A pairwise interaction model for multivariate functional and longitudinal data

Functional data vectors consisting of samples of multivariate data where each component is a random function are encountered increasingly often but have not yet been comprehensively investigated. We introduce a simple pairwise interaction model that leads to an interpretable and straightforward decomposition of multivariate functional data and of their variation into component-specific processes and pairwise interaction processes. The latter quantify the degree of pairwise interactions between the components of the functional data vectors, while the component-specific processes reflect the functional variation of a particular functional vector component that cannot be explained by the other components. Thus the proposed model provides an extension of the usual notion of a covariance or correlation matrix for multivariate vector data to functional data vectors and generates an interpretable functional interaction map. The decomposition provided by the model can also serve as a basis for subsequent analysis, such as study of the network structure of functional data vectors. The decomposition of the total variance into componentwise and interaction contributions can be quantified by an $R^2$-like decomposition. We provide consistency results for the proposed methods and illustrate the model by applying it to sparsely sampled longitudinal data from the Baltimore Longitudinal Study of Aging, examining the relationships between body mass index and blood fats.