Data Analysis on Riemannian Symmetric Spaces

• Activity Number: 92
• Type: Invited
• Date/Time: Sunday, August 4, 2013 : 8:30 PM to 10:30 PM
• Sponsor: Section on Nonparametric Statistics
• Author(s): Emil Cornea*+ and Hongtu Zhu and Joseph G. Ibrahim
• Companies: UNC-CH Biostatistics and UNC-Chapel Hill and UNC
• Keywords: Lie group action ; link function ; generalized method of moment ; regression ; Riemannian symmetric space ; medical imaging

Abstract

We present a general regression framework for the analysis of manifold-valued response in a Riemannian symmetric space (RSS) and its association with covariates of interest, such as age, in Euclidean space. Such RSS-valued data arises frequently in medical imaging, computational biology, molecular imaging, surface modeling, and computer vision, among many others. little has been done when the response is in a general RSS. We develop an intrinsic regression model solely based on an intrinsic conditional moment assumption, avoiding specifying any parametric distribution in RSS. We propose various link functions to map from the Euclidean space of covariates to the the RSS of responses. We develop a two-stage procedure to calculate the parameter estimates, and determine their asymptotic distributions. We construct the Wald and geodesic test statistics to test hypotheses of unknown parameters. We systematically investigate the geometric invariant property of these estimates and test statistics. Simulation studies are used to evaluate the finite sample properties of our methods and a real data set is analyzed to illustrate the use of our test statistics.