Topics in randomized numerical linear algebra

This thesis studies three classes of randomized numerical linear algebra algorithms, namely: (i) randomized matrix sparsification algorithms, (ii) low-rank approximation algorithms that use randomized unitary transformations, and (iii) low-rank approximation algorithms for positive-semidefinite (PSD) matrices.

Randomized matrix sparsification algorithms set randomly chosen entries of the input matrix to zero. When the approximant is substituted for the original matrix in computations, its sparsity allows one to employ faster sparsity-exploiting algorithms. This thesis contributes bounds on the approximation error of nonuniform randomized sparsification schemes, measured in the spectral norm and two NP-hard norms that are of interest in computational graph theory and subset selection applications.

Low-rank approximations based on randomized unitary transformations have several desirable properties: they have low communication costs, are amenable to parallel implementation, and exploit the existence of fast transform algorithms. This thesis investigates the tradeoff between the accuracy and cost of generating such approximations. State-of-the-art spectral and Frobenius-norm error bounds are provided.

The last class of algorithms considered are SPSD "sketching" algorithms. Such sketches can be computed faster than approximations based on projecting onto mixtures of the columns of the matrix. The performance of several such sketching schemes is empirically evaluated using a suite of canonical matrices drawn from machine learning and data analysis applications, and a framework is developed for establishing theoretical error bounds.

In addition to studying these algorithms, this thesis extends the Matrix Laplace Transform framework to derive Chernoff and Bernstein inequalities that apply to all the eigenvalues of certain classes of random matrices. These inequalities are used to investigate the behavior of the singular values of a matrix under random sampling, and to derive convergence rates for each individual eigenvalue of a sample covariance matrix.

Nonstationary response of structures and its application to earthquake engineering

This thesis presents a simplified state-variable method to solve for the nonstationary response of linear MDOF systems subjected to a modulated stationary excitation in both time and frequency domains. The resulting covariance matrix and evolutionary spectral density matrix of the response may be expressed as a product of a constant system matrix and a time-dependent matrix, the latter can be explicitly evaluated for most envelopes currently prevailing in engineering. The stationary correlation matrix of the response may be found by taking the limit of the covariance response when a unit step envelope is used. The reliability analysis can then be performed based on the first two moments of the response obtained.

The method presented facilitates obtaining explicit solutions for general linear MDOF systems and is flexible enough to be applied to different stochastic models of excitation such as the stationary models, modulated stationary models, filtered stationary models, and filtered modulated stationary models and their stochastic equivalents including the random pulse train model, filtered shot noise, and some ARMA models in earthquake engineering. This approach may also be readily incorporated into finite element codes for random vibration analysis of linear structures.

A set of explicit solutions for the response of simple linear structures subjected to modulated white noise earthquake models with four different envelopes are presented as illustration. In addition, the method has been applied to three selected topics of interest in earthquake engineering, namely, nonstationary analysis of primary-secondary systems with classical or nonclassical dampings, soil layer response and related structural reliability analysis, and the effect of the vertical components on seismic performance of structures. For all the three cases, explicit solutions are obtained, dynamic characteristics of structures are investigated, and some suggestions are given for aseismic design of structures.

Random propagation in complex systems : nonlinear matrix recursions and epidemic spread

