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

Happ and Greven (2018) developed a methodology for principal components analysis of multivariate functional data for data observed on different dimensional domains. Their approach relies on an estimation of univariate functional principal components for each univariate functional feature. In this paper, we present extensive simulations to investigate choosing the number of principal components to retain. We show empirically that the conventional approach of using a percentage of variance explained threshold for each univariate functional feature may be unreliable when aiming to explain an overall percentage of variance in the multivariate functional data, and thus we advise practitioners to be careful when using it.

On the use of the Gram matrix for multivariate functional principal components analysis

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.