## 5 Test Results and Discussion
### Test Setups
The definition and parameters setting of the FHSI are as follows [42]:  
\[\text{FHSI}=(1-\frac{1}{2}\sum_{i=1}^{N_{b}}|b_{\text{test},i}-b_{\text{ref},i}|) \tag{29}\]  
where \(N_{b}\) is the number of bins in the frequency histogram of the reference results; \(b_{\text{test},i}\) and \(b_{\text{ref},i}\) represent the heights of the \(i\)-th bin in the frequency histograms of test results and  
Figure 4: Frequency histograms of ALSC using different methods in S1: (a) MCS-based methods and (b) BASPCE-based methods. Figure 5: Frequency histograms of ALSC using different methods in S2: (a) MCS-based methods and (b) BASPCE-based methods. reference results, respectively. Note that (29) requires the two frequency histograms to have the same number of bins and identical bin positions. The numerical results in Tables 2-4 quantitatively support the conclusion that BASPCE can achieve frequency histograms very close to those obtained using the MCS method. In S1, except for the lower FHSI of SRS-MCS \(10^{3}\) and LHS-MCS \(10^{3}\), which is regarded as inaccurate, the FHSIs for the other methods are all above 97%, qualifying these methods as accurate. Moreover, the FHSIs of SRS-BASPCE \(10^{4}\) and LHS-BASPCE \(10^{4}\) are very close to those of SRS-MCS \(10^{4}\) and LHS-MCS \(10^{4}\), with only a slight reduction, indicating that BASPCE achieves significantly higher computational speed (to be discussed later) with only a minimal loss in accuracy. Furthermore, it is evident that the sample size has a significant impact on accuracy: The more samples, the higher the FHSI for any method, which explains the poorer results obtained by the LHS-MCS \(10^{3}\) method with fewer samples. Notably, the FHSIs of SRS-BASPCE \(10^{5}\) and LHS-BASPCE \(10^{5}\) already exceed those of SRS-MCS \(10^{4}\) and LHS-MCS \(10^{4}\), indicating that BASPCE-based methods have the potential to achieve higher FHSIs than MCS-based methods by increasing the sample size. The same conclusions can be drawn from the results in S2 and S3; therefore, they will not be reiterated here. Overall, both the graphical results and the numerical results support the conclusion that BASPCE achieves high accuracy in probabilistic ALSC evaluation. The high accuracy of BASPCE may primarily stem from its regularization strategy, adaptive sparse modeling capability, and efficiency in handling high-dimensional uncertainty quantification [41]. In addition to the accuracy of the frequency histograms, the accuracy of probabilistic indices is also crucial. The relative errors of the mean and standard deviation obtained by each method compared to the reference results are calculated. The calculation method for relative error is as follows:  
\[\varepsilon=\left|\frac{I_{\text{test}}-I_{\text{ref}}}{I_{\text{ref}}}\right| \times 100\% \tag{30}\]  
where \(I_{\text{test}}\) and \(I_{\text{ref}}\) denote the indices of test results and reference results, respectively. The relative errors of the mean \(\varepsilon_{E}\) and the relative errors of the standard deviation \(\varepsilon_{S}\) are shown in Tables 2-4. For the mean value, BASPCE-based methods generally exhibit high accuracy, with \(\varepsilon_{E}\) values being 0.04%-0.13% in S1, 0.05%-0.11% in S2, and 0.05%-0.11% in S3. While the MCS-based methods' \(\varepsilon_{E}\) are 0.04%-0.21% in S1, 0.05%-0.22% in S2, and 0.04%-0.21% in S3, it is evident that the BASPCE-based methods do not lag behind the MCS  
Figure 6: Frequency histograms of ALSC using different methods in S3: (**a**) MCS-based methods and (**b**) BASPCE-based methods. based methods in terms of the accuracy of mean estimation. For the standard deviation, the \(\varepsilon_{S}\) values of MCS-based methods drop in the range of 1.60%-5.29% in S1, 0.78%-5.71% in S2, and 0.41%-2.51% in S3, while the \(\varepsilon_{S}\) values of BASCE-based methods drop in the range of 3.53%-6.00% in S1, 1.86%-2.80% in S2, and 0.11%-2.54% in S3. The two methods exhibit comparable accuracy in the estimation of standard deviation.