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Epidemiological modeling of Covid-19
Schubert, Richard ; Kašpar, Jakub (referee) ; Mézl, Martin (advisor)
This thesis deals with the continuous epidemiological deterministic compartmental models and the COVID-19 pandemic modeling distinctive features. The effect of different probability distributions of individuals stay in compartments is studied numerically in relation to basic reproductive number and the final size of the epidemic, respectively. New model for a retrospective analysis of the first half of 2020 northern Italy epidemiological data is proposed. The model parameters estimation is performed using minimisation of weighted sum of squared residuals and the search through parameter space with BFGS algorithm implementation.
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Updating the capacity model of roundabouts
Patočka, Miroslav ; Bačová, Katarina (referee) ; Kumpošt, Petr (referee) ; Radimský, Michal (advisor)
The doctoral thesis deals with a new approach to the characteristics of traffic flows at roundabouts in relation to the capacity estimation. The currently used methodology, which was introduced in 1973 by Werner Siegloch, is based on imprecise assumptions and simplifications, which may ultimately distort the results of capacity assessments carried out according to the existing valid national regulation. This thesis uses not only an innovative procedure for obtaining empirical data by image analysis using so-called computer vision, but above all, the process of gap-acceptance of main-stream time clearances by minor stream vehicles is solved as a stochastic phenomenon. This approach aims to replace the deterministic values of the critical gap and follow-up headway with an appropriate probability density function that will meet empirical data in the best way.
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High-order stochastic dominance
Mikulka, Jakub ; Kopa, Miloš (advisor) ; Branda, Martin (referee)
The thesis deals with high-order stochastic dominance of random variables and portfolios. The summary of findings about high-order stochastic dominance and portfolio efficiency is presented. As a main part of the thesis it is proven that under assumption of both normal and gamma distribution the infinite-order stochastic dominance is equivalent to the second-order stochastic dominance. The necessary and sufficient condition for the infinite-order stochastic dominance portfolio efficiency is derived under the assumption of normality. The condition is used in the empirical part of the thesis where parametrical approach to the portfolio efficiency is compared to the nonparametric scenario approach. The derived necessary and sufficient condition is based on the assumption of normality; therefore we use two sets of data, one with fulfilled assumption of normality and the other for which the assumption of normality was unambigously rejected. Consequently, the influence of fulfillment of the normality assumption on the results of the necessary and sufficient condition for portfolio efficiency is estimated.
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GOF tests for gamma distribution
Klička, Petr ; Hlávka, Zdeněk (advisor) ; Kulich, Michal (referee)
The Bachelor thesis deals with the goodness of fit test for the Gamma distribution. Initially, we show several ways how to estimate the parameters of the Gamma distribution - firstly, the maximum likelihood estimator is presented, followed by estimator gained by the method of moments and fi- nally, we introduce the new estimator based on the sample covariance. The last estimator is used for constructing the goodness of fit test for the Gamma distribution. We define the test statistics V ∗ n to this test and its asymptotic normality is derived under the assumption of the null hypothesis. At the end of the thesis the simulations are realized to obtain the empirical size of the test for various values of parameter a and parameter b which equals one. 1
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Epidemiological modeling of Covid-19
Schubert, Richard ; Kašpar, Jakub (referee) ; Mézl, Martin (advisor)
This thesis deals with the continuous epidemiological deterministic compartmental models and the COVID-19 pandemic modeling distinctive features. The effect of different probability distributions of individuals stay in compartments is studied numerically in relation to basic reproductive number and the final size of the epidemic, respectively. New model for a retrospective analysis of the first half of 2020 northern Italy epidemiological data is proposed. The model parameters estimation is performed using minimisation of weighted sum of squared residuals and the search through parameter space with BFGS algorithm implementation.
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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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GOF tests for gamma distribution
Klička, Petr ; Hlávka, Zdeněk (advisor) ; Kulich, Michal (referee)
The Bachelor thesis deals with the goodness of fit test for the Gamma distribution. Initially, we show several ways how to estimate the parameters of the Gamma distribution - firstly, the maximum likelihood estimator is presented, followed by estimator gained by the method of moments and fi- nally, we introduce the new estimator based on the sample covariance. The last estimator is used for constructing the goodness of fit test for the Gamma distribution. We define the test statistics V ∗ n to this test and its asymptotic normality is derived under the assumption of the null hypothesis. At the end of the thesis the simulations are realized to obtain the empirical size of the test for various values of parameter a and parameter b which equals one. 1
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Sums of exponential random variables
Michl, Marek ; Seidler, Jan (advisor) ; Maslowski, Bohdan (referee)
Sums of exponential random variables are often found in applied mathematics. Their densities are known and are well documented in engineering articles. However, these articles usually lack detailed deductions. The purpose of this thesis is to give rigorous deductions of explicit formulas for densities of sums of independent exponential random variables, which are known. The thesis covers several cases depending on whether the variables have the same distribution or not. Furthermore, the thesis gives a summary of basic characteristics of exponential distribution and the relation between sums of identically distributed exponential random variables and a random variable with gamma distribution. Based on this relation the density of the sum of gamma random variables with the same intestity is given. Powered by TCPDF (www.tcpdf.org)
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High-order stochastic dominance
Mikulka, Jakub ; Kopa, Miloš (advisor) ; Branda, Martin (referee)
The thesis deals with high-order stochastic dominance of random variables and portfolios. The summary of findings about high-order stochastic dominance and portfolio efficiency is presented. As a main part of the thesis it is proven that under assumption of both normal and gamma distribution the infinite-order stochastic dominance is equivalent to the second-order stochastic dominance. The necessary and sufficient condition for the infinite-order stochastic dominance portfolio efficiency is derived under the assumption of normality. The condition is used in the empirical part of the thesis where parametrical approach to the portfolio efficiency is compared to the nonparametric scenario approach. The derived necessary and sufficient condition is based on the assumption of normality; therefore we use two sets of data, one with fulfilled assumption of normality and the other for which the assumption of normality was unambigously rejected. Consequently, the influence of fulfillment of the normality assumption on the results of the necessary and sufficient condition for portfolio efficiency is estimated.
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