Statistical analysis of the main parameters in the fuzzy inference process

Abstract As there are many possibilities to select the set of basic operators used in the fuzzy inference process, the search for the fuzzy operators that are most suitable for the different steps of a fuzzy system, their characterization and evaluation, can be included among the most important topics in the field of fuzzy logic. A better insight into the performances of the alternative operators would make it easier to develop a fuzzy application. In the present paper, the relevancy and relative importance of the operators involved in the fuzzy inference process are investigated by using a powerful statistical tool, the ANalysis Of the VAriance (ANOVA) [8]. The results obtained show that the defuzzifier and the T-norm operator are the most relevant factors in the fuzzy inference process. Moreover, this statistical analysis is able to establish a classification of the defuzzifiers and T-norms, according to their intrinsic characteristics. The conclusions here obtained justify the present interest, observed in many current papers, in studying both operators [6, 22–24, 33, 64, 67]. Futhermore, our results are confirmed by some experiments dealing with a real control application.

[1]  A. Baron.  Experimental Designs , 1990, The Behavior analyst.

[2]  H. Zimmermann,et al.  Comparison of fuzzy reasoning methods , 1982 .

[3]  M. Gupta,et al.  Design of fuzzy logic controllers based on generalized T -operators , 1991 .

[4]  R. Mead,et al.  The Design of Experiments: Statistical Principles for Practical Applications. , 1989 .

[5]  A. Tsoi,et al.  Importance of membership functions: a comparative study on different learning methods for fuzzy inference systems , 1994, Proceedings of 1994 IEEE 3rd International Fuzzy Systems Conference.

[6]  A. Kandel,et al.  Flexible implication operators in fuzzy reasoning , 1989, Proceedings. The Nineteenth International Symposium on Multiple-Valued Logic.

[7]  Vladik Kreinovich,et al.  How to improve mamdani's approach to fuzzy control , 1995, Int. J. Intell. Syst..

[8]  Lotfi A. Zadeh,et al.  Please Scroll down for Article International Journal of General Systems Fuzzy Sets and Systems* Fuzzy Sets and Systems* , 2022 .

[9]  Alexandre Piskunov,et al.  Fuzzy implication in fuzzy systems control , 1992 .

[10]  L. Kohout,et al.  FUZZY POWER SETS AND FUZZY IMPLICATION OPERATORS , 1980 .

[11]  Satoru Fukami.  SOME CONSIDERATIONS ON FUZZY CONDITIONAL INFERENCE , 1993 .

[12]  Rory A. Fisher,et al.  Theory of Statistical Estimation , 1925, Mathematical Proceedings of the Cambridge Philosophical Society.

[13]  William G. Cochran,et al.  Experimental Designs, 2nd Edition , 1950 .

[14]  János C. Fodor,et al.  Characterization of the ordered weighted averaging operators , 1995, IEEE Trans. Fuzzy Syst..

[15]  Rory A. Fisher,et al.  THE COMPARISON OF SAMPLES WITH POSSIBLY UNEQUAL VARIANCES , 1939 .

[16]  K M Passino,et al.  Dynamically focused fuzzy learning control , 1996, IEEE Trans. Syst. Man Cybern. Part B.

[17]  Chuen-Tsai Sun,et al.  Neuro-fuzzy modeling and control , 1995, Proc. IEEE.

[18]  A. Kandel,et al.  Applicability of some fuzzy implication operators , 1989 .

[19]  R. Yager.  Aggregation operators and fuzzy systems modeling , 1994 .

[20]  L. Kohout,et al.  FUZZY POWER SETS AND FUZZY IMPLICATION OPERATORS , 1980 .

[21]  J. Bowles,et al.  Application of fuzzy logic to reliability engineering , 1995, Proc. IEEE.

[22]  Michio Sugeno,et al.  Industrial Applications of Fuzzy Control , 1985 .

[23]  Abraham Kandel,et al.  Applicability analysis of fuzzy inference by means of generalized Dempster-Shafer theory , 1995, IEEE Trans. Fuzzy Syst..

[24]  William G. Cochran,et al.  Experimental designs, 2nd ed. , 1957 .

[25]  Li-Xin Wang,et al.  Adaptive fuzzy systems and control - design and stability analysis , 1994 .

[26]  Shun'ichi Tano,et al.  Operator tuning in fuzzy production rules using neural networks , 1993, [Proceedings 1993] Second IEEE International Conference on Fuzzy Systems.

[27]  Bart Kosko,et al.  Neural networks and fuzzy systems: a dynamical systems approach to machine intelligence , 1991 .

[28]  Li-Xin Wang,et al.  Adaptive fuzzy systems and control , 1994 .

[29]  Hans Hellendoorn.  Design and development of fuzzy systems at Siemens R&D , 1993, [Proceedings 1993] Second IEEE International Conference on Fuzzy Systems.

