Made available in DSpace on 2015-05-14T12:14:15Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 705049 bytes, checksum: 4794f7004746261efe9996d47989dd1f (MD5) Previous issue date: 2014-10-21 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this thesis, we presented models of two state stochastic systems which interact through a global coupling, in a way that each population unit contributes to the state transition rates of the other units. We presented two models of global coupling in which is possible to observe a phase transition of a regime with units equally distributed on the two states to a phase where there is an agglomeration of units in one of the states. In the first coupling model this transition occurs in a continuous way as we increase the coupling parameter. Through a mean field approximation we shown that this phase transition occurs due to a subcritical pitchfork bifurcation where one of the phases is associated to a monostable regime (units equally distributed in the two states) and the other phase to a symmetric bistable regime (majority of the units agglomerated in one of the states). On the other hand the other model presents a discontinuous phase transition as we increase the coupling parameter, the mean field approach shows that this phase transition occurs due a supercritical pitchfork bifurcation where we have a monostable regime and a tristable regime presenting symmetry in relation to the central potential well, as the coupling parameter is increased the central stability reduces while the two other states becomes more stable. It was shown that for both coupling models, when we have a finite number of oscillators the system presents a multiplicative noise structure. This noise structure turns the stable states obtained with the mean field approximation on metastable states, also the fluctuations due to a finite number of units breaks the symmetry in the multistable regimes, this symmetry break occurs due to the asymmetric intensity of the fluctuations. We also obtained a Fokker-Planck equation for this system and the probability distribution of the number of units in each state, from this distribution it was possible to build a phase diagram for the phase transition from themonostable regime to the regime that presents multistability. This transition is characterized in terms of the coupling parameter and the number of units in the system. / Nesta tese, apresentamos modelos de sistemas estocásticos de dois níveis que interagem através de um acoplamento global, de forma que o estado ocupado por cada unidade da população influi na taxa de transição de estado das demais. Apresentamos dois modelos de acoplamento global onde é possível observar uma transição de fase de um regime onde as unidades estão distribuídas igualmente entre os dois estados para uma fase onde há a aglomeração de unidades em um dos estados. Em um dos modelos de acoplamento essa transição ocorre de forma continua com o parâmetro de acoplamento. Através de uma aproximação de campo médio mostramos que essa transição de fase ocorre devido a uma bifurcação de forquilha sub-crítica onde uma das fases ´e associada a um regime monopolista (unidades igualmente divididas entre os dois estados) e a outra fase a um regime bioestavel simétrico (maior parte das unidades aglomeradas em um dos estados). J´a o outro modelo apresenta uma transic¸ ao de fase descont´ınua com o par ametro de acoplamento. A abordagem de campo m´edio revela que essa transic¸ ao de fase ocorre atrav´es de uma bifurcac¸ ao de forquilha supercr´ıtica onde temos um regimemonoest´avel e um regime triest´avel apresentando simetria com relac¸ ao ao poc¸o de potencial central e a medida que o par ametro de acoplamento ´e aumentado a estabilidade central diminui enquanto os outros dois estados se tornam mais est´aveis. Foi mostrado que para ambos os modelos de acoplamento, quando temos uma quantidade finita de osciladores o sistema apresenta uma estrutura de ru´ıdo multiplicativo. Essa estrutura de ru´ıdo torna os estados est´aveis obtidos com a aproximac¸ ao de campo m´edio em estados metaest´aveis. Tamb´em foi mostrado que as flutuac¸ oes devido a quantidade finita de unidades quebra a simetria nos regimes com multiestabilidade, essa quebra de simetria ocorre devido a assimetrias da intensidade das flutuac¸ oes. Obtemos tamb´em uma equac¸ ao de Fokker-Planck para esse sistema. A soluc¸ ao da equac¸ ao de Fokker-Planck nos d´a a distribuic¸ ao de probabilidade da quantidade de unidades em cada estado. Essa distribuic¸ ao torna poss´ıvel a construc¸ ao de um diagrama de fases para a transic¸ ao de fase dos regimes monoest ´aveis para os regimes que apresentam multiestabilidade. Essa transic¸ ao ´e caracterizada em termos do par ametro de acoplamento e da quantidade de unidades do sistema.

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.

