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Mathematics and implementations of physiologically based pharmacokinetic modeling
Rakhimov, Yestay ; Duintjer Tebbens, Erik Jurjen (advisor) ; Klemera, Petr (referee)
Charles University Faculty of Pharmacy in Hradec Kr'alov'e Department of Biophysics and Physical Chemistry Candidate: Yestay Rakhimov Supervisor: doc. Erik Jurjen Duintjer Tebbens, Ph.D. Title of diploma thesis: Mathematics and implementations of physiologically based phar- macokinetic modeling The thesis addresses some basic aspects of pharmacokinetic modeling, which is used to describe pharmacokinetic processes. Understanding these processes is important for example to determine optimal concentrations of drugs dosing. The thesis focuses on mathematical proofs of a number of pharmacokinetic equa- tions, which are often not given in standard books. The derived equations are illustrated with numerical experiments for a particular drug in the software PharmCalcCl and MAT- LAB. 4
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Exploiting numerical linear algebra to accelerate the computation of the MCD estimator
Sommerová, Kristýna ; Duintjer Tebbens, Erik Jurjen (advisor) ; Hnětynková, Iveta (referee)
This work is dealing with speeding up the algorithmization of the MCD es- timator for detection of the mean and the covariance matrix of a normally dis- tributed multivariate data contaminated with outliers. First, the main idea of the estimator and its well-known aproximation by the FastMCD algorithm is discussed. The main focus was to be placed on possibilities of a speedup of the iteration step known as C-step while maintaining the quality of the estimations. This proved to be problematic, if not impossible. The work is, therefore, aiming at creating a new implementation based on the C-step and Jacobi method for eigenvalues. The proposed JacobiMCD algorithm is compared to the FastMCD in terms of floating operation count and results. In conclusion, JacobiMCD is not found to be fully equivalent to FastMCD but hints at a possibility of its usage on larger problems. The numerical experiments suggest that the computation can indeed be quicker by an order of magnitude, while the quality of results is close to those from FastMCD in some settings. 1
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The block triangular form and its use for sparse LU-factorization
Gálfy, Ivan ; Duintjer Tebbens, Erik Jurjen (advisor) ; Tůma, Miroslav (referee)
In this thesis we will present an effective method for solving systems of linear equations with large sparse matrices using LU factorization. The goal is to avoid filling the matrix by non-zero entries during the computations. Firstly we dis- cuss the use of permutations for the matrix algorithms. Afterwards we present the maximum matching algorithm and Tarjan's algorithm, both based on graph theory. Tarjan's algorithm is used to achieve block triangular form and the max- imum matching gives us the permutation into a matrix with zero free diagonal, which is recommended as a precursor to Tarjan's algorithm. 1
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Jacobi matrices: properties and possible generalizations
Preradová, Alena ; Hnětynková, Iveta (advisor) ; Duintjer Tebbens, Erik Jurjen (referee)
This thesis summarizes basic properties of Jacobi matrices and studies their selected structural generalizations, represented by special types of band, block tridiagonal and wedge-shaped matrices. Furthermore, it describes two Krylov subspace methods connected with Jacobi matrices, namely the Lanczos iterative tridiagonalization and the Golub-Kahan iterative bidiagonalization, and their block generalizations. The thesis shows, how block methods generate in each step generalised Jacobi matrices mentioned above. Main goal is to study spectral properties of these matrices focused on ivestigation of multiplicity of eigenvalues and nonzero components of eigenvectors. Powered by TCPDF (www.tcpdf.org)
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Efficient implementation of dimension reduction methods for high-dimensional statistics
Pekař, Vojtěch ; Duintjer Tebbens, Erik Jurjen (advisor) ; Hnětynková, Iveta (referee)
The main goal of our thesis is to make the implementation of a classification method called linear discriminant analysis more efficient. It is a model of multivariate statistics which, given samples and their membership to given groups, attempts to determine the group of a new sample. We focus especially on the high-dimensional case, meaning that the number of variables is higher than number of samples and the problem leads to a singular covariance matrix. If the number of variables is too high, it can be practically impossible to use the common methods because of the high computational cost. Therefore, we look at the topic from the perspective of numerical linear algebra and we rearrange the obtained tasks to their equivalent formulation with much lower dimension. We offer new ways of solution, provide examples of particular algorithms and discuss their efficiency. Powered by TCPDF (www.tcpdf.org)
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Computation and applications of the MCD estimator for robust statistical analysis
Sommerová, Kristýna ; Duintjer Tebbens, Erik Jurjen (advisor) ; Hnětynková, Iveta (referee)
This work describes one of the basic problems of robust statistics con- cerning outlier detection and its possible solution by using the Minimum covariance determinant estimator for estimates of the mean value and the covariance matrix with multivariate data. It explains how the estimator works and analyses its properties. The work concentrates on its approximation based on the fastMCD algorithm and specifies its numerical properties with emphasis on computational costs and stability of the standard implementation in MATLAB. It also discusses possible modifications of the algorithm and its effects on numerical properties. Lastly the work shows the usage of the fastMCD algorithm on a few real data experiments. Powered by TCPDF (www.tcpdf.org)
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The small sample size problem in gene expression tasks
Athanasiadis, Savvas ; Duintjer Tebbens, Erik Jurjen (advisor) ; Kalina, Jan (referee)
Charles University in Prague Faculty of Pharmacy in Hradec Králové Department of Biophysics and Physical Chemistry Candidate: Savvas Athanasiadis Supervisor: Jurjen Duintjer Tebbens Title of diploma thesis: The small sample size problem in gene expression tasks The thesis addresses classification of genes to tumor types based on their gene expression signatures. The number of variables (amino-acids) to be inves- tigated is typically very high (in the thousands) while it is expensive and time- consuming to analyze a high number of genes; usually at most tens of them are available. The combination of a small sample size with a large number of variables makes standard statistical classification methods inappropriate. The thesis focuses on a modification of a standard classification method, Fisher's linear discriminant analysis, for the case where the number of samples is smaller than the number of variables. It proposes an improved strategy to test this modified method with leave-one-out cross validation. Using so- called low rank updates of the involved covariance matrices, the computational costs of the cross validation process can be reduced by an order of magnitude. Memory demands are reduced as well.
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Matrix-free preconditioning
Trojek, Lukáš ; Duintjer Tebbens, Erik Jurjen (advisor) ; Tůma, Miroslav (referee)
The diploma theses is focused on matrix-free preconditioning of a linear system. It gives a very brief introduction into the area of iterative methods, preconditioning and matrix-free environment. The emphasis is put on a detailed description of a variant of LU factorization which can be computed in a matrix-free manner and on a new technique connected with this factorization for preconditioning by incomplete LU factors in matrix-free environment. Its main features are storage of only one of the two incomplete factors and low memory costs during the computation of the stored factor. The thesis closes with numerical experiments demonstrating the efficiency of the proposed technique.
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