The problem of estimation has been widely investigated with all different kinds of assumptions. This study focusses on the subjective Bayesian estimation of a location vector and characteristic matrix for the univariate and multivariate elliptical model as oppose to objective Bayesian estimation that has been thoroughly discussed (see Fang and Li (1999) amongst others). The prior distributions that will be assumed is the conjugate normal-inverse Wishart prior and also the normal-Wishart prior which has not yet been considered in literature. The posterior distributions, joint and marginal, as well as the Bayes estimators will be derived. The newly developed results are applied to the multivariate normal and multivariate t-distribution. For subjective Bayesian analysis the vector-spherical matrix elliptical model is also studied. / Dissertation (MSc)--University of Pretoria, 2012. / Statistics / MSc / Unrestricted

Light scattering is a powerful characterization tool for determining shape, size, and size distribution of fine particles, as well as complex, irregular structures of their aggregates. Small angle static light scattering and elliptically polarized light scattering techniques produce accurate results and provide real time, non-intrusive, and in-situ observations on prevailing process conditions in three-dimensional systems. As such, they complement conventional characterization tools such as SEM and TEM which have their known disadvantages and limitations. In this study, we provide a thorough light scattering analysis of colloidal tungsten trioxide (WO3) nanoparticles in the shape of irregular nanospheres and cylindrical nanowires, and of the resulting aggregate morphologies. Aggregation characteristics as a function of primary particle geometry, aspect ratio of nanowires, and the change in dispersion stability in various polar solvents without the use of dispersants are monitored over different time scales and are described using the concepts of fractal theory. Using forward scattered intensities, sedimentation rates as a result of electrolyte addition and particle concentration at low solution pH are quantified, in contrast to widely reported visual observations, and are related to the aggregate structure in the dispersed phase. For nanowires of high aspect ratios, when aggregate structures cannot directly be inferred from measurements, an analytical and a quasiexperimental method are used.

Submitted by Anderson dos Santos Gonzaga null (andersongonzaga25@yahoo.com.br) on 2018-01-16T17:28:55Z No. of bitstreams: 1 Gonzaga.dissertação.pdf: 1264952 bytes, checksum: e682e5fd46c5a7d68506f3f9499cded5 (MD5) / Approved for entry into archive by Claudia Adriana Spindola null (claudia@fct.unesp.br) on 2018-01-16T17:58:17Z (GMT) No. of bitstreams: 1 gonzaga_as_me_prud.pdf: 1264952 bytes, checksum: e682e5fd46c5a7d68506f3f9499cded5 (MD5) / Made available in DSpace on 2018-01-16T17:58:17Z (GMT). No. of bitstreams: 1 gonzaga_as_me_prud.pdf: 1264952 bytes, checksum: e682e5fd46c5a7d68506f3f9499cded5 (MD5) Previous issue date: 2017-02-21 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Nesse trabalho teórico na área das equações diferenciais parciais elípticas, estudamos uma versão estacionária da equação de Schrödinger não-linear, com não-linearidade do tipo assintoticamente linear. O objetivo principal versa sobre obter resultados de existência de uma solução nodal radialmente simétrica. Ainda, sob algumas condições, buscamos também obter informações sobre o seu índice de Morse. / In this theoretical work in elliptic partial di erential equations, we study a stationary version for the nonlinear Schödinger equation with nonlinearity of the assymptotically linear type. The main objective is getting, some results of existence for a radially symmetric nodal solution. Moreover, under some conditions, we look also obtaining information about its Morse index.

