Dissertations / Theses on the topic 'Covariance'
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Kang, Xiaoning. "Contributions to Large Covariance and Inverse Covariance Matrices Estimation." Diss., Virginia Tech, 2016. http://hdl.handle.net/10919/82150.
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Estimation of covariance matrix and its inverse is of great importance in multivariate statistics with broad applications such as dimension reduction, portfolio optimization, linear discriminant analysis and gene expression analysis. However, accurate estimation of covariance or inverse covariance matrices is challenging due to the positive definiteness constraint and large number of parameters, especially in the high-dimensional cases. In this thesis, I develop several approaches for estimating large covariance and inverse covariance matrices with different applications.
In Chapter 2, I consider an estimation of time-varying covariance matrices in the analysis of multivariate financial data. An order-invariant Cholesky-log-GARCH model is developed for estimating the time-varying covariance matrices based on the modified Cholesky decomposition. This decomposition provides a statistically interpretable parametrization of the covariance matrix. The key idea of the proposed model is to consider an ensemble estimation of covariance matrix based on the multiple permutations of variables.
Chapter 3 investigates the sparse estimation of inverse covariance matrix for the highdimensional data. This problem has attracted wide attention, since zero entries in the inverse covariance matrix imply the conditional independence among variables. I propose an orderinvariant sparse estimator based on the modified Cholesky decomposition. The proposed estimator is obtained by assembling a set of estimates from the multiple permutations of variables. Hard thresholding is imposed on the ensemble Cholesky factor to encourage the sparsity in the estimated inverse covariance matrix. The proposed method is able to catch the correct sparse structure of the inverse covariance matrix.
Chapter 4 focuses on the sparse estimation of large covariance matrix. Traditional estimation approach is known to perform poorly in the high dimensions. I propose a positive-definite estimator for the covariance matrix using the modified Cholesky decomposition. Such a decomposition provides a exibility to obtain a set of covariance matrix estimates. The proposed method considers an ensemble estimator as the center" of these available estimates with respect to Frobenius norm. The proposed estimator is not only guaranteed to be positive definite, but also able to catch the underlying sparse structure of the true matrix.Ph. D.
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Cissokho, Youssouph. "Extremal Covariance Matrices." Thesis, Université d'Ottawa / University of Ottawa, 2018. http://hdl.handle.net/10393/37124.
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The tail dependence coefficient (TDC) is a natural tool to describe extremal dependence. Estimation of the tail dependence coefficient can be performed via empirical process theory. In case of extremal independence, the limit degenerates and hence one cannot construct a test for extremal independence. In order to deal with this issue, we consider an analog of the covariance matrix, namely the extremogram matrix, whose entries depend only on extremal observations. We show that under the null hypothesis of extremal independence and for finite dimension d ≥ 2, the largest eigenvalue of the sample extremogram matrix converges to the maximum of d independent normal random variables. This allows us to conduct an hypothesis testing for extremal independence by means of the asymptotic distribution of the largest eigenvalue. Simulation studies are performed to further illustrate this approach.
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Dubbs, Alexander. "Beta-ensembles with covariance." Thesis, Massachusetts Institute of Technology, 2014. http://hdl.handle.net/1721.1/90185.
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Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2014.67
Cataloged from PDF version of thesis.
Includes bibliographical references (pages 73-79).
This thesis presents analytic samplers for the [beta]-Wishart and [beta]-MANOVA ensembles with diagonal covariance. These generalize the [beta]-ensembles of Dumitriu-Edelman, Lippert, Killip-Nenciu, Forrester-Rains, and Edelman-Sutton, as well as the classical [beta] = 1, 2,4 ensembles of James, Li-Xue, and Constantine. Forrester discovered a sampler for the [beta]-Wishart ensemble around the same time, although our proof has key differences. We also derive the largest eigenvalue pdf for the [beta]-MANOVA case. In infinite-dimensional random matrix theory, we find the moments of the Wachter law, and the Jacobi parameters and free cumulants of the McKay and Wachter laws. We also present an algorithm that uses complex analysis to solve "The Moment Problem." It takes the first batch of moments of an analytic, compactly-supported distribution as input, and it outputs a fine discretization of that distribution.
by Alexander Dubbs.
Ph. D.
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Armour, Bernard. "Structured covariance autoregressive parameter estimation." Thesis, McGill University, 1989. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=59559.
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In this thesis, the problem of estimating the autoregressive (AR) parameters of a wide sense stationary process is studied for the application of spectrum estimation of short data records. The approach taken is to first estimate a structured covariance matrix satisfying an optimality criterion and then map the estimate into the AR parameter estimates. Most covariance estimators are based on a least squares prediction error criterion. The new approach taken in this thesis is the use of a maximum likelihood (ML) criterion to obtain better covariance estimates. Both approximate and exact ML algorithms are developed based on an iterative Newton-Raphson technique to maximize the loglikelihood functions. Testing reveals the symmetric centro-symmetric structured covariance provides superior estimates in comparison to the Toeplitz structure and that the exact ML AR parameter estimates are among the lowest variance. Full comparison of the ML and popular AR spectrum estimation techniques is included.
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Wilkinson, Darren James. "Bayes linear covariance matrix adjustment." Thesis, Durham University, 1995. http://etheses.dur.ac.uk/5315/.
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In this thesis, a Bayes linear methodology for the adjustment of covariance matrices is presented and discussed. A geometric framework for quantifying uncertainties about covariance matrices is set up, and an inner-product for spaces of random matrices is motivated and constructed. The inner-product on this space captures aspects of belief about the relationships between covariance matrices of interest, providing a structure rich enough to adjust beliefs about unknown matrices in the light of data such as sample covariance matrices, exploiting second-order exchangeability and related specifications to obtain representations allowing analysis. Adjustment is associated with orthogonal projection, and illustrated by examples for some common problems. The difficulties of adjusting the covariance matrices underlying exchangeable random vectors is tackled and discussed. Learning about the covariance matrices associated with multivariate time series dynamic linear models is shown to be amenable to a similar approach. Diagnostics for matrix adjustments are also discussed.
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Gent, N. D. "Scale covariance and non-triviality." Thesis, Imperial College London, 1985. http://hdl.handle.net/10044/1/37703.
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Musolas, Otaño Antoni M. (Antoni Maria). "Covariance estimation on matrix manifolds." Thesis, Massachusetts Institute of Technology, 2020. https://hdl.handle.net/1721.1/127063.
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Thesis: Ph. D., Massachusetts Institute of Technology, Department of Aeronautics and Astronautics, May, 2020Cataloged from the official PDF of thesis.
Includes bibliographical references (pages 135-150).
The estimation of covariance matrices is a fundamental problem in multivariate analysis and uncertainty quantification. Covariance matrices are an essential modeling tool in climatology, econometrics, model reduction, biostatistics, signal processing, and geostatistics, among other applications. In practice, covariances often must be estimated from samples. While the sample covariance matrix is a consistent estimator, it performs poorly when the relative number of samples is small; improved estimators that impose structure must be considered. Yet standard parametric covariance families can be insufficiently flexible for many applications, and non-parametric approaches may not easily allow certain kinds of prior knowledge to be incorporated. In this thesis, we harness the structure of the manifold of symmetric positive-(semi)definite matrices to build families of covariance matrices out of geodesic curves.
