## 3 Probabilistic Evaluation of ALSC Based on BASPCE
### Theory of Baspce [26]  
Assume that \(\mathbf{\xi}=[\xi_{1}\xi_{2},...,\xi_{m}]\) is an \(m\)-dimensional independent input random vector with a known joint probability density function. Appropriate orthogonal polynomials can be selected corresponding to each input random variable. The original model output response \(Y_{s}=g_{s}(\mathbf{\xi})\) can then be approximated by the sum of these orthogonal polynomial expansions. \[g_{s}(\mathbf{\xi})\approx\sum_{\mathbf{a}\in\Theta}\gamma_{\mathbf{a}}\Psi_{\mathbf{a}}(\mathbf{ \xi}) \tag{9}\]  
where \(\Theta\)\(\subset\mathbb{N}^{m}\) represents the truncation scheme contained within the \(m\)-dimensional natural number spherical domain; \(\Psi_{\mathbf{a}}(\mathbf{\xi})\) denotes the multivariate polynomial basis functions; \(\gamma_{\mathbf{a}}\) is the expansion coefficient corresponding to the basis function \(\Psi_{\mathbf{a}}(\mathbf{\xi})\). For the multivariate polynomial basis functions \(\Psi_{\mathbf{a}}(\mathbf{\xi})\), they are formed by the tensor product of univariate orthogonal polynomial basis functions. \[\Psi_{\mathbf{a}}(\mathbf{\xi})=\prod_{i=1}^{m}\phi_{a_{i}}^{(i)}(\xi_{i}) \tag{10}\]where \(\phi_{\alpha_{i}}^{(i)}(\xi_{i})\) represents the \(\alpha\)-th order orthogonal polynomial corresponding to the variable \(\xi_{i}\). The choice of the orthogonal polynomial basis function for a given input random variable is related to the type of probability distribution of that variable. The relationship between several classical types of input random variable probability distributions and their corresponding optimal univariate orthogonal polynomial basis functions is shown in Appendix A. For a random variable \(\xi_{i}\) following an arbitrary distribution type, the corresponding univariate orthogonal polynomials can be constructed using the recurrence relation in [27]. For the truncation scheme, the expansion terms in (3) are typically truncated based on a truncation criterion parameter \(p\), resulting in a total number of non-zero terms in the truncated PCE model given by:  
\[K^{m,p}= \begin{pmatrix}m+p\\ p\end{pmatrix} =\frac{(m+p)!}{m!p!} \tag{11}\]  
This truncation scheme is denoted as \(\Theta^{m,p}=\{\mathbf{\alpha}\in\mathbb{N}^{m}\colon\ |\mathbf{\alpha}\!\mid\!\leq p\}\), referred to as the standard truncation scheme for generalized PCE. The total number of terms in the truncated expansion is \(K^{m,p}+1\). However, not every term in the expansion has a significant impact on the output response; only a subset of the expansion terms corresponding to certain input random variables is crucial, known as the sparsity-of-effects principle. Based on this principle, a hyperbolic truncation scheme can be introduced, which effectively excludes higher-order interaction terms [28]. \[\Theta_{q}^{m,p}=\left\{\mathbf{\alpha}\in\Theta^{m,p}:\ \|\mathbf{\alpha}\|_{q}= \left(\sum_{i=1}^{m}\left(\alpha_{i}\right)^{q}\right)^{\forall q}\leq p,q\in \left(0,1\right]\right\} \tag{12}\]  
where \(q\) is the truncation norm used to adjust the sparsity. Under this truncation scheme, the total number of terms in the expansion is denoted as \(K_{q}^{m,p}+1\). When \(q=1\), the hyperbolic truncation is fully equivalent to the standard truncation scheme. When \(0<q<1\), the expansion terms under hyperbolic truncation include all high-order univariate terms but exclude higher-order interaction terms; furthermore, the smaller the value of \(q\), the fewer higher-order interaction terms are included in the expansion. Additionally, the estimated values of the expansion coefficients \(\gamma_{\mathbf{\alpha}}\) can be determined using the least squares method. By generating a sample set of input random variables \(\{\mathbf{\chi}_{1},\cdots,\mathbf{\chi}_{N_{\text{ED}}}\}\) through random sampling, this set is generally referred to as the experimental design (ED) sample set, and \(N_{\text{ED}}\) is the ED sample size.