Skip to main content
Log in

Bedeutung von Zugehörigkeitsgraden in der Fuzzy-Technologie

  • HAUPTBEITRAG
  • ZUGEHÖRIGKEITSGRADE IN DER FUZZY-TECHNOLOGIE
  • Published:
Informatik-Spektrum Aims and scope

Zusammenfassung

Der Begriff der Fuzzy-Menge erweitert den klassischen Begriff der Menge, sodass man für betrachtete Objekte nicht nur (in einer Menge) ,,enthalten“ und ,,nicht enthalten“ angeben, sondern Grade der Zugehörigkeit unterscheiden kann. Während das nur zweiwertige (Nicht-)Enthaltensein unmittelbar verständlich ist, stellt sich bei dazwischenliegenden Zugehörigkeitsgraden die Frage, was sie bedeuten. Wir geben daher in diesem Aufsatz einen kurzen Überblick über die vier am weitesten verbreiteten Ansätze, Fuzzy-Zugehörigkeitsgraden eine (präzise) Bedeutung zuzuordnen: 1. als Ähnlichkeit zu Referenzwerten, 2. als Ausdruck von Präferenz, 3. als bedingte Wahrscheinlichkeit (likelihood) und 4. als Möglichkeitsgrad (degree of possibility). Wir diskutieren die Voraussetzungen und Ausdrucksmöglichkeiten dieser vier Interpretationen und untersuchen, in welchen Anwendungsbereichen sie jeweils am nützlichsten sind, wobei wir in einigen Fällen Beispielanwendungen erwähnen.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

References

  1. Baldwin JF, Lawry J, Martin TP (1996) Mass assignment theory of the probability of fuzzy events. Fuzzy Set Syst 83:353–367

    Article  MathSciNet  Google Scholar 

  2. Bezdek JC (1981) Pattern Recognition with Fuzzy Objective Function Algorithms. Plenum Press, New York, NY, USA

    Book  MATH  Google Scholar 

  3. Bezdek JC, Keller J, Krishnapuram R, Pal N (1999) Fuzzy Models and Algorithms for Pattern Recognition and Image Processing. Kluwer, Dordrecht, Netherlands

    Book  MATH  Google Scholar 

  4. Bilgiç T and Türksen IB (2000) Measurement of membership functions: theoretical and empirical work. In: [15], pp 195–232

  5. Borgelt C, Kruse R, Steinbrecher M (2009) Graphical Models: Representations for Learning, Reasoning and Data Mining, 2nd edition. J. Wiley & Sons, Chichester, United Kingdom

    Book  Google Scholar 

  6. Cayrac D, Dubois D, Haziza M, Prade H (1996) Handling uncertainty with possibility theory and fuzzy sets in a satellite fault diagnosis application. IEEE T Fuzzy Syst 4:251–269

    Article  Google Scholar 

  7. Clarke M, Kruse R, Moral S (eds.) (1993) Symbolic and Quantitative Approaches to Reasoning and Uncertainty, LNCS vol 747. Springer, Berlin, Germany

    Book  Google Scholar 

  8. Coletti G and Scozzafava R (2004) Conditional probability, fuzzy sets, and possibility: a unifying view. Fuzzy Set Syst 144:227–249

    Article  MATH  MathSciNet  Google Scholar 

  9. Dempster AP (1967) Upper and lower probabilities induced by a multivalued mapping. Ann Math Stat 38:325–339

    Article  MATH  MathSciNet  Google Scholar 

  10. Dempster AP (1968) Upper and lower probabilities generated by a random closed interval. Ann Math Stat 39:957–966

    Article  MATH  MathSciNet  Google Scholar 

  11. Dubois D and Prade H (1988) Possibility Theory. Plenum Press, New York, NY, USA

    Book  MATH  Google Scholar 

  12. Dubois D and Prade H (1992) When upper probabilities are possibility measures. Fuzzy Set Syst 49:65–74

    Article  MATH  MathSciNet  Google Scholar 

  13. Dubois D, Fargier H, Prade H (1996) Possibility theory in constraint satisfaction problems: handling priority, preference and uncertainty. Appl Intell 6:287–309

