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Multivariate Cox point processes
Kuželová, Noemi ; Dvořák, Jiří (advisor) ; Prokešová, Michaela (referee)
The Log-Gaussian Cox process is an important example of the use of spatial modeling and spatial statistics in practice. It is useful for describing many real-world situations, from modeling tree growth in the rainforests, to trying to understand the occurrence of precipitation and earthquakes, to examining the expansion of the Greenland seal pop- ulation. In this work we focus mainly on the multivariate form of this point process. Specially in such form that allows to describe at the same time inhomogeneity, clus- tering and environmental effects in the investigated system. When the parameters of multivariate LGCP process are estimated, the minimum contrast method is usually used. However, we investigate the possibility of using composite likelihood estimation instead. We consider the composite likelihood criterion as a limit of the likelihoods in approxi- mating discrete models. We compare it with an established approach of constructing the composite likelihood based on multiplication of likelihoods for pairs of points. 1
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A point process driven by a Gaussian field
Scheib, Karel ; Beneš, Viktor (advisor) ; Šedivý, Ondřej (referee)
The thesis investigates the search for dimension reduction subspace for the Poisson point process driven by a Gaussian random eld. The work describes the method called sliced inverse regression, which is applied to a point process driven by random eld. Its functionality in mentioned context is then proved. This method is in several ways implemented and tested in R software environment on random data. The individual implementations are described and results are then compared with each other.
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Methods of modelling and statistical analysis of an extremal value process
Jelenová, Klára ; Volf, Petr (advisor) ; Branda, Martin (referee)
In the present work we deal with the problem of etremal value of time series, especially of maxima. We study times and values of maximum by an approach of point process and we model distribution of extremal values by statistical methods. We estimate parameters of distribution using different methods, namely graphical methods of data analysis and subsequently we test the estimated distribution by tests of goodness of fit. We study the stationary case and also the cases with a trend. In connection with distribution of excesess and exceedances over a threshold we deal with generalized Pareto distribution.
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Random closed sets and particle processes
Stroganov, Vladimír ; Rataj, Jan (advisor) ; Pawlas, Zbyněk (referee)
In this thesis we are concerned with representation of random closed sets in Rd with values concentrated on a space UX of locally finite unions of sets from a given class X ⊂ F. We examine existence of their repre- sentations with particle processes on the same space X, which keep invariance to rigid motions, which the initial random set was invariant to. We discuss existence of such representations for selected practically applicable spaces X: we go through the known results for convex sets and introduce new proofs for cases of sets with positive reach and for smooth k-dimensional submanifolds. Beside that we present series of general results related to representation of random UX sets. 1
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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)
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Cluster point processes in insurance mathematics
Veselá, Veronika ; Pawlas, Zbyněk (advisor) ; Dostál, Petr (referee)
Title: Cluster point processes in insurance mathematics Author: Veronika Veselá Department: Department of Probability and Mathematical Statistics Supervisor: RNDr. Zbyněk Pawlas, Ph.D. Abstract: In the present work we study point processes and their importance in insurance mathematics. With the help of cluster and marked point processes we can describe a model that considers times of claim occurence and times and hei- ghts of corresponding payments. We study two specific models which can be used to predict how much money is needed for claims which happened. The first model is chain ladder in the form of Mack's model. For this model we show chain ladder estimators of development factors, estimates of their variance and their proper- ties. We try to find one-step ahead prediction and multi-step ahead prediction, which we use for calculating prediction of reserves. We shortly review asymptotic properties of the estimators in Mack's model. The second model is the Poisson cluster model. Firstly we define this model and the variables entering the model. Then we devote attention to one-step ahead and multi-step ahead prediction. We also study prediction when some variables have specific distributions. Finally, we use both methods of prediction on simulated data and compare their average relative absolute errors....
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Estimation of the pair correlation function of a point process
Vondráček, Jakub ; Dvořák, Jiří (advisor) ; Beneš, Viktor (referee)
This thesis deals with kernel estimation of the pair correlation function of a stationary and isotropic point process. Firstly, the basics of the theory of point processes are built up. Then, the derivation of formulas for expectation and variance of a kernel estimator is provided. Also, an extension of a simple Poisson approximation of variance to the case of an estimator with more complicated edge correction compared to what is usually used in the literature is given. These formulas depend on a parameter called bandwidth. The recommendations for selecting the bandwidth that can be found in the literature are summarised and simulation experiments are performed to assess the correctness of the derived formulas. These experiments also prove that a variance approximation obtained by ignoring so called "higher order correlations" is unjustified. Lastly, bandwidth selection and the advantages and disadvantages of several approaches for bandwidth selection are discussed. 1
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Stochastic reconstruction of random point patterns
Koňasová, Kateřina ; Dvořák, Jiří (advisor) ; Prokešová, Michaela (referee)
Point procesess serve as stochastic models for locations of objects that are ran- domly placed in space, e.g. the locations of trees of a given species in a forest stand, earthquake epicenters or defect positions in industrial materials. Stochas- tic reconstruction is an algorithmic procedure providing independent replicates of point process data which may be used for various purposes, e.g. testing sta- tistical hypothesis. The main advantage of this technique is that we do not need to specify any theoretical model for the observed data, only the estimates of se- lected summary characteristics are employed. Main aim of this work is to discuss the possibility of extension of the stochastic reconstruction algorithm for inho- mogeneous point patterns. 1
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Existence and uniqueness of the distribution of a random measure given by finite dimensional projections
Jurčo, Adam ; Rataj, Jan (advisor) ; Pawlas, Zbyněk (referee)
Title: Existence and uniqueness of the distribution of a random measure given by finite dimensional projections Author: Adam Jurčo Department: Department of Probability and Mathematical Statistics Supervisor: prof. RNDr. Jan Rataj, CSc., Department of Probability and Mathe- matical Statistics Abstract: This thesis deals with the existence and uniqueness of the distribu- tion of a random measure given a system of finite-dimensional distributions. A random measure can be interpreted as a particular system of random variables. Conversely, we will want to know what conditions would allow a system of random variables to be extended to a random measure and if this extension is unique. We will start with a consistent system of finite-dimensional distributions and use Daniell-Kolmogorov theorem to find the necessary and sufficient conditions for the existence of such extension. A counterexample will be included to show that it is not possible to use this theory for random signed measures. Keywords: Random measure, point process, finite-dimensional distributions. 1
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Random closed sets and particle processes
Stroganov, Vladimír ; Rataj, Jan (advisor) ; Pawlas, Zbyněk (referee)
In this thesis we are concerned with representation of random closed sets in Rd with values concentrated on a space UX of locally finite unions of sets from a given class X ⊂ F. We examine existence of their repre- sentations with particle processes on the same space X, which keep invariance to rigid motions, which the initial random set was invariant to. We discuss existence of such representations for selected practically applicable spaces X: we go through the known results for convex sets and introduce new proofs for cases of sets with positive reach and for smooth k-dimensional submanifolds. Beside that we present series of general results related to representation of random UX sets. 1
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