This dissertation studies long-term behavior of random Riccati recursions and mathematical epidemic model. Riccati recursions are derived from Kalman filtering. The error covariance matrix of Kalman filtering satisfies Riccati recursions. Convergence condition of time-invariant Riccati recursions are well-studied by researchers. We focus on time-varying case, and assume that regressor matrix is random and identical and independently distributed according to given distribution whose probability distribution function is continuous, supported on whole space, and decaying faster than any polynomial. We study the geometric convergence of the probability distribution. We also study the global dynamics of the epidemic spread over complex networks for various models. For instance, in the discrete-time Markov chain model, each node is either healthy or infected at any given time. In this setting, the number of the state increases exponentially as the size of the network increases. The Markov chain has a unique stationary distribution where all the nodes are healthy with probability 1. Since the probability distribution of Markov chain defined on finite state converges to the stationary distribution, this Markov chain model concludes that epidemic disease dies out after long enough time. To analyze the Markov chain model, we study nonlinear epidemic model whose state at any given time is the vector obtained from the marginal probability of infection of each node in the network at that time. Convergence to the origin in the epidemic map implies the extinction of epidemics. The nonlinear model is upper-bounded by linearizing the model at the origin. As a result, the origin is the globally stable unique fixed point of the nonlinear model if the linear upper bound is stable. The nonlinear model has a second fixed point when the linear upper bound is unstable. We work on stability analysis of the second fixed point for both discrete-time and continuous-time models. Returning back to the Markov chain model, we claim that the stability of linear upper bound for nonlinear model is strongly related with the extinction time of the Markov chain. We show that stable linear upper bound is sufficient condition of fast extinction and the probability of survival is bounded by nonlinear epidemic map.

Continuous stochastic processes

A general review of stochastic processes is given in the introduction; definitions, properties and a rough classification are presented together with the position and scope of the author's work as it fits into the general scheme.

The first section presents a brief summary of the pertinent analytical properties of continuous stochastic processes and their probability-theoretic foundations which are used in the sequel.

The remaining two sections (II and III), comprising the body of the work, are the author's contribution to the theory. It turns out that a very inclusive class of continuous stochastic processes are characterized by a fundamental partial differential equation and its adjoint (the Fokker-Planck equations). The coefficients appearing in those equations assimilate, in a most concise way, all the salient properties of the process, freed from boundary value considerations. The writer’s work consists in characterizing the processes through these coefficients without recourse to solving the partial differential equations.

First, a class of coefficients leading to a unique, continuous process is presented, and several facts are proven to show why this class is restricted. Then, in terms of the coefficients, the unconditional statistics are deduced, these being the mean, variance and covariance. The most general class of coefficients leading to the Gaussian distribution is deduced, and a complete characterization of these processes is presented. By specializing the coefficients, all the known stochastic processes may be readily studied, and some examples of these are presented; viz. the Einstein process, Bachelier process, Ornstein-Uhlenbeck process, etc. The calculations are effectively reduced down to ordinary first order differential equations, and in addition to giving a comprehensive characterization, the derivations are materially simplified over the solution to the original partial differential equations.

In the last section the properties of the integral process are presented. After an expository section on the definition, meaning, and importance of the integral process, a particular example is carried through starting from basic definition. This illustrates the fundamental properties, and an inherent paradox. Next the basic coefficients of the integral process are studied in terms of the original coefficients, and the integral process is uniquely characterized. It is shown that the integral process, with a slight modification, is a continuous Markoff process.

The elementary statistics of the integral process are deduced: means, variances, and covariances, in terms of the original coefficients. It is shown that an integral process is never temporally homogeneous in a non-degenerate process.

Finally, in terms of the original class of admissible coefficients, the statistics of the integral process are explicitly presented, and the integral process of all known continuous processes are specified.

A model for energy and morphology of crystalline grain boundaries with arbitrary geometric character

It has been well-established that interfaces in crystalline materials are key players in the mechanics of a variety of mesoscopic processes such as solidification, recrystallization, grain boundary migration, and severe plastic deformation. In particular, interfaces with complex morphologies have been observed to play a crucial role in many micromechanical phenomena such as grain boundary migration, stability, and twinning. Interfaces are a unique type of material defect in that they demonstrate a breadth of behavior and characteristics eluding simplified descriptions. Indeed, modeling the complex and diverse behavior of interfaces is still an active area of research, and to the author's knowledge there are as yet no predictive models for the energy and morphology of interfaces with arbitrary character. The aim of this thesis is to develop a novel model for interface energy and morphology that i) provides accurate results (especially regarding "energy cusp" locations) for interfaces with arbitrary character, ii) depends on a small set of material parameters, and iii) is fast enough to incorporate into large scale simulations.