[pt] Consideramos o oscilador harmônico com amortecimento aleatório em presença de ruído externo. Os ruídos, representando perturbações externas e internas, são modelados pelo processo de Ornstein-Uhlenbeck ou ruído branco e pelo processo dicotômico ou ruído branco, respectivamente. Usando técnicas de sistemas dinâmicos, analisamos o valor médio e a dispersão da posição e da velocidade do oscilador harmônico estocástico, apresentando resultados analíticos e numéricos. Em particular, obtemos expressões para a expansão de baixa-ordem em relação ao tempo de correlação da perturbação interna, no caso da atuação do ruído dicotômico. Finalmente, usando o modelo de oscilador harmônico com amortecimento aleatório como referência, investigamos a série intradiária de preços do mercado brasileiro. / [en] We consider the random damping harmonic oscillator in presence of external noise. The noises, representing external and internal perturbations, are modeled as an Ornstein-Uhlenbeck process or a white noise and as a dichotomous process or a white noise, respectively. Using dynamical systems tools, we analyze the expected value as well as the dispersion of the stochastic harmonic oscillator s position and velocity, presenting analytical and numerical results. In particular, we also provide expressions for the low-order expansion in the correlation time of the internal perturbation, in the case the dichotomous noise is at play. Using random damped harmonic oscillator model as a reference, we conclude by investigating the intra-day Brazilian stock price series.

This thesis deals with the problem of estimating the velocity of a mobile station (MS)in a mobile communication system using the instantaneous frequency (IF) of the received signal at the MS antenna. This estimate is essential for satisfactory handover performance, effective dynamic channel assignment, and optimisation of adaptive multiple access wireless receivers. Conventional methods for estimating the MS velocity are based either on the statistics of the envelope or quadrature components of the received signal. In chapter 4 of the thesis, we show that their performance deteriorates in the presence of shadowing. Other velocity estimators have also been proposed which require prior estimation of the channel or the average received power. These are generally difficult to obtain due to the non-stationary nature of the received signal. An appropriate window which depends on the unknown MS velocity must first be applied in order to accurately estimate the required quantities. Using the statistics of the IF of the received signal at the MS antenna given in chapter 3, new velocity estimators are proposed in chapter 4 of this thesis. The proposed estimators are based on the moments, zero-crossing rate, and covariance of the received IF. Since the IF of the received signal is not affected by any amplitude distortion, the proposed IF-based estimators are robust to shadowing and propagation path-loss. The estimators for the MS velocity in a macro- and micro-cellular system are presented separately. A macro-cell system can be considered as a special case of a micro-cell in which there is no line-of-sight component at the receiver antenna. It follows that those estimators which are derived for micro-cells can be used in a macro-cell as well. In chapter 4, we analyse the performance of the proposed velocity estimators in the presence of additive noise, non-isotropic scattering, and shadowing. We also prove analytically that the proposed velocity estimators outperform the existing methods in the presence of shadowing and additive noise. The proposed IF-based estimators need prior estimation of both the IF of the received signal and Ricean K-factor. The IF estimation in a typical wireless environment, can be considered as a special case of a general problem of IF estimation in the presence of multiplicative and additive noise. In chapter 5, we show that current time-frequency approaches to this problem which are based on the peak of a time-frequency distribution (TFD) of the signal, fail because of the special shape of the power spectral density of the multiplicative noise in a wireless environment. To overcome this drawback, the use of the first-order moment of a TFD is studied in chapter 5. Theoretical analysis and simulations show that the IF estimator based on the first-order moment of a TFD exhibits negligible bias when the signal-to-additive noise ratio is more than 10 dB. The Ricean K-factor is not only necessary for velocity estimation in micro-cells, but also is a measure of the severity of fading and a good indicator of the channel quality. Two new methods for estimating the Ricean K-factor based on the first two moments of the envelope of the received signal, are proposed in chapter 6. Performance analysis presented in chapter 6, prove that the proposed K estimators are robust to non-isotropic scattering. Theoretical analysis and simulations which are presented in chapters 4 and 7 of this thesis, prove that the proposed velocity and K estimators outperform existing estimators in the presence of shadowing and additive noise.

