Copula Functions: Characterizing Uncertainty in Probabilistic Systems

Understanding and measuring uncertainty is central in risk analysis. Uncertainty emerges when there is less information than the total information required to describe a system and environment. Uncertainty and information are so closely associated that information provided by an experiment for instance is equal to the amount of uncertainty removed. Uncertainty prevails in several forms and various kinds of uncertainties may arise from random fluctuations, incomplete information, imprecise perception, vagueness etc. The probability theory based framework deals with the uncertainty of random phenomenon while fuzzy set concept provides an appropriate mathematical framework to deal with vagueness. We in this paper apply the concept of copula functions to characterize uncertainty associated with the probabilistic systems. Copula functions join uniform marginal distributions of random variables to form their multivariate distribution functions. Copulas are useful because they separate joint distributions into two contributions(i) marginal distributions of each variable and (ii) copula as a measure of dependence. Several families of copulas with varying shapes and simulation programs are available providing flexibility in copula based modeling.