Abstract:

In an era when the decision-making process is often based on the analysis of complex and evolving data, it is crucial to have systems which allow to incorporate human knowledge and provide valuable support to the decider. In this work, statistical modelling and mathematical optimization paradigms merge to address the problem of estimating smooth curves and hypersurfaces which verify structural properties, both in the observed domain in which data have been gathered and outwards. We assume that the smooth curve (resp. hypersurface) to be estimated is defined through a reduced-rank basis (B−splines) and fitted via a penalized splines approach (P−splines). In order to incorporate requirements about the sign, monotonicity and curvature in the fitting procedure, a conic optimization model is stated and solved which, for the first time, successfully conveys out-of-range constrained forecasting.

This approach is successfully applied to simulated and demographic data, as well as to data arising in the context of the COVID-19 pandemic. If a smooth curve is fitted using an unconstrained approach, misleading results might be depicted. That is the case of, for instance, curves depicting a negative number of infected people in certain time periods. Furthermore, forecasting the evolution of the pandemic under a different set of constrained scenarios, such as having an estimated growth rate, is also possible using the approach developed in this work.