Graphical Modeling of High-Dimensional Functional Data: Separability Structures and Unified Methodology under General Observational Designs

The statistical methodologies outlined in this project are motivated by the need to analyze multi-subject neuroimaging data sets as well as longitudinal observations from biomedical studies, among a variety of other examples. In many instances, it is of primary importance to discover interactions and dependencies between components of the data that are collected over time. In the case of neuroimaging data sets, these dependencies represent areas of the brain that coordinate during a specific task or share common features of baseline activity when the brain is at rest. This so-called functional brain connectome is known to be important biomarker for comparison across individuals or populations, provided that it can be reliably inferred from the data. The size of such data sets is typically very large, leading to practical issues in computation as well as theoretical ones related to quantifying uncertainty in outputs produced by the statistical analysis. The investigator will develop statistical methods, along with theoretical justifications and efficient computational packages, for estimating and interpreting functional connectivity networks and other large data sets of similar structure. Through both research and instructional activities, the investigator will educate and train students at both the undergraduate and graduate levels in the development and use of statistical tools related to the project aims. The data examples previously mentioned will be modeled as multivariate functional data (MFD), due to collection of multiple measurements at each time instant as well as the variability of these measurements across time. Most MFD methods, and the majority of existing computational tools for their analysis, simply apply univariate functional data methods to each component function separately, then combine the outputs for downstream analysis. Though simple, this approach ignores potentially valuable structures and properties that can be effectively harnessed in modeling and estimation. This is particularly the case for the graphical modeling of high-dimensional MFD that is the research focus of this project. The project aims to make foundational theoretical and algorithmic contributions to this nascent area of research by developing models and estimators that are flexible to different functional observation designs and manage the difficulties associated with the dual dimensionality problem of high-dimensional functional data, in which the large number of functions observed per subject is compounded with the intrinsically infinite dimension of each individual function. Specifically, the investigator will develop novel tools for a regularized inverse correlation operator estimator, underlying separability structures of the MFD, and a historical functional graphical model. The products of the project will be validated mathematically by deriving relevant statistical properties of the estimators and empirically through the analysis of real data sets. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.