Ziel dieser Arbeit ist die Untersuchung des Einflusses der stochastischen Interpretation der Langevin Gleichung mit zustandsabhängigen Diffusionskoeffizienten auf den Propagator des zugehörigen stochastischen Prozesses bzw. dessen Mittelwerte. Dies dient dem besseren Verständnis und der Interpretation von Messdaten von Diffusion in inhomogenen Systemen und geht einher mit der Frage der Form der Diffusionsgleichung in solchen Systemen. Zur Vereinfachung der Fragestellung werden in dieser Arbeit nur Systeme untersucht die vollständig durch einen ortsabhängigen Diffusionskoeffizienten und Angabe der stochastischen Interpretation beschrieben werden können. Dazu wird zunächst für mehrere experimentell relevante eindimensionale Systeme der jeweilige allgemeine Propagator bestimmt, der für jede denkbare stochastische Interpretation gültig ist. Der analytisch bestimmte Propagator wird dann für zwei exemplarisch ausgewählte stochastische Interpretationen, hier für die Itô und Klimontovich-Hänggi Interpretation, gegenübergestellt und die Unterschiede identifiziert. Für Mittelwert und Varianz der Prozesse werden die drei wesentlichen stochastischen Interpretationen verglichen, also Itô, Stratonovich und Klimontovich-Hänggi Interpretation. Diese systematische Untersuchung von inhomogenen Diffusionsprozessen kann zukünftig helfen diese Art von, in genau einer stochastischen Interpretation, driftfreien Systemen einfacher zu identifizieren. Ein weiterer wesentlicher Teil der Arbeit erweitert die Frage auf mehrdimensionale inhomogene anisotrope Systeme. Dies wird z.B. bei der Untersuchung von Diffusion in Flüssigkristallen mit inhomogenem Direktorfeld relevant. Obwohl hier, im Gegensatz zu eindimensionalen Systemen, der Propagator nicht allgemein berechnet werden kann, wird dennoch der Einfluss der Inhomogenität auf Messgrößen, wie die mittlere quadratische Verschiebung oder die Verteilung der Diffusivitäten, bestimmt. Anhand eines Beispiels wird auch der Einfluss der stochastischen Interpretation auf diese Messgrößen demonstriert. / The aim of this thesis is to investigate the influence of the stochastic interpretation of the Langevin equation with state-dependent diffusion coefficient on the propagator of the related stochastic process, or its averages, respectively. This helps to obtain a deeper understanding and to interpret measurement data of diffusion in inhomogeneous systems and is accompanied with the question of the proper form of the diffusion equation in such systems. To simplify the question, in this thesis only systems are considered which can be fully described by a spatially dependent diffusion coefficient and a given stochastic interpretation. Therefore, for several experimentally relevant one-dimensional systems, the respective general propagator is determined, which is valid for any possible stochastic interpretation. Then, the propagator for two exemplary stochastic interpretations, here the Itô and Klimontovich-Hänggi interpretation, are compared and the differences are identified. For mean and variance of the processes three major interpretations are compared, namely the Itô, the Stratonovich and the Klimontovich-Hänggi interpretation. This systematic research on inhomogeneous diffusion process may help in future to identify these kind of, in exactly one stochastic interpretation, drift-free systems more easily. Another important part of this thesis extends this question to multidimensional inhomogeneous anisotropic systems. This is of high relevance, for instance, for the research of diffusion in liquid crystalline systems with an inhomogeneous director field. Although, in contrast to one-dimensional systems, the propagator may not be calculated generally, the influence of the inhomogeneity on measurement data like the mean squared displacement or the distribution of diffusivities is determined. Based on one example, also the influence of the stochastic interpretation on these quantities is demonstrated.

