High-throughput methods systematically measure the internal state of the entire cell, but powerful computational tools are needed to infer dynamics from their raw data. Therefore, we have developed a new computational method, Eigen-genomic System Dynamic-pattern Analysis (ESDA), which uses systems theory to infer dynamic parameters from a time series of gene expression measurements. As many genes are measured at a modest number of time points, estimation of the system matrix is underdetermined and traditional approaches for estimating dynamic parameters are ineffective; thus, ESDA uses the principle of dimensionality reduction to overcome the data imbalance. We identify degradation dynamic patterns of a genomic system using ESDA. We also combine ESDA and Principal-oscillation-pattern (POP) analysis, which has been widely used in geosciences, to identify oscillation patterns. We demonstrate the first application of POP analysis to genome-wide time-series gene-expression data. Both simulation data and real-world data are used in this study to demonstrate the applicability of ESDA to genomic data. The biological interpretations of dynamic patterns are provided. We also show that ESDA not only compares favorably with previous experimental methods and existing computational methods, but that it also provides complementary information relative to other approaches. / text

In this thesis, we consider real matrix functions that depend on two parameters and study the problem of how to detect and approximate parameters' values where the singular values coalesce. We prove several results connecting the existence of coalescing points to the periodic structure of the smooth singular values decomposition computed around the boundary of a domain enclosing the points. This is further used to develop algorithms for the detection and approximation of coalescing points in planar regions. Finally, we present techniques for continuing curves of coalescing singular values of matrices depending on three parameters, and illustrate how these techniques can be used to locate coalescing singular values of complex-valued matrices depending on three parameters.

Principal Components Analysis is a standard tool in data analysis, widely used in data-rich fields such as computer vision, data mining, bioinformatics, and econometrics. For a set of vectors in n dimensions and a natural number k less than n, the method returns a subspace of dimension k whose average squared distance to that set is as small as possible. Besides saving computation by reducing the dimension, projecting to this subspace can often reveal structure that was hidden in high dimension. This thesis considers several novel extensions of PCA, which provably reveals hidden structure where standard PCA fails to do so. First, we consider Robust PCA, which prevents a few points, possibly corrupted by an adversary, from having a large effect on the analysis. When applied to learning noisy logconcave mixture models, the algorithm requires only slightly more separation between component means than is required for the noiseless case. Second, we consider Isotropic PCA, which can go beyond the first two moments in identifying ``interesting' directions in data. The method leads to the first affine-invariant algorithm that can provably learn mixtures of Gaussians in high dimensions, improving significantly on known results. Thirdly, we define the ``Subgraph Parity Tensor' of order r of a graph and reduce the problem of finding planted cliques in random graphs to the problem of finding the top principal component of this tensor.

Thesis (Electrical Engineer and M.S. in Electrical Engineering)--Naval Postgraduate School, Dec. 2004. / Thesis Advisor(s): Monique P. Fargues. Includes bibliographical references (p. 107-109). Also available online.

Eigenvectors of the Laplacian of a graph G have received increasing attention in the recent past. Here we investigate their so-called nodal domains, i.e., the connected components of the maximal induced subgraphs of G on which an eigenvector \psi does not change sign. An analogue of Courant's nodal domain theorem provides upper bounds on the number of nodal domains depending on the location of \psi in the spectrum. This bound, however, is not sharp in general. In this contribution we consider the problem of computing minimal and maximal numbers of nodal domains for a particular graph. The class of Boolean Hypercubes is discussed in detail. We find that, despite the simplicity of this graph class, for which complete spectral information is available, the computations are still non-trivial. Nevertheless, we obtained some new results and a number of conjectures. (author's abstract) / Series: Preprint Series / Department of Applied Statistics and Data Processing

A semiregular tree is a tree where all non-pendant vertices have the same degree. Among all semiregular trees with fixed order and degree, a graph with minimal (adjacency / Laplacian) spectral radius is a caterpillar. Counter examples show that the result cannot be generalized to the class of trees with a given (non-constant) degree sequence. / Series: Research Report Series / Department of Statistics and Mathematics

