Random Matrix Approach to High-Dimensional Time Series

高维时间序列在经济学、大气和环境科学、基因组学、实验化学、无线通信和许多其他学科中自然出现。在对具有大量特征的数据进行统计分析方面的最新发展表明，为定性和定量摘要制定新的范例十分重要。对一些广泛使用的描述性统计行为的探索导致了新现象的发现，而这些理论研究反过来又促进了面向分析高维数据的复杂统计程序的发展。这种追求得益于各种学科知识的融合，如概率论、优化、几何和计算机科学。随机矩阵理论对上述理论的发展作出了重大贡献。本计划的主要目标是将随机矩阵的观点引入多元时间序列的研究，并利用由此产生的理论发展来建立分析大型和复杂时间序列数据的统计方法。预计该项目将在几个方面对科学界和其他领域产生影响。这项研究有可能直接应用于计量经济学和金融学。预计研究结果还将影响气候研究、环境科学和通信理论中的模型建立和数据分析技术。这些发现将使各领域的从业人员更广泛地利用现代统计工具和概念来处理大量临时观测到的数据。在这个项目中工作的学生将通过数学，计算和数据分析技能的合并精通众多学科。这个项目的培训部分包括通过短期课程和定向的个人和小组研究，使本科生和研究生接触到现代统计和数学技术和研究问题。这将有助于他们顺利过渡到先进的学术课程和行业工作，专门从事尖端技术。在这个项目中，随机矩阵理论的技术将被扩展到分析高维时间序列的样本协方差和对称自协方差矩阵和谱密度矩阵的行为。这些统计数据是对时间相关数据进行建模和预测的主要构建块。这个提议的一个主要动机是从谱特征，如样本协方差和自协方差矩阵的特征值的经验分布，推断大维时间序列中的依赖性质。本文提出的理论将随机矩阵理论的前沿扩展到具有特殊结构的相关数据领域。该项目的另一个目的是开发高维时间序列的统计估计和预测工具，并通过混合数学和计算技术来分析这些程序的性能。这项研究也将扩大统计学、应用数学、计量经济学和工程学等学科之间的界面范围。

High-dimensional time series arise naturally in economics, atmospheric and environmental science, genomics, experimental chemistry, wireless communications, and a multitude of other disciplines. Recent developments in the statistical analysis of data with large number of features have demonstrated the importance for developing new paradigms for qualitative as well as quantitative summaries. Exploration of the behavior of some widely used descriptive statistics has resulted in the discovery of new phenomena, and these theoretical investigations in turn have contributed to the development of sophisticated statistical procedures geared towards analyzing high-dimensional data. This pursuit benefited from the confluence of knowledge from various disciplines such as probability theory, optimization, geometry, and computer science. Random matrix theory has contributed significantly to the aforementioned theoretical developments. The primary goal of this project is to introduce the random matrix perspective to the study of multivariate time series, and utilize the resulting theoretical developments to build statistical methodologies for analyzing large and complex time series data. There are several ways in which this project is expected to have an impact on the scientific community and beyond. This research has potential direct applications to econometrics and finance. The research findings are also expected to influence model building and data analysis techniques in climate studies, environmental science and communications theory. The findings will give wider access to practitioners in various fields to modern statistical tools and concepts for dealing with large volumes of temporally observed data. Students working in this project will be well-versed in a multitude of disciplines through the merger of mathematical, computational and data analytic skills. The training component of this project involves giving exposure to undergraduate and graduate students to modern statistical and mathematical techniques and research problems through short courses and directed individual and group studies. This will facilitate their smooth transition into advanced academic programs and industry jobs specializing in cutting-edge technologies. In this project, techniques of random matrix theory will be extended to analyze the behavior of sample covariance and symmetrized auto-covariance matrices and spectral density matrices for high-dimensional time series. These statistics are the primary building blocks for modeling and prediction of time-dependent data. A major motivation of this proposal is to the infer nature of dependence in large dimensional time series from the spectral characteristics, such as the empirical distribution of eigenvalues, of the sample covariance and auto-covariance matrices. The theory developed in this proposal extends the frontier of random matrix theory to the domain of dependent data with special structures. Another aim of the project is to develop tools for statistical estimation and prediction for high-dimensional time series and to analyze the performance of these procedures by blending mathematical and computational techniques. This research will also broaden the scope of interface among disciplines such as statistics, applied mathematics, econometrics and engineering.