# A High-Robust Displacement Prediction Model for Super-High Arch Dams Integrating Wavelet De-Noising and Improved Random Forest

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## Abstract

**:**

## 1. Introduction

## 2. Statistical Model Construction and Pre-Processing Technique for Dam Displacement Data

#### 2.1. The Selection of the Influencing Factors of Dam Deformation

- Water pressure component

- Temperature and aging component

#### 2.2. Wavelet De-Noising for Displacement Signal Processing

#### 2.2.1. Wavelet Threshold De-Noising Principle

#### 2.2.2. Selection of Wavelet Bases, Threshold, and Threshold Function

## 3. Random Forests for Monitoring Model Construction

#### 3.1. Random Forests

#### 3.2. The Construction Process of Random Forest

- Select samples randomly. Bootstrap is used for sampling with the replacement method, and a training set is constructed for each decision tree.
- Select features randomly to construct a decision tree. Each node of the decision tree is formed by selecting a part of the total features continuously.
- All decision trees are combined into a random forest. At the prediction stage, each decision tree votes with the weight $1/m$ Select the tree with the most votes to get the equal-weighted vote model of the basic learning model as follows:

#### 3.3. Relevant Parameters of the Random Forest Model

- Number of basic decision trees (ntree) is according to the training environment of the model.
- Maximum number of features (features) is the maximum number of the candidate features when building the basic decision tree model.
- Depth of the base decision tree (depth) controls the maximum value of the lower limit of subtree growth in a random forest model.
- Minimum sample number of leaf nodes (leaf) is used for pruning the subtree of the random forest model. This parameter determines whether to retain leaf nodes and sibling leaf nodes
- Evaluation criteria for node feature selection (criteria) include entropy, Gini coefficient, and so on.

## 4. Improved Salp Swarm Algorithm

#### 4.1. Mathematical Model of the Salp Swarm

#### 4.2. Performance-Improved Salp Swarm Algorithm

#### 4.2.1. Elite Opposition-Based Learning Strategy

#### 4.2.2. Differential Strategy

#### 4.2.3. Gaussian Mutation Strategy

## 5. Construction Process of Prediction Model and Performance Evaluation Criterions

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## 6. Engineering Example

#### 6.1. Project Background and Deformation Monitoring

^{3}. The normal pool level is 1240 m and the dead water level is 1166 m, with a regulated reservoir capacity is 9.9 billion m

^{3}[40]. The project focuses on electricity generation and has comprehensive Utility benefits. Figure 5 shows the airscape of Xiaowan Arch Dam.

#### 6.2. Performance Analysis of the Proposed Model Considering Wavelet De-Noising Technique and ISSA Algorithm

#### 6.3. Accuracy Evaluation of the Prediction Models

- (1)
- Comparison of performance between the proposed model and the ISSA-RF model: The results presented in Figure 10 and Figure 11 and Table 1 indicate that for measuring point A09-PL-02, the WD-ISSA-RF model exhibits a higher multiple correlation coefficient (R
^{2}) of 97.89% as compared to 97.48% for the ISSA-RF model. Moreover, the WD-ISSA-RF model has smaller MAE, MSE, and MAPE values than the ISSA-RF model. Similar trends were observed for the other two measuring points. Therefore, it can be concluded that the WD-ISSA-RF model outperforms the ISSA-RF model in terms of modeling accuracy, and wavelet de-noising is an effective technique for enhancing prediction capability. - (2)
- Comparison of performance between the ISSA-RF model and the SSA-RF model: The ISSA-RF model employs the elite opposition-based learning strategy, difference strategy, and Gaussian mutation strategy to enhance the global search ability and avoid local optimization. From Figure 10, it can be observed that ISSA maintains a roughly similar convergence speed but exhibits a better global search ability as compared to SSA. The results presented in Figure 11 and Table 1 indicate that the ISSA-RF model has smaller MAE, MSE, and MAPE values than the SSA-RF model at all three measuring points. Hence, the ISSA-RF model demonstrates stronger predicting, learning, and generalization ability than the SSA-RF model. Although the ME of the ISSA-RF model on A09-PL-02 is slightly higher than that of the SSA-RF model, this does not affect the overall conclusion since the maximum error is an outlier value.
- (3)
- Comparison of performance between the SSA-RF model and the GWO-RF model. As a swarm intelligence algorithm, GWO achieves global optimization according to behavior of gray wolves [41]. GWO divides the wolves into four groups, including $\alpha $, $\beta $, $\delta $ and common gray wolf according to their social rank. Wolf $\alpha $, $\beta $ and $\delta $ are the leaders of the gray wolf group, with the highest adaptability. The common gray wolf obeys wolf $\alpha $, $\beta $ and $\delta $, according to their location, and is constantly approaching the prey, which is the optimal solution. Figure 12 reveals the relationship curve (A19-PL-01) between the fitness value and the number of iterations of the SSA and GWO. Figure 12 reveals that the best fitness of SSA is better than that of GWO and the SSA algorithm has a faster convergence speed.
- (4)
- Comparison of performance between the RF model and the ELM model. The ELM model proposes a novel hidden layer forward-type network based on the theory of generalized inverse [42]. By changing the hidden layer nodes, the optimal fitness value is obtained. From Table 1 and Figure 11, the prediction accuracy of the ELM model at the three measuring points varies. Although the R2 of the ELM model on A9-PL-02 and A19-PL-01 is equal to or slightly better than that of the GWO-RF model, the R2 on A9-PL-01 is 0.893, far less than that of the GWO-RF model on A9-PL-01. This situation has also occurred in other monitoring indicators, indicating that the stability and generalization performance of the RF model surpasses the ELM model. Data samples may have the problem of multicollinearity, and this can lead to random fluctuations in the output of the ELM model. This problem is well overcome by the RF model.
- (5)
- Comparison of performance between the RF model and SR model. The SR model adaptively screens variables with high importance on the basis of multiple regression, to establish a prediction model with selected variables [1]. From Table 1 and Figure 11, it is clear that each performance index of the ISSA-RF model is better than that of the SR model at the three different measuring points. Therefore, the ISSA-RF model is more accurate in prediction than the SR model, and has stronger learning and generalization ability.

