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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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Stochastic Calculus and Its Applications in Biomedical Practice
Klimešová, Marie ; Růžičková, Miroslava (referee) ; Dzhalladova, Irada (referee) ; Baštinec, Jaromír (advisor)
V předložené práci je definována stochastická diferenciální rovnice a jsou uvedeny její základní vlastnosti. Stochastické diferenciální rovnice se používají k popisu fyzikálních jevů, které jsou ovlivněny i náhodnými vlivy. Řešením stochastického modelu je náhodný proces. Cílem analýzy náhodných procesů je konstrukce vhodného modelu, který umožní porozumět mechanismům, na jejichž základech jsou generována sledovaná data. Znalost modelu také umožňuje předvídání budoucnosti a je tak možné kontrolovat a optimalizovat činnost daného systému. V práci je nejdříve definován pravděpodobnostní prostor a Wienerův proces. Na tomto základě je definována stochastická diferenciální rovnice a jsou uvedeny její základní vlastnosti. Závěrečná část práce obsahuje příklad ilustrující použití stochastických diferenciálních rovnic v praxi.
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Generalized Moran process
Svoboda, Jakub ; Šámal, Robert (advisor) ; Balko, Martin (referee)
The Moran process is a model for simulating evolutionary dynamics. In that model, one mutant with higher fitness is introduced to a structured population. Evolution is simulated in rounds. In one round, individual is selected proportio- nally to its fitness and spreads to the place of a random neighbour. In this thesis, we motivate the Moran process, present basic results, and define our variant. We work in a vertex dependent model; every individual has fitness according to its type and occupied vertex. In the vertex dependent model we prove two theorems about the number of steps the process has to make to get to the stable state. We show that on the complete graph, the process takes only polynomially many steps and we find a graph where the process take exponentially many steps, but in the normal settings the number of steps is the same as on the complete graph. 1
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Testing time-series characteristics of prices of financial derivatives
Vdovičenko, Martin ; Kadavý, Matěj (advisor) ; Šnupárková, Jana (referee)
This work discusses Brownian motion and its basic transformations. The work describes basic properties of its trajectories and shows that Brownian motion is a martingale and a self-similar process. Next, we discuss time series analysis. We introduce graphical tools for analyzing data and we describe theoretical basics of some normality and independence tests. Finally, we consider the hypothesis that in the short run the price of financial assets can be modelled by Brownian motion. We conduct basic statistical tests on real data using the R progam and we talk through our results.
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Kolmogorov-Chentsov Theorem
Lebeda, Matěj ; Čoupek, Petr (advisor) ; Kříž, Pavel (referee)
Is there a sufficient condition for continuity of sample paths of a random process? Or, is it at least possible to modify the process so that the paths would already be continuous? An affirmative answer is given by the Kolmogorov- Chentsov theorem, whose statement and proof are the subject of this thesis. First, we introduce the notion of a random process and briefly focus on the so-called Gaussian processes. The main focus of the second chapter is the Kolmogorov- Chentsov theorem, its proof and some auxiliary assertions are given. In the final third chapter, we deal with the applications of the theorem to some well-known Gaussian processes such as the Wiener process or the Brownian bridge. Finally, we look into the Poisson process, which on the contrary does not satisfy the condition of the theorem. 1
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Stochastic Calculus and Its Applications in Biomedical Practice
Klimešová, Marie ; Růžičková, Miroslava (referee) ; Dzhalladova, Irada (referee) ; Baštinec, Jaromír (advisor)
V předložené práci je definována stochastická diferenciální rovnice a jsou uvedeny její základní vlastnosti. Stochastické diferenciální rovnice se používají k popisu fyzikálních jevů, které jsou ovlivněny i náhodnými vlivy. Řešením stochastického modelu je náhodný proces. Cílem analýzy náhodných procesů je konstrukce vhodného modelu, který umožní porozumět mechanismům, na jejichž základech jsou generována sledovaná data. Znalost modelu také umožňuje předvídání budoucnosti a je tak možné kontrolovat a optimalizovat činnost daného systému. V práci je nejdříve definován pravděpodobnostní prostor a Wienerův proces. Na tomto základě je definována stochastická diferenciální rovnice a jsou uvedeny její základní vlastnosti. Závěrečná část práce obsahuje příklad ilustrující použití stochastických diferenciálních rovnic v praxi.
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Generalized Moran process
Svoboda, Jakub ; Šámal, Robert (advisor) ; Balko, Martin (referee)
The Moran process is a model for simulating evolutionary dynamics. In that model, one mutant with higher fitness is introduced to a structured population. Evolution is simulated in rounds. In one round, individual is selected proportio- nally to its fitness and spreads to the place of a random neighbour. In this thesis, we motivate the Moran process, present basic results, and define our variant. We work in a vertex dependent model; every individual has fitness according to its type and occupied vertex. In the vertex dependent model we prove two theorems about the number of steps the process has to make to get to the stable state. We show that on the complete graph, the process takes only polynomially many steps and we find a graph where the process take exponentially many steps, but in the normal settings the number of steps is the same as on the complete graph. 1
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Autoregressive models
Rathouský, Marek ; Zichová, Jitka (advisor) ; Prášková, Zuzana (referee)
The purpose of this thesis is to compare the classic autoregressive model of order 1 to integer autoregressive model of order 1. Considering the popularity of AR(1) model, only the basics are covered within this thesis. The main focus is on the INAR(1) model. Operator ◦ necessary for INAR(1) definition is intro- duced alongside with its properties with proof. All of the non-trivial properties of INAR(1) are followed by detailed proof, stationarity condition is also derived. Common estimation techniques are described for poisson INAR(1) model. This thesis also contains simulation study, which focuses on the rate of convergence of estimates of parameters. 1
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