<p>The complexity of industrial systems and the mathematical models to describe them increases. In many cases, point sensors are no longer sufficient to provide controllers and monitoring instruments with the information necessary for operation. The need for other types of information, such as audio and video, has grown. These are examples of information-rich signals for which suitable applications range in a broad spectrum from micro-electromechanical systems and bio-medical engineering to paper making and steel production.</p><p>Recursive parameter estimation algorithms are employed to identify parameters in a mathematical model from measurements of input and output signals. For accurate parameter estimation, the input signal must be <i>persistently exciting, i.e.</i> such that important features of the modelled system are reflected in the output over a sufficient period of time.</p><p>The Stenlund-Gustafsson (SG) algorithm, a Kalman filter based method for recursive parameter estimation in linear regression models, that does not diverge under lack of excitation, is studied. The stationary properties of the algorithm and the corresponding Riccati equation are formulated in terms of the excitation space spanned by the regressor vectors.</p><p>Furthermore, it is shown that the Riccati equation of the studied algorithm can be solved element-wise. Convergence estimates for the elements of the solution to the Riccati equation are provided, directly relating convergence rate to the signal-to-noise ratio in the regression model. An element-wise form of the parameter update equation is also given, where the connection to specific elements of the solution to the Riccati equation is apparent.</p><p>The SG-algorithm is employed for two applications with audio signals. One is in an acoustic echo cancellation setting where its performance is shown to match that of other well-known estimation techniques, such as the normalized least mean squares and the Kalman filter. When the input is not sufficiently exciting, the studied method performs best of all considered schemes.</p><p>The other application is the Linz-Donawitz (LD) steel converter. The converter consists of a vessel with molten metal and foam is produced to facilitate chemical reactions. A common problem, usually referred to as slopping, arises when the foam rises above the limits of the vessel and overflows. A warning system is designed, based on the SG-algorithm and change detection methods, to give alarms before slopping occurs. A black-box model relates different sensor values of which one is the microphone signal picked up in the area above the converter. The system detected slopping correctly in 80% of the blows in field studies at SSAB Oxelösund.</p><p>A practical example of a vision-based system is provided by cavity form estimation in a water model of the steel bath. The water model is captured from the side by a video camera. The images together with a non-linear model are used to estimate important process parameters, related to the heat and mass transport in the LD-converter.</p>

Le bilan carboné d'une rotation culturale de quatre ans a été établi entre 2004 et 2008 sur le site expérimental de Lonzée, Belgique. La région se caractérise par un climat tempéré océanique avec une température moyenne de l'air de 10°C et des précipitations annuelles de l'ordre de 800 mm. Le sol de la parcelle est un Luvisol. Les cultures composant la rotation sont des cultures de betterave, froment dhiver, pomme de terre et froment dhiver. Dans le but détablir un bilan carboné, des mesures ont été effectuées à différentes échelles spatiales et temporelles. Un système de mesure par eddy covariance fournit une estimation par demiheure de l'échange net en CO2 de l'écosystème (NEE). La qualité des procédures de sélection et de traitement des flux de NEE a été contrôlée. A partir de ces mesures, la productivité primaire brute (GPP) et la respiration totale de l'écosystème (TER) sont déduites. Plusieurs procédures basées sur les mesures diurnes ou nocturnes de NEE sont comparées. L'intérêt d'utiliser un court pas de temps dans ces procédures et la température du sol comme température de référence a été mis en avant, ainsi que la nécessité de déterminer une valeur seuil de la vitesse de friction (u*) pour chaque culture et les longues inter-cultures. Pour la culture de froment d'hiver 2005, une estimation de la GPP déduite des mesures d'eddy covariance est comparée à une estimation basée sur une modélisation de mesures réalisées, une fois par semaine, à l'échelle de la feuille. La conception initiale de l'appareil de mesure a permis de réaliser les mesures sur les feuilles uniquement. Les évolutions des deux estimations sont proches à l'échelle de la saison et à l'échelle journalière. La mise en oeuvre de la méthode basée sur les mesures à léchelle de la feuille a apporté dimportantes informations en termes de réponse de la GPP aux facteurs climatiques et non climatiques et a permis une validation de lestimation basée sur les mesures deddy covariance. Toutefois, dans le cadre de létablissement dun bilan carboné, la méthode basée sur les mesures d'eddy covariance est préférée. Des mesures de la respiration de sol, réalisées à l'échelle de la miniparcelle de sol, et des mesures du contenu en carbone des plantes sont aussi réalisées. Combinées aux mesures d'eddy covariance, ces mesures permettent de déduire les parts autoet hétérotrophiques de la respiration. Sur l'ensemble de la rotation, la parcelle apparait être une source significative de carbone de 0.17 (+/- 0.14) kg C m-2. Cela suggère que, durant la rotation, le contenu en carbone du sol a diminué. Ceci peut sexpliquer par labsence dapport de fertilisation organique durant les 10 dernières années ainsi que par la récolte systématique des pailles des céréales.

