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Aspects of the notion of independence in probability theory
Anderle, Tim ; Mizera, Ivan (advisor) ; Pawlas, Zbyněk (referee)
The aim of the bachelor's thesis was to explore the independence of random events in greater depth and address less common aspects of this topic. The first part defines the independence of random events in probability theory and is illustrated with common examples. The next chapter defines and examines the Italian problem. Its already published proof in the article Balek a Mizera (1997) is explained, and a hint of an elementary proof is extended and analyzed for various values of n before being refuted. Finally, the thesis discusses the possibility of the existence of a measure of independence for a system of random events, as defined in the article Stoyanov (1995), along with its subsequent modifications for further use. 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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Tests of independence between two time series
Zdeněk, Pavel ; Pawlas, Zbyněk (advisor) ; Prášková, Zuzana (referee)
The goal of this diploma thesis is to introduce several tests of independence for time series following the ARMA model and then compare them within the simulation study. First, the basic theory of independence is reminded together with covariance and corre- lation. Asymptotic unbiasedness and consistency are derived for sample cross-covariance and also consistency for correlation. After the introduction of the ARMA model, each test is described and its advantages and disadvantages discussed. The following tests are included: Haugh test, using estimates of white noise and sample cross-correlation, modi- fied t-test, for which we assume weakly stationary series instead of random samples, and lastly distance covariance test, which uses properties of characteristic functions. These tests are compared in the simulation study together with the standard independence test using Pearson correlation coefficient. At the end, an illustrative example with finance data is presented. 1
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Causality, causality measures
Borák, Daniel ; Lachout, Petr (advisor) ; Pawlas, Zbyněk (referee)
Causality measures are useful tools when looking for causality in time series. This thesis does not only describe the theory behind the definition of measures of causality but also gives comprehensive instructions on how to use measures of causality to search for possible causal conditions. It is important to note that although measures of causality can point to possible causal relationships, they cannot confirm them. Causality is a complex relationship that cannot be captured by data alone - experiments and physical experience must also be considered. 1
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Pólya-Lundberg process
Böhm, Igor ; Pawlas, Zbyněk (advisor) ; Dvořák, Jiří (referee)
The main subject of the Bachelor's thesis is the P'olya-Lundberg process. It is a non-homogenous Markov chain that represents a generalization of the Poisson process. The main aim of the thesis is to depict some of its important features, to prove them and to put them into context. The thesis is sectioned into four chapters where the first chapter introduces basic concepts and objects that are crucial for understanding of this text. In the second chapter we define the P'olya-Lundberg process and we derive some of its main characteristics. The third chapter is devoted to the relationship between the P'olya-Lundberg process and the mixed Poisson process. Lastly, the final chapter discusses the so-called urn models, especially its generalization for which there is shown that if several conditions are fulfilled the generalized urn model converges to the P'olya-Lundberg process at a fixed time.
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Four-point problem
Hálová, Eliška ; Pawlas, Zbyněk (advisor) ; Prokešová, Michaela (referee)
In this thesis we analyze a well-known mathematical question known as the four point problem. It asks for the probability that four points taken at random in a plane form a convex quadrilateral. Since there is no concrete distribution of the random points stated in the original question, the problem does not have an unequivocal solution. In this work we consider three different probability distributions of the points, namely, continuous uniform distribution, discrete uniform distribution and bivariate normal distribution. Our assumption is that the points are mutually independent. We derive a detailed solution of the four point problem for each of the distributions. Additionally, we state some already existing results. 1
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Pólya-Aeppli process
Rada, Matej ; Pawlas, Zbyněk (advisor) ; Flimmel, Daniela (referee)
Táto práca je venovaná skúmaniu Pólyovho-Aeppliho procesu a zároveň Pólyovho- Aeppliho rozdelenia, ktoré sa v tomto procese využíva. Pri Pólyovom-Aeppliho rozdelení sú uvedené dva tvary pravdepodobnostnej funkcie - rekurzívny a explicitný. Popísané sú aj vlastnosti tohto rozdelenia. Pólyov-Aeppliho proces je zadefinovaný rôznymi spô- sobmi a odvodené sú vzťahy medzi týmito definíciami. Takisto sú popísané vlastnosti tohto procesu. Praktická časť je venovaná rôznym spôsobom, ako odhadnúť parametre Pólyovho-Aeppliho rozdelenia pre počty odohraných zápasov účastníkov grandslamových turnajoch. Nakoniec je uvedené porovnanie týchto spôsobov. 1
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Spatial epidemiology
Jalovcová, Adéla ; Dvořák, Jiří (advisor) ; Pawlas, Zbyněk (referee)
This work deals with spa al sta s cs methods that are suitable for analysing spa al epidemiological data. The work presents tests of spa al autocorrela on and applies them on data of the number of people infected by Covid 19. The main part of the work is Bayesian modelling of epidemiological data using Integrated Nested Laplace Approxima ons. We summarise the main principles of this method and present a chosen model for given data. Besides the spa al aspect of the data, the work shows how to incorporate other risk factors into the model and how to make the model spa o-temporal. Furthermore the work applies the model on the data and tests the suitability of the model with a global envelope test.
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Asymptotic inference for stochastic geometry models
Flimmel, Daniela ; Pawlas, Zbyněk (advisor) ; Schulte, Matthias (referee) ; Rataj, Jan (referee)
We compare three methods used in stochastic geometry in order to investigate asymp- totic behaviour of random geometrical structures in large domains or in a large intensity regime. Namely, we describe in detail the Malliavin-Stein method, the method of sta- bilization and the method of cumulants. Then, we discuss some of its possible variants, combinations or extensions. Each method is supplemented with numerous examples con- cerning limit behaviour of different kinds of point processes, random tessellations and graphs or particle processes. Specially, for a geometric characteristic of the typical cell in a weighted Voronoi tessellation, we use the minus-sampling technique to construct an unbiased estimator of the average value of this characteristic and using the method of stabilization, we establish variance asymptotic and the asymptotic normality of such es- timator. Next, we study asymptotic properties of a cylinder process in the plane derived by a Brillinger-type mixing point process. We prove a weak law of large numbers as well as a formula of the asymptotic variance for the area of the process. Under comparatively stronger assumptions, we also derive a central limit theorem for the cylinder process using the method of cumulants. 1
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Mixing of Markov chains - spectral methods
Hotmar, Vojtěch ; Prokešová, Michaela (advisor) ; Pawlas, Zbyněk (referee)
In this thesis, we deal with the upper and lower bounds for the mixing time of reversi- ble homogeneous Markov chains with finite state space and discrete time. The estimates are based on the spectral properties of the transition matrices belonging to these chains. Primarily, we are interested in the eigenvalues of these matrices and how they relate to the rate of convergence. Next we will describe what the product chains and the projecti- ons of Markov chains are. And also that their spectral properties can be easily derived from the properties of the chains on which these chains are built. These properties and estimates are shown on several illustrative examples. 1
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