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A Software Tool for Analyzing Stochastic Data
Lipták, Juraj ; Peringer, Petr (referee) ; Hrubý, Martin (advisor)
This thesis discusses the possibility of modeling stochastic processes. Elements of the system with the source of randomness in some cases may be represented by probability distribution. The reader will be acquainted with methods of statistical induction for selecting suitable distribution and generating random numbers. Tool developed in this project aims to propose appropriate probability distribution based on empirical data and provide random variable generating with proposed distribution.
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Models of Queueing Systems
Horký, Miroslav ; Dvořák, Jiří (referee) ; Šeda, Miloš (advisor)
The master’s thesis solves models of queueing systems, which use the property of Markov chains. The queueing system is a system, where the objects enter into this system in random moments and require the service. This thesis solves specifically such models of queueing systems, in which the intervals between the objects incomings and service time have exponential distribution. In the theoretical part of the master’s thesis I deal with the topics stochastic process, queueing theory, classification of models and description of the models having Markovian property. In the practical part I describe realization and function of the program, which solves simulation of chosen model M/M/m. At the end I compare results which were calculated in analytic way and by simulation of the model M/M/m.
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Queueing theory utilization in packet network design and optimization process
Rýzner, Zdeněk ; Zeman, Václav (referee) ; Novotný, Vít (advisor)
This master's thesis deals with queueing theory and its application in designing node models in packet-switched network. There are described general principles of designing queueing theory models and its mathematical background. Further simulator of packet delay in network was created. This application implements two described models - M/M/1 and M/G/1. Application can be used for simulating network nodes and obtaining basic network characteristics like packet delay or packet loss. Next, lab exercise was created, in that exercise students familiarize themselves with basic concepts of queueing theory and examine both analytical and simulation approach to solving queueing systems.
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Modifications of stochastic objects
Kadlec, Karel ; Štěpán, Josef (advisor) ; Dostál, Petr (referee)
In this thesis, we are concerned with the modifications of the stochastic processes and the random probability measures. First chapter is devoted to modifications of the stochastic process to the space of continuous functions, modifications of submartingale to the set of right-continuous with finite left-hand limits functions and separable modifications of stochastic process. In the second chapter is the attention on the regularization of random probability measure in Markov kernel focused. In particular, we work with random probability measures on the Borel subset of the Polish space, or Radon separable topological space.
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Stochastic methods in portfolio management
Kobulnická, Ivana ; Radová, Jarmila (advisor) ; Diviš, Martin (referee)
This master thesis aims to describe and apply in practice solutions of basic tasks in portfolio management- portfolio optimization, portfolio modelling and risk management. As value of financial assets in future is a random variable, it is necessary to use mathematic tools resulting from probability theory and statistics. Basic terms from this area are for example stochastic Wiener process or geometric Brownian motion, which are described in first part of this thesis. Next parts of thesis describe the Markowitz model or method Value at Risk. In the last part of thesis is application of calculation VaR using Monte Carlo simulation for stock portfolio constructed as optimal portfolio according to Markowitz model from real data.
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Modifications of stochastic objects
Kadlec, Karel ; Štěpán, Josef (advisor) ; Dostál, Petr (referee)
In this thesis, we are concerned with the modifications of the stochastic processes and the random probability measures. First chapter is devoted to modifications of the stochastic process to the space of continuous functions, modifications of submartingale to the set of right-continuous with finite left-hand limits functions and separable modifications of stochastic process. In the second chapter is the attention on the regularization of random probability measure in Markov kernel focused. In particular, we work with random probability measures on the Borel subset of the Polish space, or Radon separable topological space.
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A Software Tool for Analyzing Stochastic Data
Lipták, Juraj ; Peringer, Petr (referee) ; Hrubý, Martin (advisor)
This thesis discusses the possibility of modeling stochastic processes. Elements of the system with the source of randomness in some cases may be represented by probability distribution. The reader will be acquainted with methods of statistical induction for selecting suitable distribution and generating random numbers. Tool developed in this project aims to propose appropriate probability distribution based on empirical data and provide random variable generating with proposed distribution.
|
|
|
Models of Queueing Systems
Horký, Miroslav ; Dvořák, Jiří (referee) ; Šeda, Miloš (advisor)
The master’s thesis solves models of queueing systems, which use the property of Markov chains. The queueing system is a system, where the objects enter into this system in random moments and require the service. This thesis solves specifically such models of queueing systems, in which the intervals between the objects incomings and service time have exponential distribution. In the theoretical part of the master’s thesis I deal with the topics stochastic process, queueing theory, classification of models and description of the models having Markovian property. In the practical part I describe realization and function of the program, which solves simulation of chosen model M/M/m. At the end I compare results which were calculated in analytic way and by simulation of the model M/M/m.
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Queueing theory utilization in packet network design and optimization process
Rýzner, Zdeněk ; Zeman, Václav (referee) ; Novotný, Vít (advisor)
This master's thesis deals with queueing theory and its application in designing node models in packet-switched network. There are described general principles of designing queueing theory models and its mathematical background. Further simulator of packet delay in network was created. This application implements two described models - M/M/1 and M/G/1. Application can be used for simulating network nodes and obtaining basic network characteristics like packet delay or packet loss. Next, lab exercise was created, in that exercise students familiarize themselves with basic concepts of queueing theory and examine both analytical and simulation approach to solving queueing systems.
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Non-negative linear operators and their use in econometric and statistic models
Horský, Richard ; Arlt, Josef (advisor) ; Vrabec, Michal (referee) ; Klazar, Martin (referee)
Non-negative operators, in special case non-negative matrices, are an interesting topics for many scientists and scientific teams from the beginning of the 20th century. It is not suprising because there are a lot of applications in different areas of science like economy, statistics, linear programming, computer science and others. We can give as the particular example the theory of the Markov chains in which we deal with non-negative matrices, so called transition matrices. They are of the special form and we called them stochastic matrices. Another example is given by the non-negative operator on spaces of infinite dimension which is employed in the theory of stochastic processes. It is the backward shift operator called the lag operator as well. The non-negativity in these examples is considered as the piecewise non-negativity. Another type of non-negativity is that in the sense of inner products. In the case of matrices we talk about positive-definite or positive-semidefinite matrices. A typical example is the covariance matrix of a random vector or symmetrization of any linear operator, for instance the symmetrization of the difference operator. The terms inverse problem or ill-posed problem have been gaining popularity in modern science since the middle of the last century. The subjects of the first publications in this area were related to quantum scattering theory, geophysics, astronomy and others. Thanks to powerful computers the chances for applications of the theory of inverse and ill-posed problems has extended in almost all fields of science which use mathematical methods. Ill-posed problems bear the feature of instability and there is the need of regularization if we want to get some reasonable solution. A typical example of the regularization is the differencing of stochastic process with the purpose to obtain a stationary process. Another concept of regularization used for solving e.g. integral equations with compact operators consists in application of regularization method as truncated singular value decomposition, Tichonov regularization method or Landweber iteration method. Mathematical tools employed in this work are those of the functional analysis. It is the area of mathematics in which distinct mathematical structures meet each other. They are structures built within different mathematical disciplines as mathematical analysis, topology, theory of sets, algebra (mainly linear algebra) and theory of measure (probability). The functional analysis framework enables us to obtain right formulations of definitions and problems providing the general view on the notions and problems of the theory of stochastic processes.
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