We investigate the behaviour of the intensity noise in a multi-longitudinal mode Nd:YAG laser. This type of laser is a nonlinear system which exhibits complicated dynamics within the intensity noise. For example, antiphase dynamics is where there is cancellation of one or more collective modes of oscillation, which are distinct from the longitudinal modes, in the total output. Commonly lasers are studied experimentally to discriminate between models used to describe them. They are convenient since many external influences can be controlled and the oscillations of interest are low frequency (in the kHz regime) making their direct measurement relatively simple. In our laser, the collective modes of oscillation are excited by broadband ambient noise. Because the phase of the excitation is unknown, we develop the cross spectral technique to measure the antiphase dynamics directly and form a picture of the intensity noise interference for two to five mode operation. For three mode operation, we measure the contributions of the longitudinal modes to the collective modes. We also calculate power spectral densities of the individual longitudinal modes and the total intensity. We test relationships between these quantities, at the collective mode frequencies, which are derived from modal rate equation theory. For two mode operation, the theoretical relations are satisfied. For three mode operation, the relations are satisfied when the picture of interferences is taken into account. The cross spectral technique is therefore shown to be a sensitive test of the model developed by Pieroux and Mandel [T. Hill et al., Phys. Rev. A 66, 063803 (2002)]. The behaviour of the multimode laser operating near the threshold of a longitudinal mode is measured. Transitions in the cross spectrum are noted in some pairs of longitudinal modes, for an arbitrary but small pump rate above threshold of a longitudinal mode. It has been shown that longitudinal modes with a high threshold pump power may become more intense than those with a lower threshold [K. Otsuka et al., Opt. Lett. 23, 201 (1998), L. Stamatescu and M.W. Hamilton, (unpublished) (1999), N.B. Abraham et al., Phys. Rev. A 62, 013810 (2000), P.A. Khandokhin, E.A. Ovchinnikov and E.Yu. Shirokov, Phys. Rev. A 61, 053807 (2000)]. The AC noise component of the first two longitudinal modes to reach threshold, is found to exhibit similar properties to their intensity. The implications of the results of this thesis, on models used to describe the behaviour of solid state lasers, are also discussed. / Thesis (Ph.D.)--Physics, 2003.

Many empirical studies show that both equity volatility and its trading volume have long range dependence and can be modeled as fractional integrated processes. The objective of this study is to investigate relationship between volatility and volume.We adopt four estimators of volatility, which includes the squared log returns, historical volatility, iterative t estimators and $GARCH$ estimators. The results show that among the four estimators squared log returns usually have the largest integration orders and produce hightest ratios of fractional cointegration. The fractional integrated orders are estimated separately and jointly, and the cointegration parameters are estimated by ordinary least squares, a narrow band frequency domain least squares method and a semiparametric estimator of Whittle likelihood. Models are also established when volatility and volume are not fractional cointegrated.

This thesis presents the implementation and validation of a new methodology developed by Glegg et al. (2004) for solving the trailing edge noise problem. This method is based on the premises that the noise produced by a surface can be computed by the integral of the cross product between the velocity and vorticity fields, of the boundary layer and shed vorticity (Howe (1978)). To extract the source terms, proper orthogonal decomposition is applied to the velocity cross spectrum to extract modes of the unsteady velocity and vorticity. The new formulation of the trailing edge noise problem by Glegg et al. (2004) is attractive because it applies to the high frequencies of interest but does not require an excessive computational effort. Also, the nature of the formulation permits the identification of the modes producing the noise and their associated velocity fluctuations as well as the regions of the boundary layer responsible for the noise production. The source terms were obtained using the direct numerical simulation of a turbulent channel flow by Moser et al. (1998). Two-point velocity and vorticity statistics of this data set were obtained by averaging 41 instantaneous fields. For comparisons purposes, experimental boundary layer data by Adrian et al. (2000) was chosen. Statistical reduction of 50 velocity fields obtained by particle image velocimetry was performed and analysis of the two-point correlation function showed features similar to the DNS data case. Also, proper orthogonal decomposition revealed identical dominant modes and eddy structures