prikaz prve stranice dokumenta Gerrymandering - krojenje izbornih jedinica i Metropolis-Hastings algoritam
  Download
PDF 219.97 KB
master's thesis
Gerrymandering - krojenje izbornih jedinica i Metropolis-Hastings algoritam
Zagreb: University of Zagreb, Faculty of Science, 2019. urn:nbn:hr:217:320271
Pleić, Nikolina
University of Zagreb
Faculty of Science
Department of Mathematics

Institutional repository: Repository of the Faculty of Science

Cite this document

Pleić, N. (2019). Gerrymandering - krojenje izbornih jedinica i Metropolis-Hastings algoritam (Master's thesis). Zagreb: University of Zagreb, Faculty of Science. Retrieved from https://urn.nsk.hr/urn:nbn:hr:217:320271

Pleić, Nikolina. "Gerrymandering - krojenje izbornih jedinica i Metropolis-Hastings algoritam." Master's thesis, University of Zagreb, Faculty of Science, 2019. https://urn.nsk.hr/urn:nbn:hr:217:320271

Pleić, Nikolina. "Gerrymandering - krojenje izbornih jedinica i Metropolis-Hastings algoritam." Master's thesis, University of Zagreb, Faculty of Science, 2019. https://urn.nsk.hr/urn:nbn:hr:217:320271

Pleić, N. (2019). 'Gerrymandering - krojenje izbornih jedinica i Metropolis-Hastings algoritam', Master's thesis, University of Zagreb, Faculty of Science, accessed 30 December 2024, https://urn.nsk.hr/urn:nbn:hr:217:320271

Pleić N. Gerrymandering - krojenje izbornih jedinica i Metropolis-Hastings algoritam [Master's thesis]. Zagreb: University of Zagreb, Faculty of Science; 2019 [cited 2024 December 30] Available at: https://urn.nsk.hr/urn:nbn:hr:217:320271

N. Pleić, "Gerrymandering - krojenje izbornih jedinica i Metropolis-Hastings algoritam", Master's thesis, University of Zagreb, Faculty of Science, Zagreb, 2019. Available at: https://urn.nsk.hr/urn:nbn:hr:217:320271

Metadata
TitleGerrymandering - krojenje izbornih jedinica i Metropolis-Hastings algoritam
AuthorNikolina Pleić
MentorMatija Kazalicki (mentor)
Committee memberMatija Kazalicki (predsjednik povjerenstva)
Committee memberMarko Tadić (član povjerenstva)
Committee memberMarcela Hanzer (član povjerenstva)
Committee memberVjekoslav Kovač (član povjerenstva)
GranterUniversity of Zagreb
Faculty of Science
(Department of Mathematics)
Zagreb
Defense date and country2019-03-01, Croatia
Scientific / art field,
discipline and subdiscipline		NATURAL SCIENCES
Mathematics
Abstract
U prvom poglavlju definirali smo Markovljeve lance na općenitom skupu stanja. Definirali smo pojmove kao što su prijelazna jezgra Markovljevog lanca u Definiciji 1.0.1, invarijantna mjera u Definiciji 1.0.7 te detaljna ravnoteža. Pokazali smo da je uvjet detaljne ravnoteže dovoljan uvjet da bi Markovljev lanac imao invarijantnu mjeru. Nakon toga u drugom poglavlju primijenili smo tu teoriju na Markovljev lanac dobiven Metropolis-Hastings algoritmom. Naveli smo dovoljne uvjete kako bi dobiveni lanac imao stacionarnu distribuciju. U Teoremu 2.5.3 objedinili smo rezultate o egzistenciji stacionarne distribucije i konvergenciji lanca k stacionarnoj distribuciji. U trećem poglavlju promatrali smo politički problem gerrymanderinga te smo ga vizualizirali na primjeru američke države North Caroline. Uveli smo evaluacijsku funkciju J kako bismo svakom izbornom planu dodijelili vjerojatnost takvu da izborni planovi s manjom vrijednošću evaluacijske funkcije budu više vjerojatni. Pomoću evaluacijske funkcije definirali smo ciljanu vjerojatnosnu distribuciju. Konačno, predložili smo način stvaranja novih izbornih raspodjela koristeći Metropolis-Hastings algoritam.
Abstract (english)
In the first chapter we discussed the theory of Markov chains on a general state space. We defined terms such as transition kernel in Definition 1.0.1, invariant measure in 1.0.7 and detailed balance condition. We have shown that the detailed balance condition is sufficient for a Markov chain to have an invariant measure. After that, in the second chapter, we have applied that theory to a Markov chain produced by Metropolis-Hastings algorithm. We list the sufficient conditions for the produced Markov chain to have a stationary distribution. In Theorem 2.5.3 we combine results of existence of stationary distribution and the convergence of a chain to the stationary distribution. In third chapter, we examined the political problem of gerrymandering. We visualized it on an example of an US state of North Carolina. We introduced a score function J in order to assign a value to each redistricting so that redistrictings with lower scores are more likely to occur. Using this score function we have defined a target distribution. Finally, we proposed a way of creating new redistrictings using a standard Metropolis-Hastings algorithm.
Keywords
Keywords (english)
Languagecroatian
URN:NBNurn:nbn:hr:217:320271
Study programmeTitle: Mathematical Statistics
Study programme type: university
Study level: graduate
Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
Type of resourceText
File originBorn digital
Access conditionsOpen access
Terms of useLicence In copyright icon
Created on2019-09-03 12:06:23