  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.
