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，都与多个测量的大量数据相关，这些数据通常会随着时间的推移而变化。了解一段时间的进展，找到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重要组成部分。然而，两个变量之间的明显关系可能会出现在数据中，这是由于它们与其他组件共享关联造成的。主要研究者（PI）将开发一个模型，以独立的一维潜在时间序列重新表达多维时间序列。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来帮助为未来的观测找到更准确，更有效的可计算预测公式。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。研究所计划开发免费软件包，以传播研究结果。他们致力于通过研究生培训和参与REU计划来支持年轻研究人员并促进多样性。所开发的框架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。个别潜在的时间序列建模灵活，未指定的谱密度。PI将使用贝叶斯方法研究组成时间序列之间的条件独立结构以及时间序列在时间域上的因果关系。他们将独立的先验通过一个有限的随机序列之前的个人谱密度，并分解为一个稀疏矩阵和一个正交矩阵的产品的线性变换的矩阵，其中前者诱导一个图形结构的组件系列之间的条件独立性。通过这种表示，可以施加期望的平稳性和因果关系结构。通过Whittle似然近似和Hamiltonian蒙特-卡罗方法解耦将允许有效的后验采样。节点时间序列的因果关系将通过残差过程的直接非循环图建模来解决。该公式无缝地解决了混合频率采样的情况下，难以纳入竞争的方法。开发的框架有效地解决了时间和节点的因果关系，分别由舒尔互补和使用有向无环图的表征，允许一个自然的interpretation.This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。

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.