FuDGE: Functional Differential Graph Estimation with fully and discretely observed curves

Abstract

We consider the problem of estimating the difference between two functional undirected graphical models with shared structures. In many applications, data are naturally regarded as high-dimensional random function vectors rather than multivariate scalars. For example, electroencephalography (EEG) data are more appropriately treated as functions of time. In these problems, 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. In practice, curves are usually discretely observed, which makes graph structure recovery even more challenging. We formally characterize when two functional graphical models are comparable and propose a method that directly estimates the functional differential graph, which we term FuDGE. FuDGE avoids separate estimation of each graph, which allows for estimation in problems where individual graphs are dense, but their difference is sparse. We show that FuDGE consistently estimates the functional differential graph in a high-dimensional setting for both discretely observed and fully observed function paths. We illustrate finite sample properties of our method through simulation studies. In order to demonstrate the benefits of our method, we propose Joint Functional Graphical Lasso as a competitor, which is a generalization of the Joint Graphical Lasso. Finally, we apply our method to EEG data to uncover differences in functional brain connectivity between alcoholics and control subjects.

Publication
arXiv:2003.05402
Boxin Zhao
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.

Y. Samuel Wang
Y. Samuel Wang
Post-doc

Sam Wang is currently 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
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.

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