{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,26]],"date-time":"2026-03-26T08:57:32Z","timestamp":1774515452289,"version":"3.50.1"},"reference-count":35,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2022,9,26]],"date-time":"2022-09-26T00:00:00Z","timestamp":1664150400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2022,9,26]],"date-time":"2022-09-26T00:00:00Z","timestamp":1664150400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"name":"Basic Scientific Research Project of Institution of Higher Learning of Liaoning Province","award":["LJKZ0293"],"award-info":[{"award-number":["LJKZ0293"]}]},{"name":"Project by Liaoning Provincial Natural Science Foundation of China","award":["20180550700"],"award-info":[{"award-number":["20180550700"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Int. J. Fuzzy Syst."],"published-print":{"date-parts":[[2023,3]]},"DOI":"10.1007\/s40815-022-01394-w","type":"journal-article","created":{"date-parts":[[2022,9,26]],"date-time":"2022-09-26T06:02:32Z","timestamp":1664172152000},"page":"647-669","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Exponent and Logarithm Component-Wise Construction Method of FCM Clustering Validity Function Based on Subjective and Objective Weighting"],"prefix":"10.1007","volume":"25","author":[{"given":"Jia-Xu","family":"Liu","sequence":"first","affiliation":[]},{"given":"Jie-Sheng","family":"Wang","sequence":"additional","affiliation":[]},{"given":"Guan","family":"Wang","sequence":"additional","affiliation":[]},{"given":"Xiao-Rui","family":"Zhao","sequence":"additional","affiliation":[]},{"given":"Hong-Yu","family":"Wang","sequence":"additional","affiliation":[]},{"given":"Di","family":"Jin","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,9,26]]},"reference":[{"issue":"1","key":"1394_CR1","doi-asserted-by":"publisher","first-page":"51","DOI":"10.1109\/TIP.2003.818116","volume":"13","author":"SH Leung","year":"2004","unstructured":"Leung, S.H., Wang, S.L., Lau, W.H.: Lip image segmentation using fuzzy clustering incorporating an elliptic shape function. IEEE Trans. Image Process. 13(1), 51\u201362 (2004)","journal-title":"IEEE Trans. Image Process."},{"issue":"1","key":"1394_CR2","doi-asserted-by":"publisher","first-page":"150","DOI":"10.1109\/TNN.2006.884678","volume":"18","author":"J Lu","year":"2007","unstructured":"Lu, J., Yuan, X., Yahagi, T.: A method of face recognition based on fuzzy c-means clustering and associated sub-NNs. IEEE Trans. Neural Netw. 18(1), 150\u2013160 (2007)","journal-title":"IEEE Trans. Neural Netw."},{"issue":"6","key":"1394_CR3","first-page":"778","volume":"36","author":"YH Wang","year":"2010","unstructured":"Wang, Y.H., Zhao, H.C.: PolSAR image segmentation by mean shift clustering in the tensor space. Acta Automatica Sinica 36(6), 778\u2013806 (2010)","journal-title":"Acta Automatica Sinica"},{"issue":"9","key":"1394_CR4","doi-asserted-by":"publisher","first-page":"1317","DOI":"10.1016\/j.patrec.2008.02.003","volume":"29","author":"WL Hung","year":"2008","unstructured":"Hung, W.L., Yang, M.S., Chen, D.H.: Bootstrapping approach to feature-weight selection in fuzzy c-means algorithms with an application in color image segmentation. Pattern Recogn. Lett. 29(9), 1317\u20131325 (2008)","journal-title":"Pattern Recogn. Lett."},{"issue":"9","key":"1394_CR5","doi-asserted-by":"publisher","first-page":"3503","DOI":"10.1109\/TGRS.2010.2047020","volume":"48","author":"U Maulik","year":"2010","unstructured":"Maulik, U., Saha, I.: Automatic fuzzy clustering using modified differential evolution for image classification. IEEE Trans. Geosci. Remote Sens. 48(9), 3503\u20133510 (2010)","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"issue":"4","key":"1394_CR6","doi-asserted-by":"publisher","first-page":"2661","DOI":"10.1109\/TCE.2010.5681154","volume":"56","author":"SN Sulaiman","year":"2010","unstructured":"Sulaiman, S.N., Isa, N.A.M.: Adaptive fuzzy-K-means clustering algorithm for image segmentation. IEEE Trans. Consum. Electron. 