# Calculation of Joint Return Period for Connected Edge Data

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## Abstract

**:**

## 1. Introduction

## 2. Common Copula Functions and Distribution Characteristics

_{x}(x) and F

_{y}(y) are corresponding marginal distributions, there must be a correlation structure function C that enables Equation (1) true.

## 3. Examples of the Application of the Copula Function

_{i}stands for the ith observation value of X and $\overline{x}$ is the mean value of X; y

_{i}stands for the ith observation value of Y and $\overline{y}$ is the mean value of Y.

_{i}is the corresponding rank difference and n represents the number of the data in the dataset. The rank correlation coefficient reflects the correlation between the annual extreme wave height and corresponding wind speed. Figure 6 is a scatter plot of wind speed and annual extreme wave height. The scatter plot of the two is shown below:

_{1}and σ

_{2}are four undetermined parameters, g(x, y) is two-dimensional mixed Gumbel joint probability density function and G(x, y) is the corresponding distribution function.

## 4. The Joint Return Period Analysis

## 5. Conclusions

- (1)
- The multivariate joint distribution constructed by Copula function is more flexible than the common multivariate joint distribution in terms of marginal distribution. The common multivariate joint distributions usually require the marginal distribution to be specific.
- (2)
- The multivariate joint distribution constructed by the Copula function can better describe the non-normality of single variable, and combine multiple non-normal wave elements.
- (3)
- The marginal distribution of multivariate joint distribution constructed by Copula function is easy to be determined. Thus, it is easier to be realized in the joint return period analysis, especially in conditional probability analysis.

