High-dimensional Functional Graphical Model Structure Learning via Neighborhood Selection Approach

Abstract

Undirected graphical models have been widely used to model the conditional independence structure of high-dimensional random vector data for years. In many modern applications such as EEG and fMRI data, the observations are multivariate random functions rather than scalars. To model the conditional independence of this type of data, functional graphical models are proposed and have attracted an increasing attention in recent years. In this paper, we propose a neighborhood selection approach to estimate Gaussian functional graphical models. We first estimate the neighborhood of all nodes via function-on-function regression, and then we can recover the whole graph structure based on the neighborhood information. By estimating conditional structure directly, we can circumvent the need of a well-defined precision operator which generally does not exist. Besides, we can better explore the effect of the choice of function basis for dimension reduction. We give a criterion for choosing the best function basis and motivate two practically useful choices, which we justified by both theory and experiments and show that they are better than expanding each function onto its own FPCA basis as in previous literature. In addition, the neighborhood selection approach is computationally more efficient than fglasso as it is more easy to do parallel computing. The statistical consistency of our proposed methods in high-dimensional setting are supported by both theory and experiment.

Nonparametric and high-dimensional functional graphical models

“…As shown in the Corollary 4.1 in Section 4, the differences between these estimated graphs vanish asymptotically. See also [21] and [43]. We summarize the algorithm below.…”

“…[40] proposes a FGGM based on a new notion of separability for the covariance operator of multivariate functional data, termed partial separability. [43] considered a neighbourhood selection approach to estimate the FGGM, which is an extension of the popular work of [21] to the functional setting. [45] proposed a Bayesian framework for the analysis of FGGM, [42] developed a method that estimates the difference between two functional graphical models and [27] considered a dynamic functional graphical model, where the dependency structure among the Gaussian random functions is allowed to change over time.…”

The completion of covariance kernels

“…Functional generalizations of the neighborhood selection approach (Zhao et al, 2021;Lee et al, 2022) involve performing appropriately penalized functional regression on each of the random elements against every other random element. Unlike inverse thresholding, these methods do work well in the high-dimensional settings and they also possess the computational advantage of being amenable to parallel implementation.…”