# Fuzzy-Based Multivariate Analysis for Input Modeling of Risk Assessment in Wind Farm Projects
## 2 Literature Review
### Correlation and Dependence in Risk Assessment  
A common source of error in MCS is the assumption of independence between the input variables of a model [32]. If two random variables are modeled as independent probability distributions in a simulation model, the sampled random variates of the two distributions will exhibit one of the following: (1) one variable is high, while the other is low, (2) both are high, or (3) both are low [25]. Disregarding the dependence or correlation in large risk models can result in estimation errors through under- or overestimation [25]. To improve risk analysis and subsequent decision-making, relationships between random variables must be considered [33]. Multiple research studies investigating the effect of correlation in Monte Carlo risk assessment models have been conducted. [PERSON] and [PERSON] [32] presented a methodology that can account for the correlation between cost components in probabilistic cost estimation models. It is important to note, however, that their model only considered variability in the cost of and correlation between project work packages without considering external risk factors in their model. [PERSON] [34] later extended the methodology by proposing a method that can account for subjective correlation between cost components by experts when historical data are absent. [PERSON] and [PERSON] [25] proposed a method that can account for dependencies between activity durations when developing a project schedule network. The authors suggested that activities affected by the same external risk factor should be dependent and, therefore, that correlations between project activities can be determined from common risk factors that are shared. However, the correlation between cost and time impact of a risk factor was not considered in their model. [PERSON] and [PERSON] [35] developed a heuristic method for correlating schedule-risk analysis in construction schedule planning. Their model considered correlations between activities that are affected by the same risk factors, as well as correlations between the risk factors themselves. However, only the impact of risks on the schedule were modeled. [PERSON] and [PERSON] [36] presented a model that considers correlations between the costs of activities in rail line construction when calculating the cost of a rail project. They then extended this by developing a comprehensive MCS model that considers (1) variability in activity duration and cost, (2) correlations between the cost of the activities, and (3) external risk factors affecting the project [37]. However, their model did not consider the correlation between the cost and schedule impact of a risk factor. Other research studies have attempted to use multivariate distributions for presenting the output of a simulation model for integrated cost and schedule-risk analysis [38; 39]. While certain commercial software, such as @risk developed by the Palisade Group [40], have correlations implemented, dependencies between the cost and schedule impact of a risk factor have not been addressed by these previously developed models. Copulas have been used in many applications to model dependencies between random variables. A copula-based joint distribution can be constructed using assessed rank-order correlations and marginal distributions, thereby reducing the effort required for assessments and to search for conditional independence [33]. Copulas are a flexible method, as they do not have any restrictions on the type of marginal distributions that can be used [41]. Using copulas to \"couple\" the marginal distribution requires two steps, namely (1) modeling the marginal distributions and (2) modeling dependencies between random variables [33]. Copulas have been used in risk management literature to successfully model the dependencies between variables that can affect the decision to purchase a used aircraft [33], in turn determining if the purchase of a used aircraft would generate more profit than using funds for other investments.