## 2 Parameter Uncertainty Quantification Method  
### gPC Method  
The generalized chaotic polynomial method was used to solve the uncertainty of electromagnetic radiation. First, the original model was expressed as a functional \(Y\), and then the functional model was expanded in the form of a generalized chaotic polynomial. The expanded result is:  
\[\begin{array}{l}Y=c_{0}I_{0}+\sum\limits_{i_{1}=1}^{\infty}c_{i_{1}}I_{1}\big{(} \xi_{i_{1}}\big{)}+\sum\limits_{i_{1}=1}^{\infty}\sum\limits_{i_{2}=1}^{i_{1}}c_ {i_{1}i_{2}}i_{1}i_{2}I_{2}\big{(}\xi_{i_{1}},\xi_{i_{2}}\big{)}+\\ \sum\limits_{i_{1}=1}^{\infty}\sum\limits_{i_{2}=1}^{i_{1}}\sum\limits_{i_{3}= 1}^{i_{2}}c_{i_{1}i_{2}i_{3}}I_{3}\big{(}\xi_{i_{1}},\xi_{i_{2}},\xi_{i_{3}} \big{)}+\cdots=\sum\limits_{i=0}^{\infty}\xi_{i}\Phi_{i}(\xi)\end{array} \tag{1}\]  
In the above formula, \(I_{n}\big{(}\xi_{i_{1}},\cdots,\xi_{i_{n}}\big{)}\) is the nth-order representation of the mixed orthogonal polynomial, which is a function of the multidimensional standard random variable \([\xi_{i_{1}},\cdots,\xi_{i_{n}}]\). \(\Phi_{i}(\xi)\) is composed of the product of one-dimensional orthogonal polynomial basis functions. Each basis function corresponds to a different random variable. Table 1 lists the orthogonal polynomial basis functions corresponding to different distribution types. Each distribution type has its own unique orthogonal polynomial basis functions. The variables in this paper showed normal distribution and uniform distribution.  
Formula (1) shows that the chaotic polynomial expansion contains infinite items but cannot handle infinite items in the actual calculation. In order to improve the calculation efficiency, it is usually necessary to truncate the chaotic polynomial, and the truncation order is the \(P\) order. Therefore, the generalized chaotic polynomial expansion model can be expressed by truncating the expansion (2):  
\[Y=\sum\limits_{i=0}^{p}\hat{c}_{i}\Phi_{i}(\xi) \tag{2}\]  
\(\hat{c}_{i}\) is a polynomial coefficient. After the P-order truncation of the generalized chaotic polynomial expansion, the expansion contains the \(Q\) terms, and the \(p\) polynomial expansion terms remains after the first term is removed. The specific value of \(Q\) is related to the truncation order \(P\) and the dimension \(d\), which can be expressed as:  
\[Q=1+p=\frac{(d+P)!}{d!P!} \tag{3}\]  
The chaotic polynomial in Formula (3) should satisfy:  
\[\big{\langle}\Phi_{i}(\xi)\Phi_{j}(\xi)\big{\rangle}=\int\Phi_{i}(\xi)\Phi_{j} (\xi)W(\xi)d\xi=\Big{\langle}\Phi_{i}^{2}\Big{\rangle}\delta_{ij} \tag{4}\]  
In Formula (4), \(\langle\cdot\rangle\) represents the inner product operation in multidimensional space, and its weight function \(W(\xi)\) is the product of the one-dimensional weight function corresponding to each dimension random variable \(\xi_{1},\cdots,\xi_{d}\), which is the Kronecker function \(\delta_{ij}\):  
\[\delta_{ij}=\begin{cases}1,i=j\\ 0,i\
eq j\end{cases} \tag{5}\]  
Then, we used the random response surface method to solve chaotic polynomials.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Distribution Pattern** & **PDF** &
\begin{tabular}{c} **Orthogonal** \\ **Polynomial Basis** \\ \end{tabular} & **Weight Function** & **Variable Range** \\ \hline normal & \(\frac{1}{\sqrt{2\pi}}e^{-x^{2}/2}\) & Hermite \(H_{n}(x)\) & \(e^{-x^{2}/2}\) & \([-\infty,+\infty]\) \\ uniform & \(1/2\) & Legendre \(P_{n}(x)\) & \(1\) & \([-1,1]\) \\ exponential & \(e^{-x}\) & Legendre \(L_{n}(x)\) & \(e^{-x}\) & \([0,+\infty]\) \\ \(\gamma\) & \(\frac{x^{p-x}}{\Gamma(a+1)}\) & \(L_{n}^{(a,\beta)}(x)\) & \(x^{a}e^{-x}\) & \([0,+\infty]\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Orthogonal polynomial bases corresponding to the different distribution types.