Multivariate Functional Regression Via Nested Reduced-Rank Regularization

We propose a nested reduced-rank regression (NRRR) approach in fitting a regression model with multivariate functional responses and predictors to achieve tailored dimension reduction and facilitate model interpretation and visualization. Our approach is based on a two-level low-rank structure imposed on the functional regression surfaces. A global low-rank structure identifies a small set of latent principal functional responses and predictors that drives the underlying regression association. A local low-rank structure then controls the complexity and smoothness of the association between the principal functional responses and predictors. The functional problem boils down to an integrated matrix approximation task through basis expansion, where the blocks of an integrated low-rank matrix share some common row space and/or column space. This nested reduced-rank structure also finds potential applications in multivariate time series modeling and tensor regression. A blockwise coordinate descent algorithm is developed. We establish the consistency of NRRR and show through nonasymptotic analysis that it can achieve at least a comparable error rate to that of the reduced-rank regression. Simulation studies demonstrate the effectiveness of NRRR. We apply the proposed methods in an electricity demand problem to relate daily electricity consumption trajectories with daily temperatures. Supplementary files for this article are available online.