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Application of (geo)demographic methods in education
Šebestík, Libor ; Hulíková Tesárková, Klára (advisor) ; Fialová, Ludmila (referee)
Application of (geo)demographic methods in education Abstract This master's thesis presents the possibilities of application of demographic, geodemographic and statistical methods on data published by the educational sector. The methods of demographic analysis are represented by the usage of rates, the concept of multistate demography (Markov chains) and the application of life tables. The enrollment ratio at particular levels of education, the average length of schooling and the number of dropouts from school grades are evaluated by these procedures. Markov chains which are based on the probabilities of transition between grades are also examined in terms of their use for forecasting purposes. These methods analyze the situation at the preschool, primary and secondary levels and are used on data from the annual Statistical Yearbooks on Education. In the field of geodemography, the so called preferential model of migration flows is presented. This model examines how applicants for tertiary education prefer or reject the regions of the Czech Republic for their tertiary education studies. The last method is the binary logistic regression which analyzes the inequalities in access to tertiary education. Both preferential model and logistic regression are based on data files on the admission process at...
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Mixing cards and convergence of Markov chains
Drašnar, Jan ; Prokešová, Michaela (advisor) ; Beneš, Viktor (referee)
This thesis presents mixing of a deck of cards as a random walk on the group of permutations. Perfectly shuffled deck of cards is defined as uniform distribution on this group. For analysis of the distance between the uniform distribution and the current distribution of the Markov chain generated by the shuffling quite general methods are used that can be applied to many other problems - i.e. strong stacionary time, coupling and transformation to an inverse distribution. In the last chapter the riffle shuffle is studied and a rather well-known fact is proved that seven or eight shuffles should be enough to shuffle a deck of 52 cards.
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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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Analysis of number lotteries
Jedličková, Veronika ; Pawlas, Zbyněk (advisor) ; Lachout, Petr (referee)
This bachelor thesis focuses on most well-known lotteries on the Czech market, in particular Sportka and Loto. Thesis observes many aspects influencing progress of these games. Winnning prices and lottery participant's expectations are examined. Total sum of these winnings is influenced by the amount of money in jackpot. Therefore, jackpot sum modelling and period between wins is taken into account. Moreover, expected period between two jackpot hits, distribution of drawn numbers and probability of drawing the same winning sequence is examined.
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Markov chains and credit risk theory
Cvrčková, Květa ; Prokešová, Michaela (advisor) ; Lachout, Petr (referee)
Markov chains have been widely used to the credit risk measurement in the last years. Using these chains we can model movements and distribution of clients within rating grades. However, various types of markov chains could be used. The goal of the theses is to present these types together with their advan- tages and disadvantages. We focus our attention primarily on various parameter estimation methods and hypotheses testing about the parameters. The theses should help the reader with a decision, which model of a markov chain and which method of estimation should be used for him observed data. We focus our attention primarily on the following models: a discrete-time markov chain, a continuous-time markov chain (we estimate based on continuous- time observations even discrete-time observations), moreover we present an even- tuality of using semi-markov chains and semiparametric multiplicative hazard model applied on transition intensities. We illustrate the presented methods on simulation experiments and simu- lation studies in the concluding part. Keywords: credit risk, markov chain, estimates in markov chains, probability of default 1
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Statistical Problems in Markov Chains
Adamová, Markéta ; Prášková, Zuzana (advisor) ; Branda, Martin (referee)
In this thesis we study basic statistical methods in Markov chains. In the case of discrete time, this thesis is focused on estimation of transition probability matrix and some basic tests (test for a specified transition probability matrix, test for homogeneity, test for independence, test for the order of Markov chain). In the case of continuous time we will concentrate on Poisson process and birth and death process. Estimation of parameters of these processes and tests for processes with specified parameters are mentioned. Developed estimates and test statistics are applied to real data in the final chapter.
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Symmetries of transition times in complex biophysical systems
Voráč, David ; Ryabov, Artem (advisor) ; Chvosta, Petr (referee)
Conformational changes of biomolecules can be described as Markov processes on net- works of discrete states representing minima of free energy landscapes. Network states for several types of membrane proteins and molecular motors are linked into cycles, and their reaction coordinates (represented by a "particle") jump between the cycle states predominantly in one direction with rare backward jumps occurring due to thermal fluc- tuations. Assuming that interactions of the particle with other degrees of freedom (other particles) cannot be neglected, we study times that it takes to complete one cycle. In par- ticular, we compare mean times of cycle completion in and against the bias direction and show that they satisfy the universal inequality: Cycle-completion times in bias direction are never shorter than the ones against the bias. We discuss how the times depend on the interaction strength, cycle topology, quenched disorder, number of interacting par- ticles, and check validity of our findings for two-dimensional models with canonical and grand-canonical particle reservoirs.
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Counter-Example Generation in the Analysis of Markov Models
Molek, Martin ; Matyáš, Jiří (referee) ; Češka, Milan (advisor)
This thesis deals with generating counterexamples in context of probabilistic models. Counterexamples are generated for Markov models (specifically DTMC). Definitions of model properties are given by logic PCTL. Two algorithms (Best-first search and Recursive Enumration Algorithm) are used to generate these counterexamples. Thesis describes implementation of algorithms into verification tool STORM. The results of experiments show that REA is capable of handling models containg millions of states.
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Risk sensitive management in Markov chains
Vernerová, Eva ; Dostál, Petr (advisor) ; Lachout, Petr (referee)
The main topic of this bachelor thesis is Markov reward chains with finite state set. We consider a markov decision chain with a finite action space and we are concerned on finding an optimal control with respect to exponential utility function. An iterative algorithm is given. Then we prove that after finite number of steps we end up with optimal control. Afterwards we show that optimality of this cotrol is fulfilled, even if we consider an adaptive chain control strategy. In the last part of the work, there is a selection of propositions from Perron-Frobenius theory, which are essential in proofs of theorems. 1
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Roulette and its strategies
Zadražil, Tomáš ; Staněk, Jakub (advisor) ; Slavík, Antonín (referee)
Objective of this thesis is to describe history of gambling, in a context of roulette to explain basic and advanced parts of probability theory which allow to the reader to decide about function of several popular roulette systems. There was mainly used expected value of discrete random variable, homogenous discrete-time Markov chain and simulations made in programming language R. Concrete output of the thesis are in precisely calculated expected values of a profit with fixed spins and with chosen limitation and corresponding estimations provided by simulation. On the basis of that it's possible to decide which systems are functional and which are not. Main contribution of this text is in didactical approach which helps to describe popular problematics of roulette systems by using basic and advanced areas of probability theory.
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