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Qian, F., Zhang, W. and Chen, Y. (2021), Adaptive banding covariance estimation for high-dimensional multivariate longitudinal data. Can J Statistics. https://doi.org/10.1002/cjs.11598
Covariance structure plays a particularly important role in longitudinal data analysis. A large collection of fundamental statistical methods, including the regression analysis, the generalized estimating equations, linear discriminant analysis and quadratic discriminant analysis, clustering and classification, require the knowledge of the covariance structure or some aspects thereof. For multivariate longitudinal data where multiple outcomes are repeatedly measured over time for the same subjects, such as the EEG data in neuroscience, fMRI data in brain science and panel data in finance, estimating the covariance structure can be more challenging than its univariate counterpart because the covariance matrix should account for complex correlated structures: the correlation within separate responses over time,the cross-correlation between different responses at different times, and the correlation between responses at each time point. Directly using existing methods by stacking multivariate responses into vectors incurs a loss of information or misleading results.
In this paper, the authors propose a novel and efficient approach to estimate the banded covariance structure of high-dimensional multivariate longitudinal data, where the dependence of the responses all decay with time. By using the modified Cholesky block decomposition and imposing banded structure, the proposed approach can automatically guarantee the positive-definiteness constraint. To cope with the high dimensionality of the covariance matrix, the authors impose an adaptive block banded structure on the Cholesky factor and sparsity on the innovation variance matrices via a novel convex hierarchical penalty and lasso penalty, respectively. The resulting adaptive block banding regularized estimator is fully data-driven and has more flexibility than regular banding estimators. An efficient alternative convex optimization algorithm is developed using ADMM algorithms. Numerical studies demonstrate the proposed approach has satisfactory performance under a variety of situations.More Details