# Development of a Maximum Entropy-Archimedean Copula-Based Bayesian Network Method for Streamflow Frequency Analysis—A Case Study of the Kaidu River Basin, China

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## Abstract

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## 1. Introduction

## 2. Methodology

#### 2.1. Maximum Entropy Method

#### 2.2. Archimedean Copula

#### 2.3. Maximum Entropy-Archimedean Copula Method

#### 2.4. Maximum Entropy-Archimedean Copula-Based Bayesian Network

## 3. Study Area and Measures

#### 3.1. Study Area

^{2}with the mean elevation of 3100 m. The data of streamflow records are obtained from the Dashankou hydrological station for the period of 1957–2009. The unit for streamflow is in cubic meter per second (CMS). The monthly variation of streamflow is shown in Figure 3, with a maximum value of 413, a minimum value of 33.2, a mean of 110.54, a standard deviation of 68.77, a skewness of 1.37, and a kurtosis of 5.07.

#### 3.2. Dependence Measures

#### 3.3. Goodness-of-Fit (GOF)

## 4. Results and Discussion

#### 4.1. Marginal Distributions

#### 4.2. Joint Distributions

#### 4.3. Conditional Distributions

_{5|4}indicates the small change of PDF for streamflow in May when the scaled streamflow of April changes from 0.95 to 0.99. These results show that: (a) in general, high magnitudes of two adjacent streamflow corresponds to a low dependence coefficient among them, and vice versa; (b) the two minimum adjacent streamflows exhibit the highest dependence, but the two maximum streamflows do not show the minimum dependent coefficent, which indicates the interaction between streamflow and dependence coefficient is non-linear. That is mainly because (a) there are a variety of factors which affect the streamflow in the Kaidu River Basin, such as rainfall, snow melting, evapotranspiration rate, requirement of water use, and so on; (b) the conditions of streamflow are more complex under flooding season in the Kaidu River Basin, which leads to the situation that months with high streamflow values under flooding season have weak correlations with their adjacent months. In addition, the dependence coefficient between April and May is minimum mainly due to the types of streamflow contributors being different in April and May. It has been concluded before that snow melting is the main source in April in the Kaidu River Basin. However, both rainfall and snow melting make major contributions to the runoff in May. Otherwise, the conditions that the evapotranspiration rate increases with increasing temperature and the requirement of water increases due to the growth of the plants and the impact of human use may also affect the interaction of streamflow pairs between April and May.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**General framework of the maximum entropy-Archimedean copula-based Bayesian network (MECBN).

**Figure 4.**Comparison of theoretical and empirical marginal probability density function (PDF) and cumulative density function (CDF). The values are calculated monthly from 1957 to 2009. The abscissa represents streamflow values. The corresponding size varies from month to month. PDF describes the output of variables, and indicates the frequency at a given streamflow value. CDF describes the integrate of PDF, and indicates the probability of a given streamflow value. Lines in PDFs and CDFs show the performance of generated values. Bars and dots depict the empirical values in PDFs and CDFs, respectively.

**Figure 5.**Distribution of streamflow in each month given the streamflow status (0.95, 0.975, 0.99) in the previous month. The values are calculated monthly from 1957 to 2009. The abscissa represents streamflow values, which are scaled into [0,1]. f

_{K+1|K}indicates the conditional PDF of two adjacent monthly streamflows, K and K+1. 0.95, 0.975 and 0.99 mean the steamflow status of K’s streamflow.

**Table 1.**Estimation of Lagrange multipliers for maximum entropy (ME)-based marginal distribution of monthly streamflow.

Month | ${\mathit{\lambda}}_{1}$ | ${\mathit{\lambda}}_{2}$ | ${\mathit{\lambda}}_{3}$ | ${\mathit{\lambda}}_{4}$ |
---|---|---|---|---|

1 | −6.78 | 16.35 | −1.14 | −7.60 |

2 | −6.34 | 13.51 | 0.39 | −5.47 |

3 | −4.27 | 12.13 | −0.62 | −4.89 |

4 | −8.62 | 19.32 | 2.04 | −10.77 |

5 | −3.83 | 10.90 | 5.20 | −8.47 |

6 | −13.78 | 26.80 | 5.40 | −17.66 |

7 | −1.12 | 7.93 | −1.07 | −3.26 |

8 | −0.28 | 6.29 | −0.53 | −3.07 |

9 | −7.05 | 14.75 | 1.53 | −8.55 |

10 | −6.57 | 12.78 | 1.53 | −5.80 |

11 | −3.29 | 9.95 | 0.27 | −3.98 |

12 | −4.73 | 10.39 | −0.99 | −3.82 |

**Table 2.**The root means square error (RMSE) value and Kolmogorov–Smirnov (K–S) test for marginal distribution of monthly streamflow.

