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Modeling and statistics of random tessellations with applications to the study of microstructure of polycrystallic materials
Seitl, Filip ; Beneš, Viktor (advisor)
Filip Seitl Modeling and statistics of random tessellations with ap- plications to the study of microstructure of polycrystallic materials We present a parametric statistical methodology for analysing a Laguerre tessel- lation data set viewed as a realization of a marked point process of generators. We study the dependence of the marks and the point process in detail. Further, we suggest two general models. The first one is based on marked Gibbs point processes and leads to a broad class of Gibbs-Laguerre tessellations. Under mild assumptions we prove the existence of the infinite-volume Gibbs measure. Then the choice of energy function for applications is discussed in detail. The sec- ond model is hierarchical, where in the first step the point pattern is modelled and in the second step the marks are modelled conditionally on points, using exponential family models based on geometrical characteristics of the tessella- tion. Statistical tools for the parameter estimation, model selection and fitting are suggested and implemented. We apply our methodology for a 3D Laguerre tessellation data set representing the microstructure of a polycrystalline metal- lic material, where simulations under a fitted model may substitute expensive laboratory experiment. 1
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Modeling and statistics of random tessellations with applications to the study of microstructure of polycrystallic materials
Seitl, Filip ; Beneš, Viktor (advisor) ; Redenbach, Claudia (referee) ; Mrkvička, Tomáš (referee)
Filip Seitl Modeling and statistics of random tessellations with ap- plications to the study of microstructure of polycrystallic materials We present a parametric statistical methodology for analysing a Laguerre tessel- lation data set viewed as a realization of a marked point process of generators. We study the dependence of the marks and the point process in detail. Further, we suggest two general models. The first one is based on marked Gibbs point processes and leads to a broad class of Gibbs-Laguerre tessellations. Under mild assumptions we prove the existence of the infinite-volume Gibbs measure. Then the choice of energy function for applications is discussed in detail. The sec- ond model is hierarchical, where in the first step the point pattern is modelled and in the second step the marks are modelled conditionally on points, using exponential family models based on geometrical characteristics of the tessella- tion. Statistical tools for the parameter estimation, model selection and fitting are suggested and implemented. We apply our methodology for a 3D Laguerre tessellation data set representing the microstructure of a polycrystalline metal- lic material, where simulations under a fitted model may substitute expensive laboratory experiment. 1
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Point processes on the sphere
Svoboda, Willy ; Dvořák, Jiří (advisor) ; Seitl, Filip (referee)
A point process can be easily described as a random locally finite set. For example, we can model locations of arbitrary events in a city or in the world such as earthquake epicenter locations. In this thesis, we introduce basic types of point processes in a Euc- lidean space and on a sphere, describe what situations can be modelled by them, define basic properties, and lay down theoretical groundwork for the K-function (and its modi- fications for marked point processes). The main goal of this thesis is to introduce marked point processes on a sphere and to give theoretical framework, whereas the marks will give us another nontrivial information about the points, which we want to study further. In the conclusion of the thesis, we concern ourselves with testing whether those marks are mutually independent. We apply Monte Carlo permutation test using mark-weighted K-function for marked point processes on a sphere. 1
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Random tessellations modeling
Seitl, Filip ; Beneš, Viktor (advisor) ; Pawlas, Zbyněk (referee)
The motivation for this work comes from physics, when dealing with microstructures of polycrystalline materials. An adequate probabilistic model is a three-dimensional (3D) random tessellation. The original contribution of the author is dealing with the Gibbs-Voronoi and Gibbs- Laguerre tessellations in 3D, where the latter model is completely new. The energy function of the underlying Gibbs point process reflects interactions between geometrical characteristics of grains. The aim is the simulation, parameter estimation and degree-of-fit testing. Mathematical background for the methods is described and numerical results based on simulated data are presented in the form of tables and graphs. The interpretation of results confirms that the Gibbs-Laguerre model is promising for further investigation and applications.
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Simulated annealing
Seitl, Filip ; Beneš, Viktor (advisor) ; Pawlas, Zbyněk (referee)
Simulated annealing is a probabilistic optimization algorithm which is used for approximating the global extremes of a function at a large state space. We construct a homogeneous Markov chain, whose stationary distribution is depen- dent on the temperature parameter (this distribution is called the Boltzmann distribution), on this space. With declining the parameter this distribution fo- cuses on the states minimizing the function. The algorithm, on which it can be viewed as a non-homogeneous Markov chain, we use to solve the hard-core model and the graph bisection. We will also deal with the convergence of the algorithm, too rapidly decreasing sequence of the parameters can result in stucking in a lo- cal extreme of the function, therefore some requirements on this sequence will be determined. 1
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