Statistical Analysis of Complex, Highly-structured Functional Data

功能数据，其中观测单位表示曲线，表面，或图像，在现代测量系统中变得越来越普遍。虽然大多数统计方法侧重于简化独立性等假设，但现实世界的功能数据通常包含更复杂的结构。例如，在神经成像中，跨大脑区域的依赖网络需要使用基于EEG或fMRI测量的图形来表征；在蛋白质组学和光谱分析中，研究人员经常遇到具有特征区域的功能数据。这些结构表示现有分析工具无法处理的数据中的一般复杂性。这个项目的重点是发展新的统计理论和方法来模拟复杂的结构。提出的工具将应用于神经成像、质谱、基因组学和生物信息学的数据集。这项研究将与各种教育和推广活动相结合，这些活动将影响大学内外的教学。该项目有以下四个相互关联的目标：(1)开发一个功能图形模型框架来表征随机函数之间的条件独立关系，并应用该方法来估计大规模的大脑网络；(2)开发新的频率正则化策略和贝叶斯先验来选择功能数据的区域；(3)利用马尔可夫随机场表征函数间的空间依赖性，并将其应用于功能面和功能点参考数据；(4)在高维数据分析领域教育和吸引学生，开发K-12外展计划，并指导少数民族学生。

Functional data, where the observational units represent curves, surfaces, or images, are becoming increasingly prevalent in modern measurement systems. While most statistical methods focus on simplifying assumptions such as independence, real-world functional data often contain far more complex structures. For example, in neuroimaging, the dependence network across brain regions needs to be characterized using graphs based on EEG or fMRI measurements; in proteomic and spectral analysis, researchers frequently encounter functional data with featured regions. These structures represent generic complexities in data that cannot be handled by existing analytical tools. This project focuses on the development of novel statistical theory and methods to model complex structures. The proposed tools will be applied to datasets from neuroimaging, mass spectrometry, genomics, and bioinformatics. The research will be integrated with various educational and outreach activities that will impact teaching and learning both within and beyond the university. The project has the following four interrelated objectives: (1) Develop a functional graphical model framework to characterize conditional independent relationships between random functions, and apply the methods to estimate large-scale brain networks; (2) Develop novel frequentist regularization strategies and Bayesian priors to select regions of functional data; (3) Use Markov random fields to characterize spatial dependency between functions, and apply them to functional areal and functional point-reference data; and (4) Educate and engage students in the field of high-dimensional data analysis, develop K-12 outreach programs, and mentor minority students.