QC 351 A7 no. 65 / The significance of signal-dependent noise is discussed. Particular emphasis is placed on the type of multiplicative noise present in the density variations in a photographic emulsion. A theoretical treatment of the effect of multiplicative noise on signal detection and signal discrimination problems is presented. Optimum test statistics are derived for processing a sampled message to detect the presence of a known signal. Multiplicative noise described by Poisson and Gaussian statistics is considered. The expressions are extended to include the two-signal discrimination problem. Two-dimensional signal fields in the presence of multiplicative noise are simulated in a computer and processed for optimum signal detection according to the two derived methods. These results are compared to the results of processing based on the assumption of stationary noise statistics. This comparison reveals that modest improvements (20% to 30% reduction in false alarm rate) are obtained when the signal-dependent nature of the noise statistics is considered. The effects of signal-to-noise ratio, signal structure, and changing background level are also investigated. An example of optimum signal discrimination using circles and squares as signals in multiplicative noise is reported. An improvement in the percentage of correctly identified signals is again observed when the proper test statistic is used. Two examples of signal filtering in the presence of signal-dependent noise are included. The first concerns the processing of a real star field to determine the location of weak stars. The second is an illustration of the signal information contained in the noise spectrum of a message recorded on a common photographic film.

Image noise reduction, or denoising, is an active area of research, although many of the techniques cited in the literature mainly target additive white noise. With an emphasis on signal-dependent noise, this thesis presents the General Adaptive Monte Carlo Bayesian Image Denoising (GAMBID) algorithm, a model-free approach based on random sampling. Testing is conducted on synthetic images with two different signal-dependent noise types as well as on real synthetic aperture radar and ultrasound images. Results show that GAMBID can achieve state-of-the-art performance, but suffers from some limitations in dealing with textures and fine low-contrast features. These aspects can by addressed in future iterations when GAMBID is expanded to become a versatile denoising framework.

Image noise reduction, or denoising, is an active area of research, although many of the techniques cited in the literature mainly target additive white noise. With an emphasis on signal-dependent noise, this thesis presents the General Adaptive Monte Carlo Bayesian Image Denoising (GAMBID) algorithm, a model-free approach based on random sampling. Testing is conducted on synthetic images with two different signal-dependent noise types as well as on real synthetic aperture radar and ultrasound images. Results show that GAMBID can achieve state-of-the-art performance, but suffers from some limitations in dealing with textures and fine low-contrast features. These aspects can by addressed in future iterations when GAMBID is expanded to become a versatile denoising framework.

We consider a general class of finite dimensional deterministic dynamical systems with finitely many local attractors each of which supports a unique ergodic probability measure, which includes in particular the class of Morse–Smale systems in any finite dimension. The dynamical system is perturbed by a multiplicative non-Gaussian heavytailed Lévy type noise of small intensity ε > 0. Specifically we consider perturbations leading to a Itô, Stratonovich and canonical (Marcus) stochastic differential equation. The respective asymptotic first exit time and location problem from each of the domains of attractions in case of inward pointing vector fields in the limit of ε-> 0 has been investigated by the authors. We extend these results to domains with characteristic boundaries and show that the perturbed system exhibits a metastable behavior in the sense that there exits a unique ε-dependent time scale on which the random system converges to a continuous time Markov chain switching between the invariant measures. As examples we consider α-stable perturbations of the Duffing equation and a chemical system exhibiting a birhythmic behavior.

Este estudo considera o problema de controle ótimo multi-período de média-variância para sistemas em tempo discreto com saltos markovianos e ruídos multiplicativos. Inicialmente considera-se um critério de desempenho formado por uma combinação linear da variância nal e valor esperado da saída do sistema. É apresentada uma solução analítica na obtenção da estratégia ótima para este problema. Em seguida são considerados os casos onde os critérios de desempenho são minimizar a variância nal sujeito a uma restrição no valor esperado ou maximizar o valor esperado nal sujeito a uma restrição na variância nal da saída do sistema. As estratégias ótimas de controle são obtidas de um conjunto de equações de diferenças acopladas de Riccati. Os resultados obtidos neste estudo generalizam resultados anteriores da literatura para o problema de controle ótimo com saldos markovianos e ruídos multiplicativos, apresentando condições explícitas e sucientes para a otimalidade da estratégia de controle. São apresentados modelos e simulações numéricas em otimização de carteiras de investimento e estratégias de gestão de ALM (asset liabilities management). / This thesis focuses on the stochastic optimal control problem of discrete-time linear systems subject to Markov jumps and multiplicative noise under three kinds of performance criterions related to the nal value of the expectation and variance of the output. In the first problem it is desired to minimize the nal variance of the output subject to a restriction on its nal expectation, in the second one it is desired to maximize the nal expectation of the output subject to a restriction on its nal variance, and in the third one it is considered a performance criterion composed by a linear combination of the nal variance and expectation of the output of the system. The optimal control strategies are obtained from a set of interconnected Riccati dierence equations and explicit sufficient conditions are presented for the existence of an optimal control strategy for these problems, generalizing previous results in the literature. Numerical simulations of investment portfolios and asset liabilities management models for pension funds with regime switching are presented.

