Shape Analysis of Functional Data with Elastic Partial Matching
This work addresses a domain-specific problem in functional data analysis for applications like epidemiology, offering an incremental improvement over existing elastic metrics by handling unmatched boundaries.
The paper tackles the problem of analyzing functional data with variable boundaries, such as COVID-19 infection rate curves, by developing a Riemannian framework for partial matching that allows sliding boundaries and joint time-warping and scaling. It demonstrates reduced mismatch errors and variability in clustering compared to previous methods.
Elastic Riemannian metrics have been used successfully in the past for statistical treatments of functional and curve shape data. However, this usage has suffered from an important restriction: the function boundaries are assumed fixed and matched. Functional data exhibiting unmatched boundaries typically arise from dynamical systems with variable evolution rates such as COVID-19 infection rate curves associated with different geographical regions. In this case, it is more natural to model such data with sliding boundaries and use partial matching, i.e., only a part of a function is matched to another function. Here, we develop a comprehensive Riemannian framework that allows for partial matching, comparing, and clustering of functions under both phase variability and uncertain boundaries. We extend past work by: (1) Forming a joint action of the time-warping and time-scaling groups; (2) Introducing a metric that is invariant to this joint action, allowing for a gradient-based approach to elastic partial matching; and (3) Presenting a modification that, while losing the metric property, allows one to control relative influence of the two groups. This framework is illustrated for registering and clustering shapes of COVID-19 rate curves, identifying essential patterns, minimizing mismatch errors, and reducing variability within clusters compared to previous methods.