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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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Influence of injection dose and body parameters on PET image quality by means of Monte Carlo simulations
Dvořák, Jiří ; Boldyš, Jiří (advisor) ; Prokešová, Michaela (referee)
Positron emission tomography (PET) is an imaging technique allowing to determine radiotracer distribution in a patient's body. This work reviews basic principles of PET imaging. It also uses the random field theory to detect locations with increased radiotracer uptake. This procedure is tested on a collection of simulated PET images. The aim of this work is to describe the quality of simulated PET images in terms of both the patient's physical parameters and the amount of applied radiopharmaceutical. The relations are used to provide curves of constant quality determining the amount of radiopharmaceutical needed to achieve desired quality of the resulting images. The resulting curves are compared with the formula currently used in medical practice.
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Algoritmic applications of finite Markov chains
Pavlačková, Petra ; Prokešová, Michaela (advisor) ; Staňková Helisová, Kateřina (referee)
Title: Algorithmic applications of finite Markov chains Author: Petra Pavlačková Department: Department of Probability and Mathematical Statistics Supervisor: RNDr. Michaela Prokešová, Ph.D. Supervisor's e-mail address: prokesov@karlin.mff.cuni.cz In the present work we study MCMC algorithms, that we use for simulating from probability distributions on finite set of states. We apply these algorithms to two models: hard-core model and q-coloring of a graph. In this work we use the theory of stochastic processes, mainly of Markov chains and their properties. Furhter we analyze some problems, which may occur during the simulation, particularly we focus on convergence of the marginal distribution of the Markov chain to the stationary distribution. The last part of the work is a numeric illustration of the Gibbs sampler which we use in order to estimate the mean value of the number of 1 in a generalized hard-core model. Keywords: Markov chain, MCMC algorithm, hard-core model, speed of convergence
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Mixing cards and convergence of Markov chains
Drašnar, Jan ; Prokešová, Michaela (advisor) ; Beneš, Viktor (referee)
This thesis presents mixing of a deck of cards as a random walk on the group of permutations. Perfectly shuffled deck of cards is defined as uniform distribution on this group. For analysis of the distance between the uniform distribution and the current distribution of the Markov chain generated by the shuffling quite general methods are used that can be applied to many other problems - i.e. strong stacionary time, coupling and transformation to an inverse distribution. In the last chapter the riffle shuffle is studied and a rather well-known fact is proved that seven or eight shuffles should be enough to shuffle a deck of 52 cards.
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