|
|
Random marked sets and dimension reduction
Šedivý, Ondřej ; Beneš, Viktor (advisor) ; Janáček, Jiří (referee) ; Mrkvička, Tomáš (referee)
Random closed sets and random marked closed sets present an important general concept for the description of random objects appearing in a topological space, particularly in the Euclidean space. This thesis deals with two major tasks. At first, it is the dimension reduction problem where dependence of a random closed set on underlying spatial variables is studied. Solving this problem allows to find the most significant regressors or, possibly, to identify the redundant ones. This work achieves both theoretical results, based on extending the inverse regression techniques from classical to spatial statistics, and numerical justification of the methods via simulation studies. The second topic is estimation of characteristics of random marked closed sets which is primarily motivated by an application in the microstructural research. Random marked closed sets present a mathematical model for the description of ultrafine-grained microstructures of metals. Methods for statistical estimation of their selected characteristics are developed in the thesis. Correct quantitative characterization of microstructure of metals allows to better understand their macroscopic properties.
|
|
|
Interacting spatial particle systems
Zikmundová, Markéta ; Beneš, Viktor (advisor) ; Pawlas, Zbyněk (referee) ; Volf, Petr (referee)
1 Title: Interacting spatial particle systems Author: Markéta Zikmundová Department: Department of Probability and Mathematical Statistics Author's e-mail address: zikmundm@karlin.mff.cuni.cz Supervisor: Prof. RNDr. Viktor Beneš, DrSc. Supervisor's e-mail address: benesv@karlin.mff.cuni.cz Consultant: RNDr. Kateřina Helisová, Ph.D. Consultant's e-mail address: helisova@math.feld.cvut.cz Abstract: Several kinds of random union of interacting particles is studied. We define line segment process of interacting particles in R2 and process of interacting surfaces in R3 as the models with density function p with respect to some Poisson point process. The formulas for moments of the geometrical characteristics of these models are derived and the limit behaviour when the intensity tends to infinity is investigated. For time extension of such models a simulation algorithm is developed. Various estimations of parameters of density p, among them those based on sequential Monte Carlo, are studied and compare in a simulation study. Keywords: Boolean model, process with interacting particles, U−statistics, exponential family, germ-grain model, interaction, Markov properties, point process, random closed set, Markov chain Monte Carlo.
|
|
|
Two-dimensional point processes
Bakošová, Katarína ; Pawlas, Zbyněk (advisor) ; Beneš, Viktor (referee)
A point process as the special type of a random stochastic process is a theoretical model for occurrence of random events in time and space. In this thesis, we examine pairs of point processes in time and their mutual relations. The thesis acquaints the reader with the theoretical background of point processes, two- dimensional point processes and their properties based on measure theory. The purpose of this paper is to present and demonstrate methods of analyzing realizations of two point processes. Our attention is beaing focused mainly on problem of dependency of two point processes. We describe data analyses based on cross-correlation histogram, synchronization indices, and on spectral analysis using coherence. In the last chapter, we conducted these methods on nerve cell spike train data. Powered by TCPDF (www.tcpdf.org)
|
|
|
Statistical inference for spatial and space-time Cox point processes
Dvořák, Jiří ; Prokešová, Michaela (advisor) ; Beneš, Viktor (referee) ; Swart, Jan (referee)
Fitting of parametric models to spatial and space-time point patterns has been a very active research area in the last few years. Concerning clustered patterns, the Cox point process is the model of choice. To avoid the computationally demanding maximum likelihood estimation or Bayesian inference, several estimation methods based on the moment properties of the processes in question were proposed in the literature. We give overview of the state-of-the-art moment estimation methods for stationary spatial Cox point processes and compare their performance in a simulation study. We also discuss generalization of such methods for inhomogeneous spatial point processes. In the core part of the thesis we focus on minimum contrast estimation for inhomogeneous space-time shot-noise Cox point processes and investigate the possibility to use projections to the spatial and temporal domain to estimate different parts of the model separately. We propose a step-wise estimation procedure based on projection processes and also a refined method which remedies the problem of possible cluster overlapping. We establish consistency and asymptotic normality of the estimators for both methods under the increasing window asymptotics and compare their performance on middle-sized observation windows by means of a simulation study....
