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

多变量函数数据是从多个过程中连续采样的数据，在广泛的科学应用中不断涌现。分析多变量函数数据的一个中心问题是了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建模问题。然而，现有的大多数解决方案都集中在随机变量上，并且扩展到随机函数远远不是微不足道的。该项目正在为功能图形建模开发一类新的统计方法和相关理论。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和医学研究领域，包括大脑连接网络、基因调控网络和蛋白质-蛋白质相互作用网络的分析。本项目主要研究三类问题：(1)非参数泛函图建模，它放宽了高斯分布或线性结构假设，避免了维度灾难，适用于大型图；(2)功能有向无环图建模，它结合了有向图和泛函图，为推断多元函数之间的方向依赖关系提供了一种简便的解决方案；(3)条件和动态泛函图建模，它模拟了随时间等一个或多个外部变量连续变化的图。其发展的核心是基于线性运算符的统计学习，它提供了一个高度灵活和高效的平台来处理海量和复杂的功能数据。随附的估计算法和渐近理论也对多个领域的工具箱做出了有用的补充，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。

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.