Collaborative Research: Second Order Inference for High-Dimensional Time Series and Its Applications

在过去十年中，对高维独立数据的分析获得了相当大的关注。该项目涉及高维多时间序列的各种估计和统计推断问题，这是一种普遍类型的数据，在时空统计、生物医学工程、环境科学、金融和信号处理等广泛的实际应用中可见。从大量的时间序列中提取信息是一个重要的研究问题，其中二阶结构在这些应用中起着重要的作用。本课题的研究成果将为高维时间序列的时空协方差和精度矩阵及其相关泛函的估计和推断以及时变图提供理论基础。所有这些问题都联系在一起，以表征具有非线性和非平稳时间相关特征的高维时间序列的二阶性质。我们还将研究考虑时间和空间依赖结构的增强方法。本文的研究结果有助于理解高维相关数据的动态特征。特别是，这些技术适用于生物医学工程问题，如利用功能磁共振成像数据建模大脑连接网络。

During the last decade, analysis of high-dimensional independent data has gained substantial attention. The project involves a variety of estimation and statistical inference problems for high-dimensional multiple time series, a universal type of data seen in a broad spectrum of real applications in spatio-temporal statistics, biomedical engineering, environmental science, finance, and signal processing. As an important research problem, one should extract information from a large number of time series, where the second order structure plays a fundamental role in those applications.Results developed from this project will provide the theoretical foundation for estimating and inference of the space-time covariance and precision matrix, their related functionals, and time-varying graphs of high-dimensional time series. All of the problems are linked together to characterize the second order properties of the high-dimensional time series with the non-linear and non-stationary time dependent features. We will also study enhanced methods that account for the temporal and spatial dependence structures. Results from this research are useful for understanding the dynamic features of high-dimensional dependent data. In particular, the techniques are applicable to biomedical engineering problems such as modeling brain connectivity networks by using fMRI data.