In the first half of the work, a model for planar, immiscible grain boundary is formulated. By building on the assumption that anisotropic grain boundary energetics are dominated by geometry and crystallography, a construction on lattice density functions (referred to as "covariance") is introduced that provides a geometric measure of the order of an interface. Covariance forms the basis for a fully general model of the energy of a planar interface, and it is demonstrated by comparison with a wide selection of molecular dynamics energy data for FCC and BCC tilt and twist boundaries that the model accurately reproduces the energy landscape using only three material parameters. It is observed that the planar constraint on the model is, in some cases, over-restrictive; this motivates an extension of the model.

In the second half of the work, the theory of faceting in interfaces is developed and applied to the planar interface model for grain boundaries. Building on previous work in mathematics and materials science, an algorithm is formulated that returns the minimal possible energy attainable by relaxation and the corresponding relaxed morphology for a given planar energy model. It is shown that the relaxation significantly improves the energy results of the planar covariance model for FCC and BCC tilt and twist boundaries. The ability of the model to accurately predict faceting patterns is demonstrated by comparison to molecular dynamics energy data and experimental morphological observation for asymmetric tilt grain boundaries. It is also demonstrated that by varying the temperature in the planar covariance model, it is possible to reproduce a priori the experimentally observed effects of temperature on facet formation.

Finally, the range and scope of the covariance and relaxation models, having been demonstrated by means of extensive MD and experimental comparison, future applications and implementations of the model are explored.

Identificação da região periorbital utilizando a técnica SIFT em conjunto com rede neural híbrida

Nesta dissertação, foi utilizada a técnica SIFT (Scale Invariant Feature Transform) para o reconhecimento de imagens da área dos olhos (região periorbital). Foi implementada uma classificação das imagens em subgrupos internos ao banco de dados, utilizando-se das informações estatísticas provenientes dos padrões invariantes produzidos pela técnica SIFT. Procedeu-se a uma busca categorizada pelo banco de dados, ao invés da procura de um determinado padrão apresentado, através da comparação deste com cada padrão presente no banco de dados. A tais padrões foi aplicada uma abordagem estatística, através da geração da matriz de covariâncias dos padrões gerados, sendo esta utilizada para a categorização, tendo por base uma rede neural híbrida. A rede neural classifica e categoriza o banco de dados de imagens, criando uma topologia de busca. Foram obtidos resultados corretos de classificação de 76,3% pela rede neural híbrida, sendo que um algoritmo auxiliar determina uma hierarquia de busca, onde, ocorrendo uma errônea classificação, a busca segue em grupos de pesquisas mais prováveis.

Optimum linear coding for additive noise systems using information feedback

The feedback coding problem for Gaussian systems in which the noise is neither white nor statistically independent between channels is formulated in terms of arbitrary linear codes at the transmitter and at the receiver. This new formulation is used to determine a number of feedback communication systems. In particular, the optimum linear code that satisfies an average power constraint on the transmitted signals is derived for a system with noiseless feedback and forward noise of arbitrary covariance. The noisy feedback problem is considered and signal sets for the forward and feedback channels are obtained with an average power constraint on each. The general formulation and results are valid for non-Gaussian systems in which the second order statistics are known, the results being applicable to the determination of error bounds via the Chebychev inequality.

Influência do exercício físico sobre a massa e o metabolismo ósseo de indivíduos com lesão medular cervical

