Neste trabalho foram utilizadas cópulas bivariadas variantes no tempo para modelar a dependência entre séries de retornos financeiros. O objetivo deste trabalho é apresentar uma abordagem de estimação semi-paramétrica de cópulas variantes no tempo a partir de uma função de cópula paramétrica na qual o parâmetro varia no tempo. A função do parâmetro desconhecido será estimada pela aproximação de ondaleta Haar, polinômio de Taylor e Kernel. O desempenho dos três métodos de aproximação será comparado via estudos de simulação. Uma aplicação aos dados reais será apresentada para ilustrar a metodologia estudada. / In this work the bivariate Time-varying copula models have been used to model the dependence between payback. The aim of this work is to present an approach of semiparametric estimation of Time-varying copula models from a parametric copula function in which the parameter varies with the time. The function of the unknown parameter will be estimated by Haar wavelet approach, Taylor series and smoothing Kernel approximation. The measured performance of the three estimation method will be compared by simulation study. An application of the data will be presented to illustrate the studied methodology.

Neste trabalho foram utilizadas cópulas bivariadas variantes no tempo para modelar a dependência entre séries de retornos financeiros. O objetivo deste trabalho é apresentar uma abordagem de estimação semi-paramétrica de cópulas variantes no tempo a partir de uma função de cópula paramétrica na qual o parâmetro varia no tempo. A função do parâmetro desconhecido será estimada pela aproximação de ondaleta Haar, polinômio de Taylor e Kernel. O desempenho dos três métodos de aproximação será comparado via estudos de simulação. Uma aplicação aos dados reais será apresentada para ilustrar a metodologia estudada. / In this work the bivariate Time-varying copula models have been used to model the dependence between payback. The aim of this work is to present an approach of semiparametric estimation of Time-varying copula models from a parametric copula function in which the parameter varies with the time. The function of the unknown parameter will be estimated by Haar wavelet approach, Taylor series and smoothing Kernel approximation. The measured performance of the three estimation method will be compared by simulation study. An application of the data will be presented to illustrate the studied methodology.

This thesis investigates the functionality of the units used in connectionist Artificial Intelligence systems. Artificial Neural Networks form the foundation of the research and their units, Artificial Neurons, are first compared with alternative models. This initial work is mainly in the spatial-domain and introduces a new neural model, termed a Taylor Series neuron. This is designed to be flexible enough to assume most mathematical functions. The unit is based on Power Series theory and a specifically implemented Taylor Series neuron is demonstrated. These neurons are of particular usefulness in evolutionary networks as they allow the complexity to increase without adding units. Training is achieved via various traditiona and derived methods based on the Delta Rule, Backpropagation, Genetic Algorithms and associated evolutionary techniques. This new neural unit has been presented as a controllable and more highly functional alternative to previous models. The work on the Taylor Series neuron moved into time-domain behaviour and through the investigation of neural oscillators led to an examination of single-celled intelligence from which the later work developed. Connectionist approaches to Artificial Intelligence are almost always based on Artificial Neural Networks. However, another route towards Parallel Distributed Processing was introduced. This was inspired by the intelligence displayed by single-celled creatures called Protoctists (Protists). A new system based on networks of interacting proteins was introduced. These networks were tested in pattern-recognition and control tasks in the time-domain and proved more flexible than most neuron models. They were trained using a Genetic Algorithm and a derived Backpropagation Algorithm. Termed "Artificial BioChemical Networks" (ABN) they have been presented as an alternative approach to connectionist systems.

