Statistical regression analysis of functional and shape data.

Area of Science:

• Statistics
• Data Science
• Computational Geometry

Background:

• Traditional regression models assume data exists in Euclidean space.
• Nonlinear manifold data presents unique challenges for standard statistical methods.
• Analyzing complex data like shapes and spectra requires specialized approaches.

Purpose of the Study:

• To develop a robust multivariate regression model for data residing on nonlinear manifolds.
• To adapt existing regression tools for non-Euclidean data analysis.
• To demonstrate the model's effectiveness on real-world shape and functional data.

Main Methods:

• Principal Component Analysis (PCA) applied to the tangent space of manifolds.
• Utilizing principal directions derived from PCA within the regression framework.
• Employing square-root velocity function representation and parametrization-invariant metrics for shape data.

Main Results:

• Successful application of the multivariate regression model to ozone hole contour shape data.
• Effective analysis of meat absorbance spectrum functional data from the Tecator dataset.
• Demonstrated high prediction accuracy using the proposed non-Euclidean regression framework.

Conclusions:

• The developed model provides a powerful tool for regression analysis on nonlinear manifold data.
• The approach successfully handles complex shape and functional data types.
• High prediction accuracy validates the utility of the proposed methodology for non-Euclidean data.