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# Topic: Estimation of covariance matrices

###### In the News (Mon 18 Jun 18)

 Estimation of covariance matrices - Wikipedia, the free encyclopedia In multivariate statistics, the importance of the Wishart distribution stems in part from the fact that it is the probability distribution of the maximum likelihood estimator of the covariance matrix of a multivariate normal distribution. Although no one is surprised that the estimator of the population covariance matrix is simply the sample covariance matrix, the mathematical derivation is perhaps not widely known and is surprisingly subtle and elegant. is the maximum-likelihood estimator of the "population covariance matrix" Σ. en.wikipedia.org /wiki/Estimation_of_covariance_matrices   (912 words)

 Covariance matrix - Wikipedia, the free encyclopedia In statistics and probability theory, the covariance matrix is a matrix of covariances between elements of a vector. Others call it the covariance matrix, because it is the matrix of covariances between the scalar components of the vector X. The derivation of the maximum-likelihood estimator of the covariance matrix of a multivariate normal distribution is perhaps surprisingly subtle. en.wikipedia.org /wiki/Covariance_matrix   (430 words)

 [No title]   (Site not responding. Last check: 2007-10-13) Further to this embodiment of the current invention, the covariance matrices Cj of the Gaussian distributions that constitute the GMMs are constrained to be isotropic and of constrained variance V, i. Constraining the covariance matrices to be isotropic and diagonal has the effect that the individual classes will project onto this hypersphere in the form of circles 104. A system as claimed in claim 14 wherein the GMM has a covariance matrix, the elements of which remain constrained during the optimisation procedure such that the matrix is isotropic and diagonal, and the value of the non zero diagonal elements remain constant throughout the optimisation procedure. www.wipo.int /cgi-pct/guest/getbykey5?KEY=03/83831.031009&ELEMENT_SET=DECL   (8324 words)

 SSG Seminar Abstract The covariance matrix problem is framed as an intrinsic estimation problem on the space of positive definite (covariance) matrices, which has the structure of a homogeneous or quotient space, not a vector space - the necessary setting for classical Cramer-Rao bounds. Covariance matrix estimation accuracy bounds are derived from an intrinsic derivation of the Cramer-Rao bound on arbitrary Riemannian manifolds (another new development), and compared to the accuracy achieved by standard methods involving the sample covariance matrix (SCM). The accuracy bound on unbiased covariance matrix estimators is shown to be about (10/log10)*n/sqrt(K) decibels, where n is the matrix order and K is the sample support. ssg.mit.edu /cal/abs/2005_spring/stsmith020905.shtml   (438 words)

 York University: Statistics Seminar When interest lies in estimating the survival distribution, from onset, of subjects with the disease, one must take into account that the survival times of the cases identified in such a study are left truncated or length-biased; the long survivors tend to be those cases identified at the start of the study. When estimating parameters from historical data, often only the closing prices, close and open prices are used in the estimation of parameters and the subsequent analysis, although information on the extremes of the process is highly informative, particularly of the changes in volatility. A simple estimation method is proposed under the proportional hazards model for the regression analysis of such studies when the times of the occurrences of the two events defining the elapsed time are right- or interval-censored. www.math.yorku.ca /Seminars/statistics   (13408 words)

 RAND | Papers | Multivariate Empirical Bayes and Estimation of Covariance Matrices. The authors consider the problem of estimating a covariance matrix in the standard multivariate normal situation. Estimators which dominate any constant multiple of the sample covariance matrix are presented. These estimators work by shrinking the sample eigenvalues toward a central value, in much the same way as the James-Stein estimator for a mean vector shrinks the maximum likelihood estimators toward a common value. www.rand.org /pubs/papers/P5276   (298 words)

 SAMSI Program on High Dimensional Inference and Random Matrices The aim of the Program is to bring together researchers interested in the theory and applications of random matrices to share their results, discuss new research directions and develop collaborations. Furthering our understanding of the spectral properties of random matrices under various models and assumptions (this is a "direct" problem). Consistent estimation of the model structure,study of the rates of convergence. www.samsi.info /programs/2006ranmatprogram.shtml   (761 words)