The overall objective of the current study is to develop a quantitative understanding of the effects of spatial variability in physical and geochemical properties of crystalline rock on the migration of radionuclides along a single fracture in bedrock. A stochastic model was developed to describe the transport of solutes in fractured rock. The model describes the migration of radionuclides along a one-dimensional path and includes the transversal diffusion into the rock matrix and sorption kinetics. By using a Lagrangian method of description we can extend the model to the description of a two-dimensional transport problem in single fractures. This study presents the first analysis of the impact of heterogeneous mass transfer on the transport of radionuclides in rock fractures, where most of the relevant rock properties such as aperture, porosity, effective diffusivity, sorption capacity and maximum diffusion depth are defined as being spatially random. The stochastic analysis performed here reflects the uncertainty in our knowledge of the properties associated with a discrete sampling technique in site investigations. Geostatistics of the main parameters was determined experimentally on a large number of rock samples taken from the Swedish crystalline basement. The knowledge of the covariance functions of the main rock properties is then used as a basis for a stochastic analysis. By combining the small perturbation approach with the spectral method the problem could be solved in terms of closed form solutions for the central temporal moments of the residence time probability density function. In order to be able to distinguish between the effects of various mechanisms from the effects of heterogeneity on the migration of radionuclides, it was necessary to perform independent studies of the effect of the variation of the dispersion coefficient on the aspect ratio of a rectangular flow section and the effect of adsorption kinetics on the migration. Finally, the effect of the heterogeneous rock properties on the solute transport observed in a limited number of migration experiments corresponds fairly well to the theoretical effect expected on the basis of the experimentally determined auto-covariance functions.

QC 20100601

This thesis consists of six papers. The main topic of all these papers is modeling a class of linear time-invariant systems. The system class is parameterized in the context of interpolation theory with a degree constraint. In the papers included in the thesis, this parameterization is the key tool for the design of dynamical system models in fields such as spectral estimation and model reduction. A problem in spectral estimation amounts to estimating a spectral density function that captures characteristics of the stochastic process, such as covariance, cepstrum, Markov parameters and the frequency response of the process. A  model reduction problem consists in finding a small order system which replaces the original one so that the behavior of both systems is similar in an appropriately defined sense.  In Paper A a new spectral estimation technique based on the rational covariance extension theory is proposed. The novelty of this approach is in the design of a spectral density that optimally matches covariances and approximates the frequency response of a given process simultaneously.In Paper B  a model reduction problem is considered. In the literature there are several methods to perform model reduction. Our attention is focused on methods which preserve, in the model reduction phase, the stability and the positive real properties of the original system. A reduced-order model is computed employing the analytic interpolation theory with a degree constraint. We observe that in this theory there is a freedom in the placement of the spectral zeros and interpolation points. This freedom can be utilized for the computation of a rational positive real function of low degree which approximates the best a given system. A problem left open in Paper B is how to select spectral zeros and interpolation points in a systematic way in order to obtain the best approximation of a given system. This problem is the main topic in Paper C. Here, the problem is investigated in the analytic interpolation context and spectral zeros and interpolation points are obtained as solution of a optimization problem.In Paper D, the problem of modeling a floating body by a positive real function is investigated. The main focus is  on modeling the radiation forces and moment. The radiation forces are described as the forces that make a floating body oscillate in calm water. These forces are passive and usually they are modeled with system of high degree. Thus, for efficient computer simulation it is necessary to obtain a low order system which approximates the original one. In this paper, the procedure developed in Paper C is employed. Thus, this paper demonstrates the usefulness of the methodology described in Paper C for a real world application.In Paper E, an algorithm to compute the steady-state solution of a discrete-type Riccati equation, the Covariance Extension Equation, is considered. The algorithm is based on a homotopy continuation method with predictor-corrector steps. Although this approach does not seem to offer particular advantage to previous solvers, it provides insights into issues such as positive degree and model reduction, since the rank of the solution of the covariance extension problem coincides with the degree of the shaping filter. In Paper F a new algorithm for the computation of the analytic interpolant of a bounded degree is proposed. It applies to the class of non-strictly positive real interpolants and it is capable of treating the case with boundary spectral zeros. Thus, in Paper~F, we deal with a class of interpolation problems which could not be treated by the optimization-based algorithm proposed by Byrnes, Georgiou and Lindquist. The new procedure computes interpolants by solving a system of nonlinear equations. The solution of the system of nonlinear equations is obtained by a homotopy continuation method. / QC 20100721

