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

现代科学技术中收集到的大规模、复杂的数据对传统的统计方法和理论提出了巨大的挑战。函数数据分析(FDA)以随机函数为单位，对时间、空间、波长等连续分布的数据进行建模，是一个很有前途的研究领域。这样的数据可以被视为随机过程的实现，并且通常可以在许多领域中找到，例如，纵向研究、微阵列实验、医学图像、互联网商业和金融市场。函数数据建模中的一个基本问题源于“维度诅咒”，因为函数对象在概念上被定义为无限维过程。此外，大规模收集的复杂结构的功能性数据也面临着更大的挑战，这自然涉及到最近发生革命性变化的领域，如高维数据分析和压缩传感。因此，

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,