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The method of re-weighting (calibration) in survey sampling
Michálková, Anna ; Omelka, Marek (advisor) ; Antoch, Jaromír (referee)
In this thesis, we study re-weighting when estimating totals in survey sampling. The purpose of re-weighting is to adjust the structure of the sample in order to comply with the structure of the population (with respect to given auxiliary variables). We sum up some known results for methods of the traditional desin-based approach, more attention is given to the model-based approach. We generalize known asymptotic results in the model-based theory to a wider class of weighted estimators. Further, we propose a consistent estimator of asymptotic variance, which takes into consideration weights used in estimator of the total. This is in contrast to usually recommended variance estimators derived from the design-based approach. Moreover, the estimator is robust againts particular model misspecifications. In a simulation study, we investigate how the proposed estimator behaves in comparison with variance estimators which are usually recommended in the literature or used in practice. 1
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Testing equivalence and noninferiority
Rychterová, Nela ; Antoch, Jaromír (advisor) ; Omelka, Marek (referee)
This master thesis deals with topics related to the task whether customers are able to recognize a difference between products. First, testing of equivalence and non-inferiority is discussed in detail. It is an important tool when verifying that two products are equivalent or that a new product is not substantially worse than a current product. Afterwards, Thurstone's approach is introduced as a way to evaluate the impact of a stimulus on human senses. Subsequently, using the previous chapters, there is a detailed discussion dealing with three standards wi- dely used in practice in the case when someone needs to apply sensory evaluation to verify whether customers are able to recognize a difference between products. In particular, these are duo-trio, triangle and paired comparison tests. There is a thorough explanation of their statistical base and the tests are compared accor- ding to their power. Furthermore, an approach based on the Thurstone's theory is introduced as an alternative to the standard methods. Moreover, this thesis introduces Saaty's approach to the estimation of a priority vector, which is a useful tool to compare, to order or to choose the best one from n objects. We also introduce another approach to estimation of a priority vector which is based on Saaty's idea. 1
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Neighborhood components analysis and machine learning
Hanousek, Jan ; Antoch, Jaromír (advisor) ; Maciak, Matúš (referee)
In this thesis we focus on the NCA algorithm, which is a modification of k-nearest neighbors algorithm. Following a brief introduction into classification algorithms we overview KNN algorithm, its strengths and flaws and what lead to the creation of the NCA. Then we discuss two of the most widely used mod- ifications of NCA called Fast NCA and Kernel (fast) NCA, which implements the so-called kernel trick. Integral part of this thesis is also a proposed algo- rithm based on KNN (/NCA) and Linear discriminant analysis titled TSKNN (/TSNCA), respectively. We conclude this thesis with a detailed study of two real life financial problems and compare all the algorithms introduced in this thesis based on the performance in these tasks. 1
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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.
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Transformation Models
Pejřimovský, Pavel ; Hušková, Marie (advisor) ; Antoch, Jaromír (referee)
This thesis deals with a finding ideal transformation which can model data well. We focus on transformations which we know up to a parametr. We need to estimate the parametr of the transformation. The main approach of study transformation is in linear regression and in nonparametric regression. In both cases we focus on estimating the transformation parametr and properties of this estimator such as consistency and asymptotic normality. We show in linear regression that the aprroach of least squares do not work properly. Instead of this we use a generalized moment method which can estimate parametr of transformation and also a regression coefficients. We show also a different solution for our problem in specific transformation called Box-Cox. For this situation we make a simulation study for estimators and standard deviations. The standard deviation are obtained by bootstrap method. In nonparametric regression we use profile likelihood to estimate transformation parametr. We also construct an estimator of density of error terms. In both cases we know the asymptotic distribution.
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Causes of Effects and Effects of Causes
Zemánková, Lucie ; Maciak, Matúš (advisor) ; Antoch, Jaromír (referee)
The thesis deals with an associative and causal relationship between two different random phenomena and presents basic statistical methods for investigation of these relationships. Firstly it focuses on demonstrating the association between phenomena and shows that finding a causal relation between phenomena requires appropriate randomization of the system or intervention in the system. After intervening in the system, it is no longer possible to observe all situations, so-called counterfactual observation, but the causal relationship can still be demonstrated using appropriate technical procedures and theoretical assumptions. The thesis further summarizes different ways of representation of causal structures, first by means of graphs, where basic methods of estimating the causal structure are presented, and later by structural equations that already capture the quantitative measure of causal relations.
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Ensemble Kalman filter on high and infinite dimensional spaces
Kasanický, Ivan ; Hlubinka, Daniel (advisor) ; Pannekoucke, Olivier (referee) ; Antoch, Jaromír (referee)
Title: Ensemble Kalman filter on high and infinite dimensional spaces Author: Mgr. Ivan Kasanický Department: Department of Probability and Mathematical Statistics Supervisor: doc. RNDr. Daniel Hlubinka, Ph.D., Department of Probability and Mathematical Statistics Consultant: prof. RNDr. Jan Mandel, CSc., Department of Mathematical and Statistical Sciences, University of Colorado Denver Abstract: The ensemble Kalman filter (EnKF) is a recursive filter, which is used in a data assimilation to produce sequential estimates of states of a hidden dynamical system. The evolution of the system is usually governed by a set of di↵erential equations, so one concrete state of the system is, in fact, an element of an infinite dimensional space. In the presented thesis we show that the EnKF is well defined on a infinite dimensional separable Hilbert space if a data noise is a weak random variable with a covariance bounded from below. We also show that this condition is su cient for the 3DVAR and the Bayesian filtering to be well posed. Additionally, we extend the already known fact that the EnKF converges to the Kalman filter in a finite dimension, and prove that a similar statement holds even in a infinite dimension. The EnKF su↵ers from a low rank approximation of a state covariance, so a covariance localization is required in...
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Estimation of probability distribution for censored data
Teichmannová, Zuzana ; Lachout, Petr (advisor) ; Antoch, Jaromír (referee)
In this thesis, we look into estimation of probability distribution for censored data. These data are not complete, because for some reason it was impossible to observe them all. We use the Kaplan-Meier estimator and study some of its properties. We also use the Nelson-Aalen estimator. In the end we make a compa- rison of these estimators with a naive estimator, which omits the censored data. The comparison is illustrated on two numerical examples where we can see the main differences in the accuracy of the estimators. We will see that it is better to include the censored data to our estimations. 1
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Big data - extraction of key information combining methods of mathematical statistics and machine learning
Masák, Tomáš ; Antoch, Jaromír (advisor) ; Maciak, Matúš (referee)
This thesis is concerned with data analysis, especially with principal component analysis and its sparse modi cation (SPCA), which is NP-hard-to- solve. SPCA problem can be recast into the regression framework in which spar- sity is usually induced with ℓ1-penalty. In the thesis, we propose to use iteratively reweighted ℓ2-penalty instead of the aforementioned ℓ1-approach. We compare the resulting algorithm with several well-known approaches to SPCA using both simulation study and interesting practical example in which we analyze voting re- cords of the Parliament of the Czech Republic. We show experimentally that the proposed algorithm outperforms the other considered algorithms. We also prove convergence of both the proposed algorithm and the original regression-based approach to PCA. vi
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