Regularized estimation of large covariance matrices, annals of statistics, 361, 199227, 2008 peter bickel. Covariance matrices, covariance structures, and bears, oh. A multiple testing approach to the regularisation of large. Estimation of large covariance matrices liza levina. Sparse estimation of large covariance matrices via a nested lasso penalty. Regularized estimation of large covariance matrices bickel, peter j. To do this, we construct the estimator based on a k band partial autocorrelation matrix with the number of bands chosen using an exact multiple. Sparse permutation invariant covariance estimation adam j. Regularized estimation of large covariance matrices core. Permutation invariant regularization of large covariance matrices. N2 the paper proposes a new covariance estimator for large covariance matrices when the variables have a natural ordering. The banding estimator of bickel and levina 2008a and its tapering version of cai, zhang and zhou 2010, are important high dimensional covariance estimators. Vanessa smith university of york may 18, 2014 abstract this paper proposes a novel regularisation method for the estimation of large covariance matrices.
Regularized estimation of large covariance matrices. Therefore, it is important to develop a wellconditioned estimator for largedimensional covariance matrices. Both estimators require choosing a band width parameter. Covariance regularization by thresholding bickel, peter j. Gaussian graphical models explore dependence relationships between random variables, through the estimation of the corresponding inverse covariance matrices. Generalized thresholding of large covariance matrices. T1 sparse estimation of large covariance matrices via a nested lasso penalty. Covariance regularization by thresholding by peter j. A wellconditioned estimator for largedimensional covariance. Journal of multivariate analysis 36, 163174 1991 estimating covariance matrices ii weiliem loh purdue university communicated by the editors let st and sz be two independent p x p wishart matrices with s, w,e, n, and szwpe2,nz. Sparse estimation of large covariance matrices 247 here t is a lower triangular matrix with ones on the diagonal, d is a diagonal matrix, and the elements below diagonal in the ith row of t can be interpreted as regression coef. Liza levina estimating large covariance matrices 9 banding toeplitz matrices bickel and levina, 2004. Regularized estimation of large covariance matrices peter j. Optimal rates of convergence for covariance matrix estimation.
Bickel university of california berkeley, ca 947203860 email. This area has seen an upsurge in practical and theoretical approaches due. Regularized estimation of large covariance matrices liza levina. Reports reaching back to 1992 are available for download below. We propose a band width selector for the banding covariance estimator by minimizing an empirical estimate of the expected squared frobenius norms of the. Estimating a covariance matrix is essential in multivariate data analysis.
A multiple testing approach to the regularisation of large sample correlation matrices. Bickel and elizaveta levina, some theory of fishers linear discriminant function, naive bayes, and some alternatives when there are many more variables than observations, bernoulli 10 2004, no. We show that a sample of size n om log 6 p is sufficient to. Sparse permutation invariant covariance estimation.
Thresholding carries essentially no computational burden, except for crossvalidation for the tuning parameter which is also necessary for. We also establish theoretical connections between banding cholesky factors of the covariance matrix and its inverse and. A classical approach to accurately estimating the covariance matrix. Rather than just letting the matrices vary freely, we may want to model them somehow e. Suretuned tapering estimation of large covariance matrices. Bickel and elizaveta levina1 university of california, berkeley and university of michigan this paper considers estimating a covariance matrix of p variables from n observations by either banding or tapering the sample covariance matrix, or estimating a banded version of the.
Index termscovariance estimation, large p small n, shrinkage methods, robust estimation, elliptical distribution, activityintrusion detection, wireless sensor network i. Alternative estimators for large covariance matrices have therefore attracted a lot of attention recently. On fixeddomain asymptotics and covariance tapering in gaussian random field models wang, daqing and loh, wei. Covariance matrices, covariance structures, and bears, oh my.
Band width selection for high dimensional covariance. Bickel and elizaveta levina university of california, berkeley and university of michigan september 14, 2006 abstract this paper considers estimating a covariance matrix of pvariables from nobservations by either banding the sample covariance matrix or estimating a banded version of. Joint estimation of multiple graphical models biometrika. Regularized estimation of large covariance matrices by peter j.
In some cases, we may have a set of covariance matrices to deal with rather than just one, such as with the general location model e. The department faculty regularly makes its research available to the wider community in the form of tech reports. We show that a sample of size n om log6 p is sufficient to. Chapter 4 covariance, regression, and correlation corelation or correlation of structure is a phrase much used in biology, and not least in that branch of it which refers to heredity, and the idea is even more frequently present than the phrase. Pdf covariance regularization by thresholding semantic scholar. Although the sample covariance matrix is an unbiased estimator of the covariance matrix of a gaussian random vector, it has poor properties if the dimension p is large. Estimation of covariance matrices is important in a number of areas of. Rothman university of michigan ann arbor, mi 481091107 email.
Pdf estimation of large covariance and precision matrices. We show that several commonly used methods for independent observations can be applied. Estimation of large covariance matrices of longitudinal data with basis function approximations jianhua z. Worked examples 3 covariance calculations example 1 let xand y be discrete random variables with joint mass function defined by f x,y. Generalized thresholding of large covariance matrices statistics.
