Collaborative Research: Novel modeling and Bayesian analysis of high-dimensional time series

现代生活的每个方面，包括经济和金融、通信和医疗记录，都与几种测量方法的大量数据有关，这些数据通常会随着时间的推移而变化。了解随着时间的推移，发现不同变量之间的内在关系，并预测未来的观察结果是决策和政策制定的基本组成部分。然而，两个变量之间的明显关系可能出现在它们与其他组件共享关联所导致的数据中。主要研究者（pi）将开发一个模型，将多维时间序列重新表示为独立的、一维的、潜在的时间序列。该表示将解释数据随时间的演变以及组成变量中存在的内在关系。它还可以通过跨不同组件和时间提取信息，帮助找到一个更准确、更有效的预测公式，用于未来的观测。该方法的简单性和通用性将使其广泛适用于经济、金融、社会科学、通信、网络、神经成像等不同领域。pi计划开发免费软件包来传播结果。他们致力于支持年轻的研究人员，并通过研究生培训和参与REU计划来促进多样性。开发的框架是基于将观察到的多维时间序列表示为几个独立平稳潜在过程的线性组合。在不确定谱密度的情况下，灵活地对单个潜在时间序列进行建模。pi将使用贝叶斯方法研究组件时间序列之间的条件独立结构和时间序列在时域上的因果关系。他们将通过有限随机序列先验对单个谱密度进行独立先验，并对分解为稀疏矩阵和正交矩阵乘积的线性变换的矩阵进行独立先验，其中稀疏矩阵推导出成分序列之间条件独立的图形结构。通过这种表示，可以施加理想的平稳性和因果关系结构。通过惠特尔似然近似和哈密顿蒙特卡罗方法解耦将允许有效的后验抽样。节点时间序列上的因果关系将通过残差过程的直接无环图建模来解决。该配方无缝地解决了混合频率采样情况，难以纳入竞争方法。开发的框架通过舒尔互补和有向无环图的表征，有效地解决了时间和节点因果关系，从而允许自然解释。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。

Every aspect of modern life including economy and finance, communication, and medical records, is associated with large amounts of data on several measurements, often evolving over time. Understanding the progress over time, finding an intrinsic relationship among different variables, and predicting future observations are essential components of decision and policy-making. However, apparent relations between two variables can appear in data caused by their shared association with other components. The principal investigators (PIs) will develop a model to re-express the multi-dimensional time series in independent, one-dimensional, latent time series. The representation will explain the evolution of the data over time and the intrinsic relations present in the component variables. It can also help find a more accurate, efficiently computable prediction formula for future observations by pulling information across different components and time. The approach's simplicity and generality will make it widely applicable and adaptable to diverse fields in economics, finance, social sciences, communications, networks, neuroimaging, and others. The PIs plan to develop free software packages to disseminate the results. They are committed to supporting young researchers and promoting diversity through graduate student training and involvement in the REU program.The developed framework is based on representing an observed multi-dimensional time series as a linear combination of several independent stationary latent processes. The individual latent time series are modeled flexibly with unspecified spectral densities. The PIs will study the conditional independence structure among component time series and the causality of the time series over the temporal domain using a Bayesian approach. They will put independent priors on individual spectral densities through a finite random series prior, and on the matrix of the linear transformation decomposed as a product of a sparse matrix and an orthogonal matrix, the former of which induces a graphical structure for conditional independence among component series. Through this representation, desirable stationarity and causality structures can be imposed. Decoupling through the Whittle likelihood approximation and Hamiltonian Monte-Carlo methods will allow efficient posterior sampling. The causality over nodal time series will be addressed by a Direct Acyclic Graph modeling of the residual process. The formulation seamlessly addresses a mixed frequency sampling situation, difficult to incorporate into competing methods. The developed framework efficiently addresses both temporal and nodal causality respectively by characterization in terms of the Schur-complementation and using a directed acyclic graph, allowing a natural interpretation.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.