Interpretability of linguistic variables: a formal account. (English) Zbl 1249.94093

Summary: This contribution is concerned with the interpretability of fuzzy rule-based systems. While this property is widely considered to be a crucial one in fuzzy rule-based modeling, a more detailed formal investigation of what “interpretability” actually means is not available. So far, interpretability has most often been associated with rather heuristic assumptions about shape and mutual overlapping of fuzzy membership functions. In this paper, we attempt to approach this problem from a more general and formal point of view. First, we clarify what the different aspects of interpretability are in our opinion. Consequently, we propose an axiomatic framework for dealing with the interpretability of linguistic variables (in Zadeh’s original sense) which is underlined by examples and application aspects, such as, fuzzy systems design aid, data-driven learning and tuning, and rule base simplification.


94D05 Fuzzy sets and logic (in connection with information, communication, or circuits theory)
68T05 Learning and adaptive systems in artificial intelligence
68T35 Theory of languages and software systems (knowledge-based systems, expert systems, etc.) for artificial intelligence
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[1] Babuška R.: Construction of fuzzy systems - interplay between precision and transparency. Proc. Europ. Symp. on Intelligent Techniques, Aachen 2000, pp. 445-452
[2] Bikdash M.: A highly interpretable form of Sugeno inference systems. IEEE Trans. Fuzzy Systems 7 (1999), 686-696 · doi:10.1109/91.811237
[3] Bodenhofer U.: The construction of ordering-based modifiers. Fuzzy-Neuro Systems ’99 (G. Brewka, R. Der, S. Gottwald and A. Schierwagen, Leipziger Universitätsverlag 1999, pp. 55-62
[4] Bodenhofer U.: A Similarity-Based Generalization of Fuzzy Orderings. (Schriftenreihe der Johannes-Kepler-Universität Linz C26.) Universitätsverlag Rudolf Trauner, Linz 1999 · Zbl 1113.03333 · doi:10.1142/S0218488500000411
[5] Bodenhofer U.: A general framework for ordering fuzzy sets. Technologies for Constructing Intelligent Systems 1: Tasks, (B. Bouchon-Meunier, J. Guitiérrez-Ríoz, L. Magdalena, and R. R. Yager, Studies in Fuzziness and Soft Computing 89), Physica-Verlag, Heidelberg 2002, pp. 213-224 · Zbl 1009.68150
[6] Bodenhofer U., Bauer P.: Towards an axiomatic treatment of “interpretability”. Proc. 6th Internat. Conference on Soft Computing, Iizuka 2000, pp. 334-339
[7] Bodenhofer U., Bauer P.: A formal model of interpretability of linguistic variables. Interpretability Issues in Fuzzy Modeling (J. Casillas, O. Cordón, F. Herrera and L. Magdalena, Studies in Fuzziness and Soft Computing 128), Springer, Berlin 2003, pp. 524-545 · Zbl 1038.93050
[8] Bodenhofer U., Cock, M. De, Kerre E. E.: Openings and closures of fuzzy preorderings: Theoretical basics and applications to fuzzy rule-based systems. Internat. J. General Systems 4 (2003), 343-360 · Zbl 1110.03048 · doi:10.1080/0308107031000135026
[9] Bodenhofer U., Klement E. P.: Genetic optimization of fuzzy classification systems - a case study. Computational Intelligence in Theory and Practice (B. Reusch and K.-H. Temme, Advances in Soft Computing), Physica-Verlag, Heidelberg 2001, pp. 183-200 · Zbl 1002.68148
