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Bayesian inference for anisotropic cluster point processes
Pavlovičová, Diana ; Dvořák, Jiří (advisor) ; Pawlas, Zbyněk (referee)
Point processes are stochastic models widely used in biology, forestry, or astronomy. In this thesis, we are going to deal mainly with anisotropic cluster point processes. We present a new method for estimating parameters of such models. The basis of this method is the use of Bayesian statistics combined with Markov Chain Monte Carlo algorithms, which are a useful way to estimate parameters which are difficult or impossible to estimate using traditional methods. We describe the method in detail and present several examples of its application to simulated and real-life datasets and discuss the difficulties associated with it. Finally, we prove theoretical results about the convergence of the corresponding Markov chain under specific assumptions on the model and discuss the difficulties we encounter when examining these properties.
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Variance of the product of random variables
Danácsová, Michaela ; Pawlas, Zbyněk (advisor) ; Dvořák, Jiří (referee)
This thesis focuses on the variance of the product of random variables and its appli- cation in specific examples. The work is systematically divided into four chapters, where the variance for independent and dependent random variables is derived. We will focus on deriving the variance in cases of random samples and paired observations. Throughout all the chapters, we will also focus on linear approximation, which is compared throu- ghout the work to the exact form of the variance of the product of random variables. The final chapter illustrates the derived theory on examples from discrete and continuous distributions. The contribution of this bachelor's thesis is the clarification and understan- ding of the issue of the variance of the product of random variables and the application of theoretical derivations on specific distributions. 1
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Poisson cluster model
Růžička, Tomáš ; Pawlas, Zbyněk (advisor) ; Flimmel, Daniela (referee)
In this master thesis, we introduce the Poisson cluster process using the marked Poisson process. At first, we mention general definitions and then we move on to the interpretation of this model in insurance mathematics in nonlife reserving. We derive the future predictions and the mean squared errors. For a practical application of this model we propose estimators of these predictions. Then we describe alternative reserving methods that we use to compare the results in the simulation study and in the application to the real data. The chosen alternative methods are the Mack chain ladder and the generalized linear model. 1
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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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