Analysis of Functional Time Series

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In this project the investigator develops new models, methods and associated theory under a general framework of `Functional Time Series Analysis'. Analyzing time series in a functional framework is increasingly practical and is gaining importance rapidly as more and more applications involving such data sets. There are four research projects, in the board directions of functional time series driven by dynamic processes, distributional time series driven by dynamic processes, functional ARMA models, and functional regression models with functional time series errors. They are closely related but with different focuses. The combination of the projects builds a comprehensive framework for functional time series analysis. For each project, statistical properties of the underlying models, statistical inference and predictions under these models, and the theoretical properties of the inference and prediction methods are studied. Several special and important applications are studied.Functional time series analysis can be viewed as a marriage between the traditional time series analysis and the field of functional data analysis of independent functional observations. Time series analysis is mainly interested in the dependent structure of the observations over time, the understanding the dynamic nature of the underlying process and accurate predictions of the future. Modern data collection capability has lead to broader definition of `data' and more and more observations are in the form of functions, images, and distributions. The intersection between time series analysis and functional data analysis has not been systematically explored. In this project, the investigator develops a general framework of functional time series analysis that is amenable to statistical thinking and the analysis of real problems. This project paves the way for developing a completely new research area in statistics. It has broad impact in advancing our capabilities of statistical data analysis. It aims to produce advanced statistical tools for analyzing functional time series that are encountered in many important application fields including economics and finance, environmental studies, medical and neuroscience, ecology and meteorology. The project also actively engages in activities related to education and research training of graduate and undergraduate students, especially attracting minority and women students into the field of statistics and statistical applications.