single-jc.php

JACIII Vol.14 No.1 pp. 76-88
doi: 10.20965/jaciii.2010.p0076
(2010)

Paper:

Inference Based on α-Cut and Generalized Mean with Fuzzy Tautological Rules

Kiyohiko Uehara*, Takumi Koyama*, and Kaoru Hirota**

*Ibaraki University, Hitachi 316-8511, Japan

**Tokyo Institute of Technology, Yokohama 226-8502, Japan

Received:
May 12, 2009
Accepted:
August 10, 2009
Published:
January 20, 2010
Keywords:
fuzzy inference, convex fuzzy set, α-cut, generalized mean, compositional rule of inference
Abstract

Theoretical aspects are provided for inference based on α-cuts and generalized mean (α-GEMII). In order to clarify the basic properties of the inference, fuzzy tautological rules (FTRs) are focused on, which are composed by setting fuzzy sets in consequent parts identical to those in antecedent parts of initially given fuzzy rules. It is mathematically proved that the consequences deduced with FTRs are closer to given facts as the number of FTRs increases. The aspects provided in this paper are appropriate from axiomatic viewpoints and can contribute to interpretability in fuzzy systems constructed with α-GEMII. They are not obtained in conventional methods based on the compositional rule of inference. Simulations are performed by evaluating difference (mean square errors) between given facts and deduced consequences under the condition that convex and symmetric fuzzy sets are given as facts. Their results show that the difference becomes smaller as the number of FTRs increases. Thereby, it is confirmed that α-GEMII has an advantage in the interpretability with respect to FTRs over the conventional methods.

