Shape Analysis of Functional Data with Elastic Partial Matching

• URL: http://arxiv.org/abs/2105.08604v1
• Date: Tue, 18 May 2021 15:36:51 GMT
• Title: Shape Analysis of Functional Data with Elastic Partial Matching
• Abstract summary: We develop a comprehensive framework for partial matching, comparing, and clustering of functions under both phase variability and uncertain boundaries. This framework is illustrated for registering and clustering shapes of COVID-19 rate curves.
• Score: 14.391648046717073
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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