# Prediction of Maximum Tunnel Uplift Caused by Overlying Excavation Using XGBoost Algorithm with Bayesian Optimization

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

^{2}) values, which were used to assess the model’s learning and prediction abilities. Furthermore, the interpretability of the BO-XGBoost model is analyzed. The results demonstrate that the BO-XGBoost model exhibited the highest prediction performance for maximum tunnel vault uplift displacements. Moreover, the interpretability of the model will offer valuable guidance to civil engineers during their decision-making processes.

## 2. Materials and Methods

#### 2.1. Algorithm Principle for the XGBoost Model

Algorithm 1: Exact Greedy Algorithm for Split Finding |

Input: I, instance set of the current node, and d, the feature dimension |

$gain=0$ $G={\displaystyle \sum _{i=I}{g}_{i}},H={\displaystyle \sum _{i=I}{h}_{i}}$ for $k=1$$\mathbf{to}m\mathbf{do}$${G}_{L}=0,{H}_{L}=0$ for $j$$\mathbf{in}\mathbf{sorted}(I$$,\mathbf{by}{x}_{jk}$) do${G}_{L}={G}_{L}+{g}_{i},{H}_{L}={H}_{L}+{h}_{i}$ ${G}_{R}=G-{G}_{L},{H}_{R}=H-{H}_{L}$ $gain=\mathrm{max}(gain,\frac{{G}_{L}^{2}}{{H}_{L}+\lambda}+\frac{{G}_{R}^{2}}{{H}_{R}+\lambda}-\frac{{G}^{2}}{H+\lambda})$ endendOutput: Split with max gain |

#### 2.2. Hyperparameter Optimization Based on Bayesian Algorithm

## 3. Database-Building and Exploratory Analysis

#### 3.1. Database-Building

_{e}), directly affects the unloading force exerted on the soil above the tunnel, which, in turn, affects the maximum uplift of the tunnel (S

_{max}). Other factors include the radius of the shield tunnel (D), the distance between the tunnel vault and the base (H

_{t}), and the actual underpass length of the tunnel underpass (L

_{c}). Additionally, the control measures adopted in a project that control the disturbance of the soil by the excavation cannot be overlooked. They are, respectively, the excavation enclosure structure (SWM, DW, and BP), the internal support structure (OC and NC), and other excavation control measures (SM_1, SM_2, SM_3, and SM_4), respectively. It is worth noting that some of these factors are non-numerical variables, such as the soil mixing wall (SWM) in the excavation enclosure structure, which are quantified using binary values (0 and 1), with “1” indicating that the effect of the variable is considered and “0” indicating otherwise. Table 1 presents the definitions, ranges, and detailed descriptions of each variable.

#### 3.2. Data Exploratory Analysis

_{max}), while Figure 6 illustrates the corresponding box plot. It can be observed that there was a slightly higher proportion of silty clay and soft clay samples, with gravelly clay comprising a smaller portion. This can be attributed to the fact that the engineering cases primarily originated from the southeast coast, such as Shanghai, and these areas experience faster development compared to the central and western regions. Furthermore, Figure 6 shows that the overall sample distribution was relatively even, indicating that the database exhibited good representativeness and universality.

#### 3.2.1. Numeric Variables

_{max}, where no significant change was observed. The variables B, L

_{c}, He, and Ht exhibited wider distributions, as shown in Figure 7a–d, and all presented slightly positive skew values (SK > 0). Figure 7e reveals that the variable L was concentrated towards the lower end of the box plot, exhibiting a clear positive skewness. Its kurtosis (BK) was calculated as 6.6581 (after subtracting 3). While some outliers may have held valuable information for training the prediction model, only the extreme outlier of the variable L was removed, as represented by the red triangle in Figure 7e.

_{c}and B exhibited a high correlation of 0.93 (L

_{c}= B × sinα, where α is the angle of the tunnel passing through the excavation). However, when variables are highly correlated, it can reduce the interpretability of a model. Therefore, in this case, the variable L

_{c}was chosen to train the model with a higher correlation to the output S

_{max}.

#### 3.2.2. Categorical Variables

_{max}and indicates the variables as percentages of the overall sample size. For instance, the variables SM_1, SM_2, and NS, whose values were concentrated in category “1”, accounted for 95%, 98%, and 96% of the sample size, respectively. This indicated that the construction measures SM_1, SM_2, and NS were used in almost all the cases of the database, thereby making it impossible to identify the effects of these factors on the maximum tunnel uplift. In contrast, the distributions of other categorical variables in relation to S

_{max}exhibited certain patterns, indicating that they played significant guiding roles in the model training process.

#### 3.3. Data Standardized Processing

_{c}, H

_{e}, H

_{t}, SMW, DW, BP, Muc, Sof, Sil, Gra, SM_3, and SM_4 were selected as the inputs of the model, and S

_{max}was the output of the model.

