# A Prediction Model and Factor Importance Analysis of Multiple Measuring Points for Concrete Face Rockfill Dam during the Operation Period

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

^{2}) for the testing dataset being equal to 0.9578.

## 2. Materials and Methods

#### 2.1. The Multi-Factor and Multi-Monitoring Point Statistical Model

_{H}), temperature component (δ

_{S}) and time effect component (δ

_{T}) [3,11]:

_{H}, the temperature component δ

_{S}and the time effect component δ

_{T}are improved. The rheological component δε and the material component δ

_{m}are added.

#### 2.1.1. The Water Level Component δ_{H}

_{H}is the sum of the water level component of settlement measurement points δ′

_{h}and the pre-reservoir water level component f(h):

_{0}, b

_{1}, b

_{2}and b

_{3}are the coefficients of regression; h is the elevation difference between the water level and the measuring point; d

_{1}is the rockfill thickness; d

_{2}is the distance from the measuring point to the face panel (Figure 1); $\overline{h}$ represents the average water levels in the 3 days before the observation date; n is the modulus of elasticity index; and A is a constant.

#### 2.1.2. The Temperature Component δ_{S}

_{S}and temperature with the annual periodicity of the freezing period can be expressed by using the periodic function:

#### 2.1.3. The Time Effect Component δ_{T}

_{0}= the cumulative number of days since the selected monitoring date/100; and n′ is the soil porosity at the measurement point.

#### 2.1.4. The Rheological Component δ_{ε}

_{0}and a

_{1}are the coefficients of regression; D is the initial relative deformation rate; γ is the bulk density of the filled rockfill; H is height of filled rockfill above the measuring point; E

_{rc}is the tangential modulus of the rockfill and according to the Duncan–Chang model; and E

_{rc}can be presented as:

_{f}is the failure ratio; σ

_{1}, σ

_{3}are the large and small principal stresses, respectively; c is the cohesion force; φ is the friction angle; K is the tangent elastic modulus; and P

_{a}is the atmospheric pressure.

_{ε}:

#### 2.1.5. The Material Component δ_{m}

_{1}) and the distance from the measuring point to the face panel (d

_{2}). These three parameters can be used to represent the position coordinates of any point in the dam, so they can be used as variables to represent the spatial position. When the reservoir water level elevation is fixed, the deformation at different points is also related to the filling material at the location. These parameters are used to represent the comprehensive influence of the rockfill crushing characteristics, compression deformation properties or other factors in different rockfill areas, to explain the reasons for the differences in settlement values at different measuring points under the same external environmental conditions.

#### 2.2. XGBoost Model for Multiple Monitoring Points Model

#### 2.3. Hyperparameter Optimization and Performance Measures

**O**= {(a

_{1}, b

_{1}) …, (a

_{k}, b

_{k})}. A Gaussian model, GM, is fitted based on O fitting. (2) Select the hyperparameter with the best performance on the agent function: find the maximum hyperparameter a′ under GM through the collection function. (3) Apply the selected optimal hyperparameter to the objective function: the model is trained and evaluated based on the hyperparameter a′ and K-fold cross-validation, and the evaluation results are used to describe the ability of the model b′. (4) Update the proxy model and add (a′, b′) to set

**O**. (5) Repeat steps (2)~(4) until the maximum number of iterations or running time is reached.

^{2}) of different measurement points are used as quantitative indicators to evaluate the prediction ability of the model. The calculation of MAE, MAPE and RMSE and R

^{2}are presented mathematically by Equations (12)–(14):

#### 2.4. Factor Importance Analysis Based on SHAP

_{i}, the factor j of the ith sample is x

_{ij}, m is the number of factors in the model, the predicted value of the sample is y

_{i}, the baseline of the whole model (usually the mean value of the target variable of all samples) is y

_{base}, and the SHAP value is shown as follows [36]:

_{ij}) is the SHAP value of x

_{i}, e.g., f(x

_{i1}) is the contribution of the first factor in the ith sample to the final predicted value y

_{i}. When f(x

_{i1}) > 0, it indicates that this factor increases the predicted value, showing a positive correlation; otherwise, it indicates that this factor reduces the predicted value, which is a negative correlation. The new MMP model and the traditional model have 13 + 11 influencing factors, which are chosen as influencing factors (see Table 3). The factors of the new MMP model are represented by X series, while the factors of the traditional model are represented by Y series. It should be noted that all the factors are independent and have distinct meanings.