These covariance families offer more flexibility for problem-specific tailoring than classical parametric families, and are preferable to simple convex combinations. Moreover, the proposed families can be interpretable: the internal parameters may serve as explicative variables for the problem of interest. Once a covariance family has been chosen, one typically needs to select a representative member by solving an optimization problem, e.g., by maximizing the likelihood associated with a data set. Consistent with the construction of the covariance family, we propose a differential geometric interpretation of this problem: minimizing the natural distance on the covariance manifold. Our approach does not require assuming a particular probability distribution for the data. Within this framework, we explore two different estimation settings.
First, we consider problems where representative "anchor" covariance matrices are available; these matrices may result from offline empirical observations or computational simulations of the relevant spatiotemporal process at related conditions. We connect multiple anchors to build multi-parametric covariance families, and then project new observations onto this family--for instance, in online estimation with limited data. We explore this problem in the full-rank and low-rank settings. In the former, we show that the proposed natural distance-minimizing projection and maximum likelihood are locally equivalent up to second order. In the latter, we devise covariance families and minimization schemes based on generalizations of multi-linear and Bézier interpolation to the appropriate manifold.
Second, for problems where anchor matrices are unavailable, we propose a geodesic reformulation of the classical shrinkage estimator: that is, we construct a geodesic family that connects the identity (or any other target) matrix to the sample covariance matrix and minimize the expected natural distance to the true covariance. The proposed estimator inherits the properties of the geodesic distance, for instance, invariance to inversion. Leveraging previous results, we propose a solution heuristic that compares favorably with recent non-linear shrinkage estimators. We demonstrate these covariance families and estimation approaches in a range of synthetic examples, and in applications including wind field modeling and groundwater hydrology.
by Antoni Musolas.
Ph. D.
Ph.D. Massachusetts Institute of Technology, Department of Aeronautics and Astronautics
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Wegelin, Jacob A. "Latent models for cross-covariance /." Thesis, Connect to this title online; UW restricted, 2001. http://hdl.handle.net/1773/8982.
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Maillard-Teyssier, Laurence Christine. "Calcul stochastique covariant à sauts & calcul stochastique à sauts covariants." Versailles-St Quentin en Yvelines, 2003. http://www.theses.fr/2003VERS0031.
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Nous proposons un calcul stochastique covariant pour des semimartingales dans le fibré tangent TM au dessus d'une variété M. Une connexion sur M permet de définir une dérivée intrrinsèque d'une courbe (Yt), C1 dans TM, la dérivée covariante. Plus précisément, c'est la dérivée de (Yt) vue dans un repère mobile, se déplaçant parallèlement le long de sa courbe (x1)projetée sur M. Avec le principe de transfert, Norris définit l'intégration covariante le long d'une semimartingale dans TM. Nous décrivons le cas où la semimartingale saute dans TM, en utilisant les travaux de Norris et les résultats de Cohen sur le calcul stochastique à sauts sur une variété. Nous comprenons, que, selon l'ordre dans lequel on compose la fonction qui donne les sauts et la connexion, on obtient un calcul stochastique covariant à sauts covariants. Tous deux dépendent du choix de la connexion et des objets (interpolateurs et connecteurs) décrivant les sauts au sens de Stratonovich ou d'Itô. Nous étudions les choix qui rendent équivalents les deux calculs. Sous certaines conditions, on retrouve les résultats de Norris lorsque (Yt) est continue. Le cas continu est décrit par un calcul covariant continu d'ordre deux, formalisme défini à l'aide de la notion de connexion d'ordre deuxWe propose a stochastic covaraiant calculus for càdlàg semimartingales in the tangent bundle TM over a manifold M. A connexion on M allows us to define an intrinsic derivative of a C1 curve (Yt) in TM, the covariant derivative. More precisely, it is the derivative of (Yt) seen in a frame moving parallely along its projection curve (xt) on M. With the transfer principle, Norris defined the stochastic covariant integration along a continuous semimartingale in TM. We describe the case where the semimartingale jumps in TM, using Norris's work and Cohen's results about stochastic calculus with jumps on manifolds. We see that, depending on the order in which we compose the function giving the jumps and the connection, we obtain a stochastic covariant calculus with jumps or a stochastic calculus with covariant jumps. Both depend on the choice of the connection and of the tools (interpolation and connection rules) describing the jumps in the meaning of Stratonovich or Itô. We study the choices that make equivalent the two calculus. Under suitable conditions, we recover Norris's results when (Yt) is continuous. The continuous case is described by a covariant continuous calculus of order two, a formalism defined with the notion of connection of order two
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Heiderich, Karen Rachel. "Spin-two fields and general covariance." Thesis, University of British Columbia, 1991. http://hdl.handle.net/2429/31021.
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It has long been presumed that any consistent nonlinear theory of a spin-two field must be generally covariant. Using Wald's consistency criteria, we exhibit classes of nonlinear theories of a spin-two field that do not have general covariance. We consider four alternative formulations of the spin-two equations. As a first example, we consider a conformally invariant theory of a spin-two field coupled to a scalar field. In the next two cases, the usual symmetric rank-two tensor field, γab, is chosen as the potential. In the fourth case, a traceless symmetric rank-two tensor field is used as the potential. We find that consistent nonlinear generalization of these different formulations leads to theories of a spin-two field that are not generally covariant. In particular, we find types of theories which, when interpreted in terms of a metric, are invariant under the infinitesimal gauge transformation γab→γab + ∇ (a∇[symbol omitted]K[symbol omitted]), where Kab is an arbitrary two-form field. In addition, we find classes of theories that are conformally invariant.
As a related problem, we compare the types of theories obtained from the nonlinear extension of a divergence- and curl-free vector field when it is described in terms of two of its equivalent formulations. We find that nonlinear extension of the theory is quite different in each case. Moreover, the resulting types of nonlinear theories may not necessarily be equivalent. A similar analysis is carried out for three-dimensional electromagnetism.Science, Faculty of
Physics and Astronomy, Department of
Graduate
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Asgharian-Dastenaei, Masoud. "Modeling covariance in multipath changepoint problems." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1998. http://www.collectionscanada.ca/obj/s4/f2/dsk1/tape11/PQDD_0018/NQ44350.pdf.
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Payseur, Scott. "Essays in realized covariance matrix estimation /." Thesis, Connect to this title online; UW restricted, 2008. http://hdl.handle.net/1773/7410.
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Greenall, Martin James. "Covariance principles for fluids at interfaces." Thesis, Imperial College London, 2004. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.408028.
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Bell, Peter. "Full covariance modelling for speech recognition." Thesis, University of Edinburgh, 2010. http://hdl.handle.net/1842/4912.
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HMM-based systems for Automatic Speech Recognition typically model the acoustic features using mixtures of multivariate Gaussians. In this thesis, we consider the problem of learning a suitable covariance matrix for each Gaussian. A variety of schemes have been proposed for controlling the number of covariance parameters per Gaussian, and studies have shown that in general, the greater the number of parameters used in the models, the better the recognition performance. We therefore investigate systems with full covariance Gaussians. However, in this case, the obvious choice of parameters – given by the sample covariance matrix – leads to matrices that are poorly-conditioned, and do not generalise well to unseen test data. The problem is particularly acute when the amount of training data is limited. We propose two solutions to this problem: firstly, we impose the requirement that each matrix should take the form of a Gaussian graphical model, and introduce a method for learning the parameters and the model structure simultaneously. Secondly, we explain how an alternative estimator, the shrinkage estimator, is preferable to the standard maximum likelihood estimator, and derive formulae for the optimal shrinkage intensity within the context of a Gaussian mixture model. We show how this relates to the use of a diagonal covariance smoothing prior. We compare the effectiveness of these techniques to standard methods on a phone recognition task where the quantity of training data is artificially constrained. We then investigate the performance of the shrinkage estimator on a large-vocabulary conversational telephone speech recognition task. Discriminative training techniques can be used to compensate for the invalidity of the model correctness assumption underpinning maximum likelihood estimation. On the large-vocabulary task, we use discriminative training of the full covariance models and diagonal priors to yield improved recognition performance.
