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.
References
Baldwin JF, Lawry J, Martin TP (1996) Mass assignment theory of the probability of fuzzy events. Fuzzy Set Syst 83:353–367
Bezdek JC (1981) Pattern Recognition with Fuzzy Objective Function Algorithms. Plenum Press, New York, NY, USA
Bezdek JC, Keller J, Krishnapuram R, Pal N (1999) Fuzzy Models and Algorithms for Pattern Recognition and Image Processing. Kluwer, Dordrecht, Netherlands
Bilgiç T and Türksen IB (2000) Measurement of membership functions: theoretical and empirical work. In: [15], pp 195–232
Borgelt C, Kruse R, Steinbrecher M (2009) Graphical Models: Representations for Learning, Reasoning and Data Mining, 2nd edition. J. Wiley & Sons, Chichester, United Kingdom
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
Clarke M, Kruse R, Moral S (eds.) (1993) Symbolic and Quantitative Approaches to Reasoning and Uncertainty, LNCS vol 747. Springer, Berlin, Germany
Coletti G and Scozzafava R (2004) Conditional probability, fuzzy sets, and possibility: a unifying view. Fuzzy Set Syst 144:227–249
Dempster AP (1967) Upper and lower probabilities induced by a multivalued mapping. Ann Math Stat 38:325–339
Dempster AP (1968) Upper and lower probabilities generated by a random closed interval. Ann Math Stat 39:957–966
Dubois D and Prade H (1988) Possibility Theory. Plenum Press, New York, NY, USA
Dubois D and Prade H (1992) When upper probabilities are possibility measures. Fuzzy Set Syst 49:65–74
Dubois D, Fargier H, Prade H (1996) Possibility theory in constraint satisfaction problems: handling priority, preference and uncertainty. Appl Intell 6:287–309
Dubois D, Moral S, Prade H (1997) A semantics for possibility theory based on likelihoods. J Math Anal Appl 205(2):359–380
Dubois D and Prade H (eds) (2000) Fundamentals of Fuzzy Sets. Kluwer, New York, NY, USA
Dubois D, Ostasiewicz W, Prade H (2000) Fuzzy sets: history and basic notions. In: [15], pp 21–106
Fodor JC and Roubens MR (1994) Fuzzy Preference Modelling and Multicriteria Decision Support. Springer, Heidelberg, Germany
Frege G (1893) Grundgesetze der Arithmetik. Band I. Hermann Pohle, Jena, Deutschland
Frege G (1903) Grundgesetze der Arithmetik. Band II. Hermann Pohle, Jena, Deutschland
Gaines BR (1978) Fuzzy and probability uncertainty logics. Inform Control 38:154–169
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
Gebhardt J and Kruse R (1993) The context model – an integrating view of vagueness and uncertainty. Int J Approx Reason 9:283–314
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
Gebhardt J (1997) Learning from Data: Possibilistic Graphical Models. Habilitation Thesis. University of Braunschweig, Germany
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
Höppner F, Klawonn F, Kruse R, Runkler T (1999) Fuzzy Cluster Analysis. J. Wiley & Sons, Chichester, United Kingdom
Hüllermeier E (2007) Case-based Approximate Reasoning. Springer, Heidelberg/Berlin, Germany
Jensen FV (1996) An Introduction to Bayesian Networks. UCL Press, London, United Kingdom
Klement EP, Mesiar R, Pap E (2000) Triangular Norms. Kluwer, Dordrecht, Netherlands
Klir GJ, Folger TA (1988) Fuzzy Sets, Uncertainty and Information. Prentice Hall, Englewood Cliffs, NJ, USA
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
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
Lawry J (2006) Modelling and Reasoning with Vague Concepts. Springer, Heidelberg/Berlin, Germany
Loginov VJ (1966) Probability treatment of Zadeh membership functions and their use in pattern recognition. Eng Cybern 68–69
Mamdani EH, Assilian S (1975) An experiment in linguistic synthesis with a fuzzy logic controller. Int J Man Mach Stud 7:1–13
Michels K, Klawonn F, Kruse R, Nürnberger A (2006) Fuzzy Control: Fundamentals, Stability and Design of Fuzzy Controllers. Springer, Heidelberg/Berlin, Germany
Nguyen HT (1978) On random sets and belief functions. J Math Anal Appl 65:531–542
Pearl J (1988) Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan Kaufmann, San Mateo, CA, USA, 2nd edition 1992
Rommelfanger H (1996) Fuzzy linear programming and applications. Eur J Oper Res 92:512–527
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
Ruspini EH (1991) On the semantics of fuzzy logic. Int J Approx Reason 5(1):45–88
Takagi T, Sugeno M (1985) Fuzzy identification on systems and its applications to modeling and control. IEEE T Syst Man Cyb 15:116–132
Sugeno M, Kang G (1986) Structure identification of fuzzy model. Fuzzy Set Syst 28: 329–346
Tanaka H, Guo PJ (1999) Possibilistic Data Analysis for Operations Research. Physica-Verlag, Heidelberg, Germany
Wolkenhauer O (1998) Possibility Theory with Applications to Data Analysis. Research Studies Press, Chichester, United Kingdom
Zadeh LA (1965) Fuzzy sets. Inform Control 8:338–353
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
Zadeh LA (1978) Fuzzy sets as a basis for a theory of possibility. Fuzzy Set Syst 1:3–28
Zadeh LA (1995) Discussion: probability theory and fuzzy logic are complementary rather than competitive. Technometrics 37(3):271–276
Zimmermann H-J (1975) Optimale Entscheidungen bei unscharfen Problembeschreibungen. Z Betriebswirt Forsch 27:785–795
Zimmermann H-J (1976) Description and optimization of fuzzy systems. Int J Gen Syst 2:209–216
Zimmermann H-J, Zysno P (1979) Latent connectives in human decision making. Fuzzy Set Syst 3:37–51
Zimmermann H-J (1985) Application of fuzzy set theory to mathematical programming. Inform Sciences 36:29–58
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
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
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00287-015-0932-7