[30]  János C. Fodor,et al.  Nonstandard conjunctions and implications in fuzzy logic , 1995, Int. J. Approx. Reason..

[31]  J. S. Hunter,et al.  Statistics for experimenters : an introduction to design, data analysis, and model building , 1979 .

[32]  T. Runkler,et al.  Defuzzification based on fuzzy clustering , 1994, Proceedings of 1994 IEEE 3rd International Fuzzy Systems Conference.

[33]  Constantin V. Negoita,et al.  On Fuzzy Systems , 1978 .

[34]  R. Kirk.  Experimental Design: Procedures for the Behavioral Sciences , 1970 .

[35]  E. Trillas,et al.  ON IMPLICATION AND INDISTINGUISHABILITY IN THE SETTING OF FUZZY LOGIC , 1993 .

[36]  Chin-Teng Lin,et al.  Reinforcement structure/parameter learning for neural-network-based fuzzy logic control systems , 1993, [Proceedings 1993] Second IEEE International Conference on Fuzzy Systems.

[37]  Bart Kosko,et al.  Neural networks and fuzzy systems , 1998 .

[38]  Sidney Addelman,et al.  trans-Dimethanolbis(1,1,1-trifluoro-5,5-dimethylhexane-2,4-dionato)zinc(II) , 2008, Acta crystallographica. Section E, Structure reports online.

[39]  Frank Klawonn,et al.  Foundations of fuzzy systems , 1994 .

[40]  P. Breeze.  +620. Price £65 R. Mead , 1989, Animal Behaviour.

[41]  Abraham Kandel,et al.  Investigations on the applicability of fuzzy interference , 1992 .

[42]  Michio Sugeno,et al.  A fuzzy-logic-based approach to qualitative modeling , 1993, IEEE Trans. Fuzzy Syst..

[43]  Boris Kovalerchuk,et al.  Comparison of empirical and computed values of fuzzy conjunction , 1992 .

[44]  I. Turksen.  Interval-valued fuzzy sets and “compensatory AND” , 1992 .

[45]  J. Dombi.  Membership function as an evaluation , 1990 .

[46]  M. Gupta,et al.  Theory of T -norms and fuzzy inference methods , 1991 .

[47]  Ronald R. Yager,et al.  Quantifier guided aggregation using OWA operators , 1996, Int. J. Intell. Syst..

[48]  J. Dombi.  A general class of fuzzy operators, the demorgan class of fuzzy operators and fuzziness measures induced by fuzzy operators , 1982 .

[49]  L. Valverde,et al.  On Some Logical Connectives for Fuzzy Sets Theory , 1983 .

[50]  W. Pedrycz.  Why triangular membership functions , 1994 .

[51]  Ronald R. Yager,et al.  Including probabilistic uncertainty in fuzzy logic controller modeling using Dempster-Shafer theory , 1995, IEEE Trans. Syst. Man Cybern..

[52]  Joachim Weisbrod,et al.  On fuzzy implication relations , 1994 .

[53]  B. van de Walle,et al.  Implementing the generalised modus-ponens in a fuzzy rule-base for an atmospheric stability monitoring study , 1994, Proceedings of 1994 IEEE 3rd International Fuzzy Systems Conference.

[54]  Satoru Fukami,et al.  Some considerations on fuzzy conditional inference , 1980 .

[55]  Yau-Hwang Kuo,et al.  Evaluation of fuzzy operators with membership function translation approach , 1994, Proceedings of 1994 IEEE 3rd International Fuzzy Systems Conference.

[56]  Hamid R. Berenji,et al.  Learning and tuning fuzzy logic controllers through reinforcements , 1992, IEEE Trans. Neural Networks.

[57]  Gloria Bordogna,et al.  Linguistic aggregation operators of selection criteria in fuzzy information retrieval , 1995, Int. J. Intell. Syst..

[58]  G. Casella,et al.  Statistical Inference , 2003, Encyclopedia of Social Network Analysis and Mining.

[59]  János Fodor,et al.  A characterization of the Hamacher family of t -norms , 1994 .

[60]  Dimitar Filev,et al.  SLIDE: A simple adaptive defuzzification method , 1993, IEEE Trans. Fuzzy Syst..

[61]  J. Mendel.  Fuzzy logic systems for engineering: a tutorial , 1995, Proc. IEEE.

[62]  J. B. Kiszka,et al.  The influence of some fuzzy implication operators on the accuracy of a fuzzy model-part II , 1985 .

[63]  S. Mabuchi.  A proposal for a defuzzification strategy by the concept of sensitivity analysis , 1993 .

[64]  R. Mead,et al.  The Design of Experiments , 1989 .

[65]  Jon T. Butler,et al.  Multiple-valued logic , 1995 .

[66]  Margaret J. Robertson,et al.  Design and Analysis of Experiments , 2006, Handbook of statistics.

[67]  V. Kreinovich,et al.  Gaussian membership functions are most adequate in representing uncertainty in measurements , 1992 .