  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
$L^2(\mathcal{T})$. 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.
