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JACIII Vol.17 No.2 pp. 127-148
doi: 10.20965/jaciii.2013.p0127
(2013)

Paper:

Multi-Level Interpolation for Inference with Sparse Fuzzy Rules: An Extended Way of Generating Multi-Level Points

Kiyohiko Uehara* and Kaoru Hirota**

*Ibaraki University, 4-12-1 Nakanarusawa-cho, Hitachi 316-8511, Japan

**Tokyo Institute of Technology, G3-49, 4259 Nagatsuta, Midori-ku, Yokohama 226-8502, Japan

Received:
November 7, 2011
Accepted:
November 26, 2012
Published:
March 20, 2013
Keywords:
fuzzy inference, sparse rule base, nonlinear mapping, convex fuzzy set, α-cut
Abstract

As an extended way for inference based on multi-level interpolation, the number of multi-level points, generated with the α-cuts of given facts, is increased by making larger the number of levels of α. The conventional way uses the number of the levels adopted to define each given fact by α-cuts. A basic study is performed with triangular membership functions, where it is examined how the accuracy of mapping with fuzzy rules changes in increasing the number of the levels. It is also examined how deduced consequences behave when the number is increased. Moreover, convergent core sets of consequences are theoretically derived in the increase by the effective use of non-adaptive inference operations for core sets. They are used as references in simulation studies. Increasing the number of the levels provides nonlinear mapping with more precise reflection of distribution forms of sparse fuzzy rules to consequences. The basic study here contributes to improving the reflection accuracy. In simulations for the basic study, it is found that the mapping accuracy improves when the number of levels of α is increased. It is also confirmed that deduced core sets converge to those theoretically derived. Support sets are also found to converge in increasing the number of the levels. The core sets and support sets of deduced consequences do not, however, monotonically converge. This property causes difficulty in determining the optimized number of the levels so as to satisfy required mapping accuracy. In order to solve this problem, further discussions may be possible to theoretically derive convergent consequences and to use them in practical fields, in accordance with the theoretically derived core sets mentioned above.

Cite this article as:
Kiyohiko Uehara and Kaoru Hirota, “Multi-Level Interpolation for Inference with Sparse Fuzzy Rules: An Extended Way of Generating Multi-Level Points,” J. Adv. Comput. Intell. Intell. Inform., Vol.17, No.2, pp. 127-148, 2013.
Data files:
References
  1. [1] 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.
  2. [2] G. Cheng and Y. Fu, “Error Estimation of Perturbation under CRI,” IEEE Trans. Fuzzy Syst., Vol.14, No.6, pp. 709-715, Dec. 2006.
  3. [3] L. T. Kóczy and K. Hirota, “Approximate Reasoning by Linear Rule Interpolation and General Approximation,” Int. J. Approx. Reason., Vol.9, pp. 197-225, 1993.
  4. [4] L. T. Kóczy and K. Hirota, “Size Reduction by Interpolation in Fuzzy Rule Bases,” IEEE Trans. Syst., Man, Cybern. B, Cybern., Vol.27, No.1, pp. 14-33, Feb. 1997.
  5. [5] D. Tikk and P. Baranyi, “Comprehensive Analysis of a New Fuzzy Rule Interpolation Method,” IEEE Trans. Fuzzy Syst., Vol.8, No.3, pp. 281-296, June 2000.
  6. [6] P. Baranyi, L. T. Kóczy, and T. D. Gedeon, “A Generalized Concept for Fuzzy Rule Interpolation,” IEEE Trans. Fuzzy Syst., Vol.12, No.6, pp. 820-837, Dec. 2004.
  7. [7] 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.
  8. [8] 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.
  9. [9] 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.
  10. [10] L. T. Kóczy and Sz. Kovács, “On the Preservation of Convexity and Piecewise Linearity in Linear Fuzzy Rule Interpolation,” Technical Report, LIFE Chair of Fuzzy Theory, DSS, Tokyo Institute of Technology, Japan, p. 23, 1993.
  11. [11] L. T. Kóczy and Sz. Kovács, “Shape of the Fuzzy Conclusion Generated by Linear Interpolation of Trapezoidal If ... Then Rules,” Fuzzy Set Theory and its Applications, Tatra Mountains Mathematical Publications, Mathematical Institute Slovak Academy of Science, Vol.6, pp. 83-93, Bratislava, Slovakia, 1995.
  12. [12] A collection of papers related with fuzzy rule interpolation. http://fri.gamf.hu
  13. [13] K. Uehara, S. Sato, and K. Hirota, “Inference for Nonlinear Mapping with Sparse Fuzzy Rules Based on Multi-Level Interpolation,” J. of Advanced Computational Intelligence and Intelligent Informatics, Vol.15, No.3, pp. 264-287, May 2011.
  14. [14] 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.
  15. [15] 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.
  16. [16] 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.
  17. [17] K. Uehara, T. Koyama, and K. Hirota, “Suppression Effect of α-Cut Based Inference on Consequence Deviations,” J. of Advanced Computational Intelligence and Intelligent Informatics, Vol.14, No.3, pp. 256-271, April 2010.
  18. [18] L. A. Zadeh, “Fuzzy Logic = Computing with Words,” IEEE Trans. Fuzzy Syst., Vol.4, No.2, pp. 103-111, 1996.
  19. [19] 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.
  20. [20] A. Kaufmann, “Introduction to the Theory of Fuzzy Subsets,” New York: Academic, Vol.1, 1975.
  21. [21] N. R. Pal and J. C. Bezdek, “Measuring Fuzzy Uncertainty,” IEEE Trans. Fuzzy Syst., Vol.2, No.2, pp. 107-118, 1994.
  22. [22] R. R. Yager, “On the Specificity of a Possibility Distribution,” Fuzzy Sets Syst., Vol.50, pp. 279-292, 1992.
  23. [23] 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.
  24. [24] 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.

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