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Cluster analysis for functional data
Zemanová, Barbora ; Komárek, Arnošt (advisor) ; Hušková, Marie (referee)
In this work we deal with cluster analysis for functional data. Functional data contain a set of subjects that are characterized by repeated measurements of a variable. Based on these measurements we want to split the subjects into groups (clusters). The subjects in a single cluster should be similar and differ from subjects in the other clusters. The first approach we use is the reduction of data dimension followed by the clustering method K-means. The second approach is to use a finite mixture of normal linear mixed models. We estimate parameters of the model by maximum likelihood using the EM algorithm. Throughout the work we apply all described procedures to real meteorological data.
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Methods for Analyzing Change From Baseline to Final Assessment
Pekařová, Lucie ; Kulich, Michal (advisor) ; Hušková, Marie (referee)
In this thesis, we analyze treatment effect estimate in randomized clinical studies. Treatment effect estimates are constructed on the basis of three models. The first part of this thesis is about the behaviour of these estimates when the treatment effects vary with patients. We find out that all types of estimates are consistent and we derived their asymptotic distribution. The estimates are compared by their asymptotic variances. The theoretical conclusions are confirmed by a simulation study. The second part describes the case where measurements of baseline and final values contain an error. Two estimates are analyzed. We find out that both estimates are consistent. We derive their asymptotic distribution and compare their variances.
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Recursive procedures for detection of changes
Chochola, Ondřej ; Hušková, Marie (advisor) ; Černíková, Alena (referee)
In the thesis we study a sequential monitoring scheme for detecting a change in variance. We assume to have a stable historical period of length m. The goal is to propose tests with asymptotically small probability of type I error and power 1 as m tends to infinity. Two such procedures were proposed. The first uses estimates of variance from the historical period, the second uses recursive estimates. The distribution under the null hypothesis and also under the alternative hypothesis was derived for both test statistics. Furthermore a simulation study for of the finite sample performance of the monitoring schemes was conducted.
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Models of changes in autoregressive sequences
Pečánka, Jakub ; Hušková, Marie (advisor) ; Hlávka, Zdeněk (referee)
This thesis deals with the detection of change in the structure of an autoregressive time series. In the first part of the thesis we provide an overview of the main results concerning the theory of autoregressive processes (Chapter 1) and a general theory of the maximum likelihood approach towards the change point problem (Chapter 2). The second and main part of the thesis (Chapter 3) deals with various approaches to the CPP applied on the autoregressive processes and provides a comprehensive proof of a theorem that was previously published by Hušková et al. (2007a) only with a sketched proof. A third part of the thesis contains a computer simulation study of the performance of the studied statistics (Chapter 4). Two appendices contain most of our proofs and also some general results of probability theory and statistics that were used in the thesis.
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Flexibility, Robustness and Discontinuities in Nonparametric Regression Approaches
Maciak, Matúš ; Hušková, Marie (advisor) ; Hlávka, Zdeněk (referee) ; Horová, Ivanka (referee)
Thesis title: Flexibility, Robustness and Discontinuity in Nonparametric Regression Approaches Author: Mgr. Matúš Maciak, M.Sc. Department: Department of Probability and Mathematical Statistics, Charles University in Prague Supervisor: Prof. RNDr. Marie Hušková, DrSc. huskova@karlin.mff.cuni.cz Abstract: In this thesis we focus on local polynomial estimation approaches of an unknown regression function while taking into account also some robust issues like a presence of outlying observa- tions or heavy-tailed distributions of random errors as well. We will discuss the most common method used for such settings, so called local polynomial M-smoothers and we will present the main statistical properties and asymptotic inference for this method. The M-smoothers method is especially suitable for such cases because of its natural robust flavour, which can nicely deal with outliers as well as heavy-tailed distributed random errors. Another important quality we will focus in this thesis on is a discontinuity issue where we allow for sudden changes (discontinuity points) in the unknown regression function or its derivatives respectively. We will propose a discontinuity model with different variability structures for both independent and dependent random errors while the discontinuity points will be treated in a...
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