in the flow, therefore justifying considering the channel flow as an external boundary layer for noise calculations. Comparison of noise predictions with experimental data from Brooks et al. (1989) showed realistic results with the largest discrepancies, on the order of 5 dB, occurring at the lowest frequencies. The DNS results are least applicable at these frequencies, since these correspond to the longest streamwise lengthscales, which are the most affected by the periodicity conditions used in the DNS and also are the least representative of the turbulence in an external boundary layer flow. Most of the noise was shown to be produced by low-frequency streamwise velocity modes in the bottom 10% of the boundary layer and locations closest to the wall. Only 6 modes were required to obtain noise levels within 1 dB of the total noise. Finally, the method for predicting spatial velocity correlation from Reynolds stress data in wake flows, originally developed by Devenport et al. (1999, 2001) and Devenport and Glegg (2001), was adapted to boundary-layer type flows. This method, using Reynolds stresses and the prescription of a lengthscale to extrapolate the full two-point correlation, was shown to produce best results for a lengthscale prescribed as proportional to the turbulent macroscale. Noise predictions using modeled two-point statistics showed good agreement with the DNS inferred data in all but frequency magnitude, a probable consequence of the modeling of the correlation function in the streamwise direction. Other quantities associated to noise were seen to be similar to the ones obtained using the DNS. / Master of Science

Neste trabalho foram estudadas algumas formas de relação entre séries temporais multivariadas. Discutiu-se, inicialmente, a função de coerência, uma função análoga a função de correlação(que é dada no domínio do tempo) calculada no domínio da freqüência. Foram estudadas também as funções de coerência parcial e coerência parcial direcionada. A função de coerência parcial mede a relação entre duas componentes de uma série multivariada, isolados os efeitos de outra série. Em linhas gerais, a Coerência Parcial Direcionada pode ser interpredata como a decomposição da coerência parcial a partir de modelos autoregressivos multivariados. Esse conceito pode ser interpretado como uma representação do conceito de causalidade de Granger no domínio da freqüência. Finalmente, foram aplicadas as funções acima em dois conjuntos de dados: um modelo VAR(1) trivariado simulado e dados de medições de eletroencefalograma. / In this work we studied relationships between multivariate time series. We discussed the coherence function, a function similar to the correlation function(calculated in time domain) in frequency domain. Next, we discussed partial coherence and partial directed coherence. The partial coherence measures the relationship between two components of a multivariate time series, after removing the influence of another time series. Generally, the partial directed coherence can be interpreted as the decompositioin of the partial coherence from multivariate autoregressive models. We can interpret this function as a representation of the Granger causality concept in frequency domain. Finally, we applied these concepts in two situations: a simulated VAR(1) model and an electroencefalogram database.

Neste trabalho foram estudadas algumas formas de relação entre séries temporais multivariadas. Discutiu-se, inicialmente, a função de coerência, uma função análoga a função de correlação(que é dada no domínio do tempo) calculada no domínio da freqüência. Foram estudadas também as funções de coerência parcial e coerência parcial direcionada. A função de coerência parcial mede a relação entre duas componentes de uma série multivariada, isolados os efeitos de outra série. Em linhas gerais, a Coerência Parcial Direcionada pode ser interpredata como a decomposição da coerência parcial a partir de modelos autoregressivos multivariados. Esse conceito pode ser interpretado como uma representação do conceito de causalidade de Granger no domínio da freqüência. Finalmente, foram aplicadas as funções acima em dois conjuntos de dados: um modelo VAR(1) trivariado simulado e dados de medições de eletroencefalograma. / In this work we studied relationships between multivariate time series. We discussed the coherence function, a function similar to the correlation function(calculated in time domain) in frequency domain. Next, we discussed partial coherence and partial directed coherence. The partial coherence measures the relationship between two components of a multivariate time series, after removing the influence of another time series. Generally, the partial directed coherence can be interpreted as the decompositioin of the partial coherence from multivariate autoregressive models. We can interpret this function as a representation of the Granger causality concept in frequency domain. Finally, we applied these concepts in two situations: a simulated VAR(1) model and an electroencefalogram database.