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Wang, L.P.; Chen, B.Y.; Chen, C.; Chen, Z.S.; Liu, G.L. Application of linear mean-square estimation in ocean engineering. China Ocean Eng.
**2016**, 30, 149–160. [Google Scholar] [CrossRef] - Wang, L.P.; Xu, X.; Liu, G.L.; Chen, B.Y.; Chen, Z.S. A new method to estimate wave height of specified return period. Chin. J. Oceanol. Limnol.
**2017**, 35, 1002–1009. [Google Scholar] [CrossRef] - Chen, B.Y.; Liu, G.L.; Wang, L.P.; Zhang, K.Y; Zhang, S.F. Determination of Water Level Design for an Estuarine City. J. Oceanol. Limnol. 2018. [CrossRef]
- Chen, B.Y.; Zhang, K.Y.; Wang, L.P.; Jiang, S.; Liu, G.L. Generalized Extreme Value-Pareto Distribution Function and Its Applications in Ocean Engineering. China Ocean Eng.
**2019**, 2. in press. [Google Scholar] - Wang, L.P.; Chen, B.Y; Zhang, J.F.; Chen, Z.S. A new model for calculating the design wave height in typhoon-affected sea areas. Nat. Hazards
**2013**, 67, 129–143. [Google Scholar] [CrossRef] - Chen, B.Y.; Liu, G.L.; Wang, L.P. Predicting Joint Return Period Under Ocean Extremes Based on a Maximum Entropy Compound Distribution Model. Int. J. Energy Environ. Sci.
**2017**, 2, 117–126. [Google Scholar] - Chen, B.Y.; Liu, G.L.; Zhang, J.F. A Calculation Method of Design Wave Height under the Three Factors of Typhoon. China Patent CN201610972118, 29 August 2017. [Google Scholar]
- Wang, L.P.; Liu, G.L.; Chen, B.Y.; Wang, L. Typhoon Influence Considered Method for Calculating Combined Return Period of Ocean Extreme Value. China Patent CN201010595807.6, 20 March 2013. [Google Scholar]
- Liu, G.L.; Zheng, Z.J.; Wang, L.P.; Chen, B.Y.; Dong, X.J.; Xu, P.Y.; Wang, J.; Wang, C. Power-Type Wave Absorbing Device and Using Method Thereof. China Patent
**2015**. [Google Scholar] - Chen, B.Y.; Yang, Z.Y.; Huang, S.Y.; Du, X.Z.; Cui, Z.W.; Bhimani, J.; Xie, X.; Mi, N.F. Cyber-physical system enabled nearby traffic flow modelling for autonomous vehicles. In Proceedings of the IEEE 36th IEEE International Performance Computing and Communications Conference (IPCCC), San Diego, CA, USA, 10–12 December 2017; pp. 1–6. [Google Scholar]
- Escalante, H.J.; Ponce-López, V.; Wan, J.; Riegler, M.A.; Chen, B.Y.; Clapés, A.; Escalera, S.; Guyon, I.; Baró, X.; Halvorsen, P.; et al. ChaLearn joint contest on multimedia challenges beyond visual analysis: An overview. In Proceedings of the 23rd International Conference on Pattern Recognition (ICPR), Cancun, Mexico, 4–8 December 2016; pp. 67–73. [Google Scholar]
- Barrs, A.; Chen, B.Y. How Emerging Technologies Could Transform Infrastructure. Available online: http://www.governing.com/commentary/col-hyperlane-emerging-technologies-transform-infrastructure.html (accessed on 10 March 2018).
- Chen, B.Y.; Escalera, S.; Guyon, I.; Ponce-López, V.; Shah, N.; Simón, M.O. Overcoming calibration problems in pattern labeling with pairwise ratings: Application to personality traits. In Computer Vision—ECCV 2016 Workshops; Hua, G., Jégou, H., Eds.; Springer: Amsterdam, The Netherlands, 2016; pp. 419–432. [Google Scholar]
- Ponce-López, V.; Chen, B.Y.; Oliu, M.; Corneanu, C.; Clapés, A.; Guyon, I.; Baró, X.; Escalante, H.J.; Escalera, S. ChaLearn LAP 2016: First round challenge on first impressions-dataset and results. In Computer Vision-ECCV 2016 Workshops; Hua, G., Jégou, H., Eds.; Springer: Amsterdam, The Netherlands, 2016; pp. 400–418. [Google Scholar]
- Zhang, S.F.; Shen, W.; Li, D.S.; Zhang, X.W.; Chen, B.Y. Nondestructive ultrasonic testing in rod structure with a novel numerical Laplace based wavelet finite element method. Latin Am. J. Solids Struct.
**2018**, 15, e48. [Google Scholar] [CrossRef] - Chen, B.Y.; Wang, B.Y. Location Selection of Logistics Center in e-Commerce Network Environments. Am. J. Neural Netw. Appl.
**2017**, 3, 40–48. [Google Scholar] [CrossRef] - Wang, L.P.; Liu, G.L.; Chen, B.Y.; Wang, L. Typhoon Based on the Principle of Maximum Entropy Waters Affect the Design Wave Height Calculation Method. China Patent CN201010595815, 20 December 2010. [Google Scholar]
- Du, Q.; Lu, X.; Li, Y.; Wu, M.; Bai, L.; Yu, M. Carbon Emissions in China’s Construction Industry: Calculations, Factors and Regions. Int. J. Environ. Res. Public Health
