A generalized concept for fuzzy rule interpolation

The concept of fuzzy rule interpolation in sparse rule bases was introduced in 1993. It has become a widely researched topic in recent years because of its unique merits in the topic of fuzzy rule base complexity reduction. The first implemented technique of fuzzy rule interpolation was termed as /spl alpha/-cut distance based fuzzy rule base interpolation. Despite its advantageous properties in various approximation aspects and in complexity reduction, it was shown that it has some essential deficiencies, for instance, it does not always result in immediately interpretable fuzzy membership functions. This fact inspired researchers to develop various kinds of fuzzy rule interpolation techniques in order to alleviate these deficiencies. This paper is an attempt into this direction. It proposes an interpolation methodology, whose key idea is based on the interpolation of relations instead of interpolating /spl alpha/-cut distances, and which offers a way to derive a family of interpolation methods capable of eliminating some typical deficiencies of fuzzy rule interpolation techniques. The proposed concept of interpolating relations is elaborated here using fuzzy- and semantic-relations. This paper presents numerical examples, in comparison with former approaches, to show the effectiveness of the proposed interpolation methodology.

[1]  I. Burhan Türksen,et al.  An approximate analogical reasoning approach based on similarity measures , 1988, IEEE Trans. Syst. Man Cybern..

[2]  M. Mukaidono,et al.  Fuzzy resolution principle , 1988, [1988] Proceedings. The Eighteenth International Symposium on Multiple-Valued Logic.

[3]  M. Mukaidono,et al.  A new method for approximate reasoning , 1989, Proceedings. The Nineteenth International Symposium on Multiple-Valued Logic.

[4]  I. Turksen,et al.  An approximate analogical reasoning schema based on similarity measures and interval-valued fuzzy sets , 1990 .

[5]  Didier Dubois,et al.  Gradual inference rules in approximate reasoning , 1992, Inf. Sci..

[6]  László T. Kóczy,et al.  Interpolative reasoning with insufficient evidence in sparse fuzzy rule bases , 1993, Inf. Sci..

[7]  László T. Kóczy,et al.  Approximate reasoning by linear rule interpolation and general approximation , 1993, Int. J. Approx. Reason..

[8]  K. Hirota,et al.  Ordering, distance and closeness of fuzzy sets , 1993 .

[9]  L. Kóczy,et al.  Linearity and the cnf property in linear fuzzy rule interpolation , 1994, Proceedings of 1994 IEEE 3rd International Fuzzy Systems Conference.

[10]  Michel Grabisch,et al.  Gradual rules and the approximation of control laws , 1995 .

[11]  Masaharu Mizumoto,et al.  Reasoning conditions on Kóczy's interpolative reasoning method in sparse fuzzy rule bases , 1995, Fuzzy Sets Syst..

[12]  L. Kóczy,et al.  A general interpolation technique in fuzzy rule bases with arbitrary membership functions , 1996, 1996 IEEE International Conference on Systems, Man and Cybernetics. Information Intelligence and Systems (Cat. No.96CH35929).

[13]  Yan Shi,et al.  An improvement to Kóczy and Hirota's interpolative reasoning in sparse fuzzy rule bases , 1996, Int. J. Approx. Reason..

[14]  L. Kóczy,et al.  Approximate fuzzy reasoning based on interpolation in the vague environment of the fuzzy rulebase , 1997, Proceedings of IEEE International Conference on Intelligent Engineering Systems.

[15]  László T. Kóczy,et al.  Size reduction by interpolation in fuzzy rule bases , 1997, IEEE Trans. Syst. Man Cybern. Part B.

[16]  László T. Kóczy,et al.  Fuzzy rule base interpolation based on semantic revision , 1998, SMC'98 Conference Proceedings. 1998 IEEE International Conference on Systems, Man, and Cybernetics (Cat. No.98CH36218).

[17]  B. Bouchon-Meunier,et al.  Analogy and Fuzzy Interpolation in the case of Sparse Rules , 1999 .

[18]  D. Tikk,et al.  New method for avoiding abnormal conclusion for α-cut based rule interpolation , 1999 .

