Method for Uncertainty Measurement and Its Application to the Formation of Interval Type-2 Fuzzy Sets

  • Mauricio A. SanchezEmail author
  • Oscar Castillo
  • Juan R. Castro
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 401)


This paper proposes a new method for directly discovering the uncertainty from a sample of discrete data, which is then used in the formation of an Interval Type-2 Fuzzy Inference System. A Coefficient of Variation is used to measure the uncertainty on a finite sample of discrete data. Based on the maximum possible coverage area of the Footprint of Uncertainty of Gaussian membership functions, with uncertainty on the standard deviation, which then are modified according to the found index values, obtaining all antecedents in the process. Afterwards, the Cuckoo Search algorithm is used to optimize the Interval Sugeno consequents of the Fuzzy Inference System. Some sample datasets are used to measure the output interval coverage.


Membership Function Data Pair Reference Target Gaussian Membership Function Output Interval 
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.



We thank the MyDCI program of the Division of Graduate Studies and Research, UABC, and Tijuana Institute of Technology the financial support provided by our sponsor CONACYT contract grant number: 314258.


  1. 1.
    Yu, X., Mehrotra, S.: Capturing Uncertainty in Spatial Queries over Imprecise Data, pp. 192–201. Springer, Heidelberg (2003)Google Scholar
  2. 2.
    Klir, G.J.: Uncertainty and Information: Foundations of Generalized Information Theory. Wiley-IEEE Press, New York (2005)Google Scholar
  3. 3.
    Weise, K., Woger, W.: A Bayesian theory of measurement uncertainty. Measure. Sci. Technol. 3, 1–11 (1992)Google Scholar
  4. 4.
    Jurado, K., Ludvigson, S.C., Ng, S.: Measuring Uncertainty. Am. Econ. Rev. (AEA) 105(3), 1177–1216 (2013)Google Scholar
  5. 5.
    Chen, G., Ying, M., Liu, Y.: Dealing with uncertainty and fuzziness in intelligent systems. Int. J. Intell. Syst. 24, 223–225 (2009)CrossRefzbMATHGoogle Scholar
  6. 6.
    Klir, G.J., Wierman, M.J.: Uncertainty-Based Information. Physica-Verlag HD, Heidelberg (1999)CrossRefzbMATHGoogle Scholar
  7. 7.
    Mendel, J.M., John, R.I., Liu, F.: Interval Type-2 fuzzy logic systems made simple. IEEE Trans. Fuzzy Syst. 14, 808–821 (2006)CrossRefGoogle Scholar
  8. 8.
    Castillo, O., Melin, P.: Recent Advances in Interval Type-2 Fuzzy Systems. Springer, Berlin (2012)CrossRefzbMATHGoogle Scholar
  9. 9.
    Mo, H., Wang, F.-Y., Zhou, M., Li, R., Xiao, Z.: Footprint of uncertainty for type-2 fuzzy sets. Inf. Sci. (Ny) 272, 96–110 (2014)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Bezdek, J.C., Ehrlich, R., Full, W.: FCM: the fuzzy c-means clustering algorithm. Comput. Geosci. 10, 191–203 (1984)CrossRefGoogle Scholar
  11. 11.
    Yang, X.-S.: Cuckoo Search via Lévy flights. 2009 World Congress on Nature & Biologically Inspired Computing (NaBIC). pp. 210–214. IEEE (2009)Google Scholar
  12. 12.
    Ying, H.: Interval Type-2 Takagi-Sugeno fuzzy systems with linear rule consequent are universal approximators. In: NAFIPS 2009–2009 Annual Meeting of the North American Fuzzy Information Processing Society, pp. 1–5. IEEE (2009)Google Scholar
  13. 13.
    Zadeh, L.A.: Fuzzy Sets. Inf. Control 8, 338–353 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Melin, P., Castillo, O.: Fuzzy Modeling Fundamentals. Wiley Encyclopedia of Computer Science and Engineering. Wiley, New York (2007)Google Scholar
  15. 15.
    Mendel, J.: General type-2 fuzzy logic systems made simple: a tutorial. IEEE Trans. Fuzzy Syst. pp. 1–1 (2013)Google Scholar
  16. 16.
    The MathWorks, Inc., Natick, Massachusetts, U.S.: MATLAB Release 2013b, (2013)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Mauricio A. Sanchez
    • 1
    Email author
  • Oscar Castillo
    • 2
  • Juan R. Castro
    • 1
  1. 1.Autonomous University of Baja CaliforniaTijuanaMexico
  2. 2.Tijuana Institute of TechnologyTijuanaMexico

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