{"id":"https://openalex.org/W2149958903","doi":"https://doi.org/10.1109/tfuzz.2010.2048114","title":"Relational Generalizations of Cluster Validity Indices","display_name":"Relational Generalizations of Cluster Validity Indices","publication_year":2010,"publication_date":"2010-04-14","ids":{"openalex":"https://openalex.org/W2149958903","doi":"https://doi.org/10.1109/tfuzz.2010.2048114","mag":"2149958903"},"language":"en","primary_location":{"id":"doi:10.1109/tfuzz.2010.2048114","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tfuzz.2010.2048114","pdf_url":null,"source":{"id":"https://openalex.org/S134177497","display_name":"IEEE Transactions on Fuzzy Systems","issn_l":"1063-6706","issn":["1063-6706","1941-0034"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Fuzzy Systems","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5049310005","display_name":"Isaac J. Sledge","orcid":"https://orcid.org/0000-0002-7755-5886"},"institutions":[{"id":"https://openalex.org/I76835614","display_name":"University of Missouri","ror":"https://ror.org/02ymw8z06","country_code":"US","type":"education","lineage":["https://openalex.org/I76835614"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"I J Sledge","raw_affiliation_strings":["Electrical and Computer Engineering Department, University of Missouri, Columbia, MO, USA","[Electr. & Comput. Eng. Dept., Univ. of Missouri, Columbia, MO, USA]"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Electrical and Computer Engineering Department, University of Missouri, Columbia, MO, USA","institution_ids":["https://openalex.org/I76835614"]},{"raw_affiliation_string":"[Electr. & Comput. Eng. Dept., Univ. of Missouri, Columbia, MO, USA]","institution_ids":["https://openalex.org/I76835614"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5043324448","display_name":"James C. Bezdek","orcid":"https://orcid.org/0000-0003-3901-7021"},"institutions":[{"id":"https://openalex.org/I76835614","display_name":"University of Missouri","ror":"https://ror.org/02ymw8z06","country_code":"US","type":"education","lineage":["https://openalex.org/I76835614"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"J C Bezdek","raw_affiliation_strings":["Electrical and Computer Engineering Department, University of Missouri, Columbia, MO, USA","[Electr. & Comput. Eng. Dept., Univ. of Missouri, Columbia, MO, USA]"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Electrical and Computer Engineering Department, University of Missouri, Columbia, MO, USA","institution_ids":["https://openalex.org/I76835614"]},{"raw_affiliation_string":"[Electr. & Comput. Eng. Dept., Univ. of Missouri, Columbia, MO, USA]","institution_ids":["https://openalex.org/I76835614"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5060923133","display_name":"Timothy C. Havens","orcid":"https://orcid.org/0000-0002-5746-3749"},"institutions":[{"id":"https://openalex.org/I76835614","display_name":"University of Missouri","ror":"https://ror.org/02ymw8z06","country_code":"US","type":"education","lineage":["https://openalex.org/I76835614"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"T C Havens","raw_affiliation_strings":["Electrical and Computer Engineering Department, University of Missouri, Columbia, MO, USA","[Electr. & Comput. Eng. Dept., Univ. of Missouri, Columbia, MO, USA]"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Electrical and Computer Engineering Department, University of Missouri, Columbia, MO, USA","institution_ids":["https://openalex.org/I76835614"]},{"raw_affiliation_string":"[Electr. & Comput. Eng. Dept., Univ. of Missouri, Columbia, MO, USA]","institution_ids":["https://openalex.org/I76835614"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5018303984","display_name":"James M. Keller","orcid":"https://orcid.org/0000-0002-0306-7142"},"institutions":[{"id":"https://openalex.org/I76835614","display_name":"University of Missouri","ror":"https://ror.org/02ymw8z06","country_code":"US","type":"education","lineage":["https://openalex.org/I76835614"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"J M Keller","raw_affiliation_strings":["Electrical and Computer Engineering Department, University of Missouri, Columbia, MO, USA","[Electr. & Comput. Eng. Dept., Univ. of Missouri, Columbia, MO, USA]"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Electrical and Computer Engineering Department, University of Missouri, Columbia, MO, USA","institution_ids":["https://openalex.org/I76835614"]},{"raw_affiliation_string":"[Electr. & Comput. Eng. Dept., Univ. of Missouri, Columbia, MO, USA]","institution_ids":["https://openalex.org/I76835614"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I76835614"],"apc_list":null,"apc_paid":null,"fwci":5.3125,"has_fulltext":false,"cited_by_count":52,"citation_normalized_percentile":{"value":0.96522204,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"18","issue":"4","first_page":"771","last_page":"786"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10064","display_name":"Complex