Cite this article as:
Kiyohiko Uehara, Takumi Koyama, and Kaoru Hirota, “Inference Based on α-Cut and Generalized Mean with Fuzzy Tautological Rules,” J. Adv. Comput. Intell. Intell. Inform., Vol.14, No.1, pp. 76-88, 2010.
Data files:
References
  1. [1] L. A. Zadeh, “Fuzzy logic = Computing with words,” IEEE Trans. Fuzzy Syst., Vol.4, No.2, pp. 103-111, 1996.
  2. [2] 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.
  3. [3] 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.
  4. [4] G. Cheng and Y. Fu, “Error estimation of perturbation under CRI,” IEEE Trans. Fuzzy Syst., Vol.14, No.6, pp. 709-715, Dec. 2006.
  5. [5] V. G. Kaburlasos, “FINs: lattice theoretic tools for improving prediction of sugar production from populations of measurements,” IEEE Trans. Syst., Man, Cybern. B, Cybern., Vol.34, No.2, pp. 1017-1030, Apr. 2004.
  6. [6] V. G. Kaburlasos and A. Kehagias, “Novel fuzzy inference system (FIS) analysis and design based on lattice theory. part I: working principles,” Int. J. of General Systems, Vol.35, No.1, pp. 45-67, Feb. 2006.
  7. [7] V. G. Kaburlasos and A. Kehagias, “Novel fuzzy inference system (FIS) analysis and design based on lattice theory,” IEEE Trans. Fuzzy Systems, Vol.15, No.2, pp. 243-260, Apr. 2007.
  8. [8] V. G. Kaburlasos and S. E. Papadakis, “Granular self-organizing map (grSOM) for structure identification,” Neural Networks, Vol.19, pp. 623-643, 2006.
  9. [9] S.-Q. Fan, W.-X. Zhanga, and W. Xu, “Fuzzy inference based on fuzzy concept lattice,” Fuzzy Sets and Systems, Vol.157, pp. 3177-3187, Dec. 2006.
  10. [10] 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.
  11. [11] Z. Huang and Q. Shen, “Fuzzy interpolative reasoning via scale and move transformation,” IEEE Trans. Fuzzy Syst., Vol.14, No.2, pp. 340-359, Apr. 2006.
  12. [12] Z. Huang and Q. Shen, “Fuzzy interpolation and extrapolation: A practical approach,” IEEE Trans. Fuzzy Syst., Vol.16, No.1, pp. 13-28, Apr. 2008.
  13. [13] 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.
  14. [14] 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.
  15. [15] 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).
  16. [16] 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).
  17. [17] 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.
  18. [18] 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.
  19. [19] 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.
  20. [20] 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. 321-330, 2009.
  21. [21] 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.
  22. [22] A. Kaufmann, “Introduction to the theory of fuzzy subsets,” New York: Academic, Vol.1, 1975.
  23. [23] N. R. Pal and J. C. Bezdek, “Measuring Fuzzy Uncertainty,” IEEE Trans. Fuzzy Syst., Vol.2, No.2, pp. 107-118, 1994.
  24. [24] R. R. Yager, “On the specificity of a possibility distribution,” Fuzzy Sets Syst., Vol.50, pp. 279-292, 1992.
  25. [25] 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.
  26. [26] H. Jones, B. Charnomordic, D. Dubois, and S. Guillaume, “Practical inference with systems of gradual implicative rules,” IEEE Trans. Fuzzy Syst., Vol.17, No.1, pp. 61-78, Feb. 2009.
  27. [27] M. Mizumoto and H. J. Zimmermann, “Comparison of fuzzy reasoning methods,” Fuzzy Sets Syst., Vol.8, pp. 253-283, 1982.
  28. [28] S. Fukami, M. Mizumoto, and K. Tanaka, “Some considerations on fuzzy conditional inference,” Fuzzy Sets Syst., Vol.4, pp. 243-273, 1980.
  29. [29] S. J. Chen and S. M. Chen, “Fuzzy risk analysis based on similarity measures of generalized fuzzy numbers,” IEEE Trans. Fuzzy Syst., Vol.11, No.1, pp. 45-56, Feb. 2003.
  30. [30] J. Casillas, O. Cordón, M. J. del Jesus, and F. Herrera, “Genetic tuning of fuzzy rule deep structures preserving interpretability and its interaction with fuzzy rule set reduction,” IEEE Trans. Fuzzy Syst., Vol.13, No.1, pp. 13-29, Feb. 2005.
  31. [31] L. A. Zadeh, “The concept of a linguistic truth variable and its application to approximate reasoning — I, II, III,” Informat. Sci., Vol.8, pp. 199-249, pp. 301-357, Vol.9, pp. 43-80, 1975.
  32. [32] K. Uehara, “Fuzzy inference based on a weighted average of fuzzy sets and its learning algorithm for fuzzy exemplars,” in Proc. of the Int. Joint Conf. of the Fourth IEEE Int. Conf. on Fuzzy Systems and the Second Int. Fuzzy Engineering Symposium, FUZZ-IEEE⁄IFES’95 (Yokohama, Japan), 1995, Vol.IV, pp. 2253-2260.
  33. [33] 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.
  34. [34] K. Uehara and K. Hirota, “Parallel and multistage fuzzy inference based on families of α-level sets,” Int. J. of Information Sciences, Vol.106, No.1-2, Apr. 1998.
  35. [35] K. Uehara and M. Fujise, “Learning of fuzzy-inference criteria with artificial neural network,” in Proc. of Int. Conf. on Fuzzy Logic & Neural Networks, IIZUKA ’90, Vol.1, Iizuka, Japan, Jul. 1990, pp. 193-198.
  36. [36] K. Uehara, E. Taguchi, T. Watahiki, and T. Miyata, “A data converter for analog/fuzzy-logic interface,” The Trans. of the Institute of Electronics and Communication Engineers, Vol.J67-C, No.4, pp. 391-396, 1984 (in Japanese).
  37. [37] L. T. Kóczy and K. Hirota, “Approximate reasoning by linear rule interpolation and general approximation,” Int. J. Approx. Reason., Vol.9, No.3, pp. 197-225, 1993.
  38. [38] L. A. Zadeh, “Outline of a new approach to the analysis of complex systems and decision processes,” IEEE Trans. Syst. Man Cybern., Vol.SMC-3, No.1, pp. 28-44, 1973.
  39. [39] P. T. Chang and K. C. Hung, “α-cut fuzzy arithmetic: Simplifying rules and a fuzzy function optimization with a decision variable,” IEEE Trans. Fuzzy Syst., Vol.14, No.4, pp. 496-510, Aug. 2006.
  40. [40] P. T. Chang, K. C. Hung, K. P. Lin, and C. H. Chang, “A comparison of discrete algorithms for fuzzy weighted average,” IEEE Trans. Fuzzy Syst., Vol.14, No.5, pp. 809-819, 2006.

*This site is desgined based on HTML5 and CSS3 for modern browsers, e.g. Chrome, Firefox, Safari, Edge, Opera.

Last updated on Mar. 01, 2021