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

本项目中概述的统计方法的动机是需要分析多学科神经成像数据集以及来自生物医学研究的纵向观察，以及其他各种例子。在许多情况下，发现随时间收集的数据组件之间的交互和依赖关系至关重要。在神经成像数据集的情况下，这些依赖关系代表大脑中在特定任务期间进行协调的区域，或者在大脑休息时共享基线活动的共同特征。这种所谓的脑功能连接体是已知的跨个体或跨群体比较的重要生物标记物，前提是它可以从数据中可靠地推断出来。这类数据集的规模通常很大，这导致了计算中的实际问题以及与量化统计分析产生的产出的不确定性有关的理论问题。研究人员将开发统计方法，以及理论理由和有效的计算程序包，用于估计和解释功能连通性网络和其他类似结构的大型数据集。通过研究和教学活动，研究人员将教育和培训本科生和研究生开发和使用与项目目标相关的统计工具。前面提到的数据示例将被建模为多元函数数据(MFD)，因为在每个时刻收集多个测量以及这些测量在不同时间的可变性。大多数MFD方法，以及大多数现有的计算工具用于其分析，简单地将单变量函数数据方法分别应用于每个分量函数，然后将输出组合用于下游分析。虽然很简单，但这种方法忽略了可以在建模和评估中有效利用的潜在有价值的结构和属性。对于高维MFD的图形建模尤其如此，这是本项目的研究重点。该项目旨在通过开发可灵活适应不同功能观察设计的模型和估计器，并处理与高维功能数据的双重维度问题有关的困难，为这一新兴研究领域做出基础理论和算法贡献，在高维功能数据中，每个受试者观察到的大量功能与每个单独功能的本质无限维度复合在一起。具体地说，研究人员将开发新的工具，用于正则化反相关算子估计器、MFD的基本可分离性结构和历史函数图形模型。该项目的产品将通过推导估计者的相关统计属性进行数学验证，并通过对真实数据集的分析进行经验验证。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.