The main goal of this study is to propose a methodology that measures directed relations between locally stationary processes. Unlike stationary processes, locally stationary processes may present sudden pattern changes and have local characteristics in specific intervals. This behavior causes instability in measures based on Fourier transforms. The relevance of this study relies on considering these processes and propose robust methodologies that are not affected by outliers, sudden pattern changes or local behavior. We start reviewing the Partial Directed Coherence (PDC) and the Wavelet Coherence. PDC measures the directed relation between components of a multivariate stationary Vector Autoregressive (VAR) model in the frequency domain, while Wavelet Coherence is based on complex wavelets decomposition. We then propose a causal wavelet decomposition of the covariance structure for bivariate locally stationary processes: the Directed Wavelet Covariance (DWC). Compared to Fourier-based quantities, wavelet-based estimators are more appropriate for non-stationary processes and processes with local patterns, outliers and rapid regime changes like in EEG experiments with the introduction of stimuli. We then propose its estimators and calculate its expectation and analyze its variance. Next we propose a decomposition for the variance of multivariate processes with more than two components: the Partial Directed Wavelet Covariance (pDWC). Considering a N-variate locally stationary process, the pDWC calculates the Directed Wavelet Covariance of X_1(t) with X_2(t) eliminating the effect of the other components X_3(t), ... ,X_N(t). We propose two approaches to this situation. First we filter the multivariate process to remove all the exogenous influences and then we calculate the directed relation between the components. In the second case, as in Partial Directed Coherence, we consider the multivariate process as a time-varying Vector Autoregressive Model (tv-VAR) and use its coefficients in the decomposition of the covariance function to isolate the effects of the other components. We also compare results of the PDC, Wavelet Coherence and Directed Wavelet Covariance with simulated data. Finally, we present an application of the proposed Directed Wavelet Covariance and Partial Directed Wavelet Covariance on EEG data. Simulation results show that the proposed measures capture the simulated relations. The pDWC with linear filter has shown more stable estimations than the proposed pDWC considering the tv-VAR. Future studies will discuss the DWC\'s and pDWC\'s asymptotic distributions and significance tests. The proposed Directed Wavelet Covariance decomposition is a different approach to deal with non-stationary processes in the context of causality. The use of wavelets is a gain and adds to the number of studies that can be addressed when Fourier transform does not apply. The pDWC is an alternative for multivariate processes and it removes linear influences from observed external components. / O objetivo deste trabalho é propor uma metodologia para mensurar o impacto direcionado entre processos localmente estacionários. Diferente de processos estacionários, processos localmente estacionários podem apresentar mudanças bruscas e características específicas em determinados intervalos. Tal comportamento pode causar instabilidade em medidas baseadas na transformada de Fourier. A importância deste estudo se dá em englobar processos com tais características, propondo metodologias robustas que não são afetadas pela existência de mudanças bruscas, pontos discrepantes e comportamentos locais. Inicialmente apresentamos conceitos já existentes na literatura, como a Coerência Parcial Direcionada (PDC) e a Coerência de Ondaletas. A PDC mede o impacto direcionado entre componentes de um modelo vetorial autoregressivo (VAR) no domínio da frequência. A coerência de ondaletas é baseada em transformadas complexas de ondaletas. Propomos então uma decomposição no domínio de ondaletas para a estrutura de covariância de processos bivariados localmente estacionários: a Covariância Direcionada de Ondaletas (DWC). Em comparação com as quantidades baseadas na tranformada Fourier, os estimadores baseados em ondaletas são mais apropriados para processos não estacionários com padrões locais, pontos discrepantes ou mudanças rápidas de regime, como em experimentos de eletroencefalograma (EEG) com a introdução de estímulo. Ainda, propomos um estimador para a DWC, calculamos a esperança deste estimador e avaliamos sua variância. Em seguida, propomos uma quantidade análoga à DWC para processos multivariados com mais de duas componentes: a Covariância Parcial Direcionada de Ondaletas (pDWC). Considerando um processo N-variado localmente estacionário, a pDWC calcula a Covariância Direcionada de Ondaletas entre X_1(t) e X_2(t) eliminando o efeito das outras componentes X_3(t), ... , X_N(t). Propomos duas abordagens para a pDWC: na primeira, a pDWC é calculada após a aplicação de um filtro linear que remove o efeito das variáveis exógenas. No segundo caso, a exemplo da Coerência Parcial Direcionada, consideramos o processo multivariado como um Modelo Autoregressivo de Vetorial variante no tempo (tv-VAR) e usamos seus coeficientes na decomposição da função de covariância para isolar os efeitos das demais componentes. Também comparamos os resultados da PDC, Coerência de Ondaletas e Covariância Direcionada de Ondaletas com dados simulados. Por fim, apresentamos uma aplicação da DWC e da pDWC em dados de EEG. Identificamos nas simulações que tanto as medidas já existentes na literatura quanto as quantidades propostas identificaram as relações simuladas. A pDWC proposta com filtros lineares apresentou estimações mais estáveis do que a pDWC considerando os modelos tv-VAR. Estudos futuros discutirão as propriedades assintóticas e testes de significância da DWC e pDWC.

