A Framework for Analysis of Manifold-Value Data in Riemannian Symmetric Spaces

• Activity Number: 568
• Type: Invited
• Date/Time: Wednesday, August 7, 2013 : 2:00 PM to 3:50 PM
• Sponsor: IMS
• Author(s): Hongtu Zhu*+ and Emil Cornea and Joseph G. Ibrahim
• Companies: UNC-Chapel Hill and UNC-CH Biostatistics and UNC
• Keywords: Manifold-value ; Riemannian symmetric ; regression ; geodesic ; test ; covariate

Abstract

We develop a general framework for the analysis of manifold-valued response in a Riemannian symmetric (RS) space and its relationship with covariates of interest, such as age, in Euclidean space. Such manifold-valued data, such as directional data and symmetric positive-definite matrices, arises frequently in medical imaging, computational biology, molecular imaging, surface modeling, and computer vision, among many others. However, little has been done when the response is in a general RS space. We develop an intrinsic regression model solely based on an intrinsic conditional moment assumption, avoiding specifying any parametric distribution in RS space. We propose various link functions to map from the Euclidean space of covariates to the the RS space 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 on unknown parameters. Simulations studies are used to evaluate the finite sample property of our methods and a real data set is analyzed to illustrate the use of our test statistics.