## 3. Case Study

^{3}.

## 4. Results

#### 4.1. Prediction Accuracy of the New MMP Model

^{2}is used to measure the correlation between the actual value and the predicted value. The larger R

^{2}is, the more accurate the prediction of the algorithm is. In the test set, only the R

^{2}of XGBoost algorithm is above 0.90, larger than that of the CART model (around 0.6). This phenomenon shows that the XGBoost algorithm has a higher prediction accuracy than its base algorithm (CART). The RMSE, MAE and MAPE are all used to measure the difference between the actual value and the predicted value. It can be seen that the model’s MAE and RMSE values at each measurement point in the training set and the test set are smaller than 3 mm, while the RMSE and MAPE of the test set from the CART algorithms are above 3 mm. The MAPE values of the training set and test set from the XGBoost algorithm are less than 1%, much smaller than those from the CART algorithm, which are mostly higher than 1%. It can be seen that for predicting the settlement of the CFRD, the predicted value of XGBoost shows the best correlation with the actual value and the smallest error. This finding demonstrates the superiority of the XGBoost model and indicates the importance of the use of the XGBoost algorithm.

#### 4.2. Orders of Importance of Factors by SHAP

_{1}(X5) ranks second. This phenomenon is the same with V9. However, although X5 has a median value, it has the highest SHAP value; this corresponds to the purple dots of X5 in the large SHAP value areas in Figure 5. Based on the individual analysis of the MMP model, the prediction process and prediction basis of the model for each specific sample can be understood. According to the data, the point position has the largest impact on the subsidence, and the monitoring time also has significant effects.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 5.**Evaluation results of factor values based on SHAP. A redder color represents a larger impact factor, and a bluer color is a smaller impact factor.

**Figure 7.**Evaluation results of factors based on SHAP values. (

**a**) The prediction of point V9 on 30 March 2016; (

**b**) The prediction of point V10 on 30 March 2016. A redder color represents a larger impact factor, and a bluer color represents a smaller impact factor.

Water Level Component (δ_{H}) | Rheology–Soil Weight Component (δε) | Time Effect Component (δ_{T}) | Material Component (δ_{m}) | |
---|---|---|---|---|

Factors | ${h}^{1-n}({d}_{1}/{d}_{2}),\overline{h}(1-{e}^{-Dt})h$ | $(1-{e}^{-Dt}){H}^{1-n}\cdot {d}_{1}{d}_{1},{d}_{2},H,H{d}_{1}$ | $\mathrm{ln}\theta ,\mathrm{ln}{\theta}_{0}$ | ${n}^{\prime},k,\rho $ |

Parameters | Value | Range | Note |
---|---|---|---|

eta | 0.2 | [0.01, 0.3] | the shrinkage step size |

max_depth | 5 | [3, 10] | the maximum depth of the decision tree |

learning_rate | 0.1 | [0.05, 0.3] | the learning rate |

N | 160 | / | the maximum number of iterations |

Components from the MMP Model | NO. | Factors | Components from the Traditional Model | NO. | Factors |
---|---|---|---|---|---|

Water level component | X1 | ${h}^{1-n}({d}_{1}/{d}_{2})$ | Water level component | Y1 | $h$ |

X2 | $\overline{h}$ | Y2 | ${h}^{2}$ | ||

X3 | $(1-{e}^{-Dt})h$ | Y3 | ${h}^{3}$ | ||

Rheology–soil weight component | X4 | $(1-{e}^{-Dt}){H}^{1-n}\cdot {d}_{1}$ | Temperaturecomponent | Y4 | $\mathrm{sin}\frac{2\pi t}{365}$ |

X5 | ${d}_{2}$ | Y5 | $\mathrm{cos}\frac{4\pi t}{365}$ | ||

X6 | ${d}_{1}$ | Y6 | $\mathrm{cos}\frac{2\pi t}{365}$ | ||

X7 | $H$ | Y7 | $\mathrm{sin}\frac{4\pi t}{365}$ | ||

X8 | $H{d}_{1}$ | Location component | Y8 | $x$ | |

Time effect component | X9 | $\mathrm{ln}\theta $ | Y9 | $y$ | |

X10 | $\mathrm{ln}{\theta}_{0}$ | Time effect component | Y10 | $\mathrm{ln}\theta $ | |

