Nonlinear dynamic factor models and dynamic factor driven functional time series models

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Time series analysis comprises methods that allow for the discovery of dependent and dynamic structures in observations taken over time and that provide accurate predictions of the future. Time series data occur in many important application fields including economics, finance, environmental studies, neuroscience, ecology, and meteorology. In this age of Big Data, with advanced data collection capability, researchers routinely encounter large panels of time series data. How to effectively analyze the common dynamic feature of these time series, how to discover their interconnection, how to make accurate predictions, and how to assess the overall risk are important questions. The project aims to answer these questions by investigating statistical methods that extract common features from a large number of time series. The project will also describe methods to analyze data in the form of curves or images observed over time. This project provides advanced data analysis tools for solving many real world problems, and paves the way for developing a new research area in statistics. The project includes activities related to education and research training of graduate and undergraduate students, and plans for recruiting women and underrepresented minority students into the field of statistics. Results will be disseminated through conference presentations, publications, and distribution of software. The project focuses on two closely related topics: (i) developing a class of nonlinear dynamic factor models, along with associated statistical inference procedures and derivation of the theoretical properties of the proposed estimators; and (ii) developing an efficient nonparametric inference procedure for functional time series based on dimension reduction using dynamic factor models. Modeling and analyzing high-dimensional time series requires efficient dimensional reduction tools, with factor models being one of the most commonly used techniques. The project extends standard linear dynamic factor models to nonlinear models in order to capture the nonlinearity often encountered in practice. When functional or distributional observations are observed over time and exhibit dynamic behaviors, time series models in the functional space become a necessary and useful tool for analyzing such data, as well as making predictions of the future. New nonparametric approaches to modeling functional time series utilizing factor models as a dimension reduction tool will be developed. The two research topics are rapidly gaining importance as more and more applications involve such types of data. The combination of these two closely related projects builds a comprehensive framework for modern time series analysis. For each project, statistical properties of the underlying models, statistical inference and predictions for these models, and theoretical properties of the inferential and prediction methods will be studied.