We prove a law of large numbers similar to that proved in the gaussian case by bickel and levina, which implies that the spectrum of a banded empirical covariance matrix is an efficient. Computationally efficient banding of large covariance. Two broad classes of covariance estimators have emerged. When the inverse of the covariance matrix is the primary goal and the variables are ordered, regularization is usually introduced via the modi.
Large covariance estimation by thresholding principal. Sparse estimation of large covariance matrices via a nested. Example 2 let xand y be continuous random variables with joint pdf f x,yx,y 3x. This paper considers estimating a covariance matrix of p variables from n observations by. Sparse pca via covariance thresholding the journal of. Orientation multivariate statistics is longestablished. Bickel and elizaveta levina abstract this paper considers estimating a covariance matrix of p variables from n observations by either banding or tapering the sample covariance matrix, or estimating a banded version of the inverse of the covariance. In this paper we develop an estimator for such models appropriate for data from several graphical models that share the same variables and some of the dependence structure. We also introduce an analogue of the gaussian white noise model and show that if the population covariance is embeddable in that model and wellconditioned, then the banded approximations produce consistent estimates of the eigenvalues and. To do this, we construct the estimator based on a kband partial autocorrelation matrix with the number of bands chosen using an exact multiple hypothesis testing procedure. Large covariance estimation by thresholding principal orthogonal complements, journal of the royal statistical society series b, royal statistical society, vol. We consider the spectral properties of a class of regularized estimators of large empirical covariance matrices corresponding to stationary but not necessarily gaussian sequences, obtained by banding. This paper considers estimating a covariance matrix of p variables from n observations by either banding or tapering the sample covariance matrix, or estimating a banded version of the inverse of the covariance.
However, many modern applications operate with much smaller sample sizes, thus calling for estimation guarantees in the regime \n \ll p\. We also establish theoretical connections between banding cholesky factors of the covariance matrix and its inverse. Distributed algorithms for computing very large thresholded. Introduction estimating a covariance matrix or a dispersion matrix is a fundamental problem in statistical signal processing. In addition, covariance matrices are often sparse for large p. To estimate the covariance matrix when p a n or q a n, regularizing large empirical covariance matrices has been widely used in literature bickel and levina, 2008b. Permutationinvariant regularization of large covariance matrices. Optimal sample size for gaussian designs javanmard, adel and montanari, andrea, annals of statistics, 2018. Adaptive thresholding for sparse covariance matrix estimation. The results are uniform over some fairly natural wellconditioned families of covariance matrices. Posterior convergence rates for estimating large precision matrices using graphical models banerjee, sayantan and ghosal, subhashis, electronic journal of statistics, 2014.
Sep 01, 2014 estimating a covariance matrix is essential in multivariate data analysis. Sparse estimation of large covariance matrices via a. Robust shrinkage estimation of highdimensional covariance. A multiple testing approach to the regularisation of large sample correlation matrices natalia bailey queen mary, university of london m. We show that several commonly used methods for independent observations can be applied to the.
Regularized estimation of large covariance matrices department of. University of york june 27, 2018 abstract this paper proposes a regularisation method for the estimation of large covariance matrices that uses insights from the multiple testing mt literature. We also introduce an analogue of the gaussian white noise model and show that if the population covariance is. Pdf a clt for regularized sample covariance matrices. Estimation of large covariance matrices of longitudinal.
Journal of the american statistical association 106, 494 2011, 672684. When the inverse of the covariance matrix is the primary goal and the variables are ordered, regularization is usually introduced. In this article, we propose a computationally efficient approach to estimate large pdimensional covariance matrices of ordered or longitudinal data based on an independent sample of size n. If we wanted a wellconditioned estimator at any cost, we could always impose some adhoc structure on the covariance matrix to force it to be wellconditioned, such as diagonality or a factor model. We also introduce an analogue of the gaussian white noise model and show that if the population covariance is embeddable in that model and. Therefore, it is important to develop a wellconditioned estimator for large dimensional covariance matrices. University of michigan ann arbor, mi 481091107 email. Liza levina estimating large covariance matrices 2434 estimators of the inverse invariant under variable permutations inverse. If we wanted a wellconditioned estimator at any cost, we could always impose some adhoc structure on the covariance matrix to force it to be wellconditioned, such as. Covariance regularization by thresholding by peter. Search reports from berkeley and other organizations at view list of older reports back to 1981. Regularized estimation of large covariance matrices request pdf.
Hashem pesaran usc and trinity college, cambridge l. Regularized estimation of large covariance matrices sep, 2006 peter j. Sparse estimation of highdimensional covariance matrices. Regularized estimation of large covariance matrices liza.
We show that the thresholded estimate is consistent in the operator norm as. We consider the estimation of large covariance and precision matrices from highdimensional subgaussian or heaviertailed observations with slowly decaying temporal dependence. Gaussian, laguerre, jacobi ensembles contemporary multivariate statistics large p. The temporal dependence is allowed to be longrange so with longer memory than those considered in the current literature.
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