"Collaborative Research: Regression Problems in Functional Data Analysis"

现代数据收集方法现在经常返回的观察结果可以被视为数字化记录或随机函数抽样的结果。这个项目研究的回归问题，其中的响应是标量，但一些预测是功能性的。总体目标是基于部分观察到的和受错误污染的功能数据获得对模型推理的理解。将区分通常从图像中获得的密集功能数据和通常从纵向研究中获得的稀疏功能数据。具体主题包括考虑(i)使用惩罚似然方法的密集泛函数据的泛函广义线性模型，（ii）基于切片逆回归和切片平均方差估计的降维方法，以及（iii）使用近似准似然方法的稀疏泛函数据的泛函广义线性模型。在考虑这些问题时，将提出新的方法，并证明渐近理论来验证这些方法。稀疏泛函广义线性模型将在纵向生活方式剖面和终点健康结果之间的联合建模框架中考虑。这涉及到一种新的变量误差问题的研究，有望扩展纵向数据建模的范围。当前统计研究的一个重要焦点是所谓的高维数据分析。事实上，高维数据是生活中的一个事实。我们对计算机上更大存储设备的需求日益增长，证明了这一点。粗略地说，函数数据是高维数据，可以用光滑的曲线或函数来近似。在科学调查中，这类数据非常丰富，如何有效地分析这类数据至关重要。PI将调查将从根本上有助于功能数据分析实践的方法。该研究的直接应用领域包括图像分析、生物信息学和医学。基于这项研究的功能数据分析的研究性课程将在乔治亚大学和密歇根大学开设。

Modern data collection methods are now frequently returning observations that could be viewed as the results of digitized recording or sampling from random functions. This project investigates regression problems for which the response is scalar but some of the predictors are functional. The general goal is to gain understanding on the inference of the models based on partially observed and error-contaminated functional data. Distinctions will be made between dense functional data, usually obtained from images, and sparse functional data, usually obtained from longitudinal studies. The specific topics include the consideration of (i) a functional generalized linear model for dense functional data using a penalized likelihood approach, (ii) dimension reduction methodologies based on sliced inverse regression and sliced average variance estimation, and (iii) a functional generalized linear model for sparse functional data using an approximated quasi-likelihood approach. New approaches will be proposed in the consideration of these problems, and asymptotic theories will be proved to validate the approaches. The sparse functional generalized linear model will be considered in a framework of joint modeling between a longitudinal life style profile and an endpoint health outcome. This involves the study of a new type of error-in-variable problem, which is expected to extend the horizon of longitudinal-data modeling.An important current focal point of statistical research is the so-called high-dimensional data analysis. Indeed, high-dimensional data are a fact of life. This is evidenced by our increasing need for larger storage devices on our computers. Roughly speaking, functional data are high-dimensional data which can be approximated by smooth curves or functions. Such data are abundant in scientific investigations, and it is of crucial importance to be able to effectively analyze such data. The PI will investigate approaches that will fundamentally contribute to the practice of functional data analysis. Direct applications of the research can be found in areas including image analysis, bioinformatics, and medicine. Research-level classes on functional data analysis based on this research will be offered at both University of Georgia and University of Michigan.