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Random closed sets
Stroganov, Vladimír ; Honzl, Ondřej (advisor) ; Rataj, Jan (referee)
In this bachelor thesis we are concerned with basic knowledge in random set theory. We define here such terms, as capacity functional, se- lection, measurable and integrable multifunction, Castaing representation and Aumann expectation of random closed set. We present Choquet theo- rem, Himmelberg measurability theorem, theorems of properties of selections and expectation. We present also several examples which illustrate the the- ory. 1
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Nonabsolutely convergent integrals
Kuncová, Kristýna ; Malý, Jan (advisor) ; Rataj, Jan (referee)
Title: Nonabsolutely convergent integrals Author: Kristýna Kuncová Department: Department of Mathematical Analysis Supervisor: Prof. RNDr. Jan Malý, DrSc., Department of Mathematical Analysis Abstract: Our aim is to introduce an integral on a measure metric space, which will be nonabsolutely convergent but including the Lebesgue integral. We start with spaces of continuous and Lipschitz functions, spaces of Radon measures and their dual and predual spaces. We build up the so-called uniformly controlled integral (UC-integral) of a function with respect to a distribution. Then we investigate the relationship between the UC-integral with respect to a measure and the Lebesgue integral. Then we introduce another kind of integral, called UCN-integral, based on neglecting of small sets with respect to a Hausdorff measure. Hereafter, we focus on the concept of n-dimensional metric currents. We build the UC-integral with respect to a current and then we proceed to a very general version of Gauss-Green Theorem, which includes the Stokes Theorem on manifolds as a special case. Keywords: Nonabsolutely convergent integrals, Multidimensional integrals, Gauss-Green Theorem 1
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Set-indexed stochastic processes
Schenk, Martin ; Pawlas, Zbyněk (advisor) ; Rataj, Jan (referee)
This thesis deals with the problem of estimating the joint probability distribution of a marked process' parameters from a censored data. First, a Nelson-Aalen estimator of the cumulative hazard rate for one-dimensional case is constructed. This estimator is then smoothed by using a kernel function estimator. Then, a Kaplan-Meier estimator of the survival function is brought in. Further, a theory of set-indexed random processes is built up to be a base for the construction of a generalized Nelson-Aalen estimator of the cumulative hazard rate, which is then again smoothed. For a special case, a generalized Kaplan-Meier estimator of the multidimensional survival function is constructed. The application of the mentioned generalized estimators is shown on a particular case. These estimators are then used on simulated data.
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Random closed sets and particle processes
Stroganov, Vladimír ; Rataj, Jan (advisor) ; Pawlas, Zbyněk (referee)
In this thesis we are concerned with representation of random closed sets in Rd with values concentrated on a space UX of locally finite unions of sets from a given class X ⊂ F. We examine existence of their repre- sentations with particle processes on the same space X, which keep invariance to rigid motions, which the initial random set was invariant to. We discuss existence of such representations for selected practically applicable spaces X: we go through the known results for convex sets and introduce new proofs for cases of sets with positive reach and for smooth k-dimensional submanifolds. Beside that we present series of general results related to representation of random UX sets. 1
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Random measurable sets and particle processes
Jurčo, Adam ; Rataj, Jan (advisor) ; Beneš, Viktor (referee)
Random measurable sets and particle processes Adam Jurčo Abstract In this thesis we deal with particle processes on more general spaces. First we in- troduce the space of Lebesgue measurable sets represented by indicator functions with topology given by L1 loc convergence. We the explore the topological properties of this space and its subspaces of sets of finite and locally finite perimeter. As these spaces do not satisfy the usual topological assumptions needed for construction of point processes we use another approach based on measure-theoretic assumptions. This will allow us to define point processes given by finite dimensional distributions on measurable subsets of the space of Lebesgue-measurable sets. Then we will derive a formula for a volume fraction of a Boolean process defined in this more general setting. Further we introduce a Boolean process with particles of finite perimeter and derive a formula for its specific perimeter. 1
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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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Generalized random tessellations, their properties, simulation and applications
Jahn, Daniel ; Beneš, Viktor (advisor) ; Rataj, Jan (referee)
The past few years have seen advances in modelling of polycrystalline materi- als using parametric tessellation models from stochastic geometry. A promising class of tessellations, the Gibbs-type tessellation, allows the user to specify a great variety of properties through the energy function. This text focuses solely on tetrahedrizations, a three-dimensional tessellation composed of tetrahedra. The existing results for two-dimensional Delaunay triangulations are extended to the case of three-dimensional Laguerre tetrahedrization. We provide a proof of existence, a C++ implementation of the MCMC simulation and estimation of the models parameters through maximum pseudolikelihood. 1
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On the interior of a minimal convex polygon
Šplíchal, Ondřej ; Valtr, Pavel (advisor) ; Rataj, Jan (referee)
Zvolme konečnou množinu bod· P v rovině v obecné poloze, tj. žádné 3 body neleží na přímce. Konvexní n-úhelník je minimální, pokud v jeho konvexním obalu neleží jiný konvexní n-úhelník s vrcholy v P. Erd®s a Szekeres (1935) ukázali, že pro každé n ≥ 3 existuje minimální číslo ES(n) takové, že mezi libovolnými ES(n) body v rovině v obecné poloze lze vybrat n bod·, které tvoří vrcholy konvexního n-úhelníku. Z jejich tvrzení vyplývá, že v topologic- kém vnitřku minimálního konvexního n-úhelníku m·že ležet jen omezený po- čet bod· P pro libovolnou volbu P. Označíme maximální takový počet jako mci(n). V práci ukážeme horní odhad mci(n) ≤ ES(n) − n a spodní odhad 2n−3 − n + 2 ≤ mci(n) pro n ≥ 3.
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Consequences and applications of the Fock space representation theorem
Novotná, Daniela ; Beneš, Viktor (advisor) ; Rataj, Jan (referee)
Consequences and applications of the Fock space representation theorem Daniela Novotn'a Department of Probability and Mathematical Statistics, Faculty of Mathematics and Physics, Charles University Abstract In this thesis, we deal with selected applications of the Fock space rep- resentation theorem. One of the most important is the covariance identity, which can yield in an estimation of the correlation function of a point process having Papangelou conditional intensity. We used this result to generalise some asymptotic results for Gibbs particle processes. Namely, in combina- tion with Stein's method, we derived bounds for the Wasserstein distance between the standard normal distribution and the distribution of an innova- tion of a Gibbs particle process. As an application, we present a central limit theorem for a functional of a Gibbs segment process with pair potential.
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Random sets and their curvatures
Zubaľ, Andrej ; Rataj, Jan (advisor) ; Beneš, Viktor (referee)
We investigate the connection between stochastic and integral geometry. Namely, we illustrate the role of curvature measures, and the corresponding integral-geometric formulas for them, in stochastic models such as random sets or particle processes. Additionally, we work with a relatively new type of objects called DC domains and we establish their measurability in the space of closed sets of d-dimensional Euclideann space, which allows for stochastic geometry models for DC domains to be defined.
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