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Truncated counting processes
Přítel, Ondřej ; Pešta, Michal (advisor) ; Prokešová, Michaela (referee)
The aim of this thesis is the prediction of insurance events under the condition that the data related to the occurrence of the events is truncated. The nature of the truncation lies in the fact that in the present we observe only those events that were already reported to the insurance company. Occurrences and reporting are modeled by a two-dimensional non-homogeneous Poisson process. The intensity of occurrences is derived from Kingman's Displacement theorem and is computed as a convolution of the intensity of reporting and the density of the delay in between occurrences and reporting. The estimations of the parametric function of the intensity of reporting and the distribution are preformed using the maximum likelihood method. In addition, theoretical background concerning counting processes primarily directed to the Poison processes is discussed in this thesis. 1
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Coupling, transportation metrics and applications to approximate counting
Kluvancová, Rozálie ; Prokešová, Michaela (advisor) ; Swart, Jan (referee)
An important property of discrete-time Markov chains with finite state space is the rate of convergence of the marginal distribution of the chain to the stationary distribution (i.e. mixing rate). If we construct a coupling of two Markov chains with the same transition matrix, where one starts from a stationary distribution and the other from a fixed state, we can use it to estimate the mixing rate. The main goal of this thesis is to describe how we can construct such a coupling using the transportation metric, and to apply this method to approximate counting of all proper colorings of the graph. 1
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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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Random walks on the symmetric group - how many times should you shuffle a deck of cards
Hruška, Martin ; Prokešová, Michaela (advisor) ; Hlubinka, Daniel (referee)
This thesis deals with random walks on a symmetric group, namely the models that are used to describe the shuffling of a deck of cards. In this work we focus on the question of mixing speed (the speed of convergence of the marginal distribution of a random walk to its stationary distribution). We ask ourselves a basic question when shuffling cards: how many times do the cards need to be shuffled so that they are already sufficiently randomly distributed. The random walk model, which is a Markov chain, is the mathematical formalization of the card shuffling process. We transfer the card shuffling problem to the problem of estimating the distance between the marginal distribution of this Markov chain and its stationary distribution. We then use standard methods to estimate the convergence rate of the Markov chain to its stationary distribution, such as strong stationary times. 1
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Parametric estimation of the intensity function of point processes
Rybín, Jan ; Dvořák, Jiří (advisor) ; Prokešová, Michaela (referee)
The thesis introduces spatial point processes. Particularly, it focuses on Poisson process, Thomas process and intensity function, which describes those two processes. The main focus is put on processes that depend on an unknown parameter. It is shown that in order to find an estimate of the unknown parameter even for general processes, it is reasonable to use maximum likelihood function derived from Poisson processes. All new terminology is explained in detail with the help of simple examples. The new terminology is then used in simulation studies that compare qualities of estimates in different statistical models. 1
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Estimation of the K-function of a point process using global normalization
Funková, Veronika ; Dvořák, Jiří (advisor) ; Prokešová, Michaela (referee)
Point processes are random local finite sets of points in a space that are used for mod- elling and subsequent spatial data analysis. Same of their useful characteristics are the pair correlation function and also the K-function, which describe point interactions with respect to the distance between points. There are several ways to include informa- tion about the non-constant intensity function in the estimates of these characteristics for inhomogeneous processes. In the older estimate, we use information about a value of the intensity function only in places where the process points are located. However, the new estimate works with a value of the intensity function within the whole observation window. In this thesis we focus on the comparison of these two estimates. In the third chapter we theoretically present these estimates and in the fourth chapter we compare their behaviour based on simulations of 8 point process models, while finding the optimal value of bandwidth for their kernel estimates. 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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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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Statistical inference for Markov processes with continuous time
Křepinská, Dana ; Prokešová, Michaela (advisor) ; Lachout, Petr (referee)
Tato diplomová práce se zabývá odhadováním matice intenzit Markovova pro- cesu se spojitým časem na základě diskrétně pozorovaných dat. Začátek práce je věnován jednoduššímu odhadu ze spojité trajektorie pomocí metody maximální věrohodnosti. Dále je zde popsán odhad z diskrétní trajektorie přes výpočet ma- tice pravděpodobností přechodu. Následně je velmi podrobně rozebrán EM al- goritmus, který předchozí odhad zpřesňuje. Na závěr teoretické části je uvedena metoda odhadu zvaná Monte Carlo Markov Chain. Všechny postupy jsou zároveň implementovány v počítačovém softwaru a prezentace jejich výsledk· je obsahem druhé části práce. V té jsou porovnané odhady pro denní, týdenní a měsíční po- zorování a také pro pětiletou a desetiletou pozorovanou trajektorii. K výsledk·m jsou připojeny odhady rozptyl· a intervaly spolehlivosti. 1
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