## 2 Theoretical Background of the Best-Estimate Methodology
\right) \tag{3}\]  
where the \(\sigma\) terms represent the standard deviations of the two parameters and the two responses at different time instants. These matrices can be:  
* Diagonal if the correlations among the parameters/responses are unknown (most common case);
* Block diagonal if the correlations among the parameters/responses at a given time node are known;
* Full if the correlations among the parameters/response are known at each time node. The parameter and the response uncertainties are also correlated through the \(C_{\alpha r}\) matrix:  
\[C_{\alpha r}=\left(\begin{array}{cccc}\sigma_{\alpha_{1}}^{t_{1}}\cdot \sigma_{\tau_{1}}^{t_{1}}&\sigma_{\alpha_{2}}^{t_{1}}\cdot\sigma_{\tau_{1}}^{t_{1 }}&\sigma_{\alpha_{1}}^{t_{n}}\cdot\sigma_{\tau_{1}}^{t_{1}}&\sigma_{\alpha_{2 }}^{t_{n}}\cdot\sigma_{\tau_{1}}^{t_{1}}\\ \sigma_{\alpha_{1}}^{t_{1}}\cdot\sigma_{\tau_{2}}^{t_{1}}  
This matrix is usually rectangular because the number of parameters differs from that of the responses, and even if it is square, the elements on the main diagonal are not variances. Since the correlations between parameters and responses are a priori unknown, these matrix elements are null most of the time.  
In turn, the nominal values of the parameters and the responses are correlated through the sensitivity matrix. Several methodologies exist to evaluate these sensitivities, but BeSYC mostly relies on the brute force method [10]. The method consists of the execution of several code runs varying only a single parameter in each run. Each parameter is investigated twice: increasing and decreasing its value by about 3-5%. The brute force method can have a high computational cost depending on the number of considered parameters, on the size of the simulated system, and on the duration of the calculations. Under a theoretical point of view, the brute force method is a linearization via a Taylor-series expansion of a complex problem around the nominal values of the parameters. The sensitivity of a response (\(r\)) in respect to a parameter (\(\alpha\)) at a given time instant is given by:  
\[S_{r\alpha}^{t_{1}}=\frac{r_{+}^{t_{1}}-r_{-}^{t_{1}}}{\alpha_{+}^{t_{1}}- \alpha_{-}^{t_{1}}} \tag{5}\]  
where \(\alpha_{+}\) and \(\alpha_{-}\) are the values adopted by the parameter in the two sensitivity code runs, and \(r_{+}\) and \(r_{-}\) the values of the response in the same two runs. The sensitivity matrix is a block matrix having an arrangement like:  
\[S_{r\alpha}=\left( \begin{array}{cccc}S_{r_{1}{}^{t_{1}}\alpha_{1}{}^{t_{1}}} &S_{r_{1}{}^{t_{1}}\alpha_{2}{}^{t_{1}}}&0&0\\ S_{r_{2}{}^{t_{1}}\alpha_{1}{}^{t_{1}}}&S_{r_{2}{}^{t_{1}}\alpha_{2}{}^{t_{1}}} &0&0\\ \vdots&\vdots&\ddots&\vdots&\vdots\\ S_{r_{1}{}^{t_{1}}\alpha_{1}{}^{t_{1}}}&S_{r_{2}{}^{t_{1}}\alpha_{2}{}^{t_{1}}} &S_{r_{1}{}^{t_{1}}\alpha_{1}^{t_{1}}}&S_{r_{1}{}^{t_{1}}\alpha_{2}^{t_{1}}} \\ S_{r_{2}{}^{t_{2}}\alpha_{1}{}^{t_{1}}}&S_{r_{2}{}^{t_{2}}\alpha_{2}{}^{t_{1}}} &S_{r_{2}{}^{t_{2}}\alpha_{1}^{t_{1}}}&S_{r_{2}{}^{t_{2}}\alpha_{2}^{t_{2}}} \end{array} \right) \tag{6}\]  
where the block elements above the main diagonal are all null since the values of the parameters in the future cannot influence the results in the past. Very often, the blocks below the main diagonal are also null because the influence of the past parameter values on the responses at a later stage of the transient is unknown. The definition of the sensitivity matrix is the most computing-demanding task of the whole methodology considering that each parameter has to be investigated twice.