New Inference Methods For Multiway Functional Data and Multilayer Network Data

本项目的重点是为多路功能数据和多层网络数据开发有效的模型。 功能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。 多路功能数据的示例包括在空间和时间上测量的脑成像数据，以及从移动的跟踪应用程序获得的数据，其中在许多天的时间段内记录许多个体的日常活动简档。 尽管应用越来越多，但数据的复杂结构和高维性对统计建模和推理提出了重大挑战。 该项目的第二部分侧重于从多模式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。 在图像处理和时空分析的漫长历史中，许多方法都依赖于可分性，假设协方差可以分解为空间协方差和时间协方差的乘积。 最近的方法重复的功能数据和多路功能数据也调用分离，显式或隐式，以实现有效的降维。 一个新的概念，弱可分性，其中包括协方差可分性作为一种特殊情况，将被引入。 弱可分性的测试将被开发，并将提供几个开放的问题的原则性答案。 在第二部分中，我们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研究从一个新的关系k-均值的角度来看，和理论依据下提供的统计推断框架。

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.