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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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Quantile curves
Michl, Marek ; Hlávka, Zdeněk (advisor) ; Hlubinka, Daniel (referee)
Modeling of quantile curves is a common problem across various fields in today's practice. The topic of this thesis is estimating quantile curves in case of two-sample gradual change. That is, when a relationship between two continuous variables in two samples is of interest, where the relationship is the same for both samples until a certain value of the explanatory variable. From that point on the relationship can differ. The result of this thesis is a procedure for estimating quantile curves, which fulfill this concept. 1
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Statistical inference in multivariate distributions based on copula models
Kika, Vojtěch ; Omelka, Marek (advisor) ; Hlubinka, Daniel (referee)
Diploma thesis abstract Thesis title: Statistical inference in multivariate distributions based on copula models Author: Vojtěch Kika This diploma thesis aims for statistical inference in copula based models. Ba- sics of copula theory are described, followed by methods for statistical inference. These are divided into three main groups. First of them are parametric methods for copula parameter estimation which assume fully parametric structure, thus for both joint and marginal distributions. The second group consists of semi- parametric methods for copula parameter estimation which, unlike parametric methods, do not require parametric structure for marginal distributions. The last group describes goodness-of-fit tests used for testing the hypothesis that consi- dered copula belongs to some specific copula family. The thesis is accompanied by a simulation study that investigates the dependence of the observed coverage of the asymptotic confidence intervals for copula parameter on the sample size. Pseudolikelihood method was chosen for the simulation study since it is one of the most popular semiparametric methods. It is shown that sample size of 50 seems to be sufficient for the observed coverage to be close to the theoretical one. For Frank and Gumbel-Hougaard copula families even sample size of 30 gives us...
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Martingale Approach to Roulette
Fornůsková, Monika ; Večeř, Jan (advisor) ; Hlubinka, Daniel (referee)
This main aim of this thesis is to compare two different strategies in Roulette -- betting on a color and betting on a single number. Betting on a color represents a conservative strategy with diversified asset and betting on a number represents a more risky strategy without diversification. Distribution of the Maximum, the Last Exit Time and the Number of Visits of zero will be given for each strategy using Martingales or Markov Chains. The theoretical results will be supported by Monte Carlo simulations. Powered by TCPDF (www.tcpdf.org)
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Statistical Analysis of Wiener Process Based on Partial Observations
Hrochová, Magdalena ; Hlubinka, Daniel (advisor) ; Omelka, Marek (referee)
Wiener process-a random process with continuous time-plays an important role in mathematics, physics or economy. It is often good to know whether it contains any deterministic part, e.g. drift or scale. However, it is nearly impossible either observe the whole trajectory of the process or preserve its full history. This thesis deals with a statistical analysis based on partial observations, namely passage times through some given barriers. We propose several statistical methods for testing hypotheses about drift or scale using these observations. As supporting methods, we consider the maximum likelihood theory, non-parametric test against a trend, and binomial test. For testing the value of scale in the model with no drift and constant scale we recommend maximum likelihood theory. We derive the estimate and related tests in the case of observing only three barriers. The simulation study suggested observing more barriers for testing monotony of scale in a model with linear drift, or testing monotone and convex/concave drift in a model with constant scale. 1
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Models for zero-inflated data
Matula, Dominik ; Kulich, Michal (advisor) ; Hlubinka, Daniel (referee)
The aim of this thesis is to provide a comprehensive overview of the main approaches to modeling data loaded with redundant zeros. There are three main subclasses of zero modified models (ZMM) described here - zero inflated models (the main focus lies on models of this subclass), zero truncated models and hurdle models. Models of each subclass are defined and then a construction of maximum likelihood estimates of regression coefficients is described. ZMM models are mostly based on Poisson or negative binomial type 2 distribution (NB2). In this work, author has extended the theory to ZIM models generally based on any discrete distributions of exponential type. There is described a construction of MLE of regression coefficients of theese models, too. Just few of present works are interested in ZIM models based on negative binomial type 1 distribution (NB1). This distribution is not of exponential type therefore a common method of MLE construction in ZIM models cannot be used here. In this work provides modification of this method using quasi-likelihood method. There are two simulation studies concluding the work. 1
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Geometry of Linear Model
Línek, Vítězslav ; Hykšová, Magdalena (advisor) ; Nagy, Ivan (referee) ; Hlubinka, Daniel (referee)
The advantage of the geometric approach to linear model and its applications is known to many authors. In spite of that, it still remains to be rather unpopular in teaching statistics around the world and is almost missing in the Czech Republic. In this work, we use geometry of multidimensional vector spaces to derive some well-known properties of the linear model and to explain some of the most familiar statistical methods to show usefulness of this approach, also known as "free-coordinate". Besides, historical background including selected results of R. A. Fisher is briefly discussed; it follows that the geometry approach to linear model is justifiable from the historical point of view, too. Powered by TCPDF (www.tcpdf.org)
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Statistical Depth for Functional Data
Nagy, Stanislav ; Hlubinka, Daniel (advisor) ; Claeskens, Gerda (referee) ; Hušková, Marie (referee)
Statistical data depth is a nonparametric tool applicable to multivariate datasets in an attempt to generalize quantiles to complex data such as random vectors, random functions, or distributions on manifolds and graphs. The main idea is, for a general multivariate space M, to assign to a point x ∈ M and a probability distribution P on M a number D(x; P) ∈ [0, 1] characterizing how "centrally located" x is with respect to P. A point maximizing D(·; P) is then a generalization of the median to M-valued data, and the locus of points whose depth value is greater than a certain threshold constitutes the inner depth-quantile region corresponding to P. In this work, we focus on data depth designed for infinite-dimensional spaces M and functional data. Initially, a review of depth functionals available in the literature is given. The emphasis of the exposition is put on the unification of these diverse concepts from the theoretical point of view. It is shown that most of the established depths fall into the general framework of projection-driven functionals of either integrated, or infimal type. Based on the proposed methodology, characteristics and theoretical properties of all these depths can be evaluated simultaneously. The first part of the work is devoted to the investigation of these theoretical properties,...
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Dynamic fare model
Kislinger, Jan ; Lachout, Petr (advisor) ; Hlubinka, Daniel (referee)
The problem of creating dynamic fare model consists of two tasks - estimating demand for train tickets and multistage optimization of price of fare. We introduce a model of inhomogeneous Markov process for the process of selling the tickets in this thesis. Because of the complexity of the state space the optimization problem needs to be solved using simulation methods. The solution was implemented in R language for single-stage and two-stage problems. Before this application we summarize the theory of inhomogeneous Markov process with special attention to process with separable inhomogeneity. Then we propose methods for estimating the intensity using maximum likelihood theory. We also describe and compare two algorithms for simulated optimization. Powered by TCPDF (www.tcpdf.org)
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