{"id":"https://openalex.org/W2900754571","doi":"https://doi.org/10.3390/e20120991","title":"Spatial Measures of Urban Systems: from Entropy to Fractal Dimension","display_name":"Spatial Measures of Urban Systems: from Entropy to Fractal Dimension","publication_year":2018,"publication_date":"2018-12-19","ids":{"openalex":"https://openalex.org/W2900754571","doi":"https://doi.org/10.3390/e20120991","mag":"2900754571"},"language":"en","primary_location":{"id":"doi:10.3390/e20120991","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e20120991","pdf_url":"https://www.mdpi.com/1099-4300/20/12/991/pdf?version=1545313469","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":["arxiv","crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/1099-4300/20/12/991/pdf?version=1545313469","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5002167057","display_name":"Yanguang Chen","orcid":"https://orcid.org/0000-0002-7804-9993"},"institutions":[{"id":"https://openalex.org/I20231570","display_name":"Peking University","ror":"https://ror.org/02v51f717","country_code":"CN","type":"education","lineage":["https://openalex.org/I20231570"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Yanguang Chen","raw_affiliation_strings":["Department of Geography, College of Urban and Environmental Sciences, Peking University, Beijing 100871, China"],"raw_orcid":"https://orcid.org/0000-0002-7804-9993","affiliations":[{"raw_affiliation_string":"Department of Geography, College of Urban and Environmental Sciences, Peking University, Beijing 100871, China","institution_ids":["https://openalex.org/I20231570"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5067178724","display_name":"Linshan Huang","orcid":null},"institutions":[{"id":"https://openalex.org/I20231570","display_name":"Peking University","ror":"https://ror.org/02v51f717","country_code":"CN","type":"education","lineage":["https://openalex.org/I20231570"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Linshan Huang","raw_affiliation_strings":["Department of Geography, College of Urban and Environmental Sciences, Peking University, Beijing 100871, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Geography, College of Urban and Environmental Sciences, Peking University, Beijing 100871, China","institution_ids":["https://openalex.org/I20231570"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5002167057"],"corresponding_institution_ids":["https://openalex.org/I20231570"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":1.6083,"has_fulltext":false,"cited_by_count":25,"citation_normalized_percentile":{"value":0.82962511,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":99},"biblio":{"volume":"20","issue":"12","first_page":"991","last_page":"991"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10226","display_name":"Land Use and Ecosystem Services","score":0.9975000023841858,"subfield":{"id":"https://openalex.org/subfields/2306","display_name":"Global and Planetary Change"},"field":{"id":"https://openalex.org/fields/23","display_name":"Environmental Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10226","display_name":"Land Use and Ecosystem Services","score":0.9975000023841858,"subfield":{"id":"https://openalex.org/subfields/2306","display_name":"Global and Planetary Change"},"field":{"id":"https://openalex.org/fields/23","display_name":"Environmental Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T13890","display_name":"Remote Sensing and Land Use","score":0.9861000180244446,"subfield":{"id":"https://openalex.org/subfields/1902","display_name":"Atmospheric Science"},"field":{"id":"https://openalex.org/fields/19","display_name":"Earth and Planetary Sciences"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12325","display_name":"Urban Design and Spatial Analysis","score":0.967199981212616,"subfield":{"id":"https://openalex.org/subfields/2215","display_name":"Building and Construction"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/multifractal-system","display_name":"Multifractal system","score":0.9129263162612915},{"id":"https://openalex.org/keywords/fractal-dimension","display_name":"Fractal dimension","score":0.7250545024871826},{"id":"https://openalex.org/keywords/statistical-physics","display_name":"Statistical physics","score":0.721322238445282},{"id":"https://openalex.org/keywords/fractal","display_name":"Fractal","score":0.7043219804763794},{"id":"https://openalex.org/keywords/fractal-dimension-on-networks","display_name":"Fractal dimension on networks","score":0.6883869767189026},{"id":"https://openalex.org/keywords/entropy","display_name":"Entropy (arrow of time)","score":0.6149550676345825},{"id":"https://openalex.org/keywords/scaling","display_name":"Scaling","score":0.5949155688285828},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5944898724555969},{"id":"https://openalex.org/keywords/fractal-analysis","display_name":"Fractal