Paper:

# Fuzzy Rule Interpolation and Extrapolation Techniques: Criteria and Evaluation Guidelines

## Domonkos Tikk^{*1}, Zsolt Csaba Johanyák^{*2},

Szilveszter Kovács^{*3}, and Kok Wai Wong^{*4}

^{*1}Dep. of Telecommunications and Media Informatics, Budapest University of Technology and Economics, Magyar tudósok krt. 2, H-1117 Budapest, Hungary

^{*2}Institute of Information Technology, Kecskemét College, Izsáki út 10, H-6000 Kecskemét, Hungary

^{*3}Department of Information Technology, University of Miskolc, H-3515 Miskolc-Egyetemváros, Hungary

^{*4}School of Information Technology, Murdoch University, South St., Murdoch, Western Australia 6150, Australia

This paper comprehensively analyzes Fuzzy Rule Interpolation and extrapolation Techniques (FRITs). Because extrapolation techniques are usually extensions of fuzzy rule interpolation, we treat them both as approximation techniques designed to be applied where sparse or incomplete fuzzy rule bases are used, i.e., when classical inference fails. FRITs have been investigated in the literature from aspects such as applicability to control problems, usefulness regarding complexity reduction and logic. Our objectives are to create an overall FRIT standard with a general set of criteria and to set a framework for guiding their classification and comparison. This paper is our initial investigation of FRITs. We plan to analyze details in later papers on how individual techniques satisfy the groups of criteria we propose. For analysis,MATLAB FRI Toolbox provides an easy-to-use testbed, as shown in experiments.

Szilveszter Kovács, and Kok Wai Wong, “Fuzzy Rule Interpolation and Extrapolation Techniques: Criteria and Evaluation Guidelines,”

*J. Adv. Comput. Intell. Intell. Inform.*, Vol.15, No.3, pp. 254-263, 2011.