 Estimation of Polarization Parameters in Radar Meteorology (ResearchIndex)   (Site not responding. Last check: 2007-10-13) Estimation of Polarization Parameters in Radar Meteorology (1995) The analysis is carried out by using a widely-accepted model for radar meteorology -- the multivariate Gaussian model (Rayleigh fading) -- along with the assumptions that the time series of horizontally and vertically polarized echoes are acquired simultaneously, and that the autocorrelation function of the process is known. 0.1: Estimation of the Squared Modulus of the Mutual Intensity from.. citeseer.ifi.unizh.ch /83686.html   (411 words)

 Abstract   (Site not responding. Last check: 2007-10-13) A simple general model is proposed to achieve this and the consequences of the model and its implications for the the estimation of population means and covariance matrices are obtained. An estimate of the unit level covariance matrix of the grouping variable is required from some source. Data from the 1991 Census of the United Kingdom have been analysed to identify the important grouping variables and evaluate the effectiveness of the proposed adjustment methods for the estimation of covariance matrices and correlation coefficients. www.ccsr.ac.uk /staff/Marktranmer/abstract.htm   (276 words)

 Recent Papers and Preprints Estimated eigenmodes and confidence intervals for the eigenmodes and their oscillation periods and damping times can be computed from estimated model parameters. Numerical simulations indicate that, with the least squares algorithm, the AR model coefficients and the eigenmodes derived from the coefficients are estimated reliably and that the approximate 95% confidence intervals for the coefficients and eigenmodes are rough approximations of the confidence intervals inferred from the simulations. ARfit estimates the parameters of AR models from given time series data with a stepwise least squares algorithm that is computationally efficient, in particular when the data are high-dimensional. www.mat.univie.ac.at /~neum/papers.html   (7018 words)

 REML estimation: asymptotic behavior and related topics, Jiming Jiang The restricted maximum likelihood (REML) estimates of dispersion parameters (variance components) in a general (non-normal) mixed model are defined as solutions of the REML equations. In this paper, we show the REML estimates are consistent if the model is asymptotically identifiable and infinitely informative under the (location) invariant class, and are asymptotically normal (A.N.) if in addition the model is asymptotically nondegenerate. MILLER, J. Asy mptotic properties of maximum likelihood estimates in the mixed model of the analysis of variance. projecteuclid.org /Dienst/UI/1.0/Summarize/euclid.aos/1033066209   (691 words)

 Penalized Maximum-Likelihood Estimation of Covariance Matrices with Linear Structure - Schulz (ResearchIndex)   (Site not responding. Last check: 2007-10-13) Abstract: In this paper, a space-alternating generalized expectation-maximization (SAGE) algorithm is presented for the numerical computation of maximum-likelihood (ML) and penalized maximumlikelihood (PML) estimates of the parameters of covariance matrices with linear structure for complex Gaussian processes. By using a less informative hidden-data space and a sequential parameter-update scheme, a SAGE-based algorithm is derived for which convergence of the likelihood is demonstrated to be significantly... Penalized maximum-likelihood estimation of covariance matrices with linear structure. citeseer.ist.psu.edu /238228.html   (340 words)

 Nonconjugate Bayesian Estimation of Covariance Matrices and its Use in Hierarchical Models - Daniels, Kass ...   (Site not responding. Last check: 2007-10-13) This problem can be especially important in hierarchical models where the standard errors of fixed and random effects depend on estimation of the covariance matrix of the distribution of the random effects. We propose a set of hierarchical priors for the covariance matrix that produce posterior shrinkage toward a specified structure---here we examine shrinkage... Daniels, M.J. and Kass, R.E. Nonconjugate Bayesian estimation of covariance matrices and its use in hierarchical models. citeseer.ist.psu.edu /372100.html   (287 words)