56(4), 2661\u20132668 (2010)","journal-title":"IEEE Trans. Consum. Electron."},{"issue":"12","key":"1394_CR7","doi-asserted-by":"publisher","first-page":"4069","DOI":"10.1016\/j.patcog.2010.06.015","volume":"43","author":"F Tung","year":"2010","unstructured":"Tung, F., Wong, A., Clausi, D.A.: Enabling scalable spectral clustering for image segmentation. Pattern Recogn. 43(12), 4069\u20134076 (2010)","journal-title":"Pattern Recogn."},{"issue":"13","key":"1394_CR8","doi-asserted-by":"publisher","first-page":"1773","DOI":"10.1016\/j.patrec.2010.04.006","volume":"31","author":"SY Yang","year":"2010","unstructured":"Yang, S.Y., Wu, R.X., Wang, M., Jiao, L.C.: Evolutionary clustering based vector quantization and SPIHT coding for image compression. Pattern Recogn. Lett. 31(13), 1773\u20131780 (2010)","journal-title":"Pattern Recogn. Lett."},{"issue":"12","key":"1394_CR9","doi-asserted-by":"publisher","first-page":"1976","DOI":"10.1109\/TNN.2010.2079946","volume":"21","author":"ZM Wang","year":"2010","unstructured":"Wang, Z.M., Song, Q., Sohf, Y.C., Sim, K.: Robust curve clustering based on a multivariate t-distribution model. IEEE Trans. Neural Netw. 21(12), 1976\u20131984 (2010)","journal-title":"IEEE Trans. Neural Netw."},{"issue":"4","key":"1394_CR10","doi-asserted-by":"publisher","first-page":"516","DOI":"10.1109\/TFUZZ.2004.832521","volume":"12","author":"PX Liu","year":"2004","unstructured":"Liu, P.X., Meng, M.Q.H.: Online data-driven fuzzy clustering with applications to real-time robotic tracking. IEEE Trans. Fuzzy Syst. 12(4), 516\u2013523 (2004)","journal-title":"IEEE Trans. Fuzzy Syst."},{"issue":"3","key":"1394_CR11","doi-asserted-by":"publisher","first-page":"779","DOI":"10.1109\/TFUZZ.2007.905919","volume":"16","author":"A Celikyilmaz","year":"2008","unstructured":"Celikyilmaz, A., Turksen, I.B.: Enhanced fuzzy system models with improved fuzzy clustering algorithm. IEEE Trans. Fuzzy Syst. 16(3), 779\u2013794 (2008)","journal-title":"IEEE Trans. Fuzzy Syst."},{"issue":"1","key":"1394_CR12","doi-asserted-by":"publisher","first-page":"263","DOI":"10.1109\/TFUZZ.2007.903314","volume":"16","author":"L Hyong-Euk","year":"2008","unstructured":"Hyong-Euk, L., Kwang-Hyun, P., Bien, Z.Z.: Iterative fuzzy clustering algorithm with supervision to construct probabilistic fuzzy rule base from numerical data. IEEE Trans. Fuzzy Syst. 16(1), 263\u2013277 (2008)","journal-title":"IEEE Trans. Fuzzy Syst."},{"issue":"11","key":"1394_CR13","doi-asserted-by":"publisher","first-page":"1632","DOI":"10.1016\/j.patrec.2008.04.003","volume":"29","author":"J Goldberger","year":"2008","unstructured":"Goldberger, J., Tamir, T.: A hierarchical clustering algorithm based on the Hungrian method. Pattern Recogn. Lett. 29(11), 1632\u20131638 (2008)","journal-title":"Pattern Recogn. Lett."},{"key":"1394_CR14","doi-asserted-by":"publisher","first-page":"54","DOI":"10.1007\/978-1-4757-0450-1","volume-title":"Pattern Recognition with Fuzzy Object Algorithms","author":"JC Bezdek","year":"1981","unstructured":"Bezdek, J.C.: Pattern Recognition with Fuzzy Object Algorithms, pp. 54\u201357. Plenum Press, New York (1981)"},{"key":"1394_CR15","unstructured":"MacQueen J.: Some Methods of Classification and Analysis of MultiVariate Observations. Proc of Berkeley Symposium on Mathematical Statistics and Probability, 281\u2013297 (1967)."},{"issue":"13","key":"1394_CR16","doi-asserted-by":"publisher","first-page":"2597","DOI":"10.1080\/00949655.2017.1327588","volume":"87","author":"J Yodern","year":"2017","unstructured":"Yodern, J., Priebe, C.E.: Semi-supervised K-means++. J. Stat. Comput. Simul. 87(13), 2597\u20132608 (2017)","journal-title":"J. Stat. Comput. Simul."