Digital subscriber lines (DSL) have revolutionised the provision of high speed data over the ‘last mile’. Subscribers demand even more bandwidth and the penetration of the service is now nearly universal. While it is feasible to provide improved broadband services on the new very high speed DSL, such as VDSL2/3, one of the greatest challenges to further improvements in speed is the problem of crosstalk. Operating over the unused higher frequencies of the twisted pair network, this technology is subjected to electromagnetic coupling among the wires, limiting the DSL data rate and service reach. Crosstalk suppression methods such as zero-forcing or decision feedback mainly use block processing. However, to cope with the time-varying VDSL environment huge computational costs can be incurred. In contrast, adaptive processing approaches are much simpler and are more beneficial to track such a channel environment. An adaptive canceller uses a training sequence and the convergence speed depends on the number of crosstalk coefficients it has to estimate. In a populated DSL binder, only a few of the crosstalking neighbours to a particular user are significant. With the aim to reduce the computational complexity in such environments, this thesis introduces the concept of detection-guided adaptive crosstalk cancellation for DSL. We propose a least-squares test feature to detect and concentrate the adaptation only on the dominant crosstalking coefficients. In comparison to conventional adaptive cancellers, the cancellers proposed in this thesis demonstrate early convergence. Thus, by incorporating the test feature, these cancellers have to detect only the most significant canceller coefficients and therefore, the length of the training sequence is reduced. Together with enhanced adaptive cancellation with a low run-time complexity and improved convergence, the greatest advantage obtained here is in the bandwidth efficiency. While enhanced adaptive cancellation is a bandwidth-efficient approach, the frequent re-transmission of training sequences may still be required for a rapidly changing VDSL channel. Again, this can be a disadvantage in terms of bandwidth consumption. To overcome this difficulty, we propose fast-converging unsupervised cancellers with an aim to improve the bandwidth efficiency by not transmitting a training sequence. An added advantage obtained here is that this would enable Internet service providers to include multiple or improved broadband services within a single subscription. Certain properties of the DSL channel ensure the communication channel is properly conditioned. This ensures the basis vectors of the channel matrix are near-orthogonal and hence, the linear cancellers, such as zero-forcing perform near-optimally. However, this is not the case with wireless channels. We investigate user detection in wireless channels using the principle of lattice reduction. User detection can also be seen as a search for the closest vector point in the lattice of received symbols. Though a maximum likelihood (ML) detector facilitates optimal user-detection, it has exponential complexity. We identify that the closest vector problem can be cast as a non-linear optimisation problem. Using the periodicity of the maximum likelihood function, we first present a novel algorithm that approximates the ML function using the Taylor series expansion of a suitable cosine function. With the aim of minimising the approximation error, we represent the ML function as a Fourier Series expansion and later, propose another approximation using Jacobi theta functions. We study the performance of these approximations when subjected to a suitable unconstrained optimisation algorithm. Through simulations, we demonstrate that the newly-developed approximations perform better than the conventional cancellers, close to the ML and, importantly, converging in polynomial time.

Digital subscriber lines (DSL) have revolutionised the provision of high speed data over the ‘last mile’. Subscribers demand even more bandwidth and the penetration of the service is now nearly universal. While it is feasible to provide improved broadband services on the new very high speed DSL, such as VDSL2/3, one of the greatest challenges to further improvements in speed is the problem of crosstalk. Operating over the unused higher frequencies of the twisted pair network, this technology is subjected to electromagnetic coupling among the wires, limiting the DSL data rate and service reach. Crosstalk suppression methods such as zero-forcing or decision feedback mainly use block processing. However, to cope with the time-varying VDSL environment huge computational costs can be incurred. In contrast, adaptive processing approaches are much simpler and are more beneficial to track such a channel environment. An adaptive canceller uses a training sequence and the convergence speed depends on the number of crosstalk coefficients it has to estimate. In a populated DSL binder, only a few of the crosstalking neighbours to a particular user are significant. With the aim to reduce the computational complexity in such environments, this thesis introduces the concept of detection-guided adaptive crosstalk cancellation for DSL. We propose a least-squares test feature to detect and concentrate the adaptation only on the dominant crosstalking coefficients. In comparison to conventional adaptive cancellers, the cancellers proposed in this thesis demonstrate early convergence. Thus, by incorporating the test feature, these cancellers have to detect only the most significant canceller coefficients and therefore, the length of the training sequence is reduced. Together with enhanced adaptive cancellation with a low run-time complexity and improved convergence, the greatest advantage obtained here is in the bandwidth efficiency. While enhanced adaptive cancellation is a bandwidth-efficient approach, the frequent re-transmission of training sequences may still be required for a rapidly changing VDSL channel. Again, this can be a disadvantage in terms of bandwidth consumption. To overcome this difficulty, we propose fast-converging unsupervised cancellers with an aim to improve the bandwidth efficiency by not transmitting a training sequence. An added advantage obtained here is that this would enable Internet service providers to include multiple or improved broadband services within a single subscription. Certain properties of the DSL channel ensure the communication channel is properly conditioned. This ensures the basis vectors of the channel matrix are near-orthogonal and hence, the linear cancellers, such as zero-forcing perform near-optimally. However, this is not the case with wireless channels. We investigate user detection in wireless channels using the principle of lattice reduction. User detection can also be seen as a search for the closest vector point in the lattice of received symbols. Though a maximum likelihood (ML) detector facilitates optimal user-detection, it has exponential complexity. We identify that the closest vector problem can be cast as a non-linear optimisation problem. Using the periodicity of the maximum likelihood function, we first present a novel algorithm that approximates the ML function using the Taylor series expansion of a suitable cosine function. With the aim of minimising the approximation error, we represent the ML function as a Fourier Series expansion and later, propose another approximation using Jacobi theta functions. We study the performance of these approximations when subjected to a suitable unconstrained optimisation algorithm. Through simulations, we demonstrate that the newly-developed approximations perform better than the conventional cancellers, close to the ML and, importantly, converging in polynomial time.