## 4. Model Prediction Performance Discussion

#### 4.1. Model Creation and Hyperparameter Optimization

#### 4.2. Metrics of the Model Validation and Evaluation

^{2}) values as the evaluation metrics to describe the correspondence between the predicted and measured values. Smaller RMSE and MAE values reflected the higher prediction accuracy of the model, while an R

^{2}value closer to one indicated a better fit between the predictive model and the actual results. The evaluation indices were calculated as shown in Equations (7)–(9).

#### 4.3. Performance Analysis of the Prediction Model for the Maximum Tunnel Uplift Displacement

_{max}values for the SVM model, CART model, XGBoost model, and BO-XGBoost model were given. The scatter plots, which were closer to the diagonal line P = A, indicated the superior prediction results. The performances of the four established models were analyzed using the evaluation metrics. It is worth stating that model performance was evaluated based on the learning ability and prediction ability corresponding to the training and test sets, respectively.

_{max}values and smaller for larger measured S

_{max}values. The RMSE, MAE and R

^{2}values were 20.0142, 15.0773, and 0.7126 for the training set and 25.4354, 21.0216, and 0.6172 for the test set, respectively. This indicated that the prediction results of the SVM model deviated significantly from the measured S

_{max}values, and thus, they were not reliable.

_{max}value predicted by the CART model is plotted in Figure 11. Compared with the SVM model, the CART model could predict a wider range of measured S

_{max}values more accurately. The RMSE, MAE and R

^{2}values of the SVM model for the training set were 10.8984, 9.4908, and 0.9148, respectively, indicating that the predicted values for the entire training set were closer to the measured S

_{max}value. However, the deviation of the CART model was still relatively large for the test set (RMSE = 16.5961 and MAE = 14.6490).

^{2}values were 3.5153, 3.2277, and 0.9916, respectively, for the XGBoost model and 5.8232, 4.9647, and 0.9763, respectively, for the BO-XGBoost model, and they both demonstrated excellent learning abilities for the data. Figure 12a and Figure 13a show the scatter distributions over the range of 0.8 to 1.2 for the accuracy of the P = A line. For the test set, the RMSE and MAE values were reduced from 13.8493 and 11.9235 to 10.9808 and 9.2765, respectively, after the hyperparameters of the XGBoost model were optimized by the Bayesian algorithm. The prediction accuracy of the XGBoost model evaluated by the RMSE and MAE values had improved by 20% and 22%, respectively.

^{2}= 0.8865 and BO-XGBoost: R

^{2}= 0.9287). This could be explained by the fact that the unoptimized XGBoost model tended to overfit the training data, highlighting the importance of hyperparameter optimization for the XGBoost model.

_{max}values, were sorted from smallest to largest, and then the errors for the predicted and measured values were calculated using Equation (10):

_{max}was greater than the measured value while a negative value indicates the opposite.

_{max}value, with the maximum error exceeding −60 mm, particularly for the highest measured S

_{max}value. This finding highlighted the unsuitability of the SVM model for projects where S

_{max}values are expected to be relatively large, which could potentially compromise engineering safety.

_{max}values had improved, and the maximum prediction error was less than −50 mm. The XGBoost algorithm model provided a more accurate prediction for the measured S

_{max}values than the SVM and CART models, as evidenced by the maximum prediction error value of −31 mm. Meanwhile, the maximum prediction error of the BO-XGBoost algorithm was further reduced to −26 mm, and the absolute value of the maximum prediction error was controlled within 20 mm in most periods. Consequently, the BO-XGBoost algorithm demonstrated the highest prediction accuracy among the four models considered in this study.

#### 4.4. Interpretability of the Prediction Model

_{c}) of a tunnel’s under-crossing. Unexpectedly, the four soil types were ranked at the bottom according to their weights, which were far below the construction measures. This confirmed that suitable construction measures can significantly counteract the effects of geological factors resulting from underlying tunnel uplift. As a result, the interpretability of the BO-XGBoost model provided significant guidance to the tunnel engineers in the decision-making process.

## 5. Conclusions

^{2}values were selected as metrics for quantitatively evaluating the prediction performances of the SVM, CART, XGBoost, and BO-XGBoost models. The main conclusions were as follows:

