New Inference Methods For Multiway Functional Data and Multilayer Network Data

本项目致力于开发多路功能数据和多层网络数据的高效模型。功能性数据是指在连续体上记录的数据，例如许多儿童的生长曲线。多功能数据的例子包括在空间和时间上测量的脑成像数据，以及从移动跟踪应用程序获得的数据，这些应用程序记录了许多个人在许多天内的日常活动概况。尽管应用越来越多，但数据的复杂结构和高维性给统计建模和推理带来了重大挑战。该项目的第二部分侧重于从多模态和多任务大脑连接研究中获得的多层网络数据。将为这些网络结构制定严格的统计框架。在漫长的图像处理和时空分析历史中，许多方法都依赖于可分离性，即假设协方差可以被分解为空间协方差和时间协方差的乘积。最近针对重复功能数据和多路功能数据的方法也会显式或隐式地调用可分离性，以实现有效的降维。本文将引入弱可分性的一个新概念，其中协方差可分性是一个特例。将开发弱可分性的测试，并对几个悬而未决的问题提供原则性的答案。在项目的第二部分，将介绍具有社区结构的多层网络数据的生成模型。最小二乘估计的成员将从一个新的关系k-means的角度进行研究，并在统计推断框架下提供理论依据。

This project focuses on developing efficient models for multi-way functional data and multi-layer network data. Functional data refers to data recorded over a continuum, such as growth curves for many children. Examples of multi-way functional data include brain-imaging data measured over space and time, and data obtained from mobile tracking apps where daily activity profiles are recorded for a number of individuals over a period of many days. Despite the growing number of applications, the complex structure and high-dimensionality of the data pose significant challenges for statistical modeling and inference. The second part of the project focuses on multi-layer network data obtained from multi-modal and multi-task brain connectivity studies. A rigorous statistical framework for these network structures will be developed. In the long history of image processing and spatial-temporal analysis, many methods have relied on separability, the assumption that the covariance can be factorized as a product of a spatial covariance and a temporal covariance. Recent approaches for repeated functional data and multi-way functional data also invoke separability, either explicitly or implicitly, to achieve efficient dimension reduction. A new notion of weak separability, that includes covariance separability as a special case, will be introduced. Tests of weak separability will be developed, and principled answers to several open questions will be provided. In the second part of the project, generative models of multi-layer network data with community structures will be introduced. Least squares estimation of memberships will be studied from a novel relational k-means perspective, and theoretical justification provided under a statistical inference framework.