Paper:

# Inference for Nonlinear Mapping with Sparse Fuzzy Rules Based on Multi-Level Interpolation

## Kiyohiko Uehara^{*}, Shun Sato^{*}, and Kaoru Hirota^{**}

^{*}Ibaraki University, Hitachi 316-8511, Japan

^{**}Tokyo Institute of Technology, Yokohama 226-8502, Japan

An inference method is proposed for sparse fuzzy rules on the basis of interpolations at a number of points determined by α-cuts of given facts. The proposed method can perform nonlinear mapping even with sparse rule bases when each given fact activates a number of fuzzy rules which represent nonlinear relations. The operations for the nonlinear mapping are exactly the same as for the case when given facts activate no fuzzy rules due to the sparseness of rule bases. Such nonlinear mapping cannot be provided by conventional methods for sparse fuzzy rules. In evaluating the proposed method, mean square errors are adopted to indicate difference between deduced consequences and fuzzy sets transformed by nonlinear fuzzy-valued functions to be represented with sparse fuzzy rules. Simulation results show that the proposed method can follow the nonlinear fuzzy-valued functions. The proposed method contributes to both reducing the number of fuzzy rules and providing nonlinear mapping with sparse rule bases.

*J. Adv. Comput. Intell. Intell. Inform.*, Vol.15, No.3, pp. 264-287, 2011.

- [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] 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] 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] 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] 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] 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] 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] 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] 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] 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.
- [11] 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.
- [12] 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.
- [13] 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, Apr. 2010.
- [14] 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.
- [15] L. A. Zadeh, “Fuzzy Logic = Computing with Words,” IEEE Trans. Fuzzy Syst., Vol.4, No.2, pp. 103-111, 1996.
- [16] 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.
- [17] A. Kaufmann, “Introduction to the Theory of Fuzzy Subsets,” New York: Academic, Vol.1, 1975.
- [18] N. R. Pal and J. C. Bezdek, “Measuring Fuzzy Uncertainty,” IEEE Trans. Fuzzy Syst., Vol.2, No.2, pp. 107-118, 1994.
- [19] R. R. Yager, “On the Specificity of a Possibility Distribution,” Fuzzy Sets Syst., Vol.50, pp. 279-292, 1992.
- [20] 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.
- [21] 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.
- [22] 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.
- [23] 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. - [24] A collection of papers related with fuzzy rule interpolation

http://fri.gamf.hu - [25] V. Torra and Y. Narukawa, “Modeling Decisions – Information Fusion and Aggregation Operators,” Springer, 2007.
- [26] 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, April 2004.
- [27] 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.
- [28] 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, April 2007.
- [29] V. G. Kaburlasos and S. E. Papadakis, “Granular Self-Organizing Map (grSOM) for Structure Identification,” Neural Networks, Vol.19, pp. 623-643, 2006.
- [30] 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.
- [31] 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.
- [32] 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.

This article is published under a Creative Commons Attribution-NoDerivatives 4.0 Internationa License.