{"id":"https://openalex.org/W2090170322","doi":"https://doi.org/10.3390/e14122375","title":"Expectation Values and Variance Based on Lp-Norms","display_name":"Expectation Values and Variance Based on Lp-Norms","publication_year":2012,"publication_date":"2012-11-26","ids":{"openalex":"https://openalex.org/W2090170322","doi":"https://doi.org/10.3390/e14122375","mag":"2090170322"},"language":"en","primary_location":{"id":"doi:10.3390/e14122375","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e14122375","pdf_url":"https://www.mdpi.com/1099-4300/14/12/2375/pdf?version=1424784907","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"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/1099-4300/14/12/2375/pdf?version=1424784907","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5080379161","display_name":"G. Livadiotis","orcid":"https://orcid.org/0000-0002-7655-6019"},"institutions":[{"id":"https://openalex.org/I1290463931","display_name":"Southwest Research Institute","ror":"https://ror.org/03tghng59","country_code":"US","type":"nonprofit","lineage":["https://openalex.org/I1290463931"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"George Livadiotis","raw_affiliation_strings":["Southwest Research Institute, San Antonio, TX-78238, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Southwest Research Institute, San Antonio, TX-78238, USA","institution_ids":["https://openalex.org/I1290463931"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5080379161"],"corresponding_institution_ids":["https://openalex.org/I1290463931"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":1.2136,"has_fulltext":false,"cited_by_count":19,"citation_normalized_percentile":{"value":0.79273717,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":97},"biblio":{"volume":"14","issue":"12","first_page":"2375","last_page":"2396"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12261","display_name":"Statistical Mechanics and Entropy","score":0.9983000159263611,"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/T12261","display_name":"Statistical Mechanics and Entropy","score":0.9983000159263611,"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/T11270","display_name":"Complex Systems and Time Series Analysis","score":0.9954000115394592,"subfield":{"id":"https://openalex.org/subfields/2002","display_name":"Economics and Econometrics"},"field":{"id":"https://openalex.org/fields/20","display_name":"Economics, Econometrics and Finance"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},{"id":"https://openalex.org/T11871","display_name":"Advanced Statistical Methods and Models","score":0.9912999868392944,"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/euclidean-geometry","display_name":"Euclidean geometry","score":0.7456375360488892},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6976808905601501},{"id":"https://openalex.org/keywords/variance","display_name":"Variance (accounting)","score":0.6248230934143066},{"id":"https://openalex.org/keywords/euclidean-distance","display_name":"Euclidean distance","score":0.6180444955825806},{"id":"https://openalex.org/keywords/generalization","display_name":"Generalization","score":0.5695540904998779},{"id":"https://openalex.org/keywords/norm","display_name":"Norm (philosophy)","score":0.48380154371261597},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.3733707666397095},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3254004120826721},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.12861210107803345}],"concepts":[{"id":"https://openalex.org/C129782007","wikidata":"https://www.wikidata.org/wiki/Q162886","display_name":"Euclidean geometry","level":2,"score":0.7456375360488892},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6976808905601501},{"id":"https://openalex.org/C196083921","wikidata":"https://www.wikidata.org/wiki/Q7915758","display_name":"Variance (accounting)","level":2,"score":0.6248230934143066},{"id":"https://openalex.org/C120174047","wikidata":"https://www.wikidata.org/wiki/Q847073","display_name":"Euclidean distance","level":2,"score":0.6180444955825806},{"id":"https://openalex.org/C177148314","wikidata":"https://www.wikidata.org/wiki/Q170084","display_name":"Generalization","level":2,"score":0.5695540904998779},{"id":"https://openalex.org/C191795146","wikidata":"https://www.wikidata.org/wiki/Q3878446","display_name":"Norm (philosophy)","level":2,"score":0.48380154371261597},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.3733707666397095},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3254004120826721},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.12861210107803345},{"id":"https://openalex.org/C199539241","wikidata":"https://www.wikidata.org/wiki/Q7748","display_name":"Law","level":1,"score":0.0},{"id":"https://openalex.org/C121955636","wikidata":"https://www.wikidata.org/wiki/Q4116214","display_name":"Accounting","level":1,"score":0.0},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0},{"id":"https://openalex.org/C17744445","wikidata":"https://www.wikidata.org/wiki/Q36442","display_name":"Political science","level":0,"score":0.0},{"id":"https://openalex.org/C144133560","wikidata":"https://www.wikidata.org/wiki/Q4830453","display_name":"Business","level":0,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.3390/e14122375","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e14122375","pdf_url":"https://www.mdpi.com/1099-4300/14/12/2375/pdf?version=1424784907","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:doaj.org/article:270e24618c4647e8aa9fdb5cc500db37","is_oa":true,"landing_page_url":"https://doaj.org/article/270e24618c4647e8aa9fdb5cc500db37","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 14, Iss 12, Pp 2375-2396 (2012)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.3390/e14122375","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e14122375","pdf_url":"https://www.mdpi.com/1099-4300/14/12/2375/pdf?version=1424784907","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":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":35,"referenced_works":["https://openalex.org/W1494162474","https://openalex.org/W1529627758","https://openalex.org/W1593125407","https://openalex.org/W1647238455","https://openalex.org/W1672879477","https://openalex.org/W1954793731","https://openalex.org/W1978191248","https://openalex.org/W1986026923","https://openalex.org/W1988109006","https://openalex.org/W1989959848","https://openalex.org/W1993241069","https://openalex.org/W1996404747","https://openalex.org/W1997281911","https://openalex.org/W2010797828","https://openalex.org/W2012712694","https://openalex.org/W2017109395","https://openalex.org/W2025010014","https://openalex.org/W2033652368","https://openalex.org/W2034179117","https://openalex.org/W2055748047","https://openalex.org/W2061834091","https://openalex.org/W2071602085","https://openalex.org/W2072406744","https://openalex.org/W2115566940","https://openalex.org/W2120062331","https://openalex.org/W2132807882","https://openalex.org/W2142577042","https://openalex.org/W2160701304","https://openalex.org/W2329985448","https://openalex.org/W2395561311","https://openalex.org/W2489822048","https://openalex.org/W2808650095","https://openalex.org/W3102938774","https://openalex.org/W6711542536","https://openalex.org/W6806599836"],"related_works":["https://openalex.org/W2903781522","https://openalex.org/W2090152127","https://openalex.org/W1965169884","https://openalex.org/W2008939113","https://openalex.org/W3125580510","https://openalex.org/W2033213447","https://openalex.org/W14679004","https://openalex.org/W1566651525","https://openalex.org/W2977652649","https://openalex.org/W2318206461"],"abstract_inverted_index":{"This":[0],"analysis":[1],"introduces":[2],"a":[3,47,58],"generalization":[4],"of":[5,10,32,51,82],"the":[6,29,34,39,55,62,80],"basic":[7],"statistical":[8,91],"concepts":[9],"expectation":[11,44,64],"values":[12,45,65],"and":[13,66,98],"variance":[14,68],"for":[15],"non-Euclidean":[16,22],"metrics":[17],"induced":[18],"by":[19,27],"Lp-norms.":[20],"The":[21,89],"Lp":[23,35,40,43],"means":[24,52],"are":[25,69,104],"defined":[26],"exploiting":[28],"fundamental":[30],"property":[31],"minimizing":[33],"deviations":[36],"that":[37,75],"compose":[38],"variance.":[41],"These":[42],"embody":[46],"generic":[48],"formal":[49],"scheme":[50],"characterization.":[53],"Having":[54],"p-norm":[56],"as":[57],"free":[59],"parameter,":[60],"both":[61],"Lp-normed":[63],"their":[67],"flexible":[70],"to":[71],"analyze":[72],"new":[73,90],"phenomena":[74],"cannot":[76],"be":[77],"described":[78],"under":[79],"notions":[81],"classical":[83],"statistics":[84],"based":[85],"on":[86],"Euclidean":[87],"norms.":[88],"approach":[92],"provides":[93],"insights":[94],"into":[95],"regression":[96],"theory":[97],"Statistical":[99],"Physics.":[100],"Several":[101],"illuminating":[102],"examples":[103],"examined.":[105]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":2},{"year":2020,"cited_by_count":2},{"year":2019,"cited_by_count":2},{"year":2018,"cited_by_count":2},{"year":2017,"cited_by_count":2},{"year":2016,"cited_by_count":3},{"year":2015,"cited_by_count":1},{"year":2014,"cited_by_count":3},{"year":2013,"cited_by_count":1}],"updated_date":"2026-05-21T09:19:25.381259","created_date":"2025-10-10T00:00:00"}