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Bayesian probability distribution over a class of autoregression models applied to financial time series
Škerlík, Peter ; Šindelář, Jan (advisor) ; Hlávka, Zdeněk (referee)
In the present bachelor thesis we study the selection of appropriate autoregression models to forecast financial time series. We use Bayesian inference in statistics, which will be further explained. Consequently there is also given theoretical background which explains how to apply Bayesian inference to selection of models. The major contribution of the work is considered to be the application of this theoretical background to financial time series in programming environment C++ and the results of this application. The development of the probability of each autoregression model is shown in graphs. The results for Laplace and normal probability distribution of white noise in autoregression models are compared. The aim of the work is to provide the reader with enough theoretical information and to give him an practical overview of the usage of Bayesian statistics in data prediction. Also results of the work can be helpful to understand the mentioned models and to select the suitable model in practice.
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Joint Models for Longitudinal and Time-to-Event Data
Vorlíčková, Jana ; Komárek, Arnošt (advisor) ; Omelka, Marek (referee)
Title: Joint Models for Longitudinal and Time-to-Event Data Author: Jana Vorlíčková Department: Department of Probability and Mathematical Statistics Supervisor: doc. RNDr. Arnošt Komárek, Ph.D., Department of Probability and Mathematical Statistics Abstract: The joint model of longitudinal data and time-to-event data creates a framework to analyze longitudinal and survival outcomes simultaneously. A commonly used approach is an interconnection of the linear mixed effects model and the Cox model through a latent variable. Two special examples of this model are presented, namely, a joint model with shared random effects and a joint latent class model. In the thesis we focus on the joint latent class model. This model assumes an existence of latent classes in the population that we are not able to observe. Consequently, it is assumed that the longitudinal part and the survival part of the model are independent within one class. The main intention of this work is to transfer the model to the Bayesian framework and to discuss an estimation procedure of parameters using a Bayesian statistic. It consists of a definition of the model in the Bayesian framework, a discussion of prior distributions and the derivation of the full conditional distributions for all parameters of the model. The model's ability to...
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Bayesian factor analysis
Vávra, Jan ; Komárek, Arnošt (advisor) ; Maciak, Matúš (referee)
Bayesian factor analysis - abstract Factor analysis is a method which enables high-dimensional random vector of measurements to be approximated by linear combinations of much lower number of hidden factors. Classical estimation procedure of this model lies on the cho- ice of the number of factors, the decomposition of variance matrix while keeping identification conditions satisfied and on the appropriate choice of rotation for better interpretation of the model. This model will be transferred into bayesian framework which offers the usage of prior information unlike the classical appro- ach. The number of hidden factors can be considered as a random parameter and the dependency of each measurement on at most one factor can be forced by suitable specification of prior distribution. Estimates of model parameters are based on posterior distribution which is approximated by Monte Carlo Markov Chain methods. Bayesian approach solves the problem of selection of the num- ber of factors, the model estimation and the ensuring of the identifiability and the interpretability at the same time. The ability to estimate the real number of hidden factors is tested in a simulation study. 1
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MCMC methods for financial time series
Tritová, Hana ; Pawlas, Zbyněk (advisor) ; Komárek, Arnošt (referee)
This thesis focuses on estimating parameters of appropriate model for daily returns using the Markov Chain Monte Carlo method (MCMC) and Bayesian statistics. We describe MCMC methods, such as Gibbs sampling and Metropolis- Hastings algorithm and their basic properties. After that, we introduce different financial models. Particularly we focus on the lognormal autoregressive model. Later we theoretically apply Gibbs sampling to lognormal autoregressive model using principles of Bayesian statistics. Afterwards, we analyze procedu- res, that we used in simulations of posterior distribution using Gibbs sampling. Finally, we present processed output of both simulated and real data analysis.
