Dimension Reduction, Model Selection and Classification in Functional Data Analysis.

函数数据分析的目的是对数据集进行建模和分析，其中数据是随机函数，例如曲线或高维图像。由于现代数据采集方法的快速发展，这样的数据集在许多生物、医疗和工业应用中变得越来越普遍。泛函数据被看作是泛函空间中的无限维向量，通常是在离散的点上观察并有误差地测量。由于函数式数据的无穷维性，降维对于这些数据的可视化、建模和推理是必不可少的。在拟议的项目中，研究人员将研究基于样条逼近的新的、计算高效的函数数据降维方法，并使用渐近理论来开发新的统计工具，用于模型选择和推理。研究人员还将研究功能数据中的分类问题，将所提出的降维技术与现代机器学习方法相结合。拟议的研究受到结肠癌发生实验、高血压研究、艾滋病临床试验和功能磁共振成像实验的数据的推动。拟议的项目将通过促进这些科学领域的知识而造福社会。为了实现研究成果的更广泛传播，调查员将向所有科学研究人员提供免费和用户友好的软件。研究人员研究所将开设一门关于函数数据分析的新课程。新课程旨在培养学生分析真实和创新数据集的能力，并帮助他们对现代统计方法和理论有更深的理解。

Functional data analysis aims to model and analyze data sets where a datum is a random function, e.g. a curve or a high dimensional image. Due to the fast growth of modern data collection methods, such data sets become more and more prevalent in many biological, medical and industrial applications. Functional data are viewed as infinite dimensional vectors in a functional space, and are usually observed on discrete points and measured with error. Due to the infinite dimensional nature of functional data, dimension reduction is essential for visualizing, modeling and making inference on these data. In the proposed project, the investigator will study new, computationally efficient dimension reduction methods for functional data based on spline approximations, and use asymptotic theory to develop new statistical devices for model selection and inference. The investigator will also study classification problems in functional data, by combining the proposed dimension reduction techniques with modern machine learning methods.The proposed research is motivated by data from colon carcinogenesis experiments, hypertension studies, AIDS clinical trials and functional magnetic resonance imaging experiments. The proposed project will benefit the society by advancing knowledge in these scientific fields. To achieve broader dissemination of the research results, the investigator will provide free and user friendly software to all scientific researchers. A new course on functional data analysis will be developed in the investigator's institute. The new course aims to nurture the ability of students to analyze real and innovative data sets and help them gain deeper understanding of modern statistical methods and theory.