Individual Covariance Matrices Research Articles

Fast and Separable Estimation in High-Dimensional Tensor Gaussian Graphical Models

In the tensor data analysis, the Kronecker covariance structure plays a vital role in unsupervised learning and regression. Under the Kronecker covariance model assumption, the covariance of an M-way tensor is parameterized as the Kronecker product of M individual covariance matrices. With normally distributed tensors, the key to high-dimensional tensor graphical models becomes the sparse estimation of the M inverse covariance matrices. Unable to maximize the tensor normal likelihood analytically, existing approaches often require cyclic updates of the M sparse matrices. For the high-dimensional tensor graphical models, each update step solves a regularized inverse covariance estimation problem that is computationally nontrivial. This computational challenge motivates our study of whether a noncyclic approach can be as good as the cyclic algorithms in theory and practice. To handle the potentially very high-dimensional and high-order tensors, we propose a separable and parallel estimation scheme. We show that the new estimator achieves the same minimax optimal convergence rate as the cyclic estimation approaches. Numerically, the new estimator is much faster and often more accurate than the cyclic approach. Moreover, another advantage of the separable estimation scheme is its flexibility in modeling, where we can easily incorporate user-specified or specially structured covariances on any modes of the tensor. We demonstrate the efficiency of the proposed method through both simulations and a neuroimaging application. Supplementary materials for this article are available online.

Estimation of a Two-Equation Panel Model with Mixed Continuous and Ordered Categorical Outcomes and Missing Data

Summary The paper proposes an estimation approach for panel models with mixed continuous and ordered categorical outcomes based on generalized estimating equations for the mean and pseudoscore equations for the covariance parameters. A numerical study suggests that efficiency can be gained in the mean parameter estimators by using individual covariance matrices in the estimating equations for the mean parameters. The approach is applied to estimate the returns to occupational qualification in terms of income and perceived job security in a 9-year period based on the German Socio-Economic Panel. To compensate for missing data, a combined multiple imputation–weighting approach is adopted.

A mixed approach for the estimation of probit models with correlated responses: some finite sample results

In the present paper a mixed approach for the simultaneous estimation of regression and correlation structure parameters in probit models with correlated responses proposed by Spiess and Hamerle (1996a) is compared to an approach proposed by Qu, Williams, Beck and Medendorp (1992) and Qu, Piedmonte and Williams (1994) via a Monte Carlo experiment. Whereas in the former approach generalized estimating equations for the estimation of regression parameters and pseudo-score equations for the estimation of association parameters are used, in the latter approach both sets of parameters are estimated using generalized estimating equations. As a ‘reference’ estimator for an equicorrelation model, the maximum likelihood (ML) estimator of a random effects pro bit model is calculated. The results show that the mixed approach leads to the most efficient non-ML estimators for regression and correlation structure parameters if individual covariance matrices are used. Furthermore, for the equicorrelation model, the loss of efficiency is small relative to the ML estimator.

Errors in plate rotations as described by covariance matrices and their combination in reconstructions

We use covariance matrices to describe the errors in plate rotations and to combine the contrbutions of individual plate pairs for plate reconstructions. First, for the rotation of a pair of plates the covariance matrix which describes the errors in the rotation is defined. Then we show how individual covariance matrices can be combined in a circuit of successive plate rotations to assess the errors in a plate reconstruction. Using the covariance matrix for accumulated errors from a sequence of rotations, we can determine the uncertainties in the position of selected points on one plate in the plate circuit to another that is assumed to be fixed. As an example, the position of the Pacific plate relative to North America is reconstructed at chron 6 (20 Ma). This choice is made to allow a direct comparison with published error assessments. Using the accumulated covariance matrix, we find similar uncertainties in the positions of their selected points on the Pacific plate. As a further test, this reconstruction is repeated using seafloor spreading data that directly relate Antarctica to Africa, bypassing the link of India. This eliminates the largest source of error in the plate circuit, that due to the short ridge segment between Africa and India. This revised, shortened circuit results in generally smaller errors in the positions of points on the Pacific plate. The covariance matrices of the two alternative circuits are used to construct the optimal estimate and its associated error. The optimal estimate and its error region intersect the two alternative reconstructions and their error regions.