# Risk Level Assessment of Typhoon Hazard Based on Loss Utility

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area

**Definition**

**1.**

- i.
- ii.
- When α = 0, $\underset{\alpha \to 0}{\mathit{lim}}\mu \left(l\right)=-\underset{\alpha \to 0}{\mathit{lim}}\frac{ln\left(1-\alpha l\right)}{\alpha}=-\underset{\alpha \to 0}{\mathit{lim}}\frac{-l}{\left(1-\alpha l\right)}=l$.

**Definition**

**2.**

_{i}represents the probability of Typhoon i, l

_{ij}(j = 1,2, … m) represents the loss amount of item j caused by typhoon i (such as direct economic loss, casualties, farmland affected area, etc.); U

_{ij}represents the loss expectation of item j caused by the impact of typhoon i; then, according to the expected utility theory of von Neumann and Morgenstern, there are:

_{ij}can be used to measure the risk level of the typhoon disaster. The larger U

_{ij}is, the higher the risk level is; otherwise, the smaller the risk level is.

- (1)
- Gumbel Copula function:

- (2)
- Frank Copula function:

_{1}Copula, M6

_{2}Copula, and M6

_{3}Copula. The expressions of its function structure are as follows:

_{1}, θ

_{2}, θ

_{11}, θ

_{12}, θ

_{13}, θ

_{21}, θ

_{22}, and θ

_{23}are all parameters of the Copula function and can be estimated by maximum likelihood method.

_{2}Copula function is used in this paper to construct a three-dimensional joint probability distribution, and if these three variables obey the Gumbell distribution, Pearson distribution and Pearson distribution, respectively, then

_{1}) and Γ(α

_{2}) are gamma functions of α1 and α

_{2}, respectively; α

_{1}, β

_{1}, α

_{2}, β

_{2}are the shape scale parameters of the Pearson-III distribution, and their values are all greater than 0. a

_{01}and a

_{02}are positional unknown parameters of Pearson’s three-type distribution.

_{1}and L

_{2}. Let the vulnerability loss to a study area due to the combined effect of the contributing factors be L

_{j}(x,y,z), then the expected loss due to the typhoon in the study area Uj can be expressed as

_{j}(x,y,z) is the vulnerability function characterizing the risk-bearing body. It can be calculated by identifying the constituent sample from the hazard intensity-loss data recorded in the disaster. Because it is calculated based on direct economic losses or affected populations, it is the absolute loss generated by the occurrence of a typhoon in a given study area. If the value of U

_{Rj}represents the aversion utility of the absolute losses caused by the occurrence of a particular typhoon, then

_{Rj}, then we can have

- i.
- Since Equation (8) is calculated under the condition of relative loss, it is independent of the city scale. It means that the same value brings the same degree of aversion for different cities. l′ clearly has a range of [0,1].
- ii.
- The transformation of Equations (6) and (8) actually converts the random loss into a fixed loss. For instance, l′ = 30%, then it can be said that the typhoon risk faced by the city is equal to 30% of the total output loss of the city in the sense of relative loss aversion.

#### 2.2. Data Sources

_{1}) and GDP (Gross V

_{2}) of Guangdong Province for 2005–2016 are from the website of the National Bureau of Statistics (as shown in Figure 3b). Considering the availability of data and in order to ensure the accuracy of calculation, this paper selects seven typhoons landing within 100 km of Sal Sal Island from 2005 to 2016 (the typhoon names are marked on the horizontal coordinate in Figure 3a) as samples to calculate the typhoon loss expectation.

## 3. Engineering Calculation Example

#### 3.1. Model Calculation

#### 3.1.1. Probability of Typhoon Occurrence

_{2}Copula function computes the smallest AIC and RMSE values among the selected combinatorial functions. In view of this, it can be concluded that the M6

_{2}Copula function has the best-fitting effect on the three-dimensional sample sequence.

_{2}Copula function is the optimal choice for this study to establish the three-dimensional joint distribution of wave height, water increase and wind speed for calculating the probability of typhoon occurrence.

_{2}Copula function to construct the three-dimensional joint distribution of wave height, water increase and wind speed, as shown in Formula (4).

#### 3.1.2. Absolute Loss Aversion and Fixed Absolute Losses

_{1}, and R

_{2}indicate the values of loss expectation calculated based on direct economic loss and affected population indicators, respectively. U

_{R}

_{1}and U

_{R}

_{2}represent the aversion of expectation values of absolute loss calculated based on the two evaluation indicators of direct economic loss and affected population; l

_{1}and l

_{2}denote the fixed losses at an equal aversion utility, respectively.

