Conditional Functional Graphical Models

• Published in: Journal of the American Statistical Association Vol. 118; no. 541; pp. 257 - 271
• Main Authors: Lee, Kuang-Yao, Ji, Dingjue, Li, Lexin, Constable, Todd, Zhao, Hongyu
• Format: Journal Article
• Language: English
• Published: United States Taylor & Francis 02.01.2023; Taylor & Francis Ltd
• Subjects: brain; Brain connectivity analysis; Conditioning; Convergence; Correlation analysis; Efficacy; Estimators; Functional connectivity; Functional magnetic resonance imaging; Graphical model; Graphical models; Graphs; Heterogeneity; Karhunen-Loève expansion; Linear operator; Linear operators; Medical diagnosis; Modelling; Multivariate analysis; Operators; Operators (mathematics); Regression analysis; Reproducing kernel Hilbert space; sample size; Statistical methods; Statistics; Variables; Functional magnetic resonance imaging; Graphical model; Reproducing kernel Hilbert space; Linear operator; Brain connectivity analysis; Karhunen–Loève expansion
• ISSN: 0162-1459; 1537-274X; 1537-274X
• DOI: 10.1080/01621459.2021.1924178

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.