Advertisement

A new correlation coefficient of the Pythagorean fuzzy sets and its applications

  • Nguyen Xuan ThaoEmail author
Methodologies and Application
  • 11 Downloads

Abstract

In this paper, we propose a new correlation coefficient between Pythagorean fuzzy sets. We then use this new result to compute some examples through which we find that it benefits from such an outcome with some well-known results in the literature. In probability and statistical theory, the correlation coefficient indicates the strength of the linear correlation between two random variables. The correlation coefficient is equal to one in the case of a linear correlation and − 1 in the case of a linear inverse correlation. Other values in the range (− 1, 1) indicate the degree of linear dependence between variables. The closer the coefficient is to − 1 and 1, the stronger the correlation between variables. As in statistics with real variables, we refer to variance and covariance between two intuitionistic fuzzy sets. Then, we determined the formula for calculating the correlation coefficient based on the variance and covariance of the intuitionistic fuzzy set, and the value of this correlation coefficient is in [− 1, 1]. We also commented on the linear relationship between fuzzy sets affecting their correlation coefficients through examples to show the usefulness in the proposed new measure. Then, we develop this direction to build correlation coefficients between the interval-valued intuitionistic fuzzy sets and apply it in the pattern recognition problem.

Keywords

Pythagorean fuzzy set Interval-valued Pythagorean fuzzy set Variance Covariance Correlation coefficient 

Notes

Compliance with ethical standards

Conflict of interest

The authors declare that there is no conflict of interests regarding the publication of this paper.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

