{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,14]],"date-time":"2026-01-14T11:33:36Z","timestamp":1768390416375,"version":"3.49.0"},"reference-count":44,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2025,9,8]],"date-time":"2025-09-08T00:00:00Z","timestamp":1757289600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,9,8]],"date-time":"2025-09-08T00:00:00Z","timestamp":1757289600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comp. Appl. Math."],"published-print":{"date-parts":[[2026,2]]},"DOI":"10.1007\/s40314-025-03383-9","type":"journal-article","created":{"date-parts":[[2025,9,9]],"date-time":"2025-09-09T07:35:38Z","timestamp":1757403338000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Modified correlation coefficients of hesitant fuzzy sets based on proximity and their application in medical diagnosis"],"prefix":"10.1007","volume":"45","author":[{"given":"Juan","family":"Liu","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0009-0000-1941-0470","authenticated-orcid":false,"given":"Wangyong","family":"Lv","sequence":"additional","affiliation":[]},{"given":"Zhehao","family":"Sun","sequence":"additional","affiliation":[]},{"given":"Qian","family":"Song","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,9,8]]},"reference":[{"key":"3383_CR1","first-page":"1","volume":"20","author":"K Atanassov","year":"2016","unstructured":"Atanassov K (2016) Intuitionistic fuzzy sets. Int J Bioautomation 20:1","journal-title":"Int J Bioautomation"},{"issue":"2","key":"3383_CR2","doi-asserted-by":"publisher","first-page":"237","DOI":"10.1016\/0165-0114(94)00343-6","volume":"74","author":"H Bustince","year":"1995","unstructured":"Bustince H, Burillo P (1995) Correlation of interval-valued intuitionistic fuzzy sets. Fuzzy Sets Syst 74(2):237\u2013244","journal-title":"Fuzzy Sets Syst"},{"issue":"11","key":"3383_CR3","doi-asserted-by":"publisher","first-page":"2760","DOI":"10.1068\/a41323","volume":"41","author":"C Chen","year":"2009","unstructured":"Chen C, Chen J, Timmermans H (2009) Historical deposition influence in residential location decisions: a distance-based gev model for spatial correlation. Environ Plan A 41(11):2760\u20132777","journal-title":"Environ Plan A"},{"issue":"1","key":"3383_CR4","doi-asserted-by":"publisher","first-page":"234","DOI":"10.1016\/j.econmod.2008.07.004","volume":"26","author":"FY Chen","year":"2009","unstructured":"Chen FY, Liao SL (2009) Modelling var for foreign-asset portfolios in continuous time. Econ Model 26(1):234\u2013240","journal-title":"Econ Model"},{"issue":"4","key":"3383_CR5","doi-asserted-by":"publisher","first-page":"2197","DOI":"10.1016\/j.apm.2012.04.031","volume":"37","author":"N Chen","year":"2013","unstructured":"Chen N, Xu Z, Xia M (2013) Correlation coefficients of hesitant fuzzy sets and their applications to clustering analysis. Appl Math Model 37(4):2197\u20132211","journal-title":"Appl Math Model"},{"issue":"2","key":"3383_CR6","doi-asserted-by":"publisher","first-page":"209","DOI":"10.1016\/S0165-0114(98)00081-5","volume":"110","author":"DA Chiang","year":"2000","unstructured":"Chiang DA, Lin NP (2000) Partial correlation of fuzzy sets. Fuzzy Sets Syst 110(2):209\u2013215","journal-title":"Fuzzy Sets Syst"},{"issue":"11","key":"3383_CR7","first-page":"1399","volume":"82","author":"JK Conner","year":"1995","unstructured":"Conner JK, Sterling A (1995) Testing hypotheses of functional relationships: a comparative survey of correlation patterns among