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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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Tests for the Poisson distribution
Trusina, Filip ; Pawlas, Zbyněk (advisor) ; Nagy, Stanislav (referee)
In this work we deal with the question whether a sequence of independent identically distributed random variables comes from the Poisson distribution. For this task we present two different approaches and couple of tests for each appro- ach. The first approach is based on the asymptotic approximation of distribution of test statistics. The second approach uses generation of test samples. Based on simulations done by us, we discuss the power of individual tests and their advantages and disadvantages. 1
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Essential problems of random walks
Michálek, Matěj ; Hlubinka, Daniel (advisor) ; Pawlas, Zbyněk (referee)
In this paper, we cover some essential problems of (simple) random walks in one, two and three dimensions. At the begining, we work only in one dimension. We find the probability of a position on a line at particular time. Then we study returns to origin and examine if return to origin is certain. Also, we look into a theorem called the arc sine law. Furthermore, we generalise some of those problems into two and three dimensions. We investigate a probability of a position in time and space and returns to origin. 1
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Chaotic random variables in applied probability
Večeřa, Jakub ; Beneš, Viktor (advisor) ; Reitzner, Matthias (referee) ; Pawlas, Zbyněk (referee)
This thesis deals with modeling of particle processes. In the first part we ex- amine Gibbs facet process on a bounded window with discrete orientation distri- bution and we derive central limit theorem (CLT) for U-statistics of facet process with increasing intensity. We calculate all asymptotic joint moments for interac- tion U-statistics and use the method of moments for deriving the CLT. Moreover we present an alternative proof which makes use of the CLT for U-statistics of a Poisson facet process. In the second part we model planar segment processes given by a density with respect to the Poisson process. Parametric models involve reference distributions of directions and/or lengths of segments. Statistical methods are presented which first estimate scalar parameters by known approaches and then the reference distribution is estimated non-parametrically. We also introduce the Takacs-Fiksel estimate and demonstrate the use of estimators in a simulation study and also using data from actin fibres from stem cells images. In the third part we study a stationary Gibbs particle process with determin- istically bounded particles on Euclidean space defined in terms of a finite range potential and an activity parameter. For small activity parameters, we prove the CLT for certain statistics of this...
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Second-order characteristics of point processes
Gupta, Archit ; Pawlas, Zbyněk (advisor) ; Prokešová, Michaela (referee)
In this thesis we examine estimation of the K-function which is an important second-order characteristic in the theory of spatial point processes. Besides Ripley's K-function based on a spherical structuring element we also work with the multiparameter K-function where the struc- turing element is rectangular. We consider the Poisson point process model, which is the fundamental model for complete spatial randomness. We de- rive expressions for both bias and variance of the estimators. The primary goal of this thesis is the study of different edge correction methods that are available for the K-function. Using simulations we also study a few variance approximations proposed in the literature and compare them with empirical variances. 1
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Stochastic Differential Equations with Gaussian Noise
Janák, Josef ; Maslowski, Bohdan (advisor) ; Duncan, Tyrone E. (referee) ; Pawlas, Zbyněk (referee)
Title: Stochastic Differential Equations with Gaussian Noise Author: Josef Janák Department: Department of Probability and Mathematical Statistics Supervisor: Prof. RNDr. Bohdan Maslowski, DrSc., Department of Probability and Mathematical Statistics Abstract: Stochastic partial differential equations of second order with two un- known parameters are studied. The strongly continuous semigroup (S(t), t ≥ 0) for the hyperbolic system driven by Brownian motion is found as well as the formula for the covariance operator of the invariant measure Q (a,b) ∞ . Based on ergodicity, two suitable families of minimum contrast estimators are introduced and their strong consistency and asymptotic normality are proved. Moreover, another concept of estimation using "observation window" is studied, which leads to more families of strongly consistent estimators. Their properties and special cases are descibed as well as their asymptotic normality. The results are applied to the stochastic wave equation perturbed by Brownian noise and illustrated by several numerical simula- tions. Keywords: Stochastic hyperbolic equation, Ornstein-Uhlenbeck process, invariant measure, paramater estimation, strong consistency, asymptotic normality.
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Nonparametric tests of independence
Kmeťková, Diana ; Pawlas, Zbyněk (advisor) ; Hlubinka, Daniel (referee)
The main objective of this thesis is the presentation regarding the problem of testing independence between two random variables in the nonparametric model of continuous cumulative distribution functions. Firstly, the reader is informed with basic notions from the theory of independence and rank tests. Afterwards, few of the most common methods for testing independence are introduced. In the beginning, the test based on Pearson's correlation coefficient is mentioned as a representative for parametric tests, then we continue with nonparametric tests, such as test based on Spearman's, Kendall's and distance correlation coefficient. We focus in better detail on Hoeffding's test of independence, which results to be consistent against all alternatives in the model of continuous cumulative distribution functions. In the end, we compare and evaluate presented methods for testing independence using simulations in R environment.
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