The complexity of industrial systems and the mathematical models to describe them increases. In many cases, point sensors are no longer sufficient to provide controllers and monitoring instruments with the information necessary for operation. The need for other types of information, such as audio and video, has grown. These are examples of information-rich signals for which suitable applications range in a broad spectrum from micro-electromechanical systems and bio-medical engineering to paper making and steel production. Recursive parameter estimation algorithms are employed to identify parameters in a mathematical model from measurements of input and output signals. For accurate parameter estimation, the input signal must be persistently exciting, i.e. such that important features of the modelled system are reflected in the output over a sufficient period of time. The Stenlund-Gustafsson (SG) algorithm, a Kalman filter based method for recursive parameter estimation in linear regression models, that does not diverge under lack of excitation, is studied. The stationary properties of the algorithm and the corresponding Riccati equation are formulated in terms of the excitation space spanned by the regressor vectors. Furthermore, it is shown that the Riccati equation of the studied algorithm can be solved element-wise. Convergence estimates for the elements of the solution to the Riccati equation are provided, directly relating convergence rate to the signal-to-noise ratio in the regression model. An element-wise form of the parameter update equation is also given, where the connection to specific elements of the solution to the Riccati equation is apparent. The SG-algorithm is employed for two applications with audio signals. One is in an acoustic echo cancellation setting where its performance is shown to match that of other well-known estimation techniques, such as the normalized least mean squares and the Kalman filter. When the input is not sufficiently exciting, the studied method performs best of all considered schemes. The other application is the Linz-Donawitz (LD) steel converter. The converter consists of a vessel with molten metal and foam is produced to facilitate chemical reactions. A common problem, usually referred to as slopping, arises when the foam rises above the limits of the vessel and overflows. A warning system is designed, based on the SG-algorithm and change detection methods, to give alarms before slopping occurs. A black-box model relates different sensor values of which one is the microphone signal picked up in the area above the converter. The system detected slopping correctly in 80% of the blows in field studies at SSAB Oxelösund. A practical example of a vision-based system is provided by cavity form estimation in a water model of the steel bath. The water model is captured from the side by a video camera. The images together with a non-linear model are used to estimate important process parameters, related to the heat and mass transport in the LD-converter.

