We study the problem of estimating large covariance matrices under two types of structural assumptions: (i) The covariance matrix is the summation of a low rank matrix and a sparse matrix, and we have some prior information on the sparsity pattern of the sparse matrix; (ii) The data follow a transelliptical distribution. The former structure regulates the parameter space and has its roots in di...

Gaussian processes are a common analysis tool in statistics and uncertainty quantification. The covariance function of the process is generally unknown and often assumed to fall into some parameteric class. One of the scalability bottlenecks for the largescale usage of these processes is the computation of the maximum likelihood estimates of the parameters of the covariance matrix. In a classic...

Ensemble and reduced-rank approaches to prediction and assimilation rely on low-dimensional approximations of the estimation error covariances. Here stability properties of the forecast/ analysis cycle for linear, time-independent systems are used to identify factors that cause the steady-state analysis error covariance to admit a low-dimensional representation. A useful measure of forecast/ana...

Covariance estimation is a key step in many target detection algorithms. To distinguish target from background requires that the background be well-characterized. This applies to targets ranging from the precisely known chemical signatures of gaseous plumes to the wholly unspecified signals that are sought by anomaly detectors. When the background is modelled by a (global or local) Gaussian or ...

In longitudinal studies, serial dependence of repeated outcomes must be taken into account to make correct inferences on covariate effects. As such, care must be taken in modeling the covariance matrix. However, estimation of the covariance matrix is challenging because there are many parameters in the matrix and the estimated covariance matrix should be positive definite. To overcomes these li...

In this paper we propose a new regression interpretation of the Cholesky factor of the covariance matrix, as opposed to the well-known regression interpretation of the Cholesky factor of the inverse covariance, which leads to a new class of regularized covariance estimators suitable for high-dimensional problems. Regularizing the Cholesky factor of the covariance via this regression interpretat...

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. We show that these estimates are consistent in the operator norm as long as (logp)/n→ 0, and obtain explicit rates. The results are uniform over some fairly natural well-conditioned fam...

General properties of global covariance matrices representing bipartite Gaussian states can be decomposed into properties of local covariance matrices and their Schur complements. We demonstrate that given a bipartite Gaussian state ρ12 described by a 4 × 4 covariance matrix V, the Schur complement of a local covariance submatrix V1 of it can be interpreted as a new covariance matrix representi...

This work addresses the problem of covariance matrix estimation for ocean clutter modeling. For ship detection based on polarimetric synthetic aperture radar (PolSAR) imagery and constant false alarm rate (CFAR) detectors, accurate ocean clutter modeling is essential. The covariance matrix provides all the polarimetric information of the ocean clutter and its estimate is always involved in PolS...

This paper analyzes whether standard covariance matrix tests work when dimensionality is large, and in particular larger than sample size. In the latter case, the singularity of the sample covariance matrix makes likelihood ratio tests degenerate, but other tests based on quadratic forms of sample covariance matrix eigenvalues remain well-defined. We study the consistency property and limiting ...