In this study, theorem-based, generally applicable stochastic parameterisation schemes are developed and applied to a quasi-geostrophic model of extratropical atmospheric low-frequency variability (LFV). Hasselmann’s method is developed from limiting theorems for slow-fast systems of ordinary differential equations (ODEs) and applied to this high-dimensional model of intermediate complexity comprised of partial differential equations (PDEs) with complicated boundary conditions. Seamless, efficient algorithms for integrating the parameterised models are developed, which require only minimal changes to the full model algorithm. These algorithms may be readily adapted to a range of climate models of greater complexity in parameterising the effects of fast, sub-grid scale processes on the resolved scales. For comparison, the Majda-Timofeyev-Vanden-Eijnden (MTV) parameterisation method is applied to this model. The seamless algorithms are first adapted to probe the multiple regime behaviour that characterises the full model LFV. In contrast to the conclusions of a previous study, it is found that the multiple regime behaviour is not the result of a nonlinear interaction between the leading two planetary-scale modes, but rather is the result of interactions among these two modes and the leading synoptic-scale mode. The low-dimensional Hasselmann stochastic models perform well in simulating the statistics of the planetary-scale modes. In particular, a model with only one resolved (planetary-scale) mode captures the multiple regime behaviour of the full model. Although a fast-evolving synoptic-scale mode is of primary importance to the multiple regime behaviour, deterministic averaged forcing and not multiplicative noise is responsible for the regime behaviour in this model. The MTV models generate non-Gaussian statistics, but generally do not perform as well in capturing the climate statistics.

Este estudo considera o problema de controle ótimo multi-período de média-variância para sistemas em tempo discreto com saltos markovianos e ruídos multiplicativos. Inicialmente considera-se um critério de desempenho formado por uma combinação linear da variância nal e valor esperado da saída do sistema. É apresentada uma solução analítica na obtenção da estratégia ótima para este problema. Em seguida são considerados os casos onde os critérios de desempenho são minimizar a variância nal sujeito a uma restrição no valor esperado ou maximizar o valor esperado nal sujeito a uma restrição na variância nal da saída do sistema. As estratégias ótimas de controle são obtidas de um conjunto de equações de diferenças acopladas de Riccati. Os resultados obtidos neste estudo generalizam resultados anteriores da literatura para o problema de controle ótimo com saldos markovianos e ruídos multiplicativos, apresentando condições explícitas e sucientes para a otimalidade da estratégia de controle. São apresentados modelos e simulações numéricas em otimização de carteiras de investimento e estratégias de gestão de ALM (asset liabilities management). / This thesis focuses on the stochastic optimal control problem of discrete-time linear systems subject to Markov jumps and multiplicative noise under three kinds of performance criterions related to the nal value of the expectation and variance of the output. In the first problem it is desired to minimize the nal variance of the output subject to a restriction on its nal expectation, in the second one it is desired to maximize the nal expectation of the output subject to a restriction on its nal variance, and in the third one it is considered a performance criterion composed by a linear combination of the nal variance and expectation of the output of the system. The optimal control strategies are obtained from a set of interconnected Riccati dierence equations and explicit sufficient conditions are presented for the existence of an optimal control strategy for these problems, generalizing previous results in the literature. Numerical simulations of investment portfolios and asset liabilities management models for pension funds with regime switching are presented.

In this dissertation, we first prove that for a random differential equation with the multiplicative driving noise constructed from a Q-Wiener process and the Wiener shift, which is an approximation to a stochastic evolution equation, there exists a unique solution that generates a local dynamical system. There also exist a local center, unstable, stable, centerunstable, center-stable manifold, and a local stable foliation, an unstable foliation on the center-unstable manifold, and a stable foliation on the center-stable manifold, the smoothness of which depend on the vector fields of the equation. In the second half of the dissertation, we show that any two arbitrary local center manifolds constructed as above are conjugate. We also show the same conjugacy result holds for a stochastic evolution equation with the multiplicative Stratonovich noise term as u â—¦ dW

