Statistical Models, Inference, and Computation for Multidimensional Time Series Data

现在，跨多个（通常是多个）源收集数据是很常见的。例子包括功能性磁共振成像（fMRI）产生的跨越多个大脑区域的时间信号，从浮标或卫星收集的跨越多个空间位置的海浪高度序列，以及政府机构和其他各方随时间收集的多个经济指标（GPD、失业率等）。可用的技术通常要么忽略来自多个源的此类高维数据的时间依赖性，要么不适用于源数量很大的情况。该研究项目旨在开发新的统计建模工具，能够充分捕捉这些数据的时间特征以及它们在多个来源之间的依赖关系。这些工具有可能大大加强从这些数据中获得的知识。例如，对于fMRI数据，适当地考虑时间依赖性和大量的大脑区域可能有助于更好地区分各种临床类别（ADHD，自闭症等）。了解波高数据的时空相关性可以更好地预测跨洋风暴活动，并有望通过改进对多种经济指标的分析进一步了解经济活动。该项目旨在开发一种综合方法来分析大型多维时间序列数据，包括其统计模型、估计、计算（算法）和实践。研究涵盖了短期和长期相关的多维时间序列。对于短距离相关序列，重点研究了稀疏向量自回归及其相关模型、降维、变化点检测和一些非线性模型。要解决的问题涉及正则化技术、统计显著性、显示周期性变化的模型和其他问题。提出了多维远程依赖作为补充向量自回归和相关的短程依赖序列的重要类别，从而集合了现代时间序列分析中常用的两类模型。目标是开发一种新的方法，用于多维远程相关序列与所谓的一般相位，它控制了多维时间序列的对称性，在线性和非线性设置。开发的方法应该在广泛的领域有用，包括神经科学、海洋学和环境科学、地球物理学、经济学和金融学等。

It is now commonplace for data to be collected over time across multiple (often many) sources. Examples include the time signals across multiple brain regions arising from fMRI, ocean wave height series across multiple spatial locations collected from buoys or satellites, and the multiple economic indicators (GPD, unemployment, and so on) gathered over time by government agencies and other parties. Available techniques often either neglect temporal dependencies for such high-dimensional data arising from multiple sources or do not apply to situations when the number of sources is large. This research project aims to develop novel statistical modeling tools that can capture adequately both the temporal features of such data and also their dependencies across multiple sources. Such tools have the potential to greatly enhance knowledge gained from such data. With fMRI data, for example, proper accounting for temporal dependence and large number of brain regions may facilitate better distinction among various clinical categories (ADHD, autism, etc.). Understanding the temporal and spatial dependencies in wave height data can lead to better predictions of storm activity across the oceans, and further insight into economic activity is expected from improved analysis of multiple economic indicators.The project aims at developing an integrated approach to analyzing large multidimensional time series data, including their statistical models, estimation, computation (algorithms), and practice. The research covers both short-range and long-range dependent multidimensional time series. For short-range dependent series, the focus is on sparse vector autoregressive and related models, dimension reduction, change point detection and some nonlinear models. The problems to be addressed concern regularization techniques, statistical significance, models exhibiting cyclical variations and other issues. Multidimensional long-range dependence is suggested as the important class complementing vector autoregressive and related short-range dependent series, thus gathering the two general classes of models employed in modern time series analysis. The goal is to develop a new methodology for multidimensional long-range dependent series with the so-called general phase, which controls the symmetry properties of multidimensional time series, in both linear and nonlinear settings. The developed methods should be useful across a wide range of areas, including neuroscience, oceanography and environmental sciences, geophysics, economics and finance, and others.