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

本项目概述的统计方法的动机是需要分析多学科神经影像学数据集以及生物医学研究的纵向观察，以及各种其他例子。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是非常重要的。在神经成像数据集的情况下，这些依赖关系代表了在特定任务中协调的大脑区域，或者在大脑休息时共享基线活动的共同特征。这种所谓的功能性脑连接组被认为是个体或群体之间比较的重要生物标志物，前提是它可以从数据中可靠地推断出来。这种数据集的规模通常非常大，导致计算中的实际问题以及与统计分析产生的输出中的不确定性量化有关的理论问题。研究者将开发统计方法，以及理论证明和有效的计算包，用于估计和解释功能连接网络和其他类似结构的大型数据集。通过研究和教学活动，研究者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建模为多变量功能数据（multivariate functional data， MFD），因为在每个时刻收集多个测量值以及这些测量值在时间上的可变性。大多数MFD方法，以及大多数现有的用于分析的计算工具，只是简单地将单变量函数数据方法分别应用于每个组成函数，然后将输出组合起来进行下游分析。虽然简单，但是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。这对于高维MFD的图形化建模来说尤其如此，这也是本项目的研究重点。该项目旨在通过开发模型和估计器，为这一新兴研究领域做出基础理论和算法贡献，这些模型和估计器可以灵活地适应不同的功能观察设计，并管理与高维功能数据的二维问题相关的困难，其中每个主题观察到的大量功能与每个单个功能的内在无限维度相结合。具体来说，研究者将开发一种新的工具，用于正则化逆相关算子估计，MFD的潜在可分性结构和历史功能图形模型。项目的产品将通过推导估算器的相关统计属性和通过实际数据集的分析进行数学验证。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.