# Monthly Runoff Prediction by Combined Models Based on Secondary Decomposition at the Wulong Hydrological Station in the Yangtze River Basin

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## Abstract

**:**

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^{3}, MAPE of 2.57%, and RMSE of 2.266 × 10

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^{3}. The combined model proposed in this paper has the highest prediction accuracy, rendering it suitable for long-time series prediction. Accurate runoff prediction plays a pivotal role in facilitating effective watershed management and the rational allocation of water resources.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Decomposition–Prediction Model

#### 2.1.1. CEEMDAN

- (1)
- The signal $x(t)$, with the addition of adaptive white noise ${n}_{i}(t)$, is decomposed using the EMD algorithm to obtain the first IMF component:$$IM{F}_{1}(t)=\frac{1}{N}{\displaystyle \sum}_{i=1}^{N}IM{F}_{i1}$$
- (2)
- The first residual component signal ${r}_{1}(t)$ is calculated:$${r}_{1}(t)=x(t)-IM{F}_{1}(t)$$
- (3)
- ${E}_{k}$ are the k IMF components generated by EMD decomposition, and ${r}_{1}(t)+{\alpha}_{1}{E}_{1}\left[{n}_{i}(t)\right]$ is decomposed until the first IMF component is decomposed, then the second IMF component is calculated:$$IM{F}_{2}(t)=\frac{1}{N}{\displaystyle \sum}_{i=1}^{N}{E}_{i}\left[{r}_{1}(t)+{\alpha}_{1}{E}_{1}\left[{n}_{i}(t)\right]\right]$$

- (4)
- Repeat step (3) to obtain the k-th $IM{F}_{k}(t)$ and the residual component ${r}_{k}(t)$:$$IM{F}_{k}(t)=\frac{1}{N}{\displaystyle \sum}_{i=1}^{N}{E}_{k-1}\left[{r}_{k-1}(t)+{\alpha}_{k-1}{E}_{k-1}\left[{n}_{i}(t)\right]\right]$$$${r}_{k}(t)={r}_{k-1}(t)-IM{F}_{k}(t)$$
- (5)
- Continue to perform step (4) until the residual signal cannot be decomposed further; that is, there are no more than two extreme points of the residual signal. At the end of the algorithm, there are K inherent modes, and the final residual signal is:$$r(t)=x(t)-{\displaystyle \sum}_{k=1}^{K}IM{F}_{k}(t)$$

#### 2.1.2. VMD

- (1)
- Using the central frequency observation method to determine the number of variational modes (VMs) extracted by VMD, let $r=1,2,\cdots ,R$. Employ VMD for a secondary decomposition of IMF1. When $r=R+1$, the center frequencies of the VMD-extracted modes are close to 0, then $r=R$ is the final selected number of modes.
- (2)
- For the highest frequency component within the IMF components, the Hilbert transform is applied to obtain the analytical signal of the original mode, which yields the one-sided spectrum.$$\left[\delta (t)+\frac{j}{\pi t}\right]\ast {u}_{r}(t)$$
- (3)
- Shift the spectrum of each mode to the baseband by adjusting the estimated center frequencies of the entire ${u}_{r}(t)$ using the multiplication of $\left[\delta (t)+\frac{j}{\pi t}\right]\ast {u}_{r}(t)$ by ${e}^{-j{\omega}_{k}t}$, bringing all the modes to the fundamental frequency band.
- (4)
- Utilize the constrained variational model to determine the bandwidth of each mode component.$$\mathrm{min}\left\{{\displaystyle \sum _{r=1}^{R}{\Vert {\mathrm{d}}_{t}\left\{\left[\delta (t)+\frac{j}{\pi t}\right]\ast {\mu}_{r}(t)\right\}{e}^{-j{\omega}_{k}t}\Vert}_{2}^{2}}\right\}$$$$\sum}_{r=1}^{R}{u}_{r}={c}_{1}(t)$$
- (5)
- By introducing the quadratic penalty factor $\alpha $ and the Lagrange multiplier operator $\lambda (t)$, the optimal solution of the constrained variational model can be obtained, resulting in the optimal mode. The quadratic penalty factor $\alpha $ ensures signal reconstruction accuracy, while the Lagrange multiplier operator $\lambda (t)$ reinforces the constraints.

#### 2.1.3. LSTM

#### 2.1.4. Informer

_{i}under sparse evaluation, i.e., the computational complexity is reduced from $O\left({L}^{2}\right)$ to $O(L\mathrm{ln}L)$.

#### 2.2. CEEMDAN-VMD-LSTM-Informer Coupled Model

#### 2.3. Parameter Setting

^{−6}. This paper utilizes the Bayesian optimization method to determine the hyperparameters for the LSTM and Informer models [38]. The Bayesian optimization algorithm selects a set of hyperparameters for training at each iteration within predefined parameter ranges. It adjusts the next set of parameters based on the training results. Through multiple iterations, the Bayesian optimization algorithm identifies the optimal hyperparameters to improve model performance and prediction accuracy. The hyperparameter settings for the prediction model are presented in Table 1. All the decomposition and prediction models were calculated using the MATLAB R2021a.

