Local polynomial regression for symmetric positive definite matrices

• 成果类型: Article
• 署名作者: Yuan, Ying; Zhu, Hongtu; Lin, Weili; Marron, J. S.
• 署名单位: University of North Carolina; University of North Carolina Chapel Hill; University of North Carolina School of Medicine; University of North Carolina; University of North Carolina Chapel Hill; University of North Carolina School of Medicine
• 刊物名称: JOURNAL OF THE ROYAL STATISTICAL SOCIETY SERIES B-STATISTICAL METHODOLOGY
• ISSN/ISSBN: 1369-7412
• DOI: 10.1111/j.1467-9868.2011.01022.x
• 发表日期: 2012
• 页码: 697-719
• 关键词: fitting smooth paths; statistical-analysis; bandwidth selection; white-matter; tensor; splines; FRAMEWORK; hiv

摘要

Local polynomial regression has received extensive attention for the non-parametric estimation of regression functions when both the response and the covariate are in Euclidean space. However, little has been done when the response is in a Riemannian manifold. We develop an intrinsic local polynomial regression estimate for the analysis of symmetric positive definite matrices as responses that lie in a Riemannian manifold with covariate in Euclidean space. The primary motivation and application of the methodology proposed is in computer vision and medical imaging. We examine two commonly used metrics, including the trace metric and the log-Euclidean metric on the space of symmetric positive definite matrices. For each metric, we develop a cross-validation bandwidth selection method, derive the asymptotic bias, variance and normality of the intrinsic local constant and local linear estimators, and compare their asymptotic mean-square errors. Simulation studies are further used to compare the estimators under the two metrics and to examine their finite sample performance. We use our method to detect diagnostic differences between diffusion tensors along fibre tracts in a study of human immunodeficiency virus.