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A Formal Model of Interpretability of Linguistic Variables

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Interpretability Issues in Fuzzy Modeling

Part of the book series: Studies in Fuzziness and Soft Computing ((STUDFUZZ,volume 128))

Abstract

The present 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 investigation of what “interpretability” actually means is still missing. So far, interpretability has often been associated with heuristic assumptions about shape and mutual overlapping of fuzzy membership functions. In this chapter, we attempt to approach this problem from a more general and formal point of view. First, we clarify what, in our opinion, the different aspects of interpretability are. Following that, we propose an axiomatic framework for the interpretability of linguistic variables (in Zadeh’s sense) which is underlined by examples and application perspectives.

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References

  1. R. Babuška. Construction of fuzzy systems — interplay between precision and transparency. In Proc. European Symposium on Intelligent Techniques (ESIT 2000), pages 445–452, Aachen, September 2000.

    Google Scholar 

  2. M. Bikdash. A highly interpretable form of Sugeno inference systems. IEEE Trans. Fuzzy Systems, 7(6):686–696, December 1999.

    Article  Google Scholar 

  3. U. Bodenhofer. The construction of ordering-based modifiers. In G. Brewka, R. Der, S. Gottwald, and A. Schierwagen, editors, Fuzzy-Neuro Systems ’99, pages 55–62. Leipziger Universitätsverlag, 1999.

    Google Scholar 

  4. U. Bodenhofer. A Similarity-Based Generalization of Fuzzy Orderings, volume C 26 of Schriftenreihe der Johannes-Kepler-Universität Linz. Universitätsverlag Rudolf Trauner, 1999.

    Google Scholar 

  5. U. Bodenhofer. A general framework for ordering fuzzy sets. In B. BouchonMeunier, J. Guitiérrez-Ríoz, L. Magdalena, and R. R. Yager, editors, Technologies for Constructing Intelligent Systems 1: Tasks, pages 213–224. Springer, 2002. (to appear).

    Chapter  Google Scholar 

  6. U. Bodenhofer and P. Bauer. Towards an axiomatic treatment of “interpretability”. In Proc. 6th Int. Conf. on Soft Computing (IIZUKA2000), pages 334–339, Iizuka, October 2000.

    Google Scholar 

  7. U. Bodenhofer and E. P. Klement. Genetic optimization of fuzzy classification systems — a case study. In B. Reusch and K.-H. Temme, editors, Computational Intelligence in Theory and Practice, Advances in Soft Computing, pages 183— 200. Physica-Verlag, Heidelberg, 2001.

    Google Scholar 

  8. J. Casillas, O. Cordón, F. Herrera, and L. Magdalena. Finding a balance between interpretability and accuracy in fuzzy rule-based modelling: An overview. In J. Casillas, O. Cordón, F. Herrera, and L. Magdalena, editors, Trade-off between Accuracy and Interpretability in Fuzzy Rule-Based Modelling, Studies in Fuzziness and Soft Computing. Physica-Verlag, Heidelberg, 2002.

    Google Scholar 

  9. O. Cordón and F. Herrera. A proposal for improving the accuracy of linguistic modeling. IEEE Trans. Fuzzy Systems, 8(3):335–344, June 2000.

    Article  Google Scholar 

  10. B. De Baets. Analytical solution methods for fuzzy relational equations. In D. Dubois and H. Prade, editors, Fundamentals of Fuzzy Sets, volume 7 of The Handbooks of Fuzzy Sets, pages 291–340. Kluwer Academic Publishers, Boston, 2000.

    Chapter  Google Scholar 

  11. B. De Baets and R. Mesiar. T-partitions. Fuzzy Sets and Systems, 97:211–223, 1998.

    Article  MathSciNet  MATH  Google Scholar 

  12. M. De Cock, U. Bodenhofer, and E. E. Kerre. Modelling linguistic expressions using fuzzy relations. In Proc. 6th Int. Conf. on Soft Computing (IIZUKA2000), pages 353–360, Iizuka, October 2000.

    Google Scholar 

  13. M. Drobics, U. Bodenhofer, W. Winiwarter, and E. P. Klement. Data mining using synergies between self-organizing maps and inductive learning of fuzzy rules. In Proc. Joint 9th IFSA World Congress and 20th NAFIPS Int. Conf., pages 1780–1785, Vancouver, July 2001.

    Chapter  Google Scholar 

  14. D. Dubois and H. Prade. What are fuzzy rules and how to use them. Fuzzy Sets and Systems, 84:169–185, 1996.

    Article  MathSciNet  MATH  Google Scholar 

  15. D. Dubois, H. Prade, and L. Ughetto. Checking the coherence and redundancy of fuzzy knowledge bases. IEEE Trans. Fuzzy Systems, 5(6):398–417, August 1997.

    Article  Google Scholar 

  16. D. Dubois, H. Prade, and L. Ughetto. Fuzzy logic, control engineering and artificial intelligence. In H. B. Verbruggen, H.-J. Zimmermann, and R. Babuška, editors, Fuzzy Algorithms for Control, International Series in Intelligent Technologies, pages 17–57. Kluwer Academic Publishers, Boston, 1999.

    Chapter  Google Scholar 

  17. J. Espinosa and J. Vandewalle. Constructing fuzzy models with linguistic integrity from numerical data — AFRELI algorithm. IEEE Trans. Fuzzy Systems, 8(5):591–600, October 2000.

    Article  Google Scholar 

  18. J. Fodor and M. Roubens. Fuzzy Preference Modelling and Multicriteria Decision Support. Kluwer Academic Publishers, Dordrecht, 1994.

    Book  MATH  Google Scholar 

  19. A. Geyer-Schulz. Fuzzy Rule-Based Expert Systems and Genetic Machine Learning, volume 3 of Studies in Fuzziness. Physica-Verlag, Heidelberg, 1995.

