# A General Framework for Multivariate Functional Principal Component Analysis of Amplitude and Phase Variation

arXiv: https://arxiv.org/abs/1810.13321 · HTML: https://mlsanity.com/paper/1810.13321/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 31 Oct 2018.

## Built on by (most cited first, 1 of 1)

- [A Functional Approach to Curve Alignment and Shape Analysis](https://mlsanity.com/paper/2503.05632/) (2025)

## Abstract (arXiv)

Functional data typically contains amplitude and phase variation. In many data situations, phase variation is treated as a nuisance effect and is removed during preprocessing, although it may contain valuable information. In this note, we focus on joint principal component analysis (PCA) of amplitude and phase variation. As the space of warping functions has a complex geometric structure, one key element of the analysis is transforming the warping functions to L2(T). We present different transformation approaches and show how they fit into a general class of transformations. This allows us to compare their strengths and limitations. In the context of PCA, our results offer arguments in favour of the centered log-ratio transformation. We further embed existing approaches from Hadjipantelis et al. (2015) and Lee and Jung (2017) for joint PCA of amplitude and phase variation into the framework of multivariate functional PCA, where we study the properties of the estimators based on an appropriate metric. The approach is illustrated through an application from seismology.