Generalized mixture representations and combinations for additive fuzzy systems
Bart Kosko · 2017
A generalized probability mixture density governs all additive fuzzy systems. These systems sum fired if-then rules to compute an output. Their mixture structure leads to new Bayes theorems and new ways to combine or fuse fuzzy systems. Additive fuzzy rule-base systems can uniformly approximate continuous functions while their Watkins representations can exactly represent bounded real vector functions with just two rules per vector component. A new separation theorem shows how to combine both such fuzzy systems into a common rule base. A Bayes theorem specifies which rules or which of the combined fuzzy systems fired to produce an observed output from a given input. We prove that two mixed Gaussian densities with Watkins coefficients define a mixture density whose first moment equals any bounded real function. A new learning law can tune the system's mixture density with training data. The additive fuzzy system's finite rule base passes over to a rule continuum. Monte Carlo sampling can then compute fuzzy-system outputs by mixture-based sampling from the virtual rule continuum.