    Google Scholar 

  20. A. Geyer-Schulz. The MIT beer distribution game revisited: Genetic machine learning and managerial behavior in a dynamic decision making experiment. In F. Herrera and J. L. Verdegay, editors, Genetic Algorithms and Soft Computing, volume 8 of Studies in Fuzziness and Soft Computing, pages 658–682. PhysicaVerlag, Heidelberg, 1996.

    Google Scholar 

  21. S. Gottwald. Fuzzy Sets and Fuzzy Logic. Vieweg, Braunschweig, 1993.

    Book  MATH  Google Scholar 

  22. J. Haslinger, U. Bodenhofer, and M. Burger. Data-driven construction of Sugeno controllers: Analytical aspects and new numerical methods. In Proc. Joint 9th IFSA World Congress and 20th NAFIPS Int. Conf., pages 239–244, Vancouver, July 2001.

    Chapter  Google Scholar 

  23. E. E. Kerre, M. Mareš, and R. Mesiar. On the orderings of generated fuzzy quantities. In Proc. 7th Int. Conf. on Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU ’98), volume 1, pages 250–253, 1998.

    Google Scholar 

  24. E. P. Klement, R. Mesiar, and E. Pap. Triangular Norms, volume 8 of Trends in Logic. Kluwer Academic Publishers, Dordrecht, 2000.

    Book  Google Scholar 

  25. L. T. Kóczy and K. Hirota. Ordering, distance and closeness of fuzzy sets. Fuzzy Sets and Systems, 59(3):281–293, 1993.

    Article  MathSciNet  MATH  Google Scholar 

  26. R. Kruse, J. Gebhardt, and F. Klawonn. Foundations of Fuzzy Systems. John Wiley &; Sons, New York, 1994.

    Google Scholar 

  27. R. Lowen. Convex fuzzy sets. Fuzzy Sets and Systems, 3:291–310, 1980.

    Article  MathSciNet  MATH  Google Scholar 

  28. R. S. Michalski, I. Bratko, and M. Kubat. Machine Learning and Data Mining. John Wiley &; Sons, Chichester, 1998.

    Google Scholar 

  29. S. Muggleton and L. De Raedt. Inductive logic programming: Theory and methods. J. Logic Program., 19 & 20:629–680, 1994.

    Article  Google Scholar 

  30. J. R. Quinlan. Induction of decision trees. Machine Learning, 1(1):81–106, 1986.

    Google Scholar 

  31. J. R. Quinlan. Learning logical definitions from relations. Machine Learning, 5(3):239–266, 1990.

    Google Scholar 

  32. A. Ralston, E. D. Reilly, and D. Hemmendinger, editors. Encyclopedia of Computer Science. Groves Dictionaries, Williston, VT, 4th edition, 2000.

    MATH  Google Scholar 

  33. E. H. Ruspini. A new approach to clustering. Inf. Control, 15:22–32, 1969.

    Article  MATH  Google Scholar 

  34. M. Setnes, R. Babuška, and H. B. Verbruggen. Rule-based modeling: Precision and transparency. IEEE Trans. Syst. Man Cybern., Part C: Applications and Reviews, 28:165–169, 1998.

    Article  Google Scholar 

  35. M. Setnes and H. Roubos. GA-fuzzy modeling and classification: Complexity and performance. IEEE Trans. Fuzzy Systems, 8(5):509–522, October 2000.

    Article  Google Scholar 

  36. J. Yen, L. Wang, and C. W. Gillespie. Improving the interpretability of TSK fuzzy models by combining global learning and local learning. IEEE Trans. Fuzzy Systems, 6(4):530–537, November 1998.

    Article  Google Scholar 

  37. L. A. Zadeh. Fuzzy sets. Inf. Control, 8:338–353, 1965.

    Article  MathSciNet  MATH  Google Scholar 

  38. L. A. Zadeh. The concept of a linguistic variable and its application to approximate reasoning I. Inform. Sci., 8:199–250, 1975.

    Article  MathSciNet  MATH  Google Scholar 

  39. L. A. Zadeh. The concept of a linguistic variable and its application to approximate reasoning II. Inform. Sci., 8:301–357, 1975.

    Article  MathSciNet  MATH  Google Scholar 

  40. L. A. Zadeh. The concept of a linguistic variable and its application to approximate reasoning III. Inform. Sci., 9:43–80, 1975.

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Software Competence Center Hagenberg, A-4232, Hagenberg, Austria

    Ulrich Bodenhofer

  2. COMNEON Software, A-4040, Linz, Austria

    Peter Bauer

Authors
  1. Ulrich Bodenhofer
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  2. Peter Bauer
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Editor information

Editors and Affiliations

  1. Dpto. Ciencias de la Computación e Inteligencia Artificial, Escuela Técnica Superior de Ingeniería Informática, Universidad de Granada, E - 18071, Granada, Spain

    Jorge Casillas , Oscar Cordón  & Francisco Herrera ,  & 

  2. Dpto. Matemáticas Aplicadas a las Tecnologías de la Información, Escuela Técnica Superior de Ingenieros de Telecomunicación, Universidad Politécnica de Madrid, E - 28040, Madrid, Spain

    Luis Magdalena

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Bodenhofer, U., Bauer, P. (2003). A Formal Model of Interpretability of Linguistic Variables. In: Casillas, J., Cordón, O., Herrera, F., Magdalena, L. (eds) Interpretability Issues in Fuzzy Modeling. Studies in Fuzziness and Soft Computing, vol 128. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-37057-4_22

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  • DOI: https://doi.org/10.1007/978-3-540-37057-4_22

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-05702-1

  • Online ISBN: 978-3-540-37057-4

  • eBook Packages: Springer Book Archive

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