EAGER: Learning Graphical Models of High-Dimensional Time Series

无向图模型越来越多地用于探索或利用多元数据下不同随机变量之间的依赖结构，代表复杂系统。图形模型是分析多变量数据的一个重要而有用的工具。图模型是一种统计模型，其中随机变量和它们之间的条件依赖关系是通过图来指定的。图形模型最初是为具有多个独立实现的随机向量（独立且相同分布的时间序列）开发的。这些模型已被广泛研究，并被发现在生物调节网络、脑功能网络和社会网络等各种应用中都很有用。它们也被证明对聚类、半监督学习和分类任务很有用。时间相关数据（时间序列）的图形化建模是最近才出现的。依赖数据的时间序列图形模型已被应用于重症监护监测、金融时间序列、空气污染数据和功能磁共振成像数据的分析，以提供对不同大脑区域功能连接的见解。现有的关于相关时间序列的研究几乎都局限于变量数量远小于数据样本量的低维序列。为了处理变量数量超过或与样本量相当的高维时间序列，（几乎总是）假设该序列在目标函数的选择、算法设计和分析方面是独立的、同分布的，对于合成数据和真实数据都是如此。本项目旨在通过关注高维相关时间序列的图形建模方法来填补这一空白。该项目还将为研究生提供培训和研究经验。本研究探讨了高维环境下实值相关多元时间序列图形化建模的新颖、创新、通用的统计信号处理方法。重点是频域方法，而不需要对底层时间序列进行详细的参数化建模，以捕获时域中的任何依赖性。频域公式导致考虑适当的高斯随机向量的复值高斯图形模型，这是一个很少受到关注的主题。本研究的核心内容是：(1)设计、分析和优化惩罚对数似然函数以拟合图形模型。(2)分析所得解的理论性质（如一致性和稀疏性）。(3)应用于综合和实际数据，评估所考虑的方法的有效性和计算效率。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。

Undirected graphical models have been increasingly used for exploring or exploiting dependency structures among different random variables underlying multivariate data, representing complex systems. Graphical models are an important and useful tool for analyzing multivariate data. A graphical model is a statistical model where random variables and the conditional dependencies between them are specified via a graph. Graphical models were originally developed for random vectors with multiple independent realizations (independent and identically distributed time series). Such models have been extensively studied, and found to be useful in a wide variety of applications such as biological regulatory networks, functional brain networks, and social networks. They have also proved to be useful for clustering, semi-supervised learning and classification tasks. Graphical modeling of time-dependent data (time series) is more recent. Time series graphical models of dependent data have been applied to intensive care monitoring, financial time series, air pollution data, and analysis of functional magnetic resonance imaging data to provide insights into the functional connectivity of different brain regions. Almost all existing works on dependent time series are limited to low-dimensional series where number of variables is much smaller than the data sample size. To address high-dimensional time series where number of variables exceed, or are comparable to, the sample size, it is (almost always) assumed that the series is independent and identically distributed in choice of objective function, and algorithm design and analysis, for both synthetic and real data. This project aims to fill this gap by focusing on methods for graphical modeling of high-dimensional dependent time series. The project will also provide training and research experiences for graduate students.Novel, innovative, general statistical signal processing approaches to graphical modeling of real-valued dependent multivariate time series in high-dimensional settings are investigated in this research. An emphasis is on frequency-domain approaches without requiring detailed parametric modeling of the underlying time series to capture any dependencies in the time domain. Frequency-domain formulation leads to consideration of complex-valued Gaussian graphical models for proper Gaussian random vectors, a topic that has received scant attention. The following thrusts form the core of this research: (1) Design, analysis and optimization of penalized log-likelihood functions to fit graphical models. (2) Analysis of theoretical properties (such as consistency and sparsistency) of the obtained solutions. (3) Application to synthetic and real data to evaluate the efficacy and computational efficiency of the considered approaches.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.