Intrinsic Regression Models for Lie Group-Valued Data

• Activity Number: 346
• Type: Contributed
• Date/Time: Tuesday, August 2, 2011 : 10:30 AM to 12:20 PM
• Sponsor: Section on Nonparametric Statistics
• Author(s): Emil A. Cornea*+ and Hongtu Zhu and Joseph G. Ibrahim
• Companies: The University of North Carolina and The University of North Carolina and The University of North Carolina
• Keywords: semiparametric regression model ; Lie group ; link function ; conditional moment ; medical imaging

Abstract

In many fields, like medical imaging, computational biology, computer vision, there is a growing interest in analyzing various data, such as 3D rotations, symmetric positive-definite matrices, and diffusion tensors, and their association with a set of Euclidean covariates. These data do not form a vector space and may rather be viewed as lying in appropriate Lie groups, or more generally, Riemannian manifolds, thus directly applying classical multivariate regression may be inadequate. The aim of this work is to present an intrinsic regression model for the analysis Lie group-valued data and the association with a set of covariates in a Euclidean space. Our model is semiparametric, solely based on the conditional first-order moment, avoiding specifying any parametric distributions; it uses a link function to map covariates from the Euclidean space to the Lie group of responses. We develop a two-stage procedure to calculate the parameter estimates, and determine their asymptotic distributions. We construct goodness-of-fit statistics for testing possible model misspecifications, and score statistics to test hypotheses on unknown parameters. Applications in neuroimaging are presented.