The main goal of this study is to propose a methodology that measures directed relations between locally stationary processes. Unlike stationary processes, locally stationary processes may present sudden pattern changes and have local characteristics in specific intervals. This behavior causes instability in measures based on Fourier transforms. The relevance of this study relies on considering these processes and propose robust methodologies that are not affected by outliers, sudden pattern changes or local behavior. We start reviewing the Partial Directed Coherence (PDC) and the Wavelet Coherence. PDC measures the directed relation between components of a multivariate stationary Vector Autoregressive (VAR) model in the frequency domain, while Wavelet Coherence is based on complex wavelets decomposition. We then propose a causal wavelet decomposition of the covariance structure for bivariate locally stationary processes: the Directed Wavelet Covariance (DWC). Compared to Fourier-based quantities, wavelet-based estimators are more appropriate for non-stationary processes and processes with local patterns, outliers and rapid regime changes like in EEG experiments with the introduction of stimuli. We then propose its estimators and calculate its expectation and analyze its variance. Next we propose a decomposition for the variance of multivariate processes with more than two components: the Partial Directed Wavelet Covariance (pDWC). Considering a N-variate locally stationary process, the pDWC calculates the Directed Wavelet Covariance of X_1(t) with X_2(t) eliminating the effect of the other components X_3(t), ... ,X_N(t). We propose two approaches to this situation. First we filter the multivariate process to remove all the exogenous influences and then we calculate the directed relation between the components. In the second case, as in Partial Directed Coherence, we consider the multivariate process as a time-varying Vector Autoregressive Model (tv-VAR) and use its coefficients in the decomposition of the covariance function to isolate the effects of the other components. We also compare results of the PDC, Wavelet Coherence and Directed Wavelet Covariance with simulated data. Finally, we present an application of the proposed Directed Wavelet Covariance and Partial Directed Wavelet Covariance on EEG data. Simulation results show that the proposed measures capture the simulated relations. The pDWC with linear filter has shown more stable estimations than the proposed pDWC considering the tv-VAR. Future studies will discuss the DWC\'s and pDWC\'s asymptotic distributions and significance tests. The proposed Directed Wavelet Covariance decomposition is a different approach to deal with non-stationary processes in the context of causality. The use of wavelets is a gain and adds to the number of studies that can be addressed when Fourier transform does not apply. The pDWC is an alternative for multivariate processes and it removes linear influences from observed external components. / O objetivo deste trabalho é propor uma metodologia para mensurar o impacto direcionado entre processos localmente estacionários. Diferente de processos estacionários, processos localmente estacionários podem apresentar mudanças bruscas e características específicas em determinados intervalos. Tal comportamento pode causar instabilidade em medidas baseadas na transformada de Fourier. A importância deste estudo se dá em englobar processos com tais características, propondo metodologias robustas que não são afetadas pela existência de mudanças bruscas, pontos discrepantes e comportamentos locais. Inicialmente apresentamos conceitos já existentes na literatura, como a Coerência Parcial Direcionada (PDC) e a Coerência de Ondaletas. A PDC mede o impacto direcionado entre componentes de um modelo vetorial autoregressivo (VAR) no domínio da frequência. A coerência de ondaletas é baseada em transformadas complexas de ondaletas. Propomos então uma decomposição no domínio de ondaletas para a estrutura de covariância de processos bivariados localmente estacionários: a Covariância Direcionada de Ondaletas (DWC). Em comparação com as quantidades baseadas na tranformada Fourier, os estimadores baseados em ondaletas são mais apropriados para processos não estacionários com padrões locais, pontos discrepantes ou mudanças rápidas de regime, como em experimentos de eletroencefalograma (EEG) com a introdução de estímulo. Ainda, propomos um estimador para a DWC, calculamos a esperança deste estimador e avaliamos sua variância. Em seguida, propomos uma quantidade análoga à DWC para processos multivariados com mais de duas componentes: a Covariância Parcial Direcionada de Ondaletas (pDWC). Considerando um processo N-variado localmente estacionário, a pDWC calcula a Covariância Direcionada de Ondaletas entre X_1(t) e X_2(t) eliminando o efeito das outras componentes X_3(t), ... , X_N(t). Propomos duas abordagens para a pDWC: na primeira, a pDWC é calculada após a aplicação de um filtro linear que remove o efeito das variáveis exógenas. No segundo caso, a exemplo da Coerência Parcial Direcionada, consideramos o processo multivariado como um Modelo Autoregressivo de Vetorial variante no tempo (tv-VAR) e usamos seus coeficientes na decomposição da função de covariância para isolar os efeitos das demais componentes. Também comparamos os resultados da PDC, Coerência de Ondaletas e Covariância Direcionada de Ondaletas com dados simulados. Por fim, apresentamos uma aplicação da DWC e da pDWC em dados de EEG. Identificamos nas simulações que tanto as medidas já existentes na literatura quanto as quantidades propostas identificaram as relações simuladas. A pDWC proposta com filtros lineares apresentou estimações mais estáveis do que a pDWC considerando os modelos tv-VAR. Estudos futuros discutirão as propriedades assintóticas e testes de significância da DWC e pDWC.