Nonlinear function-on-scalar regression via functional universal approximation

We consider general nonlinear function-on-scalar (FOS) regression models, where the functional response depends on multiple scalar predictors in a general unknown nonlinear form. Existing methods either assume specific model forms (e.g., additive models) or directly estimate the nonlinear function in a space with dimension equal to the number of scalar predictors, which can only be applied to models with a few scalar predictors. To overcome these shortcomings, motivated by the classic universal approximation theorem used in neural networks, we develop a functional universal approximation theorem which can be used to approximate general nonlinear FOS maps and can be easily adopted into the framework of functional data analysis. With this theorem and utilizing smoothness regularity, we develop a novel method to fit the general nonlinear FOS regression model and make predictions. Our new method does not make any specific assumption on the model forms, and it avoids the direct estimation of nonlinear functions in a space with dimension equal to the number of scalar predictors. By estimating a sequence of bivariate functions, our method can be applied to models with a relatively large number of scalar predictors. The good performance of the proposed method is demonstrated by empirical studies on various simulated and real datasets.