Penalized scalar-on-functions regression with interaction term

Generalized models for scalar responses with functional covariates are extended to include linear functional interaction terms. The coefficient functions are estimated using basis expansions and maximization of a log-likelihood, which is penalized to impose smoothness upon the coefficient functions. The respective smoothing parameters for the penalties are estimated from the data, e.g. via generalized cross-validation. Further functional or scalar terms as well as functional interactions of higher order can be added within the same framework. The performance of the introduced approach is tested in simulations. Additionally, it is applied to the two motivating data sets, to spectroscopic data of a fossil fuel and to cell chip sensor data, where three functional signals are measured over time. The main aim is to predict the respective response, namely the heat value of the fossil fuel and the concentration of paracetamol in the cell chip medium.