Optimal Function-on-Function Regression With Interaction Between Functional Predictors

Abstract

We consider a functional regression model in the framework of reproducing kernel Hilbert spaces where the interaction effect of two functional predictors, as well as their main effects, over the functional response is of interest. The regression component of our model is expressed by one trivariate coefficient function whose functional ANOVA decomposition yields the main and interaction effects. The trivariate coefficient function is estimated through the optimization of a penalized least squares objective with a roughness penalty on the function estimate. The estimation procedure can be easily implemented via standard numerical tools. Asymptotic results for the proposed model with or without functional measurement errors are established under the reproducing kernel Hilbert space framework. Extensive numerical studies show the advantages of the proposed method over existing ones in terms of prediction and estimation of coefficient functions. An application to the motivating example on histone modifications and gene expressions of a liver cancer cell line further demonstrates the better prediction accuracy of the proposed method against the competitors.