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Sample Quantiles of Discrete Distributions
Štarmanová, Petra ; Prokešová, Michaela (advisor) ; Pawlas, Zbyněk (referee)
Sample quantiles for discrete distributions The classical definition of sample quantiles and their asymptotic properties for absolutely continuous distributions are well known. This no longer applies to discrete distributions. This thesis deals with implementing new quantiles based on the mid-distribution function, and their properties. The theoretical part shows the asymtotic properties of sample quantiles based on the mid-distribution function for both discrete and absolutely continuous distributions. In the absolutely continuous case it shows that the results are the same as those for the classical sample quantiles. In the discrete case the asymptotic distribution is normal. The practical part includes the exact distribution of these new sample quantiles for the binomial distribution. This thesis also includes a small simulation study for the standardized normal distribution and binomial distribution.
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Random tessellations modeling
Seitl, Filip ; Beneš, Viktor (advisor) ; Pawlas, Zbyněk (referee)
The motivation for this work comes from physics, when dealing with microstructures of polycrystalline materials. An adequate probabilistic model is a three-dimensional (3D) random tessellation. The original contribution of the author is dealing with the Gibbs-Voronoi and Gibbs- Laguerre tessellations in 3D, where the latter model is completely new. The energy function of the underlying Gibbs point process reflects interactions between geometrical characteristics of grains. The aim is the simulation, parameter estimation and degree-of-fit testing. Mathematical background for the methods is described and numerical results based on simulated data are presented in the form of tables and graphs. The interpretation of results confirms that the Gibbs-Laguerre model is promising for further investigation and applications.
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Spatio-temporal point processes
Kratochvílová, Blažena ; Beneš, Viktor (advisor) ; Volf, Petr (referee) ; Pawlas, Zbyněk (referee)
The background theory of point processes, spatio-temporal point processes, random measures and random closed sets is given in the beginning of the thesis. Then the special case of spatio-temporal Cox processes constructed from L'evy basis is studied. Formulas for theoretical characteristics are derived using the generating functional. The Cox process on the curve is defined and studied. The analysis of such a process leads to nonlinear filtering methods. Also the methods for model selection are discussed. These methods are used on simulated data, firstly on the simple discrete data and secondly on the continuous data where the curve is a spiral. Then the real data from a neurophysiology experiment is analysed. During the experiment, the spiking activity of a place cell of hippocampus of a rat moving in an arena together with the track of the rat was recorded. The track of the rat and the action potentials (spikes) present the curve and the points on it. At the end of the thesis, other approaches to neurophysiological data are discussed. The first one is an estimation of a conditional intensity of the temporal process of spikes using recursive filtering. In the second one, the track of the rat together with the random driving intensity function of the process of the spikes is viewed as a random marked set.
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Modeling of duration between financial transactions
Voráčková, Andrea ; Zichová, Jitka (advisor) ; Pawlas, Zbyněk (referee)
❆❜str❛❝t ❚❤✐s ❞✐♣❧♦♠❛ t❤❡s✐s ❞❡❛❧s ✇✐t❤ ♣r♦♣❡rt✐❡s ♦❢ ❆❈❉ ♣r♦❝❡ss ❛♥❞ ♠❡t❤♦❞s ♦❢ ✐ts ❡st✐♠❛t✐♦♥✳ ❋✐rst✱ t❤❡ ❜❛s✐❝ ❞❡☞♥✐t✐♦♥s ❛♥❞ r❡❧❛t✐♦♥s ❜❡t✇❡❡♥ ❆❘▼❆ ❛♥❞ ●❆❘❈❍ ♣r♦❝❡ss❡s ❛r❡ st❛t❡❞✳ ■♥ t❤❡ s❡❝♦♥❞ ♣❛rt ♦❢ t❤❡ t❤❡s✐s✱ t❤❡ ❆❈❉ ♣r♦❝❡ss ✐s ❞❡☞♥❡❞ ❛♥❞ t❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ ❆❘▼❆ ❛♥❞ ❆❈❉ ✐s s❤♦✇♥✳ ❚❤❡♥ ✇❡ s❤♦✇ t❤❡ ♠❡t❤♦❞s ♦❢ ❞❛t❛ ❛❞❥✉st♠❡♥t✱ ❡st✐♠❛t✐♦♥✱ ♣r❡❞✐❝t✐♦♥ ❛♥❞ ✈❡r✐☞❝❛t✐♦♥ ♦❢ t❤❡ ❆❈❉ ♠♦❞❡❧✳ ❆❢t❡r t❤❛t✱ t❤❡ ♣❛rt✐❝✉❧❛r ❝❛s❡s ♦❢ ❆❈❉ ♣r♦❝❡ss✿ ❊❆❈❉✱ ❲❆❈❉✱ ●❆❈❉✱ ●❊❱❆❈❉ ✇✐t❤ ✐ts ♣r♦♣❡rt✐❡s ❛♥❞ t❤❡ ♠♦t✐✈❛t✐♦♥❛❧ ❡①❛♠♣❧❡s ❛r❡ ✐♥tr♦❞✉❝❡❞✳ ❚❤❡ ♥✉♠❡r✐❝❛❧ ♣❛rt ✐s ♣❡r❢♦r♠❡❞ ✐♥ ❘ s♦❢t✇❛r❡ ❛♥❞ ❝♦♥❝❡r♥s t❤❡ ♣r❡❝✐s✐♦♥ ♦❢ t❤❡ ❡st✐♠❛t❡s ❛♥❞ ♣r❡❞✐❝t✐♦♥s ♦❢ t❤❡ s♣❡❝✐❛❧ ❝❛s❡s ♦❢ ❆❈❉ ♠♦❞❡❧ ❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡ ❧❡♥❣t❤ ♦❢ s❡r✐❡s ❛♥❞ ♥✉♠❜❡r ♦❢ s✐♠✉❧❛t✐♦♥s✳ ■♥ t❤❡ ❧❛st ♣❛rt✱ ✇❡ ❛♣♣❧② t❤❡ ♠❡t❤♦❞s st❛t❡❞ ✐♥ t❤❡♦r❡t✐❝❛❧ ♣❛rt ♦♥ r❡❛❧ ❞❛t❛✳ ❚❤❡ ❛❞❥✉st♠❡♥t ♦❢ t❤❡ ❞❛t❛ ❛♥❞ ❡st✐♠❛t✐♦♥ ♦❢ t❤❡ ♣❛r❛♠❡t❡rs ✐s ♣❡r❢♦r♠❡❞ ❛s ✇❡❧❧ ❛s t❤❡ ✈❡r✐☞❝❛t✐♦♥ ♦❢ t❤❡ ❆❈❉ ♠♦❞❡❧✳ ❆❢t❡r t❤❛t✱ ✇❡ ♣r❡❞✐❝t ❢❡✇ st❡♣s ❛♥❞ ❝♦♠♣❛r❡ t❤❡♠ ✇✐t❤ r❡❛❧ ❞✉r❛t✐♦♥s✳ ✶
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Measures of extremal dependence in time series
Popovič, Viktor ; Pawlas, Zbyněk (advisor) ; Hudecová, Šárka (referee)
