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Nonstationary particle processes
Jirsák, Čeněk ; Rataj, Jan (advisor) ; Beneš, Viktor (referee)
Title: Nonstacionary particle processes Author: Čeněk Jirsák Department: Department of Probability and Mathematical Statistics Supervisor: Doc. RNDr. Jan Rataj, CSc., Mathematical Institute, Charles University Supervisor's e-mail address: rataj@karlin.mff.cuni.cz Abstract: Many real phenomena can be modeled as random closed sets of different Hausdorff dimension in Rd . One of the main characteristics of such random set is its expected Hausdorff measure. In case that this measure has a density, the density is called intensity function. In present paper we define a nonparametric kernel estimation of the intensity function. The concept of Hk -rectifiable set has a key role here. Properties of kernel estimation such as unbiasness or convergence behavior are studied. As the esti- mation may be difficult to compute precisely numerical approximations are derived for practical use. Parametric models are also briefly mentioned and the kernel estimation is used with the minimum contrast method to estimate the parameters of the model. At last the suggested methods are tested on simulated data. Keywords: stochastic geometry, intensity measure, random closed set, kernel estimation 1
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Estimation of the pair correlation function of a point process
Vondráček, Jakub ; Dvořák, Jiří (advisor) ; Beneš, Viktor (referee)
This thesis deals with kernel estimation of the pair correlation function of a stationary and isotropic point process. Firstly, the basics of the theory of point processes are built up. Then, the derivation of formulas for expectation and variance of a kernel estimator is provided. Also, an extension of a simple Poisson approximation of variance to the case of an estimator with more complicated edge correction compared to what is usually used in the literature is given. These formulas depend on a parameter called bandwidth. The recommendations for selecting the bandwidth that can be found in the literature are summarised and simulation experiments are performed to assess the correctness of the derived formulas. These experiments also prove that a variance approximation obtained by ignoring so called "higher order correlations" is unjustified. Lastly, bandwidth selection and the advantages and disadvantages of several approaches for bandwidth selection are discussed. 1
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Random marked sets
Kráľová, Veronika ; Beneš, Viktor (advisor) ; Pawlas, Zbyněk (referee)
In this thesis, two models of marked point processes are investigated. One of the marks have a continuous distribution on a compact Riemannian manifold. The von Mises distribution and its properties are studied. Metropolis-Hastings algorithm of Markov chain Monte Carlo method is used for the simulation of Gibbs segment process. Takacs-Fiksel estimator and its modified version are examined. A kernel density estimator and entropy estimator are proposed and applied to simulated and real data. Powered by TCPDF (www.tcpdf.org)
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Nonstationary particle processes
Jirsák, Čeněk ; Rataj, Jan (advisor) ; Beneš, Viktor (referee)
Title: Nonstacionary particle processes Author: Čeněk Jirsák Department: Department of Probability and Mathematical Statistics Supervisor: Doc. RNDr. Jan Rataj, CSc., Mathematical Institute, Charles University Supervisor's e-mail address: rataj@karlin.mff.cuni.cz Abstract: Many real phenomena can be modeled as random closed sets of different Hausdorff dimension in Rd . One of the main characteristics of such random set is its expected Hausdorff measure. In case that this measure has a density, the density is called intensity function. In present paper we define a nonparametric kernel estimation of the intensity function. The concept of Hk -rectifiable set has a key role here. Properties of kernel estimation such as unbiasness or convergence behavior are studied. As the esti- mation may be difficult to compute precisely numerical approximations are derived for practical use. Parametric models are also briefly mentioned and the kernel estimation is used with the minimum contrast method to estimate the parameters of the model. At last the suggested methods are tested on simulated data. Keywords: stochastic geometry, intensity measure, random closed set, kernel estimation 1
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