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[{"key": "dc.contributor.advisor", "value": "Vihola, Matti", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Parkkinen, Santeri", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2019-06-04T08:48:02Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2019-06-04T08:48:02Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2019", "language": "", "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/64307", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4m\u00e4n tutkielman tavoitteena on esitell\u00e4 Markovin ketju Monte Carlo -simulointi \u00e4\u00e4rellisess\u00e4 tila-avaruudessa ja k\u00e4sitell\u00e4 simulointialgoritmien vertailuun liittyv\u00e4\u00e4 ongelmaa. Markovin ketju Monte Carlo -simuloinnissa funktion odotusarvolle lasketaan approksimaatioita simuloitua Markovin ketjua apuna k\u00e4ytt\u00e4en. Usein simulointi voidaan tehd\u00e4 useammalla kuin yhdell\u00e4 algoritmilla ja halutaan selvitt\u00e4\u00e4, mik\u00e4 algoritmi soveltuu teht\u00e4v\u00e4\u00e4n parhaiten.\n\nKun simulaatioaskelten m\u00e4\u00e4r\u00e4 l\u00e4hestyy \u00e4\u00e4ret\u00f6nt\u00e4, Markovin ketju Monte Carlo -estimaatti suppenee melkein varmasti kohti odotusarvon oikeaa arvoa k\u00e4ytetyst\u00e4 simulointialgoritmista riippumatta. K\u00e4yt\u00e4nn\u00f6ss\u00e4 Markovin ketju Monte Carlo -simulointi tuottaa kuitenkin vain odotusarvon approksimaation, koska mik\u00e4\u00e4n todellinen simulointi ei voi jatkua \u00e4\u00e4rett\u00f6m\u00e4n pitk\u00e4\u00e4n. Vain \u00e4\u00e4rellisen monen simulaatioaskeleen k\u00e4ytt\u00f6 aiheuttaa virheen, joka on luonteeltaan satunnainen: virhe kuuluu tietylle v\u00e4lille jollakin todenn\u00e4k\u00f6isyydell\u00e4. On luonnollista kysy\u00e4, miten approksimaation tarkkuus riippuu simulointialgoritmista. Kaikki algoritmit antavat odotusarvolle arvion halutulla tarkkuudella, jos simulointia jatketaan riitt\u00e4v\u00e4n kauan. Jotkut algoritmit tuottavat kuitenkin tarkkoja approksimaatioita nopeammin kuin toiset, mik\u00e4 on motivaatio algoritmien vertailulle. Ei kuitenkaan ole aivan selv\u00e4\u00e4, miten vertailu pit\u00e4isi k\u00e4yt\u00e4nn\u00f6ss\u00e4 tehd\u00e4. Halutun tarkkuuden saavuttamiseksi vaadittavaa aikaa on vaikea arvioida, eiv\u00e4tk\u00e4 pelk\u00e4t arviot edes ole riitt\u00e4v\u00e4 perusta algoritmien vertailulle. Voi esimerkiksi k\u00e4yd\u00e4 niin, ett\u00e4 arvioitu alaraja algoritmin P vaatimalle ajalle on pienempi kuin alaraja algoritmin Q vaatimalle ajalle, mutta silti algoritmi Q p\u00e4\u00e4see haluttuun tarkkuuteen nopeammin kuin algoritmi P. Pelk\u00e4t alarajat eiv\u00e4t siis kerro tarpeeksi vaadittavien aikojen todellisista arvoista. Arvioiden sijaan tarvitaan jokin eksakti riippuvuus simulointivirheen ja algoritmin v\u00e4lille.\n\nKriteeri, jonka suhteen algoritmien vertailu tehd\u00e4\u00e4n, johdetaan Markovin ketjujen keskeist\u00e4 raja-arvolausetta k\u00e4ytt\u00e4en. Keskeinen raja-arvolause mahdollistaa simulointivirheiden todenn\u00e4k\u00f6isyyksien raja-arvon eksaktin laskemisen tietyss\u00e4 erikoistapauksessa. Tarkastelu osoittaa, ett\u00e4 algoritmien vertailemiseksi kannattaa tutkia niiden asymptoottisia variansseja: kahdesta algoritmista se, jonka asymptoottinen varianssi annetulle funktiolle on pienempi, soveltuu paremmin kyseisen funktion odotusarvon simulointiin. Kriteerin ongelmana on, ett\u00e4 sen soveltaminen ei ole ihan suoraviivaista. K\u00e4yt\u00e4nn\u00f6ss\u00e4 algoritmeista tiedet\u00e4\u00e4n vain niiden siirtym\u00e4todenn\u00e4k\u00f6isyysmatriisit, joten algoritmien vertailemiseksi t\u00e4ytyy selvitt\u00e4\u00e4, miten asymptoottinen varianssi riippuu siirtym\u00e4todenn\u00e4k\u00f6isyysmatriisista. T\u00e4ss\u00e4 tutkielmassa tarkastelu rajataan k\u00e4\u00e4ntyviin Markovin ketjuihin, jolloin kyseinen riippuvuus voidaan selvitt\u00e4\u00e4 funktionaalianalyysin tuloksia hy\u00f6dynt\u00e4en. T\u00e4m\u00e4n j\u00e4lkeen vertailu onnistuu tietyss\u00e4 erikoistapauksessa Peskunin j\u00e4rjestyst\u00e4 k\u00e4ytt\u00e4m\u00e4ll\u00e4. Peskunin j\u00e4rjestyksen sovelluksena osoitetaan, ett\u00e4 Metropolis\u2013Hastings -algoritmi on parempi kuin Barkerin algoritmi.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Miia Hakanen (mihakane@jyu.fi) on 2019-06-04T08:48:02Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2019-06-04T08:48:02Z (GMT). No. of bitstreams: 0\n Previous issue date: 2019", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "64", "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": "en", "element": "rights", "qualifier": null, "schema": "dc"}, {"key": "dc.subject.other", "value": "Markovin ketju Monte Carlo -simulointi", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "siirtym\u00e4todenn\u00e4k\u00f6isyysmatriisi", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "asymptoottinen varianssi", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "Peskunin j\u00e4rjestys", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "Metropolis\u2013Hastings algoritmi", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "Barkerin algoritmi", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Markovin ketju Monte Carlo -simulointi ja Peskunin j\u00e4rjestys", "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-201906042916", "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": "Matematiikka", "language": "fi", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "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.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": "Markovin ketjut", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "Monte Carlo -menetelm\u00e4t", "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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