References

  1. Atanassov KT (1986) Intuitionistic fuzzy sets. Fuzzy Sets Syst 20(1):87–96.  https://doi.org/10.1016/S0165-0114(86)80034-3 CrossRefzbMATHGoogle Scholar
  2. Atanassov KT (1999) In intuitionistic fuzzy sets. Physica, HeidelbergCrossRefGoogle Scholar
  3. Atanassov K, Gargov G (1989) Interval valued Intuitionistic fuzzy sets. Fuzzy Sets Syst 31(3):343–349.  https://doi.org/10.1016/0165-0114(89)90205-4 MathSciNetCrossRefzbMATHGoogle Scholar
  4. Bharati SK, Singh SR (2014) Intuitionistic fuzzy optimization technique in agricultural production planning: a small farm holder perspective. Int J Comput Appl 89(6):17–23.  https://doi.org/10.5120/15507-4276 CrossRefGoogle Scholar
  5. Bustince H, Burillo P (1995) Correlation of interval-valued Intuitionistic fuzzy sets. Fuzzy Sets Syst 74(2):237–244.  https://doi.org/10.1016/0165-0114(94)00343-6 MathSciNetCrossRefzbMATHGoogle Scholar
  6. Chiang DA, Lin NP (1999) Correlation of fuzzy sets. Fuzzy Sets Syst 102(2):221–226.  https://doi.org/10.1016/S0165-0114(97)00127-9 MathSciNetCrossRefzbMATHGoogle Scholar
  7. Garg H (2016) A novel correlation coefficients between pythagorean fuzzy sets and its applications to decision-making processes. Int J Intell Syst 31(12):1234–1252.  https://doi.org/10.1002/int.21827 CrossRefGoogle Scholar
  8. Gerstenkorn T, Mańko J (1991) Correlation of intuitionistic fuzzy sets. Fuzzy Sets Syst 44(1):39–43.  https://doi.org/10.1016/0165-0114(91)90031-K MathSciNetCrossRefzbMATHGoogle Scholar
  9. Hung WL (2001) Using statistical viewpoint in developing correlation of intuitionistic fuzzy sets. Int J Uncertain Fuzziness Knowl-Based Syst 9(04):509–516.  https://doi.org/10.1142/S0218488501000910 MathSciNetCrossRefzbMATHGoogle Scholar
  10. Hung WL, Wu JW (2002) Correlation of intuitionistic fuzzy sets by centroid method. Inf Sci 144(1):219–225.  https://doi.org/10.1016/S0020-0255(02)00181-0 MathSciNetCrossRefzbMATHGoogle Scholar
  11. Hwang CM, Yang MS, Hung WL, Lee MG (2012) A similarity measure of intuitionistic fuzzy sets based on the Sugeno integral with its application to pattern recognition. Inf Sci 189:93–109.  https://doi.org/10.1016/j.ins.2011.11.029 MathSciNetCrossRefzbMATHGoogle Scholar
  12. Li J, Zeng W (2015) A new dissimilarity measure between intuitionistic fuzzy sets and its application in multiple attribute decision making. J Intell Fuzzy Syst 29(4):1311–1320.  https://doi.org/10.1002/int.21934 MathSciNetCrossRefzbMATHGoogle Scholar
  13. Li D, Zeng W (2018) Distance measure of pythagorean fuzzy sets. Int J Intell Syst 33(2):348–361.  https://doi.org/10.1002/int.21934 MathSciNetCrossRefGoogle Scholar
  14. Liu B, Shen Y, Mu L, Chen X, Chen L (2016) A new correlation measure of the intuitionistic fuzzy sets. J Intell Fuzzy Syst 30(2):1019–1028.  https://doi.org/10.3233/IFS-151824 CrossRefzbMATHGoogle Scholar
  15. Mitchell HB (2004) A correlation coefficient for intuitionistic fuzzy sets. Int J Intell Syst 19(5):483–490.  https://doi.org/10.1002/int.20004 CrossRefzbMATHGoogle Scholar
  16. Peng X, Yang Y (2015) Some results for Pythagorean fuzzy sets. Int J Intell Syst 30(11):1133–1160.  https://doi.org/10.1002/int.21738 MathSciNetCrossRefGoogle Scholar
  17. Shidpour H, Bernard A, Shahrokhi M (2013) A group decision-making method based on intuitionistic fuzzy set in the three dimensional concurrent engineering environment: a multi-objective programming approach. Procedia CIRP 7:533–538.  https://doi.org/10.1016/j.procir.2013.06.028 CrossRefGoogle Scholar
  18. Szmidt E, Kacprzyk J (2004) A similarity measure for intuitionistic fuzzy sets and its application in supporting medical diagnostic reasoning. In: International conference on artificial intelligence and soft computing, pp 388–393.  https://doi.org/10.1007/978-3-540-24844-6_56
  19. Thao NX (2018) A new correlation coefficient of the intuitionistic fuzzy sets and its application. J Intell Fuzzy Syst 35(2):1959–1968MathSciNetCrossRefGoogle Scholar
  20. Thao NX, Ali M, Smarandache F (2019) An intuitionistic fuzzy clustering algorithm based on a new correlation coefficient with application in medical diagnosis. J Intell Fuzzy Syst 36(1):189–198CrossRefGoogle Scholar
  21. Xu Z (2006) On correlation measures of intuitionistic fuzzy sets. Lect Notes Comput Sci 4224:16–24.  https://doi.org/10.1007/11875581_2 CrossRefGoogle Scholar
  22. Xu Z (2010) Choquet integrals of weighted intuitionistic fuzzy information. Inf Sci 180(5):726–736.  https://doi.org/10.1016/j.ins.2009.11.011 MathSciNetCrossRefzbMATHGoogle Scholar
  23. Yager RR (2013) Pythagorean fuzzy subsets. In: IFSA World Congress and NAFIPS Annual Meeting, pp 57–61.  https://doi.org/10.1109/ifsa-nafips.2013.6608375
  24. Yager RR (2014) Pythagorean membership grades complex numbers and decision making. Int J Intell Syst 22:958–965.  https://doi.org/10.1002/int.21584 CrossRefGoogle Scholar
  25. Zadeh LA (1965) Fuzzy sets. Inf Control 8(3):338–353.  https://doi.org/10.1016/S0019-9958(65)90241-X CrossRefzbMATHGoogle Scholar
  26. Zeng W, Wang J (2011) Correlation coefficient of interval-valued intuitionistic fuzzy sets. In: International conference on fuzzy systems and knowledge discovery, pp 98–102.  https://doi.org/10.1016/j.mcm.2009.06.010
  27. Zhang X (2016) Multicriteria Pythagorean fuzzy decision analysis: a hierarchical QUALIFLEX approach with the closeness index-based ranking methods. Inf Sci 330:104–124.  https://doi.org/10.1016/j.ins.2015.10.012 CrossRefGoogle Scholar
  28. Zhang X, Xu Z (2014) Extension of TOPSIS to multiple criteria decision making with Pythagorean fuzzy sets. Int J Intell Syst 29(12):1061–1078.  https://doi.org/10.1002/int.21676 MathSciNetCrossRefGoogle Scholar
  29. Zhou X, Zhao R, Yu F, Tian H (2016) Intuitionistic fuzzy entropy clustering algorithm for infrared image segmentation. J Intell Fuzzy Syst 30(3):1831–1840.  https://doi.org/10.3233/IFS-151894 CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Faculty of Information TechnologyVietnam National University of AgricultureHa NoiViet Nam

Personalised recommendations