The primary objective of this dissertation is to extend the classical Principal Components Analysis (PCA), aiming to reduce the dimensionality of a large number of Normal interrelated variables, in two directions. The first is to go beyond the static (contemporaneous or synchronous) covariance matrix among these interrelated variables to include certain forms of temporal (over time) dependence. The second direction takes the form of extending the PCA model beyond the Normal multivariate distribution to the Elliptically Symmetric family of distributions, which includes the Normal, the Student's t, the Laplace and the Pearson type II distributions as special cases. The result of these extensions is called the Generalized principal component analysis (GPCA). The GPCA is illustrated using both Monte Carlo simulations as well as an empirical study, in an attempt to demonstrate the enhanced reliability of these more general factor models in the context of out-of-sample forecasting. The empirical study examines the predictive capacity of the GPCA method in the context of Exchange Rate Forecasting, showing how the GPCA method dominates forecasts based on existing standard methods, including the random walk models, with or without including macroeconomic fundamentals. / Ph. D. / Factor models are employed to capture the hidden factors behind the movement among a set of variables. It uses the variation and co-variation between these variables to construct a fewer latent variables that can explain the variation in the data in hand. The principal component analysis (PCA) is the most popular among these factor models. I have developed new Factor models that are employed to reduce the dimensionality of a large set of data by extracting a small number of independent/latent factors which represent a large proportion of the variability in the particular data set. These factor models, called the generalized principal component analysis (GPCA), are extensions of the classical principal component analysis (PCA), which can account for both contemporaneous and temporal dependence based on non-Gaussian multivariate distributions. Using Monte Carlo simulations along with an empirical study, I demonstrate the enhanced reliability of my methodology in the context of out-of-sample forecasting. In the empirical study, I examine the predictability power of the GPCA method in the context of “Exchange Rate Forecasting”. I find that the GPCA method dominates forecasts based on existing standard methods as well as random walk models, with or without including macroeconomic fundamentals.

Matrix variate skew elliptically contoured distributions generalize several classes of important distributions. This paper defines and explores matrix variate skew elliptically contoured distributions. In particular, we discuss two stochastic representations of the matrix variate skew elliptically contoured distributions.

L’imagerie de filtrage en polarisation est une technique populaire largement utilisée en optique pour le biomédical pour le sondage des tissus superficiels, pour le sondage de volumes plus profonds, mais aussi pour l’examen sélectif de volumes sub-surfaciques. Du fait de l’effet de ’mémoire de polarisation’ de la lumière polarisée, l’imagerie de filtrage en polarisation elliptique est sensible á des épaisseurs de tissus différentes, depuis la surface, accessible avec la polarisation linéaire, jusqu’á une épaisseur critique accessible par la polarisation circulaire. Nous nous concentrons sur des méthodes utilisant des combinaisons de polarisations elliptiques afin de sélectionner la portion de lumière ayant maintenu son état de polarisation et éliminer le fond pour un meilleur contraste avec de plus une information sur la profondeur. Avec ce type de filtrage, il est possible d’accéder á des profondeurs de tissus biologiques bien définies selon l’ellipticité de polarisation. De plus, ces travaux ont permis d’étendre la méthode á la spectroscopie pour quantifier la concentration en chromophores á une profondeur spécifique. Les méthodes développées ont été validées in vivo á l’aide d’expériences réalisées sur des anomalies de la peau et aussi sur le cortex exposé d’un rat anesthésié. Enfin, une étude préliminaire a été réalisée pour examiner la possibilité d’étendre la méthode á l’imagerie de 'speckle'. Des tests préliminaires réalises sur fantômes montrent l’influence de l’ellipticité de polarisation sur la formation et le comportement du speckle, ce qui offre la possibilité d’accéder á des informations sur le flux sanguin á des profondeurs spécifiques dans les tissus. / Polarization gating imaging is a popular and widely used imaging technique in biomedical optics to sense tissues, deeper volumes, and also selectively probe sub-superficial volumes. Due to the ‘polarization memory’ effect of polarized light, elliptical polarization gating allows access to tissue layers between those of accessible by linear or circular polarizations. As opposed to the conventional linearly polarized illumination, we focus on polarization gating methods that combine the use of elliptically polarized light to select polarization-maintaining photons and eliminate the background while providing superior contrast and depth information. With gating, it has also become possible to access user-defined depths (dependent on optical properties) in biological tissues with the use of images at different ellipticities. Furthermore, this investigation allowed the application of polarization gating in spectroscopy to selectively quantify the concentration of tissue chromophores at user-desired depths. Polarization gating methods have been validated and demonstrated with in vivo experiments on abnormalities of human skin (nevus, burn scar) and also on the exposed cortex of an anaesthetized rat. Finally, as a first step towards the use of coherent illumination, adding the concept of polarimetry to laser-speckle imaging was demonstrated. Preliminary tests on phantoms (solid and liquid) suggested evidence of the influence of polarization ellipticity on the formation and behaviour of speckles, which could pave the way for more insight in the study of blood flow in tissues.