Este estudo considera o modelo de controle ótimo estocástico sob um critério de média-variância para sistemas lineares a tempo discreto sujeitos a saltos Markovianos e ruídos multiplicativos sob dois critérios. Inicialmente, consideramos como critério de desempenho a minimização multiperíodo de uma combinação entre a média e a variância da saída do sistema sem restrições. Em seguida, consideramos o critério de minimização multiperíodo da variância da saída do sistema ao longo do tempo com restrições sobre o valor esperado mínimo. Condições necessárias e suficientes explícitas para a existência de um controle ótimo são determinadas generalizando resultados anteriores existentes na literatura. O controle ótimo é escrito como uma realimentação de estado adicionado de um termo constante. Esta solução é obtida através de um conjunto de equações generalizadas a diferenças de Riccati interconectadas com um conjunto de equações lineares recursivas. Como aplicação, apresentamos alguns exemplos numéricos práticos para um problema de seleção de portfólio multiperíodo com mudança de regime, incluindo uma estratégia de ALM (Asset and Liability Management). Neste problema, deseja-se obter a melhor alocação de portfólio de forma a otimizar seu desempenho entre risco e retorno em cada passo de tempo até o nal do horizonte de investimento e sob um dos dois critérios citados acima. / In this work we consider the stochastic optimal control problem of discrete-time linear systems subject to Markov jumps and multiplicative noise under two criterions. First, we consider an unconstrained multiperiod mean-variance trade-off performance criterion. In the sequence, we consider a multiperiod minimum variance criterion subject to constraints on the minimum expected output along the time. We present explicit necessary and sufficient conditions for the existence of an optimal control strategy for the problems, generalizing previous results in the literature. The optimal control law is written as a state feedback added with a deterministic sequence. This solution is derived from a set of coupled generalized Riccati difference equations interconnected with a set of coupled linear recursive equations. As an application, we present some practical numerical examples on a multiperiod portfolio selection problem with regime switching, including an Asset and Liability Management strategy. In this problem it is desired to nd the best portfolio allocation in order to optimize its risk-return performance in every time step along the investment horizon, under one of the two criterions stated above.In this work we consider the stochastic optimal control problem of discrete-time linear systems subject to Markov jumps and multiplicative noise under two criterions. First, we consider an unconstrained multiperiod mean-variance trade-off performance criterion. In the sequence, we consider a multiperiod minimum variance criterion subject to constraints on the minimum expected output along the time. We present explicit necessary and sufficient conditions for the existence of an optimal control strategy for the problems, generalizing previous results in the literature. The optimal control law is written as a state feedback added with a deterministic sequence. This solution is derived from a set of coupled generalized Riccati difference equations interconnected with a set of coupled linear recursive equations. As an application, we present some practical numerical examples on a multiperiod portfolio selection problem with regime switching, including an Asset and Liability Management strategy. In this problem it is desired to nd the best portfolio allocation in order to optimize its risk-return performance in every time step along the investment horizon, under one of the two criterions stated above.

Made available in DSpace on 2016-06-02T19:06:17Z (GMT). No. of bitstreams: 1 6277.pdf: 15816665 bytes, checksum: 105661656ee67fe816f34a96605797f9 (MD5) Previous issue date: 2014-01-23 / The development of new methods and noisy images filtering techniques still attract researchers, which seek to reduce the noise with the minimal loss of details, edges, resolution and removal of fine structures of the image. Moreover, it is extremely important to expand the capacity of the filters for the different noise models present in the Image and Signal Processing literature, like the multiplicative noise speckle, present in the synthetic aperture radar (SAR) images. This Master s degree thesis aims to use a recent denoising algorithm: the nonlocal means (NLM), developed for the additive white gaussian noise (AWGN), and expand, analyze and compare its capacity for intensity SAR images denoising (despeckling), which are contaminated with the speckle. This expansion of the NLM filter is based with the use of the stochastic distances and the comparison of the estimated parameters with de G0 and the inverse Gamma distributions. Finally, this work compares the synthetic and real results of the proposed filter with some filters of the literature. / A elaboração de novos métodos e técnicas de filtragem de imagens ruidosas ainda atraem pesquisadores, que buscam a redução de ruído com a mínima perda dos detalhes, bordas, resolução e remoção de estruturas finas da imagem. Além disto, é de extrema importância ampliar a capacidade dos filtros para diversos modelos de ruído existentes na literatura de Processamento de Imagens e Sinais, como o ruído multiplicativo speckle , presente em imagens de radar de abertura sintética (SAR). Esta dissertação de Mestrado tem o objetivo de utilizar um algoritmo de filtragem recente: o nonlocal means (NLM), desenvolvido para o ruído branco aditivo gaussiano (AWGN), e ampliar, analisar e comparar a sua capacidade para a filtragem de imagens SAR de intensidade ( despeckling ), as quais são contaminadas com o speckle . Esta ampliação do filtro NLM é baseada no uso das distâncias estocásticas e na comparação dos parâmetros estimados através das distribuições G0 e da inversa da Gama. Por fim, este trabalho compara os resultados sintéticos e reais do filtro proposto com alguns filtros da literatura.