Network Analysis Techniques","score":0.9991000294685364,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10064","display_name":"Complex Network Analysis Techniques","score":0.9991000294685364,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10637","display_name":"Advanced Clustering Algorithms Research","score":0.9987000226974487,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10887","display_name":"Bioinformatics and Genomic Networks","score":0.9860000014305115,"subfield":{"id":"https://openalex.org/subfields/1312","display_name":"Molecular Biology"},"field":{"id":"https://openalex.org/fields/13","display_name":"Biochemistry, Genetics and Molecular Biology"},"domain":{"id":"https://openalex.org/domains/1","display_name":"Life Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/cluster-analysis","display_name":"Cluster analysis","score":0.758806049823761},{"id":"https://openalex.org/keywords/notation","display_name":"Notation","score":0.723997175693512},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.5333929657936096},{"id":"https://openalex.org/keywords/object","display_name":"Object (grammar)","score":0.4477541148662567},{"id":"https://openalex.org/keywords/a-priori-and-a-posteriori","display_name":"A priori and a posteriori","score":0.41728076338768005},{"id":"https://openalex.org/keywords/information-retrieval","display_name":"Information retrieval","score":0.41454723477363586},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.3469787836074829},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.33017563819885254},{"id":"https://openalex.org/keywords/arithmetic","display_name":"Arithmetic","score":0.07959464192390442}],"concepts":[{"id":"https://openalex.org/C73555534","wikidata":"https://www.wikidata.org/wiki/Q622825","display_name":"Cluster analysis","level":2,"score":0.758806049823761},{"id":"https://openalex.org/C45357846","wikidata":"https://www.wikidata.org/wiki/Q2001982","display_name":"Notation","level":2,"score":0.723997175693512},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.5333929657936096},{"id":"https://openalex.org/C2781238097","wikidata":"https://www.wikidata.org/wiki/Q175026","display_name":"Object (grammar)","level":2,"score":0.4477541148662567},{"id":"https://openalex.org/C75553542","wikidata":"https://www.wikidata.org/wiki/Q178161","display_name":"A priori and a posteriori","level":2,"score":0.41728076338768005},{"id":"https://openalex.org/C23123220","wikidata":"https://www.wikidata.org/wiki/Q816826","display_name":"Information retrieval","level":1,"score":0.41454723477363586},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.3469787836074829},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.33017563819885254},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.07959464192390442},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0},{"id":"https://openalex.org/C111472728","wikidata":"https://www.wikidata.org/wiki/Q9471","display_name":"Epistemology","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1109/tfuzz.2010.2048114","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tfuzz.2010.2048114","pdf_url":null,"source":{"id":"https://openalex.org/S134177497","display_name":"IEEE Transactions on Fuzzy Systems","issn_l":"1063-6706","issn":["1063-6706","1941-0034"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Fuzzy Systems","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[{"score":0.5699999928474426,"display_name":"Quality Education","id":"https://metadata.un.org/sdg/4"}],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":73,"referenced_works":["https://openalex.org/W597424329","https://openalex.org/W1504154695","https://openalex.org/W1508853606","https://openalex.org/W1520723760","https://openalex.org/W1527927437","https://openalex.org/W1578407488","https://openalex.org/W1793466594","https://openalex.org/W1805642456","https://openalex.org/W1898259180","https://openalex.org/W1963678392","https://openalex.org/W1971888031","https://openalex.org/W1978527081","https://openalex.org/W1989249416","https://openalex.org/W1989634533","https://openalex.org/W1990368529","https://openalex.org/W1992625510","https://openalex.org/W1994615541","https://openalex.org/W1996747841","https://openalex.org/W2000889749","https://openalex.org/W2003305541","https://openalex.org/W2004668252","https://openalex.org/W2015724019","https://openalex.org/W2025767687","https://openalex.org/W2027006788","https://openalex.org/W2028191869","https://openalex.org/W2034986427","https://openalex.org/W2040810017","https://openalex.org/W2048663205","https://openalex.org/W2061708588","https://openalex.org/W2066182660","https://openalex.org/W2070081801","https://openalex.org/W2074334456","https://openalex.org/W2084134149","https://openalex.org/W2090582154","https://openalex.org/W2090634555","https