In the present thesis we deal with dependence among extremal values within time series. Concerning this type of relations the commonly used autocorrelation function does not provide sufficient information. Moreover, autocorrelation function is suitable for Gaussian processes while nowadays we often work with heavy-tailed time series. In this thesis we cover two measures of extremal dependence that are used for this type of data. We introduce the coefficient of tail dependence, measure of extremal dependence based on tail characteristics of joint survival function. The second measure is called extremogram, which depends only on the extreme values in the sequence. In addition to the theoretical part, simulation study and application to real data of both described measures including their comparison are performed. Results are stated together with tables and graphical output.
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Statistical properties of local stereological estimators
Hájek, Tadeáš ; Pawlas, Zbyněk (advisor) ; Beneš, Viktor (referee)
In this thesis statistical properties of local stereological estimators of par- ticle volume are investigated. The emphasis is on the estimation of the va- riance of the local estimator and its components - the variance due to the variability in the particle size distribution and the variance due to the lo- cal stereological procedures. Various ways of estimation for independent and correlated particles are presented. Results of simulation studies for both inde- pendent and correlated ellipsoidal-shaped particles are presented. Described estimators are demonstrated on real biological data. Comprehensive theory that leads to the local stereological estimators of volume is presented too. 1
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Continuous Time Linear Quadratic Optimal Control
Vostal, Ondřej ; Maslowski, Bohdan (advisor) ; Pawlas, Zbyněk (referee)
We partially solve the adaptive ergodic stochastic optimal control problem where the driving process is a fractional Brownian motion with Hurst parameter H > 1/2. A formula is provided for an optimal feedback control given a strongly consistent estimator of the parameters of the controlled system is avail- able. There are some special conditions imposed on the estimator which means the results are not completely general. They apply, for example, in the case where the estimator is independent of the driving fractional Brownian motion. In the course of the thesis, construction of stochastic integrals of suitable determinis- tic functions with respect to fractional Brownian motion with Hurst parameter H > 1/2 over the unbounded positive real half-line is presented as well. 1
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Projections of space-time point processes
Nguyenová, Adéla ; Dvořák, Jiří (advisor) ; Pawlas, Zbyněk (referee)
The main topic of this bachelor's thesis is the theory of spatio-temporal point processes, focusing primarily on the derivation of the second-order moment characteristics of processes obtained through projection of the spatio-temporal process into its space, respectively its time domain. Emphasis is placed on the pair correlation function. The moment characteristics of the Thomas spatio-temporal point process model are derived. Finally the theory is complemented by simulations of its realizations and realizations of its projections into the space domain, together with estimates of the pair correlation function.
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Covering the circle by random arcs
Čelikovská, Klára ; Pawlas, Zbyněk (advisor) ; Dvořák, Jiří (referee)
In this thesis we consider the geometric probability problem of covering a circle with random arcs. We randomly place arcs of a fixed length on a circle of unit circumference. First we find the probability of covering the entire circle with a finite number of arcs of the same length and show some of its numerical values. Next we study the random variable describing the size of the covered part of the circle and the expected number of arcs needed to fully cover the circle if we place the arcs sequentially. Finally, we solve a similar problem of covering the circle by a countably infinite number of arcs of different lengths. 1
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Modelling of segment process in the plane
Pultar, Milan ; Beneš, Viktor (advisor) ; Pawlas, Zbyněk (referee)
We consider a finite planar segment process in a circle, having a density with respect to the Poisson process. This density involves unknown parameters and a reference length distribution which is not observed. The aim is to estimate these quantities semiparametrically. The segment process is inhomogeneous, but it is isotropic. Combining the relation between the observed and reference length distribution and the maximum pseudolikelihood method we suggest an estimation procedure. Its properties (bias and variability) are investigated in a simulation study. In the last part we present two more complex models. The motivation is to model stress fibers observed in cultured stem cells.
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