Conditional Functional Graphical Models

Abstract

Graphical modeling of multivariate functional data is becoming increasingly important in a wide variety of applications. The changes of graph structure can often be attributed to external variables, such as the diagnosis status or time, the latter of which gives rise to the problem of dynamic graphical modeling. Most existing methods focus on estimating the graph by aggregating samples, but largely ignore the subject-level heterogeneity due to the external variables. In this article, we introduce a conditional graphical model for multivariate random functions, where we treat the external variables as conditioning set, and allow the graph structure to vary with the external variables. Our method is built on two new linear operators, the conditional precision operator and the conditional partial correlation operator, which extend the precision matrix and the partial correlation matrix to both the conditional and functional settings. We show that their nonzero elements can be used to characterize the conditional graphs, and develop the corresponding estimators. We establish the uniform convergence of the proposed estimators and the consistency of the estimated graph, while allowing the graph size to grow with the sample size, and accommodating both completely and partially observed data. We demonstrate the efficacy of the method through both simulations and a study of brain functional connectivity network.

Keywords:

• Brain connectivity analysis
• Functional magnetic resonance imaging
• Graphical model
• Karhunen–Loève expansion
• Linear operator
• Reproducing kernel Hilbert space

Acknowledgments

We are grateful to two referees and an associate editor for their constructive comments and helpful suggestions.

Supplementary Materials

The Supplementary Appendix contains discussions of some assumptions, all the proofs of the theoretical results, and some results from additional asymptotic and numerical analyses.

Additional information

Funding

Lexin Li’s research was partially supported by NIH (grants nos. R01AG061303, R01AG062542, and R01AG034570). Hongyu Zhao’s research was partially supported by NSF (grant nos. DMS1713120 and DMS1902903), and NIH (grant no. R01 GM134005).