- [1] S. Blažič and I. Škrjanc, “Design and stability analysis of fuzzy model-based predictive control – a case study,” J. of Intelligent and Robotic Systems, Vol.49, No.3, pp. 279-292, 2007.
- [2] D. Hládek, J. Vaščák, and P. Sinčák, “Hierarchical fuzzy inference system for robotic pursuit evasion task,” Proc. of SAMI 2008, 6th Int. Symposium on Applied Machine Intelligence and Informatics, Herl’any, Slovakia, pp. 273-277, 2008.
- [3] Z. C. Johanyák and S. Kovács, “Polar-cut Based Fuzzy Model for Petrophysical Properties Prediction,” Scientific Bulletin of “Politehnica” University of Timisoara, Romania, Transactions on Automatic Control and Computer Science, Vol.57/67, No.24, pp. 195-200, 2008.
- [4] Z. C. Johanyák, R. Parthiban, and G. Sekaran, “Fuzzy Modeling for an Anaerobic Tapered Fluidized Bed Reactor,” Scientific Bulletin of “Politehnica” University of Timisoara, Romania, Transactions on Automatic Control and Computer Science, Vol.52/66, No.2, pp. 67-72, 2007.
- [5] S. Kovács and L. T. Kóczy, “Application of Interpolation-based Fuzzy Logic Reasoning in Behaviour-based Control Structures,” Proc. of the FUZZ-IEEE’04, IEEE International Conference on Fuzzy Systems, Budapest, Hungary, pp. 1543-1548, 2004.
- [6] R. E. Precup, S. Doboli, and S. Preitl, “Stability analysis and development of a class of fuzzy systems,” Engineering Applications of Artificial Intelligence, Vol.13, No.3, pp. 237-247, 2000.
- [7] K. W. Wong and T. D. Gedeon, “Petrophysical properties prediction using self-generating fuzzy rules inference system with modified alpha-cut based fuzzy interpolation,” Proc. of The Seventh Int. Conf. of Neural Information Processing ICONIP 2000, Korea, pp. 1088-1092, 2000.
- [8] L. A. Zadeh, “Outline of a new approach to the analysis of complex systems and decision processes,” IEEE Trans. on SMC, Vol.3, pp. 28-44, 1973.
- [9] P. M. Larsen, “Industrial application of fuzzy logic control,” Int. J. of Man Machine Studies, Vol.12, No.4, pp. 3-10, 1980.
- [10] E. H. Mamdani and S. Assilian, “An experiment in linguistic synthesis with a fuzzy logic controller,” Int. J. ofMan Machine Studies, Vol.7, pp. 1-13, 1975.
- [11] M. Sugeno, “An introductory survey of fuzzy control,” Information Science, Vol.36, pp. 59-83, 1985.
- [12] T. Takagi and M. Sugeno, “Fuzzy identification of systems and its applications to modeling and control,” IEEE Trans. on SMC, Vol.15, pp. 116-132, 1985.
- [13] P. Baranyi and L. T. Kóczy, “Multi-dimensional fuzzy rule interand extrapolation based on geometric solution,” Proc. of the 7th Int. Power Electronic and Motion Control Conf. (PEMC’96), Budapest, Hungary, Vol.3, pp. 443-447, 1996.
- [14] Z. Huang, “Rule Model Simplification,” Ph.D. thesis, School of Informatics, University of Edinburgh, 2005.
- [15] S. Jenei, “Interpolation and extrapolation of fuzzy quantities revisited – (I) An axiomatic approach,” Soft Computing, Vol.5, pp. 179-193, 2001.
- [16] S. Jenei, E. P. Klement, and R. Konzel, “Interpolation and extrapolation of fuzzy quantities – (II) The multiple-dimensional case,” Soft Computing, Vol.6, No.3-4, pp. 258-270, 2002.
- [17] Y. Yam and L. T. Kóczy, “Representing membership functions as points in high dimensional spaces for fuzzy interpolation and extrapolation,” Technical Report CUHK-MAE-97-03, Dept. of Mechanical and Automation Eng., The Chinese Univ. of Hong Kong, 1997.
- [18] L. T. Kóczy and K. Hirota, “Rule interpolation in approximate reasoning based fuzzy control,” In R. Lowen and M. Roubens (Eds.), Proc. of 4th IFSA World Congress, Brussels, Belgium, pp. 89-92, 1991.
- [19] L. T. Kóczy, and K. Hirota, “Size reduction by interpolation in fuzzy rule bases,” IEEE Trans. on SMC, Vol.27, pp. 14-25, 1997.
- [20] P. Baranyi, L. T. Kóczy, and T. D. Gedeon, “A generalized concept for fuzzy rule interpolation,” IEEE Trans. on Fuzzy Systems, Vol.12, pp. 820-837, 2004.
- [21] 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, 2000.
- [22] Z. C. Johanyák and S. Kovács, “A brief survey and comparison on various interpolation based fuzzy reasoning methods,” Acta Politechnica Hungarica, J. of Applied Sciences at Budapest Tech Hungary, Vol.3, No.1, pp. 91-105, 2006.
- [23] L. T. Kóczy and D. Tikk, “A survey of fuzzy interpolation techniques,” Proc. of the 1st Int. Symp. of Hungarian Researchers on Computational Intelligence (HUCI’00), Budapest, Hungary, pp. 5-13, 2000.