#### 6.4. Robustness Analysis

^{2}, were used to numerically evaluate the stability performance of the model set. Figure 14 shows the prediction accuracy, even with the addition of different types of noise. The performance indicators remained almost unchanged, demonstrating that the proposed prediction model has very stable forecast performance across different data sets and effectively eliminates the impact of outliers on prediction accuracy.

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

WD | Wavelet De-noising |

RF | Random Forests |

SSA | Salp Swarm Algorithm |

ISSA | Improved Salp Swarm Algorithm |

ANN | Artificial Neural Networks |

LSTM | Long Short-Term Memory Model |

SVM | Support Vector Machine |

PSO | Particle Swarm Optimization |

GSA | Gravity Search Algorithm |

GWO | Gray Wolf Optimization |

R^{2} | Multiple Correlation Coefficient |

ME | Maximum Error |

RMSE | Root Mean Squared Error |

MAE | Mean Absolute Error |

MAPE | Mean Absolute Percentage Error |

ELM | Extreme Learning Machine |

SR | Stepwise Regression Model |

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**Figure 1.**Schematic diagram of water components ${\delta}_{1H}$, ${\delta}_{2H}$, ${\delta}_{3H}$ for an arch dam.

**Figure 8.**The displacement and residual error plot with Savitzky-Golay de-noising or wavelet de-noising on A09-PL-01.

**Figure 9.**3D-trajectory of SSA (

**left column**) and ISSA (

**right column**) when optimizing the three measuring points.

**Figure 10.**The predicted displacements and residuals of the six different prediction models for three monitoring points. (

**a**) A09-PL-01. (

**b**) A09-PL-02. (

**c**) A19-PL-01.

**Figure 11.**Comparison of prediction performance using six prediction models for three monitoring points.

**Figure 13.**Predicting and fitting results of the proposed model in four noise modes. (

**a**) Mode 1: point-scattered type. (

**b**) Mode 2: step type. (

**c**) Mode 3: vibrating type. (

**d**) Mode 4: Mixed type.

**Table 1.**Performance indexes of six prediction models for predicting dam deformation for three monitoring points.

Statistical Indicators | WD-ISSA-RF | ISSA-RF | SSA-RF | GWO-RF | ELM | SR | |
---|---|---|---|---|---|---|---|

A09-PL-01 | R^{2} | 0.962 | 0.961 | 0.954 | 0.959 | 0.893 | 0.919 |

MAE | 1.386 | 1.451 | 1.625 | 1.521 | 2.599 | 2.248 | |

ME | 5.970 | 5.240 | 6.868 | 5.364 | 6.520 | 5.919 | |

MAPE | 0.075 | 0.082 | 0.093 | 0.088 | 0.174 | 0.165 | |

RMSE | 1.878 | 1.901 | 2.063 | 1.946 | 3.137 | 2.744 | |

A09-PL-02 | R^{2} | 0.979 | 0.975 | 0.969 | 0.972 | 0.973 | 0.971 |

MAE | 1.235 | 1.327 | 1.442 | 1.403 | 1.300 | 1.478 | |

ME | 3.977 | 4.620 | 4.454 | 4.421 | 4.311 | 3.400 | |

MAPE | 0.054 | 0.058 | 0.064 | 0.059 | 0.057 | 0.069 | |

RMSE | 1.510 | 1.649 | 1.823 | 1.713 | 1.700 | 1.761 | |

A19-PL-01 | R^{2} | 0.966 | 0.966 | 0.960 | 0.945 | 0.946 | 0.911 |

MAE | 3.464 | 3.500 | 4.038 | 4.873 | 4.766 | 6.829 | |

ME | 14.342 | 14.021 | 15.763 | 16.79 | 16.30 | 15.73 | |

MAPE | 0.058 | 0.061 | 0.073 | 0.089 | 0.092 | 0.149 | |

RMSE | 4.806 | 4.783 | 5.167 | 6.128 | 6.035 | 7.778 |

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## Share and Cite

**MDPI and ACS Style**

Gu, C.; Wu, B.; Chen, Y.
A High-Robust Displacement Prediction Model for Super-High Arch Dams Integrating Wavelet De-Noising and Improved Random Forest. *Water* **2023**, *15*, 1271.
https://doi.org/10.3390/w15071271

**AMA Style**

Gu C, Wu B, Chen Y.
A High-Robust Displacement Prediction Model for Super-High Arch Dams Integrating Wavelet De-Noising and Improved Random Forest. *Water*. 2023; 15(7):1271.
https://doi.org/10.3390/w15071271

**Chicago/Turabian Style**

Gu, Chongshi, Binqing Wu, and Yijun Chen.
2023. "A High-Robust Displacement Prediction Model for Super-High Arch Dams Integrating Wavelet De-Noising and Improved Random Forest" *Water* 15, no. 7: 1271.
https://doi.org/10.3390/w15071271