Probabilistic Flood Loss Models for CompaniesLukas Schoppa1,2*, Tobias Sieg1,2, Kristin Vogel2, Gert Zöller3, and Heidi Kreibich1

6 May 2020*lukas.schoppa@gfz-potsdam.de

Picture: DLR

1 Section Hydrology, GFZ German Research Centre for Geosciences, Potsdam, Germany 2 Institute of Environmental Science and Geography, University of Potsdam, Potsdam, Germany 3 Institute of Mathematics, University of Potsdam, Potsdam, Germany

• Commonly involves stage-damage functions

• Most stage-damage functions are univariable models: loss depends only on water depth

• Few existing flood loss models account for uncertainty in loss estimates

Status Quo in Flood Loss Modeling

06.05.2020 Background – Methods – Results – Conclusions 2

Water Depth0

1

Rel

ativ

e Lo

ss

• Towards multivariable and probabilistic models; e.g.:• Multivariate generalized regression

• Rule-based models

• Decision trees

• Bayesian networks

• Despite significant contributions of companies to total flood losses, development focused on private households

Multivariable, probabilistic flood loss models for companies are lacking

New Approaches to Flood Loss Modeling

06.05.2020 Background – Methods – Results – Conclusions 3

• Development and validation of three multivariable, probabilistic flood loss models for companies for the assets• Building (BUI)

• Equipment (EQU)

• Goods and stock (GNS)

• Comparing predictive performance of Bayesian networks, Bayesian regression and random forest

1) against an established benchmark model: probabilistic square-root stage damage function (SDF)

2) against each other

Study objectives

06.05.2020 Background – Methods – Results – Conclusions 4

• Company loss data (n=1306) stems from four telephone surveys after major floods in Germany in the period 2002 – 2013

• Collected data cover flood intensity, company characteristics, warning and emergency measures, flood experience, and private precaution

Survey Data

06.05.2020 Background – Methods – Results – Conclusions 5

Variable Abbreviation

Predictor Variables

Water depth wd

Inundation duration dur

Return period rp

Business sector sec

Company size size

Spatial situation spat

Flood experience exp

Precaution ratio pre

Response Variables

Relative loss building rloss

Relative loss equipment rloss

Relative loss goods/stock rloss

Predictor (n=8) and response (n=1) variables

Building (n=545) Equipment (n=829) Goods & Stock (n=902)

0.00

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0.50

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Variable

wd

dur

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size

spat

exp

pre

rp

rloss

Model variables

06.05.2020 Background – Methods – Results – Conclusions 6

Kernel density estimates of model data for the three company assets building, equipment, and goods and stock. For comparability, all variables are scaled from zero to one in this plot. The lines in the violin plots indicate the quartiles while the dot represents the mean. The predictor set is comprised by nominal, ordinal and continuous variables. The response variable, relative loss (rloss), is defined on the interval [0, 1] and contains significant shares of zeros and ones, which correspond to companies with no and total loss.

Multivariable Models

06.05.2020 Background – Methods – Results – Conclusions 7

+

BetaBernoulli

Bayesian Networks (BN) Bayesian Regression (BR) Random Forest (RF)

• Graphical probabilistic model

• Network structure learned from data

• Zero-and-one inflated beta distribution

• Inflation accounts for zero and one loss cases

• Quantile regression forest• Conditional inference tree

algorithm

The multivariable models and the univariable stage-damage function all return probabilistic predictions of relative flood loss in the form of samples. We developed individual models for losses to the assets building, equipment, and goods and stock.

• Water depth and precaution are dominant predictor variables for all assets and models

• The predictor importance measures of the BN, BR, and RF are plausible and agree with previous findings

• The candidate models consistently identify the same predictors as most relevant

• Damage processes differ across assets: separate models are justified

Comparing Model Fits

06.05.2020 Background – Methods – Results – Conclusions 8

pre

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BUI EQU GNS

-0.5 0.0 0.5 -0.5 0.0 0.5 -0.5 0.0 0.5

sec4

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spat2

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Parameter estimate

BUI EQU GNS

0.0000 0.0025 0.0050 0.0075 0.00 0.01 0.02 0.03 0.00 0.01 0.02 0.03 0.04 0.05

sec

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Mean decrease prediction accuracy

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BUI EQU GNS

-0.5 0.0 0.5 -0.5 0.0 0.5 -0.5 0.0 0.5

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Parameter estimate

BUI EQU GNS

0.0000 0.0025 0.0050 0.0075 0.00 0.01 0.02 0.03 0.00 0.01 0.02 0.03 0.04 0.05

sec

spat

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Mean decrease prediction accuracy

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BUI EQU GNS

-0.5 0.0 0.5 -0.5 0.0 0.5 -0.5 0.0 0.5

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Parameter estimate

BUI EQU GNS

0.0000 0.0025 0.0050 0.0075 0.00 0.01 0.02 0.03 0.00 0.01 0.02 0.03 0.04 0.05

sec

spat

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Mean decrease prediction accuracy

Building Equipment

Goods and Stock

Bayesian network structures, which were learned from the survey data, for the three assets. The BN structures show that the damage processes differ between building loss and losses to equipment, and goods and stock.

