Paper:

# Generalisation of Scale and Move Transformation-Based Fuzzy Interpolation

## Qiang Shen and Longzhi Yang

Department of Computer Science, Aberystwyth University, Aberystwyth, SY23 3DB, UK

Fuzzy interpolative reasoning has been extensively studied due to its ability to enhance the robustness of fuzzy systems and reduce system complexity. In particular, the scale and move transformation-based approach is able to handle interpolation with multiple antecedent rules involving triangular, complex polygon, Gaussian and bell-shaped fuzzy membership functions [1]. Also, this approach has been extended to deal with interpolation and extrapolation with multiple multi-antecedent rules [2]. However, the generalised extrapolation approach based on multiple rules may not degenerate back to the basic crisp extrapolation based on two rules. Besides, the approximate function of the extended approach may not be continuous. This paper therefore proposes a new approach to generalising the basic fuzzy interpolation technique of [1] in an effort to eliminate these limitations. Examples are given throughout the paper for illustration and comparative purposes. The result shows that the proposed approach avoids the identified problems, providing more reasonable interpolation and extrapolation.

*J. Adv. Comput. Intell. Intell. Inform.*, Vol.15, No.3, pp. 288-298, 2011.

- [1] Z. Huang and Q. Shen, “Fuzzy interpolative reasoning via scale and move transformations,” IEEE Trans. Fuzzy Syst., Vol.14, No.2, pp. 340-359, 2006.
- [2] Z. Huang and Q. Shen, “Fuzzy interpolation and extrapolation: a practical approach,” IEEE Trans. Fuzzy Syst., Vol.16, No.1, pp. 13-28, 2008.
- [3] 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.
- [4] L. T. Kóczy and K. Hirota, “Interpolative reasoning with insufficient evidence in sparse fuzzy rule bases,” Inf. Sci., Vol.71, No.1-2, pp. 169-201, 1993.
- [5] L. T. Kóczy and K. Hirota, “Size reduction by interpolation in fuzzy rule bases,” IEEE Trans. Syst., Man Cybern. Vol.27, No.1, pp. 14-25, 1997.
- [6] G. J. Klir, and B. Yuan, “Fuzzy sets and fuzzy logic: theory and applications,” Prentice-Hall, Inc., Upper Saddle River, NJ, USA, 1995. ISBN: 0-13-101171-5
- [7] J. A. Robinson, “A Machine-Oriented Logic Based on the Resolution Principle,” J. ACM, Vol.12, No.1, pp. 23-41, Jan. 1965.
- [8] L. A. Zadeh, “Quantitative fuzzy semantics,” Inf. Sci., Vol.3, No.2, pp. 159-176, Elsevier Science Inc., Apr. 1971.
- [9] D. Dubois and H. Prade, “On fuzzy interpolation,” Int. J. of General Systems, Vol.28, No.2, pp. 103-114, 1999.
- [10] L. Ughetto, D. Dubois, and H. Prade, “Fuzzy interpolation by convex completion of sparse rule bases,” The Ninth IEEE Int. Conf. on Fuzzy Systems 2000 (FUZZ IEEE 2000), Vol.1, pp. 465-470, May 2000.
- [11] Y.-C. Chang, S.-M. Chen, and C.-J. Liau, “Fuzzy Interpolative Reasoning for Sparse Fuzzy-Rule-Based Systems Based on the Areas of Fuzzy Sets,” IEEE Trans. on Fuzzy Systems, Vol.16, No.5, pp. 1285-1301, Oct. 2008.
- [12] D. Tikk and P. Baranyi, “Comprehensive analysis of a new fuzzy rule interpolation method,” IEEE Trans. on Fuzzy Systems, Vol.8, No.3, pp. 281-296, Jun. 2000.
- [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. on Fuzzy Systems, Vol.13, No.6, pp. 809-819, Dec. 2005.
- [14] W. Hsiao, S. Chen, and C. Lee, “A new interpolative reasoning method in sparse rule-based systems,” Fuzzy Sets Syst., Vol.93, No.1, pp. 17-22, 1998.
- [15] S. Kovács, “Extending the Fuzzy Rule Interpolation “FIVE” by Fuzzy Observation,” B. Reusch (Ed.), Computational Intelligence, Theory and Applications, pp. 485-497, Springer Berlin Heidelberg, 2006.
- [16] B. Bouchon-Meunier and L. Valverde, “A fuzzy approach to analogical reasoning,” Soft Computing – A Fusion of Foundations, Methodologies and Applications, Vol.3, No.3, pp. 141-147, 1999.
- [17] P. Baranyi, L. T. Kóczy, and T. D. Gedeo, “A generalized concept for fuzzy rule interpolation,” IEEE Trans. on Fuzzy Syst., Vol.12, No.6, pp. 820-837, 2004.
- [18] S.-M. Chen and Y.-K. Ko, “Fuzzy Interpolative Reasoning for Sparse Fuzzy Rule-Based Systems Based on α-Cuts and Transformations Techniques,” IEEE Trans. on Fuzzy Systems, Vol.16, No.6, pp. 1626-1648, Dec. 2008.
- [19] S. Jenei, “Interpolation and extrapolation of fuzzy quantities revisited – an axiomatic approach,” Soft Computing – A Fusion of Foundations, Methodologies and Applications, Vol.5, No.3, pp. 179-193, Springer Berlin/Heidelberg, 2001.
- [20] S. Jenei, E. P. Klement, and R. Konzel, “Interpolation and extrapolation of fuzzy quantities the multiple-dimensional case,” Soft Computing – A Fusion of Foundations, Methodologies and Applications Vol.6, No.3, pp. 258-270, Springer Berlin/Heidelberg, 2002.
- [21] Z. Johanyák and S. Kovács, “Fuzzy Rule Interpolation Based on Polar Cuts,” B. Reusch (Ed.), Computational Intelligence, Theory and Applications, pp. 499-511, 2006.
- [22] Z. Q. Wu, M. Mizumoto, and Y. Shi, “An improvement to Kóczy and Hirota’s interpolative reasoning in sparse fuzzy rule bases,” Int. J. of Approximate Reasoning, Vol.15, No.3, pp. 185-201, 1996.
- [23] Z. Huang and Q. Shen, “Preserving Piece-wise Linearity in Fuzzy Interpolation,” Proc. IEEE Int. Conf. Fuzzy Syst., pp. 575-580, 2009.
- [24] Q. Shen and R. Leitch, “Fuzzy qualitative simulation,” IEEE Trans. on Systems, Man and Cybernetics, Vol.23, No.4, pp. 1038-1061, Jul./Aug. 1993.
- [25] L. Yang and Q. Shen, “Towards Adaptive Interpolative Reasoning,” Proc. IEEE Int. Conf. Fuzzy Syst., pp. 542-549, 2009.
- [26] L. Yang and Q. Shen, “Adaptive Fuzzy Interpolation and Extrapolation with Multiple-antecedent Rules,” Proc. IEEE Int. Conf. Fuzzy Syst., pp. 565-572, 2010.