{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,4]],"date-time":"2026-06-04T10:40:54Z","timestamp":1780569654717,"version":"3.54.1"},"publisher-location":"Cham","reference-count":19,"publisher":"Springer International Publishing","isbn-type":[{"value":"9783030815608","type":"print"},{"value":"9783030815615","type":"electronic"}],"license":[{"start":{"date-parts":[[2021,12,9]],"date-time":"2021-12-09T00:00:00Z","timestamp":1639008000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2021,12,9]],"date-time":"2021-12-09T00:00:00Z","timestamp":1639008000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2022]]},"DOI":"10.1007\/978-3-030-81561-5_12","type":"book-chapter","created":{"date-parts":[[2021,12,8]],"date-time":"2021-12-08T10:06:34Z","timestamp":1638957994000},"page":"129-141","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Optimal Number of Classes in Fuzzy Partitions"],"prefix":"10.1007","author":[{"given":"Fabian","family":"Castiblanco","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Camilo","family":"Franco","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"J. Tinguaro","family":"Rodriguez","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Javier","family":"Montero","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2021,12,9]]},"reference":[{"key":"12_CR1","doi-asserted-by":"publisher","first-page":"58","DOI":"10.1080\/01969727308546047","volume":"3","author":"JC Bezdek","year":"1973","unstructured":"J.C. Bezdek, Cluster validity with fuzzy sets. J. Cybern. 3, 58\u201373 (1973)","journal-title":"J. Cybern."},{"key":"12_CR2","doi-asserted-by":"crossref","unstructured":"J.C. Bezdek, Pattern Recognition with Fuzzy Objective Function Algorithms (Springer, New York, 1981)","DOI":"10.1007\/978-1-4757-0450-1"},{"key":"12_CR3","doi-asserted-by":"publisher","first-page":"613","DOI":"10.1016\/0167-8655(96)00026-8","volume":"17","author":"RN Dave","year":"1996","unstructured":"R.N. Dave, Validating fuzzy partitions obtained through c-shells clustering. Pattern Recogn. Lett. 17, 613\u2013623 (1996)","journal-title":"Pattern Recogn. Lett."},{"key":"12_CR4","doi-asserted-by":"publisher","first-page":"53","DOI":"10.1016\/0377-0427(87)90125-7","volume":"20","author":"PJ Rousseeuw","year":"1987","unstructured":"P.J. Rousseeuw, Silhouettes: a graphical aid to the interpretation and validation of cluster analysis. J. Comput. Appl. Math. 20, 53\u201365 (1987)","journal-title":"J. Comput. Appl. Math."},{"key":"12_CR5","doi-asserted-by":"publisher","first-page":"2858","DOI":"10.1016\/j.fss.2006.07.006","volume":"157","author":"RJGB Campello","year":"2006","unstructured":"R.J.G.B. Campello, E.R. Hruschka, A fuzzy extension of the silhouette width criterion for cluster analysis. Fuzzy Sets Syst. 157, 2858\u20132875 (2006)","journal-title":"Fuzzy Sets Syst."},{"key":"12_CR6","doi-asserted-by":"publisher","first-page":"1285","DOI":"10.1016\/j.asoc.2016.06.037","volume":"52","author":"A Suleman","year":"2017","unstructured":"A. Suleman, Measuring the congruence of fuzzy partitions in fuzzy c-means clustering. Appl. Soft Comput. 52, 1285\u20131295 (2017)","journal-title":"Appl. Soft Comput."},{"key":"12_CR7","doi-asserted-by":"publisher","first-page":"546","DOI":"10.1109\/TFUZZ.2011.2179303","volume":"20","author":"E Hullermeier","year":"2012","unstructured":"E. Hullermeier, M. Rifqi, S. Henzgen, R. Senge, Comparing fuzzy partitions: a generalization of the rand index and related measures. IEEE Trans. Fuzzy Syst. 20, 546\u2013556 (2012)","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"12_CR8","doi-asserted-by":"publisher","first-page":"1275","DOI":"10.1016\/j.patrec.2004.11.022","volume":"26","author":"KL Wu","year":"2005","unstructured":"K.L. Wu, M.S. Yang, A cluster validity index for fuzzy clustering. Pattern Recogn. Lett. 26, 1275\u20131291 (2005)","journal-title":"Pattern Recogn. Lett."},{"key":"12_CR9","doi-asserted-by":"publisher","first-page":"107","DOI":"10.1023\/A:1012801612483","volume":"17","author":"M Halkidi","year":"2001","unstructured":"M. Halkidi, Y. Batistakis, M. Vazirgiannis, On clustering validation techniques. J. Intell. Inf. Syst. 17, 107\u2013145 (2001)","journal-title":"J. Intell. Inf. Syst."},{"key":"12_CR10","doi-asserted-by":"publisher","first-page":"2095","DOI":"10.1016\/j.fss.2007.03.004","volume":"158","author":"W Wang","year":"2007","unstructured":"W. Wang, Y. Zhang, On fuzzy cluster validity indices. Fuzzy Sets Syst. 158, 2095\u20132117 (2007)","journal-title":"Fuzzy Sets Syst."