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

在过去的十年中，对高维独立数据的分析得到了极大的关注。该项目涉及高维多时间序列的各种估计和统计推断问题，高维多时间序列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、金融和信号处理的广泛实际应用中可以看到。作为一个重要的研究问题，需要从大量的时间序列中提取信息，而二阶结构在这些应用中起着基础性的作用，其结果将为估计和推断高维时间序列的时空协方差和精度矩阵及其相关泛函和时变图提供理论基础。将这些问题联系在一起，刻画了具有非线性和非平稳时变特征的高维时间序列的二阶性质。我们还将研究解释时间和空间依赖结构的增强方法。本研究的结果对于理解高维相依数据的动态特征是有用的。特别是，这些技术适用于生物医学工程问题，如使用fMRI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.