Inference Based on α-Cut and Generalized Mean in Representing Fuzzy-Valued Functions
Kiyohiko Uehara*, Takumi Koyama*, and Kaoru Hirota**
*Ibaraki University, Hitachi 316-8511, Japan
**Tokyo Institute of Technology, Yokohama 226-8502, Japan
It is mathematically proved that inference based on α-cuts and generalized mean (α-GEMII) deduces consequences converging to fuzzy sets mapped by linear fuzzy-valued functions, to be represented with α-GEMII, as the number of fuzzy rules increases. The proof indicates that α-GEMII satisfies axiomatic properties and can contribute to presenting interpretability in designing fuzzy systems in the rule base. Such properties do not hold in conventional methods based on the compositional rule of inference. Simulation results show that the difference between deduced consequences and fuzzy sets mapped by linear fuzzyvalued functions is smaller as the number of fuzzy rules increases, in which the difference is evaluated by mean square errors. The discussions may lead to improvements of the interpretability in representing nonlinear fuzzy-valued functions by using α-GEMII.
-  I. B. Turksen and Y. Tian, “Combination of Rules or Their Consequences in Fuzzy Expert Systems,” Fuzzy Sets Syst., Vol.58, No.1, pp. 3-40, 1993.
-  G. Cheng and Y. Fu, “Error Estimation of Perturbation under CRI,” IEEE Trans. Fuzzy Syst., Vol.14, No.6, pp. 709-715, Dec. 2006.
-  K. Uehara and M. Fujise, “Multistage Fuzzy Inference Formulated as Linguistic-Truth-Value Propagation and its Learning Algorithm Based on Back-Propagating Error Information,” IEEE Trans. Fuzzy Syst., Vol.1, No.3, pp.205-221, Aug. 1993.
-  H. Maeda, M. Imuro, and S. Murakami, “A Study on the Spread of Fuzziness in Multi-Fold Multi-Stage Approximating Reasoning – Approximate Reasoning with Triangular Type Membership Function –,” J. of Japan Society for Fuzzy Theory and Systems, Vol.7, No.1, pp. 113-130, 1995 (in Japanese).
-  M. Imuro and H. Maeda, “On the Spread of Fuzziness in Multi-Fold and Multi-Stage Fuzzy Reasoning,” in Proc. 8th Fuzzy System Symposium, Hiroshima, Japan, May 1992, pp. 221-224 (in Japanese).
-  K. Uehara, T. Koyama, and K. Hirota, “Fuzzy Inference with Schemes for Guaranteeing Convexity and Symmetricity in Consequences Based on α-Cuts,” J. of Advanced Computational Intelligence and Intelligent Informatics, Vol.13, No.2, pp. 135-149, 2009.
-  K. Uehara, T. Koyama, and K. Hirota, “Inference Based on α-Cut and Generalized Mean with Fuzzy Tautological Rules,” J. of Advanced Computational Intelligence and Intelligent Informatics, Vol.14, No.1, pp. 76-88, 2010.
-  V. Torra and Y. Narukawa, “Modeling Decisions – Information Fusion and Aggregation Operators,” Springer, 2007.
-  K. Uehara, “Fuzzy Inference Based on a Weighted Average of Fuzzy Sets and its Learning Algorithm for Fuzzy Exemplars,” Proc. the Int. Joint Conf. of the Fourth IEEE Int. Conf. on Fuzzy Systems and the Second Int. Fuzzy Engineering Symposium, FUZZIEEE/IFES’95 (Yokohama, Japan), Vol.IV, pp. 2253-2260, Mar. 1995.
-  K. Uehara and K. Hirota, “Fuzzy Connection Admission Control for ATM Networks Based on Possibility Distribution of Cell Loss Ratio,” IEEE J. on Selected Areas in Communications, Vol.15, No.2, pp. 179-190, Feb. 1997.
-  Y.-Z. Zhang and H.-X. Li, “Variable Weighted Synthesis Inference Method for Fuzzy Reasoning and Fuzzy Systems,” Computers and Mathematics with Applications, Vol.52, pp. 305-322, 2006.
-  Z. Huang and Q. Shen, “Fuzzy Interpolative Reasoning via Scale and Move Transformation,” IEEE Trans. Fuzzy Syst., Vol.14, No.2, pp. 340-359, April 2006.
-  Z. Huang and Q. Shen, “Fuzzy Interpolation and Extrapolation: A Practical Approach,” IEEE Trans. Fuzzy Syst., Vol.16, No.1, pp. 13-28, April 2008.
-  K. W. Wong, D. Tikk, T. D. Gedeon, and L. T. Kóczy, “Fuzzy Rule Interpolation for Multidimensional Input Spaces with Applications: A Case Study,” IEEE Trans. Fuzzy Syst., Vol.13, No.6, pp. 809-819, Dec. 2005.
-  K. Uehara and K. Hirota, “Parallel Fuzzy Inference Based on α-Level Sets and Generalized Means,” Int. J. of Information Sciences, Vol.100, No.1-4, pp. 165-206, Aug. 1997.
-  K. Uehara, T. Koyama, and K. Hirota, “Inference with Governing Schemes for Propagation of Fuzzy Convex Constraints Based on α-Cuts,” J. of Advanced Computational Intelligence and Intelligent Informatics, Vol.13, No.3, pp. 135-149, 2009.
-  L. A. Zadeh, “Fuzzy Logic = Computing with Words,” IEEE Trans. Fuzzy Syst., Vol.4, No.2, pp. 103-111, 1996.
-  L. A. Zadeh, “Inference in Fuzzy Logic via Generalized Constraint Propagation,” Proc. 1996 26th Int. Symposium on Multi-Valued Logic (ISMVL’96), pp. 192-195, 1996.
-  A. Kaufmann, “Introduction to the theory of fuzzy subsets,” Academic Press (New York), Vol.1, 1975.
-  N. R. Pal and J. C. Bezdek, “Measuring Fuzzy Uncertainty,” IEEE Trans. Fuzzy Syst., Vol.2, No.2, pp. 107-118, 1994.
-  R. R. Yager, “On the Specificity of a Possibility Distribution,” Fuzzy Sets Syst., Vol.50, pp. 279-292, 1992.
-  R. R. Yager, “Measuring Tranquility and Anxiety in Decision Making: an Application of Fuzzy Sets,” Int. J. General Systems, Vol.8, pp. 139-146, 1982.
-  K. Uehara and M. Fujise, “Fuzzy Inference Based on Families of α-Level Sets,” IEEE Trans. Fuzzy Syst., Vol.1, No.2, pp. 111-124, May 1993.
-  S. Aja-Fernández and C. Alberola-López, “Fast Inference Using Transition Matrices: An Extension to Nonlinear Operators,” IEEE Trans. Fuzzy Syst., Vol.13, No.4, pp. 478-490, Aug. 2005.