Local Polynomial Regression for Symmetric Positive Definite Matrices.

Area of Science:

• Statistics
• Computer Vision
• Medical Imaging
• Nonparametric Statistics

Background:

• Local polynomial regression is established for Euclidean data.
• Limited methods exist for nonparametric estimation when responses are on Riemannian manifolds.
• Symmetric positive definite (SPD) matrices are common in medical imaging but require specialized analysis.

Purpose of the Study:

• To develop an intrinsic local polynomial regression method for SPD matrices as responses.
• To apply this method to analyze diffusion tensors in medical imaging, specifically for HIV studies.
• To compare different metrics and estimators for analyzing SPD matrix data.

Main Methods:

• Developed intrinsic local polynomial regression for SPD matrices in a Riemannian manifold.
• Utilized trace and Log-Euclidean metrics on the space of SPD matrices.
• Implemented cross-validation for bandwidth selection.
• Derived asymptotic properties (bias, variance, normality) of local constant and linear estimators.

Main Results:

• Compared asymptotic mean square errors of different estimators.
• Simulation studies evaluated estimator performance under various metrics.
• Demonstrated the method's ability to detect diagnostic differences in diffusion tensors.

Conclusions:

• The proposed intrinsic local polynomial regression is effective for analyzing SPD matrices.
• The method offers a valuable tool for computer vision and medical imaging applications.
• The approach successfully identified diagnostic differences in diffusion tensor imaging data related to HIV.