    Article  Google Scholar 

  14. Dubois D, Moral S, Prade H (1997) A semantics for possibility theory based on likelihoods. J Math Anal Appl 205(2):359–380

    Article  MATH  MathSciNet  Google Scholar 

  15. Dubois D and Prade H (eds) (2000) Fundamentals of Fuzzy Sets. Kluwer, New York, NY, USA

  16. Dubois D, Ostasiewicz W, Prade H (2000) Fuzzy sets: history and basic notions. In: [15], pp 21–106

  17. Fodor JC and Roubens MR (1994) Fuzzy Preference Modelling and Multicriteria Decision Support. Springer, Heidelberg, Germany

    Book  MATH  Google Scholar 

  18. Frege G (1893) Grundgesetze der Arithmetik. Band I. Hermann Pohle, Jena, Deutschland

    Google Scholar 

  19. Frege G (1903) Grundgesetze der Arithmetik. Band II. Hermann Pohle, Jena, Deutschland

    Google Scholar 

  20. Gaines BR (1978) Fuzzy and probability uncertainty logics. Inform Control 38:154–169

    Article  MATH  MathSciNet  Google Scholar 

  21. Gebhardt J, Kruse R (1992) A possibilistic interpretation of fuzzy sets in the context model. In: Proc. 1st IEEE Int. Conf. on Fuzzy Systems, FUZZ-IEEE’92, San Diego, CA, USA. IEEE Press, Piscataway, NJ, USA, pp 1089–1096

  22. Gebhardt J and Kruse R (1993) The context model – an integrating view of vagueness and uncertainty. Int J Approx Reason 9:283–314

    Article  MATH  MathSciNet  Google Scholar 

  23. Gebhardt J, Kruse R, Otte M, Schröder M (1993) A fuzzy idle speed controller. In: Proc. 26th Int. Symp. on Automotive Technology and Automation. Aachen, Germany, pp 459–463

  24. Gebhardt J (1997) Learning from Data: Possibilistic Graphical Models. Habilitation Thesis. University of Braunschweig, Germany

  25. Hisdal E (1986) Infinite-valued logic based on two-valued logic and probability. Part 1.1: Difficulties with present-day fuzzy-set theory and their resolution in the TEE model. Part 1.2: Different sources of fuzziness. Int J Man Mach Stud 25(1):89–111, 25(2):113–138

  26. Höppner F, Klawonn F, Kruse R, Runkler T (1999) Fuzzy Cluster Analysis. J. Wiley & Sons, Chichester, United Kingdom

    Google Scholar 

  27. Hüllermeier E (2007) Case-based Approximate Reasoning. Springer, Heidelberg/Berlin, Germany

    Google Scholar 

  28. Jensen FV (1996) An Introduction to Bayesian Networks. UCL Press, London, United Kingdom

    Google Scholar 

  29. Klement EP, Mesiar R, Pap E (2000) Triangular Norms. Kluwer, Dordrecht, Netherlands

    Book  MATH  Google Scholar 

  30. Klir GJ, Folger TA (1988) Fuzzy Sets, Uncertainty and Information. Prentice Hall, Englewood Cliffs, NJ, USA

    MATH  Google Scholar 

  31. Kruse R, Gebhardt J, Klawonn F (1994) Foundations of Fuzzy Systems. J. Wiley & Sons, Chichester, United Kingdom. Deutsche Ausgabe: (1993) Fuzzy Systeme, Series: Leitfäden und Monographien der Informatik. Teubner, Stuttgart, Germany

    Google Scholar 

  32. Kruse R, Borgelt C, Klawonn F, Moewes C, Ruß G, Steinbrecher M, Held P (2011) Computational Intelligence. Springer, Heidelberg/Berlin, Germany. Deutsche Ausgabe: Kruse R, Borgelt C, Klawonn F, Moewes C, Ruß G, Steinbrecher M (2013) Computational Intelligence. Springer-Vieweg, Heidelberg/Wiesbaden, Germany

