Abstract
Zadeh’s original motivation for fuzzy logic and the Fuzzy Rule-Based System (FRBS) was linguistic and hence possessed highly interpretable components. But as the complexity of a typical FRBS increases, it often becomes more like an uninterpretable neural network, and the Principle of Incompatibility predicts a degradation in interpretability for the same accuracy. This is particularly true of the so-called Takagi-Sugeno-Kang (TSK), or simply the Sugeno, approximator. We argue that imposing additional structure on the TSK system can significantly improve the tradeoff inherent in the Principle of Incompatibility. A promising structure was proposed recently in which the membership functions are local and sufficiently differentiable, and the consequent polynomials are rule-centered. This structure leads to the general interpretation that the consequent polynomials are Taylor series expansions. On this interpretation, a foundation for an algebra and a calculus of FRBSs can be built. We will illustrate these aspects of the proposed structure and discuss issues of modularity, functionality, and scalability of FRBSs.
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References
J. J. Buckley, Sugeno-type controllers are universal controllers. Fuzzy Sets & Systems, 53:299–303, 1993.
P. Bauer, E. P. Klement, A. Leikermoser, and B. Moser, Interpolation and approximation of real input—output functions using fuzzy rule bases. In Kruse, R., Gebhardt, J., and palm, R. (eds.) Fuzzy Systems in Computer Science, pages 245–254, Vieweg, Braunschweig, 1994.
M. Bikdash, A highly interpretable form of the Sugeno inference system, IEEE Transactions on Fuzzy Systems, 7(6):686–696, 1999.
J.-S. Jang, ANFIS: Adaptive network-based fuzzy inference systems, IEEE Transactions on Systems, Man, and Cybernetics, 23 (3):665–685, May 1993.
J. Y. R. Jang, C.-T. Sun, and E. Mizutani, Neuro-fuzzy and soft computing, Prentice Hall, New Jersey, 1997.
C. L. Karr, Design of an adaptive fuzzy logic controller using a genetic algorithm, Proceedings of the Fourth International Conference on Genetic Algorithms, pages 450–457, 1991.
B. Kosko, Fuzzy Engineering, Prentice Hall, 1997.
K. Hornick, M. Stinchcombe, and H. White, Multi-layer feedforward networks are universal approximators, Neural Networks, 2:359–366, 1989.
S. H. Lane, M. G. Flax, D. A. Handelmanand, and J. J. Gelfand, Multilayer perceptrons with b-spline receptive field functions, In J. E. Moody and D. Touretzky, Editors, Advances in Neural Information Processing Systems, 3:684–692, San Mateo, CA, 1991, Morgan Kaufman.
E. H., Mamdani, and Assilian, An experiment in linguistic synthesis with a fuzzy logic controller, International Journal of Man and Machine Studies, 7:1–13, 1975.
H. Nomura, I. Hayashi, and N. Wakami, A self-tuning method of fuzzy reasoning by genetic algorithm, Proceedings of the International Fuzzy Systems and Intelligent (Control Conference IFSICC ’92), Louisville, KY, pages 236–245, 1992.
K. Shimojima, T. Fukuda, F. Arai, Self—tuning fuzzy inference based on spline function, In Proceedings of the IEEE International Conference on Fuzzy Systems, pages 690–695, June 1994.
M. Sugeno, and G. T. Kang, Structure identification of fuzzy model. Fuzzy Sets and Systems, 28:15–33, North-Holland Publishing Company, 1988.
T. Takagi, and M. Sugeno, Fuzzy identification of systems and its applications to modeling and control, IEEE Transactions on Systems, Man, Cybernetics, SMC-15(1):116–132, 1985.
L. X. Wang, Fuzzy systems are universal approximators, in Proceedings of the IEEE International Conference on Fuzzy Systems, San Diego, March 1992.
L. X. Wang, and J. M. Mendel, Generating fuzzy rules by learning from examples, IEEE Transactions on Systems, Man, Cybernetics, 22(6):1414–1427, 1992.
L. X. Wang, and J. M. Mendel, Fuzzy basis function, universal approximation, and orthogonal least squares, IEEE Transactions on Neural Networks, 3(5):1414–1427, September 1992.
C.-H. Wang, T.-T. Wang, Lee, and P.-S. Tseng, Fuzzy B-spline MFs (BMF) and its applications in fuzzy neural control, IEEE Transactions on Systems, Man and Cybernetics, 25(5):841–851, May 1995.
L. A. Zadeh, Fuzzy sets, Information Control, 8:338–353, 1965.
L. A. Zadeh, Outline of a new approach to the analysis of complex systems and decision processes, IEEE Transactions on Systems, Man, and Cybernetics, SMC-3:28–44, 1973.
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Bikdash, M. (2003). Interpretability, Complexity, and Modular Structure of Fuzzy Systems. In: Casillas, J., Cordón, O., Herrera, F., Magdalena, L. (eds) Interpretability Issues in Fuzzy Modeling. Studies in Fuzziness and Soft Computing, vol 128. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-37057-4_15
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DOI: https://doi.org/10.1007/978-3-540-37057-4_15
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