## Appendix A Polynomial Chaos Expansion  
In a probabilistic framework, uncertainties in input parameters are modeled via random variables. These input uncertainties can be investigated using a Polynomial chaos expansion (PCE). This method provides the means to develop an approximation to the map between inputs and the quantities of interest (QoI). This mapping is both computationally tractable and sufficiently accurate. The main idea of PCE is to expand a QoI with a finite variance in a suitably built basis of multivariate polynomials that are orthogonal to the joint probability density functions of the inputs. It is worth noting that the random variables are assumed to be statistically independent or may be linearly correlated. The linear correlation can be handled by adequate linear transformation [58]. A PCE is a linear regression that includes linear combinations of a fixed set of nonlinear functions with respect to the input variables, known as basis functions (Appendix A). Let us consider \(\mathbf{X}\in\mathbb{R}^{M}\) as a random vector with independent components, defined by the joint probability density function \(f_{\mathbf{X}}\). Assuming that the random target variable \(\mathbf{Y}\in\mathbb{R}\) has a finite variance, its PCE representation can be cast as the following:  
\[\mathbf{Y}=\mathcal{M}(\mathbf{X})=\sum_{\mathbf{a}\in\mathbb{N}^{M}}c_{ \mathbf{a}}\Psi_{\mathbf{a}}(\mathbf{X}), \tag{1}\]  
where \(\Psi_{\mathbf{a}}(\mathbf{X})\) represents multivariate orthogonal polynomials with respect to \(f_{\mathbf{X}}\) and \(\mathbf{a}\) denotes a multi-index that encompasses the components of the multivariate polynomials \(\Psi_{\mathbf{a}}\). The \(c_{\mathbf{a}}\in\mathbb{R}\) are the corresponding coefficients (coordinates). For practical reasons, the sum in Equation (1) needs to be truncated to a finite sum, by introducing the truncated polynomial chaos expansion:  
\[\mathcal{M}(\mathbf{X})\approx\mathcal{M}^{\mathrm{PC}}(\mathbf{X})=\sum_{ \mathbf{a}\in\mathcal{A}}c_{\mathbf{a}}\Psi_{\mathbf{a}}(\mathbf{X}), \tag{2}\]where \(\mathcal{A}\subset\mathbb{N}^{M}\) denotes the set of selected multi-indices of the multivariate polynomials. A standard truncation scheme can be defined as all polynomials in the \(M\) input variables of total degree less or equal to \(p\):  
\[\mathcal{A}^{M,p}=\{\mathbf{\alpha}\in\mathbb{N}^{M}\,:\,|\mathbf{\alpha} |\leq p\}\] \[\text{card }\mathcal{A}^{M,p}\equiv P= \begin{pmatrix}M+p\\ p\end{pmatrix}  \tag{10}\]  
For more other truncation schemes, the reader is referred to Reference [59]. Polynomial basis functionsThe multivariate polynomials \(\Psi_{\mathbf{\alpha}}(\mathbf{X})\) are tensor product of the univariate polynomials:  
\[\Psi_{\mathbf{\alpha}}(\mathbf{X}):=\prod_{i=1}^{M}\psi_{x_{i}}^{(i)}(x_{i}). \tag{11}\]  
The univariate orthonormal polynomials \(\psi_{x_{i}}^{(i)}(x_{i})\) must satisfy the following:  
\[\langle\psi_{j}^{(i)}(x_{i}),\psi_{k}^{(i)}(x_{i})\rangle:=\int_{\mathcal{D}_{ X_{i}}}\psi_{j}^{(i)}(x_{i})\psi_{k}^{(i)}(x_{i})f_{X_{i}}\;(x_{i})dx_{i}= \delta_{jk}. \tag{12}\]  
where \(i\) represents the input variable with respect to which they are orthogonal as well as the corresponding polynomial family, \(j\) and \(k\) the corresponding polynomial degree, \(f_{X_{i}}(x_{i})\) is the \(i\)th-input marginal distribution and \(\delta_{jk}\) is the Kronecker delta. The classical families of univariate orthonormal polynomials are given for reference in Reference [60]. The calculation of polynomial basis via the classical families is grounded in the fact that an exact knowledge about the probability density functions is available. However, the information about the distribution of the data is distinctly restricted in engineering applications, most importantly when environmental influences or natural phenomena are of interest or when prediction is involved. [PERSON] & [PERSON] [58] demonstrate that statistical moments are the only source of information that is propagated in all polynomial expansion-based stochastic approaches.