Signal processing has been playing a key role in providing solutions to key problems encountered in communications, in general, and in wireless communications, in particular. Time-Frequency Signal Processing (TFSP) provides eective tools for analyzing nonstationary signals where the frequency content of signals varies in time as well as for analyzing linear time-varying systems. This research aimed at exploiting the advantages of TFSP, in dealing with nonstationary signals, into the fundamental issues of signal processing, namely the signal estimation and signal separation. In particular, it has investigated the problems of (i) the Instantaneous Frequency (IF) estimation of Linear Frequency-Modulated (LFM) signals corrupted in complex-valued zero-mean Multiplicative Noise (MN), and (ii) the Underdetermined Blind Source Separation (UBSS) of LFM signals, while focusing onto the fast-growing area of Wireless Communications (WCom). A common problem in the issue of signal estimation is the estimation of the frequency of Frequency-Modulated signals which are seen in many engineering and real-life applications. Accurate frequency estimation leads to accurate recovery of the true information. In some applications, the random amplitude modulation shows up when the medium is dispersive and/or when the assumption of point target is not valid; the original signal is considered to be corrupted by an MN process thus seriously aecting the recovery of the information-bearing frequency. The IF estimation of nonstationary signals corrupted by complex-valued zero-mean MN was investigated in this research. We have proposed a Second-Order Statistics approach, rather than a Higher-Order Statistics approach, for IF estimation using Time-Frequency Distributions (TFDs). The main assumption was that the autocorrelation function of the MN is real-valued but not necessarily positive (i.e. the spectrum of the MN is symmetric but does not necessary has the highest peak at zero frequency). The estimation performance was analyzed in terms of bias and variance, and compared between four dierent TFDs: Wigner-Ville Distribution, Spectrogram, Choi-Williams Distribution and Modified B Distribution. To further improve the estimation, we proposed to use the Multiple Signal Classification algorithm and showed its better performance. It was shown that the Modified B Distribution performance was the best for Signal-to-Noise Ratio less than 10dB. In the issue of signal separation, a new research direction called Blind Source Separation (BSS) has emerged over the last decade. BSS is a fundamental technique in array signal processing aiming at recovering unobserved signals or sources from observed mixtures exploiting only the assumption of mutual independence between the signals. The term "blind" indicates that neither the structure of the mixtures nor the source signals are known to the receivers. Applications of BSS are seen in, for example, radar and sonar, communications, speech processing, biomedical signal processing. In the case of nonstationary signals, a TF structure forcing approach was introduced by Belouchrani and Amin by defining the Spatial Time- Frequency Distribution (STFD), which combines both TF diversity and spatial diversity. The benefit of STFD in an environment of nonstationary signals is the direct exploitation of the information brought by the nonstationarity of the signals. A drawback of most BSS algorithms is that they fail to separate sources in situations where there are more sources than sensors, referred to as UBSS. The UBSS of nonstationary signals was investigated in this research. We have presented a new approach for blind separation of nonstationary sources using their TFDs. The separation algorithm is based on a vector clustering procedure that estimates the source TFDs by grouping together the TF points corresponding to "closely spaced" spatial directions. Simulations illustrate the performances of the proposed method for the underdetermined blind separation of FM signals. The method developed in this research represents a new research direction for solving the UBSS problem. The successful results obtained in the research development of the above two problems has led to a conclusion that TFSP is useful for WCom. Future research directions were also proposed.

Ziel dieser Arbeit ist die Untersuchung des Einflusses der stochastischen Interpretation der Langevin Gleichung mit zustandsabhängigen Diffusionskoeffizienten auf den Propagator des zugehörigen stochastischen Prozesses bzw. dessen Mittelwerte. Dies dient dem besseren Verständnis und der Interpretation von Messdaten von Diffusion in inhomogenen Systemen und geht einher mit der Frage der Form der Diffusionsgleichung in solchen Systemen. Zur Vereinfachung der Fragestellung werden in dieser Arbeit nur Systeme untersucht die vollständig durch einen ortsabhängigen Diffusionskoeffizienten und Angabe der stochastischen Interpretation beschrieben werden können. Dazu wird zunächst für mehrere experimentell relevante eindimensionale Systeme der jeweilige allgemeine Propagator bestimmt, der für jede denkbare stochastische Interpretation gültig ist. Der analytisch bestimmte Propagator wird dann für zwei exemplarisch ausgewählte stochastische Interpretationen, hier für die Itô und Klimontovich-Hänggi Interpretation, gegenübergestellt