# A Novel Intelligent Model for Monthly Streamflow Prediction Using Similarity-Derived Method

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

^{3}/s in RMSE, 6.07% in MAPE, and 8.62% in NSE, and the SVM by 53.65 m

^{3}/s, 0.24%, and 5.53%, respectively.

## 1. Introduction

## 2. Problem Formulation and Solution Techniques

#### 2.1. Data Processing

#### 2.2. Similarity Parameters

_{1}(i) and y

_{2}(i) (any combination of non-equal years in the sequences); and ${\lambda}_{t}\mathrm{and}{\beta}_{t}$ are the weights assigned to the variance of streamflow (${Q}_{t}^{(y)}$) and rainfall (${P}_{t}^{(y)}$) in month t, respectively, and they also serve to standardize the units of streamflow and rainfall.

- 1.
- The base magnitude assigned to streamflow, being no more than 1.0:$${\lambda}_{t}\le 1.0$$
- 2.
- Zero-weights assigned to streamflow unknown in the forecast period:$${\lambda}_{t}=0,\mathrm{for}m\le t\le m+\overline{M}-1$$
- 3.
- Zero-weights assigned to rainfalls unavailable in the forecast period:$${\beta}_{t}=0,\mathrm{for}m\le t\le m+\overline{M}-1$$

#### 2.3. Similarity Derivation Model

_{k}) indicates a higher similarity of the kth year to the current year.

#### 2.4. Brief Introduction of the Methods for Comparison

## 3. Case Studies

#### 3.1. Research Domain and Data

^{2}[35]. The Lancang River in China is known for its mainstream length of 2161 km and drainage area of 167,487 km

^{2}. The Lancang River, traversing the Hengduan Mountains, is characterized by its north–south orientation and is separated from the Nujiang River in the west by mountain ranges such as Bangma Mountain and Nushan, including the southern part of Luoxue Mountain, and the Jinsha River and Red River in the east by mountain ranges such as Yun Ling and Wu Liangshan [36]. Significant natural environmental differences exist between the river’s upper, middle, and lower reaches. Geographically, the basin descends in a step-like manner from north to south, with the prominent landform characterized by high mountains and deep gorges. As the mountains extends southward, the distances between them gradually widen, forming a shape resembling a broom, with a tight upper portion and a sparse lower portion. The Yunnan section accounts for more than 50% of the river’s length within China.

#### 3.2. Parameters in Calibration

#### 3.3. Results in Verification

^{3}/s, with SDM6 demonstrating the smallest RMSE value of 359.22 m

^{3}/s among the five methods. The Mean Absolute Percentage Error (MAPE) is employed to assess the accuracy of the model predictions in terms of relative errors, and the results indicate that the SDMs exhibit relative errors consistently more than 5% lower than the relative error of the Mean method, with SDM12 achieving the lowest of 18.3%. The Nash–Sutcliffe efficiency coefficient (NSE), a commonly used metric for evaluating the goodness of fit in hydrological model simulations, reveals that the SDMs outperform the Mean method, with SDM6 achieving the highest efficiency of 77.12%. The analysis of the results suggests that the SDMs, particularly SDM6, show better predictive ability than the monthly average values. Therefore, SDM6 was selected as the method with the best prediction accuracy and used for subsequent comparisons with other methods.

#### 3.4. Results in Validation

^{3}/s in RMSE, 6.73% in MAPE, and an increase of 3.09% in NSE compared to the Mean model. However, SDM6 outperforms the SVM, with the RMSE and MAPE reduced by 53.65 m

^{3}/s and 0.24%, respectively, and the NSE improved by 5.53%. The results demonstrate that both the SVM and SDM6 outperform the Mean method in terms of predictive ability, with SDM6 showing the best performance overall based on the comprehensive evaluation of the RMSE, MAPE, and NSE.

## 4. Conclusions

- The assessment during a verification period on different reference periods (3, 6, 9, and 12) reveals that SDM6 with a reference period of six months demonstrates the best performance.
- SDM6 during a validation period achieves 261.97 m3/s in RMSE, 16.01% in MAPE, and 87.74% in NSE, improving the Mean model by 79.9 m
^{3}/s in RMSE, 6.07% in MAPE, and 8.62% in NSE, and the SVM by 53.65 m^{3}/s, 0.24%, and 5.53%, respectively.

- (1)
- The model requires relatively long historical runoff data, making it unsuitable for basins with short or discontinuous data records.
- (2)
- The model’s solving process requires multiple calls to the solver, leading to slower computation speed.

