## 3 Case Study
### Kernel Function Selection  
The kernel function selection of GPR is generally based on the following considerations:  
1. Data characteristics: The selection of kernel functions should take into account the characteristics of the data, such as dimensionality, linear separability, and the presence of noise.
2. Model performance: The selection of kernel functions should also be based on the performance requirements of the model. Evaluate the performance of different kernel functions through cross-validation and other methods, and select the kernel function that can provide the best predictive performance. For example, by comparing the training error and testing error of the model, a kernel function that can fit the data well while avoiding overfitting can be selected.
3. Computational resources: When choosing a kernel function, it is necessary to balance the complexity of the model and computational feasibility.
4. Parameter adjustments: Kernel functions typically have parameters that need to be adjusted through optimization processes to maximize model performance. Therefore, when selecting kernel functions, optimization algorithms for model hyperparameters should also be considered.  
For constructing a GPR-based surrogate model, the model performance is the first and most important consideration among the four above-mentioned considerations of the kernel function selection of GPR.  
To quantitatively evaluate the performance of different kernel functions on the proposed algorithm in the test set, the average absolute percentage error (_MAPE_) and the determination coefficient (_R_2_score) are selected as the prediction and evaluation indices to measure the accuracy of the proposed algorithms. _MAPE_ is denoted by Formula (15), and \(R\)2_score is expressed by Formula (16).  
\[MAPE=\frac{100\%}{n}\sum\limits_{i=1}^{n}\left|\frac{\hat{y}_{\text{pi}}-y_{ \text{ai}}}{y_{\text{ai}}}\right| \tag{15}\]  
where \(\hat{y}_{\text{pi}}\) is prediction value; \(y_{\text{ai}}\) is actual value.  
\[R\text{2\_score}=1-\frac{\sum\limits_{i}\left(\hat{y}_{\text{pi}}-y_{\text{ai} }\right)^{2}}{\sum\limits_{i}\left(\overline{y}_{\text{ai}}-y_{\text{ai}} \right)^{2}} \tag{16}\]  
where \(\overline{y}_{\text{ai}}\) is the mean of actual data.  
Through the error analysis of elastic modulus regression results, the evaluation indices of simulation-based data test sets under six different kernel functions (_RBF_, _K1.5_, _K2.5_, _RBFK1.5_, _RBFK2.5_ and _CRBF_) were compared, as shown in Figure 4.  
\begin{table}
\begin{tabular}{c c c c c} \hline
**Random Variables** & **Distribution** & \(\mu_{X_{i}}\) & \(\sigma_{X_{i}}\) & **COV** \\ \hline \(\
u\) & Normal & 0.33 & 0.033 & 0.1 \\ \(\varphi/^{\circ}\) & Normal & 33 & 4.95 & 0.15 \\ \(E/\text{MPa}\) & Normal & 350 & 50.5 & 0.144 \\ \(c/\text{kPa}\) & Normal & 210 & 23 & 0.11 \\ \hline \end{tabular}
\end{table}
Table 2: Statistical properties of rock mechanical parameters [7].  
From Figure 4, we note that the kernel function with the strongest prediction ability is CRBF, whose \(R2\) score index is 0.954 and \(MAPE\) is only 2.232%. It is noted that the prediction ability of the model was greatly improved through the design of the combined kernel function. Therefore, CRBF is selected as the kernel function in the proposed Adam-Bayesian-GPR-based MCS reliability analysis.  
In the CRBF combined kernel function, the complexity and fitting ability of the RBF-based model can be balanced by adjusting the weight of the constant kernel function (C) to adapt to different data distributions and model requirements. The two hyperparameters of constant kernel function (C) and length scaling factor (\(l\)) in the CRBF combined kernel function can be determined using the Adam algorithm.