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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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Application of game theory on oligopoly structures
Kuzmiak, Maroš ; Kopa, Miloš (advisor) ; Lachout, Petr (referee)
This bachelor thesis is devoted to the games theory and its application to economic models of oligopolies. We focus on determining optimal strategies in non-cooperative conflict and on the issue of coalitions forming and rewards redistribution in cooperative conflict. Lessons learned and characteristics obtained are afterwards applied to different models of oligopolies. Individual chapters are supplemented by short numerical examples which are used to demonstrate the explained theory. The application of this work is to be found in areas where conflict situations are modeled using the knowledge on the choice of optimal strategies.
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Multistage Stochastic Programming Problems - Decomposition
Lapšanská, Alica ; Kaňková, Vlasta (advisor) ; Lachout, Petr (referee)
The thesis deals with a multistage stochastic model and its application to a number of practical problems. Special attention is devoted to the case where a random element follows an autoregressive sequence and the constraint sets correspond to the individual probability constraints. For this case conditions under which is the problem well-defined are specified. Further, the approximation of the problem and its convergence rate under the empirical estimate of the distribution function is analyzed. Finally, an example of the investment in financial instruments is solved, which is defined as a two-stage stochastic programming problem with the probability constraint and a random element following an autoregressive sequence. Powered by TCPDF (www.tcpdf.org)
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Numerical study on simultanious equations
Šaroch, Vojtěch ; Lachout, Petr (advisor) ; Cipra, Tomáš (referee)
Title: Numerical study on simultanious equations Author: Vojtěch Šaroch Department: Department of Probability and Mathematical Statistics Supervisor: doc. RNDr. Petr Lachout, CSc. Abstract: In this thesis we deal with the simultaneous equation model. In the first chapter we introduce theoretical aspect of this problem, especially estimation procedures and their properties. We mention issues of an identification and an inconsistency of OLS-estimates for simultaneous modeling. In th second chapter we introduce theory of estimation, especially we will focus on the interval estimation and precision. We mention empirical approach too. In the third chapter we perform a numerical study on the simple macroeconomic model of generated dates. We are interested in properties of interval estimations of parameters, the convergence rate, difference between the empirical and theoretical extimation etc. Keywords: simultaneous equations model, interval estimation, empirical estimation 1
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Non-stationary time series
Večeřa, Jakub ; Lachout, Petr (advisor) ; Cipra, Tomáš (referee)
This thesis focuses on option of omitting the stationarity assumption, which is usually used in the financial time series analysis. The theory of semi-stationary processes is introduced. This type of process has time-dependent spectra (the evolutionary spectra) in comparison with stationary process. The evolutionary spectra estimator is derived using a linear filter and then averaged in time to reduce any fluctuations caused by randomness. Predictions and variance estimates are retrieved from the estimated time dependent spectra. The semi-stationary processes theory is applied to the ARMA processes with time-dependent coefficients, a coefficient estimator based on evolutionary spectra is suggested. Calculations are performed in R software. Powered by TCPDF (www.tcpdf.org)
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Optimization problems of vertex coloring under uncertainty
Kučera, Petr ; Branda, Martin (advisor) ; Lachout, Petr (referee)
The goal of this paper is to compare two formulations of optimi- zation problems of vertex coloring under uncertainty. These two formulations are integer linear programming with constraints and integer quadratic without con- straints. First chapter introduces integer linear programming. In second chapter we learn about these two formulations. Third chapter deals with implementation of these two formulations in optimization program called GAMS. We randomly generate 20 optimization problems of vertex coloring under uncertainty and com- pare integer linear formulation with constraints and integer quadratic formulation without constraints. 1
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Optimal solutions and CLM sets
Semela, Ondřej ; Lachout, Petr (advisor) ; Branda, Martin (referee)
This thesis falls within the theory of optimization problems. In the first part, terms such as epi- convergence, lower and upper semicontinuous function, epi-continuity and CLM set are defined. For a better understanding, the definitions of the key terms are accompanied with illustrative examples and observations of their basic properties. The following part deals with searching of (local) minimizers of random or deterministic function. Using the knowledge from the first part it is showed that under a set of assumptions it is possible to transfer this search to a sequence of random functions of specific requirements. Powered by TCPDF (www.tcpdf.org)
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