floral traits in five insect-pollinated plants. Am J Bot 82(11):1399\u20131406","journal-title":"Am J Bot"},{"issue":"2","key":"3383_CR8","doi-asserted-by":"publisher","first-page":"184","DOI":"10.1002\/int.21633","volume":"29","author":"B Farhadinia","year":"2014","unstructured":"Farhadinia B (2014) Correlation for dual hesitant fuzzy sets and dual interval-valued hesitant fuzzy sets. Int J Intell Syst 29(2):184\u2013205","journal-title":"Int J Intell Syst"},{"key":"3383_CR9","doi-asserted-by":"publisher","first-page":"7009","DOI":"10.1007\/s00500-021-05629-0","volume":"25","author":"B Farhadinia","year":"2021","unstructured":"Farhadinia B, Liao H, Herrera-Viedma E (2021) A modified class of correlation coefficients of hesitant fuzzy information. Soft Comput 25:7009\u20137028","journal-title":"Soft Comput"},{"key":"3383_CR10","doi-asserted-by":"publisher","first-page":"8847","DOI":"10.1007\/s00521-019-04362-y","volume":"32","author":"H Garg","year":"2020","unstructured":"Garg H, Kaur G (2020) A robust correlation coefficient for probabilistic dual hesitant fuzzy sets and its applications. Neural Comput Appl 32:8847\u20138866","journal-title":"Neural Comput Appl"},{"issue":"1","key":"3383_CR11","doi-asserted-by":"publisher","first-page":"39","DOI":"10.1016\/0165-0114(91)90031-K","volume":"44","author":"T Gerstenkorn","year":"1991","unstructured":"Gerstenkorn T, Ma\u0144ko J (1991) Correlation of intuitionistic fuzzy sets. Fuzzy Sets Syst 44(1):39\u201343","journal-title":"Fuzzy Sets Syst"},{"issue":"2","key":"3383_CR12","doi-asserted-by":"publisher","first-page":"237","DOI":"10.15388\/Informatica.2017.128","volume":"28","author":"J Guan","year":"2017","unstructured":"Guan J, Zhou D, Meng F (2017) Distance measure and correlation coefficient for linguistic hesitant fuzzy sets and their application. Informatica 28(2):237\u2013268","journal-title":"Informatica"},{"issue":"6","key":"3383_CR13","doi-asserted-by":"publisher","first-page":"1968","DOI":"10.1007\/s40815-018-0496-1","volume":"20","author":"X Guan","year":"2018","unstructured":"Guan X, Sun G, Yi X et al (2018) Synthetic correlation coefficient between hesitant fuzzy sets with applications. Int J Fuzzy Syst 20(6):1968\u20131985","journal-title":"Int J Fuzzy Syst"},{"issue":"1","key":"3383_CR14","doi-asserted-by":"publisher","first-page":"77","DOI":"10.1016\/0165-0114(94)00330-A","volume":"75","author":"DH Hong","year":"1995","unstructured":"Hong DH, Hwang SY (1995) Correlation of intuitionistic fuzzy sets in abability spaces. Fuzzy Sets Syst 75(1):77\u201381","journal-title":"Fuzzy Sets Syst"},{"issue":"1\u20134","key":"3383_CR15","doi-asserted-by":"publisher","first-page":"219","DOI":"10.1016\/S0020-0255(02)00181-0","volume":"144","author":"WL Hung","year":"2002","unstructured":"Hung WL, Wu JW (2002) Correlation of intuitionistic fuzzy sets by centroid method. Inf Sci 144(1\u20134):219\u2013225","journal-title":"Inf Sci"},{"issue":"4","key":"3383_CR16","doi-asserted-by":"publisher","first-page":"426","DOI":"10.1002\/asna.200710892","volume":"329","author":"J Kloko\u010dn\u00edk","year":"2008","unstructured":"Kloko\u010dn\u00edk J, Kosteleck\u1ef3 J, B\u00f6hm V et al (2008) Correlation between the mayan calendar and ours: astronomy helps to answer why the most popular correlation (gmt) is wrong. Astronomische Nachrichten Astron Notes 329(4):426\u2013436","journal-title":"Astronomische Nachrichten Astron Notes"},{"key":"3383_CR17","doi-asserted-by":"publisher","first-page":"115","DOI":"10.1016\/j.knosys.2015.02.020","volume":"82","author":"H