Shape Analysis of Functional Data With Elastic Partial Matching.

Area of Science:

• Statistics
• Computational Geometry
• Dynamical Systems

Background:

• Elastic Riemannian metrics are effective for functional and curve shape data analysis.
• Existing methods require fixed and matched function boundaries, limiting their application to real-world data with variable evolution rates and unmatched endpoints, such as COVID-19 infection curves.

Purpose of the Study:

• To develop a flexible Riemannian framework for statistical analysis of functional data with phase variability and uncertain boundaries.
• To extend existing elastic metrics to accommodate partial matching and clustering of functions with unmatched endpoints.
• To apply the framework to COVID-19 infection rate curves for improved registration and clustering.

Main Methods:

• Defined a new diffeomorphism group G (semidirect product of time-warping and time-scaling groups) over positive reals.
• Introduced a novel metric invariant to the action of G.
• Imposed a Riemannian Lie group structure on G for efficient gradient-based optimization in elastic partial matching.
• Developed a modification to control boundary disparity during registration.

Main Results:

• Successfully registered and clustered COVID-19 rate curves, identifying underlying patterns.
• Minimized mismatch errors and reduced variability within clusters compared to previous methods.
• Demonstrated the framework's capability to handle functions with phase variability and uncertain boundaries.

Conclusions:

• The proposed Riemannian framework effectively addresses the limitations of fixed-boundary methods in functional data analysis.
• The framework offers a robust approach for comparing, registering, and clustering functions with variable evolution rates and unmatched boundaries.
• This methodology provides significant improvements for analyzing complex real-world data, exemplified by its application to COVID-19 infection dynamics.