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

CIF：小型：协作研究：多元函数数据的图形建模

• 批准号: 2102243
• 负责人: Kuang-Yao Lee
• 金额: $ 24.96万
• 依托单位: Temple University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-07-01 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2102243&HistoricalAwards=false
• 关键词: CIF; Small; Collaborative; Research; Graphical

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.

多元函数数据（从多个过程中进行连续观察）正在广泛的科学应用中出现。分析多元函数数据的一个中心问题是理解函数之间的相互依赖性。这可以表述为多元函数的图形建模问题。然而，大多数现有解决方案都关注随机变量，并且扩展到随机函数绝非易事。该项目正在开发一类用于函数图形建模的新颖统计方法和相关理论。这项研究非常及时，因为它满足了对功能数据分析日益增长的需求，并有望推动众多生物和医学研究领域的发展，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。该项目正在研究三组问题：（1）非参数泛函图形建模，放宽高斯分布或线性结构假设，避免维数灾难，适用于大图； (2)函数有向无环图建模，将有向图和函数图结合起来，为推断多元函数之间的方向依赖关系提供了一种易于处理的解决方案； (3)条件和动态函数图形建模，对随一个或多个外部变量（例如时间）连续变化的图形进行建模。其开发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海量和复杂的函数数据。随附的估计算法和渐近理论也为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价值和更广泛的影响审查标准。