Matrix variate skew elliptically contoured distributions generalize several classes of important distributions. This paper defines and explores matrix variate skew elliptically contoured distributions. In particular, we discuss the first two moments of the matrix variate skew elliptically contoured distributions.

Throughout stochastic data processing fields, mean and covariance matrices are commonly employed for purposes such as standardizing multivariate data through decorrelation. For practical applications, these matrices are usually estimated, and often, the data used for these estimates are non-Gaussian or may be corrupted by outliers or impulsive noise. To address this, robust estimators should be employed. However, in signal processing, where complex-valued data are common, the robust estimation techniques currently employed, such as M-estimators, provide limited robustness in the multivariate case. For this reason, this dissertation extends, to the complex-valued domain, the high-breakdown-point class of multivariate estimators called S-estimators. This dissertation defines S-estimators in the complex-valued context, and it defines their properties for complex-valued data. One major shortcoming of the leading high-breakdown-point multivariate estimators, such as the Rocke S-estimator and the smoothed hard rejection MM-estimator, is that they lack statistical efficiency at non-Gaussian distributions, which are common with real-world applications. This dissertation proposes a new tunable S-estimator, termed the Sq-estimator, for the general class of elliptically symmetric distributions—a class containing many common families such as the multivariate Gaussian, K-, W-, t-, Cauchy, Laplace, hyperbolic, variance gamma, and normal inverse Gaussian distributions. This dissertation demonstrates the diverse applicability and performance benefits of the Sq-estimator through theoretical analysis, empirical simulation, and the processing of real-world data. Through analytical and empirical means, the Sq-estimator is shown to generally provide higher maximum efficiency than the leading maximum-breakdown estimators, and it is also shown to generally be more stable with respect to initial conditions. To illustrate the theoretical benefits of the Sq for complex-valued applications, the efficiencies and influence functions of adaptive minimum variance distortionless response (MVDR) beamformers based on S- and M-estimators are compared. To illustrate the finite-sample performance benefits of the Sq-estimator, empirical simulation results of multiple signal classification (MUSIC) direction-of-arrival estimation are explored. Additionally, the optimal investment of real-world stock data is used to show the practical performance benefits of the Sq-estimator with respect to robustness to extreme events, estimation efficiency, and prediction performance. / Doctor of Philosophy / Throughout stochastic processing fields, mean and covariance matrices are commonly employed for purposes such as standardizing multivariate data through decorrelation. For practical applications, these matrices are usually estimated, and often, the data used for these estimates are non-normal or may be corrupted by outliers or large sporadic noise. To address this, estimators should be employed that are robust to these conditions. However, in signal processing, where complex-valued data are common, the robust estimation techniques currently employed provide limited robustness in the multivariate case. For this reason, this dissertation extends, to the complex-valued domain, the highly robust class of multivariate estimators called S-estimators. This dissertation defines S-estimators in the complex-valued context, and it defines their properties for complex-valued data. One major shortcoming of the leading highly robust multivariate estimators is that they may require unreasonably large numbers of samples (i.e. they may have low statistical efficiency) in order to provide good estimates at non-normal distributions, which are common with real-world applications. This dissertation proposes a new tunable S-estimator, termed the Sq-estimator, for the general class of elliptically symmetric distributions—a class containing many common families such as the multivariate Gaussian, K-, W-, t-, Cauchy, Laplace, hyperbolic, variance gamma, and normal inverse Gaussian distributions. This dissertation demonstrates the diverse applicability and performance benefits of the Sq-estimator through theoretical analysis, empirical simulation, and the processing of real-world data. Through analytical and empirical means, the Sq-estimator is shown to generally provide higher maximum efficiency than the leading highly robust estimators, and its solutions are also shown to generally be less sensitive to initial conditions. To illustrate the theoretical benefits of the Sq-estimator for complex-valued applications, the statistical efficiencies and robustness of adaptive beamformers based on various estimators are compared. To illustrate the finite-sample performance benefits of the Sq-estimator, empirical simulation results of signal direction-of-arrival estimation are explored. Additionally, the optimal investment of real-world stock data is used to show the practical performance benefits of the Sq-estimator with respect to robustness to extreme events, estimation efficiency, and prediction performance.