Digital subscriber lines (DSL) have revolutionised the provision of high speed data over the ‘last mile’. Subscribers demand even more bandwidth and the penetration of the service is now nearly universal. While it is feasible to provide improved broadband services on the new very high speed DSL, such as VDSL2/3, one of the greatest challenges to further improvements in speed is the problem of crosstalk. Operating over the unused higher frequencies of the twisted pair network, this technology is subjected to electromagnetic coupling among the wires, limiting the DSL data rate and service reach. Crosstalk suppression methods such as zero-forcing or decision feedback mainly use block processing. However, to cope with the time-varying VDSL environment huge computational costs can be incurred. In contrast, adaptive processing approaches are much simpler and are more beneficial to track such a channel environment. An adaptive canceller uses a training sequence and the convergence speed depends on the number of crosstalk coefficients it has to estimate. In a populated DSL binder, only a few of the crosstalking neighbours to a particular user are significant. With the aim to reduce the computational complexity in such environments, this thesis introduces the concept of detection-guided adaptive crosstalk cancellation for DSL. We propose a least-squares test feature to detect and concentrate the adaptation only on the dominant crosstalking coefficients. In comparison to conventional adaptive cancellers, the cancellers proposed in this thesis demonstrate early convergence. Thus, by incorporating the test feature, these cancellers have to detect only the most significant canceller coefficients and therefore, the length of the training sequence is reduced. Together with enhanced adaptive cancellation with a low run-time complexity and improved convergence, the greatest advantage obtained here is in the bandwidth efficiency. While enhanced adaptive cancellation is a bandwidth-efficient approach, the frequent re-transmission of training sequences may still be required for a rapidly changing VDSL channel. Again, this can be a disadvantage in terms of bandwidth consumption. To overcome this difficulty, we propose fast-converging unsupervised cancellers with an aim to improve the bandwidth efficiency by not transmitting a training sequence. An added advantage obtained here is that this would enable Internet service providers to include multiple or improved broadband services within a single subscription. Certain properties of the DSL channel ensure the communication channel is properly conditioned. This ensures the basis vectors of the channel matrix are near-orthogonal and hence, the linear cancellers, such as zero-forcing perform near-optimally. However, this is not the case with wireless channels. We investigate user detection in wireless channels using the principle of lattice reduction. User detection can also be seen as a search for the closest vector point in the lattice of received symbols. Though a maximum likelihood (ML) detector facilitates optimal user-detection, it has exponential complexity. We identify that the closest vector problem can be cast as a non-linear optimisation problem. Using the periodicity of the maximum likelihood function, we first present a novel algorithm that approximates the ML function using the Taylor series expansion of a suitable cosine function. With the aim of minimising the approximation error, we represent the ML function as a Fourier Series expansion and later, propose another approximation using Jacobi theta functions. We study the performance of these approximations when subjected to a suitable unconstrained optimisation algorithm. Through simulations, we demonstrate that the newly-developed approximations perform better than the conventional cancellers, close to the ML and, importantly, converging in polynomial time.

