Skip to main content
Log in

Entropy and similarity measure of Atanassov’s intuitionistic fuzzy sets and their application to pattern recognition based on fuzzy measures

  • Theoretical Advances
  • Published:
Pattern Analysis and Applications Aims and scope Submit manuscript

Abstract

In this study, we first examine entropy and similarity measure of Atanassov’s intuitionistic fuzzy sets, and define a new entropy. Meanwhile, a construction approach to get the similarity measure of Atanassov’s intuitionistic fuzzy sets is introduced, which is based on entropy. Since the independence of elements in a set is usually violated, it is not suitable to aggregate the values for patterns by additive measures. Based on the given entropy and similarity measure, we study their application to Atanassov’s intuitionistic fuzzy pattern recognition problems under fuzzy measures, where the interactions between features are considered. To overall reflect the interactive characteristics between them, we define three Shapley-weighted similarity measures. Furthermore, if the information about the weights of features is incompletely known, models for the optimal fuzzy measure on feature set are established. Moreover, an approach to pattern recognition under Atanassov’s intuitionistic fuzzy environment is developed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Zadeh LA (1965) Fuzzy sets. Inf Control 8:338–353

    Article  MathSciNet  MATH  Google Scholar 

  2. Zadeh LA (1975) The concept of a linguistic variable and its application to approximate reasoning-I. Inf Sci 8(3):199–249

    Article  MathSciNet  MATH  Google Scholar 

  3. Atanassov K (1983) Intuitionistic fuzzy sets. In: Seventh Scientific Session of ITKR, Sofia

  4. Gau WL, Buehrer DJ (1993) Vague sets. IEEE Trans Syst Man Cybern B Cybern 23(2):610–614

    Article  MATH  Google Scholar 

  5. Bustince H, Burillo P (1996) Vague sets are intuitionistic fuzzy sets. Fuzzy Sets Syst 79(3):403–405

    Article  MathSciNet  MATH  Google Scholar 

  6. Bustince H, Herrera F, Montero J (2007) Fuzzy sets and their extensions: representation, aggregation, and models. Springer-Verlag, Heidelberg

    MATH  Google Scholar 

  7. Hung WL, Yang MS (2008) On the J-divergence of intuitionistic fuzzy sets with its application to pattern recognition. Inf Sci 178(6):1641–1650

    Article  MathSciNet  MATH  Google Scholar 

  8. Kharal A (2009) Homeopathic drug selection using intuitionistic fuzzy sets. Homeopathy 98(1):35–39

    Article  Google Scholar 

  9. Li DF (2008) A note on ‘‘using intuitionistic fuzzy sets for fault-tree analysis on printed circuit board assembly’’. Microelectron Reliab 48(10):1741

    Article  Google Scholar 

  10. Mitchell HB (2003) On the Dengfeng-Chuntian similarity measure and its application to pattern recognition. Pattern Recogn Lett 24(16):3101–3104

    Article  Google Scholar 

  11. Xu ZS (2008) Intuitionistic fuzzy information: aggregation theory and applications. Science Press, Beijing

    Google Scholar 

  12. Chen TY, Li CH (2010) Determining objective weights with intuitionistic fuzzy entropy measures: a comparative analysis. Inform Sci 180(21):4207–4222

    Article  Google Scholar 

  13. Chen ZP, Yang W (2011) A new multiple attribute group decision making method in intuitionistic fuzzy setting. Appl Math Model 35(9):4424–4437

    Article  MathSciNet  MATH  Google Scholar 

  14. Burillo P, Bustince H (1996) Entropy on intuitionistic fuzzy sets and on interval valued fuzzy sets. Fuzzy Sets Syst 78(3):305–316

    Article  MathSciNet  MATH  Google Scholar 

  15. Bustince H, Barrenechea E, Pagola M, Fernandez J, Guerra C, Couto P (2011) Generalized Atanassov’s intuitionistic fuzzy index: construction of Atanassov’s fuzzy entropy from fuzzy implication operators. Int J Uncertain Fuzz 19(1):51–69

    Article  MathSciNet  MATH  Google Scholar 

  16. Szmidt E, Kacprzyk J (2001) Entropy for intuitionistic fuzzy sets. Fuzzy Sets Syst 118(3):467–477

    Article  MathSciNet  MATH  Google Scholar 

  17. De Luca A, Termini S (1972) A definition of nonprobabilistic entropy in the setting of fuzzy theory. Inf Control 20(4):301–312

    Article  MATH  MathSciNet  Google Scholar 

  18. Wang Y, Lei YJ (2007) A technique for constructing intuitionistic fuzzy entropy. Control Decis 22(12):1390–1394

    MATH  Google Scholar 

  19. Huang GS, Liu YS (2005) The fuzzy entropy of vague sets based on non-fuzzy sets. Comput Appl Soft 22(6):16–17

    Google Scholar 

  20. Wei CP, Wang P, Zhang YZ (2011) Entropy, similarity measure of interval-valued intuitionistic fuzzy sets and their applications. Inf Sci 181(19):4273–4286

    Article  MathSciNet  MATH  Google Scholar 

  21. Chen SM (1995) Measures of similarity between vague sets. Fuzzy Sets Syst 74(2):217–223

    Article  MATH  MathSciNet  Google Scholar 

  22. Chen SM (1997) Similarity measures between vague sets and between elements. IEEE Trans Syst Man Cybern B Cybern 27(1):153–158

    Article  Google Scholar 

  23. Li DF, Cheng CT (2002) New similarity measures of intuitionistic fuzzy sets and application to pattern recognitions. Pattern Recogn Lett 23(1–3):221–225

