## 3 Regionalized Sensitivity Analysis
### Monte Carlo Filtering and Regionalized Sensitivity Analysis  
To identify the key factors most responsible for the output realization of interest, regional sensitivity analysis (RSA) can be utilized. RSA performs a multi-parameter Monte Carlo simulation by sampling parameters from statistical distribution functions. To be specific, the reliability analyst proceeds in two steps: (1) qualitatively defining the system behavior (i.e., setting a target threshold for the model); and (2) classifying the model outputs into a binary set (namely, behavioral and non-behavioral). A \"behavioral set B\" or \((X_{i}|B)\) is the classification set whereby the model's output results lie within the target threshold (i.e., leading to a small magnitude of error or uncertainty) while a \"non-behavioral set \(\sim B\)\" or \((X_{i}|\sim B)\) is the classification set whereby the model's output results do not lie within the target threshold (i.e., leading to a large magnitude of error or uncertainty). The RSA procedure is a Monte Carlo filtering based procedure that was first developed and applied to identify key processes or parameters resulting in great uncertainties in environmental systems [30; 31]. By identifying or even ranking the influential parameters which are critical for producing uncertainties, this analysis can help in prioritizing research effort to further understand the system and/or reduce uncertainties of the system. In our analysis, the importance of a model parameter is determined by its role (relative to other parameters) in causing the absolute relative error of a Bayesian reliability analysis approach to exceed some pre-specified threshold. Correspondingly, in this study, the relative error between AA and DA approaches (\(RE_{AA-DA}\)) and the relative error between DAI and DA (\(RE_{DAI-DA}\)) can be computed by using the following expressions. \[RE_{AA-DA}=\frac{\left|\text{AA Posterior Mean}\right.-\left.\text{DA Posterior Mean}\right|}{\text{DA Posterior Mean}}\times 100\% \tag{15}\]  
\[RE_{DAI-DA}=\frac{\left|\text{DAI Posterior Mean}\right.-\left.\text{DA Posterior Mean}\right|}{\text{DA Posterior Mean}}\times 100\% \tag{16}\]  
In a similar way to [23], the RSA procedure in this paper is implemented as follows. We first define ranges for \(k\) input parameters \(X_{i}\) (\(i=1,2,\ldots,k\)) to represent the variation in the inputs to the model, and then classify the resulting relative error as either acceptable \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Description** & **Distribution** & **Range** \\ \hline \(a\) & Shape Parameter of Component 1 & Uniform & \([0,100]\) \\ \(b\) & Scale Parameter of Component 1 & Uniform & \([0,100]\) \\ \(c\) & Shape Parameter of Component 2 & Uniform & \([0,100]\) \\ \(d\) & Shape Parameter of Component 2 & Uniform & \([0,100]\) \\ \(t\) & observation time & Uniform & \([0,100]\) \\ \(k_{1}\) & \# of failures of Comp. 1 & Uniform & \([1,2,\ldots,100]\) \\ \(k_{2}\) & \# of failures of Comp. 2 & Uniform & \([1,2,\ldots,100]\), \(k_{2}\leq k_{1}\) \\ \(\tau\) & Value of Kendall’s tau & Uniform & \([-0.8,0.8]\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Statistical distributions of the pseudo parameters. (i.e., in the behavioral set \(B\)) or unacceptable (i.e., in the non-behavioral set \(\sim\)\(B\)). In our analysis, an acceptable realization is obtained when the relative error falls below a target threshold of 10%. This enables us to establish two subsets for each \(X_{i}\): (\(X_{i}|B\)) of m elements and \(X_{i}\):(\(X_{i}|\sim B\)) of n elements with \(m+n\) equal to the sum of pseudo parameters generated.