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Algoritmic applications of finite Markov chains
Pavlačková, Petra ; Prokešová, Michaela (advisor) ; Staňková Helisová, Kateřina (referee)
Title: Algorithmic applications of finite Markov chains Author: Petra Pavlačková Department: Department of Probability and Mathematical Statistics Supervisor: RNDr. Michaela Prokešová, Ph.D. Supervisor's e-mail address: prokesov@karlin.mff.cuni.cz In the present work we study MCMC algorithms, that we use for simulating from probability distributions on finite set of states. We apply these algorithms to two models: hard-core model and q-coloring of a graph. In this work we use the theory of stochastic processes, mainly of Markov chains and their properties. Furhter we analyze some problems, which may occur during the simulation, particularly we focus on convergence of the marginal distribution of the Markov chain to the stationary distribution. The last part of the work is a numeric illustration of the Gibbs sampler which we use in order to estimate the mean value of the number of 1 in a generalized hard-core model. Keywords: Markov chain, MCMC algorithm, hard-core model, speed of convergence
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Markov point processes
Starinská, Katarína ; Staňková Helisová, Kateřina (advisor) ; Pawlas, Zbyněk (referee)
Nazov prace: Markovske boclove procesy Autor: KaUuhia Starinska. Katcdra: Katedra, pravdepodobnosti a matematickej statistiky Veduci bakalarskcj pracc: Mgr. Katefina Helisova e-mail vedouci'ho: helisova9": karlin.inft.cimi.cz Abstrakt: Markovske bodove procesy su niodely bodovych procesov so vza- jomnym posobem'm bodov. Tioto inodoly HI'I konstmovano uva/ovani'in husto- l,y bodovehcj proccsu vzlil'a.doni k P(ji.sH(jnovriiui proccsn a pridani'ni urcitych podmienok zaisl'\ijucic;h markovsku vlastnost. Prva cast sa zaobera zaklad- nyini dcfinicianii tykajuciini sa bodovych proccsov, Poisonovyni procesoin a procesmi danyini liustotou. Druba cast obsalmjc markovske bodovc pro- cosy a v trrtoj casti sn siniuliicic markovskyt;h bodovych procesov metodou Markov Chain ATontc Carlo. Pnica je nkoncc^ia simnlaciou Stranssovho pro- ecsn. KlfcoN-a slova: Poissouov bodovy proccs. markovsky bodovy procca, Markov Chain Monte Carlo Title: Markov point processes Author: Katarina Stariiiska Department: Department of Probability and Mathematical Statistics Supervisor: Mgr. Katefina Helisova Supervisor's e-mail atldrc^.ss: heliso\w^karlin.mff.cuni.cz Abstract: Markov point processes a.ro models for1 ])oint {processes with inter- acting points. Such models are constructed by considering a density for a point, process with respect...
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Algoritmic applications of finite Markov chains
Pavlačková, Petra ; Prokešová, Michaela (advisor) ; Staňková Helisová, Kateřina (referee)
Title: Algorithmic applications of finite Markov chains Author: Petra Pavlačková Department: Department of Probability and Mathematical Statistics Supervisor: RNDr. Michaela Prokešová, Ph.D. Supervisor's e-mail address: prokesov@karlin.mff.cuni.cz In the present work we study MCMC algorithms, that we use for simulating from probability distributions on finite set of states. We apply these algorithms to two models: hard-core model and q-coloring of a graph. In this work we use the theory of stochastic processes, mainly of Markov chains and their properties. Furhter we analyze some problems, which may occur during the simulation, particularly we focus on convergence of the marginal distribution of the Markov chain to the stationary distribution. The last part of the work is a numeric illustration of the Gibbs sampler which we use in order to estimate the mean value of the number of 1 in a generalized hard-core model. Keywords: Markov chain, MCMC algorithm, hard-core model, speed of convergence
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Distributions suitable for modelling of individual losses
Janáková, Veronika ; Staňková Helisová, Kateřina (referee) ; Mazurová, Lucie (advisor)
Nazov prace: Ro/dclenia vhodne k modelovaniu vysky skod Autor: Veronika Janakova Katedra: Katcdra pravdepodobnosti a matematickcj statistiky VedoLici bakalarskej prace: RNDr. LucJc Mazurova, Ph.D e-mail vcduecho: Mazurova@karlin.nifT.cuni.cz Abstrakt: Obsahom prace jc prchfad rozdeleni vyu/ivanych v oblasti nezivotneho poistenia na modclovanic vysky skod. V iivode si definujcme zakladnc rozdclenia, ktore naslednc transfbrniujcme. Pri kazdom rozdeleni su uvcdcnc zakladne charakteristiky, ktorymi je dane rozdelenie urcenc. Ccla praca jc rozdclcna do dvoch kapitol. V prvcj kapitole sa zameriame na dcfim'cie u/ spominanych zakladnych rozdeleni a v druhcj casti prace budeme zakladne rozdelenia transibrmovaf. Vyuzijeme dva druhy transtbrmacii, pomocou ktorych ziskamc rozdelcnia s viaeerymi parametrami. Takcto rozdelenia maju sirsie pouzitie, leda su llexibilnejsie a mnoho rozdcleni ziskame ako spceialne pripady. KTucove slova: pravdepodobnostne rozdelenia, transformacia Title: Distributions suitable for modelling of individual losses Author: Veronika Janakova Department: Department of Probability and Mathematical Statistics Supervisor: RNDr. Lucie Mazurova, Ph.D Supervisor's e-mail address: Mazurova@karHn.mff.euni.cz Abstract: Subject of this paper is to provide a review of distribution used in the section of...
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