## 3 Regionalized Sensitivity Analysis
### Monte Carlo Filtering and Regionalized Sensitivity Analysis
To assess the statistical difference between the sets of parameter values that lead to acceptable and unacceptable realizations, a Kolmogorov-Smirnov two-sample test can then be performed by testing the hypotheses specified as follows:  
\[H_{0}:f_{m}(X_{i}|B)=f_{n}(X_{i}|\sim B)\ \text{vs.}\ H_{1}:f_{m}(X_{i}|B)\
eq f _{n}(X_{i}|\sim B) \tag{17}\]  
where \(f_{m}(\cdot)\) and \(f_{n}(\cdot)\) are the probability density functions defined on the two subsets determined by the threshold value. The test statistic of the Kolmogorov-Smirnov test is defined by  
\[d_{m,n}(X_{i})=\sup||F_{m}(X_{i}|B)-F_{n}(X_{i}|\sim B)||, \tag{18}\]  
where \(F_{m}(\cdot)\) and \(F_{n}(\cdot)\) are the corresponding cumulative distribution functions. The implication of the KS test results is as follows. A small p-value of the Kolmogorov-Smirnov test (and a large \(d_{m,n}\)) implies that the parameter \(X_{i}\) has a high level of importance in leading to a great magnitude of the relative error. The results of the Kolmogorov-Smirnov tests for the paired comparison analysis at 10% boundary value are shown in Tables 2 and 3, respectively. Although all parameters are used to compute the posterior means, it is not all initial parameters of the copula-based models that critically affect the magnitude of error between the results of AA and DA. Some factors were found to be more important than others. This can be noticed in Table 2. It is evident that the (shape and scale) parameters of prior probability distributions for failure rate (of Component 1) and the failure probability (of Component 2) are the most significant. Additionally, the number of failures of Component 2 (system failure) is also significant under 5% level of significance. Figure 1 graphically \begin{table}
\begin{tabular}{c c c c} \hline
**Parameter** & \(\mathbf{d_{m,n}}\) & \(\mathbf{p\text{-Value}}\) & **Significance** \\ \hline \(a\) & 0.1198 & \(2.54\times 10^{-54}\) & *** \\ \(b\) & 0.0596 & \(8.26\times 10^{-14}\) & *** \\ \(c\) & 0.0227 & 0.0230 & ** \\ \(d\) & 0.0528 & \(6.13\times 10^{-11}\) & *** \\ \(t\) & 0.0792 & 0.8972 & \\ \(k_{1}\) & 0.0891 & 0.7992 & \\ \(k_{2}\) & 0.1980 & 0.0327 & ** \\ \(\tau\) & 0.0495 & 0.9995 & \\ \hline \end{tabular} \({}^{*}p<0.1\), ** \(p<0.05\), *** \(p<0.01\).  
\end{table} 
Table 2: RSA results for AA\(-\)DA comparison. \begin{table}
\begin{tabular}{c c c c} \hline
**Parameter** & \(\mathbf{d_{m,n}}\) & \(\mathbf{p\text{-Value}}\) & **Significance** \\ \hline \(a\) & 0.0114 & 0.8354 & \\ \(b\) & 0.0110 & 0.8633 & \\ \(c\) & 0.0448 & \(1.34\times 10^{-05}\) & *** \\ \(d\) & 0.0239 & 0.0674 & * \\ \(t\) & 0.0495 & 0.9995 & \\ \(k_{1}\) & 0.1881 & 0.0489 & ** \\ \(k_{2}\) & 0.1833 & 0.0631 & * \\ \(\tau\) & 0.2202 & 0.0190 & ** \\ \hline \end{tabular} \({}^{*}p<0.1\), ** \(p<0.05\), *** \(p<0.01\).  
\end{table} 
Table 3: RSA results for DAI\(-\)DA comparison. displays the observed distribution of each parameter and enables one to visually compare whether the observed distribution of each parameter from behavioral and non-behavioral sets are different. The aggregation errors tend to be unacceptable when parameters \(a\) and \(b\) of Component 1 are small. The implication of this result is that DA is preferable when scientific uncertainty about the failure rate of Component 1 is relatively large. It should also be noted that the results obtained by AA tend to be significantly inferior to those obtained by DA for small values of \(k_{2}\) (i.e., low number of system failure). DA results are more preferable when there is insufficient evidence of system failures. On the other hand, it can be noticed that [PERSON]'s \(\tau\) was not significant in the analysis. Since both approaches (DA and AA) consider component dependence, this finding intuitively makes sense. The same comparison can be drawn between DAI and DA. However, a difference in the results obtained can be observed in Table 3. For instance, [PERSON]'s \(\tau\) was highly significant for the DAI-DA paired comparison, whereas it was the opposite case for AA \(-\)DA paired comparison.