},{"key":"12_CR11","doi-asserted-by":"publisher","first-page":"906","DOI":"10.1109\/TFUZZ.2010.2052258","volume":"18","author":"DT Anderson","year":"2010","unstructured":"D.T. Anderson, J.C. Bezdek, M. Popescu, J.M. Keller, Comparing fuzzy, probabilistic, and possibilistic partitions. IEEE Trans. Fuzzy Syst. 18, 906\u2013918 (2010)","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"12_CR12","doi-asserted-by":"publisher","first-page":"495","DOI":"10.1016\/S0377-2217(03)00002-X","volume":"156","author":"A Amo","year":"2004","unstructured":"A. Amo, J. Montero, G. Biging, V. Cutello, Fuzzy classification systems. Eur. J. Oper. Res. 156, 495\u2013507 (2004)","journal-title":"Eur. J. Oper. Res."},{"key":"12_CR13","first-page":"203","volume":"8","author":"A Amo","year":"2001","unstructured":"A. Amo, D. Gomez, J. Montero, G. Biging, Relevance and Redundancy in fuzzy classification systems. Mathware Soft Comput. 8, 203\u2013216 (2001)","journal-title":"Mathware Soft Comput."},{"key":"12_CR14","doi-asserted-by":"crossref","unstructured":"F. Castiblanco, C. Franco, J. Montero, J.T. Rodr\u00edguez, Relevance of classes in a fuzzy partition. A study from a group of aggregation operators, in Fuzzy Information Processing, ed. by G.A. Barreto, R. Coelho (Springer International Publishing, Cham, 2018), pp. 96\u2013107","DOI":"10.1007\/978-3-319-95312-0_9"},{"key":"12_CR15","doi-asserted-by":"crossref","unstructured":"F. Castiblanco, C. Franco, J. Montero, J. Tinguaro Rodr\u00edguez, Aggregation operators to evaluate the relevance of classes in a fuzzy partition, in Fuzzy Techniques: Theory and Applications. ed. by R.B. Kearfott, I. Batyrshin, M. Reformat, M. Ceberio, V. Kreinovich (Springer International Publishing, Cham, 2019), pp. 13\u201321","DOI":"10.1007\/978-3-030-21920-8_2"},{"key":"12_CR16","doi-asserted-by":"crossref","unstructured":"F. Castiblanco, D. G\u00f3mez, J. Montero, J.T. Rodr\u00edguez, Aggregation tools for the evaluation of classifications, in IFSA-SCIS 2017 - Joint 17th World Congress of International Fuzzy Systems Association and 9th International Conference on Soft Computing and Intelligent Systems (IEEE, Otsu, Japan, 2017), pp. 1\u20135","DOI":"10.1109\/IFSA-SCIS.2017.8023242"},{"key":"12_CR17","doi-asserted-by":"crossref","unstructured":"J. Fodor, M. Roubens, Preference Modelling and Multicriteria Decision Support. Theory and Decision Library. SeriesD: System Theory, Knowledge Engineering and Problem Solving (1994)","DOI":"10.1007\/978-94-017-1648-2"},{"key":"12_CR18","doi-asserted-by":"publisher","first-page":"525","DOI":"10.1016\/j.patrec.2007.11.007","volume":"29","author":"H Bustince","year":"2008","unstructured":"H. Bustince, E. Barrenechea, M. Pagola, Relationship between restricted dissimilarity functions, restricted equivalence functions and normal EN-functions: image thresholding invariant. Pattern Recogn. Lett. 29, 525\u2013536 (2008)","journal-title":"Pattern Recogn. Lett."},{"key":"12_CR19","unstructured":"D. Martin, C. Fowlkes, D. Tal, J. Malik, A database of human segmented natural images and its application to evaluating segmentation algorithms and measuring ecological statistics. Proc. IEEE Int. Conf. Comput. Vis. 2, 416\u2013423 (2001)"}],"container-title":["Advances in Intelligent Systems and Computing","Fuzzy Information Processing 2020"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/978-3-030-81561-5_12","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,10,13]],"date-time":"2022-10-13T19:05:22Z","timestamp":1665687922000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/978-3-030-81561-5_12"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,12,9]]},"ISBN":["9783030815608","9783030815615"],"references-count":19,"URL":"https:\/\/doi.org\/10.1007\/978-3-030-81561-5_12","relation":{},"ISSN":["2194-5357","2194-5365"],"issn-type":[{"value":"2194-5357","type":"print"},{"value":"2194-5365","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,12,9]]},"assertion":[{"value":"9 December 2021","order":1,"name":"first_online","label":"First Online","group":{"name":"ChapterHistory","label":"Chapter History"}},{"value":"NAFIPS","order":1,"name":"conference_acronym","label":"Conference Acronym","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"North American Fuzzy Information Processing Society Annual Conference","order":2,"name":"conference_name","label":"Conference Name","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"2020","order":5,"name":"conference_year","label":"Conference Year","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"20 August 2020","order":7,"name":"conference_start_date","label":"Conference Start Date","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"22 August 2020","order":8,"name":"conference_end_date","label":"Conference End Date","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"nafips2020","order":10,"name":"conference_id","label":"Conference ID","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"https:\/\/polytechnic.purdue.edu\/nafips2021","order":11,"name":"conference_url","label":"Conference URL","group":{"name":"ConferenceInfo","label":"Conference Information"}}]}}