Indivíduos que permanecem longo tempo em cadeira de rodas apresentam importante perda de massa óssea, principalmente nos membros inferiores, possivelmente agravada pela baixa ingestão de cálcio dietético e pelo inadequado estado nutricional de vitamina D. O exercício físico pode contribuir para a manutenção ou aumento da massa óssea em diferentes populações e nos indivíduos com lesão medular pode contribuir para atenuar a perda de massa óssea. O objetivo do presente estudo foi avaliar a influência da prática regular de exercício físico sobre a adequação da massa óssea, indicadores bioquímicos do metabolismo ósseo e estado nutricional de vitamina D em indivíduos com lesão medular cervical há pelo menos um ano. Em vinte e cinco homens de 19 a 56 anos sendo 15 fisicamente ativos e 10 sedentários, foi realizada análise sérica de cálcio, PTH, 25(OH)D, IGF-1, osteocalcina e NTx. As medidas do conteúdo mineral ósseo, densidade mineral óssea (DMO), massa magra e massa gorda foram realizadas por DXA. A pigmentação da pele (constitutiva e por bronzeamento) foi determinada por colorimetria com o objetivo de investigar sua influência sobre o estado de vitamina D. A ingestão habitual de cálcio foi registrada em um questionário de frequência alimentar direcionado para alimentos fonte. As comparações entre os dois grupos foram realizadas pela aplicação do Teste t de Student exceto para as variáveis ósseas que foram realizadas após ajustes pela massa corporal total, tempo de lesão e ingestão de cálcio utilizando-se análise de co-variância. Associações entre as variáveis estudadas foram avaliadas através de análise de correlação de Pearson. Valores de p<0.05 foram considerados significativos. Não foram observadas diferenças estatisticamente significativas entre os grupos para nenhuma variável óssea com exceção do z-score da DMO da coluna lombar, que foi significativamente maior no grupo de indivíduos sedentários (0,9 1,7 vs -0,7 0,8; p<0,05). No entanto, entre os indivíduos ativos, aqueles que iniciaram a prática de exercício físico com menos tempo decorrido após a lesão apresentaram maior DMO do fêmur (r=-0,60; p<0,05). Nos indivíduos ativos, a freqüência do exercício apresentou associação negativa com a concentração sérica de i-PTH (r = -0,50; p =0,05) e positiva com a concentração de 25(OH)D (r= 0,58; p <0,05). Após ajustes pela massa corporal total e tempo de lesão foram observadas associações positivas entre a ingestão diária de cálcio e z-score da DMO da coluna lombar (r = 0,73 e p <0,01) e DMO do rádio (r = 0,56 e p <0,05). Os resultados do presente estudo apontam para um efeito benéfico do exercício físico sobre a massa óssea e o perfil hormonal relacionado ao metabolismo ósseo. O início da prática regular de exercício físico o quanto antes após a lesão parece contribuir para atenuar a perda de massa óssea nos membros inferiores. Além disso, os resultados deste estudo sugerem uma possível potencialização do efeito osteogênico do exercício físico quando combinado a uma adequada ingestão de cálcio.

Gaussian processes for machine learning (GPML) toolbox

The GPML toolbox provides a wide range of functionality for Gaussian process (GP) inference and prediction. GPs are specified by mean and covariance functions; we offer a library of simple mean and covariance functions and mechanisms to compose more complex ones. Several likelihood functions are supported including Gaussian and heavy-tailed for regression as well as others suitable for classification. Finally, a range of inference methods is provided, including exact and variational inference, Expectation Propagation, and Laplace’s method dealing with non-Gaussian likelihoods and FITC for dealing with large regression tasks.

Assessing trends in the density of Atlantic croaker (Micropogonias undulatus): a comparison of passive acoustic and trawl methods

Using data collected simultaneously from a trawl and a hydrophone, we found that temporal and spatial trends in densities of juvenile Atlantic croaker (Micropogonias undulatus) in the Neuse River estuary in North Carolina can be identified by monitoring their sound production. Multivariate analysis of covariance (MA NCOVA) revealed that catch per unit of effort (CPUE) of Atlantic croaker had a significant relationship with the dependent variables of sound level and peak frequency of Atlantic croaker calls. Tests of between-subject correspondence failed to detect relationships between CPUE and either of the call parameters, but statistical power was low. Williamson’s index of spatial overlap indicated that call detection rate (expressed by a 0–3 calling index) was correlated in time and space with Atlantic croaker CPUE. The correspondence between acoustic parameters and trawl catch rates varied by month and by habitat. In general, the calling index had a higher degree of overlap with this species’ density than did the received sound level of their calls. Classification and regression tree analysis identified calling index as the strongest correlate of CPUE. Passive acoustics has the potential to be an inexpensive means of identifying spatial and temporal trends in abundance for soniferous fish species.

