
Depth of twodimensional data
Dočekalová, Denisa ; Šír, Zbyněk (advisor) ; Hlubinka, Daniel (referee)
In this paper we summarize the basic information about halfplane depth function. It consists of two parts. In the first part we deal with the halfplane depth based on the distribution function, we describe its basic properties and define the concepts of depth contours, central regions and the halfplane median. We also deal with these concepts in the rest of the paper with the main focus on the halfplane median. In the second part of this work we deal with the halfplane depth based on the random choice with the main focus on data visualization. The used methods for visualization are the display of depth contours and the bagplot. This work includes pictures of depth contours for specific distributions which were gained by implementation of an algorithm in the software Mathematica. 1


Nonparametric tests of independence
Kmeťková, Diana ; Pawlas, Zbyněk (advisor) ; Hlubinka, Daniel (referee)
The main objective of this thesis is the presentation regarding the problem of testing independence between two random variables in the nonparametric model of continuous cumulative distribution functions. Firstly, the reader is informed with basic notions from the theory of independence and rank tests. Afterwards, few of the most common methods for testing independence are introduced. In the beginning, the test based on Pearson's correlation coefficient is mentioned as a representative for parametric tests, then we continue with nonparametric tests, such as test based on Spearman's, Kendall's and distance correlation coefficient. We focus in better detail on Hoeffding's test of independence, which results to be consistent against all alternatives in the model of continuous cumulative distribution functions. In the end, we compare and evaluate presented methods for testing independence using simulations in R environment.


Continuous market models with stochastic volatility
Petrovič, Martin ; Maslowski, Bohdan (advisor) ; Hlubinka, Daniel (referee)
Vilela Mendes et al. (2015), based on the discovery of longrange dependence in the volatility of stock returns, proposed a stochastic volatility continuous mar ket model where the volatility is given as a transform of the fractional Brownian motion (fBm) and studied its NoArbitrage and completeness properties under va rious assumptions. We investigate the possibility of generalization of their results from fBm to a wider class of Hermite processes. We have reworked and completed the proofs of the propositions in the cited article. Under the assumption of indepen dence of the stock price and volatility driving processes the model is arbitragefree. However, apart from a case of a special relation between the drift and the volatility, the model is proved to be incomplete. Under a different assumption that there is only one source of randomness in the model and the volatility driving process is bounded, the model is arbitragefree and complete. All the above results apply to any Hermite process driving the volatility. 1


Incomplete Poisson samples
Zeman, Ondřej ; Dvořák, Jiří (advisor) ; Hlubinka, Daniel (referee)
The topic of my bachelor thesis is studying truncated Poisson sample which is a part of a sample from Poisson distribution, where zero observations are missing. The main goal is estimating the size of the original sample and the parameter λ of the Poisson distribution. In the first chapter I mainly focus on deriving three types of estimators of these parameters and I describe their basic properties. Second chapter contains simulations where the estimators from the first chapter are compared based on the estimates of relative bias and relative mean square error. Eventually in the third chapter I focus on the asymptotic properties of derived estimators with emphasis on consistency of estimators. 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.


Stochastic Evolution Equations
Čoupek, Petr ; Maslowski, Bohdan (advisor) ; GarridoAtienza, María J. (referee) ; Hlubinka, Daniel (referee)
Stochastic Evolution Equations Petr Čoupek Doctoral Thesis Abstract Linear stochastic evolution equations with additive regular Volterra noise are studied in the thesis. Regular Volterra processes need not be Gaussian, Markov or semimartingales, but they admit a certain covariance structure instead. Particular examples cover the fractional Brownian motion of H > 1/2 and, in the nonGaussian case, the Rosenblatt process. The solution is considered in the mild form, which is given by the variation of constants formula, and takes values either in a separable Hilbert space or the space Lp(D, µ) for large p. In the Hilbertspace setting, existence, spacetime regularity and largetime behaviour of the solutions are studied. In the Lp setting, existence and regularity is studied, and in concrete cases of stochastic partial differential equations, the solution is shown to be a spacetime continuous random field.

 

Bayesian optimization
Kostovčík, Peter ; Lachout, Petr (advisor) ; Hlubinka, Daniel (referee)
Optimization is an important part of mathematics and is mostly used for practical applications. For specific types of objective functions, a lot of different methods exist. A method to use when the objective is unknown and/or expensive can be difficult to determine. One of the answers is bayesian optimization, which instead of direct optimization creates a probabilistic model and uses it to constructs easily optimizable auxiliary function. It is an iterative method that uses information from previous iterations to find new point in which the objective is evaluated and tries to find the optimum within a fewer iterations. This thesis introduces bayesian optimization, suma rizes its different approaches in lower and higher dimensions and shows when to use it suitably. An important part of the thesis is my own optimization algorithm which is applied to different practical problems  e.g. parameter optimization in machine learning algorithm. 1


Statistics of directional data with an application in crystallography
Karafiátová, Iva ; Šedivý, Ondřej (advisor) ; Hlubinka, Daniel (referee)
The main purpose of this thesis is to derive methods used to estimate the mean orientation of a grain in polycrystalline material. The estimation is highly impor tant in crystallography. However, it is often expressed by only one representative value regardless of its variability within the grain. In this thesis we present the basic concepts of crystallography, including the principle of the Electron Backs catter Diffraction, which is used to explore the microstructure of a polycrystalline materials. Subsequently, we present six of the most commonly used descriptors of threedimensional orientation of a crystal lattice. This overview is completed by the summary of the particular relationships between those descriptors. In the practical part of this thesis we apply derived methods on real data within the scope of research of microstructure of an aluminum alloy. 1


Ensemble Kalman filter on high and infinite dimensional spaces
Kasanický, Ivan ; Hlubinka, Daniel (advisor) ; Pannekoucke, Olivier (referee) ; Antoch, Jaromír (referee)
Title: Ensemble Kalman filter on high and infinite dimensional spaces Author: Mgr. Ivan Kasanický Department: Department of Probability and Mathematical Statistics Supervisor: doc. RNDr. Daniel Hlubinka, Ph.D., Department of Probability and Mathematical Statistics Consultant: prof. RNDr. Jan Mandel, CSc., Department of Mathematical and Statistical Sciences, University of Colorado Denver Abstract: The ensemble Kalman filter (EnKF) is a recursive filter, which is used in a data assimilation to produce sequential estimates of states of a hidden dynamical system. The evolution of the system is usually governed by a set of di↵erential equations, so one concrete state of the system is, in fact, an element of an infinite dimensional space. In the presented thesis we show that the EnKF is well defined on a infinite dimensional separable Hilbert space if a data noise is a weak random variable with a covariance bounded from below. We also show that this condition is su cient for the 3DVAR and the Bayesian filtering to be well posed. Additionally, we extend the already known fact that the EnKF converges to the Kalman filter in a finite dimension, and prove that a similar statement holds even in a infinite dimension. The EnKF su↵ers from a low rank approximation of a state covariance, so a covariance localization is required in...