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Ball, R. D. "The effective action : Covariance and chirality." Thesis, University of Cambridge, 1986. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.373651.
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Lee, Soonmook. "Model equivalence in covariance structure modeling /." The Ohio State University, 1987. http://rave.ohiolink.edu/etdc/view?acc_num=osu1487327695623423.
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Blake, Tayler Ann Blake. "Nonparametric Covariance Estimation for Longitudinal Data." The Ohio State University, 2018. http://rave.ohiolink.edu/etdc/view?acc_num=osu15256491898913.
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Avanesov, Valeriy. "Dynamics of high-dimensional covariance matrices." Doctoral thesis, Humboldt-Universität zu Berlin, 2018. http://dx.doi.org/10.18452/18801.
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Wir betrachten die Detektion und Lokalisation von plötzlichen Änderungen in der Kovarianzstruktur hochdimensionaler zufälliger Daten. Diese Arbeit schlägt zwei neuartige Ansätze für dieses Problem vor. Die Vorgehensweise beinhaltet im Wesentlichen Verfahren zum Test von Hypothesen, welche ihrerseits die Wahl geeigneter kritischer Werte erfordern. Dafür werden Kalibrierungsschemata vorgeschlagen, die auf unterschiedlichen Nichtstandard-Bootstrap-Verfahren beruhen. Der eine der beiden Ansätze verwendet Techniken zum Schätzen inverser Kovarianzmatrizen und ist durch Anwendungen in der neurowissenschaftlichen Bildgebung motiviert. Eine Beschränkung dieses Ansatzes besteht in der für die Schätzung der „Precision matrix“ wesentlichen Voraussetzung ihrer schwachen Besetztheit. Diese Bedingung ist im zweiten Ansatz nicht erforderlich. Die Beschreibung beider Ansätze wird gefolgt durch ihre theoretische Untersuchung, welche unter schwachen Voraussetzungen die vorgeschlagenen Kalibrierungsschemata rechtfertigt und die Detektion von Änderungen der Kovarianzstruktur gewährleistet. Die theoretischen Resultate für den ersten Ansatz basieren auf den Eigenschaften der Verfahren zum Schätzen der Präzisionsmatrix. Wir können daher die adaptiven Schätzverfahren für die Präzisionsmatrix streng rechtfertigen. Alle Resultate beziehen sich auf eine echt hochdimensionale Situation (Dimensionalität p >> n) mit endlichem Stichprobenumfang. Die theoretischen Ergebnisse werden durch Simulationsstudien untermauert, die durch reale Daten aus den Neurowissenschaften oder dem Finanzwesen inspiriert sind.We consider the detection and localization of an abrupt break in the covariance structure of high-dimensional random data. The study proposes two novel approaches for this problem. The approaches are essentially hypothesis testing procedures which requires a proper choice of a critical level. In that regard calibration schemes, which are in turn different non-standard bootstrap procedures, are proposed. One of the approaches relies on techniques of inverse covariance matrix estimation, which is motivated by applications in neuroimaging. A limitation of the approach is a sparsity assumption crucial for precision matrix estimation which the second approach does not rely on. The description of the approaches are followed by a formal theoretical study justifying the proposed calibration schemes under mild assumptions and providing the guaranties for the break detection. Theoretical results for the first approach rely on the guaranties for inference of precision matrix procedures. Therefore, we rigorously justify adaptive inference procedures for precision matrices. All the results are obtained in a truly high-dimensional (dimensionality p >> n) finite-sample setting. The theoretical results are supported by simulation studies, most of which are inspired by either real-world neuroimaging or financial data.
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Sahasrabudhe, Neeraja. "Covariance Realization Problem for Spin Systems." Doctoral thesis, Università degli studi di Padova, 2013. http://hdl.handle.net/11577/3426183.
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Let (Ω, Α) be a measurable space, F be a family of measurable functions f from Ω to R, and c: F→R be a given function. A generalized moment problem consists of finding all probabilities P on (Ω, Α) such that ∫ f dP = c(f) = cf for all f є F, and in providing conditions on ( c ) f є F for the existence of at least one such probability. Generalized moment problems of this kind have been widely studied, mainly in the theoretical engineering community, for continuous random variables.
In this thesis, we consider the special case of the covariance realization problem for spin systems and discuss the necessary and sufficient conditions for the realizability of a correlation matrix of order n ≥ 2. Let Ωn = { -1, 1}ⁿ be the space of length n sequences which are denoted by σ = (σ1, σ 2, …, σn), where σi є { -1, 1 }. Define the spin random variables ξi : Ω →{ -1, 1} for 1 ≤ i ≤ n as ξi (σ ) = σ i . For a probability P on Ωn , we denote by EP the expectation with respect to P . Given a symmetric matrix C = (( c ij)), we ask the following question in this thesis: under what condition does there exist a probability P such that EP (ξi) = 0 and c ij = EP (ξi ξj) for 1 ≤ i ≤ j ≤ n ? In this case, we say that C is a spin correlation matrix.
The necessary and sufficient conditions for a symmetric matrix of order n ≤ 4 to be a spin correlation matrix are already known. In this thesis, we obtain a general set of inequalities that are necessary and sufficient for any n . We also give a minimal set of necessary and sufficient conditions for n=5,6. Finally, we discuss methods to explicitly find the measure that realizes the given spin correlations (if they are feasible). We give a deterministic algorithm as well as a stochastic version of the same algorithm to find the measure explicitly. The efficiency of different algorithms is compared and some examples are worked out to illustrate the point.Sia (Ω, Α) uno spazio misurabile, F una famiglia di funzioni misurabili f da Ω a R, e c: F→R sia una funzione assegnata. Un problema dei momenti generalizzato consiste nel trovare tutte le probabilità P su (Ω, Α) tali ∫ f dP = c(f) = cf per ogni f є F, e nel determinare le condizioni su ( c ) f є F per l'esistenza di almeno una tale probabilità. Problemi dei momenti generalizzati di questo tipo sono stati ampiamente studiati, principalmente dagli ingegneri teorici, per variabili casuali continue. In questa tesi consideriamo il caso speciale del problema di realizzazione della covarianza per sistemi di spin e discutiamo le condizioni necessarie e sufficienti per la realizzabilità di una matrice di covarianza di ordine n ≥ 2. Sia Ωn = {-1, 1}ⁿ lo spazio delle sequenze di lunghezza n, denotate con σ = (σ1, σ 2, …, σn), dove σi є {-1, 1}. Definiamo le variabili aleatorie di spin ξi : Ω →{-1, 1} per 1 ≤ i ≤ n ponendo ξi (σ ) = σ i. Data una probabilità P su Ωn , denotiamo con EP il valore atteso rispetto a P. Data una matrice simmetrica C = (( c ij)), nella tesi ci poniamo la seguente domanda: sotto quali condizioni esiste una probabilità P tale che EP (ξi) = 0 e c ij = EP (ξi ξj) for 1 ≤ i ≤ j ≤ n? In questo caso, diciamo che C è una matrice di correlazione per spin. Condizioni necessarie e sufficienti affinchè una matrice simmetrica di ordine n ≤ 4 sia una matrice di correlazione per spin sono note. In questa tesi forniamo una famiglia di disuguaglianze che costituiscono una condizione necessaria e sufficiente per ogni n. Inoltre, per n=5,6, forniamo l'insieme di condizioni necessarie e sufficienti minimali. Infine, discutiamo vari metodi per determinare una probabilità che realizza le correlazioni assegnate (se ne esiste almeno una). Forniamo per questo un algoritmo deterministico, e alcune versioni stocastiche dello stesso. Confrontiamo inoltre, su alcuni esempi, l'efficienza di tali algoritmi.
