Structural adaptation and rate accelerated estimation in bivariate functional data

We introduce directional regularity, a new definition of anisotropy for multivariate functional data. Instead of taking the conventional view, which determines anisotropy as a notion of smoothness along a dimension, directional regularity additionally views anisotropy through the lens of directions. We show that faster rates of convergence for smoothing can be obtained through a change-of-basis by adapting to the anisotropy of a bivariate process. An algorithm for the estimation and identification of the change-of-basis matrix is constructed, made possible due to the replication structure of functional data. Non-asymptotic bounds are provided for our algorithm, supplemented by numerical evidence from an extensive simulation study. Finally, a real-world rainfall measurement dataset is analyzed with our methods.