|
|
Maximum likelihood estimators in time series
Tritová, Hana ; Pawlas, Zbyněk (advisor) ; Zikmundová, Markéta (referee)
The thesis deals with maximum likelihood estimators in time series. The reader becomes familiar with three important models for time series: autoregressive model (AR), moving average model (MA) and autoregressive moving average (ARMA). Thereafter he can find out the form of their main characteristics, e.g. population mean and variance. Then there is the derivation of parameter estimates - generally and for mentioned models of times series. There are also stated two other methods for finding estimators of AR(1) and MA(1) parameters - method of moments and least squares method. The end is dedicated to examples which compares all three methods.
|
|
|
Interacting spatial particle systems
Zikmundová, Markéta ; Beneš, Viktor (advisor) ; Pawlas, Zbyněk (referee) ; Volf, Petr (referee)
1 Title: Interacting spatial particle systems Author: Markéta Zikmundová Department: Department of Probability and Mathematical Statistics Author's e-mail address: zikmundm@karlin.mff.cuni.cz Supervisor: Prof. RNDr. Viktor Beneš, DrSc. Supervisor's e-mail address: benesv@karlin.mff.cuni.cz Consultant: RNDr. Kateřina Helisová, Ph.D. Consultant's e-mail address: helisova@math.feld.cvut.cz Abstract: Several kinds of random union of interacting particles is studied. We define line segment process of interacting particles in R2 and process of interacting surfaces in R3 as the models with density function p with respect to some Poisson point process. The formulas for moments of the geometrical characteristics of these models are derived and the limit behaviour when the intensity tends to infinity is investigated. For time extension of such models a simulation algorithm is developed. Various estimations of parameters of density p, among them those based on sequential Monte Carlo, are studied and compare in a simulation study. Keywords: Boolean model, process with interacting particles, U−statistics, exponential family, germ-grain model, interaction, Markov properties, point process, random closed set, Markov chain Monte Carlo.
|
|
|
Spatial point process with interactions
Vícenová, Barbora ; Beneš, Viktor (advisor) ; Zikmundová, Markéta (referee)
This thesis deals with the estimation of model parameters of the interacting segments process in plane. The motivation is application on the system of stress fibers in human mesenchymal stem cells, which are detected by fluorescent microscopy. The model of segments is defined as a spatial Gibbs point process with marks. We use two methods for parameter estimation: moment method and Takacs-Fiksel method. Further, we implement algorithm for these estimation methods in software Mathematica. Also we are able to simulate the model structure by Markov Chain Monte Carlo, using birth-death process. Numerical results are presented for real and simulated data. Match of model and data is considered by descriptive statistics. Powered by TCPDF (www.tcpdf.org)
|
|
| |
|
|
Sequential Monte Carlo Methods
Sobková, Eva ; Zikmundová, Markéta (advisor) ; Prokešová, Michaela (referee)
Monte Carlo methods are used for stochastic systems simulations. Sequential Monte Carlo methods take advantage of the fact that observations are coming sequentially. This allows us to refine our estimate sequentially in time We introduce a State Space Model as a Hidden Markov Model. We describe Perfect Monte Carlo Sampling, Importance Sampling, Sequential Importance Sampling and discuss advantages and disadvantages of these methods. This discussion brings us to add a resampling step in Sequential Importance Sampling and introduce Particle Filter and Particle Marginal Metropolis-Hastings algorithm. We choose a Hidden Markov Model used for stochastic volatility modeling and make a simulation study in Wolfram Mathematica, version 8.
