Copula Gaussian Graphical Models for Functional Data

Abstract

We introduce a statistical graphical model for multivariate functional data, which are common in medical applications such as EEG and fMRI. Recently published functional graphical models rely on the multivariate Gaussian process assumption, but we relax it by introducing the functional copula Gaussian graphical model (FCGGM). This model removes the marginal Gaussian assumption but retains the simplicity of the Gaussian dependence structure, which is particularly attractive for large data. We develop four estimators for the FCGGM and establish the consistency and the convergence rates of one of them. We compare our FCGGM with the existing functional Gaussian graphical model by simulations, and apply our method to an EEG dataset to construct brain networks. Supplementary materials for this article are available online.

KEYWORDS:

• Conditional independence
• Covariance operator
• Karhunen–Loeve expansion
• Non-Gaussian random functions
• Precision operator
• Rank transformation

Acknowledgments

We thank two referees and an associate editor for their many insightful and constructive comments and suggestions, which helped us greatly in improving this work.

Funding

Bing Li’s research is supported in part by NSF grant DMS-1713078.