FuDGE: A Method to Estimate a Functional Differential Graph in a High-Dimensional Setting
This work addresses the challenge of analyzing functional data like EEG to uncover differential connectivity patterns, which is incremental as it extends graphical model methods to the functional domain with a focus on sparsity.
The authors tackled the problem of estimating differences between two functional graphical models in high-dimensional settings, proposing FuDGE to directly estimate a sparse functional differential graph without needing to estimate dense individual graphs, and demonstrated its consistency and application to EEG data for detecting brain connectivity differences in alcohol use disorder.
We consider the problem of estimating the difference between two undirected functional graphical models with shared structures. In many applications, data are naturally regarded as a vector of random functions rather than as a vector of scalars. For example, electroencephalography (EEG) data are treated more appropriately as functions of time. In such a problem, not only can the number of functions measured per sample be large, but each function is itself an infinite-dimensional object, making estimation of model parameters challenging. This is further complicated by the fact that curves are usually observed only at discrete time points. We first define a functional differential graph that captures the differences between two functional graphical models and formally characterize when the functional differential graph is well defined. We then propose a method, FuDGE, that directly estimates the functional differential graph without first estimating each individual graph. This is particularly beneficial in settings where the individual graphs are dense but the differential graph is sparse. We show that FuDGE consistently estimates the functional differential graph even in a high-dimensional setting for both fully observed and discretely observed function paths. We illustrate the finite sample properties of our method through simulation studies. We also propose a competing method, the Joint Functional Graphical Lasso, which generalizes the Joint Graphical Lasso to the functional setting. Finally, we apply our method to EEG data to uncover differences in functional brain connectivity between a group of individuals with alcohol use disorder and a control group.