Dynamic nonparametric functional graphical model

Connectivity between human brain regions has been proved to be highly related to phenotypical characteristics. Based on the graphical model, a novel nonparametric statistical method is proposed to estimate such dynamic connectivity among human brain regions. In the proposed model, the nodes of the graph are random functions rather than random variables, placing them within the scope of functional data analysis. Statistical operators are then exploited to capture the interdependence between these functions, which are allowed to vary with external covariates. Specifically, these random functions and operators are assumed to reside in nested Hilbert spaces, and a new precision operator–termed the nonparametric conditional additive precision operator–is constructed to capture dynamic interdependence. This operator serves as a functional counterpart to the precision matrix in the traditional Gaussian graphical model. The approach is also applicable when external covariates are high-dimensional or random functions. Furthermore, theoretical analysis establishes the consistency and optimal convergence rates for the proposed estimators. Simulation studies demonstrate that the proposed method significantly outperforms existing competitors in estimating graphs with nonlinear edges and accurately recovering how the strength of edges changes with external covariates. Finally, the framework is successfully applied to a real functional magnetic resonance imaging (fMRI) dataset.