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Distance-based testing
Solnický, Radek ; Omelka, Marek (advisor) ; Komárek, Arnošt (referee)
When analyzing ecological data, one considers traditional multivariate techniques to be unsuitable. The use of dissimilarity coefficients and distance matrices is a way, how to solve this problem. In this work we present some of these coefficients and distance-based tests: Mantel test, several versions of ANOSIM and MRPP tests and distance-based test for homogeneity of multivariate dispersions. We focus on relationships among these tests and illustrate the use with an example. We also discuss the difficulties of interpretation of the results of these tests.
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Multivariate extreme value theory
Šiklová, Renata ; Mazurová, Lucie (advisor) ; Omelka, Marek (referee)
In this thesis we will elaborate on multivariate extreme value modelling, re- lated practical and theoretical aspects. We will mainly focus on the dependence models, the extreme value copulas in particular. Extreme value copulas effec- tively unify the univariate extreme value theory and the copula framework itself in a single view. We familiarize ourselves with both of them in the first two chapters. Those chapters present generalized extreme value distribution, gen- eralized Pareto distribution and Archimedean copulas, that are suitable for the multivariate maxima and the threshold exceedances description. These two top- ics will be addressed in the third chapter in detail. Taking into consideration rather practical focus of this thesis, we examine the methods of data analysis extensively. Furthermore, we will employ these methods in a comprehensive case study, that will aim to reveal the importance of extreme value theory application in the Catastrophe Insurance. 1
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Applications of EM-algorithm
Komora, Antonín ; Omelka, Marek (advisor) ; Kulich, Michal (referee)
EM algorithm is a very valuable tool in solving statistical problems, where the data presented is incomplete. It is an iterative algorithm, which in its first step estimates the missing data based on the parameter estimate from the last iteration and the given data and it does so by using the conditional expectation. In the second step it uses the maximum likelihood estimation to find the value that maximizes the logarithmic likelihood function and passes it along to the next iteration. This is repeated until the point, where the value increment of the logarithmic likelihood function is small enough to stop the algorithm without significant errors. A very important characteristic of this algorithm is its monotone convergence and that it does so under fairly general conditions. However the convergence itself is not very fast, and therefore at times requires a great number of iterations.
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Random walk
Baňasová, Barbora ; Omelka, Marek (advisor) ; Dostál, Petr (referee)
Random walk is a well-known mathematical model used in various scientific fields. The aim of this thesis is to explain and to show the relation between the basic characteristics of simple random walk. The paper summarizes theoretical knowledge concerning this mathematical model in terms of its symmetrical or asymmetrical version. It deals with the derivation of absorbing probabilities, probability of the first and repeated return to origin and clasification of simple random walk states. The final part presents random walk in a wider perspective as a martingale. The conditions under which a random walk equals a martingale are established as well. It is also shown how it is possible to apply this more general mathematical structure on the model of random walk.
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Properties of two-phase testing procedures
Krausová, Eliška ; Kulich, Michal (advisor) ; Omelka, Marek (referee)
In the present work we study properties of two-phase testing procedures which formally verify assumptions by performing some test ( first phase) and subsequently calculate a test statistic selected according to the results of the previous test (second phase). In the beginning we describe two-phase testing procedures and mention some literature, in which they are recommended. We try to derive formula for the combined level and power of the whole two-phase testing procedure. After that we illustrate their properties through simulation studies.
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Analysis of variance when the assumption of normality is violated
Kika, Vojtěch ; Omelka, Marek (advisor) ; Kulich, Michal (referee)
Attention is restricted to a method called Analysis of variance (ANOVA) that is used to compare expected values of several independent random samples. The clas- sic ANOVA theory with all its assumptions, including the assumption of normality, is presented at the beginning. Afterwards, an instance when the assumption of nor- mality of input data is violated is exemplified. The asymptotic distribution of test statistic under the hypothesis of the equality of the expected values is derived. The distribution is used to test the equality. Subsequently, it is shown that Tukey's range test and Scheffé's method of multiple comparison in case of non-normality could be used in the same way as for normal samples. The methods serve for compa- ring expected values of pairs of random samples. Thus, they can determine expected values which are different. Finally, a simulation study is presented which is to verify the proved theoretical results and to describe situations with data from non-normal distributions.
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Archimedean copulas
Vedyushenko, Anna ; Pešta, Michal (advisor) ; Omelka, Marek (referee)
The thesis deals with Archimedean copulas which are very popular nowadays due to easy construction and their appealing properties. At first it introduces the general definition of a copula and also shows its fundamental properties. After that the definition and the basic properties of an Archimedean copula are discussed. The paper also describes some of the commonly used families of Archi- medean copulas. Then several methods of parameter estimation for Archimedean copulas are shown. Finally, we make a study of two real datasets where the distri- bution of the data is estimated based on the procedures described in the thesis. 1
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