Local Polynomial Regression for Symmetric Positive Definite Matrices.

TitleLocal Polynomial Regression for Symmetric Positive Definite Matrices.
Publication TypeJournal Article
Year of Publication2012
AuthorsYuan, Ying, Hongtu Zhu, Weili Lin, and J S. Marron
JournalJ R Stat Soc Series B Stat Methodol
Volume74
Issue4
Pagination697-719
Date Published2012 Sep 01
ISSN1369-7412
Abstract

Local polynomial regression has received extensive attention for the nonparametric 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 (SPD) matrices as responses that lie in a Riemannian manifold with covariate in Euclidean space. The primary motivation and application of the proposed methodology 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 SPD 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 fiber tracts in a study of human immunodeficiency virus.

DOI10.1111/j.1467-9868.2011.01022.x
Alternate JournalJ R Stat Soc Series B Stat Methodol
Original PublicationLocal polynomial regression for symmetric positive definite matrices.
PubMed ID23008683
PubMed Central IDPMC3448376
Grant ListR01 CA074015-04A1 / CA / NCI NIH HHS / United States
UL1 RR025747-01 / RR / NCRR NIH HHS / United States
R01 MH086633 / MH / NIMH NIH HHS / United States
UL1 RR025747-02S3 / RR / NCRR NIH HHS / United States
R01 CA074015-12 / CA / NCI NIH HHS / United States
R01 NS054079-01A1 / NS / NINDS NIH HHS / United States
R01 CA074015-08A2 / CA / NCI NIH HHS / United States
R01 CA074015-11A1 / CA / NCI NIH HHS / United States
R01 NS054079-04S1 / NS / NINDS NIH HHS / United States
R01 CA074015-09 / CA / NCI NIH HHS / United States
UL1 RR025747-01S1 / RR / NCRR NIH HHS / United States
P01 CA142538-01 / CA / NCI NIH HHS / United States
R01 NS054079-05 / NS / NINDS NIH HHS / United States
R01 NS054079-03 / NS / NINDS NIH HHS / United States
P01 CA142538-03 / CA / NCI NIH HHS / United States
R01 CA074015-06 / CA / NCI NIH HHS / United States
R01 CA074015-05 / CA / NCI NIH HHS / United States
TL1 RR025745-02 / RR / NCRR NIH HHS / United States
R01 CA074015-03 / CA / NCI NIH HHS / United States
P01 CA142538-02 / CA / NCI NIH HHS / United States
R01 NS054079-04 / NS / NINDS NIH HHS / United States
P50 CA058223 / CA / NCI NIH HHS / United States
UL1 RR025747-02 / RR / NCRR NIH HHS / United States
R01 CA074015-10 / CA / NCI NIH HHS / United States
R01 NS054079 / NS / NINDS NIH HHS / United States
R01 NS054079-02 / NS / NINDS NIH HHS / United States
P01 CA142538 / CA / NCI NIH HHS / United States
R01 CA074015-07 / CA / NCI NIH HHS / United States
R01 CA074015 / CA / NCI NIH HHS / United States
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