Function-on-function regression with thousands of predictive curves

With the advance of technology, thousands of curves can be simultaneously recorded by electronic devices, such as simultaneous EEG and fMRI data. To study the relationship between these curves, we consider a functional linear regression model with functional response and functional predictors, where the number of predictive curves is much larger than the sample size. The high dimensionality of this problem poses theoretical and practical difficulties for the existing methods, including estimation inconsistency and prediction inaccuracy. Motivated by the simultaneous EEG and fMRI data, we focus on models with sparsity structures where most of the coefficient functions of the predictive curves have small norms. To take advantage of this sparsity structure and the smoothness of coefficient functions, we propose a simultaneous sparse-smooth penalty which is incorporated into a generalized functional eigenvalue problem to obtain estimates of the model. We establish the asymptotic upper bounds for the prediction and estimation errors as both the sample size and the number of predictive curves go to infinity. We implement the proposed method in the R package FRegSigComp. Simulation studies show that the proposed method has good predictive performance for models with sparsity structures. The proposed method is applied to a simultaneous EEG and fMRI dataset.