Material component | X11 | ${n}^{\prime}$ | Y11 | $\mathrm{ln}{\theta}_{0}$ | |

X12 | $k$ | Note: Time effect components are same in the two models; X5(d _{2}) and Y9(y) are the same. | |||

X13 | r |

**Table 4.**Settlement fitting and prediction performance analysis of new multiple points monitoring model.

Measuring Point | Training Set | Test Set by XGboost | Test Set by CART | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

MAE /mm | RMSE /mm | MAPE/% | R^{2} | MAE /mm | RMSE /mm | MAPE/% | R^{2} | MAE /mm | RMSE /mm | MAPE/% | R^{2} | |

V2 | 3.41 | 4.18 | 1.54 | 0.88 | 7.31 | 7.74 | 2.65 | 0.85 | 9.54 | 8.65 | 5.88 | 0.49 |

V3 | 1.34 | 1.57 | 0.34 | 0.97 | 1.95 | 2.18 | 0.47 | 0.94 | 6.87 | 8.64 | 3.76 | 0.56 |

V4 | 2.24 | 2.58 | 0.54 | 0.95 | 4.00 | 4.19 | 0.88 | 0.93 | 5.88 | 6.00 | 1.56 | 0.58 |

V5 | 1.49 | 1.92 | 0.24 | 0.98 | 3.17 | 3.54 | 0.50 | 0.92 | 4.67 | 4.99 | 1.09 | 0.58 |

V6 | 0.93 | 1.37 | 0.17 | 0.99 | 1.35 | 2.05 | 0.24 | 0.96 | 4.56 | 3.77 | 1.88 | 0.64 |

V7 | 1.27 | 2.02 | 0.39 | 0.94 | 1.97 | 2.49 | 0.55 | 0.92 | 3.64 | 5.85 | 1.55 | 0.56 |

V9 | 2.39 | 2.86 | 0.62 | 0.95 | 1.77 | 2.09 | 0.41 | 0.93 | 5.47 | 6.16 | 1.2 | 0.59 |

V10 | 1.80 | 2.23 | 0.26 | 0.98 | 1.67 | 2.23 | 0.24 | 0.95 | 2.67 | 4.73 | 0.89 | 0.65 |

V11 | 2.44 | 2.68 | 0.49 | 0.96 | 2.14 | 2.48 | 0.42 | 0.93 | 4.34 | 4.53 | 1.14 | 0.67 |

V12 | 0.68 | 2.04 | 0.22 | 0.97 | 1.22 | 3.15 | 0.34 | 0.94 | 1.54 | 3.86 | 0.39 | 0.73 |

V14 | 1.23 | 2.22 | 0.34 | 0.94 | 4.12 | 4.79 | 1.00 | 0.90 | 8.12 | 8.42 | 3.55 | 0.62 |

V15 | 0.61 | 2.46 | 0.23 | 0.95 | 7.08 | 7.55 | 2.03 | 0.85 | 10.57 | 10.05 | 7.89 | 0.50 |

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## Share and Cite

**MDPI and ACS Style**

Shao, L.; Wang, T.; Wang, Y.; Wang, Z.; Min, K.
A Prediction Model and Factor Importance Analysis of Multiple Measuring Points for Concrete Face Rockfill Dam during the Operation Period. *Water* **2023**, *15*, 1081.
https://doi.org/10.3390/w15061081

**AMA Style**

Shao L, Wang T, Wang Y, Wang Z, Min K.
A Prediction Model and Factor Importance Analysis of Multiple Measuring Points for Concrete Face Rockfill Dam during the Operation Period. *Water*. 2023; 15(6):1081.
https://doi.org/10.3390/w15061081

**Chicago/Turabian Style**

Shao, Lei, Ting Wang, Youde Wang, Zilong Wang, and Kaiyi Min.
2023. "A Prediction Model and Factor Importance Analysis of Multiple Measuring Points for Concrete Face Rockfill Dam during the Operation Period" *Water* 15, no. 6: 1081.
https://doi.org/10.3390/w15061081