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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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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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Bodové procesy v čase a prostoru
Koubek, Antonín ; Beneš, Viktor (advisor) ; Pawlas, Zbyněk (referee)
In this work we present an introduction to the theory of point processes in space and time with focus on space--time shot--noise Cox process. Further from theoretical point of view we study its simulation, space--time separability, kernel estimate of intensity function and non--parametric estimation of some summary statistics using edge corrections. For two ambit models and one space--time separable model we do numerical calculations using the presented theory and software Wolfram Mathematica 9.0. For these three models we do simulations, we select the best bandwidth for kernel estimate of the intensity function and we also calculate some theoretical summary statistics including the pair correlation function. Powered by TCPDF (www.tcpdf.org)
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Tree spatial pattern in secondary tropical forest - study within the SAFE Project led by European scientists in Borneo
Kvasnica, Jakub
This study, conducted in secondary tropical rainforest in Sabah, Malaysia, is based on data collection using FieldMap technology. All positions of trees (DBH bigger 1 cm) and other attributes were recorded within 24 research plots (25 x 25 m). The aim of this study is to determine any deviation from a random arrangement of trees and nonrandom spatial structures in tree mortality using second order statistics (pair correlation function). All analyses were carried out with the comprehensive software package for conducting spatial point pattern analysis, Programita (developed by Wiegand 2012).
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