Collaborative Research: Learning Graphical Models for Nonstationary Time Series

在生物科学和社会科学中，许多核心问题都要求人们了解复杂系统中的相互作用如何随着时间的推移而演变。例如，在神经科学中，监测不同大脑区域之间随时间变化的连接对于研究神经退行性疾病的进展非常重要。另一个例子是，相互关联的金融市场的风险管理和监测往往需要了解不同公司之间的联系如何随着时间的推移而演变。这些例子表明，需要开发严格的和可扩展的统计方法，能够从大规模复杂的时间序列数据中学习连通性的演变。这些方法可以帮助洞察复杂系统的工作，并指导数据驱动的政策制定。虽然图形模型（GM）为数据驱动的网络架构发现提供了一个强大的框架，但该领域现有的统计研究主要集中在从平稳时间序列建模时不变的连接。该项目将为称为NonStGM的非平稳图形模型框架开发估计和推理方法。该框架以傅立叶域中的稀疏算子的形式捕获多变量系统中的非平稳动态，其结构又可以使用正则化回归方法从数据中估计。重点将给出两类结构非平稳性，这是在许多应用中普遍存在的：（a）本地平稳性，允许突然变化和平滑演变的时间动态，和（B）周期平稳性。从大规模时间序列数据中学习的非StGM结构将用于构建具有时变向量自回归（VAR）模型的有向图。该项目中开发的算法将通过广泛的数值实验和真实的脑电波（EEG）数据集进行验证。所有产品都将以开放源码软件包的形式公开提供。这些产品有望帮助临床研究人员了解神经系统疾病患者大脑中的连接体异常。研究成果将被纳入研究生课程的教育模块。该项目将提供大量的机会，培养研究生在大规模时间序列建模的专题研究领域，并将积极致力于提高统计科学的多样性和包容性。该奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

In the biological and social sciences, many central questions require one to understand how interactions within a complex system evolve over time. For example, in neurosciences, monitoring time-varying connections among different brain regions is important for studying the progression of neurodegenerative diseases. As another example, risk management and monitoring in interconnected financial markets often require learning how the linkages among different firms evolve over time. These examples demonstrate the need to develop rigorous and scalable statistical methods that are able to learn the evolution of connectivity from large-scale complex time series data. Such methods can help offer insights into the working of a complex system and guide data-driven policy making.While graphical models (GM) offer a powerful framework for data-driven discovery of network architecture, existing statistical research in this area has focused primarily on modeling time-invariant connections from stationary time series. This project will develop estimation and inference methods for a nonstationary graphical model framework called NonStGM. This framework captures nonstationary dynamics in a multivariate system in the form of a sparse operator in the Fourier domain, whose structure can in turn be estimated from data using regularized regression methods. Key emphasis will be given on two classes of structured nonstationarity which are prevalent in many applications: (a) local stationarity that allows both abrupt changes and smooth evolution of the temporal dynamics, and (b) periodic stationarity. NonStGM structures learned from large-scale time series data will be used to build directed graphs with time-varying vector autoregressive (VAR) models. Algorithms developed in this project will be validated with extensive numerical experiments and real electroencephalogram (EEG) data sets. All products will be made publicly available in the form of open-source software packages. These products are expected to aid clinical researchers, amongst others, in their understanding of connectome abnormalities in the brains of patients suffering from neurological disorders. Research outcomes will be integrated into educational modules of graduate level courses. The project will provide numerous opportunities to train graduate students in a topical research area of large-scale time series modeling and will actively focus on enhancing diversity and inclusion in statistical sciences.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.