**2018**, 15, 1220. [Google Scholar] [CrossRef] - Yang, A.M.; Li, S.S.; Lin, H.L.; Jin, D.H. Edge Extraction of Mineralogical Phase Based on Fractal Theory. Chaos Solitions Fractals
**2018**, 117, 215–221. [Google Scholar] - Liu, X.J.; He, Y.Q.; Fu, H.L.; Chen, B.Y.; Wang, M.; Wang, Z. How Environmental Protection Motivation Influences on Residents’ Recycled Water Reuse Behaviors: A Case Study in Xi’an City. Water
**2018**, 10, 1282. [Google Scholar] [CrossRef] - Liu, G.L.; Chen, B.Y.; Wang, L.P.; Zhang, S.F.; Zhang, K.Y.; Lei, X. Wave height statistical characteristic analysis. Oceanol. Limnol.
**2018**. [Google Scholar] [CrossRef] - Yue, S. The Gumbel Mixed Model Applied to Storm Frequency Analysis. Water Resour. Manag.
**2000**, 14, 377–389. [Google Scholar] [CrossRef] - Zhou, D.C.; Duan, Z.D. The Gumbel-logistic model for joint probability distribution of extreme-value wind speeds and effective wave heights. Ocean Eng.
**2003**, 21, 45–51. [Google Scholar] - Dong, S.; Cong, J.S.; Yu, H.J. Design Parameter Estimation of Joint Extreme Significant Wave Height and Wind Speed at Weizhoudao Observation Station. Period Ocean Univ. China
**2006**, 36, 489–492. [Google Scholar] - Liu, X.; Wang, M.; Fu, H. Visualized analysis of knowledge development in green building based on bibliographic data mining. J. Supercomput.
**2018**. [Google Scholar] [CrossRef] - Jiang, S.; Lian, M.J.; Lu, C.W.; Ruan, S.L.; Wang, Z.; Chen, B.Y. SVM-DS fusion based soft fault detection and diagnosis in solar water heaters. Energy Explor. Exploit.
**2018**. [Google Scholar] [CrossRef] - Jiang, S.; Lian, M.J.; Lu, C.W; Gu, Q.H; Ruan, S.L.; Xie, X.C. Ensemble Prediction Algorithm of Anomaly Monitoring Based on Big Data Analysis Platform of Open-Pit Mine Slope. Complexity
**2018**, 2018, 1048756. [Google Scholar] [CrossRef] - Chen, B.; Liu, G.; Wang, L.; Zhang, K.; Zhang, S. Determination of Water Level Design for an Estuarine City. J. Oceanol. Limnol.
**2019**. [Google Scholar] [CrossRef] - Yang, A.M.; Yang, X.L.; Wu, W.R.; Liu, H.X.; Zhuansun, Y.X. Research on Feature Extraction of Tumor Image Based on Convolutional Neural Network. IEEE Access
**2019**. [Google Scholar] [CrossRef] - Skla, A. Fonctions de repartition a n dimensions et leurs marges. Publ. Inst. Statist. Univ. Paris
**1959**, 8, 229–231. [Google Scholar] - Deng, W.; Yao, R.; Zhao, H.M.; Yang, X.H.; Li, G.Y. A novel intelligent diagnosis method using optimal LS-SVM with improved PSO algorithm. Soft Comput.
**2017**. [Google Scholar] [CrossRef] - Deng, W.; Zhao, H.M.; Yang, X.H.; Xiong, J.X.; Sun, M.; Li, B. Study on an improved adaptive PSO algorithm for solving multi-objective gate assignment. Appl. Soft Comput.
**2017**, 59, 288–302. [Google Scholar] [CrossRef] - Deng, W.; Zhao, H.M.; Zou, L.; Li, G.Y.; Yang, X.H.; Wu, D.Q. A novel collaborative optimization algorithm in solving complex optimization problems. Soft Comput.
**2017**, 21, 4387–4398. [Google Scholar] [CrossRef] - Liu, G.; Chen, B.; Jiang, S.; Fu, H.; Wang, L.; Jiang, W. Double Entropy Joint Distribution Function and Its Application in Calculation of Design Wave Height. Entropy
**2019**, 21, 64. [Google Scholar] [CrossRef] - Kang, L.; Du, H.L.; Zhang, H.; Ma, W.L. Systematic research on the application of steel slag resources under the background of big data. Complexity
**2018**, 2018, 6703908. [Google Scholar] [CrossRef] - Song, J.; Feng, Q.; Wang, X.; Fu, H.; Jiang, W.; Chen, B. Spatial Association and Effect Evaluation of CO
_{2}Emission in the Chengdu-Chongqing Urban Agglomeration: Quantitative Evidence from Social Network Analysis. Sustainability**2019**, 11, 1. [Google Scholar] [CrossRef] - Deng, W.; Zhang, S.J.; Zhao, H.M.; Yang, X.H. A novel fault diagnosis method based on integrating empirical wavelet transform and fuzzy entropy for motor bearing. IEEE Access
**2018**, 6, 35042–35056. [Google Scholar] [CrossRef] - Zhao, H.M.; Yao, R.; Yu, L.X.; Xu, L.; Li, G.Y.; Deng, W. Study on a novel fault damage degree identification method using high-order differential mathematical morphology gradient spectrum entropy. Entropy
**2018**, 20, 682. [Google Scholar] [CrossRef] - Zhao, H.M.; Sun, M.; Deng, W.; Yang, X.H. A new feature extraction method based on EEMD and multi-scale fuzzy entropy for motor bearing. Entropy
**2017**, 19, 14. [Google Scholar] [CrossRef]

