Collaborative Research: New Directions in Multidimensional and Multivariate Functional Data Analysis

函数数据分析是处理函数或曲线样本的一种方法，在现代数据分析中占有重要地位。在“大数据”时代，多维、多元的功能数据越来越普遍，特别是在生物、医学、工程等领域的应用。这些数据的巨大维度和复杂结构带来了重大挑战。拟议的研究将大大缩小在实践中处理此类数据的需求日益增加与统计方法和计算工具发展不足之间的差距。这项研究应用于神经科学、气候科学和工程学。它将为科学家、工程师和医生提供工具，帮助他们了解各自领域的问题，并加强跨学科合作。本项目提供了一个全面的研究计划，以促进对多维和多元函数数据的理解和应用。研究将集中在以下三个子项目：(1)开发多维函数数据的数据自适应和可解释的协方差函数表示；(2)开发一种新的无模型过程来检测多元函数数据组件之间的依赖关系；(3)解决多元函数时间序列的建模和预测问题。由此产生的方法将应用于神经成像和气候数据。这三个子项目的整合将促进多维和多元功能数据的创新方向和策略。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。

Functional data analysis, which deals with a sample of functions or curves, plays an important role in modern data analysis. Nowadays in the era of "Big Data", multidimensional and multivariate functional data are becoming increasingly common, especially in biological, medical, and engineering applications. There are significant challenges posed by the very large dimension and complex structure of these data. The proposed research will substantially narrow the gap between the increasing demand for handling such data in practice, and the insufficient development of statistical methods and computational tools. This research has applications to neuroscience, climate science, and engineering. It will provide scientists, engineers, and doctors with tools to help understand problems in their area, and enhance interdisciplinary collaborations.This project offers a comprehensive research plan to advance the understanding and applicability of multidimensional and multivariate functional data. The research will focus on the following three sub-projects: (1) Develop data-adaptive and interpretable representation of the covariance function for multidimensional functional data, (2) Develop a novel model-free procedure to detect dependency between components of multivariate functional data, and (3) Address the modeling and prediction of multivariate functional time series. The resulting methods will be applied to neuroimaging and climate data. The integration of these three sub-projects will foster creative directions and strategies for multidimensional and multivariate functional data.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.