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Locally adaptive splines
Dian, Patrik ; Maciak, Matúš (advisor) ; Hlávka, Zdeněk (referee)
The aim of this thesis is to present some semiparametric methods used for estimating an unknown regression function. All approaches are based on a minimization of an ob- jective function, which is formulated as the sum of a loss and a penalization function. We present a reformulation of unsolvable problems due to infinite dimensionality as problems which are of finite dimensions, in the form of classical ridge or lasso regression. A crucial part in these methods plays a penalization parameter used to obtain a balance between the bias and the variability of the estimate. Techniques devoted to finding the optimal values of the penalization parameter are shown. Finally, applications of the mentioned methods on two simulated datasets are displayed. We focus on the local adaptivity of in- dividual approaches, as well as the computational intensity, which we shall compare. In addition, we analyse a new method proposed to find the optimal value of the penali- zation parameter. 1
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Sparse regression model
Bessisso, Samir ; Maciak, Matúš (advisor) ; Mizera, Ivan (referee)
In sparse linear regression models, the effect of the majority of explanatory variables on the conditional expected value of the response is null. The estimates produced by the adaptive lasso method are sparse and possess the oracle properties; meaning they provide asymptotically accurate identification of null elements within the regression coefficients vector while also being √ n-consistent estimates of the non-zero regression coefficients. In the first chapter of this diploma thesis, we revise the properties of the ordinary least squares estimate and we present arguments favoring the adoption of biased regularized estimates. In the second and third chapters, we examine the lasso and adaptive lasso methods. In the fourth and concluding chapter of this diploma thesis, we discuss the challenges of the post-model-selection inference and we derive a method for constructing exact confidence intervals in a linear regression model whose set of the explanatory vari- ables was chosen as a support of the lasso estimate. 1
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Wild binary segmentation
Lasota, Jakub ; Pešta, Michal (advisor) ; Maciak, Matúš (referee)
The goal of this work is to describe some (multiple) change-point detection methods that aim to estimate the total number and locations of structural changes in the data. From the variety of all change-point detection methods, only binary segmentation and wild binary segmentation are explained. To enhance the understanding, the work contains a few illustrative examples that try to show the strengths and weaknesses of each method. The practical part of the work focuses on using and comparing both methods with various parameter choices on daily logarithmic returns of the Zoom Video Communications stock. 1
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Bayesian classification and regression trees
Dvořák, Martin ; Antoch, Jaromír (advisor) ; Maciak, Matúš (referee)
The bachelor's thesis is devoted to classification and regression trees, their con- struction, and interpretation. In the first part, the reader gets acquainted with the structure of decision trees, basic definitions, and methodology. In the second part, more advanced and efficient methods for creating such trees using a Bayesian approach to the whole problem are presented. The last part of the work is focused on a practical task, where knowledge from this work is used. The entire text is accompanied by pictures, explanations, and derivations to make it easier for the reader to understand the whole problem in more depth. The thesis Bayesian classification and regression trees can serve all those interested who want to learn more about the issue of decision trees. 1
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Theoretical and empirical quantiles and their use for prediction interval construction
Šimičák, Jakub ; Maciak, Matúš (advisor) ; Omelka, Marek (referee)
The purpose of the bachelor thesis is to introduce the reader to two approaches to the construction of prediction intervals. The first procedure assumes a probabilistic model and leads to a frequentist prediction interval that uses the relevant theoretical quantiles of probability distributions. The second procedure assumes no probabilistic model and leads to a conformal prediction interval that uses empirical quantiles of the relevant random sample. In the course of the paper, both approaches will be derived in general terms and then illustrated with concrete examples. The thesis also includes a simulation study comparing the empirical coverage of frequentist and conformal prediction inter- vals for random selections from different distributions. 1
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Classification based on mixture models
Janečková, Lucie ; Komárek, Arnošt (advisor) ; Maciak, Matúš (referee)
This thesis deals with classification based on mixture models, mainly on models finite normal. At first, there are introduced basic definitions and characteristics of finite mix- tures. Afterwards there is described the maximum likelihood method and her obstacles in context of finite mixtures, which we are using for unknown parameters estimation. Then there is described EM algorithm, that is used to obtain the maximum likelihood estimator and there are calculated the formulae for one iteration of EM algorithm. In the last part there is shown, how can finite normal mixtures be used for classification. 1
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James-Stein Estimator
Novotný, Vojtěch ; Maciak, Matúš (advisor) ; Nagy, Stanislav (referee)
In this thesis, we will introduce the James-Stein estimator, we will study its properties and compare them with other estimation methods. We will explain, what is admissibility of an estimator and figure out if our estimators are admissable. We will introduce the Bayes estimators and the empirical Bayes estimators. Furthermore, we will analyse how their properties can be examined differently. Finally, we will perform a simulation study and we will compare the quality of estimations on its results and see if they follow the explained theory. Using this, we will try to decide when is using the James-Stein estimator appropriate. 1
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Statistical tests of normality
Krupa, Tomáš ; Maciak, Matúš (advisor) ; Omelka, Marek (referee)
The aim of this paper is to present the well-known normality tests used in practice and to compare them. The first chapter consists of the basic concepts and properties of the nor- mal distribution. In the second chapter 6 normality tests are treated, namely Kolmogorov- Smirnov, Lilliefors, Shapiro-Wilk, Anderson-Darling, D'Agostino-Pearson and Jarque- Bera. For each test, test statistic and shape of critical region are given, among others. The third chapter, with empirical study, contains two parts. In the first part, nature of the study is briefly explained and level of significance declared by tests is empirically-checked. In the second part, power of tests is empirically compared against various alternatives and the results are discussed. 1
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Likelihood based estimation
Březinová, Eva ; Maciak, Matúš (advisor) ; Kříž, Pavel (referee)
In this thesis we will describe the maximum likelihood method, method of estima- ting unknown parameters that determine the probability distribution of the observed data. We will also introduce other methods derived from the likelihood. We focus pri- marily on a quasi-likelihood and a pseudo-likelihood approach. Then we briefly describe profile likelihood, empirical likelihood, and conditional likelihood. The thesis includes a simulation study which compares the quality of the estimators based on the maximum likelihood and the quasi-likelihood or the maximum likelihood and the pseudo-likelihood using the mean squared error. 1
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Conformal prediction
Krynická, Michaela ; Maciak, Matúš (advisor) ; Týbl, Ondřej (referee)
The main objective of this work is to formalize the concept of conformal prediction. This robust, nonparametric method allows the construction of an accurate prediction interval at a specified level, for which it is sufficient to assume that the input data are independent, equally distributed. In the context of random sampling from a one- dimensional continuous distribution, we expose the theoretical foundations of the method. Subsequently, we define the key concept of the degree of nonconformance and present the algorithmic design, first for random sampling and then in the context of regression ana- lysis. At the end of the work, we compare the reliability and effectiveness of conformal prediction with a specific frequency method on randomly generated data. 1
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