# Fuzzy-Based Multivariate Analysis for Input Modeling of Risk Assessment in Wind Farm Projects
## 4 Application and Results
### Data Processing
#### 4.2.2 Validation of the Marginal Distribution
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{1}{c}{\multirow{2}{*}{**Element of**}} & \multicolumn{5}{c}{**Frequency of Occurrence (\(F\))**} \\ \cline{2-6}  & **Very Unlikely** & **Unlikely** & **Somewhat Likely** & **Likely** & **Very Likely** \\ \hline
0.1 & 1 & 0 & 0 & 0 & 0 \\
0.2 & 0.5 & 0.5 & 0 & 0 & 0 \\
0.3 & 0 & 1 & 0 & 0 & 0 \\
0.4 & 0 & 0.5 & 0.5 & 0 & 0 \\
0.5 & 0 & 0 & 1 & 0 & 0 \\
0.6 & 0 & 0 & 0.5 & 0.5 & 0 \\
0.7 & 0 & 0 & 0 & 1.0 & 0 \\
0.8 & 0 & 0 & 0 & 0.5 & 0.5 \\
0.9 & 0 & 0 & 0 & 0 & 1 \\
1.0 & 0 & 0 & 0 & 0 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: New membership function for frequency of occurrence (\(F\)). Figure 9: Marginal Beta distribution of cost-risk impact after changing the membership function of (\(F\)). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Element of**} & \multicolumn{5}{c}{**Adverse Consequence (C)**} \\ \cline{2-5}  & **Very Small** & **Small** & **Medium** & **Large** & **Very Large** \\ \hline  
\hline \hline \end{tabular}
\end{table} 
Table 11: New membership function for adverse consequence (C). The final sensitivity experiment investigated the influence of changing the membership values for calibrating the risk impact range. This was conducted by experimenting with different mapping values, while keeping all other values (i.e., membership functions for frequency of occurrence and adverse consequences, boundaries of the risk impact, and root causes and their evaluation) unchanged and set to their original values. The impact mapping values for each impact subset (i.e., \"small impact\", \"medium impact\", or \"large impact\", as in Figure 6) were changed for each trial. Specific changes are summarized in Table 12. The parameters of the resulting distributions are detailed in Table 13, and the resulting probability density functions are shown in Figure 11a-f.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Trial**} & \multicolumn{3}{c}{**Impact Mapping Values 1**} & \multirow{2}{*}{**PDF 2**} & \multirow{2}{*}{**Distribution**} \\ \cline{2-2} \cline{5-6}  & **Small** & **Medium** & **Large** & & \\ \hline A & 1 & 0 & 0 & Figure 11a &
\begin{tabular}{c} Skews right, as values for small impact subset are equal to 1  
Results of the sensitivity analysis demonstrate that the proposed method is not particularly sensitive to changes in the membership function of the frequency of occurrence (\(F\)) nor the adverse consequence (\(C\)). In contrast, the model was shown to be sensitive to changes in the mapping values of the risk impact range. This finding indicates the importance of capturing an expert's perceived impact values as precisely as possible. As discussed previously, decomposition of (1) a risk factor into its root causes and/or (2) a risk impact range into smaller subsets can be used to reduce biases, \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Trial**} & \multicolumn{4}{c}{**Statistical Parameters of Beta Distribution**} \\ \cline{2-5}  & **Minimum (\$)** & **Maximum (\$)** & **\(\alpha\)** & **\(\delta\)** \\ \hline A & 10,000 & 40,000 & 1.5 & 7.5 \\ B & 10,000 & 40,000 & 8.5 & 8.5 \\ C & 10,000 & 40,000 & 7.5 & 1.5 \\ D & 10,000 & 40,000 & 1.3 & 1.3 \\ E & 10,000 & 40,000 & 12.035 & 32.785 \\ F & 10,000 & 40,000 & 35.175 & 13.44 \\ \hline \hline \end{tabular}
\end{table} 
Table 13: Results of sensitivity analysis of mapping values. Figure 11: Marginal Beta distributions after changing the mapping value of the risk impact range in (**a**) Trial A, (**b**) Trial B, (**c**) Trial C, (**d**) Trial D, (**e**) Trial E, and (**f**) Trial F.  
thereby enhancing the comprehensiveness and accuracy of model results. This explains the shape of the distribution observed in the illustrative case study (Figures 9 and 10), where the expert was not confident about the mapping values for the risk impact, resulting in distribution with a uniform-like appearance.