Abstract: In this work we develop a new foundational framework for analyzing Riemannian functional data, including intrinsic Riemannian functional principal component analysis (iRFPCA) and intrinsic Riemannian functional linear regression (iRFLR). The key concept in our development is a novel tensor Hilbert space along a curve on the manifold, based on which Karhunen-Loeve expansion for a Riemannian random process is established for the first time. This framework also features a proper comparison of objects from different tensor Hilbert spaces, which paves the way for asymptotic analysis in Riemannian functional data analysis. Built upon  intrinsic geometric concepts such as vector field, Levi-Civita connection and parallel transport on Riemannian manifolds, the proposed framework embraces full generality of applications and proper handle of intrinsic geometric concepts. We then provide estimation procedures for iRFPCA and iRFLR that are distinct from their traditional and/or extrinsic counterparts, and investigate their asymptotic properties within the intrinsic geometry. Numerical performance is illustrated by simulated and real examples.

个人简介：

姚方，北京大学讲席教授（ 数学科学学院概率统计系，统计科学中心），多伦多大学统计科学系教授，中组部“计划”入选专家，数理统计学会（IMS）Fellow，美国统计学会（ASA）Fellow，国际统计学会（ISI）Elected Member。

2000年本科毕业于中国科技大学统计专业，2003获得加利福尼亚大学戴维斯分校统计学博士学位，主要研究方向包括无限维空间的函数型数据分析，具有高维或者流形结构的函数型数据的方法和理论以及在大型复杂数据中的应用。由于在函数型数据分析领域所做出的奠基性和开创性的贡献，2012年获加拿大自然与工程科研基金颁发给统计学科的 DAS奖，2014年获得由加拿大统计学会和数学研究中心联合颁发的授予博士毕业15年内在加拿大做出突出贡献的统计学者的 CRM-SSC奖。至今为止担任9个国际统计学核心期刊编委，包括统计学顶级期刊Journal of the American Statistical Association和 Annals of Statistics。