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.
