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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Atanassov KT (1986) Intuitionistic fuzzy sets. Fuzzy Sets Syst 20(1):87–96. https://doi.org/10.1016/S0165-0114(86)80034-3
Atanassov KT (1999) In intuitionistic fuzzy sets. Physica, Heidelberg
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Thao NX (2018) A new correlation coefficient of the intuitionistic fuzzy sets and its application. J Intell Fuzzy Syst 35(2):1959–1968
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):189–198
Xu Z (2006) On correlation measures of intuitionistic fuzzy sets. Lect Notes Comput Sci 4224:16–24. https://doi.org/10.1007/11875581_2
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
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
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
Zadeh LA (1965) Fuzzy sets. Inf Control 8(3):338–353. https://doi.org/10.1016/S0019-9958(65)90241-X
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
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
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
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
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
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.
Additional information
Communicated by V. Loia.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Thao, N.X. A new correlation coefficient of the Pythagorean fuzzy sets and its applications. Soft Comput 24, 9467–9478 (2020). https://doi.org/10.1007/s00500-019-04457-7
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00500-019-04457-7