## 2 Parameter Uncertainty Quantification Method
### Random Response Surface Method  
In the calculation process of the PC method, calculating the expansion coefficient is a crucial step. [PERSON] first proposed a method based on linear regression to solve the PC coefficient [20], which mainly constructs the response surface based on random probability space and can only use the Hermite orthogonal polynomial to construct the chaotic polynomial expansion model at the earliest. In this paper, it was extended to the gPC expansion model, and the calculation process of the stochastic response surface method adopted in this paper was as follows:  
* Construction of the generalized chaotic polynomial expansion model  
Through the determination of the input variables and output variables of the system, according to their different distribution types and truncation processing, we obtained a chaotic polynomial model that can be expressed in a truncated polynomial form; the model is  
\[Y=\sum_{i=0}^{p}\hat{b}_{i}\Phi_{i}(\xi) \tag{6}\]  
To distinguish it from Equation (2), the polynomial coefficients to be found are expressed as \(\hat{b}_{i}\), where \(\xi\) is a set of D-dimensional standard random variables \(\xi_{1},\cdots,\xi_{d}\). * Estimation of the coefficient of interest  
(1) A certain number of sample points were selected by the MC method, and n valid samples \(\xi^{S}=\left[\xi_{1}^{S},\cdots,\xi_{j}^{S},\cdots,\xi_{N}^{S}\right]^{T}\) were selected from the standard random space, where each sample point was denoted as S, i.e., \(\xi_{j}^{S}\) is \(\xi_{j}^{S}=\left[\xi_{j1}^{S},\cdots,\xi_{jd}^{S}\right]\). (2) The sample points were transformed from the standard random space to the original random space, obtaining:  
\[X_{j1}^{S}=T_{1}\left(\xi_{j1}^{S}\right),\cdots,X_{ji}^{S}=T_{i}\left(\xi_{j \mu}^{S}\right),\cdots,X_{jd}^{S}=T_{d}\left(\xi_{j\mu}^{S}\right) \tag{7}\]  
\(\xi_{ji}^{S}=T_{i}\left(\xi_{ji}^{S}\right)\) represents the value of the one-dimensional sample point \(\xi_{ji}^{S}\) in the \(j\)-th sample point in the original random space. For example, \(X_{i}\) in the formula obeys a normal distribution, and \(\xi_{i}\) is a standard normal random variable, then:  
\[T_{i}\left(\xi_{ji}^{S}\right)=\sigma_{X_{i}}\xi_{ji}^{S}+\mu_{X_{i}} \tag{8}\]  
\(\sigma_{X_{i}}\) and \(\mu_{X_{i}}\) represent the mean and standard deviation of the one-dimensional random input variable c, respectively. (3) The real response function value obtained after calculating the selected effective sample points through the original model to obtain the response function \(g(X)\) can be expressed as:  
\[G=\left[g\left(X_{1}^{S}\right),\cdots,g\left(X_{N}^{S}\right)\right]^{T} \tag{9}\]  
\(g\left(X_{N}^{S}\right)\) indicates the true response function value of the Nth sample point. (4) Using least-squares regression to estimate the coefficients and then substituting the sample \(\xi_{j}^{S}=\left[\xi_{1}^{S},\cdots,\xi_{j}^{S},\cdots,\xi_{N}^{S}\right]^{T}\) and its true response function value \(G=\left[g\left(X_{1}^{S}\right),\cdots,g\left(X_{N}^{S}\right)\right]^{T}\) into the gPC expansion model, respectively, we obtained the following expressions:  
\[ \begin{bmatrix}\[PERSON]_{0}\left(\xi_{1}^{S}\right)&\[PERSON]_{1}\left(\xi_{1}^{S} \right)&\cdots&\[PERSON]_{P}\left(\xi_{1}^{S}\right)\\ \[PERSON]_{0}\left(\xi_{2}^{S}\right)&\[PERSON]_{1}\left(\xi_{2}^{S}\right)&\cdots&\[PERSON] _{P}\left(\xi_{2}^{S}\right)\\ \vdots&\vdots&\ddots&\vdots\\ \[PERSON]_{0}\left(\xi_{N}^{S}\right)&\[PERSON]_{1}\left(\xi_{N}^{S}\right)&\cdots&\[PERSON] _{P}\left(\xi_{N}^{S}\right)\end{bmatrix}\begin{bmatrix}b_{0}\\ b_{1}\\ \vdots\\ b_{p}\end{bmatrix} = \begin{bmatrix}g\left(X_{1}^{S}\right)\\ g\left(X_{2}^{S}\right)\\ \vdots\\ g\left(X_{N}^{S}\right)\end{bmatrix}  \tag{10}\]The above formula can be abbreviated as \(Ab=G\), where the matrix A can be expressed as:  
\[A=