Probabilistic Fuzzy Systems as Additive Fuzzy Systems

Probabilistic fuzzy systems combine a linguistic description of the system behaviour with statistical properties of data. It was originally derived based on Zadeh’s concept of probability of a fuzzy event. Two possible and equivalent additive reasoning schemes were proposed, that lead to the estimation of the output’s conditional probability density. In this work we take a complementary approach and derive a probabilistic fuzzy system from an additive fuzzy system. We show that some fuzzy systems with universal approximation capabilities can compute the same expected output value as probabilistic fuzzy systems and discuss some similarities and differences between them. A practical relevance of this functional equivalence result is that learning algorithms, optimization techniques and design issues can, under certain circumstances, be transferred across different paradigms.