
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.


Applications of bootstrap methods to time series
Baumová, Tereza ; Prášková, Zuzana (advisor) ; Hlávka, Zdeněk (referee)
Práce se vìnuje studiu variant metody bootstrap vhodných pro vy¹etøování vlastností autoregresních procesù s náhodnými koe cienty. Ètenáø je nejprve se známen s pùvodní metodou bootstrap navr¾enou pro nezávislé stejnì rozdìlené náhodné velièiny a se základními variantami této metody bì¾nì pou¾ívanými pro analýzu èasových øad. Poté je pøedstaven autoregresní proces s náhodnými koe  cienty øádu p (RCA(p)). Jsou popsány základní vlastnosti tohoto procesu a blí¾e prozkoumány vlastnosti procesu RCA(1). V dal¹í èásti jsou uvedeny varianty me tody bootstrap, které jsou v pøípadì procesu RCA(1) konzistentní, a pro metodu wild bootstrap je odvozena konzistence pro proces RCA(2). V poslední kapitole jsou na simulovaných datech ovìøeny vlastnosti popsaných metod. 1


Median in some statistical methods
Bejda, Přemysl ; Cipra, Tomáš (advisor) ; Hlávka, Zdeněk (referee) ; Víšek, Jan Ámos (referee)
Median in some statistical methods Abstract: This work is focused on utilization of robust properties of median. We propose variety of algorithms with respect to their breakdown point. In addition, other properties are studied such as consistency (strong or weak), equivariance and computational complexity. From practical point of view we are looking for methods balancing good robust properties and computational complexity, be cause these two properties do not usually correspond to each other. The disser tation is divided to two parts. In the first part, robust methods similar to the exponential smoothing are suggested. Firstly, the previous results for the exponential smoothing with ab solute norm are generalized using the regression quantiles. Further, the method based on the classical sign test is introduced, which deals not only with outliers but also detects change points. In the second part we propose new estimators of location. These estimators select a robust set around the geometric median, enlarge it and compute the (iterative) weighted mean from it. In this way we obtain a robust estimator in the sense of the breakdown point which exploits more information from observations than standard estimators. We apply our approach on the concepts of boxplot and bagplot. We work in a general normed vector...


Model for shortterm forecasting of photovoltaic energy production
Kotlorz, Lukáš ; Pelikán, Emil (advisor) ; Hlávka, Zdeněk (referee)
Nowadays, electricity production from photovoltaics power plants is becoming important increasingly. In order to set production to other power plants, it is necessary to predict the generation of electricity from these sources. The thesis is mainly devoted to models for shortterm prediction, which is based on weather forecast. The models were designated by beta regression and linear regression with transformed explanatory variable. One part of thesis is devoted to Clear sky model, which is used to estimated the maximum possible production at given hour. 1
