  Existing approaches for multivariate functional principal component analysis
are restricted to data on the same one-dimensional interval. The presented
approach focuses on multivariate functional data on different domains that may
differ in dimension, e.g. functions and images. The theoretical basis for
multivariate functional principal component analysis is given in terms of a
Karhunen-Lo\`eve Theorem. For the practically relevant case of a finite
Karhunen-Lo\`eve representation, a relationship between univariate and
multivariate functional principal component analysis is established. This
offers an estimation strategy to calculate multivariate functional principal
components and scores based on their univariate counterparts. For the resulting
estimators, asymptotic results are derived. The approach can be extended to
finite univariate expansions in general, not necessarily orthonormal bases. It
is also applicable for sparse functional data or data with measurement error. A
flexible R-implementation is available on CRAN. The new method is shown to be
competitive to existing approaches for data observed on a common
one-dimensional domain. The motivating application is a neuroimaging study,
where the goal is to explore how longitudinal trajectories of a
neuropsychological test score covary with FDG-PET brain scans at baseline.
Supplementary material, including detailed proofs, additional simulation
results and software is available online.
