A New Axiomatic Approach to Interval-Valued Entropy

  • Humberto BustinceEmail author
  • Javier Fernandez
  • Iosu Rodriguez
  • Borja de la Osa
  • Cédric Marco-Detchart
  • Jose Antonio Sanz Delgado
  • Zdenko Takáč
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 1000)


In this work we propose a new definition of interval-valued entropy taking into account the width of the considered membership intervals. We build these new entropies by aggregating normal \(E_N\) functions.



This work has been suported by research project TIN2016-77356-P (AEI/UE,FEDER) of the Spanish Government and by Project VEGA 1/0614/18.


  1. 1.
    Asiain, M.J., Bustince, H., Mesiar, R., Kolesárová, A., Takáč, Z.: Negations with respect to admissible orders in the interval-valued fuzzy set theory. IEEE Trans. Fuzzy Syst. 26, 556–568 (2018)CrossRefGoogle Scholar
  2. 2.
    Barrenechea, E., Bustince, H., De Baets, B., Lopez-Molina, C.: Construction of interval-valued fuzzy relations with application to the generation of fuzzy edge images. IEEE Trans. Fuzzy Syst. 19(5), 819–830 (2011)CrossRefGoogle Scholar
  3. 3.
    Barrenechea, E., Fernandez, J., Pagola, M., Chiclana, F., Bustince, H.: Construction of interval-valued fuzzy preference relations from ignorance functions and fuzzy preference relations. Appl. Decis. Making Knowl.-Based Syst. 58, 33–44 (2014)CrossRefGoogle Scholar
  4. 4.
    Bentkowska, U., Bustince, H., Jurio, A., Pagola, M., Pekala, B.: Decision making with an interval-valued fuzzy preference relation and admissible orders. Appl. Soft Comput. 35, 792–801 (2015)CrossRefGoogle Scholar
  5. 5.
    Burillo, P., Bustince, H.: Construction theorems for intuitionistic fuzzy sets. Fuzzy Sets Syst. 84, 271–281 (1996)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bustince, H.: Indicator of inclusion grade for interval-valued fuzzy sets. Application to approximate reasoning based on interval-valued fuzzy sets. Int. J. Approximate Reasoning 23(3), 137–209 (2000)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bustince, H., Barrenechea, E., Pagola, M.: Relationship between restricted dissimilarity functions, restricted equivalence functions and normal \(E_N\)-functions: image thresholding invariant. Pattern Recogn. Lett. 29(4), 525–536 (2008)CrossRefGoogle Scholar
  8. 8.
    Bustince, H., Barrenechea, E., Pagola, M., Fernández, J.: Interval-valued fuzzy sets constructed from matrices: application to edge detection. Fuzzy Sets Syst. 160, 1819–1840 (2009)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Bustince, H., Barrenechea, E., Pagola, M., Fernández, J., Xu, Z., Bedregal, B., Montero, J., Hagras, H., Herrera, F., De Baets, B.: A historical account of types of fuzzy sets and their relationship. IEEE Trans. Fuzzy Syst. 24(1), 179–194 (2016)CrossRefGoogle Scholar
  10. 10.
    Bustince, H., Fernandez, J., Kolesárová, A., Mesiar, R.: Generation of linear orders for intervals by means of aggregation functions. Fuzzy Sets Syst. 220, 69–77 (2013)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Bustince, H., Marco-Detchart, C., Fernandez, J., Wagner, C., Garibaldi, J., Takáč, Z.: Similarity between interval-valued fuzzy sets taking into account the width of the intervals and admissible orders. Fuzzy Sets Syst. (Submitted )Google Scholar
  12. 12.
    Castillo, O., Melin, P.: A review on interval type-2 fuzzy logic applications in intelligent control. Inf. Sci. 279, 615–631 (2014)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Cornelis, C., Deschrijver, G., Kerre, E.E.: Implication in intuitionistic fuzzy and interval-valued fuzzy set theory: construction, classification, application. Int. J. Approximate Reasoning 35(1), 55–95 (2004)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Choi, H.M., Mun, G.S., Ahn, J.Y.: A medical diagnosis based on interval-valued fuzzy sets. Biomed. Eng.-Appl. Basis Commun. 24(4), 349–354 (2012)CrossRefGoogle Scholar
  15. 15.
    Couto, P., Jurio, A., Varejao, A., Pagola, M., Bustince, H., Melo-Pinto, P.: An IVFS-based image segmentation methodology for rat gait analysis. Soft Comput. 15(10), 1937–1944 (2011)CrossRefGoogle Scholar
  16. 16.
    Jurio, A., Pagola, M., Mesiar, R., Beliakov, G., Bustince, H.: Image magnification using interval information. IEEE Trans. Image Process. 20(11), 3112–3123 (2011)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Sambuc, R.: Function phi-flous application a l’aide au diagnostic en pathologie thyroidienne. Ph.D. thesis, University of Marseille (1975)Google Scholar
  18. 18.
    Sanz, J.A., Fernández, A., Bustince, H., Herrera, F.: A genetic tuning to improve the performance of fuzzy rule-based classification systems with interval-valued fuzzy sets: degree of ignorance and lateral position. Int. J. Approximate Reasoning 52(6), 751–766 (2011)CrossRefGoogle Scholar
  19. 19.
    Sanz, J.A., Fernández, A., Bustince, H., Herrera, F.: Improving the performance of fuzzy rule-based classification systems with interval-valued fuzzy sets and genetic amplitude tuning. Inf. Sci. 180(19), 3674–3685 (2010)CrossRefGoogle Scholar
  20. 20.
    Sanz, J.A., Fernandez, A., Bustince, H., Herrera, F.: IVTURS: a linguistic fuzzy rule-based classification system based on a new interval-valued fuzzy reasoning method with tuning and rule selection. IEEE Trans. Fuzzy Syst. 21(3), 399–411 (2013)CrossRefGoogle Scholar
  21. 21.
    Wang, J., Guo, Q.: Ensemble interval-valued fuzzy cognitive maps. IEEE Access 6, 38356–38366 (2018)CrossRefGoogle Scholar
  22. 22.
    Xu, Z.S., Yager, R.R.: Some geometric aggregation operators based on intuitionistic fuzzy sets. Int. J. Gen. Syst. 35, 417–433 (2006)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Humberto Bustince
    • 1
    Email author
  • Javier Fernandez
    • 1
  • Iosu Rodriguez
    • 1
  • Borja de la Osa
    • 1
  • Cédric Marco-Detchart
    • 1
  • Jose Antonio Sanz Delgado
    • 1
  • Zdenko Takáč
    • 2
  1. 1.Departamento de Estadistica, Informatica y MatematicasUniversidad Publica de NavarraPamplonaSpain
  2. 2.Institute of Information Engineering, Automation and MathematicsThe Slovak University of TechnologyBratislavaSlovakia

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