Simultaneous Confidence Regions for Functional Data Analysis: Theory and Methods

本研究项目为功能数据分析（functional data analysis， FDA）中各种功能特征提供同步置信区域，具有渐近理论和指导实际实施的意义。具体而言，将对(1)变系数纵向回归模型中函数数据的均值函数和系数函数构造渐近正确的置信区域；(2)函数数据的协方差函数和函数线性模型中的回归函数。对于(1)中较简单的函数，研究者将采用回归样条和局部多项式方法，以便为稀疏和密集函数数据建立严格的渐近理论。利用布朗运动的部分和强逼近的结果和非平稳高斯过程序列的先进极值理论，得到最大偏差过程的分布性质。对于(2)中更复杂的函数，研究者将提出两步估计，并证明它与一些不可行的函数一样渐近有效。类似物。最大偏差的渐近分布建立在？不可行估计?然后由两步估计器继承。功能数据，也称为曲线数据，由数字记录的曲线或曲面的集合组成，通常带有随机误差。这样的数据在几乎所有的科学学科中都大量存在，包括但不限于气候学、临床研究、流行病学、进化生物学和食品工程/科学。从曲线样本中提取信息的需求，加上现代计算能力的释放，使得功能数据分析（FDA）成为当代统计研究中最活跃的领域之一。多元统计是关于未知的向量和矩阵，而FDA关注未知的曲线和曲面，这是最自然的置信区域。研究者开发的方法填补了目前FDA方法的一个主要空白，该方法缺乏在具有可量化不确定性的整个曲线上得出结论的程序。用Matlab或R等常用软件包编写的代码将免费分发，以便学术界和工业界的从业者实时分析功能数据，并选择自己的显著性水平。完成这个项目关键取决于几个有能力的博士生在研究者的指导下工作。因此，最先进的研究与培养研究生成为未来的研究人员相结合，这与NSF的教育目标是一致的。

This research project provides simultaneous confidence regions for various functional features in functional data analysis (FDA), with asymptotic theory and guide to practical implementation. Specifically, asymptotically correct confidence regions will be constructed for (1) the mean function of functional data and the coefficient function in varying coefficient longitudinal regression model; and (2) the covariance function of functional data and the regression function in functional linear model. For the simpler functions in (1), the investigator will employ both regression spline and local polynomial methods in order to establish rigorous asymptotic theory for both sparse and dense function data. Results on partial sum strong approximation by Brownian motions and advanced extreme value theory for sequences of non-stationary Gaussian processes will be applied to obtain distributional properties of the maximal deviation processes. For the more complicated functions in (2), the investigator will propose two-step estimators and show that it is asymptotically as efficient as some ?infeasible? analogs. Asymptotic distributions for maximal deviations are established for the ?infeasible estimators? which are then inherited by the two-step estimators.Functional data, also known as curve data, consist of collections of digitally recorded curves or surfaces, often with random errors. Such data abound in virtually all scientific disciplines, including but not limited to, climatology, clinical studies, epidemiology, evolutionary biology and food engineering/science. The need to draw information out of a sample of curves, coupled with the unleashing of modern computing power, has made functional data analysis (FDA) one of the most active areas of contemporary statistics research. While multivariate statistics is about unknown vectors and matrices, FDA concerns unknown curves and surfaces, which is most naturally done with confidence regions. The methods developed by the investigator fill a major gap in the current FDA methodology, which lacks procedures to make conclusions on an entire curve with quantifiable uncertainty. Codes written in common software packages such as Matlab or R will be freely distributed so practitioners from academia and industry for analyzing functional data in real time, with own chosen significance levels. Completing this project depends crucially on several capable Ph. D. students working under the investigator?s supervision, so state-of-the-art research is integrated with the training of graduate students as future researchers, consistent with NSF's education goal.