[10] Casillas J., Cordón O., Herrera, F., Magdalena L.: Interpretability improvements to find the balance interpretability-accuracy in fuzzy modeling: an overview. Interpretability Issues in Fuzzy Modeling (J. Casillas, O. Cordón, F. Herrera and L. Magdalena, Studies in Fuzziness and Soft Computing 128), Springer-Verlag, Berlin 2003, pp. 3-24 · Zbl 1048.93003
[11] Casillas J., Cordón O., Herrera, F., (eds.) L. Magdalena: Interpretability Issues in Fuzzy Modeling (Studies in Fuzziness and Soft Computing 128). Springer-Verlag, Berlin 2003
[12] Cordón O., Herrera F.: A proposal for improving the accuracy of linguistic modeling. IEEE Trans. Fuzzy Systems 8 (2000), 335-344 · doi:10.1109/91.855921
[13] Baets B. De: Analytical solution methods for fuzzy relational equations. Fundamentals of Fuzzy Sets (D. Dubois and H. Prade, The Handbooks of Fuzzy Sets 7), Kluwer Academic Publishers, Boston 2000, pp. 291-340 · Zbl 0970.03044
[14] Baets B. De, Mesiar R.: \(T\)-partitions. Fuzzy Sets and Systems 97 (1998), 211-223 · Zbl 0930.03070 · doi:10.1016/S0165-0114(96)00331-4
[15] Cock M. De, Bodenhofer, U., Kerre E. E.: Modelling linguistic expressions using fuzzy relations. Proc. 6th Internat. Conference on Soft Computing, Iizuka 2000, pp. 353-360
[16] Drobics M., Bodenhofer U.: Fuzzy modeling with decision trees. Proc. 2002 IEEE Inernat. Conference on Systems, Man and Cybernetics, Hammamet 2002
[17] Drobics M., Bodenhofer, U., Klement E. P.: FS-FOIL: An inductive learning method for extracting interpretable fuzzy descriptions. Internat. J. Approx. Reason. 32 (2003), 131-152 · Zbl 1026.68111 · doi:10.1016/S0888-613X(02)00080-4
[18] Dubois D., Prade H.: What are fuzzy rules and how to use them. Fuzzy Sets and Systems 84 (1996), 169-185 · Zbl 0905.03008 · doi:10.1016/0165-0114(96)00066-8
[19] Dubois D., Prade, H., Ughetto L.: Checking the coherence and redundancy of fuzzy knowledge bases. IEEE Trans. Fuzzy Systems 5 (1997), 398-417 · doi:10.1109/91.618276
[20] Dubois D., Prade, H., Ughetto L.: Fuzzy logic, control engineering and artificial intelligence. Fuzzy Algorithms for Control (H. B. Verbruggen, H.-J. Zimmermann, and R. Babuška, International Series in Intelligent Technologies), Kluwer Academic Publishers, Boston 1999, pp. 17-57
[21] Espinosa J., Vandewalle J.: Constructing fuzzy models with linguistic integrity from numerical data - AFRELI algorithm. IEEE Trans. Fuzzy Systems 8 (2000), 591-600 · doi:10.1109/91.873582
[22] Fodor J., Roubens M.: Fuzzy Preference Modelling and Multicriteria Decision Support. Kluwer Academic Publishers, Dordrecht 1994 · Zbl 0827.90002
[23] Geyer-Schulz A.: Fuzzy Rule-based Expert Systems and Genetic Machine Learning. (Studies in Fuzziness 3.) Physica-Verlag, Heidelberg 1995 · Zbl 0914.68166
[24] Geyer-Schulz A.: The MIT beer distribution game revisited: Genetic machine learning and managerial behavior in a dynamic decision making experiment. Genetic Algorithms and Soft Computing (F. Herrera and J. L. Verdegay, Studies in Fuzziness and Soft Computing 8, Physica-Verlag, Heidelberg 1996, pp. 658-682
[25] Gottwald S.: Fuzzy Sets and Fuzzy Logic. Vieweg, Braunschweig 1993 · Zbl 1088.03024 · doi:10.1016/j.fss.2005.05.031
