Flood Damage and Inuencing Factors: A Bayesian Network Perspective

Transcription

1 Flood Damage and Inuencing Factors: A Bayesian Network Perspective Kristin Vogel 1, Carsten Riggelsen 1, Bruno Merz 2, Heidi Kreibich 2, Frank Scherbaum 1 1 University of Potsdam, 2 GFZ Potsdam

2 Problem Natural Hazards Flood water Earthquake Tsunami Landslide complex processes, not well understood many inuencing factors uncertainty in (potentially observable) variables and in modelling frameworks

3 Problem Bayesian Networks describe non-deterministic systems and processes, capturing (in-)dependencies between the variables of interest allow inclusion of domain knowledge can be updated for new observations allow reasoning under and propagation of uncertainty

4 Problem often faced problems discretization: many variables are continuous (functional form sometimes unknown) or discrete with high number of states; for distribution free learning the variables are discretized high resolution of target variable: main interest is often in the prediction of target variable; the discretization used for network learning is quite coarse

5 Problem Flood damage assessment - dataset 2002 and 2005/2006 oods in Elbe and Danube catchments (Germany) 28 variables describing ood event, aected buildings, precaution, warning, socio-economic factors target variable: relative building loss 1135 partly missing observations

6 Scoring metric BN MAP score for BN learning: searching for the pair of network structure (DAG) and parameter (Θ), that maximize the joint posterior P(DAG, Θ d) P(d DAG, Θ)P(DAG, Θ) P(d DAG, Θ)P(Θ DAG )P(DAG ) P(d DAG, Θ) P(Θ DAG ) P(DAG ) multinomial distribution for discrete variables dened to be a product Dirichlet distribution dened to be uniform over DAGs ignored

7 Scoring metric BN MAP score for BN learning: searching for the pair of network structure (DAG) and parameter (Θ), that maximize the joint posterior P(DAG, Θ d) P(d DAG, Θ)P(DAG, Θ) BN MAP score for BN learning and variable discretization: searching for the triple of network structure (DAG) and parameter (Θ) and discretization (Λ), that maximize the joint posterior P(DAG, Θ, Λ d c ) P(d c DAG, Θ, Λ)P(DAG, Θ, Λ)

8 P(DAG, Θ, Λ d c ) P(d c DAG, Θ, Λ)P(DAG, Θ, Λ) = P(d DAG, Θ, Λ)P(d c d, Λ) P(Θ DAG, Λ)P(Λ DAG )P(DAG )

9 P(DAG, Θ, Λ d c ) P(d c DAG, Θ, Λ)P(DAG, Θ, Λ) = P(d DAG, Θ, Λ)P(d c d, Λ) P(Θ DAG, Λ)P(Λ DAG )P(DAG ) Monti, Cooper; 1998

10 P(DAG, Θ, Λ d c ) P(d c DAG, Θ, Λ)P(DAG, Θ, Λ) P(d c d, Λ) = i = P(d DAG, Θ, Λ)P(d c d, Λ) xi ( independent on DAG and Θ P(Θ DAG, Λ)P(Λ DAG )P(DAG ) ) n(xi ) 1 λ x i λ xi

11 P(DAG, Θ, Λ d c ) P(d c DAG, Θ, Λ)P(DAG, Θ, Λ) as in original BN MAP score = P(d DAG, Θ, Λ)P(d c d, Λ) P(Θ DAG, Λ)P(Λ DAG )P(DAG )

12 P(DAG, Θ, Λ d c ) P(d c DAG, Θ, Λ)P(DAG, Θ, Λ) = P(d DAG, Θ, Λ)P(d c d, Λ) P(Θ DAG, Λ)P(Λ DAG )P(DAG ) P(Λ DAG ) dened to be uniform over Λs ignored

13 learned Bayesian Network

14 approximate continuous target variable distribution target variable, relative building loss, is discretized into 5 intervals resolution might be not sucient (e.g. for decision makers) aim to approximate the continuous conditional distribution functions rediscretization of target variable into many (e.g. 512) bins and adoption of parameter estimates conditional probability of building loss log loss

15 approximate continuous target variable distribution target variable, relative building loss, is discretized into 5 intervals resolution might be not sucient (e.g. for decision makers) aim to approximate the continuous conditional distribution functions rediscretization of target variable into many (e.g. 512) bins and adoption of parameter estimates conditional probability of building loss log loss

16 adaption of parameter estimates high number of states few observations per state usual maximum likelihood estimator leads to weak estimates alternative: Gaussian kernel density estimator considers observations of neighbouring states as well calculate kernel density for each midpoint of the 512 intervals probability of each state is estimated as density of midpoint width of interval

17 Mixtures of truncated exponential approximation with kernel density estimator requires large number of parameters; inference becomes time and space consuming alternative with less parameters: Mixtures of truncated exponentials allows simultaneous approximation of several continuous variable distributions

18 Results Comparison with currently used models stage-damage-function: root function tted to damage data of certain object classes; function of water depth only FLEMOps+r: developed from same dataset (Elmer et al.,2010), dividing data into subsamples according to inundation depth, ood frequency, building types, building quality, contamination, private precaution; calculating average loss for each class

19 Results compare predictions of target variable, building loss draw 100 bootstrap samples, each with 100 observation RMSE correlation coefficiant estimation of building loss is quantied by root mean squared error and Pearson correlation coecient sd f FL BN sd f FL BN The Bayesian Network performs comparable to FLEMOps+r best method currently in use), eventhough it is designed to give optimal prediction on the target variable, but on the joint distribution

20 Results Further work interpretation of the learned model structure include expert knowledge better handling of missing values use MTEs for approximation of continuous distributions