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Covariance estimation for filtering in high dimension
Turčičová, Marie ; Mandel, Jan (advisor) ; van Leeuwen, Peter Jan (referee) ; Pawlas, Zbyněk (referee)
Estimating large covariance matrices from small samples is an important problem in many fields. Among others, this includes spatial statistics and data assimilation. In this thesis, we deal with several methods of covariance estimation with emphasis on regula- rization and covariance models useful in filtering problems. We prove several properties of estimators and propose a new filtering method. After a brief summary of basic esti- mating methods used in data assimilation, the attention is shifted to covariance models. We show a distinct type of hierarchy in nested models applied to the spectral diagonal covariance matrix: explicit estimators of parameters are computed by the maximum like- lihood method and asymptotic variance of these estimators is shown to decrease when the maximization is restricted to a subspace that contains the true parameter value. A similar result is obtained for general M-estimators. For more complex covariance mo- dels, maximum likelihood method cannot provide explicit parameter estimates. In the case of a linear model for a precision matrix, however, consistent estimator in a closed form can be computed by the score matching method. Modelling of the precision ma- trix is particularly beneficial in Gaussian Markov random fields (GMRF), which possess a sparse precision matrix. The...
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Exponential distribution and its generalizations
Vočadlo, Vojtěch ; Pawlas, Zbyněk (advisor) ; Nagy, Stanislav (referee)
This bachelor thesis deals with the research and comparison of two proposed two- parameter generalizations of the exponential distribution. It studies the basic properties of densities and gives relations for moments of the first four orders. Furthermore, pa- rameter estimators are derived using the moment method and the maximum likelihood method. Subsequently, a simulation study is performed, on which differences between the methods used can be observed. At the end of the work, an example of approximation of data densities from real-world situations is presented using generalized exponential distributions under investigation. 1
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Detection of causality in time series using extreme values
Bodík, Juraj ; Pawlas, Zbyněk (advisor) ; Komárek, Arnošt (referee)
Juraj Bodík Abstract This thesis is dealing with the following problem: Let us have two stationary time series with heavy- tailed marginal distributions. We want to detect whether they have a causal relation, i.e. if a change in one of them causes a change in the other. The question of distinguishing between causality and correlation is essential in many different science fields. Usual methods for causality detection are not well suited if the causal mechanisms only manifest themselves in extremes. In this thesis, we propose a new method that can help us in such a nontraditional case distinguish between correlation and causality. We define the so-called causal tail coefficient for time series, which, under some assumptions, correctly detects the asymmetrical causal relations between different time series. We will rigorously prove this claim, and we also propose a method on how to statistically estimate the causal tail coefficient from a finite number of data. The advantage is that this method works even if nonlinear relations and common ancestors are present. Moreover, we will mention how our method can help detect a time delay between the two time series. We will show how our method performs on some simulations. Finally, we will show on a real dataset how this method works, discussing a cause of...
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Voronoi tessellations
Pohly, Jakub ; Pawlas, Zbyněk (advisor) ; Beneš, Viktor (referee)
In the presented work we deal with the theory of Voronoi tessellations. We deal with the properties of general Voronoi tessellations, but we focus mainly on those tessellations that are randomly generated. We study the point processes that create random Voronoi tessellations. We define the most common Poisson process. We focus on the renewal pro- cesses, specifically the ordinary renewal process, the delayed process and the equilibrium renewal process. With the help of these processes, we build a one-dimensional version of the Poisson process. We examine Voronoi tessellations primarily on a semi-straight line. Later, we generalize the obtained results for the line and the plane. In the conclusion of the work we deal with Voronoi tessellations in space. 1
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Stochastic Methods in Crystallography
Kulich, Damián ; Beneš, Viktor (advisor) ; Pawlas, Zbyněk (referee)
First we define marked tessellations to use as a model for polycrystalline structure. Then we list the necessary descriptions of orientations to use as marks for the tessellation. We formulate the necessary theory of Markov chains, so that we can use MCMC algo- rithms. The main goal is to simulate possible distributions of misorientations between neighboring cells of a tessellation. For that we formulate a parametric stochastic model and show, that we can simulate from the target distribution using an MCMC method. In the final chapter, we discuss how the results depend on the parameter and geometry of the tessellation. 1
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Measures of dependence
Matoušková, Monika ; Pawlas, Zbyněk (advisor) ; Dvořák, Jiří (referee)
The most common measure of dependence is the correlation coefficient. Its problem is that it can be zero for two dependent random variables. We will discuss two measures of dependence, which are equal to zero if and only if the two random variables are inde- pendent. We will compare them with Pearson's correlation coefficient. The first one will be the maximal correlation, which is often difficult to calculate. That is why we define the maximal polynomial correlation, which is easier to calculate and is non-decreasing in a degree of a polynomial. We also define the distance correlation and we discuss other ways of the expression of distance correlation, which can be used in the calculation. We deal with the case of normal distribution and we show some calculations of these measures of dependence. 1
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Random walks on networks and mixing of Markov chains
Gemrotová, Kateřina ; Prokešová, Michaela (advisor) ; Pawlas, Zbyněk (referee)
The thesis presents the study of deriving upper bounds of the speed of convergence of reversible Markov chains with discrete time and discrete finite space state to their stationary distributions. We express the derived upper bound in terms of several variables and we make use of the theory of electrical networks, which will help us to represent random walks on a graph. The result of this thesis will be simply obtainable upper bound of mixing time of random walks on connected graphs with an arbitrary number of vertices and edges. Partial results will be demonstrated on simple examples and counterexamples. 1
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Random marked tessellations with applications in research of polycrystalline materials
Karafiátová, Iva ; Pawlas, Zbyněk (advisor) ; Dvořák, Jiří (referee)
Experimental data obtained from polycrystalline microstructure can be in certain situations viewed as a realization of a random field or as a realization of a random marked tessellation with marks such as grain volume or grain orientation. A natural question is, whether there are dependencies within the random field or whether the marks are assigned to each grain independently on the tessellation. In this work characteristics quantifying measure of spatial dependence between marks are presented and based on them non- parametric tests of independent marking of a tessellation are proposed. We investigate power of the tests on newly introduced models of dependently marked tessellations with marks from space representing grain orientation. Proposed methods are applied on real data of microstructure with cubic crystal lattice. 1
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Orthogonal series density estimation
Zheng, Ci Jie ; Dvořák, Jiří (advisor) ; Pawlas, Zbyněk (referee)
There exist many ways to estimate the shape of the underlying density. Generally, we can categorize them into a parametric and a nonparametric methodology. Examples of a nonparametric density estimation are histogram and kernel density estimation. Another example of the nonparametric methodology is orthogonal series density estimation. In this work, we will describe the fundamental idea behind this methodology. We will also show how Kronmal-Tarter method estimates the density of known underlying data.
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Random measurable sets
Fojtík, Vít ; Rataj, Jan (advisor) ; Pawlas, Zbyněk (referee)
The aim of this thesis is to compare two major models of random sets, the well established random closed sets (RACS) and the more recent and more general random measurable sets (RAMS). First, we study the topologies underlying the models, showing they are very different. Thereafter, we introduce RAMS and RACS and reformulate prior findings about their relationship. The main result of this thesis is a characterization of those RAMS that do not induce a corresponding RACS. We conclude by some examples of such RAMS, including a construction of a translation invariant RAMS. 1
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