- [24] L. T. Kóczy, D. Tikk, and L. Muresan, “Fuzzy systems with interpolation. An overview,” Proc. of the Joint 9th Int. Fuzzy Systems Association World Congress and 20th Int. Conf. of North American Fuzzy Information Processing Society (IFSA/NAFIPS’01), Vancouver, BC, Canada, pp. 2494-2498, 2001.
- [25] S. Mizik, D. Szabó, and P. Korondi, “Survey on fuzzy interpolation techniques,” Proc. of the IEEE Int. Conf. on Intelligent Engineering Systems (INES’99), Poprad, Slovakia, pp. 587-592, 1999.
- [26] Z. C. Johanyák, D. Tikk, S. Kovács, and K. W. Wong, “Fuzzy rule interpolation Matlab toolbox – FRI toolbox,” Proc. of the IEEE World Congress on Computational Intelligence (WCCI’06), 15th Int. Conf. on Fuzzy Systems (FUZZ-IEEE’06), Vancouver, BC, Canada, pp. 1427-1433, 2006.
- [27] G. Klir and B. Yuan, “Fuzzy Sets and Fuzzy Logic, Theory and Applications,” Prentice-Hall, Upper Saddle River, 1995.
- [28] R. Babuška, “Construction of fuzzy systems – interplay between precision and transparency,” Proc. of European Symp. on Intelligent Techniques (ESIT’00), Aachen, Germany, pp. 445-452, 2000.
- [29] D. Dubois and H. Prade, “Gradual rules in approximate reasoning,” Information Science, Vol.61, pp. 103-122, 1992.
- [30] J. V. de Oliveira, “Toward neuro – linguistic modeling: constraint for optimization of membership functions,” Fuzzy Sets and Systems, Vol.106, No.3, pp. 357-380, 1999.
- [31] L. T. Kóczy and K. Hirota, “Ordering, distance and closeness of fuzzy sets,” Fuzzy Sets and Systems, Vol.59, pp. 281-293, 1993.
- [32] Z. Huang and Q. Shen, “Fuzzy interpolative reasoning via scale and move transformations,” IEEE Trans. on Fuzzy Systems, Vol.14, No.2, pp. 340-359, 2006.
- [33] Z. C. Johanyák and S. Kovács, “Fuzzy rule interpolation based on polar cuts,” In B. Reusch (Ed.), Computational Intelligence, Theory and Applications, Springer, Berlin-Heidelberg, Proc. of the Int. Conf. of 9th Fuzzy Days in Dortmund, Germany, pp. 499-511, 2006.
- [34] D. Tikk, P. Baranyi, Y. Yam, and L. T. Kóczy, “Stability of a new interpolation method,” Proc. of the IEEE Int. Conf. on System, Man, and Cybernetics (IEEE SMC’99), Tokyo, Japan, III, pp. 7-9, 1999.
- [35] L. T. Kóczy and S. Kovács, “Shape of the fuzzy conclusion generated by linear interpolation in trapezoidal fuzzy rule bases,” Proc. of the 2nd European Congress on Intelligent Techniques and Soft Computing, Aachen, Germany, pp. 1666-1670, 1994.
- [36] G. Vass, L. Kalmár, and L. T. Kóczy, “Extension of the fuzzy rule interpolation method,” Proc. of the Int. Conf. on Fuzzy Sets Theory and its Applications, Liptovský, Jan. 1992.
- [37] S. Kovács, “New aspect of interpolative reasoning,” Proc. of Int. Conf. on Information Processing and Management of Uncertainty in Knowledge-based Systems (IPMU’96), Granada, pp. 477-482, 1996.
- [38] S. Kovács, “Extending the fuzzy rule interpolation FIVE by fuzzy observation,” In B. Reusch (Ed.), Advances in Soft Computing, Computational Intelligence, Theory and Applications, Springer, Heidelberg-Berlin, pp. 485-497, 2006.
- [39] 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, 2005.
- [40] Z. C. Johanyák and S. Kovács, “Fuzzy rule interpolation by the least squares method,” Proc. of the 7th Int. Symp. of Hungarian Researchers on Computational Intelligence (HUCI’06), Budapest, Hungary, pp. 495-506, 2006.
- [41] Z. C. Johanyák and S. Kovács, “Vague Environment-based Twostep Fuzzy Rule Interpolation Method,” Proc. of the 5th Slovakian-Hungarian Joint Symposium on Applied Machine Intelligence and Informatics (SAMI 2007), Poprad, Slovakia, pp. 189-200, 2007.
- [42] I. Perfilieva and S. Lehmke, “Correct models of fuzzy IF–THEN rules are continuous,” Fuzzy Sets and Systems, Vol.157, No.24, pp. 3188-3197, 2006.
- [43] D. Tikk, I. Joó, L. T. Kóczy, P. Várlaki, B.Moser, and T. D. Gedeon, “Stability of interpolative fuzzy KH-controllers,” Fuzzy Sets and Systems, Vol.125, No.1, pp. 105-119, 2002.
- [44] D. Shepard, “A two dimensional interpolation function for irregularly spaced data,” Proc. of the 23rd ACM Int. Conf., pp. 517-524, 1968.
- [45] L. T. Kóczy, K. Hirota, and T. D. Gedeon, “Fuzzy rule interpolation by the conservation of relative fuzziness,” Technical Report 97/2, Hirota Lab, Dept. of Comp. Intelligent and Sys. Sci., Tokyo Institute of Technology, Yokohama, Japan, 1997.

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