Journal of Mathematical Chemistry

, Volume 57, Issue 5, pp 1252–1267 | Cite as

Approximation of fuzzy functions by fuzzy interpolating bicubic splines: 2018 CMMSE conference

  • P. GonzálezEmail author
  • H. Idais
  • M. Pasadas
  • M. Yasin
Original Paper


One of the most interesting and important techniques in applied mathematics is interpolation and approximation. It can be also shown that basic theory of fuzzy sets and operations with fuzzy numbers can be used, and result very convenient, for solving many analytical chemistry problems: as depth profile comparison or calibration with errors in signals and/or concentrations, for spectra interpretation, or even for automatic qualitative analysis or expert systems with X-ray spectroscopy. In this special context, we introduce in this paper a new fuzzy interpolation method of fuzzy data or functions, specially indicated in all these chemical applications. We will use bicubic splines (cubic splines of two variables) of class \(\mathcal {C}^2\) as linear combinations of a basis constructed by a tensor product of univariate B-splines in each of these variables. First we establish a bicubic interpolation spline method of a given fuzzy data set or a given fuzzy bivariate function. We proof the existence of a unique solution of this problem and we show a convergence result. Finally, we test the effectiveness of this method by some numerical and graphical examples.


Interpolation and approximation Fuzzy data Bicubic splines Analytical chemistry Calibration Spectroscopy 



We would like to thanks the financial and logistic aid granted by the Department of Applied Mathematics of the University of Granada for the accomplishment of this research work and the possibility to present it at the 2018 CMMSE International Conference.


  1. 1.
    S. Abbasbandy, M. Amirfakhrian, Numerical approximation of fuzzy functions by fuzzy polynomials. Appl. Math. Comput. 174, 1001–1006 (2006)Google Scholar
  2. 2.
    S. Abbasbandy, R. Ezzati, H. Behforooz, Interpolation of fuzzy data by using fuzzy splines. Int. J. Uncertain. Fuzziness Knowl. Based Syst. 16(1), 107–115 (2008)CrossRefGoogle Scholar
  3. 3.
    A. Anile, B. Falcidieno, G. Gallo, M. Spagnuolo, S. Spinello, Modeling uncertain data with fuzzy b-splines. Fuzzy Sets Syst. 113, 397–410 (2000)CrossRefGoogle Scholar
  4. 4.
    H. Bandemer, M. Otto, Fuzzy theory in analytical chemistry. Mikrochim. Acta 89, 93–124 (1986)CrossRefGoogle Scholar
  5. 5.
    T. Blaffert, Computer-assisted multicomponent spectral analysis with fuzzy data sets. Anal. Chim. Acta 161, 135–148 (1984)CrossRefGoogle Scholar
  6. 6.
    S. Chen, New methods for subjective mental workload assessment and fuzzy risk analysis. Int. J. Cybern. Syst. 27(5), 449–472 (1996)CrossRefGoogle Scholar
  7. 7.
    S. Chen, S. Chen, Fuzzy risk analysis based on similarity measures of generalized fuzzy numbers. IEEE Trans. Fuzzy Syst. 11, 45–56 (2003)CrossRefGoogle Scholar
  8. 8.
    C. Hsieh, S. Chen, Similarity of generalized fuzzy numbers with graded mean integration representation. Proc. Int. Fuzzy Syst. Assoc. World Congr. Taipei 2, 551–555 (1999)Google Scholar
  9. 9.
    H. Hsu, C. Chen, Aggregation of fuzzy opinions under group decison making. Fuzzy Sets Syst. 79(3), 279–285 (1996)CrossRefGoogle Scholar
  10. 10.
    O. Kaleva, Interpolation of fuzzy data. Fuzzy Sets Syst. 61, 63–70 (1994)CrossRefGoogle Scholar
  11. 11.
    P.J. King, E.H. Mamdani, The application of fuzzy control systems to industrial processes. Automatica 13, 235–242 (1977)CrossRefGoogle Scholar
  12. 12.
    G. Klir, U. Clair, B. Yuan, Fuzzy Set Theory: Foundations and Applications (Prentice-Hall, Upper Saddle River, 1997)Google Scholar
  13. 13.
    H. Lee, An optimal aggregation method for fuzzy opinions of group decision. Proc. IEEE Int. Conf. Syst. Man Cybern. 3, 314–319 (1999)Google Scholar
  14. 14.
    R.F. Liao, C.W. Chan, J. Hromek, G.H. Huang, L. He, Fuzzy logic control for a petroleum separation process. Eng. Appl. Artif. Intel. 21, 835–845 (2008)CrossRefGoogle Scholar
  15. 15.
    R. Lowen, A fuzzy Lagrange interpolation theorem. Fuzzy Sets Syst. 34, 33–38 (1990)CrossRefGoogle Scholar
  16. 16.
    M. Otto, Fuzzy theory explained. Chemom. Intell. Lab. Syst. 4, 101–120 (1988)CrossRefGoogle Scholar
  17. 17.
    P.M. Prenter, Splines and Variational Methods (Wiley, New York, 1989), pp. 118–135Google Scholar
  18. 18.
    K. Schmucker, Fuzzy Sets Natural Language Computations and Risk Analysis (Computer Science, Rockville, MD, 1984)Google Scholar
  19. 19.
    O. Valenzuela, M. Pasadas, A new approach to estimate the interpolation error of fuzzy data using similarity measures of fuzzy numbers. Comput. Math. Appl. 61, 1633–1654 (2011)CrossRefGoogle Scholar
  20. 20.
    L. Zadeh, Fuzzy sets. Inf. Control 8, 338–353 (1965)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Dpto. de Matemática AplicadaUniv. de GranadaGranadaSpain

Personalised recommendations