 

Modelling dependence between hydrological and meteorological variables measured on several stations
Turčičová, Marie ; Jarušková, Daniela (advisor) ; Hlávka, Zdeněk (referee)
Title: Modelling dependence between hydrological and meteorological variables measured on several stations Author: Bc. Marie Turčičová Department: Department of Probability and Mathematical Statistics Supervisor: Prof. RNDr. Daniela Jarušková CSc., Czech Technical University in Prague, Faculty of Civil Engineering, Department of Mathematics Abstract: The aim of the thesis is to explore the dependence of daily discharge averages of the Opava river on high daily precipitation values in its basin. Three methods are presented that can be used for analyzing the dependence between high values of random variables. Their application on the studied data is also given. First it is the taildependence coefficient that measures the dependence between high values of two continuous random variables. The model for the high quantiles of the discharge at a given precipitation value was first determined nonparametrically by quantile regression and then parametrically through the peaksoverthreshold (POT) method. Keywords: extremal dependence, taildependence coefficient, quantile regression, peaks over threshold method


Prediction of transformed time series
Polák, Tomáš ; Anděl, Jiří (advisor) ; Jarušková, Daniela (referee)
The aim of this thesis is to find prediction for nonlinear transformation of time series. First, under certain assumptions regarding the original time series, the autocovariance function and spectral density of the transformed time series are studied. General theorems are applied to concrete ARMA processes. Then general formulas for predictions of the transformed time series, which do not require knowledge of the autocovariance function of the transformed series nor its spectral density are presented. These formulas are applied to three concrete transformations and explicit formulas for ARMA processes are derived. Three types of predictions (optimal, naive and linear) are compared in the terms of proportional increase of mean square prediction error. Explicit formulas for ARMA processes are verified by a simulation.

 

Stochastical inference in the model of extreme events
Dienstbier, Jan ; Picek, Jan (advisor) ; Jurečková, Jana (referee) ; Jarušková, Daniela (referee)
Title: Stochastical inference in the model of extreme events Author: Jan Dienstbier Department/Institute: Department of probability and mathematical statistics Supervisor of the doctoral thesis: Doc. RNDr. Jan Picek, CSc. Abstract: The thesis deals with extremal aspects of linear models. We provide a brief explanation of extreme value theory. The attention is then turned to linear models Yn×1 = Xn×pβp×1 + En×1 with the errors Ei ∼ F, i = 1, . . . , n fulfilling the do main of attraction condition. We examine the properties of the regression quantiles of Koenker and Basset (1978) under this setting we develop theory dealing with extremal characteristics of linear models. Our methods are based on an approximation of the regression quantile process for α ∈ [0, 1] expanding older results of Gutenbrunner et al. (1993). Our result holds in [α∗ n, 1 − α∗ n] with a better rate of α∗ n → 0 than the other approximations described previously in the literature. Consecutively we provide an ap proximation of the tails of regression quantile. The approximations of the tails enable to develop theory of the smooth functionals, which are used to establish a new class of estimates of extreme value index. We prove T(F−1 n (1 − knt/n)) is consistent and asymp totically normal estimate of extreme for any T member of the class....


Testing Structural Changes Using Ratio Type Statistics
Peštová, Barbora ; Hušková, Marie (advisor) ; Prášková, Zuzana (referee) ; Jarušková, Daniela (referee)
Testing Structural Changes Using Ratio Type Statistics Barbora Peštová Charles University in Prague, Faculty of Mathematics and Physics, Department of Probability and Mathematical Statistics, Czech Republic Abstract of the doctoral thesis We deal with sequences of observations that are naturally ordered in time and assume various underlying stochastic models. These models are parametric and some of the parameters are possibly subject to change at some unknown time point. The main goal of this thesis is to test whether such an unknown change has occurred or not. The core of the change point methods presented here is in ratio type statistics based on maxima of cumulative sums. Firstly, an overview of thesis' starting points is given. Then we focus on methods for detecting a gradual change in mean. Consequently, procedures for detection of an abrupt change in mean are generalized by considering a score function. We explore the possibility of applying the bootstrap methods for obtaining critical values, while disturbances of the change point model are considered as weakly dependent. Procedures for detection of changes in parameters of linear regression models are shown as well and a permutation version of the test is derived. Then, a related problem of testing a change in autoregression parameter is studied....


