Functional Data Analysis of Amplitude and Phase Variation

TL;DR

This paper discusses the challenges of phase variability in functional data analysis, reviews current methods for separating phase and amplitude, and highlights the importance of addressing phase variation for accurate data interpretation.

Contribution

It summarizes and contrasts various approaches for separating phase and amplitude in functional data, emphasizing the importance of proper phase handling.

Findings

Phase variability inflates data variance and distorts principal components.

Existing alignment methods based on minimizing the L2 norm are often inadequate.

Different approaches vary in phase representation, objective functions, and algorithms.

Abstract

The abundance of functional observations in scientific endeavors has led to a significant development in tools for functional data analysis (FDA). This kind of data comes with several challenges: infinite-dimensionality of function spaces, observation noise, and so on. However, there is another interesting phenomena that creates problems in FDA. The functional data often comes with lateral displacements/deformations in curves, a phenomenon which is different from the height or amplitude variability and is termed phase variation. The presence of phase variability artificially often inflates data variance, blurs underlying data structures, and distorts principal components. While the separation and/or removal of phase from amplitude data is desirable, this is a difficult problem. In particular, a commonly used alignment procedure, based on minimizing the $\mathbb{L}^2$ norm between functions, does not provide satisfactory results. In this paper we motivate the importance of dealing with the phase variability and summarize several current ideas for separating phase and amplitude components. These approaches differ in the following: (1) the definition and mathematical representation of phase variability, (2) the objective functions that are used in functional data alignment, and (3) the algorithmic tools for solving estimation/optimization problems. We use simple examples to illustrate various approaches and to provide useful contrast between them.