Fuzzy Systems Are Universal Approximators for Random Dependencies: A Simplified Proof

In many real-life situations, we do not know the actual dependence y = f(x1, ..., xn) between the physical quantities xi and y, we only know expert rules describing this dependence. These rules are often described by using imprecise ( fuzzy ) words from natural language. Fuzzy techniques have been invented with the purpose to translate these rules into a precise dependence y = f(x1, ..., xn). For deterministic dependencies y = f(x1, ..., xn), there are universal approximation results according to which for each continuous function on a bounded domain and for every ε \u3e 0, there exist fuzzy rules for which the resulting approximate dependence y = F(x1, ..., xn) is ε-close to the original function f(x1, ..., xn). In practice, many dependencies are random, in the sense that for each combination of the values x1, ..., xn, we may get different values y with different probabilities. It has been proven that fuzzy systems are universal approximators for such random dependencies as well. However, the existing proofs are very complicated and not intuitive. In this paper, we provide a simplified proof of this universal approximation property