://openalex.org/W2099495349","https://openalex.org/W2100546446","https://openalex.org/W2109970232","https://openalex.org/W2113076747","https://openalex.org/W2115364272","https://openalex.org/W2120688485","https://openalex.org/W2125836384","https://openalex.org/W2127385182","https://openalex.org/W2136753665","https://openalex.org/W2137686028","https://openalex.org/W2140798815","https://openalex.org/W2149218391","https://openalex.org/W2150593711","https://openalex.org/W2150926065","https://openalex.org/W2151297804","https://openalex.org/W2153649609","https://openalex.org/W2156718197","https://openalex.org/W2158358659","https://openalex.org/W2166698530","https://openalex.org/W2167428023","https://openalex.org/W2167865654","https://openalex.org/W2170590171","https://openalex.org/W2171975443","https://openalex.org/W2319660501","https://openalex.org/W2319794630","https://openalex.org/W2539182710","https://openalex.org/W2611736676","https://openalex.org/W2797746357","https://openalex.org/W2916147039","https://openalex.org/W3015962962","https://openalex.org/W3017143921","https://openalex.org/W3120740533","https://openalex.org/W4235169531","https://openalex.org/W4244000367","https://openalex.org/W4256358309","https://openalex.org/W4285719527","https://openalex.org/W4301891957","https://openalex.org/W6682755970"],"related_works":["https://openalex.org/W2504004674","https://openalex.org/W2498744856","https://openalex.org/W763418848","https://openalex.org/W1595229445","https://openalex.org/W322408318","https://openalex.org/W149041114","https://openalex.org/W4388311650","https://openalex.org/W2963177394","https://openalex.org/W1965815883","https://openalex.org/W2055474767"],"abstract_inverted_index":{"Numerous":[0],"computational":[1],"schemes":[2],"have":[3,44,75],"arisen":[4],"over":[5],"the":[6,17,52,67,79,112,116,119,148,154,199,203,223],"years":[7],"that":[8,122],"attempt":[9],"to":[10,24,63,74,77,158,168,174,219],"learn":[11],"information":[12],"about":[13],"objects":[14],"based":[15],"upon":[16],"similarity":[18],"or":[19,107],"dissimilarity":[20],"of":[21,34,51,54,87,97,114,118,202],"one":[22],"object":[23,190],"another.":[25],"One":[26],"such":[27],"scheme,":[28],"clustering,":[29],"looks":[30],"for":[31,65,83,135],"self-similar":[32],"groups":[33],"objects.":[35],"To":[36,138],"use":[37],"clustering":[38,105],"algorithms,":[39],"an":[40],"investigator":[41],"must":[42],"often":[43,72],"<emphasis":[45],"emphasistype=\"italic\"":[46],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[47,59,90,100,125,210],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">a":[48],"priori</emphasis>":[49],"knowledge":[50],"number":[53],"clusters,":[55],"i.e.,":[56],"<formula":[57,88,98,123,208],"formulatype=\"inline\"":[58,89,99,124,209],"xmlns:xlink=\"http://www.w3.org/1999/xlink\"><tex":[60,91,101,126,211],"Notation=\"TeX\">$c$</tex></formula>":[61,102,127],",":[62,94],"search":[64],"in":[66],"data.":[68,160],"Moreover,":[69],"it":[70],"is":[71,132],"convenient":[73],"ways":[76],"rank":[78],"returned":[80],"results,":[81,120],"either":[82],"a":[84,95,166,175],"single":[85],"value":[86],"Notation=\"TeX\">$c$</tex>":[92,212],"</formula>":[93,213],"range":[96],"\u2019s":[103],"different":[104],"methods,":[106],"any":[108],"combination":[109],"thereof.":[110],"However,":[111],"task":[113],"assessing":[115],"quality":[117],"so":[121],"may":[128],"be":[129],"determined":[130],"objectively,":[131],"currently":[133],"ill-defined":[134],"object\u2013object":[136],"relationships.":[137],"bridge":[139],"this":[140],"gap,":[141],"we":[142,164],"generalize":[143],"three":[144],"well-known":[145],"validity":[146,172],"indices:":[147],"modified":[149],"Hubert\u2019s":[150],"Gamma,":[151],"Xie\u2013Beni,":[152],"and":[153,192,206,221],"generalized":[155],"Dunn\u2019s":[156],"indices,":[157],"relational":[159,176,200],"In":[161],"doing":[162],"so,":[163],"develop":[165],"framework":[167],"convert":[169],"many":[170],"other":[171],"indices":[173],"form.":[177],"Numerical":[178],"examples":[179],"on":[180],"12":[181],"datasets":[182],"(samples":[183],"from":[184],"four":[185,188,193],"normal":[186],"mixtures,":[187],"real-world":[189,194],"datasets,":[191],"\u201cpure":[195],"relational\u201d":[196],"datasets)":[197],"using":[198],"duals":[201],"hard,":[204],"fuzzy,":[205],"possibilistic":[207],"-means":[214],"cluster":[215],"algorithms":[216],"are":[217],"offered":[218],"illustrate":[220],"evaluate":[222],"new":[224],"indices.":[225]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":2},{"year":2021,"cited_by_count":3},{"year":2020,"cited_by_count":3},{"year":2019,"cited_by_count":4},{"year":2018,"cited_by_count":3},{"year":2017,"cited_by_count":3},{"year":2016,"cited_by_count":1},{"year":2015,"cited_by_count":2},{"year":2014,"cited_by_count":5},{"year":2013,"cited_by_count":2},{"year":2012,"cited_by_count":7}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}