Paper:

# Suppression Effect of α-Cut Based Inference on Consequence Deviations

## Kiyohiko Uehara^{*}, Takumi Koyama^{*}, 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

This paper clarifies that inference based on α-cut and generalized mean (α-GEMII) is effective in suppressing consequence deviations. The suppression effect of α-GEMII is numerically evaluated in comparison to conventional inference based on the Compositional Rule of Inference (CRI). CRI-based parallel inference causes discontinuous deviations in the least upper and greatest lower bounds of deduced fuzzy sets even when it models the continuous input-output relation of a system and given facts change continuously. In contrast, α-GEMII can suppress the deviations because of its schemes originally developed for constraint propagation control. In simulations, indices are defined for numerically evaluating the degree to which deduced consequences follow the change in fuzzy outputs of given systems. Simulation results show that α-GEMII is effective in suppressing the deviations, compared to CRI-based parallel inference. In effective use of the schemes for suppressing the consequence deviations, α-GEMII can be applied to nonlinear prediction filters for complex time series, especially with fluctuations that do not always originate from a correlation between time series data.

*J. Adv. Comput. Intell. Intell. Inform.*, Vol.14, No.3, pp. 256-271, 2010.

- [1] S. J. Chen and S. M. Chen, “Fuzzy Risk Analysis Based on Similarity Measures of Generalized Fuzzy Numbers,” IEEE Trans. on Fuzzy Systems, Vol.11, No.1, pp. 45-56, Feb. 2003.
- [2] 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. on Fuzzy Systems, Vol.13, No.1, pp. 13-29, Feb. 2005.
- [3] H.-X. Li, L. Zhang, K.-Y. Cai, and G. Chen, “An Improved Robust Fuzzy-PID Controller with Optimal Fuzzy Reasoning,” IEEE Trans. on Systems, Man, and Cybernetics Part B: Cybernetics, Vol.35, No.6, pp. 1283-1294, Dec. 2005.
- [4] F. Farbiz, M. B. Menhaj, S. A. Motamedi, and M. T. Hagan, “A New Fuzzy Logic Filter for Image Enhancement,” IEEE Trans. on Systems, Man, and Cybernetics Part B: Cybernetics, Vol.30, No.1, pp. 110-119, Feb. 2000.
- [5] S. Guillaume and B. Charnomordic, “Generating an Interpretable Families of Fuzzy Partitions from Data,” IEEE Trans. on Fuzzy Systems, Vol.12, No.3, pp. 324-335, June 2004.
- [6] I. B. Turksen and Y. Tian, “Combination of Rules or Their Consequences in Fuzzy Expert Systems,” Fuzzy Sets and Systems, Vol.58, No.1, pp. 3-40, Aug. 1993.
- [7] G. Cheng and Y. Fu, “Error Estimation of Perturbation under CRI,” IEEE Trans. on Fuzzy Systems, Vol.14, No.6, pp. 709-715, Dec. 2006.
- [8] 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.
- [9] V. G. Kaburlasos, “FINs: Lattice Theoretic Tools for Improving Prediction of Sugar Production from Populations of Measurements,” IEEE Trans. on Systems, Man, and Cybernetics Part B: Cybernetics, Vol.34, No.2, pp. 1017-1030, April 2004.
- [10] 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.
- [11] V. G. Kaburlasos and A. Kehagias, “Novel Fuzzy Inference System (FIS) Analysis and Design Based on Lattice Theory,” IEEE Trans. on Fuzzy Systems, Vol.15, No.2, pp. 243-260, April 2007.
- [12] V. G. Kaburlasos and S. E. Papadakis, “Granular Self-Organizing Map (grSOM) for Structure Identification,” Neural Networks, Vol.19, No.5, pp. 623-643, June 2006.
- [13] 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.