#### 3.1.3. Ranking of Typhoon Risks

_{1}′ and the proportion of the affected population L

_{2}′, where L

_{1}′ = L

_{1}/GrossV

_{1}, and L

_{2}′ = L

_{2}/GrossV

_{2}. The relative loss aversion effect formula as well as the relative loss aversion conversion formula are then calculated and the results are presented in Table 5. UR1′ and UR2′ represent the aversion utility values of relative loss calculated based on the two evaluation indicators of direct economic loss and affected population, respectively. The l1′ and l2′ are their respective fixed relative losses corresponding to the same degree of aversion.

_{1}and L

_{2}caused by these two typhoons are similar, but in terms of typhoon occurrence probability, the occurrence probability of Chanthu is slightly bigger than that of Prapiroon (from the extreme value data, the wave height and wind speed data of Chanthu after the landing of Naozhou region are slightly higher than those of Prapiroon). Therefore, the loss expectation aversion model calculates that the relative loss aversion value and fixed relative loss of Chanthu are relatively high. The relative loss aversion values of these two typhoons are also much higher than those of the subsequent high-ranking typhoons, which may be related to the increased awareness of disaster prevention, the improved ability to avoid high-risk events, and the improved accuracy of typhoon warnings and forecasting systems.

_{2}′ is on the interval [0,0.6%]. Therefore, we can divide the l

_{1}′ and l

_{2}′ intervals into several sections to represent different risk levels, and establish a typhoon risk level classification method according to their values. As shown in Figure 8 and Figure 9:

_{1}′ in Table 5. Then, the fixed direct economic loss rate l

_{1}′ of typhoon disasters solved in the sense of relative loss aversion is divided into five segments in the range of [0,0.02%]: [0,0.004%,0.008%,0.012%,0.016%,0.02%]. According to the conversion of the degree of equal disgust, the above l

_{1}′ values (l

_{1}′ takes 0.004%, 0.008%, 0.012%, 0.016%, 0.02% ) are brought into the formula (8), that is, U (l

_{1}′) = μ (L

_{1}′) (j = 1 in formula 8, that is, each typhoon disaster is calculated separately), and each l

_{1}′ is worth a set of (F, L

_{1}′) values. The values of each group (F, L

_{1}′) are traced, and the curves in the graph are obtained. Each curve is an isoline of expected loss aversion. Finally, according to the expected loss aversion value of each typhoon disaster in different aversion intervals, the risk level is divided. Figure 9 is the same.

_{1}′ + L

_{2}′)/2. If (L

_{1}′ + L

_{2}′)/2 is not an integer, it is rounded off.

#### 3.2. Discussion

_{L}-L

_{P}type). Typhoon Kai-tak and Rumbia, on the contrary, belong to low loss and high probability (L

_{L}-H

_{P}) risk events. According to the classification results in Figure 10b, Hagupit, Prapiroon, Rammasun and Mujigae belong to the H

_{L}-L

_{P}type of risk events. Chanthu, Kai-tak and Rumbia are L

_{L}-H

_{P}risk events. According to the classification results of the two charts, typhoon disasters of H

_{L}-L

_{P}type are Hagupit and Prapiroon. The L

_{L}-H

_{P}typhoons are Kai-tak and Rumbia.

_{1}= P × L

_{1}, R′

_{2}= P × L

_{2}), and (b) and (e) are listed as expected loss rates considering utility. In addition, e

_{1}and e

_{2}represent the ratio of the difference between the loss expectation of considering utility and the loss expectation of classical quantitative algorithms (e

_{1}= (R

_{1}− P × L

_{1})/P × L

_{1}, e

_{2}= (R

_{2}− P × L

_{2})/P × L

_{2}). The subscript of the variable based on the disaster population rate is 1, and the subscript based on the direct economic loss rate is 2.

_{i}(I = 1,2) is slightly higher than R′

_{i}(I = 1,2), mainly because the classical loss expectation formula does not consider the risk attitude (i.e., the case of utility coefficient α = 0), and the loss utility function reflects the aversion and dissatisfaction of the disaster-stricken groups to typhoon disasters. Further comparative analysis found that this high situation is not linear, but has a certain relationship with the risk level and typhoon wind speed level. As can be seen from Figure 11, the interpolation ratio e

_{1}, which takes the affected population rate as an indicator, increases roughly with the increase in the typhoon’s comprehensive risk level. The interpolation ratio e

_{2}, which takes the direct economic loss rate as the index, basically increases with the increase in typhoon wind speed level. In addition, e

_{1}> e

_{2}indicates to a certain extent that the risk assessment model newly constructed in this paper can better reflect the disaster-affected people’s dissatisfaction with typhoon disaster in the risk assessment model by taking the disaster-affected population as the index.