Liao","year":"2015","unstructured":"Liao H, Xu Z, Zeng XJ (2015) Novel correlation coefficients between hesitant fuzzy sets and their application in decision making. Knowl-Based Syst 82:115\u2013127","journal-title":"Knowl-Based Syst"},{"issue":"7","key":"3383_CR18","doi-asserted-by":"publisher","first-page":"2609","DOI":"10.1007\/s40815-023-01509-x","volume":"25","author":"M Liu","year":"2023","unstructured":"Liu M, Zhang X, Mo Z (2023) A modified correlation coefficient of probabilistic hesitant fuzzy sets and its applications of decision making, medical diagnosis, cluster analysis. Int J Fuzzy Syst 25(7):2609\u20132627","journal-title":"Int J Fuzzy Syst"},{"issue":"2","key":"3383_CR19","doi-asserted-by":"publisher","first-page":"1393","DOI":"10.2991\/ijcis.d.191119.001","volume":"12","author":"P Liu","year":"2019","unstructured":"Liu P, Cheng S (2019) Interval-valued probabilistic dual hesitant fuzzy sets for multi-criteria group decision-making. Int J Comput Intell Syst 12(2):1393\u20131411","journal-title":"Int J Comput Intell Syst"},{"key":"3383_CR20","doi-asserted-by":"crossref","unstructured":"Maassen MA (2020) Recent developments in the sustainability field: Energy and economic indicators in the european union in 2015. In: Eurasian Economic Perspectives: Proceedings of the 23rd Eurasia Business and Economics Society Conference, Springer, pp 145\u2013154","DOI":"10.1007\/978-3-030-40375-1_10"},{"issue":"1","key":"3383_CR21","doi-asserted-by":"publisher","first-page":"17","DOI":"10.1002\/int.21741","volume":"31","author":"F Meng","year":"2016","unstructured":"Meng F, Wang C, Chen X et al (2016) Correlation coefficients of interval-valued hesitant fuzzy sets and their application based on the Shapley function. Int J Intell Syst 31(1):17\u201343","journal-title":"Int J Intell Syst"},{"issue":"5","key":"3383_CR22","doi-asserted-by":"publisher","first-page":"483","DOI":"10.1002\/int.20004","volume":"19","author":"H Mitchell","year":"2004","unstructured":"Mitchell H (2004) A correlation coefficient for intuitionistic fuzzy sets. Int J Intell Syst 19(5):483\u2013490","journal-title":"Int J Intell Syst"},{"issue":"1","key":"3383_CR23","doi-asserted-by":"publisher","first-page":"23","DOI":"10.1016\/0165-0114(85)90004-1","volume":"17","author":"C Murthy","year":"1985","unstructured":"Murthy C, Pal S, Majumder DD (1985) Correlation between two fuzzy membership functions. Fuzzy Sets Syst 17(1):23\u201338","journal-title":"Fuzzy Sets Syst"},{"issue":"2","key":"3383_CR24","doi-asserted-by":"publisher","first-page":"177","DOI":"10.1046\/j.1365-2540.2000.00735.x","volume":"85","author":"FM Norry","year":"2000","unstructured":"Norry FM, Vilardi JC, Hasson E (2000) Negative genetic correlation between traits of the drosophila head, and interspecific divergence in head shape. Heredity 85(2):177\u2013183","journal-title":"Heredity"},{"issue":"9\u201310","key":"3383_CR25","doi-asserted-by":"publisher","first-page":"1279","DOI":"10.1016\/j.mcm.2009.06.010","volume":"50","author":"DG Park","year":"2009","unstructured":"Park DG, Kwun YC, Park JH et al (2009) Correlation coefficient of interval-valued intuitionistic fuzzy sets and its application to multiple attribute group decision making problems. Math Comput Model 50(9\u201310):1279\u20131293","journal-title":"Math Comput Model"},{"key":"3383_CR26","doi-asserted-by":"publisher","first-page":"355","DOI":"10.1007\/s40815-018-0578-0","volume":"21","author":"C