- [14] K. Uehara, “Fuzzy Inference Based on a Weighted Average of Fuzzy Sets and its Learning Algorithm for Fuzzy Exemplars,” Proceedings of 1995 IEEE International Conference on Fuzzy Systems, 1995. International Joint Conference of the Fourth IEEE International Conference on Fuzzy Systems and The Second International Fuzzy Engineering Symposium, FUZZ-IEEE/IFES’95 (Yokohama, Japan), Vol.4, pp. 2253-2260, March 20-24 1995.
- [15] 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, No.3-4, pp. 305-322, Aug. 2006.
- [16] Z. Huang and Q. Shen, “Fuzzy Interpolative Reasoning via Scale and Move Transformation,” IEEE Trans. on Fuzzy Systems, Vol.14, No.2, pp. 340-359, April 2006.
- [17] Z. Huang and Q. Shen, “Fuzzy Interpolation and Extrapolation: A Practical Approach,” IEEE Trans. on Fuzzy Systems, Vol.16, No.1, pp. 13-28, Feb. 2008.
- [18] 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. on Fuzzy Systems, Vol.13, No.6, pp. 809-819, Dec. 2005.
- [19] 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. on Fuzzy Systems, Vol.1, No.3, pp. 205-221, Aug. 1993.
- [20] 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)
- [21] 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)
- [22] S. Aja-Fernández and C. Alberola-López, “Fast Inference Using Transition Matrices: An Extension to Nonlinear Operators,” IEEE Trans. on Fuzzy Systems, Vol.13, No.4, pp. 478-490, Aug. 2005.
- [23] 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.
- [24] 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.
- [25] 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.
- [26] 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.
- [27] L. A. Zadeh, “Fuzzy Logic = Computing with Words,” IEEE Trans. on Fuzzy Systems, Vol.4, No.2, pp. 103-111, May 1996.
- [28] L. A. Zadeh, “Inference in Fuzzy Logic via Generalized Constraint Propagation,” Proc. 1996 26th International Symposium on Multi-Valued Logic (ISMVL’96), pp. 192-195, 1996.
- [29] A. Kaufmann, Introduction to the Theory of Fuzzy Subsets. New York: Academic, Vol.1, 1975.
- [30] N. R. Pal and J. C. Bezdek, “Measuring Fuzzy Uncertainty,” IEEE Trans. on Fuzzy Systems, Vol.2, No.2, pp. 107-118, May 1994.
- [31] R. R. Yager, “On the Specificity of a Possibility Distribution,” Fuzzy Sets and Systems, Vol.50, pp. 279-292, Sep. 1992.
- [32] R. R. Yager, “Measuring Tranquility and Anxiety in Decision Making: an Application of Fuzzy Sets,” Int. J. of General Systems, Vol.8, No.3, pp. 139-146, 1982.
- [33] H. Jones, B. Charnomordic, D. Dubois, and S. Guillaume, “Practical Inference with Systems of Gradual Implicative Rules,” IEEE Trans. on Fuzzy Systems, Vol.17, No.1, pp. 61-78, Feb. 2009.
- [34] K. Uehara and M. Fujise, “Fuzzy Inference Based on Families of α-Level Sets,” IEEE Trans. on Fuzzy Systems, Vol.1, No.2, pp. 111-124, May 1993.
- [35] M. Mas, M. Monserrat, J. Torrens, and E. Trillas, “A Survey on Fuzzy Implication Functions,” IEEE Trans. on Fuzzy Systems, Vol.15, No.6, pp. 1107-1121, Dec. 2007.
- [36] D. Dubois and H. Prade, “Fuzzy Sets in Approximate Reasoning, Part 1: Inference with Possibility Distributions,” Fuzzy Sets and Systems, Vol.40, pp. 143-202, March 1991.
- [37] I. B. Turksen, “Approximate Reasoning for Production Planning,” Fuzzy Sets and Systems, Vol.26, No.1 pp. 23-37, April 1988.
- [38] J. B. Kiszka, M. E. Kochańska, and D. S. Sliwińska, “The Influence of Some Fuzzy Implication Operators on the Accuracy of a Fuzzy Model – Part II,” Fuzzy Sets and Systems, Vol.15, No.3 April pp. 223-240, 1985.
- [39] F. Janabi-Sharifi, “A Neuro-Fuzzy System for Looper Tension Control in Rolling Mills,” Control Engineering Practice, Vol.13, pp. 1-13, Jan. 2005.
- [40] 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, April 1998.

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