On constrained smoothing and out-of-range prediction using P-splines: A conic optimization approach

Decision-making is often based on the analysis of complex and evolving data. Thus, having systems which allow to incorporate human knowledge and provide valuable support to the decider becomes crucial. In this work, statistical modelling and mathematical optimization paradigms merge to address the problem of estimating smooth curves which verify structural properties, both in the observed domain in which data have been gathered and outwards. We assume that the curve to be estimated is defined through a reduced-rank basis (B-splines) and fitted via a penalized splines approach (P-splines). To incorporate requirements about the sign, monotonicity and curvature in the fitting procedure, a conic programming approach is developed which, for the first time, successfully conveys out-of-range constrained prediction. In summary, the contributions of this paper are fourfold: first, a mathematical optimization formulation for the estimation of non-negative P-splines is proposed; second, previous results are generalized to the out-of-range prediction framework; third, these approaches are extended to other shape constraints and to multiple curves fitting; and fourth, an open source Python library is developed: cpsplines. We use simulated instances, data of the evolution of the COVID-19 pandemic and of mortality rates for different age groups to test our approaches.