## 2 Methodology
### Uncertainty Propagation Analysis
#### 2.4.1 Polynomial Chaos Expansion  
The Polynomial Chaos Expansion approximates a stochastic model \(\mathcal{M}(\mathbf{X})\), where the stochastic nature comes from the uncertainty of the inputs (\(\mathbf{X}\)), through a series of orthogonal polynomials. The random vector \(\mathbf{X}\) is represented by a joint probability distribution which, in the case of independent variables (as in this work), is simply the product of the single marginal probability density functions associated with each input. It is possible to express the model \(\mathcal{M}\) in terms of an expansion of orthogonal polynomials \(\Psi\) selected according to the joint distribution of the input:  
\[\mathcal{M}(\mathbf{X})=\sum_{c=0}^{\infty}\mathbf{a}_{\mathbf{c}}\Psi_{ \mathbf{c}}(\mathbf{X})\approx\sum_{c=0}^{C}\mathbf{a}_{\mathbf{c}}\Psi_{ \mathbf{c}}(\mathbf{X}), \tag{5}\]  
where \(\mathbf{a}_{\mathbf{c}}\) denotes the coefficients of the multivariate orthogonal polynomials. The infinite sum in Equation (5) must be truncated at the first \(C\) terms due to practical reasons. The total number of terms \(C\) in the polynomial expansion depends on the polynomial order \(p\) and on the dimension of the input \(d\) and, according to the total degree truncation scheme, it is as follows:  
\[C=\frac{(p+d)!}{p!d!}. \tag{6}\]  
There are several techniques for the evaluation of the polynomial coefficients, which are the unknowns of the PCE method. Some of these techniques are intrusive, requiring the modification of the model equations (e.g., the stochastic Galerkin method [24]); others are non-intrusive, where the computational model is treated as a black box (e.g., pseudo-spectral projection, point collocation [25], and stochastic testing [26]). In this work, the pseudo-spectral projection method implemented in _chaospy_ has been used.  
In the pseudo-spectral projection method, the coefficient of Equation (5) is computed through a quadrature integration scheme as follows:  
\[\mathbf{a}_{\mathbf{c}}=\int_{-\infty}^{+\infty}d\mathbf{X}\mathcal{M}( \mathbf{X})\Psi_{\mathbf{c}}(\mathbf{X})\mathbf{w}(\mathbf{X})\approx\sum_{i= 1}^{I}\mathcal{M}(\mathbf{X}_{i})\Psi_{\mathbf{c}}(\mathbf{X}_{i})w(\mathbf{ X}_{i}), \tag{7}\]  
where \(\mathbf{X}=X_{1},X_{2},\ldots,X_{I}\) is theuniform distributions, respectively; see [5] for a more detailed description about the relation between the type of distribution and the most appropriate orthogonal polynomial set. Once the Is exact model simulations have been performed, the polynomial metamodel can be used to evaluate the first moments of the output distributions (i.e., mean and variance), as well as the complete distributions and other quantities like Sobol's sensitivity indices, which allow us to rank the input parameters based on how much they affect the uncertainty of the output.  
In principle, the surrogate model is able to reproduce the results of the exact model inside the range of validity of the input distributions, with the advantage of being much faster and simpler. It is this peculiar characteristic which justifies the use of the metamodel for uncertainty quantification purposes.