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Robust Student estimator
Hlavinka, Radek ; Friml, Dominik (referee) ; Dokoupil, Jakub (advisor)
This Master's thesis deals with Bayesian approach to robust parameter estimation for ARX models. Robustness is achieved by assuming the measurement noise to be generated by Student-t distribution. The asumption of Student-t noise renders the model's posterior intractable and requires utilization of approximation techniques. This thesis considers algorithms using Gibbs sampler and Variational approximation and compares them with Ordinary Least Squares. The algorithms are compared based on their Maximum Likelihood estimation. It is shown that approaches assuming the Student-t noise perform better in simulation. The results from data acquired from physical system are however similar for all algorithms considered.
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Convergence of the Markov chain Monte Carlo method
Dzurilla, Matúš ; Beneš, Viktor (advisor) ; Dostál, Petr (referee)
This thesis deals with the problem of random q-coloring from graph theory, in which goal is to color all vertices of graph by q colors so that no adjacent vertices have the same color. The aim is to generate random q-coloring from uniform distribution on the set of relevant solutions. The problem was expres- sed through Markov chain and approach was done through Markov Chain Monte Carlo method, namely the Gibbs sampler. The aim was to modify theorem of fast convergence of Gibbs sampler from systematic sweep to random sweep. It was ne- cessary to prove several auxiliary theorems, and in the proof of main theorem the "coupling" method was used. We managed to estimate the number of iterations needed to make the distance, in terms of total variation,from the distribution on the chain to the target distribution sufficiently small. The meaning og the the- orem was demonstrated in numerical examples and example od simulation was also added. 24
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Monte Carlo Potts model
Vlachovský, Karel ; Beneš, Viktor (advisor) ; Dvořák, Jiří (referee)
Potts model is a generalisation of the Ising model which is used in statistical mechanics. Our goal is to sample from the distribution of that model. However, the state space is too large, so we cannot sample from it directly. We will use Markov Chain Monte Carlo methods instead. It means that the markov chain would have Potts distribution as its stationary distribution. We will compare Gibbs sampler, Metropolis algorithm, Swendsen-Wang algorithm and significantly we will introduce a new mixing algorithm. We will show that all these algorithms are uniformly ergodic. We will implement them and show that it is wise to use only mixing algorithm and Swendsen-Wang algorithm for larger parameter of temperature for Potts model. 1
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Convergence of the Markov chain Monte Carlo method
Dzurilla, Matúš ; Beneš, Viktor (advisor) ; Dostál, Petr (referee)
This thesis deals with the problem of random q-coloring from graph theory, in which goal is to color all vertices of graph by q colors so that no adjacent vertices have the same color. The aim is to generate random q-coloring from uniform distribution on the set of relevant solutions. The problem was expres- sed through Markov chain and approach was done through Markov Chain Monte Carlo method, namely the Gibbs sampler. The aim was to modify theorem of fast convergence of Gibbs sampler from systematic sweep to random sweep. It was ne- cessary to prove several auxiliary theorems, and in the proof of main theorem the "coupling" method was used. We managed to estimate the number of iterations needed to make the distance, in terms of total variation,from the distribution on the chain to the target distribution sufficiently small. The meaning og the the- orem was demonstrated in numerical examples and example od simulation was also added. 24
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