    Google Scholar 

  33. Lawry J (2006) Modelling and Reasoning with Vague Concepts. Springer, Heidelberg/Berlin, Germany

    MATH  Google Scholar 

  34. Loginov VJ (1966) Probability treatment of Zadeh membership functions and their use in pattern recognition. Eng Cybern 68–69

  35. Mamdani EH, Assilian S (1975) An experiment in linguistic synthesis with a fuzzy logic controller. Int J Man Mach Stud 7:1–13

    Article  MATH  Google Scholar 

  36. Michels K, Klawonn F, Kruse R, Nürnberger A (2006) Fuzzy Control: Fundamentals, Stability and Design of Fuzzy Controllers. Springer, Heidelberg/Berlin, Germany

    Google Scholar 

  37. Nguyen HT (1978) On random sets and belief functions. J Math Anal Appl 65:531–542

    Article  MATH  MathSciNet  Google Scholar 

  38. Pearl J (1988) Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan Kaufmann, San Mateo, CA, USA, 2nd edition 1992

    Google Scholar 

  39. Rommelfanger H (1996) Fuzzy linear programming and applications. Eur J Oper Res 92:512–527

    Article  MATH  Google Scholar 

  40. Ruspini EH (1977) A theory of fuzzy clustering. In: Proc. 16th IEEE Conf. on Decision and Control, New Orleans, LA. IEEE Press, Piscataway, NJ, USA, pp 1378–1383

  41. Ruspini EH (1991) On the semantics of fuzzy logic. Int J Approx Reason 5(1):45–88

    Article  MATH  MathSciNet  Google Scholar 

  42. Takagi T, Sugeno M (1985) Fuzzy identification on systems and its applications to modeling and control. IEEE T Syst Man Cyb 15:116–132

    Article  MATH  Google Scholar 

  43. Sugeno M, Kang G (1986) Structure identification of fuzzy model. Fuzzy Set Syst 28: 329–346

  44. Tanaka H, Guo PJ (1999) Possibilistic Data Analysis for Operations Research. Physica-Verlag, Heidelberg, Germany

    MATH  Google Scholar 

  45. Wolkenhauer O (1998) Possibility Theory with Applications to Data Analysis. Research Studies Press, Chichester, United Kingdom

    MATH  Google Scholar 

  46. Zadeh LA (1965) Fuzzy sets. Inform Control 8:338–353

    Article  MATH  MathSciNet  Google Scholar 

  47. Zadeh LA (1975) The concept of a linguistic variable and its application to approximate reasoning I–III. Inform Sciences 8:199–249, 8:301–357, 9:43–80

  48. Zadeh LA (1978) Fuzzy sets as a basis for a theory of possibility. Fuzzy Set Syst 1:3–28

    Article  MATH  MathSciNet  Google Scholar 

  49. Zadeh LA (1995) Discussion: probability theory and fuzzy logic are complementary rather than competitive. Technometrics 37(3):271–276

    Article  Google Scholar 

  50. Zimmermann H-J (1975) Optimale Entscheidungen bei unscharfen Problembeschreibungen. Z Betriebswirt Forsch 27:785–795

    Google Scholar 

  51. Zimmermann H-J (1976) Description and optimization of fuzzy systems. Int J Gen Syst 2:209–216

    Article  MATH  Google Scholar 

  52. Zimmermann H-J, Zysno P (1979) Latent connectives in human decision making. Fuzzy Set Syst 3:37–51

    Google Scholar 

  53. Zimmermann H-J (1985) Application of fuzzy set theory to mathematical programming. Inform Sciences 36:29–58

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Christian Borgelt.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Borgelt, C., Kruse, R. Bedeutung von Zugehörigkeitsgraden in der Fuzzy-Technologie. Informatik Spektrum 38, 490–499 (2015). https://doi.org/10.1007/s00287-015-0932-7

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00287-015-0932-7

Navigation