- (1)
- Among the four algorithmic models for S
_{max}prediction, the SVM model’s predicted results deviated the most from the measured values (RMSE = 25.4354, MAE = 21.0216, and R^{2}= 0.6172). The highest prediction accuracy was achieved by the BO-XGBoost model. In addition, compared to the unoptimized XGBoost model, the RMSE and MAE values of the BO-XGBoost model improved from 13.8493 and 11.9235 to 10.9808 and 9.2765, respectively. - (2)
- According to the prediction results for the test set, the prediction errors of the four models showed tendencies to grow larger as S
_{max}values grew larger in certain time periods. These tendencies were particularly reflected in the SVM and CART models. The maximum prediction error of the SVM-based model was more than −60 mm, whereas it was near −50 mm for the CART model. However, the maximum prediction error of the BO-XGBoost model was controlled to within ±2 mm for most time periods, which was superior to the ±31 mm error of the XGBoost model. - (3)
- The BO-XGBoost model had better interpretability, including its ability to visualize the decision trees and calculate the feature importance. According to the weights of the characteristic importance elements, the three measures of the excavation enclosure, in order of the SWM, BP, and DW, were the most critical factors in predicting the maximum tunnel uplift displacement.
- (4)
- Expanding a dataset is beneficial for improving the prediction accuracy of a model. Currently, cases of actual projects with excavations over existing tunnels are insufficient. Apart from this, the input parameters of the training prediction model were still divided in inadequate detail. These play vital roles in achieving higher levels of accuracy for predictive models.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**Relative position of the excavation and the tunnel: (

**a**) bird’s eye view, and (

**b**) sectional view.

**Figure 7.**Numerical variable distribution: (

**a**) B, (

**b**) L

_{c}, (

**c**) H

_{e}, (

**d**) H

_{t}, (

**e**) L, and (

**f**) R.

Symbol | Category | Unit | Parameter Description | Note | |
---|---|---|---|---|---|

Min-Max | Mean | ||||

R | Input | m | 6–11 | 6.36 | Circular tunnel diameter |

L | Input | m | 8.2–867 | 90.04 | Lateral excavation length of an excavation |

B | Input | m | 9.7–200 | 48.14 | Longitudinal excavation length of an excavation |

H_{e} | Input | m | 4–24.3 | 8.85 | Excavation depth |

H_{t} | Input | m | 0.35–12.4 | 5.20 | The vertical distance between a tunnel vault and the bottom of an excavation |

L_{c} | Input | m | 10–203 | 53.24 | The actual undercrossing length of a tunnel |

Mud | Input | - | 0–1 | 0.25 | Mucky clay |

Sof | Input | - | 0–1 | 0.24 | Soft clay |

Sil | Input | - | 0–1 | 0.25 | Silty clay |

Gra | Input | - | 0–1 | 0.26 | Gravelly soil |

SWM | Input | - | 0–1 | 0.28 | Soil mixing wall |

DW | Input | - | 0–1 | 0.15 | Diaphragm wall |

BP | Input | - | 0–1 | 0.48 | Bored cast-in-place pile |

OC | Input | - | 0–1 | 0.21 | Sloping excavation |

NC | Input | - | 0–1 | 0.70 | Internal support of an excavation |

SM_1 | Input | - | 0–1 | 0.95 | Excavation bottom reinforcement |

SM_2 | Input | - | 0–1 | 0.98 | Layered excavation of an excavation |

SM_3 | Input | - | 0–1 | 0.68 | Uplift pile |

SM_4 | Input | - | 0–1 | 0.21 | Excavation bottom weight anti-floating |

S_{max} | Output | mm | 20–205 | 87.89 | Maximum displacement of a tunnel’s vault uplift |

**Table 2.**Search scopes and optimal hyperparameters for each of the XGBoost parameters in the BO tuning process.

XGBoost Hyperparameter | Search Scope | Default Value | Optimal Value |
---|---|---|---|

n_estimators | 1–50 | 100 | 43 |

max_depth | 1–20 | 6 | 4 |

learning_rate | 0.00001–1 | 0.3 | 0.6633 |

subsample | 0.1–1 | 1 | 0.9131 |

gamma | 0–20 | 0 | 14.6795 |

reg_alpha | 0–20 | 0 | 1.5322 |

reg_lambda | 0–20 | 1 | 16.5579 |

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## Share and Cite

**MDPI and ACS Style**

Zhao, H.; Wang, Y.; Li, X.; Guo, P.; Lin, H.
Prediction of Maximum Tunnel Uplift Caused by Overlying Excavation Using XGBoost Algorithm with Bayesian Optimization. *Appl. Sci.* **2023**, *13*, 9726.
https://doi.org/10.3390/app13179726

**AMA Style**

Zhao H, Wang Y, Li X, Guo P, Lin H.
Prediction of Maximum Tunnel Uplift Caused by Overlying Excavation Using XGBoost Algorithm with Bayesian Optimization. *Applied Sciences*. 2023; 13(17):9726.
https://doi.org/10.3390/app13179726

**Chicago/Turabian Style**

Zhao, Haolei, Yixian Wang, Xian Li, Panpan Guo, and Hang Lin.
2023. "Prediction of Maximum Tunnel Uplift Caused by Overlying Excavation Using XGBoost Algorithm with Bayesian Optimization" *Applied Sciences* 13, no. 17: 9726.
https://doi.org/10.3390/app13179726