Intrinsic regression models for manifold-valued data.

Area of Science:

• Medical imaging analysis
• Computer vision
• Computational geometry
• Biostatistics

Background:

• Growing interest in analyzing manifold-valued data (e.g., 3D rotations, SPD matrices, m-reps) in population studies.
• Need for robust regression frameworks to associate covariates (age, gender, diagnostic status) with manifold-valued data for characterizing brain structure and shape differences.
• Classical multivariate regression is inadequate for non-Euclidean manifold data.

Purpose of the Study:

• To develop an intrinsic regression model for analyzing manifold-valued data as responses on a Riemannian manifold.
• To establish associations between manifold-valued data and covariates in Euclidean space.
• To provide a statistical framework for characterizing morphological changes in population studies.

Main Methods:

• Developed a semiparametric intrinsic regression model using a link function to map covariates to the Riemannian manifold.
• Created an estimation procedure for an intrinsic least squares estimator and established its limiting distribution.
• Developed score statistics for testing linear hypotheses on model parameters.

Main Results:

• The intrinsic regression model effectively handles manifold-valued data, overcoming limitations of classical methods.
• The estimation procedure and statistical tests provide a valid framework for analyzing associations.
• Demonstrated application in detecting morphological differences in hippocampi between schizophrenia patients and controls.

Conclusions:

• The proposed intrinsic regression model is a powerful tool for analyzing manifold-valued data in medical imaging and population studies.
• This framework enables robust characterization of brain structure and shape differences associated with covariates.
• The method has practical implications for understanding neurological conditions like schizophrenia through shape analysis.