
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 Gibbstype tessellation, allows the user to specify a great variety of properties through the energy function. This text focuses solely on tetrahedrizations, a threedimensional tessellation composed of tetrahedra. The existing results for twodimensional Delaunay triangulations are extended to the case of threedimensional 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


Kernel Methods in Particle Filtering
Coufal, David ; Beneš, Viktor (advisor)
Kernel Methods in Particle Filtering David Coufal Doctoral thesis  abstract The thesis deals with the use of kernel density estimates in particle filtering. In particular, it examines the convergence of the kernel density estimates to the filtering densities. The estimates are constructed on the basis of an out put from particle filtering. It is proved theoretically that using the standard kernel density estimation methodology is effective in the context of particle filtering, although particle filtering does not produce random samples from the filtering densities. The main theoretical results are: 1) specification of the upper bounds on the MISE error of the estimates of the filtering densities and their partial derivatives; 2) specification of the related lower bounds and 3) providing a suitable tool for checking persistence of the Sobolev character of the filtering densities over time. In addition, the thesis also focuses on designing kernels suitable for practical use. 1


Kernel Methods in Particle Filtering
Coufal, David ; Beneš, Viktor (advisor) ; Klebanov, Lev (referee) ; Studený, Milan (referee)
Kernel Methods in Particle Filtering David Coufal Doctoral thesis  abstract The thesis deals with the use of kernel density estimates in particle filtering. In particular, it examines the convergence of the kernel density estimates to the filtering densities. The estimates are constructed on the basis of an out put from particle filtering. It is proved theoretically that using the standard kernel density estimation methodology is effective in the context of particle filtering, although particle filtering does not produce random samples from the filtering densities. The main theoretical results are: 1) specification of the upper bounds on the MISE error of the estimates of the filtering densities and their partial derivatives; 2) specification of the related lower bounds and 3) providing a suitable tool for checking persistence of the Sobolev character of the filtering densities over time. In addition, the thesis also focuses on designing kernels suitable for practical use. 1


Selected topics of random walks
Filipová, Anna ; Hlubinka, Daniel (advisor) ; Beneš, Viktor (referee)
The theme of this thesis are symmetric random walks. We define different types of paths and prove the reflection principle. Then, based on the paths, we define random walks. The thesis also deals with probabilities of returns to the origin and first returns to the origin, further with probabilities of number of changes of sign or returns to the origin up to a certain time. We also define the maximum of the random walk and the first passage through a certain point. In the second chapter, we solve several problems, which form the proofs of some theorems from the first chapter or complement the first chapter in a different way. For example, we prove geometrically that the number of paths of one type equals the number of paths of another type or we compute the probability that there occurs a certain number of changes of sign up to a given time.


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 threedimensional (3D) random tessellation. The original contribution of the author is dealing with the GibbsVoronoi 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 degreeoffit 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 GibbsLaguerre model is promising for further investigation and applications.


Spatiotemporal point processes
Kratochvílová, Blažena ; Beneš, Viktor (advisor) ; Volf, Petr (referee) ; Pawlas, Zbyněk (referee)
The background theory of point processes, spatiotemporal point processes, random measures and random closed sets is given in the beginning of the thesis. Then the special case of spatiotemporal Cox processes constructed from L'evy basis is studied. Formulas for theoretical characteristics are derived using the generating functional. The Cox process on the curve is defined and studied. The analysis of such a process leads to nonlinear filtering methods. Also the methods for model selection are discussed. These methods are used on simulated data, firstly on the simple discrete data and secondly on the continuous data where the curve is a spiral. Then the real data from a neurophysiology experiment is analysed. During the experiment, the spiking activity of a place cell of hippocampus of a rat moving in an arena together with the track of the rat was recorded. The track of the rat and the action potentials (spikes) present the curve and the points on it. At the end of the thesis, other approaches to neurophysiological data are discussed. The first one is an estimation of a conditional intensity of the temporal process of spikes using recursive filtering. In the second one, the track of the rat together with the random driving intensity function of the process of the spikes is viewed as a random marked set.


Normal approximation for statistics of Gibbs point processes
Maha, Petr ; Beneš, Viktor (advisor) ; Dvořák, Jiří (referee)
In this thesis, we deal with finite Gibbs point processes, especially the processes with densities with respect to a Poisson point process. The main aim of this work is to investigate a fourparametric marked point process of circular discs in three dimensions with two and three way point interactions. In the second chapter, our goal is to simulate such a process. For that purpose, the birth death MetropolisHastings algorithm is presented including theoretical results. After that, the algorithm is applied on the disc process and numerical results for different choices of parameters are presented. The third chapter consists of two approaches for the estimation of parameters. First is the TakacsFiksel estimation procedure with a choice of weight functions as the derivatives of pseudolikelihood. The second one is the estimation procedure aiming for the optimal choice of weight functions for the estimation in order to provide better quality estimates. The theoretical background for both of these approaches is derived as well as detailed calculations for the disc process. The numerical results for both methods are presented as well as their comparison. 1


Statistical properties of local stereological estimators
Hájek, Tadeáš ; Pawlas, Zbyněk (advisor) ; Beneš, Viktor (referee)
In this thesis statistical properties of local stereological estimators of par ticle volume are investigated. The emphasis is on the estimation of the va riance of the local estimator and its components  the variance due to the variability in the particle size distribution and the variance due to the lo cal stereological procedures. Various ways of estimation for independent and correlated particles are presented. Results of simulation studies for both inde pendent and correlated ellipsoidalshaped particles are presented. Described estimators are demonstrated on real biological data. Comprehensive theory that leads to the local stereological estimators of volume is presented too. 1


Exact envelope tests
Maděřičová, Soňa ; Dvořák, Jiří (advisor) ; Beneš, Viktor (referee)
In this work we are focusing on Monte Carlo simulation tests, in particular we are dealing with envelope and deviation tests. We describe the development of envelope tests from standard envelope tests, in which we can not control significance level, through refined envelope tests, where we can control the significance level indirectly, to exact envelope tests, for which the significance level can be chosen in advance. We will show how the exact envelope tests are related to deviation tests. Further we compare individual kinds of tests using examples and describe their advantages and disadvantages.


Modelling of segment process in the plane
Pultar, Milan ; Beneš, Viktor (advisor) ; Pawlas, Zbyněk (referee)
We consider a finite planar segment process in a circle, having a density with respect to the Poisson process. This density involves unknown parameters and a reference length distribution which is not observed. The aim is to estimate these quantities semiparametrically. The segment process is inhomogeneous, but it is isotropic. Combining the relation between the observed and reference length distribution and the maximum pseudolikelihood method we suggest an estimation procedure. Its properties (bias and variability) are investigated in a simulation study. In the last part we present two more complex models. The motivation is to model stress fibers observed in cultured stem cells.