Orientador: Prof. Dr. Vinicius Cifú Lopes / Dissertação (mestrado) - Universidade Federal do ABC, Programa de Pós-Graduação em Mestrado Profissional em Matemática em Rede Nacional, 2016. / Inicia-se o presente trabalho apresentando ao leitor a necessidade de se apropriar profundamente dos conceitos relacionados às funções lineares e quadráticas, seu crescimento e decrescimento, estudos dos sinais e construção de seus respectivos gráficos, da resolução pelo método do varal para inequações-produto e inequações-quociente, que auxiliam na construção de gráficos de funções de graus maiores que dois, das variáveis e substituição de variáveis, bem como calcular e operar com polinômios, especialmente a divisão euclidiana e o algoritmo de Briot-Ruffini, para então estudar as sequências e séries numéricas. O estudo das séries de potências, desenvolvido no capítulo 5, é de fundamental importância na expansão do polinômio de Taylor, com suas aproximações sucessivas para as funções seno, cosseno entre outras, definidas por séries de potências e para a perfeita compreensão dos resultados presentes no capítulo final. / The present work begins by presenting to the reader the need to take a firm hold of the concepts related to linear and quadratic functions, their growth and decrement, studies of their signal and construction of their respective graphs, resolution by the factor method for product inequalities and quotient inequalities, which help constructing graphs of functions of degrees greater than two, variables and variable substitution, as well as to calculate and operate with polynomials, especially the Euclidean division and the Briot-Ruffini algorithm, and then to study the sequences and series numbers. The study of power series, developed in Chapter 5, has fundamental importance for the expansion of the Taylor polynomial, with its successive approximations for the functions sine, cosine among others, defined by power series, and for the perfect understanding of the results shown in the final chapter.

Submitted by Geandra Rodrigues (geandrar@gmail.com) on 2018-09-25T13:33:58Z No. of bitstreams: 1 laurademesquitaquintanilha.pdf: 5886814 bytes, checksum: 7bcc03d59c68d5b4124de87183557234 (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2018-10-01T18:13:33Z (GMT) No. of bitstreams: 1 laurademesquitaquintanilha.pdf: 5886814 bytes, checksum: 7bcc03d59c68d5b4124de87183557234 (MD5) / Made available in DSpace on 2018-10-01T18:13:34Z (GMT). No. of bitstreams: 1 laurademesquitaquintanilha.pdf: 5886814 bytes, checksum: 7bcc03d59c68d5b4124de87183557234 (MD5) Previous issue date: 2018-07-20 / A análise de fluxo de potência visa calcular as tensões nas barras e as correntes nos ramos, para um dado cenário pré-estabelecido de geração e carga. É uma ferramenta essencial na operação e no controle dos sistemas elétricos de potência. Na análise tradicional, os parâmetros são tratados como quantidades determinísticas. Contudo, na prática, esses parâmetros podem apresentar incertezas associadas à medição ou à variação inerente ao longo do tempo. Em adição, o crescimento da participação de fontes intermitentes, como eólica e solar, em redes elétricas, aumenta o grau de incerteza e, portanto, estudos específicos de fluxo de potência devem ser desenvolvidos no sentido de tratar esta possível variabilidade de dados. Neste contexto, este trabalho investiga um método, publicado na literatura, que modela o fluxo de potência sujeito a incertezas associadas às cargas ativa e reativa das barras. A idéia básica deste método é proceder a expansão completa, em termos da série de Taylor, das equações de potência expressas em coordenadas retangulares das tensões nas barras. O método é implementado em MATLAB, considerando diferentes incertezas aplicadas aos sistemas IEEE 57 barras e brasileiro de 107 barras. Os resultados são, então, comparados com aqueles gerados pela matemática intervalar e pela simulação de Monte Carlo. De forma geral, a qualidade dos intervalos gerados pelo método em estudo é melhor que aquela apresentada pela matemática intervalar. / The power flow analysis aims to calculate bus voltage and current in branches, for a given pre-established scenario of generation and load. It is an essential tool in electrical power systems operation and control In traditional analysis, the parameters are treated as deterministic values. However, in practice, these parameters may present uncertainties associated with measurement as well as their inherent variation over the time. In addition, the growth of intermittent sources participation, such as wind and solar, into power grids has increased the uncertainties level, which demands the development of specific power flow studies in order to deal with data variability. In this context, this work investigates a method published in literature, that models the power flow subject to uncertainties associated with active and reactive bus loads. The basic idea of this method is to carry out the complete expansion of power equations, in terms of Taylor series, expressed in rectangular coordinates of bus voltages. The method is implemented in MATLAB, considering different uncertainties applied to IEEE 57 bus and Brazilian 107 bus. The results are then compared with those generated by interval mathematics and Monte Carlo simulation. In general, the quality of this method generated intervals is better than that presented by interval mathematics.

