Approximation of Membership Functions and Aggregation Operators Using Splines

  • Gleb Beliakov
Part of the Studies in Fuzziness and Soft Computing book series (STUDFUZZ, volume 90)


A good choice of membership functions and aggregation operators is crucial to the behavior fuzzy systems. In many cases there are no theoretical criteria that would justify the use of one or another function, and they are selected based on their goodness of fit to empirical data. This paper discusses a general non-parametric approach to construction of membership functions and aggregation operators based on empirical data. This method is computer oriented: it does not produce operators in closed algebraic form, but the quality of fit and flexibility are superior to other methods. The method is also general, since it can produce membership functions and operators from any class. Restrictions to a particular class can be easily introduced. Examples based on published empirical data are provided.


Membership Function Aggregation Operator Algebraic Form Quadratic Spline Triangular Norm 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Aczel, J. (1969) On applications and theory of functioonal equations, Academic Press, New York.Google Scholar
  2. [2]
    Amaya Cruz, G.P. and Beliakov, G. (1995), Approximate reasoning and interpretation of laboratory tests in medical diagnostics, Cybernetics and Systems, 26, 713–729.MATHCrossRefGoogle Scholar
  3. [3]
    Beatson, R.K. and Ziegler, Z. (1985). Monotonicity preserving surface interpolation, SIAM J. Numer. Anal. 22, 401–411.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    Beliakov, G. (2000). Definition of general aggregation operators through similarity relations, Fuzzy Sets and Systems 114, 437–453.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    Beliakov, G. (1999). Aggregation operators as similarity relations, in Information, Uncertainty, Fusion., R. Yager, B. Bouchon-Meunier and L. Zadeh, eds, Kluwer, Boston, 331–342.Google Scholar
  6. [6]
    Beliakov, G. (2001) Shape preserving approximation using least squares splines, Approximation theory and applications,in press.Google Scholar
  7. [7]
    de Boor C. (1978). A practical guide to splines, Springer, Berlin — New York.MATHCrossRefGoogle Scholar
  8. [8]
    Chen, J. and Otto, K. (1995). Constructing membership functions using interpolation and measurement theory, Fuzzy Sets and Systems 73, 313–327.MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    Dierckx P. (1995). Curve and surface fitting with splines, Clarendon press, Oxford.MATHGoogle Scholar
  10. [10]
    Dombi, J. (1982). Basic concepts for a theory of evaluation: The aggregative operator, Europ. J. Oper. Res. 10, 282–293.MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    Dombi, J. (1990). Membership function as evaluation, Fuzzy Sets and Systems, 35, 1–21.MathSciNetMATHCrossRefGoogle Scholar
  12. [12]
    Dongarra, J.J. and Grosse, E. (1987). Distribution of mathematical software via electronic mail, Comm. ACM 30, 403–440.CrossRefGoogle Scholar
  13. [13]
    Dubois, D. and Prade, H. (1985) A review of fuzzy set aggregation connectives, Information Sciences 36, 85–121.MathSciNetMATHCrossRefGoogle Scholar
  14. [14]
    Dyckhoff, H. and Pedrycz, W. (1984). Generalized means as model of compensative connectives, Fuzzy Sets and Systems 14, 143–154.MathSciNetMATHCrossRefGoogle Scholar
  15. [15]
    Elfving, T. and Andersson, L.-E. (1989). An algorithm for computing constrained smoothing spline functions, Numer.Math. 52, 583–595.MathSciNetCrossRefGoogle Scholar
  16. [16]
    Eubank, R. L. (1999) Nonparametric regression and spline smoothing, Marcel Dekker, New York.MATHGoogle Scholar
  17. [17]
    Filev, D. and Yager, R. (1998). On the issue of obtaining OWA operator weights, Fuzzy Sets and Systems 94, 157–169.MathSciNetCrossRefGoogle Scholar
  18. [18]
    Halmos, P. (1958). Finite-dimensional vector spaces, D.van Nostrand Company, Princeton.MATHGoogle Scholar
  19. [19]
    Hanson, R.J. and Haskell, K.H. (1982). Algorithm 587. Two algorithms for the linearly constrained least squares problem, ACM Trans. Math. Software 8, 323–333.MATHCrossRefGoogle Scholar
  20. [20]
    Hisdal, E. (1998) Logical structures for representation of knowledge and uncertainty, Physica-Verlag, New York.MATHGoogle Scholar
  21. [21]
    Klir, G. and Folger, T. (1992). Fuzzy sets, uncertainty, and information, Prentice Hall, Singapore.Google Scholar
  22. [22]
    Lawson, C. and Hanson, R. (1995). Solving least squares problems, SIAM, Philadelphia.MATHCrossRefGoogle Scholar
  23. [23]
    Mizumoto, M. (1989). Pictorial representations of fuzzy connectives, Part I: Cases of t-norms, t-conorms and averaging operators, Fuzzy Sets and Systems 31, 217–242.MathSciNetCrossRefGoogle Scholar
  24. [24]
    Norwich, A.M. and Turksen, I.B. (1984). A model for the measurement of membership and the consequences of its empirical implementation, Fuzzy Sets and Systems, 12, 1–25.MathSciNetMATHCrossRefGoogle Scholar
  25. [25]
    Ramsay, J.O. (1988). Monotone regression splines in action (with comments), Stat. Science 3, 425–461.CrossRefGoogle Scholar
  26. [26]
    Schweizer, B. and Sklar, A. (1961) Associative functions and statistical triangle inequalities, Publ. Mathematicae Debrecen, 8, 169–186.MathSciNetMATHGoogle Scholar
  27. [27]
    Yager, R. (1978). Fuzzy decision making including unequal objectives, Fuzzy Sets and Systems 1 87–95.MATHCrossRefGoogle Scholar
  28. [28]
    Yager, R. (1988). On ordered weighted averaging aggregation operators in multicriteria decision making, IEEE Trans. Syst.,Man, Cybern. 18, 183–190.MathSciNetMATHGoogle Scholar
  29. [29]
    Yager, R. (1998). Including importances in OWA aggregations using fuzzy systems modeling, IEEE Trans. Fuzzy Syst 6, 286–294.CrossRefGoogle Scholar
  30. [30]
    Zimmermann, H.-J. (1996). Fuzzy set theory - and its applications, Kluwer, Boston.MATHGoogle Scholar
  31. [31]
    Zimmermann, H.-J. and Zysno, P. (1980). Latent connectives in human decision making, Fuzzy Sets and Systems, 4, 37–51.MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Gleb Beliakov
    • 1
  1. 1.School of Computing and MathematicsDeakin UniversityClaytonAustralia

Personalised recommendations