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[{"key": "dc.contributor.advisor", "value": "Juutinen, Petri", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Hirvel\u00e4, Juulia", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2023-06-13T08:42:32Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2023-06-13T08:42:32Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2023", "language": "", "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/87698", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4m\u00e4n tutkielman tarkoituksena on esitell\u00e4 matematiikkaa, johon Googlen hakutulosten j\u00e4rjest\u00e4miseen k\u00e4ytt\u00e4m\u00e4 PageRank-algoritmi perustuu. Tutkielmassa hy\u00f6dynnet\u00e4\u00e4n lineaarialgebran, graafien ja Markovin ketjujen teorian yhteyksi\u00e4, joiden avulla algoritmin matemaattinen muoto saadaan esitetty\u00e4. \n\tHyv\u00e4 hakukone tarjoaa hakutuloksissaan ensimm\u00e4isen\u00e4 sellaisia sivuja, jotka ovat netinselaajan mielest\u00e4 hy\u00f6dyllisi\u00e4. Google-hakukone k\u00e4ytt\u00e4\u00e4 sivun t\u00e4rkeyden selvitt\u00e4miseen omaa PageRank-algoritmiaan, joka laskee jokaiselle nettisivulle t\u00e4rkeysarvon eli PageRankin. Hakutulokset j\u00e4rjestet\u00e4\u00e4n t\u00e4m\u00e4n arvon perusteella suuruusj\u00e4rjestykseen. Sivun PageRank m\u00e4\u00e4r\u00e4ytyy sivulle johtavien linkkien m\u00e4\u00e4r\u00e4st\u00e4 ja viittaavien sivujen t\u00e4rkeydest\u00e4. \n\tAlgoritmi perustuu World Wide Webin linkkirakenteen esitt\u00e4miseen suunnattuna graafina. Graafin informaatiosta muodostetaan matriisi, jonka itseisarvoltaan suurinta ominaisarvoa vastaava ominaisvektori on niin sanottu PageRank-vektori. Algoritmin tarkoituksena on saada selville t\u00e4m\u00e4 vektori, koska se pit\u00e4\u00e4 sis\u00e4ll\u00e4\u00e4n sivujen PageRankit. Ominaisarvon ja sit\u00e4 vastaavaa vektorin laskeminen suoraan matriisin ominaisarvoyht\u00e4l\u00f6st\u00e4 olisi kuitenkin liian ty\u00f6l\u00e4st\u00e4, joten t\u00e4t\u00e4 varten ty\u00f6ss\u00e4 esitell\u00e4\u00e4n iteratiivinen prosessi nimelt\u00e4 potenssimenetelm\u00e4, jossa mielivaltaisesti valittua aloitusvektoria kerrotaan toistuvasti edell\u00e4 muodostetulla matriisilla. \n\tJotta menetelm\u00e4\u00e4 voidaan hy\u00f6dynt\u00e4\u00e4, t\u00e4ytyy ensin varmistua siit\u00e4, ett\u00e4 sen avulla laskettava vektorijono suppenee. T\u00e4m\u00e4 asettaa edell\u00e4 muodostetulle matriisille tietyt vaatimukset. Tutkielmassa huomataan, ett\u00e4 mik\u00e4li matriisi on muistittoman stokastisen prosessin primitiivinen siirtym\u00e4matriisi, potenssimenetelm\u00e4ll\u00e4 muodostettu vektorijono suppenee kohti PageRank-vektoria, joka vastaa itse asiassa kyseisen stokastisen prosessin tasapainotilaa. Jonon suppenemisen ehtoja selvitt\u00e4ess\u00e4 tutustutaan muun muassa redusoituviin matriiseihin, Perronin ja Frobeniuksen lauseeseen sek\u00e4 Markovin ketjuihin.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Paivi Vuorio (paelvuor@jyu.fi) on 2023-06-13T08:42:32Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2023-06-13T08:42:32Z (GMT). No. of bitstreams: 0\n Previous issue date: 2023", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "34", "language": "", "element": "format", "qualifier": "extent", "schema": "dc"}, {"key": "dc.language.iso", "value": "fin", "language": null, "element": "language", "qualifier": "iso", "schema": "dc"}, {"key": "dc.rights", "value": "In Copyright", "language": null, "element": "rights", "qualifier": null, "schema": "dc"}, {"key": "dc.title", "value": "Googlen PageRankin matematiikka", "language": "", "element": "title", "qualifier": null, "schema": "dc"}, {"key": "dc.type", "value": "master thesis", "language": null, "element": "type", "qualifier": null, "schema": "dc"}, {"key": "dc.identifier.urn", "value": "URN:NBN:fi:jyu-202306133766", "language": "", "element": "identifier", "qualifier": "urn", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Master\u2019s thesis", "language": "en", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Pro gradu -tutkielma", "language": "fi", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Matemaattis-luonnontieteellinen tiedekunta", "language": "fi", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Faculty of Sciences", "language": "en", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Matematiikan ja tilastotieteen laitos", "language": "fi", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Department of Mathematics and Statistics", "language": "en", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "Jyv\u00e4skyl\u00e4n yliopisto", "language": "fi", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "University of Jyv\u00e4skyl\u00e4", "language": "en", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Matematiikan opettajankoulutus", "language": "fi", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Teacher education programme in Mathematics", "language": "en", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "yvv.contractresearch.funding", "value": "0", "language": "", "element": "contractresearch", "qualifier": "funding", "schema": "yvv"}, {"key": "dc.type.coar", "value": "http://purl.org/coar/resource_type/c_bdcc", "language": null, "element": "type", "qualifier": "coar", "schema": "dc"}, {"key": "dc.rights.copyright", "value": "\u00a9 The Author(s)", "language": null, "element": "rights", "qualifier": "copyright", "schema": "dc"}, {"key": "dc.rights.accesslevel", "value": "openAccess", "language": null, "element": "rights", "qualifier": "accesslevel", "schema": "dc"}, {"key": "dc.type.publication", "value": "masterThesis", "language": null, "element": "type", "qualifier": "publication", "schema": "dc"}, {"key": "dc.subject.oppiainekoodi", "value": "4041", "language": "", "element": "subject", "qualifier": "oppiainekoodi", "schema": "dc"}, {"key": "dc.subject.yso", "value": "Google", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "verkkoteoria", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "Markovin ketjut", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "ominaisarvot", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "algoritmit", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "lineaarialgebra", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.rights.url", "value": "https://rightsstatements.org/page/InC/1.0/", "language": null, "element": "rights", "qualifier": "url", "schema": "dc"}]
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