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.

Boxin Zhao

PhD student

Boxin Zhao is a PhD student in Econometrics and Statistics at University of Chicago, Booth School of Business. His research interests include probabilistic graphical models, functional data analysis and distributed learning, with a focus on developing novel methodologies with both practical applications and theoretical guarantees.

Percy Zhai

PhD Student

Y. Samuel Wang

Post-doc (2018-2021)

Sam Wang is currently an assistant professor at Cornell University. He was a post-doc at the University of Chicago Booth School of Business advised by Mladen Kolar. His theoretical and methodological research interests are focused on causal discovery, graphical models, and high-dimensional statistical methods; his applied research interests lie in the social sciences. He completed his PhD in 2018 at the University of Washington advised by Mathias Drton and received his undergraduate degree from Rice University. Prior to embarking on his PhD studies, he worked in management consulting.

Mladen Kolar

Associate Professor of Econometrics and Statistics

Mladen Kolar is an Associate Professor of Econometrics and Statistics at the University of Chicago Booth School of Business. His research is focused on high-dimensional statistical methods, graphical models, varying-coefficient models and data mining, driven by the need to uncover interesting and scientifically meaningful structures from observational data.