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Gossner, Jesse Ross. "An analytic method of propagating a covariance matrix to a maneuver condition for linear covariance analysis during redezvous." Thesis, Massachusetts Institute of Technology, 1991. http://hdl.handle.net/1721.1/42498.
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Chen, Min. "Modeling covariance structure in unbalanced longitudinal data." [College Station, Tex. : Texas A&M University, 2008. http://hdl.handle.net/1969.1/ETD-TAMU-3073.
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Asgharian, Dastenaei Masoud. "Modeling covariance in multi-path changepoint problems." Thesis, McGill University, 1998. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=34909.
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Although the single-path changepoint problem has been extensively treated in the statistical literature, the multi-path changepoint problem has been largely ignored.In the multi-path changepoint setting it is often of interest to assess the impact of covariates on the changepoint itself as well as on the parameters before and after the changepoint. This thesis is concerned with including covariates in the changepoint distribution, a topic never before addressed in the literature. The model we introduce, based on the hazard of change, enjoys features which allow one to establish asymptotic results needed for estimation and testing. Indeed, we establish consistency of the maximum likelihood estimators of the parameters of our model.
As the proposed model is a mixture model, two of the difficulties associated with such models are addressed. They are identifiability, and positive definiteness of the information matrix. It is shown that under suitable conditions the set of zeros of the determinant of the information matrix is a nowhere dense set, thus partially compensating for the impossibility of directly establishing positive definiteness.
A limited simulation, using simulated annealing, is carried out to assess how the estimation procedure works in practice. In the example presented, the estimators appear to follow an approximately normal distribution even for moderate sample sizes. The maximum likelihood estimators appear to approximate their parameter counterparts well.
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Ohlson, Martin. "Studies in Estimation of Patterned Covariance Matrices." Doctoral thesis, Linköpings universitet, Matematisk statistik, 2009. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-18519.
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Many testing, estimation and confidence interval procedures discussed in the multivariate statistical literature are based on the assumption that the observation vectors are independent and normally distributed. The main reason for this is that often sets of multivariate observations are, at least approximately, normally distributed. Normally distributed data can be modeled entirely in terms of their means and variances/covariances. Estimating the mean and the covariance matrix is therefore a problem of great interest in statistics and it is of great significance to consider the correct statistical model. The estimator for the covariance matrix is important since inference on the mean parameters strongly depends on the estimated covariance matrix and the dispersion matrix for the estimator of the mean is a function of it. In this thesis the problem of estimating parameters for a matrix normal distribution with different patterned covariance matrices, i.e., different statistical models, is studied. A p-dimensional random vector is considered for a banded covariance structure reflecting m-dependence. A simple non-iterative estimation procedure is suggested which gives an explicit, unbiased and consistent estimator of the mean and an explicit and consistent estimator of the covariance matrix for arbitrary p and m. Estimation of parameters in the classical Growth Curve model when the covariance matrix has some specific linear structure is considered. In our examples maximum likelihood estimators can not be obtained explicitly and must rely on numerical optimization algorithms. Therefore explicit estimators are obtained as alternatives to the maximum likelihood estimators. From a discussion about residuals, a simple non-iterative estimation procedure is suggested which gives explicit and consistent estimators of both the mean and the linearly structured covariance matrix. This thesis also deals with the problem of estimating the Kronecker product structure. The sample observation matrix is assumed to follow a matrix normal distribution with a separable covariance matrix, in other words it can be written as a Kronecker product of two positive definite matrices. The proposed estimators are used to derive a likelihood ratio test for spatial independence. Two cases are considered, when the temporal covariance is known and when it is unknown. When the temporal covariance is known, the maximum likelihood estimates are computed and the asymptotic null distribution is given. In the case when the temporal covariance is unknown the maximum likelihood estimates of the parameters are found by an iterative alternating algorithm and the null distribution for the likelihood ratio statistic is discussed.
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McLeod, Christopher W. "Effect of nonlinearities on orbit covariance propagation." Thesis, Monterey, California: Naval Postgraduate School, 2013. http://hdl.handle.net/10945/37675.
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Approved for public release; distribution is unlimitedThis thesis will examine the effect of nonlinearities on the propagation of orbit uncertainties in order to gain insight into the accurateness of the estimation of covariance with time. Many real-world applications rely on a first-order approximation of nonlinear equations of motion for propagation of orbit uncertainty. The nonlinear effects that are ignored during the linearization process can greatly influence the accuracy of the solution. A comparative analysis of linear and nonlinear orbit uncertainty propagation is presented in order to attempt to determine when linearized uncertainty becomes non-Gaussian. An examination of performance metrics is then accomplished to compare linearly propagated uncertainty to uncertainty propagated using a second-order approximation. An attempt is then made to develop a performance metric that determines when propagated uncertainty is no longer Gaussian. The results show it is difficult to determine a clear method of defining when the linear approximated uncertainty is no longer Gaussian, but there are metrics that can be implemented given a user-defined threshold of performance.
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Lopes, Kim Samejima Mascarenhas. "Directed wavelet covariance for locally stationary processes." Universidade de São Paulo, 2018. http://www.teses.usp.br/teses/disponiveis/45/45133/tde-14032018-174950/.
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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.
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Yekollu, Srikar. "Graph Based Regularization of Large Covariance Matrices." The Ohio State University, 2009. http://rave.ohiolink.edu/etdc/view?acc_num=osu1237243768.
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Ge, Ming. "Noise covariance identification for filtering and prediction." Thesis, Imperial College London, 2016. http://hdl.handle.net/10044/1/31434.
