## 1 Introduction
### Sensitivity Index Analysis Process Based on the Kriging Model and Surrogate Sampling  
Based on the above derivation, the flowchart for solving the sensitivity indices \(S_{i}\) and \(S_{Ti}\) using the ALK-SS method is shown in Figure 3. The specific steps are as follows:  
1. Generate \(N_{K}\) samples in the distribution parameter sample space according to the marginal probability density function \(f_{\theta}(\theta)\), forming a sample pool \(S_{SS}=\left[\theta_{1},\theta_{2},\cdots,\theta_{N_{K}}\right]\);
2. Extract \(N_{X}\) distribution parameter samples \(\left[x_{1},x_{2},\cdots,x_{N_{X}}\right]\) according to the surrogate sampling probability density function \(h_{X}(x\left|\theta^{*}\right)\), and substitute them into the simulation model to obtain the corresponding output statistical characteristic values \(\left[g(x_{1}),g(x_{2}),\cdots,g(x_{N_{X}})\right]\);
3. Randomly extract \(N_{T}\) distribution parameter samples \(\left[\theta_{1},\theta_{2},\cdots,\theta_{N_{T}}\right]\) from the sample pool \(S_{SS}\). For each distribution parameter sample point \(\theta_{j}(j=1,2,\cdots,N_{T})\), calculate the corresponding output statistical characteristic values \(m_{j}(j=1,2,\cdots,N_{T})\) based on \(g(x_{t})\) and \(f_{X}\left(x_{t}|\theta_{j}\right)/h_{X}(x_{t}|\theta^{*})\) according to Equations (19) and (20), forming the initial training sample set \(T_{SS}=\left[\theta_{i},m_{j}\right]\);
4. Establish the initial Kriging surrogate model \(\Psi_{K}(\Theta)\) based on the current \(T_{SS}\);
5. Select the update sample point \(\theta_{u}\) from \(S_{SS}\) as: \[\theta_{u}=\arg\min_{\theta\in S_{SS}}U(\theta)\] (21) Calculate the predicted value \(\Psi_{K}(\theta_{u})\) and the true response value \(\Psi(\theta_{u})\). Stop the active learning process and proceed to step 6 when Equation (21) is satisfied; otherwise, add \(\left[\theta_{u},\Psi(\theta_{u})\right]\) to the training set \(T_{SS}\) and return to step 4;
6. Randomly extract \(N_{\Theta}\) test distribution parameter samples \(\left[\theta_{1},\theta_{2},\cdots,\theta_{N_{\Theta}}\right]\) from the sample pool \(S_{SS}\), and substitute them into the Kriging model to obtain the corresponding output performance characteristic value samples \(\left[m_{1},m_{2},\cdots,m_{N_{\Theta}}\right]\), from which the total variance \(V(M)\) can be calculated;
7. Fix the distribution parameter \(\Theta_{\sim i}\) at its sample point \(\theta_{i}^{i}(j=1,2,\cdots,N_{\Theta})\). Combine it with other distribution parameter samples \(\theta_{\sim i}^{k}(k=1,2,\cdots,N_{\Theta})\) and, based on the established Kriging model, solve the conditional output statistical characteristic values \(M|\left(\theta_{i}^{i},\theta_{\sim i}^{k}\right)\left(k=1,2,\cdots,N_{\Theta}\right)\) to obtain their conditional mean \(E\left(M|\theta_{i}^{i}\right)\). Traverse \(j(j=1,2,\cdots,N_{\Theta})\) to obtain \(N_{\Theta}\) sample values of \(E\left(M|\Theta_{i}\right)\) under the \(i\)-th dimension distribution parameter conditions \(\left[E\left(m|\theta_{i}^{1}\right),E\left(m|\theta_{i}^{2}\right),\cdots,E \left(m|\theta_{i}^{N_{\Theta}}\right)\right]\). Calculate  
Figure 3: Sensitivity Index Solving Process Based on Kriging and Surrogate Sampling. the variance in the conditional expectation of the output performance characteristic values \(V(E(M|\Theta_{i}))\), and use Equation (5) to obtain the primary sensitivity index \(S_{i}\) of the distribution parameter \(\Theta_{i}(i=1,2,\cdots,n)\);
8. Fix the distribution parameter \(\Theta_{\sim i}\) at its sample point \(\theta_{\sim i}^{j}(j=1,2,\cdots,N_{\Theta})\). Combine it with other distribution parameter samples \(\theta_{i}^{k}(k=1,2,\cdots,N_{\Theta})\) and, based on the established Kriging model, solve the conditional output statistical characteristic values \(M|\left(\theta_{\sim i}^{j},\theta_{i}^{k}\right)(k=1,2,\cdots,N_{\Theta})\) to obtain its conditional mean \(E\left(M|\theta_{\sim i}^{j}\right)\).