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Point processes on linear networks
Moravec, Jan ; Prokešová, Michaela (advisor) ; Pawlas, Zbyněk (referee)
The central theme of this thesis is the theory of point processes on linear net- works, in particular two kinds of the network K-function. The first part is devoted to the theory of stationary point processes in the plane, including the K-function and its estimator. The second part is concerned with the theory of point proces- ses on linear networks. There is defined the Okabe-Yamada network K -function and its estimator, the geometrically corrected network K-function, including its estimator, and there are explained their theoretical properties. In the third part we examine the ability of these two kinds of the network K-function to detect clustering or regularity in point processes on linear networks. There is explained the envelope test, the refined envelope test and the deviation tests. The software environment R with library spatstat is used for simulations.
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Strong stationary times and convergence of Markov chains
Suk, Luboš ; Prokešová, Michaela (advisor) ; Kříž, Pavel (referee)
In this thesis we study the estimation of speed of convergence of Markov chains to their stacionary distributions. For that purpose we will use the method of strong stationary times. We focus on irreducible and aperiodic chains only since in that case the existence of exactly one stationary distribution is guaranteed. We introduce the mixing time for a Markov chain as the time needed for the marginal distribution of the chain to be sufficiently close to the stationary dis- tribution. The distance between two distributions is measured by the total variation distance. The main goal of this thesis is to construct an appropriate strong stationary time for selected chains and then use it for obtaining an upper bound for the mixing time.
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Estimation in continuous time Markov chains
Nemčovič, Bohuš ; Prokešová, Michaela (advisor) ; Kadlec, Karel (referee)
Title: Estimation in continuous time Markov chains Author: Bohuš Nemčovič Department: Department of Probability and Mathematical Statistics Supervisor: RNDr. Michaela Prokešová, Ph.D., Department of Probability and Mathematical Statistics Abstract: In this work we deal with estimating the intensity matrices of continu- ous Markov chains in the case of complete observation and observation at selected discrete time points. To obtain an estimate we use the maximum likelihood met- hod. In the second chapter we first introduce the general EM algorithm and then adjust it for finding the intensity matrix estimate based on observations at disc- rete time points. In the last chapter we will illustrate the impact of the discrete step size on the quality of intensity matrix estimate. Keywords: Markov chains, intensity matrix, maximum likelihood estimation, EM algorithm 1
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Coupling and speed of convergence of discrete MCMC algorithms.
Kalaš, Martin ; Prokešová, Michaela (advisor) ; Dvořák, Jiří (referee)
Convergence of the marginal distribution of a Markov chain to its stationary distribution is an essential property of this model with many applications in different fields of modern mathematics. Such typical applications are for example the Markov Chain Monte Carlo algorithms, which are useful for sampling from complicated probability distributions. A crucial point for usefulness of such algorithms is the so called mixing time of corresponding Markov chain, i.e. the number of steps the chain has to make for the difference between its current marginal distribution and stationary distribution to be sufficiently small. The main goal of this thesis is to describe a method for estimation of the mixing time based on a probability technique called coupling. In the first part we collect some definitions and propositions to show how the method works. Later the method is demonstrated on several traditional examples of Markov chains including e.g. random walk on a graph. In the end we study Metropolis chain on the set of proper colorings of a graph as a specific example of MCMC algorithm and show how to estimate its mixing time.
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Random Processes in Reliability Analysis
Chovanec, Kamil ; Volf, Petr (advisor) ; Prokešová, Michaela (referee)
Title: Random Processes in Reliability Analysis Author: Kamil Chovanec Department: Department of Probability and Mathematical Statistics Supervisor: Doc. Petr Volf, CSc. Supervisor's e-mail address: volf@utia.cas.cz Abstract: The thesis is aimed at the reliability analysis with special em- phasis at the Aalen additive model. The result of hypothesis testing in the reliability analysis is often a process that converges to a Gaussian martingale under the null hypothesis. We can estimate the variance of the martingale using a uniformly consistent estimator. The result of this estimation is a new hypothesis about the process resulting from the original hypothesis. There are several ways to test for this hypothesis. The thesis presents some of these tests and compares their power for various models and sample sizes using Monte Carlo simulations. In a special case we derive a point that maximizes the asymptotic power of two of the tests. Keywords: Martingale, Aalen's additive model, hazard function 1
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