Full textAbstract:
In this thesis, we introduce two different methods for determining noise covariance matrices in order to improve the stability and accuracy in state estimation and output prediction of discrete-time linear time varying (LTV) and nonlinear state space systems. The first method is based on the auto-covariance least squares (ALS) method, where the noise covariance matrices can be estimated by establishing a linear relationship between noise covariances and correlations of innovation sequence, hence solving a linear least squares problem. For LTV systems, we propose a new ALS algorithm that does not involve any approximations in the formulation. Our new ALS algorithm has fewer parameters to determine and can provide more accurate noise covariance estimation even when the historical output measurement window is not sufficiently long, comparing to an existing method. In addition to the noise covariance estimates, our ALS algorithm can also provide the estimate of the initial state error covariance, which is required by most state estimation methods. For higher-order systems, we also provide a much faster and less memory demanding formulation by splitting large Kronecker products with sums of smaller Kronecker or Schur products. For nonlinear systems, we have to approximate nonlinear parts as time-varying matrices by linearizing the nonlinear function around current state estimates. In addition to the extended Kalman Filter (EKF), our ALS algorithm also uses moving horizon estimation (MHE) to estimate the system state. MHE guarantees stability, is able to add state constraints and provides more accurate state estimates and local linearizations around the current state than the EKF. The second method is based on expectation maximization (EM), where the noise covariance matrices are determined by recursively maximizing the likelihood of covariance matrices, given output measurements. In our method, the noise covariance matrices are estimated using a semi-definite programming (SDP) solver, so that the results are more accurate and guaranteed to be positive definite. We propose a new EM algorithm that, combined with MHE and full information estimation (FIE) rather than a Kalman-based filter/smoother, allows the addition of state constraints, provides stable and more accurate estimates, so that the performance of noise covariance estimation can be significantly improved. Finally, we apply our noise covariance estimation methods to ocean wave prediction for the control of a wave energy converter (WEC), in order to approach optimal efficiency of wave energy extraction. We use a state space model representation for an autoregressive (AR) process, combined with noise covariance estimation, to simulate wave height forecasting based on data recorded at Galway Bay, Ireland. The simulation returns good wave predictions. Compared to existing wave prediction methods, our model has fewer parameters to tune and is able to provide more stable and accurate wave predictions by using a Kalman-based filter combined with the ALS or EM method.
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Jackson, J. Michael. "Nonparametric analysis of covariance based on residuals /." free to MU campus, to others for purchase, 1997. http://wwwlib.umi.com/cr/mo/fullcit?p9841154.
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Christensen, Randall S. "Linear Covariance Analysis For Gimbaled Pointing Systems." DigitalCommons@USU, 2013. https://digitalcommons.usu.edu/etd/1766.
Full textAbstract:
Linear covariance analysis has been utilized in a wide variety of applications. Historically, the theory has made significant contributions to navigation system design and analysis. More recently, the theory has been extended to capture the combined effect of navigation errors and closed-loop control on the performance of the system. These advancements have made possible rapid analysis and comprehensive trade studies of complicated systems ranging from autonomous rendezvous to vehicle ascent trajectory analysis. Comprehensive trade studies are also needed in the area of gimbaled pointing systems where the information needs are different from previous applications. It is therefore the objective of this research to extend the capabilities of linear covariance theory to analyze the closed-loop navigation and control of a gimbaled pointing system. The extensions developed in this research include modifying the linear covariance equations to accommodate a wider variety of controllers. This enables the analysis of controllers common to gimbaled pointing systems, with internal states and associated dynamics as well as actuator command filtering and auxiliary controller measurements. The second extension is the extraction of power spectral density estimates from information available in linear covariance analysis. This information is especially important to gimbaled pointing systems where not just the variance but also the spectrum of the pointing error impacts the performance. The extended theory is applied to a model of a gimbaled pointing system which includes both flexible and rigid body elements as well as input disturbances, sensor errors, and actuator errors. The results of the analysis are validated by direct comparison to a Monte Carlo-based analysis approach. Once the developed linear covariance theory is validated, analysis techniques that are often prohibitory with Monte Carlo analysis are used to gain further insight into the system. These include the creation of conventional error budgets through sensitivity analysis and a new analysis approach that combines sensitivity analysis with power spectral density estimation. This new approach resolves not only the contribution of a particular error source, but also the spectrum of its contribution to the total error. In summary, the objective of this dissertation is to increase the utility of linear covariance analysis for systems with a wide variety of controllers and for whom the spectrum of the errors is critical to performance.
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Farne', Matteo <1988>. "Large Covariance Matrix Estimation by Composite Minimization." Doctoral thesis, Alma Mater Studiorum - Università di Bologna, 2016. http://amsdottorato.unibo.it/7250/1/Farn%C3%A8_Matteo_tesi.pdf.
Full textAbstract:
The present thesis concerns large covariance matrix estimation via composite minimization under the assumption of low rank plus sparse structure.
Existing methods like POET (Principal Orthogonal complEment Thresholding) perform estimation
by extracting principal components and then applying a soft thresholding algorithm.
In contrast, our method recovers the low rank plus sparse decomposition of the covariance matrix
by least squares minimization under nuclear norm plus $l_1$ norm penalization.
This non-smooth convex minimization procedure is based on semidefinite programming and subdifferential methods,
resulting in two separable problems solved by a singular value thresholding plus soft thresholding algorithm.
The most recent estimator in literature is called LOREC (Low Rank and sparsE Covariance estimator) and provides non-asymptotic error rates as well as identifiability conditions in the context of algebraic geometry.
Our work shows that the unshrinkage of the estimated eigenvalues of the low rank component improves the performance of LOREC considerably.
The same method also recovers covariance structures with very spiked latent eigenvalues like in the POET setting, thus overcoming the necessary condition $p\leq n$.
In addition, it is proved that our method recovers structures with intermediate degrees of spikiness, obtaining a loss which is bounded accordingly.
Then, an ad hoc model selection criterion which detects the optimal point in terms of composite penalty is proposed. Empirical results coming from a wide original simulation study where various low rank plus sparse settings are simulated according to different parameter values are described outlining in detail the improvements upon existing methods. Two real data-sets are finally explored highlighting the usefulness of our method in practical applications.
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Farne', Matteo <1988>. "Large Covariance Matrix Estimation by Composite Minimization." Doctoral thesis, Alma Mater Studiorum - Università di Bologna, 2016. http://amsdottorato.unibo.it/7250/.
Full textAbstract:
The present thesis concerns large covariance matrix estimation via composite minimization under the assumption of low rank plus sparse structure.
Existing methods like POET (Principal Orthogonal complEment Thresholding) perform estimation
by extracting principal components and then applying a soft thresholding algorithm.
In contrast, our method recovers the low rank plus sparse decomposition of the covariance matrix
by least squares minimization under nuclear norm plus $l_1$ norm penalization.
This non-smooth convex minimization procedure is based on semidefinite programming and subdifferential methods,
resulting in two separable problems solved by a singular value thresholding plus soft thresholding algorithm.
The most recent estimator in literature is called LOREC (Low Rank and sparsE Covariance estimator) and provides non-asymptotic error rates as well as identifiability conditions in the context of algebraic geometry.
Our work shows that the unshrinkage of the estimated eigenvalues of the low rank component improves the performance of LOREC considerably.
The same method also recovers covariance structures with very spiked latent eigenvalues like in the POET setting, thus overcoming the necessary condition $p\leq n$.
In addition, it is proved that our method recovers structures with intermediate degrees of spikiness, obtaining a loss which is bounded accordingly.
Then, an ad hoc model selection criterion which detects the optimal point in terms of composite penalty is proposed. Empirical results coming from a wide original simulation study where various low rank plus sparse settings are simulated according to different parameter values are described outlining in detail the improvements upon existing methods. Two real data-sets are finally explored highlighting the usefulness of our method in practical applications.
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MBOUSSA, ANGA Gael. "Essays on exploding processes and covariance estimation." Doctoral thesis, Scuola Normale Superiore, 2020. http://hdl.handle.net/11384/91155.
