Smoothing splines with some discontinuities

Matteo Pelagatti, Paolo Maranzano and Giulio Cesare Gatto Ronchieri
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Matteo Pelagatti: University of Milano-Bicca, Italy
Paolo Maranzano: University of Milano-Bicca, Italy,Fondazione Eni Enrico Mattei, Italy
Giulio Cesare Gatto Ronchieri: University of Milano-Bicca, Italy

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Abstract: We generalize cubic smoothing splines to signals with a finite number of discontinuities at unknown positions. Rather than changing the roughness penalty, we deform the domain over which it is computed: every interval between consecutive observations can be stretched by a non-negative warp, subject to a total budget, and the optimizer spends the budget only where a jump makes stretching cheaper than bending. The estimator inherits the state-space form of smoothing splines, so it can be fitted by maximum likelihood with the Kalman filter, and it also admits a fast alternating algorithm that solves a banded linear system and takes a projected-gradient step on the warps. The budget is selected by an extended Bayesian information criterion whose constant is calibrated by simulation. In a Monte Carlo comparison the new estimators reconstruct a piecewise-smooth signal more accurately than the cubic smoothing spline with discontinuities of Storath and Weinmann, at the cost of more false alarms. Two energy applications illustrate the method: the temperature response of electricity demand in Northern Italy, and retail fuel prices, whose discontinuities, changes in excise duties, are dated by public record. The three supplements follow the references.

Date: 2026-10
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