analysis","score":0.5439857840538025},{"id":"https://openalex.org/keywords/binary-entropy-function","display_name":"Binary entropy function","score":0.5376447439193726},{"id":"https://openalex.org/keywords/correlation-dimension","display_name":"Correlation dimension","score":0.4649105966091156},{"id":"https://openalex.org/keywords/homogeneity","display_name":"Homogeneity (statistics)","score":0.4327653646469116},{"id":"https://openalex.org/keywords/principle-of-maximum-entropy","display_name":"Principle of maximum entropy","score":0.2722884714603424},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.21974638104438782},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.20262864232063293},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.15973907709121704},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.15572935342788696},{"id":"https://openalex.org/keywords/thermodynamics","display_name":"Thermodynamics","score":0.06905195116996765}],"concepts":[{"id":"https://openalex.org/C133905733","wikidata":"https://www.wikidata.org/wiki/Q2629238","display_name":"Multifractal system","level":3,"score":0.9129263162612915},{"id":"https://openalex.org/C26546657","wikidata":"https://www.wikidata.org/wiki/Q1412452","display_name":"Fractal dimension","level":3,"score":0.7250545024871826},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.721322238445282},{"id":"https://openalex.org/C40636538","wikidata":"https://www.wikidata.org/wiki/Q81392","display_name":"Fractal","level":2,"score":0.7043219804763794},{"id":"https://openalex.org/C60396315","wikidata":"https://www.wikidata.org/wiki/Q5477489","display_name":"Fractal dimension on networks","level":5,"score":0.6883869767189026},{"id":"https://openalex.org/C106301342","wikidata":"https://www.wikidata.org/wiki/Q4117933","display_name":"Entropy (arrow of time)","level":2,"score":0.6149550676345825},{"id":"https://openalex.org/C99844830","wikidata":"https://www.wikidata.org/wiki/Q102441924","display_name":"Scaling","level":2,"score":0.5949155688285828},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5944898724555969},{"id":"https://openalex.org/C162494671","wikidata":"https://www.wikidata.org/wiki/Q2845227","display_name":"Fractal analysis","level":4,"score":0.5439857840538025},{"id":"https://openalex.org/C44415725","wikidata":"https://www.wikidata.org/wiki/Q4913893","display_name":"Binary entropy function","level":3,"score":0.5376447439193726},{"id":"https://openalex.org/C110601934","wikidata":"https://www.wikidata.org/wiki/Q5172849","display_name":"Correlation dimension","level":4,"score":0.4649105966091156},{"id":"https://openalex.org/C142259097","wikidata":"https://www.wikidata.org/wiki/Q5891314","display_name":"Homogeneity (statistics)","level":2,"score":0.4327653646469116},{"id":"https://openalex.org/C9679016","wikidata":"https://www.wikidata.org/wiki/Q1417473","display_name":"Principle of maximum entropy","level":2,"score":0.2722884714603424},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.21974638104438782},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.20262864232063293},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.15973907709121704},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.15572935342788696},{"id":"https://openalex.org/C97355855","wikidata":"https://www.wikidata.org/wiki/Q11473","display_name":"Thermodynamics","level":1,"score":0.06905195116996765}],"mesh":[],"locations_count":5,"locations":[{"id":"doi:10.3390/e20120991","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e20120991","pdf_url":"https://www.mdpi.com/1099-4300/20/12/991/pdf?version=1545313469","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":"pmh:oai:arXiv.org:1811.07657","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1811.07657","pdf_url":"https://arxiv.org/pdf/1811.07657","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"pmh:oai:doaj.org/article:657cb35906e144329c230bb44a9ec5c1","is_oa":true,"landing_page_url":"https://doaj.org/article/657cb35906e144329c230bb44a9ec5c1","pdf_url":null,"source":{"id":"https://openalex.org/S112646816","display_name":"SHILAP Revista de lepidopterolog\u00eda","issn_l":"0300-5267","issn":["0300-5267","2340-4078"],"is_oa":true,"is_in_doaj":true,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"journal"},"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 20, Iss 12, p 991 (2018)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/1099-4300/20/12/991/","is_oa":true,"landing_page_url":"http://dx.doi.org/10.3390/e20120991","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:7512591","is_oa":true,"landing_page_url":"https://www.ncbi.nlm.nih.gov/pmc/articles/7512591","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/e20120991","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e20120991","pdf_url":"https://www.mdpi.com/1099-4300/20/12/991/pdf?version=1545313469","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":[{"score":0.8299999833106995,"display_name":"Sustainable