Adaptive Regression for Dependent Data by Combining Different Procedures

这一建议涉及随机误差相依时的自适应回归的研究和教育。对于基于不同假设的非参数回归，已经（并将）提出了许多程序。在应用程序中，用户经常面临的一个困难是为数据处理选择最佳方法。特别是对于高维函数估计，为了克服维数的诅咒，根据目标函数的不同表征，提出了各种简约模型，如投影寻踪、CART、神经网络、加性模型、MARS等。本研究的主要兴趣是通过组合候选过程的集合来构建自适应估计器。组合过程的目标是在不知道最佳原始过程是哪一个的情况下自动执行得与最佳原始过程一样好（或几乎一样好）。我们假定随机误差通常是相关的，包括短期和长期的情况。本文将研究依赖对适应能力的影响。预计将提出理论证明和计算可行的算法，以结合针对回归函数的各种特征和随机误差的不同依赖结构的回归过程。函数估计是一种重要的统计工具，它试图准确地理解基于数据的变量之间的函数关系，它在许多学科中都有应用，可以成功地解决科学问题。在现实中，观察总是受到来自不同来源的随机噪声（误差）的影响。当随机误差相互依赖时，这种依赖关系可能会掩盖兴趣的函数关系。长距离依赖是指即使误差发生在彼此相距很远的时间或地点，误差仍然高度相关的情况。众所周知，这种长期的依赖关系使得对目标函数的估计更加困难。在应用中，误差之间的依赖程度通常是未知的，这使得函数估计问题更加困难。在本提案中，我们打算开发自适应处理误差之间不同程度的依赖的方法，以便在不知道误差的依赖结构的情况下，可以最优地估计感兴趣的函数。在几门统计学课程中，将向不同层次的学生介绍长期相关数据的研究成果和相关工作。将与爱荷华州立大学的几位教授和他们在大气科学、电气工程、农学和可能的其他领域的学生合作，以适当地解决长期依赖现象，这些现象经常遇到，并且已知会在现有统计方法的数据分析中造成问题。

This proposal concerns research and education on adaptive regressionwhen the random errors are dependent. Many procedures have been (and will be) proposed for nonparametric regression based ondifferent assumptions. In applications, a difficulty a user oftenfaces is the choice of the best method for the data athand. This is specially the case for high-dimensional functionestimation, where to overcome the curse of dimensionality, various parsimonious models such as projection pursuit, CART, neural nets, additive models, MARS, etc. are proposed according to different characterizations of the target function. A main interest in this research is to construct adaptive estimators by combining acollection of candidate procedures. The goal for the combinedprocedure is to perform automatically as well as (or nearly as wellas) the best original procedure without knowing which one it is. The random errors will be assumed to be generally dependent,including both short- and long-range cases. The effects ofdependence on adaptation capability will be studied. It isanticipated that theoretically proven and computationally feasible algorithms will be proposed to combine regression procedurestargeted at various characteristics of the regression function and different dependence structures for the random errors.Function estimation is an important statistical tool that tries tounderstand accurately the functional relationships between variables based on data and it has applications in many disciplines for successfully addressing scientific questions. In reality, observations are alwayssubject to random noise (error) from different sources. When therandom errors are dependent on each other, the dependence maydisguise the functional relationship of interest. Long-rangedependence refers to a situation where the errors are still highlycorrelated even when they occur at times or locations that are faraway from each other. It is known that such a long-range dependence makesthe estimation of the target function much harder. In applications,the degree of dependence between the errors is usually unknown,which makes the function estimation problem even harder. In thisproposal, we intend to develop methods that adaptively handledifferent degrees of dependence among the errors so that thefunction of interest can be estimated optimally without knowing thedependence structure of the errors. The research results and relatedwork by others on long-range dependent data will be brought tostudents at various levels in several statisticscourses. Collaborations will be conducted with several professors atIowa State University and their students in atmospheric science, electrical engineering, agronomy and possibly other fields toappropriately address long-range dependence phenomena, which havebeen encountered often and known to cause problems in data analysis with the existing statistical methods.