Modelling the covariance matrix of multiple responses in longitudinal data plays a vital role. It is more challenging than its univariate counterpart due to the presence of correlations among multiple responses. Using the modified Cholesky block decomposition, we 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. We develop an efficient alternative convex optimization algorithm using the Alternating Direction Method of Multipliers (ADMM) algorithm. The resulting estimators converge optimally in the Frobenius norm. We establish row-specific support recovery for the precision matrix. Simulations and real data analysis show that the proposed estimator is better able to reveal banding sparsity patterns in data.

中文翻译：

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高维多元纵向数据的自适应条带协方差估计

对纵向数据中多个响应的协方差矩阵进行建模起着至关重要的作用。由于多个响应之间存在相关性，因此它比其单变量对应项更具挑战性。使用改进的 Cholesky 块分解，我们分别通过新颖的凸层次惩罚和套索惩罚对 Cholesky 因子和创新方差矩阵的稀疏性施加了自适应块带结构。由此产生的自适应块带正则化估计器是完全数据驱动的，并且比常规带估计器具有更大的灵活性。我们使用乘法器交替方向法 (ADMM) 算法开发了一种高效的替代凸优化算法。结果估计量在 Frobenius 范数中最优收敛。我们为精度矩阵建立了特定于行的支持恢复。模拟和真实数据分析表明，所提出的估计器能够更好地揭示数据中的条带稀疏模式。