    MATH  Google Scholar 

  24. Liang ZZ, Shi PF (2003) Similarity measures on intuitionistic fuzzy sets. Pattern Recogn Lett 24(15):2687–2693

    Article  MATH  Google Scholar 

  25. Szmidt E, Kacprzyk J (2009) Analysis of similarity measures for Atanassov’s intuitionistic fuzzy sets. In: Proceedings IFSA/EUSFLAT, the DBLP Computer Science Bibliography, Lisbon, Portugal, pp 1416–1421

  26. Bustince H, Barrenechea E, Pagola M (2007) Image thresholding using restricted equivalence functions and maximizing the measures of similarity. Fuzzy Sets Syst 158(5):496–516

    Article  MathSciNet  MATH  Google Scholar 

  27. Bustince H, Barrenechea E, Pagola M (2008) Relationship between restricted dissimilarity functions, restricted equivalence functions and normal EN-functions: image thresholding invariant. Pattern Recogn Lett 29(4):525–536

    Article  Google Scholar 

  28. Grabisch M (1995) Fuzzy integral in multicriteria decision making. Fuzzy Sets Syst 69(3):279–298

    Article  MathSciNet  MATH  Google Scholar 

  29. Grabisch M (1996) The application of fuzzy integrals in multicriteria decision making. Eur J Oper Res 89(3):445–456

    Article  MATH  Google Scholar 

  30. Sugeno M (1974) Theory of fuzzy integral and its application. Doctorial Dissertation, Tokyo Institute of Technology

  31. Grabisch M (1997) k-order additive discrete fuzzy measures and their representation. Fuzzy Sets Syst 92(2):167–189

    Article  MathSciNet  MATH  Google Scholar 

  32. Grabisch M, Roubens M (1999) An axiomatic approach to the concept of interaction among players in cooperative games. Int J Game Theory 28(4):547–565

    Article  MathSciNet  MATH  Google Scholar 

  33. Kojadinovic I (2005) Relevance measures for subset variable selection in regression problems based on k-additive mutual Information. Comput Stat Data Anal 49(4):205–1227

    Article  MathSciNet  MATH  Google Scholar 

  34. Marichal JL, Kojadinovic I, Fujimoto K (2007) Axiomatic characterizations of generalized values. Discret Appl Math 155(1):26–43

    Article  MathSciNet  MATH  Google Scholar 

  35. Li SJ, Zhang Q (2008) The measure of interaction among players in games with fuzzy coalitions. Fuzzy Sets Syst 159(2):119–137

    Article  MATH  MathSciNet  Google Scholar 

  36. Grabisch M, Murofushi T, Sugeno M (2000) Fuzzy measure and integrals. Physica-Verlag, New York

    MATH  Google Scholar 

  37. Grabisch M, Labreuche C (2008) A decade of application of the Choquet and Sugeno integrals in multi-criteria decision aid. 4OR-Q J Oper Res 6(1):1–44

    Article  MathSciNet  MATH  Google Scholar 

  38. Tan CQ, Chen XH (2010) Intuitionistic fuzzy Choquet integral operator for multi-criteria decision making. Expert Syst Appl 37(1):149–157

    Article  Google Scholar 

  39. Tan CQ (2011) Generalized intuitionistic fuzzy geometric aggregation operator and its application to multi-criteria group decision making. Soft Comput 15(5):867–876

    Article  MATH  Google Scholar 

  40. Tan CQ, Chen XH (2011) Induced intuitionistic fuzzy Choquet integral operator for multi-criteria decision making. Int J Intell Syst 26(7):659–686

    Article  Google Scholar 

  41. Xu ZS (2010) Choquet integrals of weighted intuitionistic fuzzy information. Inf Sci 180(5):726–736

    Article  MATH  MathSciNet  Google Scholar 

  42. Li JQ, Deng GN, Li HX, Zeng WY (2012) The relationship between similarity measure and entropy of intuitionistic fuzzy sets. Inf Sci 188(1):314–321

    Article  MathSciNet  MATH  Google Scholar 

  43. Hung WL, Yang MS (2004) Similarity measures of intuitionistic fuzzy sets based on Hausdorff distance. Pattern Recogn Lett 25(14):1603–1611

    Article  Google Scholar 

  44. Shapley LS (1953) A value for n-person game. In: Kuhn H, Tucker A (eds) Contributions to the theory of games. Princeton University Press, Princeton

    Google Scholar 

  45. Zhang QS, Jiang SY (2008) A note on information entropy measures for vague sets and its applications. Inf Sci 178(21):4184–4191

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

The authors gratefully thank the Editor-in-Chief and two anonymous referees for their valuable comments, which have much improved the paper. This work was supported by the Funds for Creative Research Groups of China (No. 71221061), the Projects of Major International Cooperation NSFC (No. 71210003), the National Natural Science Foundation of China (Nos. 71201089, 71201110, 27127117 and 71271029), the Natural Science Foundation Youth Project of Shandong Province, China (ZR2012GQ005), the Specialized Research Fund for the Doctoral Program of Higher Education (No. 20111101110036), and the Program for New Century Excellent Talents in University of China (No. NCET-12-0541).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Fanyong Meng.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Meng, F., Chen, X. Entropy and similarity measure of Atanassov’s intuitionistic fuzzy sets and their application to pattern recognition based on fuzzy measures. Pattern Anal Applic 19, 11–20 (2016). https://doi.org/10.1007/s10044-014-0378-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10044-014-0378-6

Keywords

Navigation