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Puccio, Elena. "COVARIANCE AND CORRELATION ESTIMATORS IN BIPARTITE SYSTEMS." Doctoral thesis, Università degli Studi di Palermo, 2017. http://hdl.handle.net/10447/221177.
Full textAbstract:
We present a weighted estimator of the covariance and correlation in bipartite complex systems with a double layer of heterogeneity. The advantage provided by the weighted estimators lies in the fact that the unweighted sample covariance and correlation can be shown to possess a bias. Indeed, such a bias affects real bipartite systems, and, for example, we report its effects on two empirical systems, one social and the other biological. On the contrary, our newly proposed weighted estimators remove the bias and are better suited to describe such systems.
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de, Oliveira Fredi André Roberto. "New applications of covariance NMR and experimental development for measurements of homonuclear coupling constants in overlapping signals." Doctoral thesis, Universitat Autònoma de Barcelona, 2018. http://hdl.handle.net/10803/565884.
Full textAbstract:
Los resultados experimentales obtenidos en esta tesis están presentados en la forma de 3 artículos científicos publicados en revistas científicas especializadas na área de Resonancia Magnética Nuclear (RMN). Dos artículos con el uso de Covariancia de RMN como un general método para obtener espectros de pRMN. El último trabajo describe un nuevo experimento llamado de selTOCSY G-SERF para medir JHH en señales solapadas.
Publicación 1 describe un método general para generar espectros de psRMN por Covariancia de RMN. Este nuevo enfoque es único en la espectroscopia de RMN; dando una manera barata, rápida e fácil de reconstruir espectros de psRMN sin despender tiempo en el espectrómetro.
El concepto de Covariancia de psRMN ha sido extendido a la publicación 2 para añadir la información de Multiplicidad-Editada (ME) en experimentos 2D que son difíciles o mismo imposible de lograr experimentalmente. Se muestra cómo la información puede transferirse a un conjunto de espectros de RMN 2D homonucleares y heteronucleares por procesamiento de covarianza, reconstruyendo nuevos espectros de psME de una manera rápida.
Finalmente, G-SERF y los métodos relacionados solo funcionan para señales de 1H aisladas en las que se puede aplicar con éxito la excitación selectiva. Desafortunadamente, como sucede en otros experimentos selectivos de frecuencia, este enfoque falla para las señales solapadas. En la publicación 3 se presenta un esquema TOCSY G-SERF doblemente selectivo para eludir esta limitación, midiendo JHH de manera eficiente incluso para protones que resonantes en regiones atestadas.The experimental results obtained in this thesis are presented in the form of three papers published in NMR specialised scientific peer-reviewed journals. Two articles dela with the use of covariance NMR as a general method to generate novel psNMR spectra. The last work describes a new selTOCSY G-SERF experiment, for accurately measuring JHH in overlapped regions. Publication 1 describes a novel general protocol to generate psNMR spectra by Covariance NMR. This new approach is unique in NMR spectroscopy; giving a cheap, fast an easy way to reconstruct psNMR spectra without spending time in the spectrometer. This new strategy has been referenced to as psNMR Covariance. The concept of psNMR Covariance has been extended in Publication 2 by inserting Multiplicity-Edited (ME) information into 2D experiments that are difficult or even impossible to achieve experimentally. It is shown how the ME information can be efficiently transferred to a set of homonuclear and heteronuclear 2D NMR spectra by Covariance processing, reconstructing new psME spectra in a fast way. Finally, G-SERF and related methods only work for isolated 1H signals on which selective excitation can be successfully applied. Unfortunately, as it happens in other frequency-selective experiments, this approach fails for overlapped signals. A doubly-selective TOCSY G-SERF scheme is presented in the Publication 3 to circumvent this limitation, by measuring JHH efficiently even for protons resonating in crowded regions.
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Villares, Piera Javier. "Sample Covariance Based Parameter Estimation For Digital Communications." Doctoral thesis, Universitat Politècnica de Catalunya, 2005. http://hdl.handle.net/10803/6895.
Full textAbstract:
En aquesta tesi s'estudia el problema d'estimació cega de segon ordre en comunicacions digitals. En aquest camp, els símbols transmesos esdevenen paràmetres no desitjats (nuisance parameters) d'estadística no gaussiana que degraden les prestacions de l'estimador. En aquest context, l'estimador de màxima versemblança (ML) és normalment desconegut excepte si la relació senyal-soroll (SNR) és prou baixa. En aquest cas particular, l'estimador ML és una funció quadràtica del vector de dades rebudes o, equivalentment, una transformació lineal de la matriu de covariància mostral. Aquesta característica es compartida per altres estimadors importants basats en el principi de màxima versemblança com ara l'estimador ML gaussià (GML) i l'estimador ML condicional (CML). Així mateix, l'estimador MUSIC, i altres mètodes de subespai relacionats amb ell, es basen en la diagonalització de la matriu de covariància mostral. En aquest marc, l'aportació principal d'aquesta tesi és la deducció i avaluació de l'estimador òptim de segon ordre per qualsevol SNR i qualsevol distribució dels nuisance parameters.El disseny d'estimadors quadràtics en llaç obert i llaç tancat s'ha plantejat de forma unificada. Pel que fa als estimadors en llaç obert, s'han derivat els estimadors de mínim error quadràtic mig i mínima variància considerant que els paràmetres d'interès són variables aleatòries amb una distribució estadística coneguda a priori però, altrament, arbitrària. A partir d'aquest plantejament Bayesià, els estimadors en llaç tancat es poden obtenir suposant que la distribució a priori dels paràmetres és altament informativa. En aquest model de petit error, el millor estimador quadràtic no esbiaixat, anomenat BQUE, s'ha formulat sense convenir cap estadística particular pels nuisance parameters. Afegit a això, l'anàlisi de l'estimador BQUE ha permès calcular quina és la fita inferior que no pot millorar cap estimador cec que utilitzi la matriu de covariància mostral.
Probablement, el resultat principal de la tesi és la demostració de què els estimadors quadràtics són capaços d'utilitzar la informació estadística de quart ordre dels nuisance parameters. Més en concret, s'ha demostrat que tota la informació no gaussiana de les dades que els mètodes de segon ordre són capaços d'aprofitar apareix reflectida en els cumulants de quart ordre dels nuisance parameters. De fet, aquesta informació de quart ordre esdevé rellevant si el mòdul dels nuisance parameters és constant i la SNR és moderada o alta. En aquestes condicions, es demostra que la suposició gaussiana dels nuisance parameters dóna lloc a estimadors quadràtics no eficients.
Un altre resultat original que es presenta en aquesta memòria és la deducció del filtre de Kalman estès de segon ordre, anomenat QEKF. L'estudi del QEKF assenyala que els algoritmes de seguiment (trackers) de segon ordre poden millorar simultàniament les seves prestacions d'adquisició i seguiment si la informació estadística de quart ordre dels nuisance parameters es té en compte. Una vegada més, aquesta millora és significativa si els nuisance parameters tenen mòdul constant i la SNR és prou alta.
Finalment, la teoria dels estimadors quadràtics plantejada s'ha aplicat en alguns problemes d'estimació clàssics en l'àmbit de les comunicacions digitals com ara la sincronització digital no assistida per dades, el problema de l'estimació del temps d'arribada en entorns amb propagació multicamí, la identificació cega de la resposta impulsional del canal i, per últim, l'estimació de l'angle d'arribada en sistemes de comunicacions mòbils amb múltiples antenes. Per cadascuna d'aquestes aplicacions, s'ha realitzat un anàlisi intensiu, tant numèric com asimptòtic, de les prestacions que es poden aconseguir amb mètodes d'estimació de segon ordre.