[26] Haslinger J., Bodenhofer, U., Burger M.: Data-driven construction of Sugeno controllers: Analytical aspects and new numerical methods. Proc. Joint 9th IFSA World Congress and 20th NAFIPS Internat. Conference, Vancouver 2001, pp. 239-244
[27] Kerre E. E., Mareš, M., Mesiar R.: On the orderings of generated fuzzy quantities. Proc. 7th Internat. Conference on Information Processing and Management of Uncertainty in Knowledge-based Systems, Paris 1998, pp. 250-253
[28] Klement E. P., Mesiar, R., Pap E.: Triangular Norms (Trends in Logic 8). Kluwer Academic Publishers, Dordrecht 2000 · Zbl 0972.03002
[29] Kóczy L. T., Hirota K.: Ordering, distance and closeness of fuzzy sets. Fuzzy Sets and Systems 59 (1993), 281-293 · Zbl 1002.03530 · doi:10.1016/0165-0114(93)90473-U
[30] Kruse R., Gebhardt, J., Klawonn F.: Foundations of Fuzzy Systems. Wiley, New York 1994 · Zbl 0843.68109
[31] Lowen R.: Convex fuzzy sets. Fuzzy Sets and Systems 3 (1980), 291-310 · Zbl 0439.52001 · doi:10.1016/0165-0114(80)90025-1
[32] Michalski R. S., Bratko, I., Kubat M.: Machine Learning and Data Mining. Wiley, Chichester 1998
[33] Muggleton S., Raedt L. De: Inductive logic programming: Theory and methods. J. Logic Program. 19/20 (1994), 629-680 · Zbl 0816.68043 · doi:10.1016/0743-1066(94)90035-3
[34] Pedrycz W., Sosnowski Z. A.: Designing decision trees with the use of fuzzy granulation. IEEE Trans. Systems Man Cybernet. A 30 (2000), 151-159 · doi:10.1109/3468.833095
[35] Quinlan J. R.: Induction of decision trees. Mach. Learning 1 (1986), 81-106 · doi:10.1007/BF00116251
[36] Quinlan J. R.: Learning logical definitions from relations. Mach. Learning 5 (1990), 239-266 · doi:10.1007/BF00117105
[37] Ralston A., Reilly E. D., (eds.) D. Hemmendinger: Encyclopedia of Computer Science. Fourth edition. Groves Dictionaries, Williston 2000 · Zbl 0954.68002
[38] Ruspini E. H.: A new approach to clustering. Inform. and Control 15 (1969), 22-32 · Zbl 0192.57101 · doi:10.1016/S0019-9958(69)90591-9
[39] Setnes M., Babuška, R., Verbruggen H. B.: Rule-based modeling: Precision and transparency. IEEE Trans. Systems Man Cybernet. C 28 (1998), 165-169 · doi:10.1109/5326.661100
[40] Setnes M., Roubos H.: GA-fuzzy modeling and classification: Complexity and performance. IEEE Trans. Fuzzy Systems 8 (2000), 509-522 · doi:10.1109/91.873575
[41] Yen J., Wang, L., Gillespie C. W.: Improving the interpretability of TSK fuzzy models by combining global learning and local learning. IEEE Trans. Fuzzy Systems 6 (1998), 530-537 · doi:10.1109/91.728447
[42] Zadeh L. A.: Fuzzy sets. Inform. and Control 8 (1965), 338-353 · Zbl 0139.24606 · doi:10.1016/S0019-9958(65)90241-X
[43] Zadeh L. A.: The concept of a linguistic variable and its application to approximate reasoning I. Inform. Sci. 8 (1975), 199-250 · Zbl 0397.68071 · doi:10.1016/0020-0255(75)90036-5
[44] Zadeh L. A.: The concept of a linguistic variable and its application to approximate reasoning II. Inform. Sci. 8 (1975), 301-357 · Zbl 0404.68074 · doi:10.1016/0020-0255(75)90046-8
[45] Zadeh L. A.: The concept of a linguistic variable and its application to approximate reasoning III. Inform. Sci. 9 (1975), 43-80 · Zbl 0404.68075 · doi:10.1016/0020-0255(75)90017-1
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