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Skorokhod's representation theorem
Paulová, Nikol ; Nagy, Stanislav (advisor) ; Hlubinka, Daniel (referee)
We know that almost sure convergence of random variables implies their convergence in distribution. Are there any conditions that would allow us to obtain al- most sure convergence from convergence in distribution? The Skorokhod representation theorem answers this question. We can find representations of the weakly convergent random variables such that they converge almost surely. First, we introduce the needed definitions and lemmata. The main focus of the second chapter is the Skorokhod repre- sentation theorem on the real numbers, its proof and some auxiliary assertions are given. In the final third chapter, we deal with the applications of the theorem to prove some well known and commonly used theorems and to prove some less known theorems. 1
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Analysis of shape of random functions
Fürst, Matouš ; Nagy, Stanislav (advisor) ; Maciak, Matúš (referee)
Registration of functional data is a part of functional data analysis which focuses on the transformation of a sample of functions such that the shapes of these functions are aligned and undesired variation between them removed. This thesis describes the general theory of functional registration and compares two selected methods. These two methods are thoroughly described, with a focus on their theoretical properties. Moreover, a modification of one of the methods is proposed. A comparison is performed on both real and simulated datasets, and an extension for registration into multiple groups is utilized. 1
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The Depth of Functional Data.
Nagy, Stanislav ; Hlubinka, Daniel (advisor) ; Omelka, Marek (referee)
The depth function (functional) is a modern nonparametric statistical analysis tool for (finite-dimensional) data with lots of practical applications. In the present work we focus on the possibilities of the extension of the depth concept onto a functional data case. In the case of finite-dimensional functional data the isomorphism between the functional space and the finite-dimensional Euclidean space will be utilized in order to introduce the induced functional data depths. A theorem about induced depths' properties will be proven and on several examples the possibilities and restraints of it's practical applications will be shown. Moreover, we describe and demonstrate the advantages and disadvantages of the established depth functionals used in the literature (Fraiman-Muniz depths and band depths). In order to facilitate the outcoming drawbacks of known depths, we propose new, K-band depth based on the inference extension from continuous to smooth functions. Several important properties of the K-band depth will be derived. On a final supervised classification simulation study the reasonability of practical use of the new approach will be shown. As a conclusion, the computational complexity of all presented depth functionals will be compared.
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Modelování hry tenis
Tsapparellas, Kyriakos ; Lachout, Petr (advisor) ; Nagy, Stanislav (referee)
This thesis introduces three methods/models in forecasting the winner of a tennis match, analyzes them, studies their effectiveness under certain circumstances and detects their advantages or disadvantages using sufficient amount of previous data and results. Moreover, a personal fourth model is being introduced and tested which aims to give an answer to a question posted by Franc Klaassen and Jan Magnus, whether the forecast error can be reduced by not assuming that points during a match are independent and identically distributed and allows changes to happen as the match unfolds. If there is an actual improvement it will be showed and discussed subsequently.
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Functional ANOVA
Dolník, Viktor ; Dvořák, Jiří (advisor) ; Nagy, Stanislav (referee)
We introduce the concept of functional data and the problem of functional analysis of variance, which differs from the univariate case in the fact that random functions, not random variables, are the subject of comparison. We continue by deriving an asymptotic test for functional one-way ANOVA from the elementary univariate F-test. We describe the simulation envelope test, whose global version suffers from the multiple comparisons problem. Then, an ordering is defined, based on which we create the rank envelope test, a stronger alternative to the simulation envelope test. We also describe how the rank test can be interpreted graphically. Using the rank envelope test, we devise another test for functional one-way ANOVA, which is also graphically interpretable and thus does not need a post-hoc analysis to identify which groups caused rejection of the null hypothesis. We compare the one-way ANOVA tests on a real-case study and a simulation study. 1
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Chebyshev inequality and some its modifications
Drabinová, Adéla ; Anděl, Jiří (advisor) ; Nagy, Stanislav (referee)
In the presented thesis we describe some improvements of Chebyshev inequa- lity. In the first chapter we introduce inequalities for random variables with uni- modal distributions. We prove Gauss and Camp-Meidell inequality and we deduce Vysochanskii-Petunin inequality. We describe inequalities for variables with mode 0 and with unspecified mode. In the second chapter we consider constants C(r), for which the approximations are the best. We are interested in finding optimal parameter r or its approximation. In the third chapter we state inequalities from the first chapter for specific distributions, calculation of their constants, appli- cations and graphic presentations of the results. 1
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Robust linear regression
Rábek, Július ; Maciak, Matúš (advisor) ; Nagy, Stanislav (referee)
Regression analysis is one of the most extensively used statistical tools applied across different fields of science, with linear regression being its most well-known method. How- ever, the traditional procedure to obtain the linear model estimates, the least squares approach, is highly sensitive to even slight departures from the assumed modelling frame- work. This is especially pronounced when atypical values occur in the observed data. This lack of stability of the least squares approach is a serious problem in applications. Thus, the focus of this thesis lies in assessing the available robust alternatives to least squares estimation, which are not so easily affected by any outlying values. First, we introduce the linear regression model theory and derive the least squares method. Then, we char- acterise different types of unusual observations and outline some fundamental robustness measures. Next, we define and examine the robust alternatives to the classical estimation in the linear regression models. Finally, we conduct a comprehensive simulation study comparing the performance of robust methods under different scenarios. 1
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Symmetry of random vectors
Říha, Adam ; Nagy, Stanislav (advisor) ; Hušková, Marie (referee)
In this thesis we introduce the spherical, central, angular, halfspace and regression symmetry of random vectors and their measures. Firstly we deal with their mutual relations and equivalent expressions. We also study the uniqueness of the center of individual symmetries and other interesting properties. Then we define the halfspace, projection, spatial and regression multidimensional median and show their properties. Finally we look at the relationships between these medians and symmetric distributions. 1
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Depth of variance matrices
Brabenec, Tomáš ; Nagy, Stanislav (advisor) ; Hlubinka, Daniel (referee)
The scatter halfspace depth is a quite recently established concept which extends the idea of the location halfspace depth for positive definite matrices. It provides an interest- ing insight into the problem of suitability quantification of a matrix for the description of the covariance structure of the multivariate distribution. The thesis focuses on the investigation of theoretical properties of the depth for both general and more specific probability distributions which can be used for data analysis. It turns out that the es- timators of scatter parameters based on the empirical scatter depth are quite effective even under relatively weak assumptions. These estimators are useful especially for dealing with a sample containing outliers or contaminating observations. 1
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