Copulas and t-norms: Mathematical tools for combining probabilistic and fuzzy information, with application to error propagation and interaction

Copulas are the mathematical tool for combining probability distributions, t-norms are the mathematical tool for combining membership functions of fuzzy sets. Both are similar, thus probability theory and fuzzy set theory are not so different. After introducing into copulas and t-norms their role in error propagation and in modeling interactions is discussed. Especially, soft error propagation and Gaussian error propagation are investigated. For example, t-norms can be used to model the strength of soft error propagation and copulas give some insight into Gaussian error propagation. Interactions are modeled classically by regression models. Here a fuzzy-measure-based model is introduced and proposed, where the type of fuzzy measure is modeled by t-conorms.