# A Robust and Transferable Model for the Prediction of Flood Losses on Household Contents

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## Abstract

**:**

## 1. Introduction

## 2. Material and Methods

#### 2.1. Data

#### 2.1.1. Quality Check

#### 2.1.2. Data Distribution

#### 2.2. Regression Model

#### 2.2.1. Data Transformation and Fitting

#### 2.2.2. Cross-Validation

#### 2.2.3. Assessment of Transferability

## 3. Results

#### 3.1. On the Role of Household Contents

#### 3.2. Model Fitting

#### 3.2.1. Data Transformation

#### 3.2.2. Regression Model

#### 3.3. Cross-Validation

#### 3.4. Transferability

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

BP | Breusch-Pagan (test) |

CI | Confidence Interval |

DoL | Degree of Loss |

FOEN | Federal Office of Environment |

FOWG | Federal Office of Water and Geology |

PSL | Pound Sterling Live |

PTBS | Pseudo-Transform-Both-Sides |

PTBS.seplam | Pseudo-Transform-Both-Sides with separate transformation parameters $\lambda $ for x and y |

SW | Shapiro-Wilks (test) |

TBS | Transform-Both-Sides |

## Appendix A

#### Appendix A.1

**Figure A1.**Diagnostic plots for the residuals (transformed degrees of loss) shown by Figure 3a. (

**Top left**) Normal-Q-Q-plot, points along diagonal line don’t reject normality (SW: normality with a p-value = 0.385). All other figures show the fitted values on X-axis against the standardised residuals on the Y-axis as normal (

**top right**), absolute (

**bottom left**) and the square root (

**bottom right**) of the absolute values. Blue borders indicate high leverage points, red filled circles indicate high values for Cook’s distance and the orange dashed lines indicate the borders to the definition of large residuals. Heteroscedasticity is not evident. One outlier (Bonferroni outlier test) is not shown in the plot.

#### Appendix A.2

**Figure A2.**Leverage (X-axis) vs Cook’s distance (

**left**) and standardised residuals (

**right**) for the relative loss model on the vertical axis. Blue borders indicate high leverage points, red filled circles indicate high values for Cook’s distance and the orange dashed lines indicate the borders to the definition of large residuals. One outlier (Bonferroni outlier test) is not shown in the plot.

## Appendix B

#### Appendix B.1

**Figure A3.**Diagnostic plots for the regression showed in Figure 3c. (

**Top left**) Normal-Q-Q-plot, points along diagonal line don’t reject normality (SW: normality with a p-value = 0.245). All other figures show the fitted values on X-axis against the standardised residuals on the Y-axis as normal (

**top right**), absolute (

**bottom left**) and the square root (

**bottom right**) of the absolute numbers. Blue borders indicate high leverage points, red filled circles indicate high values for Cook’s distance and the orange dashed lines indicate the borders to the definition of large residuals. Heteroscedasticity is not evident. One outlier (Bonferroni outlier test) is not shown in the plot.

#### Appendix B.2

**Figure A4.**Leverage (X-axis) vs Cook’s distance (

**left**) and standardised residuals (

**right**) of the relative loss model on the vertical axis. Blue borders indicate high leverage points, red filled circles indicate high values for Cook’s distance and the orange dashed lines indicate the borders to the definition of large residuals. The outlier found for the relative loss model (Bonferroni outlier test) is not shown in the plot and was not used during the model fitting procedure.

## Appendix C

**Figure A5.**Dependence of single errors on ranking. Absolute errors show high variability in higher ranks of target loss (original scale), whereas relative errors are more variable in lower ranks.

## Appendix D

**Figure A6.**Non-random cross-validation. K-fold = 2, (

**A**): Obwalden + Ticino vs. Schwyz + Valais + Uri; (

**B**): Obwalden + Schwyz vs. Ticino + Valais + Uri. K-fold = 5: One group per canton. K-Fold = 10/20: Split Obwalden, Ticino, Schwyz and Uri into multiple groups such that all groups have approximately the same size.

