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Random measurable sets
Fojtík, Vít ; Rataj, Jan (advisor) ; Pawlas, Zbyněk (referee)
The aim of this thesis is to compare two major models of random sets, the well established random closed sets (RACS) and the more recent and more general random measurable sets (RAMS). First, we study the topologies underlying the models, showing they are very different. Thereafter, we introduce RAMS and RACS and reformulate prior findings about their relationship. The main result of this thesis is a characterization of those RAMS that do not induce a corresponding RACS. We conclude by some examples of such RAMS, including a construction of a translation invariant RAMS. 1
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Discrete linear dynamical systems with control
Procházková, Zuzana ; Tůma, Jiří (advisor) ; Růžička, Pavel (referee)
Discrete linear dynamical systems with control Author: Zuzana Procházková Department: Department of Algebra Supervisor: doc. RNDr. Jiří Tůma, DrSc., Department of Algebra Abstract: In this thesis we describe elementary property of discrete linear dyna- mical system. We define discrete linear dynamical system with control and its controllability and then we define descrete linear dynamical system with output and its observability. After that we show the duality of observability and con- trollability with definition of dual system and its description. There are three problems solved in the last chapter. 1
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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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