|
|
|
Spatial point process with interactions
Vícenová, Barbora ; Beneš, Viktor (advisor) ; Zikmundová, Markéta (referee)
This thesis deals with the estimation of model parameters of the interacting segments process in plane. The motivation is application on the system of stress fibers in human mesenchymal stem cells, which are detected by fluorescent microscopy. The model of segments is defined as a spatial Gibbs point process with marks. We use two methods for parameter estimation: moment method and Takacs-Fiksel method. Further, we implement algorithm for these estimation methods in software Mathematica. Also we are able to simulate the model structure by Markov Chain Monte Carlo, using birth-death process. Numerical results are presented for real and simulated data. Match of model and data is considered by descriptive statistics. Powered by TCPDF (www.tcpdf.org)
|
|
|
Random fields of facets
Novotná, Daniela ; Beneš, Viktor (advisor) ; Pawlas, Zbyněk (referee)
Facet process is a special example of a point process in Euclidean space Rd , where points are in this case represented by compact subsets of hyperplanes in Rd with given orientation, size and shape. We focus on finite facet processes with density from exponential family with respect to the distribution of Poisson point process. Its submodel is simulated using the Metropolis-Hastings birth death algorithm, which gives us a homogeneous Markov chain. Specially in R2 space we derive its stationary distribution. In spaces R2 and R4 we perform numerical simulations to show behavior of the chain for various parameters in such model. 1
|
|
|
Black-Scholes models of option pricing
Čekal, Martin ; Maslowski, Bohdan (advisor) ; Beneš, Viktor (referee)
Title: Black-Scholes Models of Option Pricing Author: Martin Cekal Department: Department of Probability and Mathematical Statistics Supervisor: prof. RNDr. Bohdan Maslowski, DrSc., Charles University in Prague, Faculty of Mathematics and Physics, Department of Probability and Mathematical Statistics. Abstract: In the present master thesis we study a generalization of Black-Scholes model using fractional Brownian motion and jump processes. The main goal is a derivation of the price of call option in a fractional jump market model. The first chapter introduces long memory and its modelling by discrete and continuous time models. In the second chapter fractional Brownian motion is defined, appropriate stochastic analysis is developed and we generalize the notion of Lévy and jump processes. The third chapter introduces fractional Black-Scholes model. In the fourth chapter, tools developed in the second chapter are used for the construction of jump fractional Black-Scholes model and derivation of explicit formula for the price of european call option. In the fifth chapter, we analyze long memory contained in simulated and empirical time series. Keywords: Black-Scholes model, fractional Brownian motion, fractional jump process, long- memory, options pricing.
|
|
|
On Selected Geometric Properties of Brownian Motion Paths
Honzl, Ondřej ; Rataj, Jan (advisor) ; Beneš, Viktor (referee) ; Mrkvička, Tomáš (referee)
Title: On Selected Geometric Properties of Brownian Motion Paths Author: Mgr. Ondřej Honzl E-mail Address: honzl@karlin.mff.cuni.cz Department: Department of Probability and Mathematical Statistics Supervisor: Prof. RNDr. Jan Rataj, CSc. E-mail Address: rataj@karlin.mff.cuni.cz Department: Mathematical Institute, Charles University Abstract: Our thesis is focused on certain geometric properties of Brownian motion paths. Firstly, it deals with cone points of Brownian motion in the plane and we show some connections between cone points and critical points of Brownian motion. The motivation of the study of critical points is provided by a pleasant behavior of the distance function outside of the set of these points. We prove the theorem on a non-existence of π+ cone points on fixed line. This statement leads us to the conjecture that there are only countably many critical points of the Brownian motion path in the plane. Next, the thesis discusses an asymptotic behavior of the surface area of r-neigh- bourhood of Brownian motion, which is called Wiener sausage. Using the proper- ties of a Kneser function, we prove the claim about the relation of the Minkowski content and S-content. As the consequence, we obtain a limit behavior of the surface area of the Wiener sausage almost surely in dimension d ≥ 3. Finally,...
|
|
|
Sekvenční metody Monte Carlo
Coufal, David ; Beneš, Viktor (advisor) ; Prokešová, Michaela (referee)
Title: Sequential Monte Carlo Methods Author: David Coufal Department: Department of Probability and Mathematical Statistics Supervisor: prof. RNDr. Viktor Beneš, DrSc. Abstract: The thesis summarizes theoretical foundations of sequential Monte Carlo methods with a focus on the application in the area of particle filters; and basic results from the theory of nonparametric kernel density estimation. The summary creates the basis for investigation of application of kernel meth- ods for approximation of densities of distributions generated by particle filters. The main results of the work are the proof of convergence of kernel estimates to related theoretical densities and the specification of the development of approx- imation error with respect to time evolution of a filter. The work is completed by an experimental part demonstrating the work of presented algorithms by simulations in the MATLABR⃝ computational environment. Keywords: sequential Monte Carlo methods, particle filters, nonparametric kernel estimates
|
|
|
Applications of Markov chains
Berdák, Vladimír ; Beneš, Viktor (advisor) ; Kadlec, Karel (referee)
The goal of the thesis is the use of Markov chains and applying them to algorithms of the method Monte Carlo. Necessary theory of Markov chains is introduced and we are aiming to understand stationary distribution. Among MCMC methods the thesis is focused on Gibbs sampler which we apply to the hard-core model. We subsequently simulate distribution of ones and zeros on vertices of a graph. Statistical characteristics of the number of ones are estimated from realizations of MCMC and presented in figures.
|
guest ::
RSS feed.