Parameter estima-tion and model selection for mixtures of truncatedexponentials. International Journal of Approxi-mate Reasoning, 51(5):485–498.

Bruno Merz, Heidi Kreibich, Reimund Schwarze,and Annegret H. Thieken. 2010. Review article”Assessment of economic flood damage”. NaturalHazards and Earth System Science, 10(8):1697–1724.

Stefano Monti and Gregory F. Cooper. 1998. Amultivariate discretization method for learningBayesian networks from mixed data. In Proceed-ings of the Fourteenth conference on Uncertaintyin artificial intelligence UAI’98, pages 404–413.

Serafın Moral, Rafael Rumı, and Antonio Salmeron.2001. Mixtures of Truncated Exponentials in Hy-brid Bayesian Networks. In Salem Benferhat andPhilippe Besnard, editors, Symbolic and Quanti-tative Approaches to Reasoning with Uncertainty,pages 156–167.

Carsten Riggelsen, Matthias Ohrnberger, and FrankScherbaum. 2007. Dynamic bayesian networksfor real-time classification of seismic signals. InProceedings of the 11th European conference onPrinciples and Practice of Knowledge Discoveryin Databases, pages 565–572.

Carsten Riggelsen. 2006. Learning BayesianNetworks from Incomplete Data: An EfficientMethod for Generating Approximate PredictiveDistributions. In SIAM International conf. ondata mining, pages 130–140.

Carsten Riggelsen. 2008. Learning Bayesian Net-works: A MAP Criterion for Joint Selection ofModel Structure and Parameter. In ICDM, 2008Eighth IEEE International Conference on DataMining, pages 522–529.

Rafael Rumı, Antonio Salmeron, and Serafın Moral.2006. Estimating mixtures of truncated ex-ponentials in hybrid bayesian networks. Test,15(2):397–421.

Bernard W. Silverman. 1986. Density Estimationfor Statistics and Data Analysis, volume 26.

Annegret H. Thieken, Meike Muller, Heidi Kreibich,and Bruno Merz. 2005. Flood damage and in-fluencing factors: New insights from the August2002 flood in Germany. Water resources research,41.

Kristin Vogel, Carsten Riggelsen, Nicolas Kuehn,and Frank Scherbaum. 2012. Graphical Modelsas Surrogates for Complex Ground Motion Mod-els. Proceedings of the 11th International Confer-ence on Artificial Intelligence and Soft Comput-ing.