_{L}-L

_{P}” and “L

_{L}-H

_{P}” typhoons are as follows. Taking the calculation results of the disaster population rate index in Table 7 as an example, the calculation results of Hagupit and Prapiroon of the “H

_{L}-L

_{P}” type are similar to those of Kai-tak and Rumbia of the “L

_{L}-H

_{P}” type, respectively (see Figure 12). The $\Delta {R}_{1}$ and $\Delta {R}_{1}^{\prime}$ between Hagupit of the “H

_{L}-L

_{P}” type and Kai-tak of the “L

_{L}-H

_{P}” type are both less than 0.05%, and $\Delta {R}_{2}$ and $\Delta {R}_{2}^{\prime}$ are both less than 0.01%. The $\Delta {R}_{1}$ and $\Delta {R}_{1}^{\prime}$ between Prapiroon of the “H

_{L}-L

_{P}” type and Rumbia of the “L

_{L}-H

_{P}” type are both less than 0.08% and $\Delta {R}_{2}$ and $\Delta {R}_{2}^{\prime}$ are both less than 0.02%.

_{L}-L

_{P}” and “L

_{L}-H

_{P}” in traditional calculation.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Topographic map of Naozhou Island (the scene in the figure represents the geographic elevation information).

**Figure 3.**Statistics of 22 Typhoon Disasters and Resident Population and GDP Data that Attacked Guangdong Province from 2005 to 2016.

**Figure 4.**Distribution test diagram of sample data. (

**a**) based on wave height samples; (

**b**) based on water increment samples; (

**c**) based on wind speed samples.

**Figure 5.**Comparison of fitting results of wave height-water increase-wind speed samples based on different Copula functions.

**Figure 7.**Comparison of loss expectations based on direct economic losses (

**a**); Comparison of loss expectations based on affected population (

**b**).

**Figure 8.**Typhoon risk rating based on direct economic loss rate with logarithmic aversion utility function.

**Figure 9.**Typhoon risk rating based on logarithmic aversion utility function for the proportion of affected population.

**Figure 10.**Risk zoning diagram (

**a**) The disaster population rate L

_{1}and typhoon probability P diagram; (

**b**) The chart of the disaster-affected population rate L

_{2}and the probability P of typhoon occurrence; “H

_{L}-L

_{P}” represents high loss-low probability risk region; “L

_{L}-H

_{P}” indicates the low-loss-high-probability risk area.

**Figure 11.**(

**a**) Relationship between typhoon risk level and e

_{1}; (

**b**) The relationship between typhoon wind speed level and e

_{2}.

Distributions | Wave Height (m) | Water Increment (m) | Wind Speed (m/s) |
---|---|---|---|

Gamma distribution | 0.11364 (0.9511) | 0.10621 (0.9405) | 0.17019 (0.4639) |

Gumbel distribution | 0.11006 (0.9225) | 0.097616 (0.9908) | 0.15869 (0.5548) |

Pearson-III distribution | 0.11716 (0.8825) | 0.093219 (0.9816) | 0.11475 (0.8970) |

Copula Function | Frank | Gumbel | M6_{1} | M6_{2} | M6_{3} |
---|---|---|---|---|---|

Estimate | 2.67 | 1.415 | (1.836, 1.272) | (1.211, 1.433) | (1.399, 1.382) |

AIC | −180.258 | −184.665 | −177.824 | −185.179 | −183.523 |

RMSE | 0.0261 | 0.0239 | 0.0274 | 0.0237 | 0.0245 |

Number | Name | Landing Time | Level | Typhoon Occurrence Probability F |
---|---|---|---|---|

0606 | Prapiroon | 8.1 | 12 | 0.0881 |

0814 | Hagupit | 9.24 | 15 | 0.0347 |

1003 | Chanthu | 7.22 | 12 | 0.1248 |

1213 | Kai-tak | 8.17 | 13 | 0.1116 |

1306 | Rumbia | 7.2 | 11 | 0.2710 |

1409 | Rammasun | 7.18 | 15 | 0.0249 |

1522 | Mujigae | 10.4 | 15 | 0.0152 |

(a) | (b) | (c) | (d) | (e) | (f) | ||
---|---|---|---|---|---|---|---|

Number | Name | R_{1} | R_{2} | U_{R}_{1} | U_{R}_{2} | l_{1} | l_{2} |

0606 | Prapiroon | 5.112 | 45.959 | 5.251 | 62.056 | 5.238 | 60.355 |

0814 | Hagupit | 2.693 | 26.993 | 2.791 | 46.434 | 2.788 | 45.477 |

1003 | Chanthu | 7.014 | 53.277 | 7.197 | 67.232 | 7.174 | 65.239 |

1213 | Kai-tak | 2.857 | 23.849 | 2.890 | 26.487 | 2.887 | 26.173 |

1306 | Rumbia | 2.873 | 44.986 | 2.886 | 48.724 | 2.883 | 47.671 |

1409 | Rammasun | 3.820 | 5.162 | 4.110 | 5.713 | 4.103 | 5.698 |

1522 | Mujigae | 4.115 | 6.241 | 4.716 | 7.791 | 4.706 | 7.764 |

**Table 5.**Table of expected relative loss aversions and fixed relative losses with equal aversions for seven typhoons.

(a) | (b) | (c) | (d) | (e) | (f) | ||
---|---|---|---|---|---|---|---|

Name | Level | L_{1}′ (%) | L_{2}′ (%) | U_{R}_{1}′
| U_{R}_{2}′
| l_{1}′ (%) | l_{2}′ (%) |

