A penalized spline estimator for functional linear regression with functional response

Abstract In many scientific studies in recent years, data have been collected at such a high frequency that they can be considered as functional data. In the case when both the response variable to be predicted and the covariates are functions, we provide a novel and easy-to-implement method addressing function-on-function linear modelling and obtain interpretable parameters. Two main types of models are considered: (i) the concurrent model which explains the response curve $$Y_i(t)$$ at time t from the values at same time t of the covariates $$X_i^l(t)$$ ; (ii) the (feed-forward) integral model which explains $$Y_i(t)$$ based on the values of covariate curves $$X_i^l(s)$$ observed at any times $$s\le t$$ . A regularized inference approach is proposed, which accurately selects an appropriate set of basis functions that can be used for functional data reconstruction and at the same time provides smooth and interpretable functional parameters. A functional confidence interval procedure is also proposed which uses the conformalization framework. Numerical studies on simulated data with different scenarios illustrate the good performance of our method to capture the relationship between covariates and response. The method is finally applied to well-known data and compared to a baseline: on Canadian weather data (predicting precipitations from temperature measurements) and on Hawaii ocean data (predicting ocean salinity from temperature, oxygen, chloropigments and density measurements). Our method shows significant improvements on prediction error.