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Geometric approach to the estimation of scatter
Bodík, Juraj ; Nagy, Stanislav (advisor) ; Antoch, Jaromír (referee)
In this thesis we describe improved methods of estimating mean and scatter from multivariate data. As we know, the sample mean and the sample variance matrix are non-robust estimators, which means that even a small amount of measurement errors can seriously affect the resulting estimate. We can deal with that problem using MCD estimator (minimum covariance determinant), that finds a sample variance matrix only from a selection of data, specifically those with the smallest determinant of this matrix. This estimator can be also very helpful in outlier detection, which is used in many applications. Moreover, we will introduce the MVE estimator (minimum volume ellipsoid). We will discuss some of the properties and compare these two estimators.
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Edgeworth expansion
Dzurilla, Matúš ; Omelka, Marek (advisor) ; Nagy, Stanislav (referee)
This thesis is focused around Edgeworths expansion for aproximation of distribution for parameter estimation. Aim of the thesis is to introduce term Edgeworths expansion, its assumptions and terminology associeted with it. Afterwords demonstrate process of deducting first term of Edgeworths expansion. In the end demonstrate this deduction on examples and compare it with different approximations (mainly central limit theorem), and show strong and weak points of Edgeworths expansion.
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Principal components
Zavadilová, Anna ; Hlávka, Zdeněk (advisor) ; Nagy, Stanislav (referee)
This thesis presents principal components as a useful tool for data dimensio- nality reduction. In the first part, the basic terminology and theoretical properties of principal components are described and a biplot construction is derived there as well. Besides, heuristic methods for a choice of the optimum number of prin- cipal components are summarised there. Subsequently, asymptotical properties of sample eigenvalues of covariance and white Wishart matrices are described and cases of equality of some eigenvalues are distinguished at the same time. In the second part of the thesis, asymptotic distribution of the largest eigenva- lue of white Wishart matrices is described, completed with graphic illustrations. A test of the number of significant eigenvalues is suggested on the basis of this limiting distribution, and the connection of this test to the number of suitable principal components is presented. The final part of the thesis provides an over- view of advanced computational methods for the choice of an adequate number of principal components. The thesis is completed with graphical illustrations and a simulation study using Wolfram Mathematica and R.
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Calibration Estimators in Survey Sampling
Klička, Petr ; Omelka, Marek (advisor) ; Nagy, Stanislav (referee)
V této práci se zabýváme odhady populačního úhrnu s využitím pomoc- ných informací. V práci je popsán obecný regresní odhad a předpoklady, za kterých je splněna asymptotická normalita tohoto odhadu. Dále jsou zde po- psány kalibrační odhady a zmínka o jejich asymptotické ekvivalenci s obec- ným regresním odhadem. Odvozené závěry aplikujeme na data z RADIO- PROJEKTu a porovnáme je s výsledky získanými společnostmi, které tento projekt realizovali. Na závěr pomocí simulací porovnáme skutečné pravdě- podobnosti pokrytí interval· spolehlivosti pro populační úhrn spočítané na základě teorie uvedené v této práci a na základě metod společností realizu- jících RADIOPROJEKT. 1
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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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Statistical Depth for Functional Data
Nagy, Stanislav ; Hlubinka, Daniel (advisor) ; Claeskens, Gerda (referee) ; Hušková, Marie (referee)
Statistical data depth is a nonparametric tool applicable to multivariate datasets in an attempt to generalize quantiles to complex data such as random vectors, random functions, or distributions on manifolds and graphs. The main idea is, for a general multivariate space M, to assign to a point x ∈ M and a probability distribution P on M a number D(x; P) ∈ [0, 1] characterizing how "centrally located" x is with respect to P. A point maximizing D(·; P) is then a generalization of the median to M-valued data, and the locus of points whose depth value is greater than a certain threshold constitutes the inner depth-quantile region corresponding to P. In this work, we focus on data depth designed for infinite-dimensional spaces M and functional data. Initially, a review of depth functionals available in the literature is given. The emphasis of the exposition is put on the unification of these diverse concepts from the theoretical point of view. It is shown that most of the established depths fall into the general framework of projection-driven functionals of either integrated, or infimal type. Based on the proposed methodology, characteristics and theoretical properties of all these depths can be evaluated simultaneously. The first part of the work is devoted to the investigation of these theoretical properties,...
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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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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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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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