cities and communities","id":"https://metadata.un.org/sdg/11"}],"awards":[{"id":"https://openalex.org/G7754769808","display_name":null,"funder_award_id":"41671167","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/W2900754571.pdf","grobid_xml":"https://content.openalex.org/works/W2900754571.grobid-xml"},"referenced_works_count":45,"referenced_works":["https://openalex.org/W419962475","https://openalex.org/W600251820","https://openalex.org/W1620133492","https://openalex.org/W1834408334","https://openalex.org/W1962391267","https://openalex.org/W1975056775","https://openalex.org/W1981070902","https://openalex.org/W1991038884","https://openalex.org/W2017413556","https://openalex.org/W2017587126","https://openalex.org/W2018810248","https://openalex.org/W2030888688","https://openalex.org/W2033750748","https://openalex.org/W2044586796","https://openalex.org/W2046969942","https://openalex.org/W2047646383","https://openalex.org/W2068142485","https://openalex.org/W2072648611","https://openalex.org/W2081357582","https://openalex.org/W2084625851","https://openalex.org/W2087786641","https://openalex.org/W2089422350","https://openalex.org/W2094369723","https://openalex.org/W2120025875","https://openalex.org/W2125833549","https://openalex.org/W2132227723","https://openalex.org/W2139364442","https://openalex.org/W2158304989","https://openalex.org/W2302493635","https://openalex.org/W2434905021","https://openalex.org/W2528165859","https://openalex.org/W2580019464","https://openalex.org/W2593007382","https://openalex.org/W2734767870","https://openalex.org/W2761670133","https://openalex.org/W2949116535","https://openalex.org/W2994441007","https://openalex.org/W3098018196","https://openalex.org/W3098083750","https://openalex.org/W3098776669","https://openalex.org/W3101430794","https://openalex.org/W3148918439","https://openalex.org/W4233497874","https://openalex.org/W4294698888","https://openalex.org/W6771358752"],"related_works":["https://openalex.org/W2069237854","https://openalex.org/W1868966746","https://openalex.org/W1965262908","https://openalex.org/W4323967758","https://openalex.org/W2109345986","https://openalex.org/W2004745477","https://openalex.org/W3098334782","https://openalex.org/W3148344277","https://openalex.org/W2062900467","https://openalex.org/W3098018196"],"abstract_inverted_index":{"One":[0,95],"type":[1],"of":[2,67,89,103,112,135,142,156,176,196],"fractal":[3,16,38,61,90,121],"dimension":[4],"definition":[5],"is":[6,96,117],"based":[7],"on":[8,53,108],"the":[9,32,64,72,97,100,109,124,132,139,148,154,173,193],"generalized":[10,149],"entropy":[11,14,36,58,88,101,118,126,129,160],"function.":[12],"Both":[13],"and":[15,29,37,45,83,138,159,191],"dimensions":[17,62],"can":[18,162,179],"be":[19,80,163,180,185],"employed":[20],"to":[21,56,79,165],"characterize":[22],"complex":[23],"spatial":[24,65,87,113,140,170,194],"systems":[25,77,105],"such":[26],"as":[27],"cities":[28,91],"regions.":[30],"Despite":[31],"inherent":[33],"connection":[34],"between":[35],"dimensions,":[39],"they":[40],"have":[41],"different":[42,120],"application":[43],"scopes":[44],"directions":[46],"in":[47,189],"urban":[48,69,104,136,143,169,177,197],"studies.":[49],"This":[50,182],"paper":[51],"focuses":[52],"exploring":[54],"how":[55],"convert":[57,147],"measurements":[59],"into":[60,151],"for":[63,168,187],"analysis":[66],"scale-free":[68],"phenomena":[70],"using":[71],"ideas":[73],"from":[74],"scaling.":[75],"Urban":[76],"proved":[78],"random":[81],"prefractal":[82],"multifractal":[84,152],"systems.":[85],"The":[86,115],"bears":[92],"two":[93],"properties.":[94],"scale":[98,157],"dependence:":[99],"values":[102],"always":[106],"depend":[107],"linear":[110],"scales":[111],"measurement.":[114],"other":[116],"conservation:":[119],"parts":[122],"bear":[123],"same":[125],"value.":[127],"Thus,":[128],"cannot":[130],"reflect":[131],"simple":[133],"rules":[134],"processes":[137],"heterogeneity":[141],"patterns.":[144],"If":[145],"we":[146],"entropies":[150],"spectrums,":[153],"problems":[155],"dependence":[158],"homogeneity":[161],"solved":[164],"a":[166],"degree":[167],"analysis.":[171],"Especially,":[172],"geographical":[174],"analyses":[175],"evolution":[178],"simplified.":[181],"study":[183],"may":[184],"helpful":[186],"students":[188],"describing":[190],"explaining":[192],"complexity":[195],"evolution.":[198]},"counts_by_year":[{"year":2026,"cited_by_count":2},{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":4},{"year":2022,"cited_by_count":6},{"year":2021,"cited_by_count":2},{"year":2020,"cited_by_count":7},{"year":2019,"cited_by_count":2}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}
