Paper:

# Inference with Fuzzy Rule Interpolation at an Infinite Number of Activating 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

An inference method for sparse fuzzy rules is proposed which interpolates fuzzy rules at an infinite number of activating points and deduces consequences based on α-GEMII (α-level-set and generalized-mean-based inference). The activating points, proposed in this paper, are determined so as to activate interpolated fuzzy rules by each given fact. The proposed method is named α-GEMINAS (α-GEMII-based inference with fuzzy rule interpolation at an infinite number of activating points). α-GEMINAS solves the problem in infinite-level interpolation where fuzzy rules are interpolated at the least upper and greatest lower bounds of an infinite number of α-cuts of each given fact. The infinite-level interpolation can nonlinearly transform the shapes of given membership functions to those of deduced ones in accordance even with sparse fuzzy rules under some conditions. These conditions are, however, strict from a practical viewpoint. α-GEMINAS can deduce consequences without these conditions and provide nonlinear mapping comparable with infinite-level interpolation. Simulation results demonstrate these properties of α-GEMINAS. Thereby, it is found that α-GEMINAS is practical and applicable to a wide variety of fields.

*J. Adv. Comput. Intell. Intell. Inform.*, Vol.19, No.1, pp. 74-90, 2015.

- [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] 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.
- [4] 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.
- [5] 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.
- [6] 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, Jun. 2000.
- [7] 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.
- [8] 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.
- [9] 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.
- [10] 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.
- [11] L. T. Kóczy and S. 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.
- [12] L. T. Kóczy and S. 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.
- [13] D. Tikk, Z. C. Johanyák, S. Kovács, and K. W. Wong, “Fuzzy Rule Interpolation and Extrapolation Techniques: Criteria and Evaluation Guidelines,” J. of Advanced Computational Intelligence and Intelligent Informatics, Vol.15, No.3, pp. 254-263, 2011.
- [14] Q. Shen, and L. Yang, “Generalisation of Scale and Move Transformation-Based Fuzzy Interpolation,” J. of Advanced Computational Intelligence and Intelligent Informatics, Vol.15, No.3, pp. 288-298, 2011.
- [15] L. Kovács, “Compound Distance Function for Similarity Measurement Between Fuzzy Sets,” J. of Advanced Computational Intelligence and Intelligent Informatics, Vol.15, No.3, pp. 299-303, 2011.
- [16] S. Kato and K. W. Wong, “Intelligent Automated Guided Vehicle Controller with Reverse Strategy,” J. of Advanced Computational Intelligence and Intelligent Informatics, Vol.15, No.3, pp. 304-312, 2011.
- [17] D. Vincze and S. Kovács, “Performance Optimization of the Fuzzy Rule Interpolation Method “FIVE”,” J. of Advanced Computational Intelligence and Intelligent Informatics, Vol.15, No.3, pp. 313-320, 2011.
- [18] K. Uehara and K. Hirota, “Multi-Level Interpolation for Inference with Sparse Fuzzy Rules: An Extended Way of Generating Multi-Level Points,” J. of Advanced Computational Intelligence and Intelligent Informatics, Vol.17, No.2, pp. 127-148, Mar. 2013.
- [19] K. Uehara and K. Hirota, “Infinite-Level Interpolation for Inference with Sparse Fuzzy Rules: Fundamental Analysis Toward Practical Use,” J. of Advanced Computational Intelligence and Intelligent Informatics, Vol.17, No.1, pp. 44-59, Jan. 2013.
- [20] 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.
- [21] 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.
- [22] 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.
- [23] 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.
- [24] K. Uehara, T. Koyama, and K. Hirota, “Inference Based on α-Cut and Generalized Mean in Representing Fuzzy-Valued Functions,” J. of Advanced Computational Intelligence and Intelligent Informatics, Vol.14, No.6, pp. 581-592, 2010.
- [25] K. Uehara and K. Hirota, “Multi-Level Control of Fuzzy-Constraint Propagation in Inference Based on α-Cuts and Generalized Mean,” J. of Advanced Computational Intelligence and Intelligent Informatics, Vol.17, No.4, pp. 647-662, 2013.

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