Anisotropic expansion and SNIa: An open issue

We review the appropriateness of using SNIa observations to detect potential signatures of anisotropic expansion in the Universe. We focus on Union2 and SNLS3 SNIa datasets and use the hemispherical comparison method to detect possible anisotropic features. Unlike some previous works where nondiagonal elements of the covariance matrix were neglected, we use the full covariance matrix of the SNIa data, thus obtaining more realistic and not underestimated errors. As a matter of fact, the significance of previously claimed detections of a preferred direction in the Union2 dataset completely disappears once we include the effects of using the full covariance matrix. Moreover, we also find that such apreferred direction is aligned with the orthogonal direction of the SDSS observational plane and this suggests a clear indication that the SDSS subsample of the Union2 dataset introduces a significant bias, making the detected preferred direction unphysical. We thus find that current SNIa surveys are inappropriate to test anisotropic features due to their highly non-homogeneous angular distribution in the sky. In addition, after removal of the highest in homogeneous sub-samples, the number of SNIa is too low. Finally, we take advantage of the particular distribution of SNLS SNIa sub- sample in the SNLS3 data set, in which the observations were taken along four different directions. We fit each direction independently and find consistent results at the 1 sigma level. Although the likelihoods peak at relatively different values of Omega(m), the low number of data along each direction gives rise to large errors so that the likelihoods are sufficiently broad as to overlap within 1 sigma. (C) 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http:// creativecommons. org/licenses/by/4.0/).

Length-weight relationships of fishes from tributaries of the Volta River, Ghana: Part II and Conclusion

Slopes and intercepts of length-weight relationships obtained from 37 populations from the rivers Oti, Pru and Black Volta in Ghana were compared using a one way analysis of covariance with fixed effects. Although no significant differences were obtained from this analysis, an ANOVA comparing the magnitudes of mean condition factors (Wx100/SL3) found 9 out of 37 populations significantly different at the 0.05 level. A two-way nested ANOVA using all populations combined, however, did not yield any significant differences between the three rivers. Thus, pooling the data to obtain the results presented in Part I (see Entsua-Mensah et al. Naga 1995) is justified here.

Em busca de um índice alternativo à relação Book to Market para a construção de carteiras mais rentáveis

Muitos estudos buscam tentar prever o retorno potencial sobre portfólios de ações, com intuito de obter melhor rentabilidade sobre o capital aplicado. Diversas modelagens já foram utilizadas, sendo que as mais conhecidas são as que relacionam o risco com o retorno. Nesta linha destacam-se a Teoria de Carteiras proposta por Markowitz, e o CAPM de Sharpe. Através destas teorias entende-se a questão da influência da covariância dos retornos e que para um melhor desempenho de uma carteira, não é suficiente avaliar cada ativo individualmente. Por outro lado, diversas críticas em relação ao CAPM, vêm ensejando estudos complementares na busca de outras variáveis que melhorem os métodos de seleção de ativos. Fama e French (1993) fizeram um estudo com variáveis complementares em relação ao beta do CAPM, utilizando o tamanho e a relação Book to Market, conseguindo resultados melhores que o CAPM tradicional. O presente estudo leva em conta a questão do reinvestimento do lucro gerado e utilizando o modelo de Gordon propõe uma variável de classificação de empresas de crescimento e empresas valor, conceito já utilizado na literatura de finanças.Com base nesta variável montam-se carteiras de ações entre os anos de 2005 e 2012 e observa-se que é possível obter ganhos com a lógica proposta. Ao longo do período seria possível obter com as carteiras selecionadas ganhos de até 107,85% contra os retornos de 55,58% das carteiras com todos os ativos. Organizamos os mesmos ativos pela ótica da relação Book to Market as quais obtiveram retorno total do período de 90,42%. Apesar de notar uma mudança clara de comportamento, onde apenas nos quatro primeiros anos do estudo as carteiras com empresas value são superiores e nos quatro últimos períodos as carteiras de empresas growth são as melhores. Estes resultados são compatíveis com os resultados de Braga e Leal (2000), e Mescolin, Martinelli Braga e da Costa Jr. (1997), verificando um melhor desempenho para as empresas value.