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Community ecology from the perspective of classic and bayesian statistics
Klimeš, Adam ; Keil, Petr (advisor) ; Herben, Tomáš (referee)
Community ecology from the perspective of classic and Bayesian statistics Ekologie společenstev z hlediska klasické a Bayesovské statistiky Řešitel: Adam Klimeš Vedoucí práce: Mgr. Petr Keil, Ph.D. Abstract Quantitative evaluation of evidence through statistics is a central part of present-day science. Bayesian approach represents an emerging but rapidly developing enrichment of statistical analysis. The approach differs in its foundations from the classic methods. These differences, such as the different interpretation of probability, are often seen as obstacles for acceptance of Bayesian approach. In this thesis I outline ways to deal with the assumptions of Bayesian approach, and I address the main objections against it. I present Bayesian approach as a new way to handle data to answer scientific questions. I do this from a standpoint of community ecology: I illustrate the novelty that Bayesian approach brings to data analysis of typical community ecology data, specifically, the analysis of multivariate datasets. I focus on principal component analysis, one of the typical and frequently used analytical techniques. I execute Bayesian analyses that are analogical to the classic principal components analysis, I report the advantages of the Bayesian version, such as the possibility of working with...
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Bayesian Statistics - Limits and its Application in Sociology
Krčková, Anna ; Soukup, Petr (advisor) ; Hendl, Jan (referee)
The purpose of this thesis is to find how we can use Bayesian statistics in analysis of sociological data and to compare outcomes of frequentist and Bayesian approach. Bayesian statistics uses probability distributions on statistical parameters. In the beginning of the analysis in Bayesian approach a prior probability (that is chosen on the basis of relevant information) is attached to the parameters. After combining prior probability and our observed data, posterior probability is computed. Because of the posterior probability we can make statistical conclusions. Comparison of approaches was made from the view of theoretical foundations and procedures and also by means of analysis of sociological data. Point estimates, interval estimates, hypothesis testing (on the example of two-sample t-test) and multiple linear regression analysis were compared. The outcome of this thesis is that considering its philosophy and thanks to the interpretational simplicity the Bayesian analysis is more suitable for sociological data analysis than common frequentist approach. Comparison showed that there is no difference between outcomes of frequentist and objective Bayesian analysis regardless of the sample size. For hypothesis testing we can use Bayesian credible intervals. Using subjective Bayesian analysis on...
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Design of dynamic decision-making strategies for futures trading
Vosáhlo, Jaroslav ; Guy, Tatiana Valentine (advisor) ; Lachout, Petr (referee)
This thesis deals with an issue of futures derivative trading from a perspective of a minor speculator. The aim of this work is to find and design an optimal trading strategy using dynamic programming and approximate dynamic programming. We use means of Bayesian statistics to obtain predictions of variate's behavior and risk indicators to form a rate of carefulness. Effectivity of algorithm is afterwards tested in Matlab program. Available data for testing the success of the method offer more then 15.000 trading days.
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Studium negaussovských světelných křivek pomocí Karhunenova-Loveho rozvoje
Greškovič, Peter ; Pecháček, Tomáš (advisor) ; Mészáros, Attila (referee)
We present an innovative Bayesian method for estimation of statistical parameters of time series data. This method works by comparing coefficients of Karhunen-Lo\`{e}ve expansion of observed and synthetic data with known parameters. We show one new method for generating synthetic data with prescribed properties and we demonstrate on a numerical example how this method can be used for estimation of physically interesting features in power spectra calculated from observed light curves of some X-ray sources.
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Bayesian probability distribution over a class of autoregression models applied to financial time series
Škerlík, Peter ; Šindelář, Jan (advisor) ; Hlávka, Zdeněk (referee)
In the present bachelor thesis we study the selection of appropriate autoregression models to forecast financial time series. We use Bayesian inference in statistics, which will be further explained. Consequently there is also given theoretical background which explains how to apply Bayesian inference to selection of models. The major contribution of the work is considered to be the application of this theoretical background to financial time series in programming environment C++ and the results of this application. The development of the probability of each autoregression model is shown in graphs. The results for Laplace and normal probability distribution of white noise in autoregression models are compared. The aim of the work is to provide the reader with enough theoretical information and to give him an practical overview of the usage of Bayesian statistics in data prediction. Also results of the work can be helpful to understand the mentioned models and to select the suitable model in practice.
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