Bayesian functional graphical models

TL;DR

This paper introduces a Bayesian framework for modeling dynamic graphical structures in multivariate functional data, allowing graphs to vary over the domain and capturing complex correlations.

Contribution

It develops a novel nonparametric Bayesian approach combining basis functions and regularization to estimate evolving graphs in functional data, with theoretical and scalable computational methods.

Findings

Effective reconstruction of functionally-evolving graphs demonstrated in simulations.

Method scales to large datasets with fine grid observations.

The approach captures within-function correlations and graph dynamics.

Abstract

We develop a Bayesian graphical modeling framework for functional data for correlated multivariate random variables observed over a continuous domain. Our method leads to graphical Markov models for functional data which allows the graphs to vary over the functional domain. The model involves estimation of graphical models that evolve functionally in a nonparametric fashion while accounting for within-functional correlations and borrowing strength across functional positions so contiguous locations are encouraged but not forced to have similar graph structure and edge strength. We utilize a strategy that combines nonparametric basis function modeling with modified Bayesian graphical regularization techniques, which induces a new class of hypoexponential normal scale mixture distributions that not only leads to adaptively shrunken estimators of the conditional cross-covariance but also facilitates a thorough theoretical investigation of the shrinkage properties. Our approach scales up to large functional datasets collected on a fine grid. We show through simulations and real data analysis that the Bayesian functional graphical model can efficiently reconstruct the functionally-evolving graphical models by accounting for within-function correlations.