Estimation of Large Covariance and Precision Matrices from Temporally Dependent Observations
This addresses the challenge of statistical estimation in fields like neuroscience where data exhibit slow-decaying temporal correlations, though it is incremental as it extends known methods to a broader dependency context.
The paper tackles the problem of estimating large covariance and precision matrices from high-dimensional data with long-range temporal dependence, showing that existing methods for independent data can be applied with established convergence rates and consistency properties, and demonstrates this with fMRI data.
We consider the estimation of large covariance and precision matrices from high-dimensional sub-Gaussian or heavier-tailed observations with slowly decaying temporal dependence. The temporal dependence is allowed to be long-range so with longer memory than those considered in the current literature. We show that several commonly used methods for independent observations can be applied to the temporally dependent data. In particular, the rates of convergence are obtained for the generalized thresholding estimation of covariance and correlation matrices, and for the constrained $\ell_1$ minimization and the $\ell_1$ penalized likelihood estimation of precision matrix. Properties of sparsistency and sign-consistency are also established. A gap-block cross-validation method is proposed for the tuning parameter selection, which performs well in simulations. As a motivating example, we study the brain functional connectivity using resting-state fMRI time series data with long-range temporal dependence.