The Discrete Nodal Domain Theorem states that an eigenfunction of the k-th largest eigenvalue of a generalized graph Laplacian has at most k (weak) nodal domains. We show that the number of strong nodal domains cannot exceed the size of a maximal induced bipartite subgraph and that this bound is sharp for generalized graph Laplacians. Similarly, the number of weak nodal domains is bounded by the size of a maximal bipartite minor. (author's abstract) / Series: Preprint Series / Department of Applied Statistics and Data Processing

Orientador: Renato da Rocha Lopes / Dissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia Elétrica e de Computação / Made available in DSpace on 2018-08-21T12:31:48Z (GMT). No. of bitstreams: 1 Barros_TiagoTavaresLeite_M.pdf: 6650333 bytes, checksum: 1abff9ddc2ee8bb4334ec96ba469f422 (MD5) Previous issue date: 2012 / Resumo: Nesta dissertação, nós discutimos o cálculo de funções de coerência de alta resolução para a estimação dos parâmetros de empilhamento em processamento de sinais sísmicos. Nosso foco é o algoritmo de estimação por Classificação de Sinais Múltiplos (MUSIC, do inglês MUltiple SIgnal Classification). Este pode superar a tradicional função de coerência semblance em casos em que há reflexões próximas ou interferentes. Nossa principal contribuição é a proposta de diversas simplificações para sua implementação. Primeiro, mostramos como obter os valores da função MUSIC a partir do subespaço de sinais do dado sísmico, que possui dimensão menor do que o subespaço de ruído, usualmente empregado. Depois disso, mostramos como obter o subespaço de sinais a partir do método da potência. Chamamos esta técnica de MUSIC com Método da Potência (PM-MUSIC). Também propusemos uma nova maneira de obtenção do espectro de MUSIC, baseada na decomposição em autovalores e autovetores da matriz de correlação temporal do dado sísmico. Este método contrasta com os algoritmos presentes na literatura, que se baseiam na correlação espacial. A partir do uso do Método da Potência, obtivemos reduções de complexidade tanto para a variante espacial quanto para a temporal do algoritmo MUSIC. Finalmente, também propusemos uma nova função de normalização para o cálculo de MUSIC, a qual chamamos de ponderação por semblance. Esta função leva em conta o espectro de velocidades obtido com a função de coerência semblance e lida com a alta variação dinâmica produzida pelo espectro de velocidades calculado com MUSIC. Nós comparamos a implementação de PM-MUSIC, a partir das correlações temporal e espacial. Exemplos numéricos com dados sísmicos sintéticos e de levantamentos reais demonstraram que o algoritmo PM-MUSIC supera o semblance e que sua variante temporal possui alta resolução, assim como sua variante espacial. Além disso, PM-MUSIC obtido a partir da correlação temporal mostrou-se extremamente robusto ao lidar com sinais correlacionados / Abstract: In this dissertation we discuss high-resolution coherence functions for the estimation of the stacking parameters in seismic signal processing. We focus on the MUltiple SIgnal Classification (MUSIC) algorithm, which uses the eigendecomposition of the seismic data to measure the coherence. MUSIC can outperform the traditional semblance in cases of close or interfering reflections. Our main contribution is to propose several simplifications to the implementation of MUSIC. First, we show how to compute MUSIC coherence measure in terms of the signal subspace of seismic data, which has lower dimension than the one currently used, the noise subspace. After that, we show how to obtain the signal subspace, iteratively, with the power method. We called this technique of Power Method MUSIC (PM-MUSIC). We also propose a new way to obtain the MUSIC spectrum, based on the eigendecomposition of the temporal correlation matrix of the seismic data. This is in contrast to the algorithms in the literature, which are based on the spatial correlation. Complexity reductions are obtained and discussed with the use of Power Method for both spatial and temporal variant of MUSIC. Finally, we propose a new normalization function for MUSIC, which we called semblance weighting. This function takes into account semblance coefficient and deals with high dynamic range in MUSIC velocity spectrum. We compared spatial and temporal correlation matrices, implemented with PM-MUSIC. Numerical examples with synthetic and real seismic data indicated that PM-MUSIC outperforms semblance and that the temporal variant of PM-MUSIC can present the same high-resolution as its spatial counterpart. Moreover, temporal PM-MUSIC is particularly useful when dealing with correlated signals / Mestrado / Telecomunicações e Telemática / Mestre em Engenharia Elétrica

