4. Discussion
Motivated by the analysis of functional time series of demand and offer curves in the Italian natural gas market, we have proposed a model for functional time series that preserves particular features of the curves, such as monotonicity and the equality and inequality constraints at the edges of the domain. A bijective map that associates each possible constrained function with an unconstrained one is introduced, and an isometry between the space of the constrained functions and L 2 , a suitable geometry, is developed in order to achieve this. Specifically, we provide the constrained functions with a suitable pre-Hilbert structure. The transformed curves are then modeled by means of a functional autoregressive model. The autoregressive lagged operators and the non-centrality function of the model are obtained by minimizing the squared L 2 distance between the functional data and functional predictions, with a penalty term based on the Hilbert–Schmidt squared norm of autoregressive lagged operators. We have proved that a unique solution always exists, and that it is linear on the data with respect to the introduced geometry, thus guaranteeing that the plug-in predictions of future functional data satisfy all required constraints. We also provide explicit expressions for estimates and predictions. The model can be generalized easily to the modeling of multivariate functional data or the inclusion of scalar covariates, or other functional predictors that are available at the time of prediction. The inclusion of a trend and/or seasonality term in the model would be tricky in terms of estimation but might lead to further improvements in the predictiveperformance, and would certainly be worthy of future investigation.