Estudo de associação entre os níveis de coordenação motora, adiposidade e atividade física de crianças em idade escolar

Há um crescente conjunto de evidências que têm indicado associações significativas entre os níveis de coordenação motora e outros atributos relacionados à saúde, tais como os níveis de adiposidade corporal e atividade física. Entretanto, as associações entre os níveis de coordenação motora, adiposidade e atividade física têm sido analisadas de forma bivariada, sem considerar a influência recíproca que essas variáveis exercem entre si, o que pode ser a causa da produção de dados enviesados. Assim sendo, o objetivo geral do presente estudo foi analisar o inter-relacionamento entre os níveis de coordenação motora, adiposidade e atividade física de crianças entre 12 e 14 anos de idade. Cento e dezenove participantes (51 meninos e 68 meninas) foram recrutados. O questionário Physical Activity Questionnaire for Older Children, o teste de coordenação motora Körperkoordinationstest für Kinder e um plicômetro clínico foram utilizados para estimar, respectivamente, os níveis de atividade física, coordenação motora e adiposidade corporal dos participantes. Coeficientes de Correlação de Pearson foram usados para examinar as associações bivariadas entre níveis de atividade física e coordenação motora e entre níveis de adiposidade corporal e coordenação motora. Correlações parciais foram usadas para analisar as associações entre os níveis de atividade física e coordenação motora, controlando pelos níveis de adiposidade corporal, e entre os níveis de adiposidade corporal e coordenação motora, controlando pelos níveis de atividade física. O teste de Análise de Covariância Multivariada (MANCOVA) foi utilizado para testar diferenças entre os grupos formados de acordo com o status de adiposidade e atividade física com o intuito de examinar a influência combinada dessas variáveis sobre os níveis de coordenação motora. De um modo geral, os resultados deste estudo indicaram que as associações dos níveis de coordenação motora com os níveis de adiposidade corporal e atividade física podem sofrer alterações de acordo com as covariáveis consideradas nas análises em meninos, mas não em meninas. O fato de tal fenômeno não ter sido observado em meninas pode estar relacionado aos baixos níveis de atividade física apresentados por elas. Foram sugeridas pesquisas adicionais nas quais sejam recrutadas meninas com maiores níveis de atividade física com o intuito de testar a hipótese acima aludida. Por fim, embora nesta oportunidade não se tenha estabelecido qualquer relação de causalidade entre as variáveis estudadas, não temos dúvidas de que crianças devem ser encorajadas a desenvolverem adequados níveis de coordenação motora, pois tal variável está associada com atributos relacionados ao estado de saúde.

Motor task variation induces structural learning.

When we have learned a motor skill, such as cycling or ice-skating, we can rapidly generalize to novel tasks, such as motorcycling or rollerblading [1-8]. Such facilitation of learning could arise through two distinct mechanisms by which the motor system might adjust its control parameters. First, fast learning could simply be a consequence of the proximity of the original and final settings of the control parameters. Second, by structural learning [9-14], the motor system could constrain the parameter adjustments to conform to the control parameters' covariance structure. Thus, facilitation of learning would rely on the novel task parameters' lying on the structure of a lower-dimensional subspace that can be explored more efficiently. To test between these two hypotheses, we exposed subjects to randomly varying visuomotor tasks of fixed structure. Although such randomly varying tasks are thought to prevent learning, we show that when subsequently presented with novel tasks, subjects exhibit three key features of structural learning: facilitated learning of tasks with the same structure, strong reduction in interference normally observed when switching between tasks that require opposite control strategies, and preferential exploration along the learned structure. These results suggest that skill generalization relies on task variation and structural learning.