A semiregular tree is a tree where all non-pendant vertices have the same degree. Belardo et al. (MATCH Commun. Math. Chem. 61(2), pp. 503-515, 2009) have shown that among all semiregular trees with a fixed order and degree, a graph with index is caterpillar. In this technical report we provide a different proof for this theorem. Furthermore, we give counter examples that show that this result cannot be generalized to the class of trees with a given (non-constant) degree sequence. / Series: Research Report Series / Department of Statistics and Mathematics

Orientadores: Jose Pissolato Filho, Afonso Jose do Prado / Tese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Eletrica e de Computação / Made available in DSpace on 2018-08-14T11:17:10Z (GMT). No. of bitstreams: 1 Campos_JoseCarlosdaCosta_D.pdf: 4388913 bytes, checksum: c565c846bae4e32b03dcef712a2494df (MD5) Previous issue date: 2009 / Resumo: As matrizes de transformação reais e constantes são aplicadas como matrizes de transformação fase-modo características de um sistema simétrico com circuito trifásico duplo transposto e de duas linhas de transmissão paralelas transpostas com circuito trifásico duplo. Essas matrizes de transformação reais e constantes são baseadas na matriz de Clarke. Usando a combinação linear dos elementos da matriz de Clarke, as técnicas aplicadas para linhas trifásicas simples são ampliadas para sistemas com 6 e 12 condutores de fase. Para uma linha trifásica dupla transposta, as matrizes Z e Y são convertidas em matrizes diagonais no domínio dos modos. Considerando um caso não transposto de uma linha trifásica dupla, os resultados não são exatos e as análises de erros são realizadas mediante os autovalores. No caso de duas linhas trifásicas paralelas duplas e transpostas, a matriz de transformação exata com elementos reais e constantes não foi obtida ainda. Para esse caso, como sugestão para desenvolvimento futuro, a determinação da matriz de transformação modal real e constante provavelmente deverá ser baseada em uma única referência homopolar. Tal sugestão se deve ao fato de que, neste trabalho, a estrutura das matrizes de transformação utilizadas tem como base a aplicação do modo homopolar como única referência homopolar para todos condutores de fase do sistema estudado. / Abstract: Single real transformation matrices are applied as phase-mode transformation matrices of typical symmetrical systems with double three-phase and two parallel double three-phase transmission lines. These single real transformation matrices are achieved from eigenvector matrices of the mentioned systems and they are based on Clarke's matrix. Using linear combinations of the Clarke's matrix elements, the techniques applied to the single three-phase lines are extended to systems with 6 or 12 phase conductors. For transposed double three-phase lines, phase Z and Y matrices are changed into diagonal matrices in mode domain. Considering non-transposed cases of double three-phase lines, the results are not exact and the error analyses are performed using the eigenvalues. In case of two parallel double three-phase lines, the exact single real transformation matrix has not been obtained yet. Probably, for two parallel double three-phase lines, considering future development and searching for the exact single real transformation matrix, the analyses are based on a single homopolar reference. This suggestion is related to that in the all analyses carried out in this work, the homopolar mode is used as the only homopolar reference for all phase conductors of the studied system. / Doutorado / Energia Eletrica / Doutor em Engenharia Elétrica