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Nonparametric tests of independence between animal movement trajectories
Veselý, Martin ; Dvořák, Jiří (advisor) ; Omelka, Marek (referee)
In this thesis, we assume observing a pair of trajectories of two objects which could interact with one another and we want to propose a way to test their independence. We formulate basic point process definitions and discuss ways to describe trajectory data. We formulate the theory behind Monte Carlo tests and global envelope testing. In Chapter 2, we propose a parametric model to represent trajectories and derive Maximum Likelihood estimates of its model. We conclude the chapter by exploring the performance of these estimates. In Chapter 3, we propose test statistics used to test for independence using a nonparametric Monte Carlo test based on a random shift approach. We perform a simulation study to assess the performance of these statistics under various conditions and discuss the selection of fine-tuning parameters. Finally, in Chapter 4, we study real data provided by the Voyageurs Wolf Project and apply the proposed tests on real wolf trajectories. 1
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Glivenko-Cantelli theorem and its generalization
Pustějovský, Zdeněk ; Omelka, Marek (advisor) ; Pawlasová, Kateřina (referee)
In this thesis we look at Glivenko-Cantelli theorem and its generalization. Firstly we prove the classical version of this theorem with empirical ditribution function and as its corollary we show uniform convergence of sample quantiles to the actual ones. Next we give definition of bracketing number and prove generalized version of Glivenko-Cantelli theorem for function classes with finite bracketing number. Then we show how from the generalized version of the theorem follows the classical one not only for real random variables, but also for random vectors. Lastly we give examples of some Glivenko-Cantelli classes of functions. Throughout the work we also have applications of proven theorems in mind. 1
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Confidence intervals for the correlation coefficient
Farda, Martin ; Kulich, Michal (advisor) ; Omelka, Marek (referee)
The main goal of the thesis is to introduce methods used for the construction of confidence intervals for correlation coefficient in detail and to show their performances on various examples. In the first chapter is an introduction of basic properties of correlation coefficient and Fisher's z-transformation. The second chapter is about a method based on generalized pivotal quantities. It also contains an explanation why is an assumption of bivariate normal distribution necessary for this method. In the third chapter there is a description of two methods based on empirical likelihood. These methods are approptiate also for non-normal bivariate distributions. In the last chapter are all mentioned methods applied on several examples and compared with each other. 1
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Confidence intervals for ratios
Krett, Jakub ; Omelka, Marek (advisor) ; Antoch, Jaromír (referee)
This thesis is devoted to the derivation of various types of confidence intervals for ratios of mean values. The inspiration for this work is applying the acquired theoretical knowledge to the problem of waste sorting, such as estimating the weight of the unsor- ted waste components concerning the total weight of the mixture. Firstly, the confidence intervals based on the standard asymptotic inference are derived, such as the standard asymptotic confidence interval and the interval estimate derived using the logit trans- formation. Furthermore, the thesis introduces the bootstrap method, which leads to the derivation of the basic, percentile, and studentized bootstrap confidence interval. Finally, the end of the thesis explores the properties of these interval estimates using two simu- lation models. 1
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Sensometric discriminant testing - comparison of paired comparison test and ranking test
Švarcová, Karolína ; Antoch, Jaromír (advisor) ; Omelka, Marek (referee)
Sensometric tests are useful for deciding if there exists a perceivable sensory difference between two or more products. The tests can be divided into two main groups - first for determining the existence of a difference based on a known and specified sensory attribute, and second for determination when the variance in the sensory attribute is not known. In this thesis we deal with sensometric tests from the first group and describe the contrast in the statistical approach of a test method that evaluates two samples and a test method evaluating more samples at once. Particularly, within the paired comparison test the calculations are based on the binomial distribution, whereas for the ranking test, statistical methods working with the ranks of random samples are used. To illustrate the process of a sensometric test, we execute a paired comparison test, and we show how a good test method for a given problem is chosen.
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Generalized estimating equaitons
Sotáková, Martina ; Omelka, Marek (advisor) ; Antoch, Jaromír (referee)
In this thesis we are interested in generalized estimating equations (GEE). First, we introduce the term of generalized linear model, on which generalized estimating equations are based. Next we present the methos of pseudo maximum likelyhood and quasi-pseudo maximum likelyhood, from which we move on to the methods of generalized estimating equations. Finally, we perform simulation studies, which demonstrates the theoretical results presented in the thesis. 1
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Two-sample Wilcoxon test in the presence of ties
Kulla, Filip ; Omelka, Marek (advisor) ; Maciak, Matúš (referee)
This work is devoted to the Wilcoxon rank-sum test in the presence of ties. In this work, asymptotic distribution of the Wilcoxon test statistic in the presence of ties is derived using well-known results about U-statistics. With the aid of this result, a corrected test statistic for data containing ties is proposed. Furthermore, the thesis examines the relation between the derived correction and conditional variance of the test statistic (and conditional expectation). Finally, by means of simulation, the effect of the derived correction on the actual significance level is examined as the number of tied observations increases. 1
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Joint Models for Longitudinal and Time-to-Event Data
Vorlíčková, Jana ; Komárek, Arnošt (advisor) ; Omelka, Marek (referee)
Title: Joint Models for Longitudinal and Time-to-Event Data Author: Jana Vorlíčková Department: Department of Probability and Mathematical Statistics Supervisor: doc. RNDr. Arnošt Komárek, Ph.D., Department of Probability and Mathematical Statistics Abstract: The joint model of longitudinal data and time-to-event data creates a framework to analyze longitudinal and survival outcomes simultaneously. A commonly used approach is an interconnection of the linear mixed effects model and the Cox model through a latent variable. Two special examples of this model are presented, namely, a joint model with shared random effects and a joint latent class model. In the thesis we focus on the joint latent class model. This model assumes an existence of latent classes in the population that we are not able to observe. Consequently, it is assumed that the longitudinal part and the survival part of the model are independent within one class. The main intention of this work is to transfer the model to the Bayesian framework and to discuss an estimation procedure of parameters using a Bayesian statistic. It consists of a definition of the model in the Bayesian framework, a discussion of prior distributions and the derivation of the full conditional distributions for all parameters of the model. The model's ability to...
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Testing independence in two-by-two tables
Obukhov, Andrey ; Omelka, Marek (advisor) ; Kulich, Michal (referee)
The main purpose of this work is to describe three well-known statistical tests of independence in two-by-two contingency tables. We will deeply study chi- squared test of independence, Fisher's exact test and Barnard's test and apply them on examples. Also we will describe, in general, categorical variables, which are often analysed using a multinomial distribution. At the end we will apply tests on the examples, using data simulated from a multinomial and binomial distribution. 1
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Testing independence in two-by-two tables
Obukhov, Andrey ; Omelka, Marek (advisor) ; Kulich, Michal (referee)
The main purpose of this work is to describe three well-known statistical tests of independence in two-by-two contingency tables. We will deeply study chi- squared test of independence, Fisher's exact test and Barnard's test and apply them on examples. Also we will describe, in general, categorical variables, which are often analysed using a multinomial distribution. At the end we will apply tests on the examples, using data simulated from a multinomial and binomial distribution. 1
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