{"id":"https://openalex.org/W2088252355","doi":"https://doi.org/10.1109/tit.1976.1055570","title":"Backwards Markovian models for second-order stochastic processes (Corresp.)","display_name":"Backwards Markovian models for second-order stochastic processes (Corresp.)","publication_year":1976,"publication_date":"1976-07-01","ids":{"openalex":"https://openalex.org/W2088252355","doi":"https://doi.org/10.1109/tit.1976.1055570","mag":"2088252355"},"language":"en","primary_location":{"id":"doi:10.1109/tit.1976.1055570","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tit.1976.1055570","pdf_url":null,"source":{"id":"https://openalex.org/S4502562","display_name":"IEEE Transactions on Information Theory","issn_l":"0018-9448","issn":["0018-9448","1557-9654"],"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 Information Theory","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/A5078405221","display_name":"Lennart Ljung","orcid":"https://orcid.org/0000-0003-4881-8955"},"institutions":[{"id":"https://openalex.org/I187531555","display_name":"Lund University","ror":"https://ror.org/012a77v79","country_code":"SE","type":"education","lineage":["https://openalex.org/I187531555"]}],"countries":["SE"],"is_corresponding":false,"raw_author_name":"L. Ljung","raw_affiliation_strings":["Department of Automatic Control, Lund Institute of Technology, Lund, Sweden","Information Systems Laboratory and the Department of Electrical Engineering, University of Stanford, Stanford, CA, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Automatic Control, Lund Institute of Technology, Lund, Sweden","institution_ids":["https://openalex.org/I187531555"]},{"raw_affiliation_string":"Information Systems Laboratory and the Department of Electrical Engineering, University of Stanford, Stanford, CA, USA","institution_ids":[]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5014088470","display_name":"T. Kailath","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"T. Kailath","raw_affiliation_strings":["Information Systems Laboratory and the Department of Electrical Engineering, University of Stanford, Stanford, CA, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Information Systems Laboratory and the Department of Electrical Engineering, University of Stanford, Stanford, CA, USA","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":9.8476,"has_fulltext":false,"cited_by_count":65,"citation_normalized_percentile":{"value":0.9816417,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":97},"biblio":{"volume":"22","issue":"4","first_page":"488","last_page":"491"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10876","display_name":"Fault Detection and Control Systems","score":0.9987000226974487,"subfield":{"id":"https://openalex.org/subfields/2207","display_name":"Control and Systems Engineering"},"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/T10876","display_name":"Fault Detection and Control Systems","score":0.9987000226974487,"subfield":{"id":"https://openalex.org/subfields/2207","display_name":"Control and Systems Engineering"},"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/T10711","display_name":"Target Tracking and Data Fusion in Sensor Networks","score":0.9955999851226807,"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/T11236","display_name":"Control Systems and Identification","score":0.9945999979972839,"subfield":{"id":"https://openalex.org/subfields/2207","display_name":"Control and Systems Engineering"},"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/representation","display_name":"Representation (politics)","score":0.7523320317268372},{"id":"https://openalex.org/keywords/markov-process","display_name":"Markov process","score":0.694809079170227},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6062277555465698},{"id":"https://openalex.org/keywords/stochastic-process","display_name":"Stochastic process","score":0.5369683504104614},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.5132083892822266},{"id":"https://openalex.org/keywords/smoothing","display_name":"Smoothing","score":0.49294304847717285},{"id":"https://openalex.org/keywords/state","display_name":"State (computer science)","score":0.4809824228286743},{"id":"https://openalex.org/keywords/state-space","display_name":"State space","score":0.4298357665538788},{"id":"https://openalex.org/keywords/set","display_name":"Set (abstract data type)","score":0.4221712350845337},{"id":"https://openalex.org/keywords/state-space-representation","display_name":"State-space representation","score":0.4163908064365387},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.2999407649040222},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.2933196723461151},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.09460785984992981}],"concepts":[{"id":"https://openalex.org/C2776359362","wikidata":"https://www.wikidata.org/wiki/Q2145286","display_name":"Representation (politics)","level":3,"score":0.7523320317268372},{"id":"https://openalex.org/C159886148","wikidata":"https://www.wikidata.org/wiki/Q176645","display_name":"Markov