
Zero inflated Poisson model
Veselý, Martin ; Komárek, Arnošt (advisor) ; Hlávka, Zdeněk (referee)
This paper deals with the zeroinflated Poisson distribution. First the Poisson model is defined and generalized to a zeroinflated model. The basic properties of this generalized model are derived. After wards the basics of the method of moments and the maximum likelihood method are described. Both of these are used to derive parameter estimates of such distribution. The feasibility of calculating the distribution of moment method estimates is analyzed. Then the asymptotic distribution of maximum likelihood estimates is derived and used to create confidence intervals. In the last chapter a numeric si mulation of the derived asymptotic properties is performed. Special attention is paid to situations where regularity conditions are not met. 1


Boxplot for multivariate data
Brabenec, Tomáš ; Nagy, Stanislav (advisor) ; Hlávka, Zdeněk (referee)
We will introduce three methods of extension of the classical Tukey's Boxplot for multivariate data. These are the Rangefinder, the Relplot and the Bagplot. To implement the methods, we will need the notions like Mahalanobis distance, elliptically symmetric distributions and halfspace depth. A big part of the thesis is focused on the construction of the Relplot and the Bagplot. We will also discuss, how do these methods detect outliers and what are their advantages and disadvantages. This work contains many examples and illustrating images. 1


Statistical machine learning with applications in music
Janásková, Eliška ; Večeř, Jan (advisor) ; Hlávka, Zdeněk (referee)
The aim of this thesis is to review the current state of machine learning in music composition and to train a computer on Beatles' songs using research project Magenta from the Google Brain Team to produce its own music. In order to explore the qualities of the generated music more thoroughly, we restrict our selves to monophonic melodies only. We train three deep learning models with three different configurations (Basic, Lookback, and Attention) and compare generated results. Even though the generated music is not as interesting as the original Beatles, it is quite likable. According to our analysis based on musically informed metrics, generated melodies differ from the original ones especially in lengths of notes and in pitch differences between consecutive notes. Generated melodies tend to use shorter notes and higher pitch differences. In theoretical background, we cover the most commonly used machine learning algorithms, introduce neural networks and review related work of music generation. 1


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.


ksample problem with ordered alternative
Nováková, Martina ; Hlávka, Zdeněk (advisor) ; Pešta, Michal (referee)
In this thesis we deal with ksample problem with ordered alternative. At the beginning of the thesis isotonic regression is introduced. We use isotonic regression for maximum likelihood estimation of ordered parameters. In the second chapter, we describe the χ2 and E 2 tests that use the knowledge of isotonic regression and are based on the likelihood ratio. The exact null hypothesis distributions of their test statistics are derived in detail. The onesided studentized range test is also further described. At the end of the thesis, we show the use of the E 2 test on the real data. 1


Ratio estimators
Klyuchevskiy, Iakov ; Hlávka, Zdeněk (advisor) ; Antoch, Jaromír (referee)
The aim of the bachelor thesis is to estimate the incidence of fractures in women from 0 to 20 years in the Czech Republic. In the introductory chapter we will introduce the concept of incidence and show the statistical data that we will continue to work with. In the second and third chapters we define statistical models for estimating the incidence and also the unit estimation by which we estimate the incidence, we will examine its properties. In the fourth chapter, we will show the real data to estimate the incidence of fractures in women for each age category.


Prediction error for mixed models
Šlampiak, Tomáš ; Komárek, Arnošt (advisor) ; Hlávka, Zdeněk (referee)
A Linear mixedeffects model (LME) is one of the possible tools for longitudinal or groupdependent data. This thesis deals with evaluating of prediction error in LME. Firstly, it is derived the mean square error of prediction (MSEP) by direct calculation. Then the covariance penalty method and crossvalidation is presented for evaluation of MSEP in LME. Further, it is shown how Akaike information criterion (AIC) can be used in mixedeffects models. Because of the model's properties two types of AIC are distinguished  marginal and conditional one. Subsequently, the procedures of AIC's calculation and its basic asymptotic properties are described. Finally, the thesis contains simulation study of behaviour of marginal and conditional AIC with the goal to choose the right variance structure of random effects. It turns out that the marginal criterion tends to select models with smaller number of random effects than conditional criterion.


Joinpoint Regression
Lain, Michal ; Maciak, Matúš (advisor) ; Hlávka, Zdeněk (referee)
The theme of this thesis is the joinpoint regression, the description of model, its properties and its construction. We are interested in methods of estimating parameters. We show practical use of the model. In the first chapter we define the model, we describe alternative forms and properties. In the second chapter we focus on estimating parameters of model. We briefly mention of Hudson method, profile likelihood, grid search and LASSO. We mention likelihood ratio for testing hypotheses about values of parameters. The third chapter deals with comparison of models by number of break points by permutation tests and information cri terions. In the fourth chapter we deal with practical examples. We show diverse application of the model. We compare methods using simulations and show model application. 1


Parameter estimation of gamma distribution
Zahrádková, Petra ; Kulich, Michal (advisor) ; Hlávka, Zdeněk (referee)
It is wellknown that maximum likelihood (ML) estimators of the two parame ters in a Gamma distribution do not have closed forms. The Gamma distribution is a special case of a generalized Gamma distribution. Two of the three likeli hood equations of the generalized Gamma distribution can be used as estimating equations for the Gamma distribution, based on which simple closedform estima tors for the two Gamma parameters are available. Intuitively, performance of the new estimators based on likelihood equations should be close to the ML estima tors. The study consolidates this conjecture by establishing the asymptotic beha viours of the new estimators. In addition, the closedforms enable biascorrections to these estimators. 1


Multiple comparison with controls
Sychova, Maryna ; Hlávka, Zdeněk (advisor) ; Komárek, Arnošt (referee)
The main theme of the diploma thesis is description of multiple comparison methods, which are used to compare pairs of means or medians. At the beggining we define multiple testing and describe methods that control the probability of first type error at level α. The Šidák method and the prerequi sites required for its use are described in detail. The work also includes a brief description of analysis of variance and an overview of several methods of multiple comparison. Additionally, the method of multiple comparison with control, its modifications and practical implementation is presented.
