Statistica Sinica 29 (2019), 719-739
Abstract: We consider a nonlinear function-on-function additive regression model with multiple functional predictors. The forms of the nonlinear functions are unspecified, and offer great flexibility to model various relationships between the response curve and predictor curves. We clarify the identifiability issue of the model and identify the best decompositions of the nonlinear functions in the model in terms of prediction. To estimate this expansion, we solve a penalized functional generalized eigenvalue problem followed by a penalized least squares procedure. With the minimum prediction error of the proposed decomposition, our approach has good prediction accuracy. Our approach converts the estimation of three-dimensional nonlinear functions to the estimation of two- and one-dimensional functions, which considerably reduces computational costs. Asymptotic results are provided, and simulations and a data application show that the proposed method has good predictive performance and is efficient in dimension reduction and computation. This method is implemented in the R package FRegSigCom.
Key words and phrases: Additive model, function-on-function regression, generalized functional eigenvalue problem, nonlinear functional regression model, signal function, the Karhunen-Loève expansion.