A fuzzy system approximates a function by covering the graph of the function with fuzzy rule patches and averaging patches that overlap. But the number of rules grows exponentially with the total number of input and output variables. The best rules cover the extrema or bumps in the function-they patch the bumps. For mean-squared approximation this follows from the mean value theorem of calculus. Optimal rules can help reduce the computational burden. To find them we can find or learn the zeroes of the derivative map and then center input fuzzy sets at these points. Neural systems can then both tune these rules and add rules to improve the function approximation. o 1995 John Wiley & Sons, Inc.
I. FUZZY FUNCTION APPROXIMATION AND THE CURSE OF DIMENSIONALITY
A fuzzy system needs too many fuzzy rules to approximate most functions. The number of rules grows exponentially with the number of input and output variables. In the end this "curse of dimensionality" can defeat an expert who guesses at the rules or a neural system that tries to learn the rules from data.
The rule geometry shows the problem. The rules define fuzzy patches that can cover part of the graph of the function. An additive fuzzy system adds or averages patches that overlap and can always approximate a continuous function on a compact set with a finite number of rules.' Forf: R + R it takes k rule patches in the plane to cover the graph. Forf: R 2 -+ R it takes on the order of k 2 rules to cover the surface in some 3-D rectangle. In general forf: R" .--) RP it takes on the order of k ' ' + p -l rules to cover the graph off.
Optimal rules can reduce the number of rules used to approximate a function. Neural learning tends to find some of these and so can prune the rule set as well as tune it. In theory we can find the best rules by minimizing the mean-squared error of the approximation for a given fuzzy architecture. A complete closed-form solution depends on the shape of the fuzzy sets and how the system converts inputs to outputs.