Talk Abstract: 

Functional data analysis is an important research field in statistics which treats  data as random functions drawn from some infinite-dimensional functional space, and functional principal component analysis (FPCA) based on eigen-decomposition plays a central role for data reduction and representation. After nearly three decades of research, there remains a key problem unsolved, namely, the perturbation analysis of covariance operator for diverging number of eigencomponents obtained from noisy and discretely observed data. This is fundamental for studying models and methods based on FPCA, while there has not been substantial progress since the result obtained by Hall et al. (2006) for a fixed number of eigenfunction estimates. In this work, we establish a unified theory for this problem, obtaining upper bounds for eigenfunctions with diverging indices in both the L2 and supremum norms, and deriving the asymptotic distributions of eigenvalues for a wide range of sampling schemes. Our results provide insight into the phenomenon when the L2 bound of eigenfunction estimates with diverging indices is minimax optimal as if the curves are fully observed, and reveal the transition of convergence rates from nonparametric to parametric regimes in connection to sparse or dense sampling.  The technical arguments in this work are useful for handling the perturbation series with noisy and discretely observed data and can be applied in models or those involving inverse problems based on FPCA as regularization, such as functional linear regression.

Bio:
姚方，国家高层次人才，北京大学讲席教授，北大统计科学中心主任、概率统计系主任，数理统计学会与美国统计学会会士。2000年本科毕业于中国科学技术大学，2003获得加利福尼亚大学戴维斯分校统计学博士学位，曾任职于多伦多大学统计科学系长聘正教授。获加拿大CRM-SSC奖、第六届科学探索奖（数学物理学）。至今担任9个国际统计学核心期刊主编或编委，包括《加拿大统计学期刊》主编、顶级期刊《北美统计学会会刊》和 《统计年刊》的编委。