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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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Marked particle processes
Kovář, Matěj ; Pawlas, Zbyněk (advisor) ; Beneš, Viktor (referee)
This diploma thesis delves into the study of marked particle processes, a relatively unexplored field in stochastic geometry and spatial statistics. The main aim of the thesis is to lay some basic foundations of the marked particle processes and to present a suitable second-order characteristic that would appropriately evaluate an interaction between particles and corresponding marks. A primary focus is devoted to introducing the notion of the particle-weighted f-mark correlation function defined via standard second- order factorial measure. The thesis presents some of the potential marked particle process models and some of the forms of the particle-weighted f-mark correlation function. In the last two chapters the simulation study discussing the models and its f-mark correlation function is presented. 1
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Point processes on linear networks and analysis of traffic accidents
Zemčík, Jan ; Dvořák, Jiří (advisor) ; Beneš, Viktor (referee)
In this thesis we shall study point processes on linear networks. We introduce the definition of first-order intensity function, which represents the intensity of occurrence of points of the point process in individual points of the linear net- work. We require the first-order intensity function estimate utilising a smooth- ing kernel to preserve mass on the linear network. We define the KD (u|x) function, which represents the smoothing kernel. Our next objective is to state theorems which show us, when the integral of the function KD (u|x) evaluates to 1. The first will be stated for linear networks without cycles of length smaller than the width of the kernel and the second for linear networks which may con- tain cycles. We believe that the relevant literature on the subject contains an erroneous statement of the latter version of the theorem. We present detailed proofs for both versions of the theorem. At the end of the thesis we present two analyses based on real data (traffic accidents on D1 highway and Chicago Crime Data). 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 networks
Zahrádková, Petra ; Beneš, Viktor (advisor) ; Dvořák, Jiří (referee)
The aim of this master thesis is to develop the background for point processes on both the linear network and the planar network in R3 . The linear, planar network is formed by the system of edges, faces of a 3D tessellation, respectively. Using the Gibbs-Laguerre tessellation model we can investigate the case of a regular or irregular tessellation. We consider cluster and hard core point processes and we compare two types of distances between points, the Euclidean distance and the shortest path distance within the net- work. The algorithms for simulation of point processes on both networks are developed. Using the simulated realizations of point processes we estimate some functional summary characteristics. Based on plotted graphs of these estimates, the influence of the choice of tessellations, point process models and distances is discussed. 1
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Computations of Laguerre tessellations with given cell volumes
Kornijčuk, Oleksandr ; Beneš, Viktor (advisor) ; Dvořák, Jiří (referee)
Given a finite set of pairs in Rd ×R, nuclei and weights, Laguerre tessellations allow us to subdivide Euclidean space Rd into finitely many polyhedral cells using the power dis- tance. We are interested in the problem of finding weights so that the Laguerre cells have prescribed volumes and nuclei. Our primary aim is to present the theoretical background leading to the problem's solution. Here we complete some proofs that are shortened in the literature, while other theorems are cited. Then, we demonstrate two own computer programs and the corresponding numerical results. First, we compute the desired set of weights that generates the Laguerre tessellation with prescribed cell volumes and apply it to a unit cube in R3 . The application of this method relies on the Barzilai-Borwein gradient descent and Voro++ library, which computes the volumes of cells in each itera- tion. Furthermore, an iterative approach approximates a centroidal Laguerre tessellation, where the nuclei coincide with the centroids of Laguerre cells. 1
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Non-orphan cluster point processes
Hájková, Eliška ; Dvořák, Jiří (advisor) ; Beneš, Viktor (referee)
In this work we introduce some basic concepts from the theory of spatial point pro- cesses and two methods of estimation of the parametres for Thomas process. Firstly the method of minimum contrast, that is used for situations when we do not know the position of parent points and secondly our proposed method using information about the parent points location. Using simulations in program R, we find out that our method estima- ted mentioned parameters better considering the relative bias and relative mean squared error. Subsequently, we estimate the parameters of the real data using both methods. Fi- nally we test by Global Envelope Test, whether our data match Thomas process with para- meters estimated from data with mentioned methods. For the combinations of parameters obtained by discussed methods we do not reject the hypothesis. 1
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Propp-Wilson algorithm
Urx, Vojtěch ; Beneš, Viktor (advisor) ; Antoch, Jaromír (referee)
This thesis deals with the theory leading up to the Propp-Wilson Algorithm and the application of this algorithm on the Ising model. The goal is to sum- marize the relevant theory from the book H¨aggstr¨om (2002) and solve some problems it gives to the reader. These problems consist of, among others, parts of the proof of correctness for a specific version of the Propp-Wilson algorithm on the Ising model and the Implementation of this algorithm in Python. 1
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Point processes of objects with random lifetime
Kulla, Filip ; Dvořák, Jiří (advisor) ; Beneš, Viktor (referee)
The thesis deals with point processes of objects with random lifetime. The form of the likelihood function of an observed spatial-temporal pattern with random lifetimes is derived, where the formula is subsequently generalised to the case of censored life- times. Moreover, some simple parametric models are introduced and conditions under which they are non-explosive are derived. In addition, aspects of our implementation of the algorithm which generates a realisation of a given spatial-temporal point process with random lifetimes and of the likelihood-based estimation are discussed. The thesis contains a simulation study in which the use of the (partial) likelihood on simulated data is demonstrated and properties of resulting estimates are discussed. Furthermore, it contains an application of the partial likelihood to the real data, where the question of interest is the spatial dynamics of propagation of an observed population of flowers. 1
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