Song","year":"2019","unstructured":"Song C, Xu Z, Zhao H (2019) New correlation coefficients between probabilistic hesitant fuzzy sets and their applications in cluster analysis. Int J Fuzzy Syst 21:355\u2013368","journal-title":"Int J Fuzzy Syst"},{"key":"3383_CR27","doi-asserted-by":"publisher","first-page":"65714","DOI":"10.1109\/ACCESS.2019.2916564","volume":"7","author":"Z Su","year":"2019","unstructured":"Su Z, Xu Z, Zhao H et al (2019) Entropy measures for probabilistic hesitant fuzzy information. IEEE Access 7:65714\u201365727","journal-title":"IEEE Access"},{"issue":"4","key":"3383_CR28","doi-asserted-by":"publisher","first-page":"529","DOI":"10.1007\/s12559-019-9623-z","volume":"11","author":"G Sun","year":"2019","unstructured":"Sun G, Guan X, Yi X et al (2019) Improvements on correlation coefficients of hesitant fuzzy sets and their applications. Cognit Comput 11(4):529\u2013544","journal-title":"Cognit Comput"},{"issue":"6","key":"3383_CR29","first-page":"529","volume":"25","author":"V Torra","year":"2010","unstructured":"Torra V (2010) Hesitant fuzzy sets. Int J Intell Syst 25(6):529\u2013539","journal-title":"Int J Intell Syst"},{"issue":"22","key":"3383_CR30","doi-asserted-by":"publisher","first-page":"7082","DOI":"10.1016\/j.apm.2015.02.046","volume":"39","author":"SK Tyagi","year":"2015","unstructured":"Tyagi SK (2015) Correlation coefficient of dual hesitant fuzzy sets and its applications. Appl Math Model 39(22):7082\u20137092","journal-title":"Appl Math Model"},{"issue":"5","key":"3383_CR31","first-page":"388","volume":"25","author":"O Voltsit","year":"1991","unstructured":"Voltsit O (1991) The correlation variability of the morphometric traits in 4 species of ixodid tick species (ixodidae). Parazitologiia 25(5):388\u2013395","journal-title":"Parazitologiia"},{"issue":"1","key":"3383_CR32","doi-asserted-by":"publisher","first-page":"169","DOI":"10.1016\/S0165-0114(97)00303-5","volume":"103","author":"G Wang","year":"1999","unstructured":"Wang G, Li X (1999) Correlation and information energy of interval-valued fuzzy numbers. Fuzzy Sets Syst 103(1):169\u2013175","journal-title":"Fuzzy Sets Syst"},{"key":"3383_CR33","volume":"1","author":"L Wang","year":"2013","unstructured":"Wang L, Ni M, Zhu L (2013) Correlation measures of dual hesitant fuzzy sets. J Appl Math 1:593739","journal-title":"J Appl Math"},{"key":"3383_CR34","doi-asserted-by":"crossref","unstructured":"Wang Zx, Li J (2017) Correlation coefficients of probabilistic hesitant fuzzy elements and their applications to evaluation of the alternatives. Symmetry 9(11):259","DOI":"10.3390\/sym9110259"},{"issue":"8","key":"3383_CR35","doi-asserted-by":"publisher","first-page":"120","DOI":"10.3390\/a11080120","volume":"11","author":"W Wu","year":"2018","unstructured":"Wu W, Li Y, Ni Z et al (2018) Probabilistic interval-valued hesitant fuzzy information aggregation operators and their application to multi-attribute decision making. Algorithms 11(8):120","journal-title":"Algorithms"},{"key":"3383_CR36","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-319-04711-9","volume-title":"Hesitant fuzzy sets theory","author":"Z Xu","year":"2014","unstructured":"Xu Z (2014) Hesitant fuzzy sets theory, vol 314. Springer, Berlin"},{"issue":"5","key":"3383_CR37","doi-asserted-by":"publisher","first-page":"410","DOI":"10.1002\/int.20474","volume":"26","author":"Z Xu","year":"2011","unstructured":"Xu Z, Xia M (2011) On distance and correlation measures