{"id":"https://openalex.org/W2963724574","doi":"https://doi.org/10.3390/e21080724","title":"A Copula Entropy Approach to Dependence Measurement for Multiple Degradation Processes","display_name":"A Copula Entropy Approach to Dependence Measurement for Multiple Degradation Processes","publication_year":2019,"publication_date":"2019-07-25","ids":{"openalex":"https://openalex.org/W2963724574","doi":"https://doi.org/10.3390/e21080724","mag":"2963724574","pmid":"https://pubmed.ncbi.nlm.nih.gov/33267438"},"language":"en","primary_location":{"id":"doi:10.3390/e21080724","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e21080724","pdf_url":"https://www.mdpi.com/1099-4300/21/8/724/pdf?version=1564135166","source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj","pubmed"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/1099-4300/21/8/724/pdf?version=1564135166","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5088929641","display_name":"Fuqiang Sun","orcid":"https://orcid.org/0000-0002-7667-7363"},"institutions":[{"id":"https://openalex.org/I82880672","display_name":"Beihang University","ror":"https://ror.org/00wk2mp56","country_code":"CN","type":"education","lineage":["https://openalex.org/I82880672"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Fuqiang Sun","raw_affiliation_strings":["Science and Technology on Reliability and Environmental Engineering Laboratory, School of Reliability and Systems Engineering, Beihang University, Beijing 100191, China"],"raw_orcid":"https://orcid.org/0000-0002-7667-7363","affiliations":[{"raw_affiliation_string":"Science and Technology on Reliability and Environmental Engineering Laboratory, School of Reliability and Systems Engineering, Beihang University, Beijing 100191, China","institution_ids":["https://openalex.org/I82880672"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5108735255","display_name":"Wendi Zhang","orcid":null},"institutions":[{"id":"https://openalex.org/I82880672","display_name":"Beihang University","ror":"https://ror.org/00wk2mp56","country_code":"CN","type":"education","lineage":["https://openalex.org/I82880672"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Wendi Zhang","raw_affiliation_strings":["Science and Technology on Reliability and Environmental Engineering Laboratory, School of Reliability and Systems Engineering, Beihang University, Beijing 100191, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Science and Technology on Reliability and Environmental Engineering Laboratory, School of Reliability and Systems Engineering, Beihang University, Beijing 100191, China","institution_ids":["https://openalex.org/I82880672"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5100387085","display_name":"Ning Wang","orcid":"https://orcid.org/0000-0001-6938-0521"},"institutions":[{"id":"https://openalex.org/I1311643757","display_name":"Commercial Aircraft Corporation of China (China)","ror":"https://ror.org/05gxmms51","country_code":"CN","type":"company","lineage":["https://openalex.org/I1311643757"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Ning Wang","raw_affiliation_strings":["Beijing Aeronautical Science & Technology Research Institute, Commercial Aircraft Corporation of China Ltd., Beijing 102211, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Beijing Aeronautical Science & Technology Research Institute, Commercial Aircraft Corporation of China Ltd., Beijing 102211, China","institution_ids":["https://openalex.org/I1311643757"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5101657644","display_name":"Wei Zhang","orcid":"https://orcid.org/0000-0002-7754-6058"},"institutions":[{"id":"https://openalex.org/I82880672","display_name":"Beihang University","ror":"https://ror.org/00wk2mp56","country_code":"CN","type":"education","lineage":["https://openalex.org/I82880672"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Wei Zhang","raw_affiliation_strings":["Science and Technology on Reliability and Environmental Engineering Laboratory, School of Reliability and Systems Engineering, Beihang University, Beijing 100191, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Science and Technology on Reliability and Environmental Engineering Laboratory, School of Reliability and Systems Engineering, Beihang University, Beijing 100191, China","institution_ids":["https://openalex.org/I82880672"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5101657644"],"corresponding_institution_ids":["https://openalex.org/I82880672"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":1.9626,"has_fulltext":true,"cited_by_count":15,"citation_normalized_percentile":{"value":0.84818358,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"21","issue":"8","first_page":"724","last_page":"724"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10780","display_name":"Reliability and Maintenance Optimization","score":0.9998000264167786,"subfield":{"id":"https://openalex.org/subfields/2213","display_name":"Safety, Risk, Reliability and Quality"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10780","display_name":"Reliability and Maintenance Optimization","score":0.9998000264167786,"subfield":{"id":"https://openalex.org/subfields/2213","display_name":"Safety, Risk, Reliability and Quality"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11357","display_name":"Risk and Safety Analysis","score":0.9936000108718872,"subfield":{"id":"https://openalex.org/subfields/1804","display_name":"Statistics, Probability