Testing Structural Changes Using Ratio Type Statistics
Peštová, Barbora ; Hušková, Marie (advisor) ; Prášková, Zuzana (referee) ; Jarušková, Daniela (referee)
Testing Structural Changes Using Ratio Type Statistics Barbora Peštová Charles University in Prague, Faculty of Mathematics and Physics, Department of Probability and Mathematical Statistics, Czech Republic Abstract of the doctoral thesis We deal with sequences of observations that are naturally ordered in time and assume various underlying stochastic models. These models are parametric and some of the parameters are possibly subject to change at some unknown time point. The main goal of this thesis is to test whether such an unknown change has occurred or not. The core of the change point methods presented here is in ratio type statistics based on maxima of cumulative sums. Firstly, an overview of thesis' starting points is given. Then we focus on methods for detecting a gradual change in mean. Consequently, procedures for detection of an abrupt change in mean are generalized by considering a score function. We explore the possibility of applying the bootstrap methods for obtaining critical values, while disturbances of the change point model are considered as weakly dependent. Procedures for detection of changes in parameters of linear regression models are shown as well and a permutation version of the test is derived. Then, a related problem of testing a change in autoregression parameter is studied....


Statistical analysis of long hydrological and climatological data series
Ledvinka, Ondřej ; Kotvalt, Václav (advisor) ; Fošumpaur, Pavel (referee) ; Jarušková, Daniela (referee)
2 Abstract Although entitled more generally, the thesis deals primarily with trend analyses applied to the instrumental records of hydrometeorological variables measured over the territory of Czechia, sometimes specializing in particular river basins extending to the neighbouring countries such as Germany and Poland. The hydrological data (namely discharge or spring yield) predominate, to which also the climatological ones such as precipitation, snow cover depth and air temperature can be added since they significantly influence the water cycle in Czechia. Under ideal circumstances, the trend analysis might answer the question whether frequently discussed climate change has its important role in the development of quantitative water resources characteristics. However, due to the record length, starting usually in the 1960s here, it is really hard to conclude if the discovered patterns are a result of deterministic relationships or if they are rather of random origins (e.g. they are a part of climate fluctuation cycle). Trends were identified mainly using the MannKendall test and its modifications whose rationales are studied here in detail. Special focus was on longterm persistence that, besides shortterm persistence, may adversely influence the variance of the test statistic. The detection of longterm...


Multivariate extreme value models and their application in hydrology
Drápal, Lukáš ; Jarušková, Daniela (advisor) ; Hušková, Marie (referee)
Present thesis deals with the multivariate extreme value theory. First, concepts of modelling block maxima and threshold excesses in the univariate case are reviewed. In the multivariate setting the point process approach is chosen to model dependence. The dependence structure of multivariate extremes is provided by a spectral measure or an exponent function. Models for asymptotically dependent variables are provided. A construction principle from Ballani and Schlather (2011) is discussed. Based on this discussion the pairwise beta model introduced by Cooley et al. (2010) is modified to provide higher flexibility. Models are applied to data from nine hydrological stations from northern Moravia previously analysed by Jarušková (2009). Usage of the new pairwise beta model is justified as it brought a substantial improvement of loglikelihood. Models are also compared with Bayesian model selection introduced by Sabourin et al. (2013). Powered by TCPDF (www.tcpdf.org)


Permutation Tests of Statistical Hypotheses
Veselý, Zdeněk ; Jurečková, Jana (advisor) ; Jarušková, Daniela (referee)
Title: Permutation Tests of Statistical Hypotheses Author: Zdeněk Veselý Department: Department of Probability and Mathematical Statistics Supervisor: prof. RNDr. Jana Jurečková DrSc., Department of Probability and Mathematical Statistics Abstract: This thesis presents permutation tests concept. Permutation test is demonstrated as response to testing problems where it is inconvenient to make any deeper presumptions on data probability distribution. For some of these problems it is even the only exact solution. The construction of permutation test is described in the thesis as well as approach to search of the most powerful tests to specific alternatives. In the second part of the thesis there are comparisons of powers of parametric, permutation and rank test using simulations. The result is that power of parametric and permutation test are very similar most of the times and that confirms that permutation tests are useful tool for praxis. Keywords: Permutation tests, Exact tests, Hypothesis testing, Power of tests