of hesitant fuzzy information. Int J Intell Syst 26(5):410\u2013425","journal-title":"Int J Intell Syst"},{"key":"3383_CR38","doi-asserted-by":"publisher","first-page":"481","DOI":"10.1007\/s10700-016-9257-5","volume":"16","author":"Z Xu","year":"2017","unstructured":"Xu Z, Zhou W (2017) Consensus building with a group of decision makers under the hesitant probabilistic fuzzy environment. Fuzzy Optim Decis Making 16:481\u2013503","journal-title":"Fuzzy Optim Decis Making"},{"issue":"6","key":"3383_CR39","first-page":"6427","volume":"35","author":"J Yang","year":"2018","unstructured":"Yang J, Tang X, Yang S (2018) Novel correlation coefficients for hesitant fuzzy sets and their applications to supplier selection and medical diagnosis. J Intell Fuzzy Syst 35(6):6427\u20136441","journal-title":"J Intell Fuzzy Syst"},{"key":"3383_CR40","doi-asserted-by":"publisher","first-page":"935","DOI":"10.1016\/j.asoc.2017.08.011","volume":"61","author":"X Yang","year":"2017","unstructured":"Yang X, Xu Z, Liao H (2017) Correlation coefficients of hesitant multiplicative sets and their applications in decision making and clustering analysis. Appl Soft Comput 61:935\u2013946","journal-title":"Appl Soft Comput"},{"issue":"2","key":"3383_CR41","doi-asserted-by":"publisher","first-page":"659","DOI":"10.1016\/j.apm.2013.07.010","volume":"38","author":"J Ye","year":"2014","unstructured":"Ye J (2014) Correlation coefficient of dual hesitant fuzzy sets and its application to multiple attribute decision making. Appl Math Model 38(2):659\u2013666","journal-title":"Appl Math Model"},{"issue":"1","key":"3383_CR42","doi-asserted-by":"publisher","first-page":"154","DOI":"10.1007\/s40815-023-01581-3","volume":"26","author":"L Ying","year":"2024","unstructured":"Ying L, Xin G (2024) Mixed correlation coefficient between probability hesitation fuzzy sets and applications. Int J Fuzzy Syst 26(1):154\u2013167","journal-title":"Int J Fuzzy Syst"},{"key":"3383_CR43","doi-asserted-by":"crossref","unstructured":"Zadeh LA (1965) Fuzzy sets. Inf Control 8(3):338\u2013353","DOI":"10.1016\/S0019-9958(65)90241-X"},{"key":"3383_CR44","volume-title":"Decision method for research and application based on preference relation","author":"B Zhu","year":"2014","unstructured":"Zhu B (2014) Decision method for research and application based on preference relation. Southeast University, Nanjing"}],"container-title":["Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-025-03383-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s40314-025-03383-9","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-025-03383-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,1,14]],"date-time":"2026-01-14T07:02:40Z","timestamp":1768374160000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s40314-025-03383-9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,9,8]]},"references-count":44,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2026,2]]}},"alternative-id":["3383"],"URL":"https:\/\/doi.org\/10.1007\/s40314-025-03383-9","relation":{},"ISSN":["2238-3603","1807-0302"],"issn-type":[{"value":"2238-3603","type":"print"},{"value":"1807-0302","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,9,8]]},"assertion":[{"value":"25 March 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"1 July 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"7 August 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 September 2025","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"6"}}