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Scenario generation for multidimensional distributions
Olos, Marek ; Dupačová, Jitka (advisor) ; Kaňková, Vlasta (referee)
Some methods for generating scenarios from multidimensional distribution assume we are able to generate scenarios from the one-dimensional distribution. We dedicate chapter 3 to this problem. At the end of the chapter, we provide references for applicable algorithms. Chapter 4 is focused on selected methods for generating scenarios from multidimensional distributions. In chapter 4.3, we introduce an algorithm for generating scenarios, which do not use any assumption about the distribution, except the first four moments and correlations to be specified. A method of generating scenarios based on approximation of multivariate normal distribution by the binomial distribution is described in chapter 4.5. Dimension reduction technique using principal components is presented in chapter 4.4. The algorithm is presented under the assumption of normal distribution. In chapter 4.6, we introduce the basics of the copula theory and a method for generating scenarios by C-vine copula. In chapter 5, we implement selected methods for generating scenarios for the estimation of daily value at risk for selected indexes and we discuss the results. Powered by TCPDF (www.tcpdf.org)
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Principal components analysis and its applications
Dubová, Mária ; Hendrych, Radek (advisor) ; Prášková, Zuzana (referee)
In the present thesis, we deal with the principal components analy- sis. In the first of this text, we study different aspects of principals components, for instance, their derivation for a multidimensional random vector from general distribution or their calculation based on a covariance or correlation matrix. It is also important to choose the proper number of principal components for reducing the dimensionality of data in order to preserve most of information. Theoretical knowledge are illustrated with several examples. In the second part of the thesis, we focus on the value at risk. This term is defined in the text also with seve- ral usual formulas to calculate it. Then, we deal with a practical application of this concept and the principal component analysis. Concretely, we analyse the portfolio of some different interest rates to obtain the value at risk in some cases. 1
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Principal components
Zavadilová, Anna ; Hlávka, Zdeněk (advisor) ; Nagy, Stanislav (referee)
This thesis presents principal components as a useful tool for data dimensio- nality reduction. In the first part, the basic terminology and theoretical properties of principal components are described and a biplot construction is derived there as well. Besides, heuristic methods for a choice of the optimum number of prin- cipal components are summarised there. Subsequently, asymptotical properties of sample eigenvalues of covariance and white Wishart matrices are described and cases of equality of some eigenvalues are distinguished at the same time. In the second part of the thesis, asymptotic distribution of the largest eigenva- lue of white Wishart matrices is described, completed with graphic illustrations. A test of the number of significant eigenvalues is suggested on the basis of this limiting distribution, and the connection of this test to the number of suitable principal components is presented. The final part of the thesis provides an over- view of advanced computational methods for the choice of an adequate number of principal components. The thesis is completed with graphical illustrations and a simulation study using Wolfram Mathematica and R.
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Scenario generation for multidimensional distributions
Olos, Marek ; Dupačová, Jitka (advisor) ; Kaňková, Vlasta (referee)
Some methods for generating scenarios from multidimensional distribution assume we are able to generate scenarios from the one-dimensional distribution. We dedicate chapter 3 to this problem. At the end of the chapter, we provide references for applicable algorithms. Chapter 4 is focused on selected methods for generating scenarios from multidimensional distributions. In chapter 4.3, we introduce an algorithm for generating scenarios, which do not use any assumption about the distribution, except the first four moments and correlations to be specified. A method of generating scenarios based on approximation of multivariate normal distribution by the binomial distribution is described in chapter 4.5. Dimension reduction technique using principal components is presented in chapter 4.4. The algorithm is presented under the assumption of normal distribution. In chapter 4.6, we introduce the basics of the copula theory and a method for generating scenarios by C-vine copula. In chapter 5, we implement selected methods for generating scenarios for the estimation of daily value at risk for selected indexes and we discuss the results. Powered by TCPDF (www.tcpdf.org)
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Principal components analysis and its applications
Dubová, Mária ; Hendrych, Radek (advisor) ; Prášková, Zuzana (referee)
In the present thesis, we deal with the principal components analy- sis. In the first of this text, we study different aspects of principals components, for instance, their derivation for a multidimensional random vector from general distribution or their calculation based on a covariance or correlation matrix. It is also important to choose the proper number of principal components for reducing the dimensionality of data in order to preserve most of information. Theoretical knowledge are illustrated with several examples. In the second part of the thesis, we focus on the value at risk. This term is defined in the text also with seve- ral usual formulas to calculate it. Then, we deal with a practical application of this concept and the principal component analysis. Concretely, we analyse the portfolio of some different interest rates to obtain the value at risk in some cases. 1
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Robust Knowledge Discovery from High-Dimensional Data
Kalina, Jan
The paper is devoted to advanced robust methods for information extraction from highdimensional data. The concept of knowledge discovery is discussed together with its two important aspects: high dimensionality of the data and sensitivity to the presence of outlying data values. We propose new robust methods for knowledge discovery suitable for highdimensional data. They are based on the idea of implicit weighting, which is inspired by the least weighted squares regression estimator. We propose a highly robust method for a dimension reduction, which can be described as a robust alternative of the principal component analysis based on implicit down-weighting of less reliable data values. Further, we propose a novel robust approach to cluster analysis, which is a popular knowledge discovery method of unsupervised learning. A two-stage cluster analysis method tailor-made for highdimensional data is obtained by combining the robust principal component analysis with the robust cluster analysis. The procedure can be interpreted as a robust knowledge discovery method tailor made for high-dimensional data.
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