On the estimation of the number of components in multivariate functional principal component analysis

Happ and Greven [2018] develop innovative theory and methodology for the dimension reduction of multivariate functional data on possibly different dimensional domains (e.g., curves and images), which extends existing methods that were limited to either univariate functional data or multivariate functional data on a common one-dimensional domain. Recent research has shown a growing presence of data defined on different dimensional domains in diverse fields such as biomechanics, e.g., Warmenhoven et al. [2019] and neuroscience, e.g., Song and Kim [2022], so we expect the work to have significant practical impact. We aim to provide commentary on the estimation of the number of principal components utilising the methodology devised in Happ and Greven [2018]. To achieve this, we conduct an extensive simulation study and subsequently propose practical guidelines for practitioners to adeptly choose the appropriate number of components for multivariate functional datasets. For ease of presentation, we use the same notation as in Happ and Greven [2018].