Continuously dynamic additive models for functional data

In this article, we propose the continuously dynamic additive model (CDAM), in which both the predictor and response are random functions. In continuously dynamic additive modeling, we assume that additivity occurs in the time domain rather than in spectral domain, and characterize this model through a time-dependent smooth surface that reflects the underlying nonlinear dynamic relationships between functional predictor and functional response. We use tensor product basis expansion with varying coefficient functions to approximate the time-varying smooth surface, and then estimate varying-coefficient functions by combining functional principal components analysis with penalized least squares method. In a theoretical investigation, we show that the predictions obtained from the fitted CDAM are asymptotically consistent under some mild conditions. Finally, we demonstrate the superiority of the proposed model and method through extensive simulation studies as well as a real data example.