This Article 
 Bibliographic References 
 Add to: 
Fuzzy Systems as Universal Approximators
November 1994 (vol. 43 no. 11)
pp. 1329-1333

An additive fuzzy system can uniformly approximate any real continuous function on a compact domain to any degree of accuracy. An additive fuzzy system approximates the function by covering its graph with fuzzy patches in the input-output state space and averaging patches that overlap. The fuzzy system computes a conditional expectation E|Y|X| if we view the fuzzy sets as random sets. Each fuzzy rule defines a fuzzy patch and connects commonsense knowledge with state-space geometry. Neural or statistical clustering systems can approximate the unknown fuzzy patches from training data. These adaptive fuzzy systems approximate a function at two levels. At the local level the neural system approximates and tunes the fuzzy rules. At the global level the rules or patches approximate the function.

[1] J. A. Dickerson and B. Kosko, "Fuzzy Function approximation with supervised ellipsoidal learning," inProc. World Congress on Neural Netw. (INNS WCNN-93), vol. 2, July 1993, pp. 9-17.
[2] D. Dubois and H. Prade,Fuzzy Sets and Systems: Theory and Applications. New York: Academic, 1980.
[3] E. Hartman, J. D. Keeler, and J. Kowalski, "Layered neural networks with gaussian hidden units as universal approximators,"Neural Computat., vol. 2, pp. 210-215, 1990.
[4] K. Hornik, M. Stinchcombe, and H. White, "Multilayer feedforward networks are universal approximators,"Neural Networks, vol. 2, pp. 359-366, 1989.
[5] B. Kosko, "Fuzzy knowledge combination,"Int. J. Intell. Syst., vol. 1, pp. 293-320, 1986.
[6] B. kosko, "Stochastic competitive learning,"IEEE Trans. Neural Netw., vol. 2, no. 5, pp. 522-529, Sept. 1991.
[7] B. Kosko,Neural Networks and Fuzzy Systems: A Dynamical Systems Approach to Machine Intelligence, Englewood Cliffs, NJ: Prentice Hall, 1991.
[8] C. A. Mead,Analog VLSI and Neural Systems. Reading, MA: Addison-Wesley, 1989.
[9] W. Rudin,Functional Analysis. New York: McGraw-Hill, 1973.
[10] W. Rudin,Real and Complex Analysis, second ed. New York: McGraw-Hill, 1974.
[11] L. Wang and J. M. Mendel, "Fuzzy basis functions, universal approximation, and orthogonal least-squares learning,"IEEE Trans. Neural Netw., vol. 3, no. 5, pp. 807-814, Sept. 1992.
[12] L. A. Zadeh, "Fuzzy sets,"Inform. Contr., vol. 8, pp. 338-353, 1965.

Index Terms:
neural nets; function approximation; curve fitting; fuzzy set theory; universal approximators; additive fuzzy system; fuzzy patches; input-output state space; conditional expectation; commonsense knowledge; state-space geometry; statistical clustering systems; training data; neural system; fuzzy rules.
B. Kosko, "Fuzzy Systems as Universal Approximators," IEEE Transactions on Computers, vol. 43, no. 11, pp. 1329-1333, Nov. 1994, doi:10.1109/12.324566
Usage of this product signifies your acceptance of the Terms of Use.