and Uncertainty"},"field":{"id":"https://openalex.org/fields/18","display_name":"Decision Sciences"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},{"id":"https://openalex.org/T10968","display_name":"Statistical Distribution Estimation and Applications","score":0.9919000267982483,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/copula","display_name":"Copula (linguistics)","score":0.7789926528930664},{"id":"https://openalex.org/keywords/entropy","display_name":"Entropy (arrow of time)","score":0.5554149746894836},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.5487310886383057},{"id":"https://openalex.org/keywords/statistical-physics","display_name":"Statistical physics","score":0.5301875472068787},{"id":"https://openalex.org/keywords/joint-entropy","display_name":"Joint entropy","score":0.5049052834510803},{"id":"https://openalex.org/keywords/multivariate-statistics","display_name":"Multivariate statistics","score":0.4458080232143402},{"id":"https://openalex.org/keywords/information-theory","display_name":"Information theory","score":0.4398382902145386},{"id":"https://openalex.org/keywords/transfer-entropy","display_name":"Transfer entropy","score":0.42497774958610535},{"id":"https://openalex.org/keywords/principle-of-maximum-entropy","display_name":"Principle of maximum entropy","score":0.3445061147212982},{"id":"https://openalex.org/keywords/data-mining","display_name":"Data mining","score":0.32174524664878845},{"id":"https://openalex.org/keywords/econometrics","display_name":"Econometrics","score":0.3153967261314392},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.3129141330718994},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.25511762499809265},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.23650607466697693},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.183098703622818},{"id":"https://openalex.org/keywords/machine-learning","display_name":"Machine learning","score":0.18305063247680664},{"id":"https://openalex.org/keywords/thermodynamics","display_name":"Thermodynamics","score":0.1255655586719513}],"concepts":[{"id":"https://openalex.org/C17618745","wikidata":"https://www.wikidata.org/wiki/Q207509","display_name":"Copula (linguistics)","level":2,"score":0.7789926528930664},{"id":"https://openalex.org/C106301342","wikidata":"https://www.wikidata.org/wiki/Q4117933","display_name":"Entropy (arrow of time)","level":2,"score":0.5554149746894836},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.5487310886383057},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.5301875472068787},{"id":"https://openalex.org/C106752470","wikidata":"https://www.wikidata.org/wiki/Q1364826","display_name":"Joint entropy","level":3,"score":0.5049052834510803},{"id":"https://openalex.org/C161584116","wikidata":"https://www.wikidata.org/wiki/Q1952580","display_name":"Multivariate statistics","level":2,"score":0.4458080232143402},{"id":"https://openalex.org/C52622258","wikidata":"https://www.wikidata.org/wiki/Q131222","display_name":"Information theory","level":2,"score":0.4398382902145386},{"id":"https://openalex.org/C182049051","wikidata":"https://www.wikidata.org/wiki/Q17147155","display_name":"Transfer entropy","level":3,"score":0.42497774958610535},{"id":"https://openalex.org/C9679016","wikidata":"https://www.wikidata.org/wiki/Q1417473","display_name":"Principle of maximum entropy","level":2,"score":0.3445061147212982},{"id":"https://openalex.org/C124101348","wikidata":"https://www.wikidata.org/wiki/Q172491","display_name":"Data mining","level":1,"score":0.32174524664878845},{"id":"https://openalex.org/C149782125","wikidata":"https://www.wikidata.org/wiki/Q160039","display_name":"Econometrics","level":1,"score":0.3153967261314392},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.3129141330718994},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.25511762499809265},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.23650607466697693},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.183098703622818},{"id":"https://openalex.org/C119857082","wikidata":"https://www.wikidata.org/wiki/Q2539","display_name":"Machine learning","level":1,"score":0.18305063247680664},{"id":"https://openalex.org/C97355855","wikidata":"https://www.wikidata.org/wiki/Q11473","display_name":"Thermodynamics","level":1,"score":0.1255655586719513}],"mesh":[],"locations_count":5,"locations":[{"id":"doi:10.3390/e21080724","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e21080724","pdf_url":"https://www.mdpi.com/1099-4300/21/8/724/pdf?version=1564135166","source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy","raw_type":"journal-article"},{"id":"pmid:33267438","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/33267438","pdf_url":null,"source":{"id":"https://openalex.org/S4306525036","display_name":"PubMed","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy (Basel, Switzerland)","raw_type":null},{"id":"pmh:oai:doaj.org/article:86804729d97e4d80beb77f85fb0e6fbe","is_oa":true,"landing_page_url":"https://doaj.org/article/86804729d97e4d80beb77f85fb0e6fbe","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Entropy, Vol 21, Iss 8, p 724 (2019)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/1099-4300/21/8/724/","is_oa":true,"landing_page_url":"http://dx.doi.org/10.3390/e21080724","pdf_url":null,"source":{"id":"https://openalex.org/S4306400947","display_name":"MDPI (MDPI AG)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210097602","host_organization_name":"Multidisciplinary Digital Publishing Institute (Switzerland)","host_organization_lineage":["https://openalex.org/I4210097602"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Entropy","raw_type":"Text"},{"id":"pmh:oai:pubmedcentral.nih.gov:7515253","is_oa":true,"landing_page_url":"https://www.ncbi.nlm.nih.gov/pmc/articles/7515253","pdf_url":null,"source":{"id":"https://openalex.org/S2764455111","display_name":"PubMed Central","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Entropy (Basel)","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/e21080724","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e21080724","pdf_url":"https://www.mdpi.com/1099-4300/21/8/724/pdf?version=1564135166","source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G7343125525","display_name":null,"funder_award_id":"61603018 and 51875019","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G92370711","display_name":null,"funder_award_id":"61603018","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"}],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2963724574.pdf","grobid_xml":"https://content.openalex.org/works/W2963724574.grobid-xml"},"referenced_works_count":53,"referenced_works":["https://openalex.org/W157032409","https://openalex.org/W614670535","https://openalex.org/W1457220277","https://openalex.org/W1482605500","https://openalex.org/W1517766493","https://openalex.org/W1582142628","https://openalex.org/W1593125407","https://openalex.org/W1932372884","https://openalex.org/W1977267557","https://openalex.org/W1989705587","https://openalex.org/W1994044089","https://openalex.org/W1995341919","https://openalex.org/W1995875735","https://openalex.org/W2004839758","https://openalex.org/W2014268383","https://openalex.org/W2019841176","https://openalex.org/W2030931858","https://openalex.org/W2031235730","https://openalex.org/W2040287221","https://openalex.org/W2046188839","https://openalex.org/W2055281720","https://openalex.org/W2056911559","https://openalex.org/W2064115435","https://openalex.org/W2071554058","https://openalex.org/W2078178705","https://openalex.org/W2092086030","https://openalex.org/W2099551891","https://openalex.org/W2105828028","https://openalex.org/W2118020555","https://openalex.org/W2125150867","https://openalex.org/W2296274909","https://openalex.org/W2345145213","https://openalex.org/W2362701603","https://openalex.org/W2507039649","https://openalex.org/W2512716701","https://openalex.org/W2513769405","https://openalex.org/W2518518020","https://openalex.org/W2578540448","https://openalex.org/W2606400548","https://openalex.org/W2615162136","https://openalex.org/W2745759370","https://openalex.org/W2759138613","https://openalex.org/W2805738088","https://openalex.org/W2808208847","https://openalex.org/W2964235834","https://openalex.org/W3099837250","https://openalex.org/W4240463847","https://openalex.org/W6633032110","https://openalex.org/W6658530583","https://openalex.org/W6670324742","https://openalex.org/W6738554657","https://openalex.org/W6757598146","https://openalex.org/W6989407874"],"related_works":["https://openalex.org/W3105581519","https://openalex.org/W3032013634","https://openalex.org/W2889261288","https://openalex.org/W2492100837","https://openalex.org/W1595266364","https://openalex.org/W1519338247","https://openalex.org/W2949821122","https://openalex.org/W4386352335","https://openalex.org/W1972053074","https://openalex.org/W2388154903"],"abstract_inverted_index":{"Degradation":[0],"analysis":[1],"has":[2],"been":[3],"widely":[4],"used":[5,124],"in":[6,53],"reliability":[7],"modeling":[8,65],"problems":[9],"of":[10,28,34,43,49,62,89,116,129,140,156],"complex":[11,16,113],"systems.":[12],"A":[13],"system":[14,55],"with":[15],"structure":[17,115],"and":[18,26,46,69,93,119],"various":[19],"functions":[20],"may":[21,30],"have":[22],"multiple":[23,50,74,157],"degradation":[24,51,64,103,158],"features,":[25,118],"any":[27],"them":[29],"be":[31,57],"a":[32,54,87],"cause":[33],"product":[35],"failure.":[36],"Typically,":[37],"these":[38],"features":[39],"are":[40],"not":[41],"independent":[42],"each":[44],"other,":[45],"the":[47,60,71,90,99,112,127,138,141,153],"dependence":[48,72,100,114,154],"processes":[52],"cannot":[56],"ignored.":[58],"Therefore,":[59],"premise":[61],"multivariate":[63],"is":[66,86,150],"to":[67,97,110,125,136],"capture":[68],"measure":[70,98],"among":[73,101],"features.":[75],"To":[76],"address":[77],"this":[78,80,148],"problem,":[79],"paper":[81],"adopts":[82],"copula":[83,91,106],"entropy,":[84],"which":[85],"combination":[88],"function":[92,107],"information":[94,120],"entropy":[95,121],"theory,":[96],"different":[102],"processes.":[104,159],"The":[105,144],"was":[108,123,134],"employed":[109],"identify":[111],"performance":[117],"theory":[122],"quantify":[126],"degree":[128],"dependence.":[130],"An":[131],"engineering":[132],"case":[133],"utilized":[135],"illustrate":[137],"effectiveness":[139],"proposed":[142],"method.":[143],"results":[145],"show":[146],"that":[147],"method":[149],"valid":[151],"for":[152],"measurement":[155]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":2},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":2},{"year":2021,"cited_by_count":5},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":1}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}