Mathematical Sciences: Bayesian and Nonparametric Methods for Time Series Analysis with Environmental & Economic Applications

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9623884 This project develops and implements new statistical methods for modeling complicated features of time series using computationally computationally intensive techniques. First the Principal Investigator developes applications of Markov Chain Monte Carlo methods to analyze multiple time series in a Bayesian framework. She shows how to incorporate features such as seasonal variation, random interventions, endogenous variable relationships, and long-range dependence in a Bayesian linear model for multivariate time series. The Bayesian framework allows for incorporation of prior information, such as knowledge about the covariance structure between series, and results in computationally feasible likelihood evaluations. The Bayesian model proposed can be used to describe some of the time series features observed in practice, but is restricted by its linear structure. The second area of research focuses on methods for modeling complicated nonlinear time series relationships in a non-Bayesian setup. The theoretical framework is based on recent advances in nonparametric function and density estimation in the regression setting. First, a test is developed for nonlinear periodic behavior and show how to model that behavior. Next simplified linear relationships among multiple nonlinear series using a nonparametric test applied to linear combinations of nonlinear series are explored. The simplified relationships can be used to improve forecasts of the individual series. Finally, full multivariate nonlinear models are developed using multivariate kernel density methods and local fitting of vector autoregressions. To help promote use of the new methods in practice, part of the effort is devoted to developing algorithms that are computationally feasible. The third project area focuses on educational innovations to improve statistical training and communication skills among engineers and applied statisticians. Improvement of these skills will help foster interactions with investigators in other discip lines. The overall goal of the research is to develop methods that can be used to better understand complex relationships among real-world processes. Statistical tools are developed that can be used, for example, to investigate long-range dependent relationships between environmental series in order to assess global changes in the earth's atmosphere, or to analyze important nonlinear relationships among economic series. To help insure that future statistics practitioners have sufficient training to use such methods, educational innovations are developed, in parallel, aimed at improving data analysis, computing, and communication skills at both the undergraduate and graduate levels. By better educating students in the application of statistics to real-world problems, they will be better equipped to use statistical methods for understanding complex real-world dynamics and making intelligent policy decisions.