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.

Bio: Fang Yao is Bo-Ya Distinguished Professor in Statistics at Peking University, and Professor in the Department of Statistical Sciences at University of Toronto. He received his B.S. degree in 2000 from University of Science & Technology in China, and his M.S. and Ph.D. degrees in Statistics in 2002 and 2003, respectively, at UC Davis. Dr. Yao’s research primarily focuses on functional and longitudinal data, complex data structures such as high dimensions and manifolds, and their applications in various disciplines. In 2014, he received the CRM-SSC Prize that recognizes a statistical scientist’s professional accomplishments in research primarily conducted in Canada during the first 15 years after receiving a doctorate. He serves on editorial boards for nine statistical journals, including Annals of Statistics and JASA, and will be the editor for Canadian Journal of Statistics starting from January 2019. He is a Fellow of IMS, ASA and an elected member of ISI.