Large-scale and Complex Functional Data: Foundation, Regression and Inference

现代科学技术收集的大量复杂数据对传统的统计方法和理论提出了巨大的挑战。函数型数据分析（FDA）是一个新兴的研究领域，它以随机函数为单位，对分布在时间、空间、波长等连续体上的数据进行建模，这些数据可以看作是随机过程的实现，在许多领域都很常见，例如：纵向研究、微阵列实验、医学图像、互联网商务和金融市场。函数数据建模中的一个基本问题来自“维数灾难”，因为函数对象在概念上被框定为无限维过程。此外，对于大规模收集的具有复杂结构的功能数据，已经出现了更大的挑战，这自然涉及最近革命性的领域，例如高维数据分析和压缩传感。因此，我们认为，

Large-scale and complex data collected in modern science and technology impose tremendous challenges for traditional statistical methods and theory. Functional data analysis (FDA) has emerged as a promising field that employs random functions as units and is designed to model data distributed over continua such as time, space, wavelength and so on. Such data may be viewed as realizations of random processes and are commonly found in many fields, e.g., longitudinal studies, microarray experiments, medical images, internet commerce and financial markets. A fundamental issue in modelling functional data arises from the “curse of dimensionality”, as functional objects are conceptually framed as infinite-dimensional processes. Further, greater challenges have emerged for functional data that are collected on a large scale and with complex structures, which naturally relates to the recently revolutionized fields, such as high-dimensional data analysis and compressed sensing. Therefore,