},{"key":"1394_CR17","doi-asserted-by":"publisher","first-page":"289","DOI":"10.1016\/j.knosys.2018.01.031","volume":"145","author":"Z Geng","year":"2018","unstructured":"Geng, Z., Chengchang, Z., Huayu, Z.: Improved K-means algorithm based on density Canopy. Knowl.-Based Syst. 145, 289\u2013297 (2018)","journal-title":"Knowl.-Based Syst."},{"key":"1394_CR18","doi-asserted-by":"publisher","first-page":"60","DOI":"10.1016\/j.cose.2018.06.003","volume":"78","author":"NH Hiep","year":"2018","unstructured":"Hiep, N.H.: Privacy-preserving mechanisms for k-modes clustering. Comput. Secur. 78, 60\u201375 (2018)","journal-title":"Comput. Secur."},{"key":"1394_CR19","doi-asserted-by":"crossref","unstructured":"J. C. Bezdek, N. R. Pal.: Some new indexes of cluster validity. IEEE Transactions on Systems, Man and Cybernetics, Part B(Cybernetics), 28(3), 301\u2013315 (1998).","DOI":"10.1109\/3477.678624"},{"key":"1394_CR20","doi-asserted-by":"crossref","unstructured":"D. A. Simovici, S. Jaroszewicz.: An axiomatization of partition entropy. IEEE Transactions on Information Theory, 48(7), 2138\u20132142 (2002).","DOI":"10.1109\/TIT.2002.1013159"},{"issue":"6","key":"1394_CR21","doi-asserted-by":"publisher","first-page":"2174","DOI":"10.1109\/TFUZZ.2015.2407901","volume":"23","author":"L Silva","year":"2015","unstructured":"Silva, L., Moura, R., Canuto, A.M.P., Santiago, R.H.N., Bedregal, B.: An Interval-Based Framework for Fuzzy Clustering Applications. IEEE Trans. Fuzzy Syst. 23(6), 2174\u20132187 (2015)","journal-title":"IEEE Trans. Fuzzy Syst."},{"issue":"2","key":"1394_CR22","doi-asserted-by":"publisher","first-page":"243","DOI":"10.1016\/S0165-0114(03)00160-X","volume":"142","author":"MY Chen","year":"2004","unstructured":"Chen, M.Y., Linkens, D.A.: Rule-base self-generation and simplification for data-driven fuzzy models. Fuzzy Sets Syst. 142(2), 243\u2013265 (2004)","journal-title":"Fuzzy Sets Syst."},{"key":"1394_CR23","doi-asserted-by":"crossref","unstructured":"Chen J.S., Pi D. C.: A cluster validity index for fuzzy clustering based on non-distance. In 2013 International Conference on Computational and Information Science. IEEE, 880\u2013883 (2013).","DOI":"10.1109\/ICCIS.2013.236"},{"issue":"32","key":"1394_CR24","doi-asserted-by":"publisher","first-page":"325","DOI":"10.3182\/20131218-3-IN-2045.00117","volume":"46","author":"S Joopudi","year":"2013","unstructured":"Joopudi, S., Rathi, S.S., Narasimhan, S., Rengaswamy, R.: A new cluster validity index for fuzzy clustering. IFAC Proc. Vol. 46(32), 325\u2013330 (2013)","journal-title":"IFAC Proc. Vol."},{"issue":"6","key":"1394_CR25","doi-asserted-by":"publisher","first-page":"841","DOI":"10.1109\/34.85677","volume":"17","author":"XL Xie","year":"1991","unstructured":"Xie, X.L., Beni, G.: A validity measure for fuzzy clustering. IEEE Trans. Pattern Anal. Mach. Intell. 17(6), 841\u2013847 (1991)","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"issue":"9","key":"1394_CR26","doi-asserted-by":"publisher","first-page":"1275","DOI":"10.1016\/j.patrec.2004.11.022","volume":"26","author":"K-L Wu","year":"2005","unstructured":"Wu, K.-L., Yang, M.-S.: A cluster validity index for fuzzy clustering. Pattern Recogn. Lett. 26(9), 1275\u20131291 (2005)","journal-title":"Pattern Recogn. Lett."},{"key":"1394_CR27","doi-asserted-by":"crossref","unstructured":"C. Wu, C. Ouyang, L. Chen and L. Lu.: A New Fuzzy Clustering Validity Index With a Median Factor for Centroid-Based Clustering. IEEE Transactions on Fuzzy Systems, 23(3): 701\u2013718 (2004).","DOI":"10.1109\/TFUZZ.2014.2322495"},{"issue":"11","key":"1394_CR28","first-page":"15","volume":"33","author":"L Meng","year":"2007","unstructured":"Meng, L., Hu, C.: Cluster Validity Index Based on Measure of Fuzzy Partition. Comput. Eng. 33(11), 15\u201317 (2007)","journal-title":"Comput. Eng."