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

9623884本项目开发并实现了新的统计方法，用于使用计算密集型技术对时间序列的复杂特征进行建模。首先，首席研究员开发了马尔可夫链蒙特卡罗方法在贝叶斯框架中分析多个时间序列的应用。她展示了如何将季节性变化、随机干预、内生变量关系和长期依赖等特征纳入多变量时间序列的贝叶斯线性模型。贝叶斯框架允许合并先验信息，例如关于序列之间协方差结构的知识，并在计算上可行的似然评估结果。所提出的贝叶斯模型可以用来描述在实践中观察到的一些时间序列特征，但受其线性结构的限制。第二个研究领域集中在非贝叶斯设置中复杂非线性时间序列关系的建模方法。理论框架是基于非参数函数和密度估计在回归设置的最新进展。首先，开发了非线性周期行为的测试，并展示了如何对该行为进行建模。其次，利用非线性序列线性组合的非参数检验，探讨了多个非线性序列之间的简化线性关系。简化的关系可用于改进对单个序列的预测。最后，利用多元核密度方法和向量自回归的局部拟合建立了全多元非线性模型。为了促进新方法在实践中的应用，部分工作致力于开发计算上可行的算法。第三个项目领域侧重于教育创新，以改善工程师和应用统计学家之间的统计培训和沟通技巧。这些技能的提高将有助于促进与其他学科线的研究人员的互动。该研究的总体目标是开发可用于更好地理解现实世界过程之间复杂关系的方法。开发了统计工具，例如可用于调查环境序列之间的长期依赖关系，以便评估地球大气的全球变化，或分析经济序列之间的重要非线性关系。为了确保未来的统计从业人员有足够的培训来使用这些方法，教育创新也在发展，旨在提高本科生和研究生的数据分析、计算和沟通技能。通过更好地教育学生将统计学应用于现实世界的问题，他们将更好地利用统计方法来理解复杂的现实世界动态并做出明智的政策决策。

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.