Prapiroon | 12 | 0.2183 | 5.5250 | 0.0192 | 0.4872 | 0.0193 | 0.5117 |

Hagupit | 15 | 0.2109 | 7.8628 | 0.0073 | 0.2732 | 0.0073 | 0.2947 |

Chanthu | 12 | 0.1221 | 4.0887 | 0.0152 | 0.5105 | 0.0153 | 0.5285 |

Kai-tak | 13 | 0.0449 | 2.0172 | 0.0050 | 0.2252 | 0.0050 | 0.2290 |

Rumbia | 11 | 0.0170 | 1.5596 | 0.0046 | 0.4227 | 0.0046 | 0.4273 |

Rammasun | 15 | 0.2262 | 1.9330 | 0.0056 | 0.0481 | 0.0056 | 0.0490 |

Mujigae | 15 | 0.3718 | 3.7847 | 0.0057 | 0.0576 | 0.0057 | 0.0597 |

Number | Name | Classification | ||
---|---|---|---|---|

L_{1}′
| L_{2}′
| (L_{1}′+ L_{2}′)/2 | ||

0606 | Prapiroon | Level 5 | Level 5 | Level 5 |

0814 | Hagupit | Level 2 | Level 3 | Level 3 |

1003 | Chanthu | Level 4 | Level 5 | Level 5 |

1213 | Kai-tak | Level 2 | Level 2 | Level 2 |

1306 | Rumbia | Level 2 | Level 4 | Level 3 |

1409 | Rammasun | Level 2 | Level 1 | Level 2 |

1522 | Mujigae | Level 2 | Level 1 | Level 2 |

(a) | (b) | (c) | (d) | (e) | (f) | Rank of Risk | Rank of Wind | ||
---|---|---|---|---|---|---|---|---|---|

Name | R′_{1} (%) | R_{1} (%) | e_{1} (%) | R′_{2} (%) | R_{2} (%) | e_{2} (%) | |||

H_{L}-L_{P} | Prapiroon | 0.4868 | 0.4880 | 0.2495 | 0.019232 | 0.019234 | 0.0984 | 5 | 15 |

Hagupit | 0.2728 | 0.2738 | 0.3555 | 0.007318 | 0.007319 | 0.0950 | 5 | 12 | |

L_{L}-H_{P} | Kai-tak | 0.2251 | 0.2253 | 0.0909 | 0.005011 | 0.005011 | 0.0202 | 2 | 12 |

Rumbia | 0.4227 | 0.4229 | 0.0702 | 0.004607 | 0.004607 | 0.0077 | 3 | 15 | |

Chanthu | 0.5103 | 0.5112 | 0.1844 | 0.015238 | 0.015239 | 0.0550 | 2 | 13 | |

Rammasun | 0.0481 | 0.0482 | 0.0871 | 0.005632 | 0.005633 | 0.1019 | 2 | 15 | |

Mujigae | 0.0575 | 0.0576 | 0.1707 | 0.005651 | 0.005652 | 0.1677 | 2 | 11 |

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## Share and Cite

**MDPI and ACS Style**

Liu, G.; Yang, B.; Nong, X.; Kou, Y.; Wu, F.; Zhao, D.; Yu, P.
Risk Level Assessment of Typhoon Hazard Based on Loss Utility. *J. Mar. Sci. Eng.* **2023**, *11*, 2177.
https://doi.org/10.3390/jmse11112177

**AMA Style**

Liu G, Yang B, Nong X, Kou Y, Wu F, Zhao D, Yu P.
Risk Level Assessment of Typhoon Hazard Based on Loss Utility. *Journal of Marine Science and Engineering*. 2023; 11(11):2177.
https://doi.org/10.3390/jmse11112177

**Chicago/Turabian Style**

Liu, Guilin, Bokai Yang, Xiuxiu Nong, Yi Kou, Fang Wu, Daniel Zhao, and Pubing Yu.
2023. "Risk Level Assessment of Typhoon Hazard Based on Loss Utility" *Journal of Marine Science and Engineering* 11, no. 11: 2177.
https://doi.org/10.3390/jmse11112177