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Parameter Estimation in Stochastic Differential Equations
Pacák, Daniel ; Maslowski, Bohdan (advisor) ; Hlubinka, Daniel (referee)
In the Thesis the problem of estimating an unknown parameter in a stochastic dif- ferential equation is studied. Linear equations with Volterra process as the source of noise are considered. Firstly, the properties of Volterra processes and the properties of stochastic integral with respect to a Volterra process are presented. Secondly, the prop- erties of the solution to the equation under consideration are discussed. This includes the existence of the strictly stationary solution, the properties of such solution and ergodic results. These results are then generalized to equations with a mixed noise. Ergodic results are used to derive strongly consistent estimators of the unknown parameter. 1
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Localization Of Gamma Radiation Sources
Lazna, Tomas
The paper describes a method for acquiring and processing data concerned with a radiation situation in a pre-defined region of interest with a goal of localizing point sources present in the region. The acquisition underlies the robotic platform Orpheus-X4 equipped with a precise navigation module and gamma radiation detectors, a path planning involves Boustrophedon decomposition. The processing is based on the Gauss-Newton method, a contribution of the paper consists in a way to provide the method correct input data. Introduced algorithms were verified experimentally. Although not all sources can be found using the chosen equipment, in case of successful localization the accuracy is not worse than 0.1 m.
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Light Airplane Flight Parameters Estimation
Dittrich, Petr ; Pačes, Pavel (referee) ; Fiľakovský, Karol (referee) ; Chudý, Peter (advisor)
Tato práce je zaměřena na odhad letových parametrů malého letounu, konkrétně letounu Evektor SportStar RTC. Pro odhad letových parametrů jsou použity metody "Equation Error Method", "Output Error Method" a metody rekurzivních nejmenších čtverců. Práce je zaměřena na zkoumání charakteristik aerodynamických parametrů podélného pohybu a ověření, zda takto odhadnuté letové parametry odpovídají naměřeným datům a tudíž vytvářejí předpoklad pro realizaci dostatečně přesného modelu letadla. Odhadnuté letové parametry jsou dále porovnávány s a-priorními hodnotami získanými s využitím programů Tornado, AVL a softwarovéverze sbírky Datcom. Rozdíly mezi a-priorními hodnotami a odhadnutými letovými paramatery jsou porovnány s korekcemi publikovanými pro subsonické letové podmínky modelu letounu F-18 Hornet.
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Segway driver parameter estimation and its use for optimizing the control algorithm
Dobossy, Barnabás ; Zouhar, František (referee) ; Brablc, Martin (advisor)
Táto práca sa zaoberá vývojom, testovaním a implementáciou adaptívneho riadiaceho systému pre dvojkolesové samobalancujúce vozidlo. Adaptácia parametrov vozidla sa uskutoční na základe parametrov vodiča. Parametre sústavy sa nemerajú priamo, ale sú odhadované na základe priebehu stavových premenných a odozvy sústavy. Medzi odhadované parametre patrí hmotnosť a poloha ťažiska vodiča. Cieľom práce je zabezpečiť adaptáciu jazdných vlastností vozidla k rôznym vodičom s rôznou hmotnosťou, kvôli zlepšeniu stability vozidla. Táto práca je pokračovaním predchádzajúcich projektov z roku 2011 a 2015.
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Stochastic Differential Equations with Gaussian Noise
Janák, Josef ; Maslowski, Bohdan (advisor)
Title: Stochastic Differential Equations with Gaussian Noise Author: Josef Janák Department: Department of Probability and Mathematical Statistics Supervisor: Prof. RNDr. Bohdan Maslowski, DrSc., Department of Probability and Mathematical Statistics Abstract: Stochastic partial differential equations of second order with two un- known parameters are studied. The strongly continuous semigroup (S(t), t ≥ 0) for the hyperbolic system driven by Brownian motion is found as well as the formula for the covariance operator of the invariant measure Q (a,b) ∞ . Based on ergodicity, two suitable families of minimum contrast estimators are introduced and their strong consistency and asymptotic normality are proved. Moreover, another concept of estimation using "observation window" is studied, which leads to more families of strongly consistent estimators. Their properties and special cases are descibed as well as their asymptotic normality. The results are applied to the stochastic wave equation perturbed by Brownian noise and illustrated by several numerical simula- tions. Keywords: Stochastic hyperbolic equation, Ornstein-Uhlenbeck process, invariant measure, paramater estimation, strong consistency, asymptotic normality.
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Stochastic Differential Equations with Gaussian Noise
Janák, Josef ; Maslowski, Bohdan (advisor) ; Duncan, Tyrone E. (referee) ; Pawlas, Zbyněk (referee)
Title: Stochastic Differential Equations with Gaussian Noise Author: Josef Janák Department: Department of Probability and Mathematical Statistics Supervisor: Prof. RNDr. Bohdan Maslowski, DrSc., Department of Probability and Mathematical Statistics Abstract: Stochastic partial differential equations of second order with two un- known parameters are studied. The strongly continuous semigroup (S(t), t ≥ 0) for the hyperbolic system driven by Brownian motion is found as well as the formula for the covariance operator of the invariant measure Q (a,b) ∞ . Based on ergodicity, two suitable families of minimum contrast estimators are introduced and their strong consistency and asymptotic normality are proved. Moreover, another concept of estimation using "observation window" is studied, which leads to more families of strongly consistent estimators. Their properties and special cases are descibed as well as their asymptotic normality. The results are applied to the stochastic wave equation perturbed by Brownian noise and illustrated by several numerical simula- tions. Keywords: Stochastic hyperbolic equation, Ornstein-Uhlenbeck process, invariant measure, paramater estimation, strong consistency, asymptotic normality.
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Parameter estimation of gamma distribution
Zahrádková, Petra ; Kulich, Michal (advisor) ; Hlávka, Zdeněk (referee)
It is well-known that maximum likelihood (ML) estimators of the two parame- ters in a Gamma distribution do not have closed forms. The Gamma distribution is a special case of a generalized Gamma distribution. Two of the three likeli- hood equations of the generalized Gamma distribution can be used as estimating equations for the Gamma distribution, based on which simple closed-form estima- tors for the two Gamma parameters are available. Intuitively, performance of the new estimators based on likelihood equations should be close to the ML estima- tors. The study consolidates this conjecture by establishing the asymptotic beha- viours of the new estimators. In addition, the closed-forms enable bias-corrections to these estimators. 1
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