#### Appendix D.1

Relative Loss Model | Monetary Loss Model | |
---|---|---|

Spearman’s $\rho $ | 0.746 | 0.720 |

Kendall’s $\tau $ | 0.556 | 0.527 |

$\widehat{\lambda}$ Maximum Likelihood Estimate | 0.205 | 0.131 |

$\lambda $ CI * | (0.144, 0.265) | (0.068, 0.193) |

$\widehat{\sigma}$ | 0.495 | 2.745 |

${\widehat{\beta}}_{0}$ | −0.098 | 3.798 |

${\beta}_{0}$ CI * | (−0.255, 0.060) | (2.179, 5.416) |

${\widehat{\beta}}_{1}$ | 0.817 | 0.618 |

${\beta}_{1}$ CI * | (0.750, 0.884) | (0.560, 0.676) |

adjusted R${}^{2}$ | 0.668 | 0.618 |

Shapiro-Wilks p-value | 0.385 | 0.245 |

Breusch-Pagan p-value | 0.742 | 0.221 |

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**Figure 1.**Canton-wise distribution of records in the data sets used for the analyses. (

**Left**): Degree of loss [-], (

**right**): monetary loss [CHF]; both on log-scale. Sample size is given by “n:” and illustrated by the width of the box plots. Due to the low number of claim records, data recorded in Geneva and Appenzell Inner-Rhodes are not presented.

**Figure 2.**(

**a**)

**Left**: Share of the household content loss (${L}_{hc}$) on the total building loss (${L}_{tot}$ = structure loss + content loss).

**Right**: Share of the insurance sum of household contents (${V}_{hc}$) on the total insurance sum of a building (${V}_{tot}$). (

**b**) Ratio of the degree of loss on household contents ($Do{L}_{hc}$) to the degree of loss on structure ($Do{L}_{bs}$) for the same building. The red diamond symbols indicate the corresponding mean values.

**Figure 3.**Loss models based on degrees of loss (

**a**,

**b**) and monetary loss in CHF (

**c**,

**d**). In plots (

**a**,

**c**), the models based on transformed input values are presented, whereas in (

**b**,

**d**) the model results are shown on the original scale after back-transformation. CI: Confidence Interval; IQR: Inter-Quartile Range; PI: Prediction Interval. Note that the mean in plots (

**a**,

**c**) coincides with the median. Cross symbol: Outlier excluded before model fitting.

**Figure 4.**Cross-validation results for the model combinations based on leave-one-out cross-validation. ∘: Aggregated prediction error with median prediction; +: Aggregated prediction error with mean prediction. The blue line indicates the standard deviation of the individual prediction errors; (

**i**) Bias [CHF]; (

**ii**) Relative bias [%]; (

**iii**) Mean absolute error [CHF]; (

**iv**) Relative absolute error [CHF].

**Table 1.**The role of household contents in five Swiss Cantons. OW: Obwalden, SZ: Schwyz, TI: Ticino, UR: Uri, VS: Valais. * Share of the summed content loss on the summed total (structure + content) loss per canton. ** Means and medians of observed shares of content loss on total loss. *** Means and medians of the observed ratios of degrees of loss of contents to degrees of loss of structure

OW | SZ | TI | UR | VS | |
---|---|---|---|---|---|

Share of content loss on total building loss * | 0.22 | 0.23 | 0.32 | 0.21 | 0.26 |

Mean/median loss fraction ** | 0.27/0.22 | 0.31/0.28 | 0.33/0.28 | 0.25/0.24 | 0.28/0.24 |

Mean/median DoL rati o *** | 2.8/1.67 | 3.81/1.94 | 6.22/2.62 | 2.69/1.81 | 2.32/1.49 |

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## Share and Cite

**MDPI and ACS Style**

Mosimann, M.; Frossard, L.; Keiler, M.; Weingartner, R.; Zischg, A.P.
A Robust and Transferable Model for the Prediction of Flood Losses on Household Contents. *Water* **2018**, *10*, 1596.
https://doi.org/10.3390/w10111596

**AMA Style**

Mosimann M, Frossard L, Keiler M, Weingartner R, Zischg AP.
A Robust and Transferable Model for the Prediction of Flood Losses on Household Contents. *Water*. 2018; 10(11):1596.
https://doi.org/10.3390/w10111596

**Chicago/Turabian Style**

Mosimann, Markus, Linda Frossard, Margreth Keiler, Rolf Weingartner, and Andreas Paul Zischg.
2018. "A Robust and Transferable Model for the Prediction of Flood Losses on Household Contents" *Water* 10, no. 11: 1596.
https://doi.org/10.3390/w10111596