[19]  D. Tikk,et al.  A new method for avoiding abnormal conclusion for /spl alpha/-cut based rule interpolation , 1999, FUZZ-IEEE'99. 1999 IEEE International Fuzzy Systems. Conference Proceedings (Cat. No.99CH36315).

[20]  D. Tikk,et al.  Stability of a new interpolation method , 1999, IEEE SMC'99 Conference Proceedings. 1999 IEEE International Conference on Systems, Man, and Cybernetics (Cat. No.99CH37028).

[21]  D. Dubois,et al.  ON FUZZY INTERPOLATION , 1999 .

[22]  L.T. Koczy,et al.  Interpolation in hierarchical fuzzy rule bases , 2000, Ninth IEEE International Conference on Fuzzy Systems. FUZZ- IEEE 2000 (Cat. No.00CH37063).

[23]  C. Marsala,et al.  Interpolative reasoning based on graduality , 2000, Ninth IEEE International Conference on Fuzzy Systems. FUZZ- IEEE 2000 (Cat. No.00CH37063).

[24]  Vladik Kreinovich,et al.  Extracting fuzzy sparse rules by Cartesian representation and clustering , 2000, Smc 2000 conference proceedings. 2000 ieee international conference on systems, man and cybernetics. 'cybernetics evolving to systems, humans, organizations, and their complex interactions' (cat. no.0.

[25]  Kok Wai Wong,et al.  Using modified alpha-cut based fuzzy interpolation in petrophysical properties prediction , 2000 .

[26]  László T. Kóczy,et al.  Representing membership functions as points in high-dimensional spaces for fuzzy interpolation and extrapolation , 2000, IEEE Trans. Fuzzy Syst..

[27]  Mayuka F. Kawaguchi,et al.  A fuzzy rule interpolation technique based on bi-splines in multiple input systems , 2000, Ninth IEEE International Conference on Fuzzy Systems. FUZZ- IEEE 2000 (Cat. No.00CH37063).

[28]  Kok Wai Wong,et al.  An improved multidimensional alpha-cut based fuzzy interpolation technique , 2000 .

[29]  Péter Baranyi,et al.  Comprehensive analysis of a new fuzzy rule interpolation method , 2000, IEEE Trans. Fuzzy Syst..

[30]  Yeung Yam,et al.  Interpolation as Mappings Between Cartesian Spaces , 2000 .

[31]  Masaaki Miyakoshi,et al.  Fuzzy Spline Interpolation in Sparse Fuzzy Rule Bases , 2001 .

[32]  Sándor Jenei,et al.  Interpolation and extrapolation of fuzzy quantities revisited – an axiomatic approach , 2001, Soft Comput..

[33]  B. Bouchon-Meunier,et al.  Interpolative reasoning with multi-variable rules , 2001, Proceedings Joint 9th IFSA World Congress and 20th NAFIPS International Conference (Cat. No. 01TH8569).

[34]  Kok Wai Wong,et al.  Fuzzy rule interpolation for multidimensional input space with petroleum engineering application , 2001, Proceedings Joint 9th IFSA World Congress and 20th NAFIPS International Conference (Cat. No. 01TH8569).

[35]  Y. Yam,et al.  Fuzzy interpolation with Cartesian representation and extensibility functions , 2001, Proceedings Joint 9th IFSA World Congress and 20th NAFIPS International Conference (Cat. No. 01TH8569).

[36]  H. Prade,et al.  A comparative view of interpolation methods between sparse fuzzy rules , 2001, Proceedings Joint 9th IFSA World Congress and 20th NAFIPS International Conference (Cat. No. 01TH8569).

[37]  Peter Baranyi,et al.  A fuzzy interpolation algorithm closed over CNF sets , 2002 .

[38]  Sándor Jenei,et al.  Interpolation and extrapolation of fuzzy quantities – the multiple-dimensional case , 2002, Soft Comput..

[39]  László T. Kóczy,et al.  Stability of interpolative fuzzy KH controllers , 2002, Fuzzy Sets Syst..

[40]  P. Baranyi.  Rule Interpolation by Spatial Geometric Representation , 2022 .

[41]  P. Baranyi,et al.  A general method for fuzzy rule interpolation: specialised for crisp triangular and trapezoidal rules , 2022 .