- (1)
- Future work can explore optimizing additional variables, including the number of similar years and forecast months, to further investigate the model’s performance.
- (2)
- Further investigation is warranted to understand the influence of historical rainfall on identifying the similar years used to derive the forecast streamflow.
- (3)
- The model and procedures may also be extended to encompass daily, weekly, and annual streamflow predictions.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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Station | Period | Max (m^{3}/s) | Min (m^{3}/s) | Mean (m^{3}/s) | Cv | Cs |
---|---|---|---|---|---|---|

Xiaowan | 1954–2020 | 4948.0 | 275.0 | 1201.55 | 0.73 | 1.15 |

Ref (t) | ${\mathit{\lambda}}_{\mathit{t}}^{\ast}$ | ${\mathit{\beta}}_{\mathit{t}}^{\ast}$ | ||||||
---|---|---|---|---|---|---|---|---|

Jan | Apr | Jul | Oct | Jan | Apr | Jul | Oct | |

m-6 | 0 | 0 | 0 | 0 | 0.7 | 3296.6 | 5954 | 0 |

m-5 | 1.10 × 10^{−5} | 2.90 × 10^{−3} | 7.60 × 10^{−1} | 0 | 0.4 | 0 | 0 | 0 |

m-4 | 1.60 × 10^{−5} | 0 | 0 | 7.50 × 10^{−4} | 1.4 | 0 | 0 | 0 |

m-3 | 8.80 × 10^{−5} | 0 | 0 | 1.20 × 10^{−2} | 0 | 0 | 0 | 0 |

m-2 | 0 | 0 | 0 | 4.50 × 10^{−4} | 0 | 0 | 0 | 26.8 |

m-1 | 4.10 × 10^{−3} | 3.30 × 10^{−1} | 7.60 × 10^{−3} | 9.40 × 10^{−4} | 51.1 | 772.1 | 1401.9 | 97.4 |

Ref (t) | ${\mathit{\lambda}}_{\mathit{t}}^{\ast}$ | ${\mathit{\beta}}_{\mathit{t}}^{\ast}$ | ||||||
---|---|---|---|---|---|---|---|---|

Jan | Apr | Jul | Oct | Jan | Apr | Jul | Oct | |

m-9 | 1.80 × 10^{−3} | 0 | 0 | 0 | 0 | 0 | 0 | 1491.6 |

m-8 | 1.00 × 10^{−3} | 0 | 0 | 9.90 × 10^{−2} | 17.2 | 0 | 0 | 1505.1 |

m-7 | 0 | 3.60 × 10^{−5} | 0 | 0 | 0 | 34.5 | 60,073.3 | 559 |

m-6 | 0 | 0 | 0 | 0 | 3.6 | 337.9 | 11,402.2 | 0 |

m-5 | 8.10 × 10^{−5} | 4.10 × 10^{−1} | 9.40 × 10^{−1} | 0 | 1.7 | 0 | 0 | 0 |

m-4 | 1.60 × 10^{−4} | 0 | 0 | 2.70 × 10^{−3} | 11.7 | 0 | 0 | 0 |

m-3 | 5.50 × 10^{−4} | 0 | 0 | 3.20 × 10^{−3} | 0 | 0 | 0 | 0 |

m-2 | 0 | 0 | 0 | 1.30 × 10^{−3} | 0 | 0 | 0 | 31.5 |

m-1 | 2.40 × 10^{−2} | 3.10 × 10^{−2} | 9.90 × 10^{−3} | 3.00 × 10^{−3} | 156.7 | 69 | 1804.5 | 330.9 |

Indicators | MEAN | SDM3 | SDM6 | SDM9 | SDM12 |
---|---|---|---|---|---|

RMSE (m^{3}/s) | 374.47 | 371.31 | 359.22 | 370.57 | 373.14 |

MAPE (%) | 23.82 | 18.59 | 18.46 | 18.39 | 18.3 |

NSE (%) | 75.13 | 75.55 | 77.12 | 75.65 | 75.31 |

Indicators | Mean | SVM | SDM6 |
---|---|---|---|

RMSE (m^{3}/s) | 341.87 | 315.62 | 261.97 |

MAPE (%) | 22.98 | 16.25 | 16.01 |

NSE (%) | 79.12 | 82.21 | 87.74 |

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**MDPI and ACS Style**

Xu, Z.; Cheng, M.; Zhang, H.; Xia, W.; Luo, X.; Wang, J.
A Novel Intelligent Model for Monthly Streamflow Prediction Using Similarity-Derived Method. *Water* **2023**, *15*, 3270.
https://doi.org/10.3390/w15183270

**AMA Style**

Xu Z, Cheng M, Zhang H, Xia W, Luo X, Wang J.
A Novel Intelligent Model for Monthly Streamflow Prediction Using Similarity-Derived Method. *Water*. 2023; 15(18):3270.
https://doi.org/10.3390/w15183270

**Chicago/Turabian Style**

Xu, Zifan, Meng Cheng, Hong Zhang, Wang Xia, Xuhan Luo, and Jinwen Wang.
2023. "A Novel Intelligent Model for Monthly Streamflow Prediction Using Similarity-Derived Method" *Water* 15, no. 18: 3270.
https://doi.org/10.3390/w15183270