# Functional Registration and Local Variations: Identifiability, Rank, and Tuning

## TL;DR

This paper establishes conditions for the identifiability of phase and amplitude variations in functional data, and introduces a parameter-free registration method based on local variation, with theoretical guarantees.

## Contribution

It provides sharp nonparametric conditions for identifiability and proposes a novel local variation-based registration method without tuning parameters.

## Key findings

- Identifiability depends on the underlying rank of the data.
- The proposed method is free of tuning parameters and avoids over/under-registration.
- Asymptotic theory quantifies bias under mild departures from identifiability.

## Abstract

We develop theory and methodology for the problem of nonparametric registration of functional data that have been subjected to random deformation (warping) of their time scale. The separation of this phase variation ("horizontal" variation) from the amplitude variation ("vertical" variation) is crucial in order to properly conduct further analyses, which otherwise can be severely distorted. We determine precise nonparametric conditions under which the two forms of variation are identifiable. These show that the identifiability delicately depends on the underlying rank. By means of several counterexamples, we demonstrate that our conditions are sharp if one wishes a genuinely nonparametric setup; and in doing so we caution that popular remedies such as structural assumptions or roughness penalties can easily fail. We then propose a nonparametric registration method based on a "local variation measure", the main element in elucidating identifiability. A key advantage of the method is that it is free of any tuning or penalisation parameters regulating the amount of alignment, thus circumventing the problem of over/under-registration often encountered in practice. We provide asymptotic theory for the resulting estimators under the identifiable regime, but also under mild departures from identifiability, quantifying the resulting bias in terms of the amplitude variation's spectral gap.