New algorithms for the matrix exponential and its Fréchet derivative are presented. First, we derive a new scaling and squaring algorithm (denoted expm[new]) for computing eA, where A is any square matrix, that mitigates the overscaling problem. The algorithm is built on the algorithm of Higham [SIAM J.Matrix Anal. Appl., 26(4): 1179-1193, 2005] but improves on it by two key features. The first, specific to triangular matrices, is to compute the diagonal elements in the squaring phase as exponentials instead of powering them. The second is to base the backward error analysis that underlies the algorithm on members of the sequence {||Ak||1/k} instead of ||A||. The terms ||Ak||1/k are estimated without computing powers of A by using a matrix 1-norm estimator. Second, a new algorithm is developed for computing the action of the matrix exponential on a matrix, etAB, where A is an n x n matrix and B is n x n₀ with n₀ << n. The algorithm works for any A, its computational cost is dominated by the formation of products of A with n x n₀ matrices, and the only input parameter is a backward error tolerance. The algorithm can return a single matrix etAB or a sequence etkAB on an equally spaced grid of points tk. It uses the scaling part of the scaling and squaring method together with a truncated Taylor series approximation to the exponential. It determines the amount of scaling and the Taylor degree using the strategy of expm[new].Preprocessing steps are used to reduce the cost of the algorithm. An important application of the algorithm is to exponential integrators for ordinary differential equations. It is shown that the sums of the form $\sum_{k=0}^p\varphi_k(A)u_k$ that arise in exponential integrators, where the $\varphi_k$ are related to the exponential function, can be expressed in terms of a single exponential of a matrix of dimension $n+p$ built by augmenting $A$ with additional rows and columns. Third, a general framework for simultaneously computing a matrix function, $f(A)$, and its Fréchet derivative in the direction $E$, $L_f(A,E)$, is established for a wide range of matrix functions. In particular, we extend the algorithm of Higham and $\mathrm{expm_{new}}$ to two algorithms that intertwine the evaluation of both $e^A$ and $L(A,E)$ at a cost about three times that for computing $e^A$ alone. These two extended algorithms are then adapted to algorithms that simultaneously calculate $e^A$ together with an estimate of its condition number. Finally, we show that $L_f(A,E)$, where $f$ is a real-valued matrix function and $A$ and $E$ are real matrices, can be approximated by $\Im f(A+ihE)/h$ for some suitably small $h$. This approximation generalizes the complex step approximation known in the scalar case, and is proved to be of second order in $h$ for analytic functions $f$ and also for the matrix sign function. It is shown that it does not suffer the inherent cancellation that limits the accuracy of finite difference approximations in floating point arithmetic. However, cancellation does nevertheless vitiate the approximation when the underlying method for evaluating $f$ employs complex arithmetic. The complex step approximation is attractive when specialized methods for evaluating the Fréchet derivative are not available.

This work describes numerical integration and solution for ordinary differential equations by the Taylor series by different types of integrators. The next part is a description of floating point and fixed point arithmetic. Subsequently, we are presenting designs and calculation methods for parallels multiplication and division integrators in floating point and fixed point arithmetic. The designs were realized in VHDL and implemented on FPGA. Finally we summarizes the proposed solution and compare time complexity with another numerical methods.