α-separability and adjustable combination of amplitude and phase model for functional data

We consider separating and joint modelling amplitude and phase variations for functional data in an identifiable manner. To rigorously address this separability issue, we introduce the notion of α-separability upon constructing a family of α-indexed metrics. We bridge α-separability with the uniqueness of Fréchet mean, leading to the proposed adjustable combination of amplitude and phase model. The parameter α allows user-defined modelling emphasis between vertical and horizontal features and provides a novel viewpoint on the identifiability issue. We prove the consistency of the sample Fréchet mean and variance, and the proposed estimators. Our method is illustrated in simulations and COVID-19 infection rate data.