# Hazard Assessment under Multivariate Distributional Change-Points: Guidelines and a Flood Case Study

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## Abstract

**:**

## 1. Introduction

## 2. Materials

## 3. Methods

- changes of any of the marginal distributions ${F}_{i}$s, or
- changes of the copula $\mathbf{C}$, or
- both of the previous cases.

#### 3.1. Hazard Scenarios

**Definition**

**1.**

**Definition**

**2.**

#### 3.2. The Failure Probability Approach

## 4. Results and Discussion

- a change of the univariate distribution ${F}_{Q}$ (respectively, ${F}_{V}$), or
- a change of the copula ${\mathbf{C}}_{QV}$ associated with $(Q,V)$, or
- both the previous instances.

`npcp`[43].

## 5. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Time series of the available Q and V data—see text. The vertical dashed line indicates the possible Change-Point year (1971).

**Figure 2.**Observed pairs $(Q,V)$s (markers), and selected isolines of the fitted distribution ${\mathbf{F}}_{QV}$—see text: (

**Top**panel) all data; (

**Bottom-left**panel) the data before the Change-Point year; (

**Bottom-right**panel) the data after the Change-Point year.

**Figure 3.**Observed pairs $(Q,V)$s (full circles), “OR” Hazard Scenarios (the regions “above” the dashed lines), the design pairs indicated in Table 3 (respectively, empty square, circle, and diamond), and isolines of the fitted ${\mathbf{F}}_{QV}$ crossing the design pairs—see text: (

**Top**panel) all data; (

**Bottom-left**panel) the data before the Change-Point year; (

**Bottom-right**panel) the data after the Change-Point year.

**Figure 4.**Confidence bands (at a 90% level) for the Failure Probabilities ${p}_{T}^{\vee}$s associated with the design pairs $\tilde{\mathbf{x}}$s plotted in Figure 3—see text: (

**Top**panel) all data; (

**Bottom-left**panel) before the Change-Point year; (

**Bottom-right**panel) after the Change-Point year.

**Figure 5.**Observed pairs $(Q,V)$s (full circles), and mean “Most Likely” design pairs ${\delta}^{*}=({Q}^{*},{V}^{*})$s (markers) for different design Failure Probabilities—see text: (

**Top**panel) all data; (

**Bottom-left**panel) before the Change-Point year; (

**Bottom-right**panel) after the Change-Point year. Also shown are Monte Carlo Confidence Intervals at a 90% level.

**Figure 6.**The available $(Q,V)$ observations—see text: (

**Top**panel) the data; (

**Bottom**panel) the pseudo-observations, viz. the normalized ranks. The occurrences before and after the possible Change-Point year are indicated via different markers.

**Table 1.**Maximum Likelihood estimates of the parameters of the GEV distributions fitting the variables Q (in m${}^{3}$/s) and V (in 10${}^{6}$ m${}^{3}$), either considering all the data, or only those collected, respectively, before and after the Change-Point year (1971)—see text. Also shown are estimated standard errors and approximate Monte Carlo Goodness-of-Fit test p-Values (based on Kolmogorov–Smirnov statistics).

Variable | Shape | Scale | Position | p-Value |
---|---|---|---|---|

All data | ||||

Q | 0.37 | 36.21 | 59.36 | 77% |

s.e. | 0.11 | 5.04 | 5.71 | |

V | 0.61 | 1.52 | 1.72 | 91% |

s.e. | 0.13 | 0.25 | 0.24 | |

Before Change-Point | ||||

Q | 0.02 | 24.46 | 50.05 | 87% |

s.e. | 0.11 | 3.79 | 5.36 | |

V | 0.12 | 0.95 | 1.39 | 99% |

s.e. | 0.16 | 0.16 | 0.21 | |

After Change-Point | ||||

Q | 0.71 | 42.96 | 71.74 | 98% |

s.e. | 0.30 | 11.75 | 10.92 | |

V | 1.07 | 2.03 | 2.20 | 92% |

s.e. | 0.34 | 0.67 | 0.50 |

**Table 2.**Maximum Likelihood estimates of the survival-Clayton 2-copula parameter $\theta $ fitting the pairs $(Q,V)$s, either considering all the data, or only those collected, respectively, before and after the Change-Point year (1971)—see text. Also shown are estimated standard errors and approximate p-Values (via a Multiplier Method) of the Cramér–von Mises Goodness-of-Fit test for Copulas based on the copula empirical process [32].

All Data | Before Change-Point | After Change-Point | |
---|---|---|---|

$\theta $ | 4.33 | 1.53 | 11.69 |

s.e. | 1.37 | 0.72 | 5.31 |

p-Value | 9% | 47% | 44% |

**Table 3.**The design pairs $\tilde{\mathbf{x}}$s—see text: the units are m${}^{3}$/s for Q, and 10${}^{6}\xb7$m${}^{3}$ for V. Also shown are the corresponding estimates of the “OR” HS levels ${\alpha}^{\vee}$s.

Quantile (%) | Q | V | ${\mathit{\alpha}}^{\vee}$ | |
---|---|---|---|---|

${\tilde{\mathbf{x}}}_{1}$ | 90% | 197 | 14 | 0.0880 |

${\tilde{\mathbf{x}}}_{2}$ | 95% | 282 | 17 | 0.0453 |

${\tilde{\mathbf{x}}}_{3}$ | 99% | 439 | 31 | 0.0172 |

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## Share and Cite

**MDPI and ACS Style**

Salvadori, G.; Durante, F.; De Michele, C.; Bernardi, M.
Hazard Assessment under Multivariate Distributional Change-Points: Guidelines and a Flood Case Study. *Water* **2018**, *10*, 751.
https://doi.org/10.3390/w10060751

**AMA Style**

Salvadori G, Durante F, De Michele C, Bernardi M.
Hazard Assessment under Multivariate Distributional Change-Points: Guidelines and a Flood Case Study. *Water*. 2018; 10(6):751.
https://doi.org/10.3390/w10060751

**Chicago/Turabian Style**

Salvadori, Gianfausto, Fabrizio Durante, Carlo De Michele, and Mauro Bernardi.
2018. "Hazard Assessment under Multivariate Distributional Change-Points: Guidelines and a Flood Case Study" *Water* 10, no. 6: 751.
https://doi.org/10.3390/w10060751