# Fuzzy-Based Multivariate Analysis for Input Modeling of Risk Assessment in Wind Farm Projects
## 3 Materials and Methods
### Data Processing
#### 3.2.1 Marginal Distributions
Union \(U\), between the fuzzy relation matrices \(R\) (\(F\), \(C\)), is then computed as:  
\[U=max\;[(F_{1}\times\;C_{1})\;\cup\;(F_{2}\times\;C_{2})\;\cup\;(F_{3}\times\; C_{3})\;\dots\dots\dots\cup\;(F_{k}\times\;C_{k})], \tag{5}\]  
where \(max\) = the maximum value of the two relations. The next step uses expert knowledge to subjectively assess the relationship between the adverse consequence of a root cause and the overall cost or schedule impact of a risk factor. For example, \"if the adverse consequence of root 1 is large, then the overall risk impact is medium.\" The range of a risk factor impact (i.e., between minimum and maximum) is then calibrated by mapping the range to a predefined scale using the concept of membership values. The mapping values represent the confidence level with which the expert believes that a particular value belongs to the set [58] and that the impact will be in a certain range. To facilitate mapping, the risk impact range is divided into three equidistant subsets. Notably, Beta distributions can take multiple shapes, of which three cases, as shown in Figure 5, are of interest for calibration. These are shapes that are (1) skewed to the upper limit, (2) skewed to the lower limit, or (3) symmetric. Cases where the distribution is skewed toward the lower limit are mapped to the small impact range; cases where the distribution is symmetric are mapped to the medium impact range; while cases where the distribution is skewed toward the upper limit are mapped to the large impact range. Once all of the adverse consequences of root causes are related to the overall risk impact, a fuzzy relation Q (\(C\), \(I\)), which is the Cartesian product \(C\times\;I\) between fuzzy subset \(C\), representing the adverse consequence, and fuzzy subset \(I\), representing the overall risk impact as per Equation (3), is computed. After forming all fuzzy relations matrices, a union matrix, \(V\), of all relations is calculated using Equation (5). Finally, a fuzzy composition of the union matrices \(U\) and \(V\) is calculated to assess the overall combined impact of all root causes. The composition is defined by Equation (6), which is the standard max-min composition as follows:  
\[U\circ V(x_{i},z_{k})=\max_{y_{j}}\{min\big{[}\mu_{U}(x_{i},y_{i}),\mu_{V} \big{(}y_{j},z_{k}\big{)}\big{]}\}, \tag{6}\]  
where \(U\circ V(x_{i},z_{k})\) = membership value of element \((x_{i},z_{k})\) in composition matrix between \(U\) and \(V\); \(\mu_{U}(x_{i},y_{i})\) = membership value of element \((x_{i},y_{i})\) in union matrix \(U\); and \(\mu_{V}\big{(}y_{j},z_{k}\big{)}\) = membership value of element \(\big{(}y_{j},z_{k}\big{)}\) in union matrix \(V\). A fuzzy subset from the composition matrix is used to represent the overall uncertain variable under study [46]. A fuzzy subset (e.g., subset = \(O\)), or one row from the matrix, will be selected such  
Figure 5: Calibration of risk impact range. that the product of the row summation and the corresponding frequency of occurrence is the maximum. Then, the selected fuzzy subset (e.g., subset = \(O\)) will be used to calculate the mean, \(\mu_{I}\), and the variance, \(\sigma^{2}_{I}\), of the marginal risk impact according to the following equations derived from [50; 59; 60; 61]:  
\[P(R_{I}=z_{k})=\frac{\mu_{o}(z_{k})}{\sum_{1}^{m}\mu_{o}(z_{k})}, \tag{7}\]  
\[\mu_{I}=\sum\
olimits_{k=1}^{m}(z_{k})*\ P(R_{I}=z_{k}), \tag{8}\]  
\[A\ =\ \sum\
olimits^{n}_{1=1}\frac{\mu_{i}}{x_{i}}=\ \frac{\mu_{1}}{x_{1}}+\ \frac{\mu_{2}}{x_{2}}+\ldots\ldots+\frac{\mu_{n}}{x_{n}}=\ \sum\mu_{A}(x)/x, \tag{2}\]0  
where \(R_{I}\) = risk impact; \(z_{k}\) = element of the risk impact; \(P(R_{I}=z_{k})\) = probability of occurrence of the risk impact to be element \(z_{k}\); \(\mu_{o}(z_{k})\) = membership value of element \(z_{k}\) in subset \(O\); and m = number of risk impact elements in subset \(O\). Once the mean and variance are calculated, they are used, along with the minimum (\(A\)) and the maximum (\(B\)), to derive a generalized Beta distribution.