|
|
|
Interacting spatial particle systems
Zikmundová, Markéta ; Beneš, Viktor (advisor) ; Pawlas, Zbyněk (referee) ; Volf, Petr (referee)
1 Title: Interacting spatial particle systems Author: Markéta Zikmundová Department: Department of Probability and Mathematical Statistics Author's e-mail address: zikmundm@karlin.mff.cuni.cz Supervisor: Prof. RNDr. Viktor Beneš, DrSc. Supervisor's e-mail address: benesv@karlin.mff.cuni.cz Consultant: RNDr. Kateřina Helisová, Ph.D. Consultant's e-mail address: helisova@math.feld.cvut.cz Abstract: Several kinds of random union of interacting particles is studied. We define line segment process of interacting particles in R2 and process of interacting surfaces in R3 as the models with density function p with respect to some Poisson point process. The formulas for moments of the geometrical characteristics of these models are derived and the limit behaviour when the intensity tends to infinity is investigated. For time extension of such models a simulation algorithm is developed. Various estimations of parameters of density p, among them those based on sequential Monte Carlo, are studied and compare in a simulation study. Keywords: Boolean model, process with interacting particles, U−statistics, exponential family, germ-grain model, interaction, Markov properties, point process, random closed set, Markov chain Monte Carlo.
|
|
|
Spatial point process with interactions
Vícenová, Barbora ; Beneš, Viktor (advisor) ; Zikmundová, Markéta (referee)
This thesis deals with the estimation of model parameters of the interacting segments process in plane. The motivation is application on the system of stress fibers in human mesenchymal stem cells, which are detected by fluorescent microscopy. The model of segments is defined as a spatial Gibbs point process with marks. We use two methods for parameter estimation: moment method and Takacs-Fiksel method. Further, we implement algorithm for these estimation methods in software Mathematica. Also we are able to simulate the model structure by Markov Chain Monte Carlo, using birth-death process. Numerical results are presented for real and simulated data. Match of model and data is considered by descriptive statistics. Powered by TCPDF (www.tcpdf.org)
|
|
|
Sequential Monte Carlo Methods
Sobková, Eva ; Zikmundová, Markéta (advisor) ; Prokešová, Michaela (referee)
Monte Carlo methods are used for stochastic systems simulations. Sequential Monte Carlo methods take advantage of the fact that observations are coming sequentially. This allows us to refine our estimate sequentially in time We introduce a State Space Model as a Hidden Markov Model. We describe Perfect Monte Carlo Sampling, Importance Sampling, Sequential Importance Sampling and discuss advantages and disadvantages of these methods. This discussion brings us to add a resampling step in Sequential Importance Sampling and introduce Particle Filter and Particle Marginal Metropolis-Hastings algorithm. We choose a Hidden Markov Model used for stochastic volatility modeling and make a simulation study in Wolfram Mathematica, version 8.
|
|
|
Stochastic games
Holková, Klára ; Dostál, Petr (advisor) ; Zikmundová, Markéta (referee)
Title: Stochastic games Author: Klára Holková Department: Department of Probability and Mathematical Statistics Supervisor: Mgr. Petr Dostál, Ph.D., Department of Probability and Mathema- tical Statistics Abstract: In the text of this thesis we deal with the classification of stochastic games, which is closely related to investing in various financial instruments. With the help of utility functions, we are looking for the optimal and relatively safe bet on a mathematically convenient game. It will be demonstrated, why it does not pay off to take part in fair games and games of chance in the long run, as well as how such games relate to moral hazard. The thesis is written in a way that should make it accessible to the wide public, therefore there are numerous examples and interpretations in order to facilitate the understanding of the mathematical theory. Some parts can also be used by teachers as a direct source for teaching financial literacy in primary and secondary schools. Keywords: Optimal and relatively safe bet, CARA a HARA utility functions, moral hazard, financial literacy 1
|
|
|
Maximum likelihood estimators in time series
Tritová, Hana ; Pawlas, Zbyněk (advisor) ; Zikmundová, Markéta (referee)
The thesis deals with maximum likelihood estimators in time series. The reader becomes familiar with three important models for time series: autoregressive model (AR), moving average model (MA) and autoregressive moving average (ARMA). Thereafter he can find out the form of their main characteristics, e.g. population mean and variance. Then there is the derivation of parameter estimates - generally and for mentioned models of times series. There are also stated two other methods for finding estimators of AR(1) and MA(1) parameters - method of moments and least squares method. The end is dedicated to examples which compares all three methods.
|
guest ::
RSS feed.