Estimation of large covariance and precision matrices from temporally dependent observations

2014/12/31 by Hai Shu, Bin Nan
Computer Science · Mathematics · Neuroscience · #Algorithm #Applied mathematics #Blind Source Separation Techniques #Consistency (knowledge bases) #Convergence (economics) #Covariance #Covariance function #Covariance matrix #Estimation of covariance matrices #Functional Brain Connectivity Studies #Gaussian #Mathematics #Model selection #Neural dynamics and brain function #Range (aeronautics) #Series (stratigraphy) #Statistics #math.ST #stat.ML #stat.TH

paper · pdf · doi:10.1214/18-aos1716

published as The Annals of Statistics, 2019, 47(3): 1321-1350 · The result for banding estimator of covariance matrix is given in the version 2 of this article. See arXiv:1412.5059v2

arxiv created 2017/07/18 · openalex publication_date 2019/02/13 · arxiv updated 2019/12/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06

Abstract

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 ℓ₁ minimization and the ℓ₁ 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.

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