Probabilistic Predictions

06.05.2020 Background – Methods – Results – Conclusions 9

Obs. Loss: 0.4 | ID: 330 Obs. Loss: 1 | ID: 104 Obs. Loss: 1 | ID: 136

Obs. Loss: 0.04 | ID: 392 Obs. Loss: 0.21 | ID: 165 Obs. Loss: 0.21 | ID: 450

Obs. Loss: 0 | ID: 183 Obs. Loss: 0 | ID: 261 Obs. Loss: 0 | ID: 37

0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00

Relative Loss Building [-]

Model BN BR RF SDF

• Prediction accuracy and sharpness appear to be higher for minor losses than for severe losses

• Stage-damage function (SDF) predictions are often bimodal (high density at 0/1)

• Predictive densities of the multivariable models (BN, BR, RF) are more flexible; e.g. de-/inflation of predictive densities at 0/1

Examples of predictive densities from the four models (color-coded) for the building loss of nine randomly selected companies (identified by ID). The observed loss is indicated by the black lines.

Predictive Performance

06.05.2020 Background – Methods – Results – Conclusions 10

MAE Mean CRPS MBE

BUI EQU GNS BUI EQU GNS BUI EQU GNS

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Model BN BR RF SDF

MAE Mean CRPS MBE

BUI EQU GNS BUI EQU GNS BUI EQU GNS

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Model BN BR RF SDF

Performance metrics mean average error (MAE), mean continuous ranked probability score (CRPS), and mean bias error (MBE) for the four models (color-coded) and assets (x-axis). Each boxplot summarizes 100 repetitions of a 10-fold cross-validation with varying data partitioning.

* CRPS: generalization of the MAE; optimum at zero

* • Multivariable models outperform stage-damage functions for all assets/performance metrics

• Performance differences among multivariable models are small

Predictive Performance

06.05.2020 Background – Methods – Results – Conclusions 11

mean CRPS:

0.099

mean CRPS:

0.161

mean CRPS:

0.162

mean CRPS:

0.103

mean CRPS:

0.165

mean CRPS:

0.158

mean CRPS:

0.099

mean CRPS:

0.159

mean CRPS:

0.154

mean CRPS:

0.125

mean CRPS:

0.204

mean CRPS:

0.208

BN BR RF SDF

BU

IE

QU

GN

S

0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00

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Observed Relative Loss [-]

CR

PS

[-]

mean CRPS:

0.099

mean CRPS:

0.161

mean CRPS:

0.162

mean CRPS:

0.103

mean CRPS:

0.165

mean CRPS:

0.158

mean CRPS:

0.099

mean CRPS:

0.159

mean CRPS:

0.154

mean CRPS:

0.125

mean CRPS:

0.204

mean CRPS:

0.208

BN BR RF SDF

BU

IE

QU

GN

S

0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00

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Observed Relative Loss [-]

CR

PS

[-]

Scatter plots of observed relative loss versus cross-validated continuous ranked probability scores (CRPS) for all models (columns, color-coded) and building loss (BUI). Each symbol represents the prediction error incurred by the respective model for one company. The black, step-wise lines show the average CRPS in different intervals of observed relative loss. The labels in the top-left corner of each panel contain the mean CRPS over all predictions of the respective model. We observed similar predictive errors for the assets equipment, and goods and stock.

• The scatter plots confirm that model errors are larger for severe losses

• The variation in the predictions and, hence, the errors is larger for multivariable models than for the stage-damage function

• Still, on average, the multivariable models perform better

Bias-variance tradeoff

Advantages of Multivariable Models

06.05.2020 Background – Methods – Results – Conclusions 12

• Water depth insufficiently discriminates between minor and major losses (see figure); additional predictors contain valuable information

• The structural complexity of the multivariable models allows for large flexibility in the predictive distributions, which is required to model the large shares of 0/1-cases in the data

BUI EQU GNS

0 250 500 750 1000 0 250 500 750 1000 0 250 500 750 1000

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Water Depth [cm]

Re

lative

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-]

Scatter plots of water depth versus relative loss for the assets building (BUI), equipment (EQU), and goods and stock (GNS). The relationship between water depth and relative loss is characterized by pronounced noise.

Conclusions

06.05.2020 Background – Methods – Results – Conclusions 13

• Bayesian networks, Bayesian regression, and random forest outperform the stage-damage function

• Performance differences among multivariable models are small; model choice depends on data availability and study task

• Large predictive errors for severe losses could be caused by imbalances in the data: extreme losses are less frequent than minor losses

• Wide predictive densities suggest that the uncertainty in the predictive distributions is generally high and, hence, requires quantification through probabilistic loss models (especially for large losses)