This thesis deals with the problem of blind second-order estimation in digital communications. In this field, the transmitted symbols appear as non-Gaussian nuisance parameters degrading the estimator performance. In this context, the Maximum Likelihood (ML) estimator is generally unknown unless the signal-to-noise (SNR) is very low. In this particular case, if the SNR is asymptotically low, the ML solution is quadratic in the received data or, equivalently, linear in the sample covariance matrix. This significant feature is shared by other important ML-based estimators such as, for example, the Gaussian and Conditional ML estimators. Likewise, MUSIC and other related subspace methods are based on the eigendecomposition of the sample covariance matrix. From this background, the main contribution of this thesis is the deduction and evaluation of the optimal second-order parameter estimator for any SNR and any distribution of the nuisance parameters.
A unified framework is provided for the design of open- and closed-loop second-order estimators. In the first case, the minimum mean square error and minimum variance second-order estimators are deduced considering that the wanted parameters are random variables of known but arbitrary prior distribution. From this Bayesian approach, closed-loop estimators are derived by imposing an asymptotically informative prior. In this small-error scenario, the best quadratic unbiased estimator (BQUE) is obtained without adopting any assumption about the statistics of the nuisance parameters. In addition, the BQUE analysis yields the lower bound on the performance of any blind estimator based on the sample covariance matrix.
Probably, the main result in this thesis is the proof that quadratic estimators are able to exploit the fourth-order statistical information about the nuisance parameters. Specifically, the nuisance parameters fourth-order cumulants are shown to provide all the non-Gaussian information that is utilizable for second-order estimation. This fourth-order information becomes relevant in case of constant modulus nuisance parameters and medium-to-high SNRs. In this situation, the Gaussian assumption is proved to yield inefficient second-order estimates.
Another original result in this thesis is the deduction of the quadratic extended Kalman filter (QEKF). The QEKF study concludes that second-order trackers can improve simultaneously the acquisition and steady-state performance if the fourth-order statistical information about the nuisance parameters is taken into account. Once again, this improvement is significant in case of constant modulus nuisance parameters and medium-to-high SNRs.
Finally, the proposed second-order estimation theory is applied to some classical estimation problems in the field of digital communications such as non-data-aided digital synchronization, the related problem of time-of-arrival estimation in multipath channels, blind channel impulse response identification, and direction-of-arrival estimation in mobile multi-antenna communication systems. In these applications, an intensive asymptotic and numerical analysis is carried out in order to evaluate the ultimate limits of second-order estimation.
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Gunay, Melih. "Representation Of Covariance Matrices In Track Fusion Problems." Master's thesis, METU, 2007. http://etd.lib.metu.edu.tr/upload/12609026/index.pdf.
Full textAbstract:
Covariance Matrix in target tracking algorithms has a critical role at multi-
sensor track fusion systems. This matrix reveals the uncertainty of state es-
timates that are obtained from diferent sensors. So, many subproblems of
track fusion usually utilize this matrix to get more accurate results. That is
why this matrix should be interchanged between the nodes of the multi-sensor
tracking system. This thesis mainly deals with analysis of approximations of
the covariance matrix that can best represent this matrix in order to efectively
transmit this matrix to the demanding site. Kullback-Leibler (KL) Distance
is exploited to derive some of the representations for Gaussian case. Also com-
parison of these representations is another objective of this work and this is
based on the fusion performance of the representations and the performance
is measured for a system of a 2-radar track fusion system.
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Thomaz, Carlos Eduardo. "Maximum entropy covariance estimate for statistical pattern recognition." Thesis, Imperial College London, 2004. http://hdl.handle.net/10044/1/8755.
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Porteous, B. T. "Properties of log linear and covariance selection models." Thesis, University of Cambridge, 1985. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.372900.
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Scotiniadis, Dimitris. "Covariance approximations with a value at risk application." Thesis, Imperial College London, 2002. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.397976.
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Strasser, Helmut. "The covariance structure of conditional maximum likelihood estimates." Oldenbourg Verlag, 2012. http://epub.wu.ac.at/3619/1/covariance_final.pdf.
Full textAbstract:
In this paper we consider conditional maximum likelihood (cml) estimates for
item parameters in the Rasch model under random subject parameters. We give
a simple approximation for the asymptotic covariance matrix of the cml-estimates.
The approximation is stated as a limit theorem when the number of item parameters
goes to infinity. The results contain precise mathematical information on the order
of approximation.
The results enable the analysis of the covariance structure of cml-estimates when
the number of items is large. Let us give a rough picture. The covariance matrix has
a dominating main diagonal containing the asymptotic variances of the estimators.
These variances are almost equal to the efficient variances under ml-estimation when
the distribution of the subject parameter is known. Apart from very small numbers
n of item parameters the variances are almost not affected by the number n. The
covariances are more or less negligible when the number of item parameters is large.
Although this picture intuitively is not surprising it has to be established in precise
mathematical terms. This has been done in the present paper.
The paper is based on previous results [5] of the author concerning conditional
distributions of non-identical replications of Bernoulli trials. The mathematical background
are Edgeworth expansions for the central limit theorem. These previous results
are the basis of approximations for the Fisher information matrices of cmlestimates.
The main results of the present paper are concerned with the approximation
of the covariance matrices.
Numerical illustrations of the results and numerical experiments based on the
results are presented in Strasser, [6]. (author's abstract)
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Orchard, Peter Raymond. "Sparse inverse covariance estimation in Gaussian graphical models." Thesis, University of Edinburgh, 2014. http://hdl.handle.net/1842/9955.
Full textAbstract:
One of the fundamental tasks in science is to find explainable relationships between observed phenomena. Recent work has addressed this problem by attempting to learn the structure of graphical models - especially Gaussian models - by the imposition of sparsity constraints. The graphical lasso is a popular method for learning the structure of a Gaussian model. It uses regularisation to impose sparsity. In real-world problems, there may be latent variables that confound the relationships between the observed variables. Ignoring these latents, and imposing sparsity in the space of the visibles, may lead to the pruning of important structural relationships. We address this problem by introducing an expectation maximisation (EM) method for learning a Gaussian model that is sparse in the joint space of visible and latent variables. By extending this to a conditional mixture, we introduce multiple structures, and allow side information to be used to predict which structure is most appropriate for each data point. Finally, we handle non-Gaussian data by extending each sparse latent Gaussian to a Gaussian copula. We train these models on a financial data set; we find the structures to be interpretable, and the new models to perform better than their existing competitors. A potential problem with the mixture model is that it does not require the structure to persist in time, whereas this may be expected in practice. So we construct an input-output HMM with sparse Gaussian emissions. But the main result is that, provided the side information is rich enough, the temporal component of the model provides little benefit, and reduces efficiency considerably. The GWishart distribution may be used as the basis for a Bayesian approach to learning a sparse Gaussian. However, sampling from this distribution often limits the efficiency of inference in these models. We make a small change to the state-of-the-art block Gibbs sampler to improve its efficiency. We then introduce a Hamiltonian Monte Carlo sampler that is much more efficient than block Gibbs, especially in high dimensions. We use these samplers to compare a Bayesian approach to learning a sparse Gaussian with the (non-Bayesian) graphical lasso. We find that, even when limited to the same time budget, the Bayesian method can perform better. In summary, this thesis introduces practically useful advances in structure learning for Gaussian graphical models and their extensions. The contributions include the addition of latent variables, a non-Gaussian extension, (temporal) conditional mixtures, and methods for efficient inference in a Bayesian formulation.
