## 3 Uncertainty Representation of Uncertain Distribution Parameters
### Fusion of Uncertain Distribution Parameters  
Using the Gaussian interpolation method, the insufficient data from Source 1 and evidence theory information from Source 2 are fused to represent all of the uncertain distribution parameters \(\xi\) for the epistemic uncertainty variables \(\mathbf{y}\). Since the uncertainty representation data from two sources are not sufficient to cover the whole uncertainty design region of distribution parameters \(\xi\), the design domain of \(\xi\) is firstly determined using a method similar to that for triangle fuzzy variables ([PERSON] and [PERSON], 2017). After the available point data and random sampling points of interval data are acquired, the mean value \(\mu_{1}\) and standard deviation \(\sigma_{1}\) of insufficient data from Source 1 are calculated using Equations (5) and (6), respectively. \[\mu_{1}=\frac{1}{m_{1}+ns_{1}\times n_{1}}\sum_{k=1}^{m_{1}+ns_{1}\times n_{1 }}\chi_{k}^{1}, \tag{5}\]  
\[\sigma_{1}=\sqrt{\frac{1}{m_{1}+ns_{1}\times n_{1}-1}\sum_{k=1}^{m_{1}+ns_{1 }\times n_{1}}\big{(}\chi_{k}^{1}-\mu^{1}\big{)}^{2}}. \tag{6}\]  
\(\chi_{k}^{1}\) is the data set, which contains \(m_{1}\) sampling points \(a_{1}^{1}(i=1,\cdots,m_{1})\) and \(ns_{1}\times n_{1}\) random sampling points from \(n_{1}\) sampling intervals \(\big{[}lb_{j}^{1},ub_{j}^{1}\big{]}(j=1,\cdots,n_{1})\). The lower and upper bound of insufficient data from Source 1 are determined based on available point and/or interval data and standard deviation \(\sigma_{1}\) of insufficient data in Equation (6). The lower and upper bound of evidence theory information from Source 2 are determined based on available interval data. The design domain \([l,u]\) of \(\xi\) is determined considering data of lower bound and upper bound from both Source 1 and Source 2, which are shown in Equations (7) and (8):  
\[l=\min\Big{(}\text{Min}_{1}-\psi\times\sigma_{1},L^{1}\Big{)}, \tag{7}\]  
\[u=\max(\text{Max}_{1}+\psi\times\sigma_{1},U^{\text{max}}), \tag{8}\]  
where \(\text{Max}_{1}\) and \(\text{Min}_{1}\) are the maximum and minimum values of sampling data from Source 1, \(L^{1}\) and \(U^{\text{max}}\) are the lower bound and upper bound of the interval information from Source 2, respectively. \(\psi\) is the extension parameter with a value equal to 0.4 ([PERSON] and [PERSON], 2017). The design domain \([l,u]\) of \(\xi\) is discretized into \(Q\) points \(\{\lambda_{i}|\lambda_{i}=l+(i-1)(u-l)/(Q-1),(i=1,\cdots,Q)\}\) uniformly. The corresponding optimum PDF values \(p_{i}^{1}\) for \(\lambda_{i}\) in Gaussian model \(G_{1}\) for Source 1 are calculated through maximizing the likelihood function \(L_{1}(\mathbf{p})\) of insufficient data in Equation (9):  
\[\max L_{1}(\mathbf{p})=\sum_{i=1}^{m_{1}}\log\Bigl{[}f_{\xi} \left(a_{i}^{1}\Big{|}\mathbf{p}\right)\Bigr{]}+\sum_{j=1}^{n_{1}}\log\Bigl{[} f_{\xi}\left(ub_{j}^{1}\Big{|}\mathbf{p}\right)-f_{\xi}\left(lb_{j}^{1}\Big{|} \mathbf{p}\right)\Bigr{]} \tag{9}\] \[\text{s.t. }\left\{ \begin{array}{l}f_{\xi}(\xi=\lambda_{i}| \mathbf{p})=p_{i},(i=1,\cdots,Q)\\ p_{i}\geq 0\\ f_{\xi}^{*}(\xi|\mathbf{p})\geq 0\\ \int_{l}^{n_{i}}f_{\xi}(\xi|\mathbf{p})\mathrm{d}\xi=1\end{array} \right..\]  
\(f_{\xi}\left(a_{i}^{1}|\mathbf{p}\right)\) are the probability density values of sampling point \(a_{i}^{1}\) from Source 1. \(f_{\xi}\left(ub_{j}^{1}\Big{|}\mathbf{p}\right)\) and \(f_{\xi}\left(lb_{j}^{1}\Big{|}\mathbf{p}\right)\) are the cumulative probability density values of the sampling interval's upper bound \(ub_{j}^{1}\) and lower bound \(lb_{j}^{1}\) from Source 1, respectively. Using Equation (9), the optimum PDF values at discrete points \(\lambda_{i}(i=1,\cdots,Q)\) are gained and named as \(p_{i}^{1}\), then the Gaussian interpolation model \(G_{1}\) for Source 1 is constructed.