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].

Dimension reduction is crucial in functional data analysis (FDA). The key tool to reduce the dimension of the data is functional principal component analysis. Existing approaches for functional principal component analysis usually involve the diagonalization of the covariance operator. With the increasing size and complexity of functional datasets, estimating the covariance operator has become more challenging. Therefore, there is a growing need for efficient methodologies to estimate the eigencomponents. Using the duality of the space of observations and the space of functional features, we propose to use the inner-product between the curves to estimate the eigenelements of multivariate and multidimensional functional datasets. The relationship between the eigenelements of the covariance operator and those of the inner-product matrix is established. We explore the application of these methodologies in several FDA settings and provide general guidance on their usability.