## 3 Case Study
### Results and Discussion
#### 3.4.3 Comparison of Tunnel Reliability Based on ABGPR, Kriging, and XGBoost  
The specific parameter values used by ABGPR, Kriging, and XGBoost in this study are as follows, respectively. 1. ABGPR parameter values: constant kernel function (C) = 30.5\({}^{2}\), length scaling factor (I) = 0.0882. 2. The DACE [33] (Design and Analysis o I Computer Experiments) toolbox function [dmodel. perf] = dacefit (S, Y, regr, corr, theta) in MATLAB is used to construct Kriging surrogate model. Kriging parameter values: theta = 1. 3. XGBoost parameter values: max_depth = 1, learning_rate = 0.5, n_estimators = 4000, gamma = 0. Figure 6: Comparison of numerical calculation and surrogate model calculation for monitoring point displacement (Unit: cm). Note: The numbers in parentheses for each monitoring point in Figure 6 represent the numerical calculated values and surrogate model calculation values based on ABGPR, Kriging, and XGBoost, respectively. The kernel function used in ABGPR is denoted by Formula (12). The covariance function used in the Kriging surrogate model is denoted by Formula (18). \[K_{\text{Kriging}}\big{(}x,x^{\prime}\big{)}=\exp\left(-\frac{1}{2}\big{(}x-x^{ \prime}\big{)}^{\text{T}}\Lambda^{-1}\big{(}x-x^{\prime}\big{)}\right) \tag{18}\]  
where \(\Lambda=\text{diag}(\lambda_{1},\lambda_{2},\dots,\lambda_{m})\). The convergence criteria for the surrogate models based on ABGPR, Kriging, and XGBoost are as follows: if there is no underfitting or overfitting, it can be considered that the surrogate model has converged when the change in the loss function value of the surrogate model is very small (such as less than 0.001) or fluctuates within a small range. Using the same sample dataset as the ABGPR calculation, the nonlinear mapping surrogate models between geo-mechanical parameters and tunnel displacement or convergence response are constructed by using the ABGPR, Kriging, and XGBoost algorithms, respectively. Using these optimized surrogate models, the displacement or convergence values of point A, line BC, AB, AC, AD and AE are obtained, corresponding to 30,000 sets of geo-mechanical parameters generated by the MCS method. For point A, line BC, AB, AC, AD and AE, the \(R2\_\)score indices and MPAE of optimized surrogate models based on ABGPR, Kriging, and XGBoost are shown in Figures 7 and 8, respectively, and their reliability indices are calculated by Formulas (14) and (15), as shown in Figure 9.  
From Figure 7, we note that for monitoring point A and monitoring lines BC, AB, AC, AD and AE, the \(R2\_\)score indices of ABGPR-based surrogate models are all greater than 0.956 and fluctuate between 0.956 and 0.975; the \(R2\_\)score indices of Kriging-based surrogate models fluctuate between 0.899 and 0.939; the \(R2\_\)score indices of XGBoost-based surrogate models fluctuate between 0.94 and 0.962. Furthermore, the ABGPR-based \(R2\_\)score index of each measuring point/line is greater than that based on Kriging or XGBoost. The fluctuation in ABGPR-based \(R2\_\)score values (0.019) is the least among them, which is close to that of XGBoost-based \(R2\_\)score values (0.022), and that of Kriging-based \(R2\_\)score values (0.04) is the largest among them. Figure 7: Comparison of \(R2\) for models based on ABGPR, Kriging and XGBoost. From Figure 8, we note that for monitoring point A and monitoring lines BC, AB, AC, AD and AE, the _MAPE_ values of ABGPR-based surrogate models fluctuate between 2.471% and 2.908%; the _MAPE_ values of Kriging-based surrogate models fluctuate between 2.885% and 3.796%; the _MAPE_ values of XGBoost-based surrogate models fluctuate between 2.8% and 3.6%. Furthermore, the ABGPR-based _MAPE_ values of each measuring point/line are much smaller than the Kriging-based results or the XGBoost-based results. From Figure 9, we note that the ABGPR-based reliability indices are relatively close to the Kriging-based results and differ significantly from the XGBoost-based results. The  
Figure 8: Comparison of _MAPE_ of models based on ABGPR, Kriging and XGBoost.