On constrained smoothing and out-of-range prediction using [formula omitted]-splines: A conic optimization approach

• We study the problem of estimating smooth curves which verify structural properties. • We propose a mathematical optimization formulation to build constrained P-splines. • An open-source Python library is developed: cpsplines. • We estimate constrained curves in simulated, COVID-19 and demographic data. 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. [ FROM AUTHOR]

On sparse ensemble methods: An application to short-term predictions of the evolution of COVID-19.

Since the seminal paper by Bates and Granger in 1969, a vast number of ensemble methods that combine different base regressors to generate a unique one have been proposed in the literature. The so-obtained regressor method may have better accuracy than its components, but at the same time it may overfit, it may be distorted by base regressors with low accuracy, and it may be too complex to understand and explain. This paper proposes and studies a novel Mathematical Optimization model to build a sparse ensemble, which trades off the accuracy of the ensemble and the number of base regressors used. The latter is controlled by means of a regularization term that penalizes regressors with a poor individual performance. Our approach is flexible to incorporate desirable properties one may have on the ensemble, such as controlling the performance of the ensemble in critical groups of records, or the costs associated with the base regressors involved in the ensemble. We illustrate our approach with real data sets arising in the COVID-19 context.