#### 2.4. Seasonal Exponential Smoothing Model

#### 2.5. Evaluation Indicators

## 3. Case Study

#### 3.1. Study Area

#### 3.2. Runoff Characteristics Analysis

#### 3.2.1. Monthly Scale Analysis

#### 3.2.2. Time Series Analysis

## 4. Results and Discussion

#### 4.1. Results

#### 4.1.1. Predicted Results of CEEMDAN-VMD-LSTM-Informer Coupled Model

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^{3}, MAPE was 2.57%, and RMSE was 1.327 × 10

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#### 4.1.2. Predicted Results of Seasonal Exponential Smoothing Model

^{2}between the original value and the fitting value was 0.693, thus the model basically meets the requirements. It can be seen from the ACF and PACF of the residual sequence in Figure 13 that the residual basically fluctuated evenly around 0, and there was no autocorrelation when the order was 1–120, and all of them were within the confidence interval. Therefore, the residual sequence was classified as a white noise sequence, and the model fully extracted the information from the modeling sequence, and the model was effective. The prediction results of the seasonal exponential smoothing model are shown in Figure 14. By comparing the predicted value with the measured value, the NSE was calculated as 0.516, MAE as 8.629 × 10

^{8}m

^{3}, MAPE as 15.49%, and RMSE as 9.658 × 10

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^{3}.

#### 4.2. Discussion

## 5. Conclusions

- (1)
- Monthly scale runoff data at the Wulong hydrological station was a nonstationary and nonrandom sequence. The sequence showed a long-term decreasing trend and a pattern with an annual cycle. The predicted results of the seasonal exponential smoothing model showed that the prediction accuracy of the traditional statistical model is far inferior to that of machine learning models.
- (2)
- The application of the decomposition method provides a partial remedy for modal aliasing, resulting in more comprehensive and robust time series decomposition. This approach achieves a smoother monthly runoff series and reduces the interference of stochastic components with deterministic ones, ultimately enhancing the model’s predictive capabilities.
- (3)
- The combined model, which incorporates secondary decomposition, yields superior runoff prediction results. Specifically, the utilization of VMD for secondary decomposition of the highest-frequency component addresses the limitations of single-decomposition methods, resulting in a notable improvement in the accuracy of runoff predictions.
- (4)
- The CEEMDAN-VMD-LSTM-Informer model surpasses EEMD-VMD-LSTM-Informer, CEEMDAN-LSTM-Informer, CEEMDAN-LSTM, and EMD-LSTM in terms of prediction accuracy. The application of the CEEMDAN-VMD-LSTM-Informer model to monthly runoff time series prediction proves to be reliable and offers a novel approach to enhance research in monthly runoff prediction.
- (5)
- It is worth noting that this model solely considers runoff as a predictive factor. In future research endeavors, it would be advantageous to incorporate additional variables such as precipitation, evaporation, and temperature to further refine prediction accuracy.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 15.**Taylor diagram for comparison of prediction results of different machine learning models.

Models | Hyperparameters | Values |
---|---|---|

LSTM | Time step | 24 |

Batch size | 32 | |

Layers | 2 | |

Hidden size | 128 | |

Learning rate | 0.000001 | |

Informer | Encoder layers | 2 |

Decoder layers | 1 | |

Multi-head attention | 8 | |

Batch size | 16 | |

Epochs | 20 | |

Dropout | 0.1 |

Models | NSE | MAE (10^{8} m^{3}) | MAPE (%) | RMSE (10^{8} m^{3}) |
---|---|---|---|---|

CEEMDAN-VMD-LSTM-Informer | 0.997 | 1.327 | 2.57 | 2.266 |

EEMD-VMD-LSTM-Informer | 0.946 | 1.756 | 2.86 | 2.752 |

CEEMDAN-LSTM-Informer | 0.926 | 2.285 | 3.15 | 3.058 |

CEEMDAN-LSTM | 0.915 | 2.689 | 3.48 | 3.269 |

EMD-LSTM | 0.867 | 2.952 | 4.20 | 3.681 |

Seasonal exponential smoothing | 0.516 | 8.629 | 15.49 | 9.658 |

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## Share and Cite

**MDPI and ACS Style**

Wei, H.; Wang, Y.; Liu, J.; Cao, Y.
Monthly Runoff Prediction by Combined Models Based on Secondary Decomposition at the Wulong Hydrological Station in the Yangtze River Basin. *Water* **2023**, *15*, 3717.
https://doi.org/10.3390/w15213717

**AMA Style**

Wei H, Wang Y, Liu J, Cao Y.
Monthly Runoff Prediction by Combined Models Based on Secondary Decomposition at the Wulong Hydrological Station in the Yangtze River Basin. *Water*. 2023; 15(21):3717.
https://doi.org/10.3390/w15213717

**Chicago/Turabian Style**

Wei, Huaibin, Yao Wang, Jing Liu, and Yongxiao Cao.
2023. "Monthly Runoff Prediction by Combined Models Based on Secondary Decomposition at the Wulong Hydrological Station in the Yangtze River Basin" *Water* 15, no. 21: 3717.
https://doi.org/10.3390/w15213717