},{"key":"1394_CR29","doi-asserted-by":"publisher","first-page":"152289","DOI":"10.1109\/ACCESS.2019.2946599","volume":"7","author":"LF Zhu","year":"2019","unstructured":"Zhu, L.F., Wang, J.S., Wang, H.Y.: A novel clustering validity function of FCM clustering algorithm. IEEE Access 7, 152289\u2013152315 (2019)","journal-title":"IEEE Access"},{"issue":"6","key":"1394_CR30","doi-asserted-by":"publisher","first-page":"12411","DOI":"10.3233\/JIFS-210555","volume":"40","author":"HY Wang","year":"2021","unstructured":"Wang, H.Y., Wang, J.S., Zhu, L.F.: A new validity function of FCM clustering algorithm based on the intra-class compactness and inter-class separation. J. Intell. Fuzzy Syst. 40(6), 12411\u201312432 (2021)","journal-title":"J. Intell. Fuzzy Syst."},{"key":"1394_CR31","first-page":"1","volume":"1","author":"G Wang","year":"2022","unstructured":"Wang, G., Wang, J.S., Wang, H.Y.: Fuzzy C-means clustering validity function based on multiple clustering performance evaluation components. Int. J. Fuzzy Syst. 1, 1\u201329 (2022)","journal-title":"Int. J. Fuzzy Syst."},{"issue":"6","key":"1394_CR32","doi-asserted-by":"publisher","first-page":"1156","DOI":"10.1109\/TSMCB.2005.850173","volume":"35","author":"WG Sheng","year":"2005","unstructured":"Sheng, W.G., Swift, S., Zhang, L.: A weighted sum validity function for clustering with a hybrid niching genetic algorithm. IEEE Trans. Syst. Man Cybern. B 35(6), 1156\u20131167 (2005)","journal-title":"IEEE Trans. Syst. Man Cybern. B"},{"key":"1394_CR33","doi-asserted-by":"crossref","unstructured":"Dong H.B, Hou W, Ying G.S.: An evolutionary clustering algorithm based on adaptive fuzzy weighted sum validity function. In: 2010 Third International Joint Conference on Computational Science and Optimization, IEEE, 357\u2013361 (2010).","DOI":"10.1109\/CSO.2010.204"},{"key":"1394_CR34","doi-asserted-by":"crossref","unstructured":"Wu Z. F, Huang H. K.: A dynamic weighted sum validity function for fuzzy clustering with an adaptive differential evolution algorithm. In: 2010 Third International Joint Conference on Computational Science and Optimization, IEEE, 362366 (2010).","DOI":"10.1109\/CSO.2010.149"},{"key":"1394_CR35","doi-asserted-by":"publisher","first-page":"27239","DOI":"10.1109\/ACCESS.2021.3058264","volume":"9","author":"HY Wang","year":"2021","unstructured":"Wang, H.Y., Wang, J.S.: Combination evaluation method of fuzzy C-means clustering validity based on hybrid weighted strategy. IEEE Access 9, 27239\u201327261 (2021)","journal-title":"IEEE Access"}],"container-title":["International Journal of Fuzzy Systems"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40815-022-01394-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s40815-022-01394-w\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40815-022-01394-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,14]],"date-time":"2023-03-14T20:17:35Z","timestamp":1678825055000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s40815-022-01394-w"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,9,26]]},"references-count":35,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2023,3]]}},"alternative-id":["1394"],"URL":"https:\/\/doi.org\/10.1007\/s40815-022-01394-w","relation":{},"ISSN":["1562-2479","2199-3211"],"issn-type":[{"value":"1562-2479","type":"print"},{"value":"2199-3211","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,9,26]]},"assertion":[{"value":"31 May 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"29 July 2022","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 August 2022","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"26 September 2022","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 there is no conflict of interests regarding the publication of this article.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}