CIF: Small: Collaborative Research: Graphical Modeling of Multivariate Functional Data

多元函数数据，即从多个过程中采样连续观察的数据，正在广泛的科学应用中出现。分析多元函数数据的一个核心问题是理解函数之间的相互依赖关系。这可以表述为多变量函数的图形化建模问题。然而，大多数现有的解决方案都集中在随机变量上，而扩展到随机函数也绝非易事。这个项目是开发一类新的统计方法和相关理论的功能图形建模。这项研究是及时的，因为它响应了对功能数据分析日益增长的需求，并有望推进许多生物学和医学研究领域，包括分析大脑连接网络，基因调控网络和蛋白质-蛋白质相互作用网络。本课题主要研究三组问题：(1)非参数泛函图形化建模，它放宽了高斯分布或线性结构假设，避免了维数诅咒，适用于大型图；(2)函数有向无环图建模，将有向图与函数图相结合，为多元函数间的方向依赖关系推断提供了一种易于处理的解决方案；(3)条件和动态功能图形化建模，对随时间等一个或多个外部变量连续变化的图形进行建模。其发展的核心是基于线性算子的统计学习，它提供了一个高度灵活和高效的平台来处理大量复杂的功能数据。随附的估计算法和渐近理论也为多个领域的工具箱提供了有用的补充，包括功能数据分析，网络和图形建模以及统计机器学习。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。

Multivariate functional data, where continuous observations are sampled from multiple processes, are emerging in a wide range of scientific applications. A central problem in analyzing multivariate functional data is to understand the interdependency among the functions. This can be formulated as the problem of graphical modeling of multivariate functions. The majority of existing solutions, however, focus on random variables, and extension to random functions is far from trivial. This project is developing a class of novel statistical methods and associated theory for functional graphical modeling. This research is timely in that it responds to the growing demand for functional data analysis, and is expected to advance numerous biological and medical research areas, including the analyses of brain connectivity networks, gene regulatory networks, and protein-protein interaction networks. This project is studying three sets of problems: (1) nonparametric functional graphical modeling, which relaxes the Gaussian distribution or the linear structural assumptions, and avoids the curse of dimensionality and works for large graphs; (2) functional directed acyclic graphical modeling, which combines directed graph and functional graph, and offers a tractable solution for inferring directional dependency among multivariate functions; and (3) conditional and dynamic functional graphical modeling, which models graph that varies continuously with one or multiple external variables such as time. At the heart of its development is linear-operator-based statistical learning, which provides a highly flexible and efficient platform to handle massive and complex functional data. The accompanying estimation algorithms and asymptotic theory also make useful additions to the toolbox of multiple fields, including functional data analysis, network and graphical modeling, and statistical machine learning.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.