Function-on-Function Regression Models With Nonlinear Dynamic Effect and Linear Concurrent Effect

Abstract We propose a function-on-function regression model that predicts a functional response by both a nonlinear dynamic effect of a functional predictor and a linear concurrent effect of another functional predictor. The nonlinear dynamic effect is characterized by taking an integral of a time-dependent two-dimensional smooth surface and the linear concurrent effect is modeled through a time-varying coefficient. The model structure combines the flexibility of nonlinear modeling with the interpretability of the linear concurrent effect. To approximate the two-dimensional surface, we use tensor product basis expansions, and for the time-varying coefficient in the concurrent effect, we employ B-spline expansions. The expansion parameters for each effect are estimated iteratively to account for the mutual dependencies between these two estimated effects. Each iteration of parameter estimation involves solving a penalized least squares problem. We establish the asymptotic properties of our estimator. The numerical performance of the proposed method is illustrated by simulation studies and two real data applications.