Test of Partial Separability for Multivariate Functional Data

For multivariate functional data, it is quite challenging to model the cross-covariance structure which consists of dual aspects of multivariate and functional features. To simplify the cross-covariance analysis, the assumption of partial separability is widely used to decompose the data into an additive form of multivariate random variables and functional components. In this article, we propose hypothesis testing procedures to examine the validity of partial separability. We study the asymptotic properties of the l2 and l infinity norm of the test statistic, resulting in a chi-square type mixture test and a high-dimensional test that are suitable to finite-or high-dimensional multivariate functional data with diverse multivariate dependence. We assess the empirical performance of the proposed tests through two simulation studies for multivariate functional data and graphical functional data, followed by two corresponding real examples: multichannel tonnage data and electroencephalography data.