und die Unterschiede identifiziert. Für Mittelwert und Varianz der Prozesse werden die drei wesentlichen stochastischen Interpretationen verglichen, also Itô, Stratonovich und Klimontovich-Hänggi Interpretation. Diese systematische Untersuchung von inhomogenen Diffusionsprozessen kann zukünftig helfen diese Art von, in genau einer stochastischen Interpretation, driftfreien Systemen einfacher zu identifizieren. Ein weiterer wesentlicher Teil der Arbeit erweitert die Frage auf mehrdimensionale inhomogene anisotrope Systeme. Dies wird z.B. bei der Untersuchung von Diffusion in Flüssigkristallen mit inhomogenem Direktorfeld relevant. Obwohl hier, im Gegensatz zu eindimensionalen Systemen, der Propagator nicht allgemein berechnet werden kann, wird dennoch der Einfluss der Inhomogenität auf Messgrößen, wie die mittlere quadratische Verschiebung oder die Verteilung der Diffusivitäten, bestimmt. Anhand eines Beispiels wird auch der Einfluss der stochastischen Interpretation auf diese Messgrößen demonstriert. / The aim of this thesis is to investigate the influence of the stochastic interpretation of the Langevin equation with state-dependent diffusion coefficient on the propagator of the related stochastic process, or its averages, respectively. This helps to obtain a deeper understanding and to interpret measurement data of diffusion in inhomogeneous systems and is accompanied with the question of the proper form of the diffusion equation in such systems. To simplify the question, in this thesis only systems are considered which can be fully described by a spatially dependent diffusion coefficient and a given stochastic interpretation. Therefore, for several experimentally relevant one-dimensional systems, the respective general propagator is determined, which is valid for any possible stochastic interpretation. Then, the propagator for two exemplary stochastic interpretations, here the Itô and Klimontovich-Hänggi interpretation, are compared and the differences are identified. For mean and variance of the processes three major interpretations are compared, namely the Itô, the Stratonovich and the Klimontovich-Hänggi interpretation. This systematic research on inhomogeneous diffusion process may help in future to identify these kind of, in exactly one stochastic interpretation, drift-free systems more easily. Another important part of this thesis extends this question to multidimensional inhomogeneous anisotropic systems. This is of high relevance, for instance, for the research of diffusion in liquid crystalline systems with an inhomogeneous director field. Although, in contrast to one-dimensional systems, the propagator may not be calculated generally, the influence of the inhomogeneity on measurement data like the mean squared displacement or the distribution of diffusivities is determined. Based on one example, also the influence of the stochastic interpretation on these quantities is demonstrated.

Le problème de restauration d'images dégradées par des flous variables connaît un attrait croissant et touche plusieurs domaines tels que l'astronomie, la vision par ordinateur et la microscopie à feuille de lumière où les images sont de taille un milliard de pixels. Les flous variables peuvent être modélisés par des opérateurs intégraux qui associent à une image nette u, une image floue Hu. Une fois discrétisé pour être appliqué sur des images de N pixels, l'opérateur H peut être vu comme une matrice de taille N x N. Pour les applications visées, la matrice est stockée en mémoire avec un exaoctet. On voit apparaître ici les difficultés liées à ce problème de restauration des images qui sont i) le stockage de ce grand volume de données, ii) les coûts de calculs prohibitifs des produits matrice-vecteur. Ce problème souffre du fléau de la dimension. D'autre part, dans beaucoup d'applications, l'opérateur de flou n'est pas ou que partialement connu. Il y a donc deux problèmes complémentaires mais étroitement liés qui sont l'approximation et l'estimation des opérateurs de flou. Cette thèse a consisté à développer des nouveaux modèles et méthodes numériques permettant de traiter ces problèmes. / The restoration of images degraded by spatially varying blurs is a problem of increasing importance. It is encountered in many applications such as astronomy, computer vision and fluorescence microscopy where images can be of size one billion pixels. Variable blurs can be modelled by linear integral operators H that map a sharp image u to its blurred version Hu. After discretization of the image on a grid of N pixels, H can be viewed as a matrix of size N x N. For targeted applications, matrices is stored with using exabytes on the memory. This simple observation illustrates the difficulties associated to this problem: i) the storage of a huge amount of data, ii) the prohibitive computation costs of matrix-vector products. This problems suffers from the challenging curse of dimensionality. In addition, in many applications, the operator is usually unknown or only partially known. There are therefore two different problems, the approximation and the estimation of blurring operators. They are intricate and have to be addressed with a global overview. Most of the work of this thesis is dedicated to the development of new models and computational methods to address those issues.