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Young, Daniel Laurence. "Hebbian covariance learning and self-tuning optimal control." Thesis, Massachusetts Institute of Technology, 1997. http://hdl.handle.net/1721.1/42813.
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Small, Todd V. (Todd Vincent). "Optimal trajectory-shaping with sensitivity and covariance techniques." Thesis, Massachusetts Institute of Technology, 2010. http://hdl.handle.net/1721.1/67176.
Full textAbstract:
Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 2010.Cataloged from PDF version of thesis.
Includes bibliographical references (p. 129-130).
Traditional trajectory design approaches apply optimal control techniques to maximize desired performance, subject to specified constraints. Normal metrics and constraints are composed of the deterministic states and controls in the plant dynamics, so classical design methods do not directly address trajectory robustness in the presence of system uncertainties. This work explores the introduction of uncertainty directly into the trajectory design process. The state transition (sensitivity) and covariance matrices both measure the impact of plant uncertainty, and each of these mathematical constructs can be adjoined to the trajectory optimization problem to generate solutions that are less sensitive to prevalent uncertainties. A simple Zermelo boat problem is used to compare the methodologies for any combination of state initialization errors, state process noise, parametric biases, and parametric process noise, under any predefined feedback control law. The covariance technique is shown to possess several advantages over the sensitivity technique. Subsequently, the covariance method is used to simultaneously design reference trajectories and feedback control laws with closed-loop performance constraints for the Zermelo problem. The covariance trajectory-shaping technique is then applied to a generic hypersonic recoverable reentry vehicle. The trajectories include uncertainties in atmospheric density, axial and normal force coefficients, commanded attitude, and initial position and velocity. Reachability footprints with uncertainty bounds are generated by the trajectory-shaping methodology, and shown to extend the vehicle's range of confidence. Relative to a fixed recovery site within the footprint boundary, the covariance technique improves the circular error probable (CEP) radius by almost 50%. Lastly, by segmenting the problem, trajectory designs successfully reach the recovery site using a balance of dispersion penalties and maximum intermediate maneuvers. Improvements in final CEP are shown to require sacrifices in planned maneuvering.
by Todd V. Small.
S.M.
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Sivapalan, Ajani. "Estimating covariance matrices in a portfolio allocation problem." Thesis, Imperial College London, 2012. http://hdl.handle.net/10044/1/39390.
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Berg, Linda. "Evolutionary Covariance Among DNA Replication Restart Primosome Genes." University of Dayton / OhioLINK, 2012. http://rave.ohiolink.edu/etdc/view?acc_num=dayton1342115880.
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Wu, Hao. "An Empirical Bayesian Approach to Misspecified Covariance Structures." The Ohio State University, 2010. http://rave.ohiolink.edu/etdc/view?acc_num=osu1282058097.
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Kim, Myung Geun. "Models for the covariance matrices of several groups /." The Ohio State University, 1991. http://rave.ohiolink.edu/etdc/view?acc_num=osu1487758680162533.
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O'Neill, Robert. "Essays on forecasting the multivariate variance-covariance matrix." Thesis, University of Manchester, 2011. https://www.research.manchester.ac.uk/portal/en/theses/essays-on-forecasting-the-multivariate-variancecovariance-matrix(717082a1-e464-40ba-a0be-b16ef82f1621).html.
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This thesis is concerned with forecasting the variance covariance matrix (VCM) for a range of financial assets and investigating whether combining the elements of such forecasts result in more accurate predictions of portfolio volatility than those obtained from univariate models of aggregate volatility. There are three substantive chapters in the thesis; two introduce new methods for forecasting the VCM, while the third examines the accuracy of the techniques available for forecasting overall portfolio volatility. The first chapter introduces a model labelled the CD-MIDAS model, designed to improve the forecasting of the VCM at frequencies lower than a single day, for example we focus on predicting the VCM for a 22 day (monthly) horizon. The CD-MIDAS model uses the approach of Chiriac and Voev (2010) in forecasting the elements of the Cholesky decomposition of the VCM, rather than attempting to directly forecast elements of the matrix which are subject to restrictions ensuring that the forecast VCM is symmetric and positive definite. The elements of the Cholesky decomposition are modelled using the mixed data sampling (MIDAS) methodology introduced in Ghysels, Santa-Clara and Valkanov (2004,2006) which allows for the use of data observed at a high frequency (i.e. daily) to forecast the same variable observed at a lower frequency (i.e. monthly). The forecasting performance of this model is compared to that of other popular multivariate models and evidence is found, in both simulations and applied experiments, that the CD-MIDAS model is able to produce forecasts of the monthly VCM that are more accurate than its competitors. The second substantive chapter builds on findings in the univariate volatility forecasting literature that the level of return volatility for financial assets can be related to observations of certain economic variables. The kernel technique introduced in this chapter uses a multiplicative kernel to compare the characteristics of past periods with those at the point when the forecast of the VCM is being made. A weighting is then assigned to each point of time depending on how the historical economic and VCM characteristics compare to those at the point of forecast, the more similar the two points are, the higher the weight will be. All weights are positive and are applied to historical realizations of the VCM, thus the resulting forecast is guaranteed to be symmetric and positive definite, while the calculation method avoids the curse of dimensionality. In applied investigations it is shown that versions of the kernel technique produce the most accurate forecasts of those considered at horizons of 1, 5 and 22 days. In addition it is shown that the addition of the economic data to the kernel produces a statistically significant improvement in the accuracy of the forecasts generated. The final chapter considers which models provide the best forecasts when we are interested in forecasting overall portfolio volatility. This question can be seen as an extension of the aggregation vs. disaggregation literature in which we are essentially testing whether the aggregation error, cause by modelling an aggregate of several time series, is more or less important than the misspecification error caused by having a disaggregated model. While the latter can potentially capture idiosyncrasies of individual component series, it also may contain a larger number of misspecified representations and may suffer from increased parameter uncertainty due to the large number of parameters requiring estimation. Hence this chapter examines whether it is best to use multivariate models, using individual stock data, or univariate models, using portfolio level data, when the aim is to generate forecasts of total portfolio return volatility. An applied experiment shows that the best performing models are univariate models based on realized measures of portfolio variance. It is also apparent that any model, univariate or multivariate that does not make use of realized data, computed from high frequency returns data is significantly handicapped in terms of forecasting performance when compared to those that do. Hence the results imply that the misspecification errors in currently available multivariate models are of more concern to those wishing to forecast total portfolio return volatility than the misspecification inherent in modelling the aggregate of a number of variables.
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Das, Barnali. "Global covariance modeling : a deformation approach to anisotropy /." Thesis, Connect to this title online; UW restricted, 2000. http://hdl.handle.net/1773/8955.
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Nadadur, Desikachari. "Noise covariance estimation in low-level computer vision /." Thesis, Connect to this title online; UW restricted, 2001. http://hdl.handle.net/1773/5988.
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