Collaborative Research: Spectral Functional Principal Components on Abelian Groups with Applications to Spatial Functional Data

最近，通过计算机气候模型输出、卫星遥感记录和脑部扫描等方式，可以获得网格化二维和三维域的大量数据集。这些数据集具有时间和空间两个维度。例如，每隔一定的时间间隔（每天或每周）在覆盖农业区域的网格上观察植被状况。这些数据可以看作是时间的函数，每个空间网格单元一个函数。它们的主要特征是在网格节点上观察到的曲线的空间依赖性。越来越需要开发统计工具，使研究人员能够从这些数据中提取有用的信息。pi将开发这样的工具。激发这项研究的数据和问题出现在几个对社会有重要影响的科学领域。例如，从未来气候模型中得出的结论有助于政府和企业规划各种资产的分配。对退伍军人经历的创伤和阿尔茨海默病的大脑研究被认为是重要的社会目标。pi将进行的统计研究将为这些领域的科学家提供有用的定量工具。将创建新方法的数学基础，以及特定于领域的方法。新方法将在R包中实现，并提供给研究社区、政府机构和商业企业。在拟进行的研究过程中，将培养两名博士生。pi将创建一个新的框架，用于在具有加性组结构的域上定义的功能数据的推理。新的降维方法将具有多尺度、数据驱动表示的特征，它考虑了对群元素（例如空间网格节点）定义的函数的依赖性。pi将使用阿贝尔群的傅立叶分析方法，功能数据的谱理论，希尔伯特空间的不变性原理，计算效率的时空样条表示，下载和操作大量数据集的例程。pi将开发几个推理程序，包括基于自举的推理，空间和分布结构的测试，以及应用于评估计算机气候模式的准确性。pi还将开发相应的计算技术，这将导致计算快速表示大到大规模的各种数据结构。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。

Massive data sets on gridded 2D and 3D domains have recently become available through computer climate model outputs, records from satellite remote sensing and brain scans, among others. These data sets have both temporal and spatial dimension. For example, a state of vegetation is observed on a grid covering an agricultural area at regular time intervals, every day or every week. Such data can be viewed as functions of time, one function per spatial grid unit. Their chief characteristic is the spatial dependence of curves observed at the grid nodes. There is an increasing need to develop statistical tools, which will allow researchers to extract useful information from such data. The PIs will develop such tools. The data and problems that motivate this research arise in several science fields, which have important impacts on society. For example, conclusions drawn from future climate models help the government and corporations plan for the allocation of various assets. Brain research on trauma experienced by military veterans and on Alzheimer's disease are recognized as important societal goals. The statistical research the PIs will conduct will provide useful quantitative tools to help scientists in these fields. Mathematical foundations of the new approach will be created, together with domain-specific approaches. The new methods will be implemented in R packages and made available to research community, government agencies and commercial enterprises. In the course of the proposed research, two Ph.D. students will be trained. The PIs will create a new framework for inference for functional data defined on domains with an additive group structure. The new dimension reduction approach will have characteristics of a multi-scale, data-driven representation, which takes into account the dependence of the functions defined on group elements, for example spatial grid nodes. The PIs will use methods of Fourier analysis on Abelian groups, spectral theory for functional data, invariance principles in Hilbert spaces, computationally efficient spatio-temporal spline representations, routines for downloading and manipulating massive data sets. The PIs will develop several inferential procedures, including bootstrap-based inference, tests for the spatial and distributional structure, and applications to the evaluation of the accuracy of computer climate models. The PIs will also develop corresponding computational techniques, which will lead to the computationally fast representation of various data structures of large to massive size.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.