**Figure 1.**Scatter diagrams of the two Copula functions. (

**a**)the Gumbel-Hougaard (GH) Copula function; (

**b**) the Clayton Copula function.

**Figure 7.**Gumbel distribution of the annual extreme wave height: (

**a**) Probability, (

**b**) quantile, (

**c**) return level, and (

**d**) density.

**Figure 8.**Gumbel distribution of the wind speed: (

**a**) Probability, (

**b**) quantile, (

**c**) return level, and (

**d**) density.

**Figure 9.**Weibull distribution of the annual extreme wave height: (

**a**) Probability, (

**b**) quantile, (

**c**) Weibull probability distribution, and (

**d**) density.

**Figure 10.**Weibull distribution of the wind speed: (

**a**) Probability, (

**b**) quantile, (

**c**) Weibull probability distribution, and (

**d**) density.

**Figure 11.**Pearson-III distribution of the annual extreme wave height: (

**a**) Probability, (

**b**) quantile, (

**c**) Pearson-III probability distribution, and (

**d**) density.

**Figure 12.**Pearson-III distribution of the wind speed: (

**a**) Probability, (

**b**) quantile, (

**c**) Pearson-III probability distribution, and (

**d**) density.

**Figure 13.**The above four joint distribution function plots: (

**a**) mixed Gumbel distribution, (

**b**) Gumbel-Logistic distribution, (

**c**) joint distribution based on G-H Copula function, and (

**d**) joint distribution based on Clayton Copula function.

**Figure 14.**The above four joint distribution function contour plots: (

**a**) mixed Gumbel distribution, (

**b**) Gumbel-Logistic distribution, (

**c**) joint distribution based on G-H Copula function, and (

**d**) joint distribution based on Clayton Copula function.

Correlation Coefficients | Linear Correlation Coefficient ρ | Rank Correlation Coefficient τ |
---|---|---|

The annual extreme wave height and corresponding wind speed | 0.4992 | 0.383 |

Distribution Functions | Gumbel Distribution | Weibull Distribution | Person-Ⅲ Distribution | Maximum Entropy Distribution |
---|---|---|---|---|

P-Value | 0.8158 | 0.7854 | 0.7728 | 0.8200 |

Ksstat | 0.1127 | 0.1163 | 0.1178 | 0.1122 |

Distribution Functions | Gumbel Distribution | Weibull Distribution | Person-Ⅲ Distribution | Maximum Entropy Distribution |
---|---|---|---|---|

P-Value | 0.9496 | 0.7313 | 0.9464 | 0.9031 |

Ksstat | 0.0916 | 0.1225 | 0.0923 | 0.1005 |

**Table 4.**Design wave height (m) and design wind speed (m/s) in different return periods based on the mixed Gumbel distribution.

Return Period | Marginal Distribution Design Value | ||
---|---|---|---|

T (a Single Variable) | T_{0} | T (a Single Variable) | T_{0} |

100 | 258 | 100 | 258 |

200 | 518 | 200 | 518 |

500 | 1302 | 500 | 1302 |

1000 | 2605 | 1000 | 2605 |

**Table 5.**Design wave height (m) and design wind speed (m/s) in different return periods based on the Gumbel-Logistic distribution.

Return Period | Marginal Distribution Design Value | ||
---|---|---|---|

T (a Single Variable) | T_{0} | T (a Single Variable) | T_{0} |

100 | 269 | 100 | 269 |

200 | 540 | 200 | 540 |

500 | 1358 | 500 | 1358 |

1000 | 2718 | 1000 | 2718 |

**Table 6.**Design wave height (m) and design wind speed (m/s) in different return periods based on the G-H Copula distribution.

Return Period | Marginal Distribution Design Value | ||
---|---|---|---|

T (a Single Variable) | T_{0} | T (a Single Variable) | T_{0} |

100 | 198 | 100 | 198 |

200 | 381 | 200 | 381 |

500 | 966 | 500 | 966 |

1000 | 2087 | 1000 | 2087 |

**Table 7.**Design wave height (m) and design wind speed (m/s) in different return periods based on the Clayton Copula distribution.

Return Period | Marginal Distribution Design Value | ||
---|---|---|---|

T (a Single Variable) | T_{0} | T (a Single Variable) | T_{0} |

100 | 201 | 100 | 201 |

200 | 390 | 200 | 390 |

500 | 981 | 500 | 981 |

1000 | 2072 | 1000 | 2072 |

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**MDPI and ACS Style**

Liu, G.; Chen, B.; Gao, Z.; Fu, H.; Jiang, S.; Wang, L.; Yi, K.
Calculation of Joint Return Period for Connected Edge Data. *Water* **2019**, *11*, 300.
https://doi.org/10.3390/w11020300

**AMA Style**

Liu G, Chen B, Gao Z, Fu H, Jiang S, Wang L, Yi K.
Calculation of Joint Return Period for Connected Edge Data. *Water*. 2019; 11(2):300.
https://doi.org/10.3390/w11020300

**Chicago/Turabian Style**

Liu, Guilin, Baiyu Chen, Zhikang Gao, Hanliang Fu, Song Jiang, Liping Wang, and Kou Yi.
2019. "Calculation of Joint Return Period for Connected Edge Data" *Water* 11, no. 2: 300.
https://doi.org/10.3390/w11020300