Month | RMSE | K–S test | |
---|---|---|---|

$T$ | p-Value | ||

1 | 2.28 | 0.135 | 0.145 |

2 | 1.46 | 0.099 | 0.347 |

3 | 1.20 | 0.081 | 0.490 |

4 | 2.31 | 0.074 | 0.546 |

5 | 8.62 | 0.100 | 0.340 |

6 | 5.06 | 0.055 | 0.706 |

7 | 10.28 | 0.086 | 0.447 |

8 | 5.41 | 0.058 | 0.684 |

9 | 4.44 | 0.096 | 0.371 |

10 | 1.52 | 0.054 | 0.722 |

11 | 1.82 | 0.087 | 0.436 |

12 | 2.06 | 0.107 | 0.290 |

**Table 3.**Comparison of Cramér von Mises statistic for joint distribution of different streamflow pairs.

Month | Gumbel-Hougaard | Frank | Clayton | |||
---|---|---|---|---|---|---|

${\mathit{S}}_{\mathit{n}}^{\left(\mathit{B}\right)}$ | p-Value | ${\mathit{S}}_{\mathit{n}}^{\left(\mathit{B}\right)}$ | p-Value | ${\mathit{S}}_{\mathit{n}}^{\left(\mathit{B}\right)}$ | p-Value | |

1–2 | 34.03 | 0.383 | 32.30 | 0.782 | 35.79 | 0.053 |

2–3 | 33.89 | 0.368 | 32.31 | 0.762 | 35.28 | 0.063 |

3–4 | 32.62 | 0.552 | 32.46 | 0.667 | 32.40 | 0.063 |

4–5 | 34.31 | 0.303 | 34.22 | 0.408 | 37.56 | 0.023 |

5–6 | 33.83 | 0.437 | 33.32 | 0.612 | 33.43 | 0.068 |

6–7 | 33.15 | 0.542 | 32.93 | 0.662 | 33.96 | 0.048 |

7–8 | 33.53 | 0.482 | 33.00 | 0.672 | 33.20 | 0.093 |

8–9 | 33.27 | 0.507 | 32.38 | 0.752 | 33.46 | 0.088 |

9–10 | 34.70 | 0.288 | 32.53 | 0.697 | 36.36 | 0.043 |

10–11 | 33.39 | 0.527 | 32.47 | 0.722 | 34.08 | 0.078 |

11–12 | 32.49 | 0.637 | 32.16 | 0.752 | 33.63 | 0.083 |

12–1 | 33.39 | 0.437 | 32.34 | 0.792 | 34.15 | 0.063 |

**Table 4.**Comparison of Akaike information criterion (AIC) and RMSE values for joint distribution of different streamflow pairs.

Month | Gumbel–Hougaard | Frank | ||
---|---|---|---|---|

AIC | RMSE | AIC | RMSE | |

1–2 | −311.52 | 0.0462 | −202.87 | 0.1342 |

2–3 | −321.77 | 0.0418 | −223.97 | 0.1091 |

3–4 | −347.22 | 0.0329 | −284.88 | 0.0601 |

4–5 | −327.10 | 0.0397 | −326.69 | 0.0399 |

5–6 | −341.56 | 0.0345 | −325.32 | 0.0404 |

6–7 | −360.95 | 0.0285 | −320.81 | 0.0422 |

7–8 | −341.83 | 0.0344 | −277.97 | 0.0643 |

8–9 | −321.79 | 0.0418 | −224.64 | 0.1084 |

9–10 | −338.00 | 0.0357 | −257.98 | 0.0782 |

10–11 | −358.19 | 0.0293 | −239.11 | 0.0941 |

11–12 | −299.66 | 0.0520 | −215.47 | 0.1186 |

12–1 | −294.01 | 0.0549 | −229.85 | 0.1030 |

Month | Spearman’s Rho | Kendall’s Tau | Upper Tail Dependence Coefficient |
---|---|---|---|

${\widehat{\mathit{\rho}}}_{\mathit{s}}$ | $\widehat{\mathit{\tau}}$ | ${\mathit{\lambda}}_{\mathit{U}}^{\mathit{G}\mathit{H}}$ | |

1–2 | 0.931 | 0.784 | 0.838 |

2–3 | 0.856 | 0.678 | 0.750 |

3–4 | 0.541 | 0.392 | 0.476 |

4–5 | 0.292 | 0.180 | 0.235 |

5–6 | 0.435 | 0.312 | 0.389 |

6–7 | 0.410 | 0.280 | 0.353 |

7–8 | 0.595 | 0.435 | 0.521 |

8–9 | 0.716 | 0.527 | 0.612 |

9–10 | 0.853 | 0.678 | 0.750 |

10–11 | 0.753 | 0.603 | 0.683 |

11–12 | 0.856 | 0.691 | 0.761 |

12–1 | 0.775 | 0.597 | 0.678 |

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## Share and Cite

**MDPI and ACS Style**

Kong, X.; Zeng, X.; Chen, C.; Fan, Y.; Huang, G.; Li, Y.; Wang, C.
Development of a Maximum Entropy-Archimedean Copula-Based Bayesian Network Method for Streamflow Frequency Analysis—A Case Study of the Kaidu River Basin, China. *Water* **2019**, *11*, 42.
https://doi.org/10.3390/w11010042

**AMA Style**

Kong X, Zeng X, Chen C, Fan Y, Huang G, Li Y, Wang C.
Development of a Maximum Entropy-Archimedean Copula-Based Bayesian Network Method for Streamflow Frequency Analysis—A Case Study of the Kaidu River Basin, China. *Water*. 2019; 11(1):42.
https://doi.org/10.3390/w11010042

**Chicago/Turabian Style**

Kong, Xiangming, Xueting Zeng, Cong Chen, Yurui Fan, Guohe Huang, Yongping Li, and Chunxiao Wang.
2019. "Development of a Maximum Entropy-Archimedean Copula-Based Bayesian Network Method for Streamflow Frequency Analysis—A Case Study of the Kaidu River Basin, China" *Water* 11, no. 1: 42.
https://doi.org/10.3390/w11010042