Distinguishability quantification of fuzzy sets

Distinguishability is a semantic property of fuzzy sets that has a great relevance in the design of interpretable fuzzy models. Distinguishability has been mathematically defined through different measures, which are addressed in this paper. Special emphasis is given to similarity, which exhibits sound theoretical properties but its calculation is usually computationally intensive, and possibility, whose calculation can be very efficient but it does not exhibit the same properties of similarity. It is shown that under mild conditions – usually met in interpretable fuzzy modeling – possibility can be used as a valid measure for assessing distinguishability, thus overcoming the computational inefficiencies of similarity measures. Moreover, procedures that minimize possibility also minimize similarity and, consequently, improve distinguishability. In this sense, the use of possibility is fully justified in interpretable fuzzy modeling.

[1]  Magne Setnes,et al.  Rule-based modeling: precision and transparency , 1998, IEEE Trans. Syst. Man Cybern. Part C.

[2]  Didier Dubois,et al.  Possibility Theory - An Approach to Computerized Processing of Uncertainty , 1988 .

[3]  Uzay Kaymak,et al.  Similarity measures in fuzzy rule base simplification , 1998, IEEE Trans. Syst. Man Cybern. Part B.

[4]  Ian Witten,et al.  Data Mining , 2000 .

[5]  Yaochu Jin,et al.  Fuzzy modeling of high-dimensional systems: complexity reduction and interpretability improvement , 2000, IEEE Trans. Fuzzy Syst..

[6]  Brigitte Charnomordic,et al.  Generating an interpretable family of fuzzy partitions from data , 2004, IEEE Transactions on Fuzzy Systems.

[7]  Mo-Yuen Chow,et al.  Heuristic constraints enforcement for training of and knowledge extraction from a fuzzy/neural architecture. I. Foundation , 1999, IEEE Trans. Fuzzy Syst..

[8]  Herbert Toth Towards Fixing Some ‘Fuzzy’ Catchwords: A Terminological Primer , 1999 .

[9]  Giovanna Castellano,et al.  A neuro-fuzzy network to generate human-understandable knowledge from data , 2002, Cognitive Systems Research.

[10]  Witold Pedrycz,et al.  Data Mining Methods for Knowledge Discovery , 1998, IEEE Trans. Neural Networks.

[11]  Janusz Kacprzyk,et al.  Computing with Words in Information/Intelligent Systems 1 , 1999 .

[12]  J. Casillas Interpretability issues in fuzzy modeling , 2003 .

[13]  J. Valente de Oliveira On the optimization of fuzzy systems using bio-inspired strategies , 1998 .

[14]  Bernhard Sendhoff,et al.  Extracting Interpretable Fuzzy Rules from RBF Networks , 2003, Neural Processing Letters.

[15]  Magne Setnes,et al.  Compact and transparent fuzzy models and classifiers through iterative complexity reduction , 2001, IEEE Trans. Fuzzy Syst..

[16]  José Valente de Oliveira,et al.  Towards neuro-linguistic modeling: Constraints for optimization of membership functions , 1999, Fuzzy Sets Syst..

[17]  Rudolf Kruse,et al.  A neuro-fuzzy approach to obtain interpretable fuzzy systems for function approximation , 1998, 1998 IEEE International Conference on Fuzzy Systems Proceedings. IEEE World Congress on Computational Intelligence (Cat. No.98CH36228).

[18]  L. Zadeh Fuzzy sets as a basis for a theory of possibility , 1999 .

[19]  Gary G. Yen,et al.  Quantitative measures of the accuracy, comprehensibility, and completeness of a fuzzy expert system , 2002, 2002 IEEE World Congress on Computational Intelligence. 2002 IEEE International Conference on Fuzzy Systems. FUZZ-IEEE'02. Proceedings (Cat. No.02CH37291).

[20]  W. Pedrycz,et al.  An introduction to fuzzy sets : analysis and design , 1998 .

[21]  Robert Babuška,et al.  A multi-objective evolutionary algorithm for fuzzy modeling , 2001, Proceedings Joint 9th IFSA World Congress and 20th NAFIPS International Conference (Cat. No. 01TH8569).

[22]  José Valente de Oliveira,et al.  Semantic constraints for membership function optimization , 1999, IEEE Trans. Syst. Man Cybern. Part A.

[23]  Joos Vandewalle,et al.  Constructing fuzzy models with linguistic integrity from numerical data-AFRELI algorithm , 2000, IEEE Trans. Fuzzy Syst..

[24]  Piero P. Bonissone,et al.  Hybrid soft computing systems: industrial and commercial applications , 1999, Proc. IEEE.

[25]  Luis Magdalena,et al.  Interpretability Improvements to Find the Balance Interpretability-Accuracy in Fuzzy Modeling: An Overview , 2003 .

[26]  Moshe Sipper,et al.  Fuzzy CoCo: Balancing Accuracy and Interpretability of Fuzzy Models by Means of Coevolution , 2003 .

[27]  Rami Zwick,et al.  Measures of similarity among fuzzy concepts: A comparative analysis , 1987, Int. J. Approx. Reason..