We study Gaussian sample covariance matrices with population covariance a bounded-rank perturbation of the identity, as well as Wigner matrices with bounded-rank additive perturbations. The top eigenvalues are known to exhibit a phase transition in the large size limit: with weak perturbations they follow Tracy-Widom statistics as in the unperturbed case, while above a threshold there are outliers with independent Gaussian fluctuations. Baik, Ben Arous and Péché (2005) described the transition in the complex case and conjectured a similar picture in the real case, the latter of most relevance to high-dimensional data analysis. Resolving the conjecture, we prove that in all cases the top eigenvalues have a limit near the phase transition. Our starting point is the work of Rámirez, Rider and Virág (2006) on the general beta random matrix soft edge. For rank one perturbations, a modified tridiagonal form converges to the known random Schrödinger operator on the half-line but with a boundary condition that depends on the perturbation. For general finite-rank perturbations we develop a new band form; it converges to a limiting operator with matrix-valued potential. The low-lying eigenvalues describe the limit, jointly as the perturbation varies in a fixed subspace. Their laws are also characterized in terms of a diffusion related to Dyson's Brownian motion and in terms of a linear parabolic PDE. We offer a related heuristic for the supercritical behaviour and rigorously treat the supercritical asymptotics of the ground state of the limiting operator. In a further development, we use the PDE to make the first explicit connection between a general beta characterization and the celebrated Painlevé representations of Tracy and Widom (1993, 1996). In particular, for beta = 2,4 we give novel proofs of the latter. Finally, we report briefly on evidence suggesting that the PDE provides a stable, even efficient method for numerical evaluation of the Tracy-Widom distributions, their general beta analogues and the deformations discussed and introduced here. This thesis is based in part on work to be published jointly with Bálint Virág.

We study Gaussian sample covariance matrices with population covariance a bounded-rank perturbation of the identity, as well as Wigner matrices with bounded-rank additive perturbations. The top eigenvalues are known to exhibit a phase transition in the large size limit: with weak perturbations they follow Tracy-Widom statistics as in the unperturbed case, while above a threshold there are outliers with independent Gaussian fluctuations. Baik, Ben Arous and Péché (2005) described the transition in the complex case and conjectured a similar picture in the real case, the latter of most relevance to high-dimensional data analysis. Resolving the conjecture, we prove that in all cases the top eigenvalues have a limit near the phase transition. Our starting point is the work of Rámirez, Rider and Virág (2006) on the general beta random matrix soft edge. For rank one perturbations, a modified tridiagonal form converges to the known random Schrödinger operator on the half-line but with a boundary condition that depends on the perturbation. For general finite-rank perturbations we develop a new band form; it converges to a limiting operator with matrix-valued potential. The low-lying eigenvalues describe the limit, jointly as the perturbation varies in a fixed subspace. Their laws are also characterized in terms of a diffusion related to Dyson's Brownian motion and in terms of a linear parabolic PDE. We offer a related heuristic for the supercritical behaviour and rigorously treat the supercritical asymptotics of the ground state of the limiting operator. In a further development, we use the PDE to make the first explicit connection between a general beta characterization and the celebrated Painlevé representations of Tracy and Widom (1993, 1996). In particular, for beta = 2,4 we give novel proofs of the latter. Finally, we report briefly on evidence suggesting that the PDE provides a stable, even efficient method for numerical evaluation of the Tracy-Widom distributions, their general beta analogues and the deformations discussed and introduced here. This thesis is based in part on work to be published jointly with Bálint Virág.

This thesis addresses three topics in the area of statistics and probability, with applications in risk management. First, for the testing problems in the high-dimensional (HD) data analysis, we present a novel method to formulate empirical likelihood tests and jackknife empirical likelihood tests by splitting the sample into subgroups. New tests are constructed to test the equality of two HD means, the coefficient in the HD linear models and the HD covariance matrices. Second, we propose jackknife empirical likelihood methods to formulate interval estimations for important quantities in actuarial science and risk management, such as the risk-distortion measures, Spearman's rho and parametric copulas. Lastly, we introduce the theory of completely mixable (CM) distributions. We give properties of the CM distributions, show that a few classes of distributions are CM and use the new technique to find the bounds for the sum of individual risks with given marginal distributions but unspecific dependence structure. The result partially solves a problem that had been a challenge for decades, and directly leads to the bounds on quantities of interest in risk management, such as the variance, the stop-loss premium, the price of the European options and the Value-at-Risk associated with a joint portfolio.

The focus in this thesis is on the calculations of an empirical null distributionfor likelihood ratio tests testing either separable or double separable covariancematrix structures versus an unstructured covariance matrix. These calculationshave been performed for various dimensions and sample sizes, and are comparedwith the asymptotic χ2-distribution that is commonly used as an approximative distribution. Tests of separable structures are of particular interest in cases when data iscollected such that more than one relation between the components of the observationis suspected. For instance, if there are both a spatial and a temporalaspect, a hypothesis of two covariance matrices, one for each aspect, is reasonable.