# Functional Data Analysis of Amplitude and Phase Variation

arXiv: https://arxiv.org/abs/1512.03216 · HTML: https://mlsanity.com/paper/1512.03216/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Dec 2015.

## Built on by (most cited first, 16 of 16)

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- [Shape Analysis of Functional Data with Elastic Partial Matching](https://mlsanity.com/paper/2105.08604/) (2021)
- [DeepFRC: An End-to-End Deep Learning Model for Functional Registration and Classification](https://mlsanity.com/paper/2501.18116/) (2025)
- [A Functional Approach to Curve Alignment and Shape Analysis](https://mlsanity.com/paper/2503.05632/) (2025)
- [NeuralFLoC: Neural Flow-Based Joint Registration and Clustering of Functional Data](https://mlsanity.com/paper/2602.03169/) (2026)
- [Accurate shape and phase averaging of time series through Dynamic Time Warping](https://mlsanity.com/paper/2109.00978/) (2021)
- [Data-Driven, Soft Alignment of Functional Data Using Shapes and Landmarks](https://mlsanity.com/paper/2203.14810/) (2022)
- [Joint Alignment of Multivariate Quasi-Periodic Functional Data Using Deep Learning](https://mlsanity.com/paper/2312.09422/) (2023)
- [Joint Registration and Conformal Prediction for Partially Observed Functional Data](https://mlsanity.com/paper/2502.15000/) (2025)
- [Functional BART with Shape Priors: A Bayesian Tree Approach to Constrained Functional Regression](https://mlsanity.com/paper/2502.16888/) (2025)
- [Supervised Manifold Learning for Functional Data](https://mlsanity.com/paper/2503.17943/) (2025)
- [Functional Random Forest with Adaptive Cost-Sensitive Splitting for Imbalanced Functional Data Classification](https://mlsanity.com/paper/2512.07888/) (2025)
- [Neural Operators Can Discover Functional Clusters](https://mlsanity.com/paper/2602.23528/) (2026)
- [Why Directly Learning Periodic Trajectories Can Fail](https://mlsanity.com/paper/2609.32254/) (2026)
- [Functional Autoencoders for Amplitude-Phase Representation Learning](https://mlsanity.com/paper/2609.34207/) (2026)

## Abstract (arXiv)

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 mathbbL2 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.