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Second-order characteristics of point processes
Gupta, Archit ; Pawlas, Zbyněk (advisor) ; Prokešová, Michaela (referee)
In this thesis we examine estimation of the K-function which is an important second-order characteristic in the theory of spatial point processes. Besides Ripley's K-function based on a spherical structuring element we also work with the multiparameter K-function where the struc- turing element is rectangular. We consider the Poisson point process model, which is the fundamental model for complete spatial randomness. We de- rive expressions for both bias and variance of the estimators. The primary goal of this thesis is the study of different edge correction methods that are available for the K-function. Using simulations we also study a few variance approximations proposed in the literature and compare them with empirical variances. 1
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Multivariate point processes and their application on neurophysiological data
Bakošová, Katarína ; Pawlas, Zbyněk (advisor) ; Prokešová, Michaela (referee)
This thesis examines a multivariate point process in time with focus on a mu- tual relations of its marginal point processes. The first chapter acquaints the re- ader with the theoretical background of multivariate point processes and their properties, especially the higher-order cumulant-correlation measures. Later on, several models of multivariate point processes with different dependence structu- res are characterized, such as the random superposition model, a Poisson depen- dent superposition point process, a jitter Poisson dependent superposition point process orrenewal processes models. Simulations of each of them are provided. Furthermore, two statistical methods for higher-order correlations are presented; the cumulant based inference of higher-order correlations, and the extended til- ling coefficient. Finally, the introduced methods are applied not only on the data from simulations, but also on the real, simultaneously recorded nerve cells spike train data. The results are discussed. 1
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Buffon needle problem and its generalizations
Hledík, Jakub ; Pawlas, Zbyněk (advisor) ; Prokešová, Michaela (referee)
This thesis contains detailed derivation of results of several generalizations of the Buffon needle problem. Next to the original problem we study grids composed of rectangles, known as Buffon-Laplace needle problem, then grids composed of parallelograms, triangles or hexagons. The application of this problem is briefly shown on the estimation of π, additional references are mentioned. We provide a proof of the theorem computing the area of a polygon, if the Cartesian coordi- nates of its vertices are known. Finally, we show how to solve grids composed of several different shapes, this is demonstrated on the grid composed of a regular hexagon and a diamond. 1
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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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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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Alternative K-functions for stationary point processes
Koňasová, Kateřina ; Dvořák, Jiří (advisor) ; Prokešová, Michaela (referee)
The main theme of this thesis is the theory of stationary point processes, in particular the directional K-function. In the first chapter we explain the essentials of planar point process theory including the classical definition of K-function and its estimator. The second chapter introduces two types of the directional K-function: cylindrical K-function whose structural element is a cylinder and directional K-function using double spherical cones. The third chapter presents the comparison of directional K-function and its estimator on an anisotropic version of Thomas process. We also illustrate the major contribution of directional K-function in orientation analysis of point patterns. We introduce a heuristic method for detecting anisotropies in clustered or regular data. 1
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Long range dependence in time series
Till, Alexander ; Prokešová, Michaela (advisor) ; Hurt, Jan (referee)
Title: Long range dependence in time series Author: Alexander Till Department: Department of Probability and Mathematical Statistics Supervisor: RNDr. Michaela Prokešová, Ph.D. Abstract: The diploma thesis demonstrates the necessity of a study of long range dependence, introduces fractional Gaussian noise and discusses possi- ble definitions of long memory. It is done by notions of ergodic theory and by second moment characteristics and spectral density. These definitions are confronted with the model of fractional Gaussian noise and with intuitive un- derstanding of long range memory. Relations and connections between these criteria are studied as well. The work is restricted to the study of discrete time processes. Method for Hurst index estimation for fractional Gaussian noise and it's application on logarithmic returns of shares of selected produ- cers of beer are included in this work. 1
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Estimation in continuous time Markov chains
Nemčovič, Bohuš ; Prokešová, Michaela (advisor) ; Kadlec, Karel (referee)
Title: Estimation in continuous time Markov chains Author: Bohuš Nemčovič Department: Department of Probability and Mathematical Statistics Supervisor: RNDr. Michaela Prokešová, Ph.D., Department of Probability and Mathematical Statistics Abstract: In this work we deal with estimating the intensity matrices of continu- ous Markov chains in the case of complete observation and observation at selected discrete time points. To obtain an estimate we use the maximum likelihood met- hod. In the second chapter we first introduce the general EM algorithm and then adjust it for finding the intensity matrix estimate based on observations at disc- rete time points. In the last chapter we will illustrate the impact of the discrete step size on the quality of intensity matrix estimate. Keywords: Markov chains, intensity matrix, maximum likelihood estimation, EM algorithm 1
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Long range dependence in time series
Till, Alexander ; Prokešová, Michaela (advisor) ; Hurt, Jan (referee)
Title: Long range dependence in time series Author: Alexander Till Department: Department of Probability and Mathematical Statistics Supervisor: RNDr. Michaela Prokešová, Ph.D. Abstract: The diploma thesis demonstrates the necessity of a study of long range dependence, introduces fractional Gaussian noise and discusses possible definitions of long memory. It is done by notions of ergodic theory and by second moment characteristics and spectral density. These definitions are confronted with the model of fractional Gaussian noise and with intuitive understanding of long range memory. Relations and connections between these criteria are studied as well. The work is restricted to the study of discrete time processes. 1
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