 ICASSP-99 -> Technical Program -> Technical Sessions In some applications the covariance matrix of the observations is not only symmetric with respect to its main diagonal but also with respect to the anti-diagonal. An explicit expression for the difference between the estimation error covariance matrices of the two sample estimates is given. It is shown that Capon based on the forward-only sample covariance (F-Capon) underestimates the power spectrum, and also that the bias for Capon based on the forward-backward sample covariance is half that of F-Capon. www.eas.asu.edu /~icassp99/technical/sessions/abstracts-SAM-03.html   (1230 words)

 Journal of the American Statistical Association: REML estimation of covariance matrices with restricted parameter ...   (Site not responding. Last check: 2007-10-13) Journal of the American Statistical Association: REML estimation of covariance matrices with restricted parameter spaces. The problem of restricted maximum likelihood (REML) estimation of covariance matrices with restricted parameter spaces is examined. Maximum likelihood (ML) estimation of covariance matrices with a constrained parameter space, as in multivariate variance components estimation where the... highbeam.com /library/doc0.asp?docid=1G1:16679098&refid=ink_tptd_mag   (204 words)

 [No title]   (Site not responding. Last check: 2007-10-13) The % matrix Mxx_h is the regularized inverse of the covariance matrix % Cxx, % % Mxx_h = inv(Cxx + h^2 * I). The normal % equations are solved via an eigendecomposition of the covariance % matrix Cxx. However, if the data matrices X and Y are directly % available, a method based on a direct factorization of the data % matrices will usually be more efficient and more accurate. www.gps.caltech.edu /~tapio/imputation/mridge.m   (315 words)

 Antwerpian Group of Robust & Applied Statisticians A general discussion of robustness and optimality problems within the framework of estimation of a single parameter. Unified treatment of estimators and tests, and univariate and multivariate problems. Estimation and testing in multiple linear regression models. www.agoras.ua.ac.be /books/Robsta86.htm   (359 words)

 SSI - Scientific Software International, Inc. Anderson T.W. Estimation of Covariance Matrices which are Linear Combinations or whose Inverses are Linear Combination of given Matrices. Anderson T.W. Asymptotically Efficient Estimation of Covariance Matrices with Linear Structure. Lee, S.Y. Analysis of covariance and correlation structures. www.ssicentral.com /lisrel/references.html   (1427 words)

 Research Log I would like to estimate the states of ten or twenty thousand variables (to process a single scanline). Most applications find that the covariance matrix is the major bottleneck (since it is n^2 the size of the state). Weather forecasting is one field that routinely deals with estimating the states of huge numbers of variables. www.cs.duke.edu /~mark/research/log   (2587 words)

 SSRN-Bootstrap Estimation of Covariance Matrices via the Percentile Method by José Machado, Paulo Parente Consistency of the bootstrap second moments does not usually follow from the proofs of consistency of the distribution of the bootstrap. Here it is shown that the convergence of the bootstrap distribution to a normal variate implicitly defines a consistent estimator for the asymptotic second moments. The estimator is based on the L-estimation of the scale parameter of arbitrary linear combinations of the bootstrap sequence and uses Classical Minimum Distance techniques to impose the positive semi-definiteness restrictions. papers.ssrn.com /sol3/papers.cfm?abstract_id=682846   (215 words)

 Mario Micheli's Research The sequence of estimation error covariance matrices (which measure the effectiveness of the estimation algorithm) is not deterministic as for the ordinary Kalman Filter, but is a stochastic process itself: in fact we show that it is a homogeneous Markov process. In the one-dimensional case we compute a complete statistical description of this process: such description depends on the Poisson sampling rate (which is proportional to the number of sensors on a network) and on the dynamics of the continuous-time system represented by the state equation. As far as stable systems are concerned, when prior knowledge on state is poor it is convenient to wait until state is sufficiently close to the origin before starting state estimation: this explains the apparently paradoxical fact that in some situations increasing the sampling rate does not lower the estimation error variance. www.dam.brown.edu /people/mariom/research.html   (878 words)