Beaucoup de résultats sur l’estimation d’une matrice d’échelle en analyse multidimensionnelle sont obtenus sous l’hypothèse de normalité (condition sous laquelle il s’agit de la matrice de covariance). Or il s’avère que, dans des domaines tels que la gestion de portefeuille en finance, cette hypothèse n’est pas très appropriée. Dans ce cas, la famille des distributions à symétrie elliptique, qui contient la distribution gaussienne, est une alternative intéressante. Nous considérons dans cette thèse le problème d’estimation de la matrice d’échelle Σ du modèle additif Yp_m = M + E, d’un point de vue de la théorie de la décision. Ici, p représente le nombre de variables, m le nombre d’observations, M une matrice de paramètres inconnus de rang q < p et E un bruit aléatoire de distribution à symétrie elliptique, avec une matrice de covariance proportionnelle à Im x Σ. Ce problème d’estimation est abordé sous la représentation canonique de ce modèle où la matrice d’observation Y est décomposée en deux matrices, à savoir, Zq x p qui résume l’information contenue dans M et une matrice Un x p, où n = m - q, qui résume l’information suffisante pour l’estimation de Σ. Comme les estimateurs naturels de la forme Σa = a S (où S = UT U et a est une constante positive) ne sont pas de bons estimateurs lorsque le nombre de variables p et le rapport p=n sont grands, nous proposons des estimateurs alternatifs de la forme ^Σa;G = a(S + S S+G(Z; S)) où S+ est l’inverse de Moore-Penrose de S (qui coïncide avec l’inverse S-1 lorsque S est inversible). Nous fournissons des conditions sur la matrice de correction SS+G(Z; S) telles que ^Σa;G améliore^Σa sous le coût quadratique L(Σ; ^Σ) = tr(^ΣΣ‾1 - Ip)² et sous une modification de ce dernier, à savoir le coût basé sur les données LS (Σ; ^Σ) = tr(S+Σ(^ΣΣ‾1 - Ip)²). Nous adoptons une approche unifiée des deux cas où S est inversible et S est non inversible. À cette fin, une nouvelle identité de type Stein-Haff et un nouveau calcul sur la décomposition en valeurs propres de S sont développés. Notre théorie est illustrée par une grande classe d’estimateurs orthogonalement invariants et par un ensemble de simulations. / Numerous results on the estimation of a scale matrix in multivariate analysis are obtained under Gaussian assumption (condition under which it is the covariance matrix). However in such areas as Portfolio management in finance, this assumption is not well adapted. Thus, the family of elliptical symmetric distribution, which contains the Gaussian distribution, is an interesting alternative. In this thesis, we consider the problem of estimating the scale matrix _ of the additif model Yp_m = M + E, under theoretical decision point of view. Here, p is the number of variables, m is the number of observations, M is a matrix of unknown parameters with rank q < p and E is a random noise, whose distribution is elliptically symmetric with covariance matrix proportional to Im x Σ. It is more convenient to deal with the canonical forme of this model where Y is decomposed in two matrices, namely, Zq_p which summarizes the information contained in M, and Un_p, where n = m - q which summarizes the information sufficient to estimate Σ. As the natural estimators of the form ^Σ a = a S (where S = UT U and a is a positive constant) perform poorly when the dimension of variables p and the ratio p=n are large, we propose estimators of the form ^Σa;G = a(S + S S+G(Z; S)) where S+ is the Moore-Penrose inverse of S (which coincides with S-1 when S is invertible). We provide conditions on the correction matrix SS+G(Z; S) such that ^Σa;G improves over ^Σa under the quadratic loss L(Σ; ^Σ) = tr(^ΣΣ‾1 - Ip)² and under the data based loss LS (Σ; ^Σ) = tr(S+Σ(^ΣΣ‾1 - Ip)²).. We adopt a unified approach of the two cases where S is invertible and S is non-invertible. To this end, a new Stein-Haff type identity and calculus on eigenstructure for S are developed. Our theory is illustrated with the large class of orthogonally invariant estimators and with simulations.