process","level":2,"score":0.694809079170227},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6062277555465698},{"id":"https://openalex.org/C8272713","wikidata":"https://www.wikidata.org/wiki/Q176737","display_name":"Stochastic process","level":2,"score":0.5369683504104614},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5132083892822266},{"id":"https://openalex.org/C3770464","wikidata":"https://www.wikidata.org/wiki/Q775963","display_name":"Smoothing","level":2,"score":0.49294304847717285},{"id":"https://openalex.org/C48103436","wikidata":"https://www.wikidata.org/wiki/Q599031","display_name":"State (computer science)","level":2,"score":0.4809824228286743},{"id":"https://openalex.org/C72434380","wikidata":"https://www.wikidata.org/wiki/Q230930","display_name":"State space","level":2,"score":0.4298357665538788},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.4221712350845337},{"id":"https://openalex.org/C52918065","wikidata":"https://www.wikidata.org/wiki/Q230945","display_name":"State-space representation","level":2,"score":0.4163908064365387},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.2999407649040222},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.2933196723461151},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.09460785984992981},{"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/C199539241","wikidata":"https://www.wikidata.org/wiki/Q7748","display_name":"Law","level":1,"score":0.0},{"id":"https://openalex.org/C94625758","wikidata":"https://www.wikidata.org/wiki/Q7163","display_name":"Politics","level":2,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1109/tit.1976.1055570","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tit.1976.1055570","pdf_url":null,"source":{"id":"https://openalex.org/S4502562","display_name":"IEEE Transactions on Information Theory","issn_l":"0018-9448","issn":["0018-9448","1557-9654"],"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 Information Theory","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320338279","display_name":"Air Force Office of Scientific Research","ror":"https://ror.org/011e9bt93"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":19,"referenced_works":["https://openalex.org/W89970292","https://openalex.org/W190672911","https://openalex.org/W1519304378","https://openalex.org/W1593981226","https://openalex.org/W2001779022","https://openalex.org/W2002325250","https://openalex.org/W2012992565","https://openalex.org/W2013890716","https://openalex.org/W2018295752","https://openalex.org/W2019704783","https://openalex.org/W2037799710","https://openalex.org/W2058433513","https://openalex.org/W2095337693","https://openalex.org/W2107856431","https://openalex.org/W2154136811","https://openalex.org/W2155137986","https://openalex.org/W4249925951","https://openalex.org/W4253691539","https://openalex.org/W6635405258"],"related_works":["https://openalex.org/W2579634771","https://openalex.org/W1560664277","https://openalex.org/W3013517953","https://openalex.org/W1765957141","https://openalex.org/W2756401664","https://openalex.org/W1993731165","https://openalex.org/W2338837772","https://openalex.org/W1538101308","https://openalex.org/W1627116927","https://openalex.org/W2547148345"],"abstract_inverted_index":{"A":[0],"state-space":[1],"model":[2,115],"of":[3,15,18,41,91],"a":[4,9,12,16,45,50,94],"second-order":[5],"random":[6],"process":[7,30],"is":[8,32,47,59],"representation":[10,46],"as":[11],"linear":[13,23],"combination":[14],"set":[17],"state-variables":[19],"which":[20,57],"obey":[21],"first-order":[22],"differential":[24],"equations":[25],"driven":[26],"by":[27,86],"an":[28],"input":[29],"that":[31],"both":[33],"white":[34],"and":[35,66,109],"uncorrelated":[36],"with":[37],"the":[38,42,89,113],"initial":[39],"values":[40],"state-variables.":[43],"Such":[44],"often":[48],"called":[49],"Markovian":[51,71,96],"representation.":[52,97],"There":[53],"are":[54],"applications":[55],"in":[56,74,93],"it":[58],"useful":[60],"to":[61,67,119],"consider":[62],"time":[63,92],"running":[64],"backwards":[65,70,80,114],"obtain":[68],"corresponding":[69],"representations.":[72],"Except":[73],"one":[75],"very":[76],"special":[77],"circumstance,":[78],"these":[79],"representations":[81],"cannot":[82],"be":[83,104,117],"obtained":[84],"simply":[85],"just":[87],"reversing":[88],"direction":[90],"forwards":[95],"We":[98],"show":[99],"how":[100,112],"this":[101],"problem":[102],"can":[103,116],"solved,":[105],"give":[106],"some":[107],"examples,":[108],"also":[110],"illustrate":[111],"used":[118],"clarify":[120],"certain":[121],"least":[122],"squares":[123],"smoothing":[124],"formulas.":[125]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2021,"cited_by_count":2},{"year":2019,"cited_by_count":3},{"year":2018,"cited_by_count":1},{"year":2015,"cited_by_count":2},{"year":2013,"cited_by_count":1},{"year":2012,"cited_by_count":2}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}