Dynamic Covariance Models

ABSTRACT

An important problem in contemporary statistics is to understand the relationship among a large number of variables based on a dataset, usually with p, the number of the variables, much larger than n, the sample size. Recent efforts have focused on modeling static covariance matrices where pairwise covariances are considered invariant. In many real systems, however, these pairwise relations often change. To characterize the changing correlations in a high-dimensional system, we study a class of dynamic covariance models (DCMs) assumed to be sparse, and investigate for the first time a unified theory for understanding their nonasymptotic error rates and model selection properties. In particular, in the challenging high-dimensional regime, we highlight a new uniform consistency theory in which the sample size can be seen as n^4/5 when the bandwidth parameter is chosen as h∝n^{− 1/5} for accounting for the dynamics. We show that this result holds uniformly over a range of the variable used for modeling the dynamics. The convergence rate bears the mark of the familiar bias-variance trade-off in the kernel smoothing literature. We illustrate the results with simulations and the analysis of a neuroimaging dataset. Supplementary materials for this article are available online.

Keywords:

• Covariance model
• Dynamic covariance
• Functional connectivity
• High dimensionality
• Marginal independence
• Rate of convergence
• Sparsity
• Uniform consistency

Supplementary materials

The online supplementary materials for this article contain the lower bound theory for estimating sparse nonparametric covariance matrices, and the article appendices.

Acknowledgments

We thank the Joint Editor, an Associate Editor, and two reviewers for their constructive comments. We also acknowledge the Neuro Bureau and the ADHD-200 consortium for making the fMRI dataset used in this article freely available.

Funding

Chen's research is supported in part by National Nature Science Foundation of China (no. 11401593), Specialized Research Fund for the Doctoral Program of Higher Education of China (no. 20130162120086), China Postdoctoral Science Foundation (no. 2013M531796), and China Postdoctoral Science Foundation (no. 2014T70778).