 Analysis of incomplete datasets: Estimation of mean values and covariance matrices and imputation of missing values   (Site not responding. Last check: 2007-10-13) Analysis of incomplete climate data: Estimation of mean values and covariance matrices and imputation of missing values. In the regularized EM algorithm, ridge regression with generalized cross-validation replaces the conditional maximum likelihood estimation of regression parameters in the conventional EM algorithm. The implementation of the regularized EM algorithm is modular, so that the modules that perform the ridge regression and the generalized cross-validation can be exchanged for other regularization methods and other methods of determining a regularization parameter. www.gps.caltech.edu /~tapio/imputation   (531 words)

 [No title]   (Site not responding. Last check: 2007-10-13) Most estimators attempt to correct the distortion of the eigenstructure by the sample covariance matrix: the largest sample eigenvalue tends to be too big, and the smallest tends to be too small. In hierarchical models poor estimation of covariance matrices can affect inferences about both fixed and random effects. Simulation studies demonstrate that the new estimators can be very effective in lowering small sample risk in estimating the covariance matrix and increasing the the coverage probability of confidence intervals (or posterior probability intervals) for random effects. www.biostat.umn.edu /seminar/sem991026.txt   (134 words)

 Michael Daniels   (Site not responding. Last check: 2007-10-13) Daniels M, Kass R. (1999) Nonconjugate Bayesian estimation of covariance matrices in hierarchical models. Daniels M.J., Kass R.E. (2001) Shrinkage estimators for covariance matrices. Daniels, M. Pourahmadi, M. (2002) Bayesian analysis of covariance matrices and dynamic models for longitudinal data. www.stat.ufl.edu /personnel/usrpages/daniels.shtml   (102 words)

 Ruan's Home Page We also discussed several implemental issues, such as non-Gaussian distribution noise due to quantization, out-of-sequence measurement due to communication delay, and etc. We proposed solutions to these issues in the framework of a compander/particle-filter combination, and we showed that quite good performance is achievable with only 2-3 bits per dimension per observation. (b) Many distributed multi-sensor tracking systems are based on some form of track fusion, in which local track estimates and their associated covariances are shared among sensors. The scheme involves intelligent scalar and vector quantization of the local state estimates and of the associated estimation error covariance matrices. www.wright.edu /~yanhua.ruan/research/r2.html   (182 words)

 Michael J. Daniels Daniels M. (2006) Bayesian modeling of several covariance matrices and some results on propriety of the posterior for linear regression with correlated and/or heterogeneous errors. Botts, C. and Daniels M. (2006) A shrinkage estimator of spectral densities. Daniels, M., Pourahmadi, M. (2002) "Bayesian analysis of covariance matrices and dynamic models for longitudinal data", Biometrika. www.stat.ufl.edu /~mdaniels/research.html   (595 words)

 CEPR Discussion Paper Abstracts   (Site not responding. Last check: 2007-10-13) The method is based on the generalized dynamic factor model proposed in Forni, Hallin, Lippi, and Reichlin (2000), and takes advantage of the information on the dynamic covariance structure of the whole panel. We first use our previous method to obtain an estimation for the covariance matrices of common and idiosyncratic components. The generalized eigenvectors of this couple of matrices are then used to derive a consistent estimate of the optimal forecast, which is constructed as a linear combination of present and past observations only (one-sided filter). www.cepr.org /PUBS/NEW-DPS/dplist.asp?dpno=3432   (288 words)

 Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation This paper is concerned with the estimation of covariance matrices in the presence of heteroskedasticity and autocorrelation of unknown forms. Currently available estimators that are designed for this context depend upon the choice of a lag truncation parameter and a weighting scheme. No results are available regarding the choice of lag truncation parameter for a fixed sample size, regarding data-dependent automatic lag truncation parameters, or regarding the choice of weighting scheme. ideas.repec.org /a/ecm/emetrp/v59y1991i3p817-58.html   (278 words)

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