An Adaptive Curvature-Guided Approach for the Knot-Placement Problem in Fitted Splines

Abstract

This paper presents an adaptive and computationally efficient curvature-guided algorithm for localizing optimum knot locations in fitted splines based on the local minimization of an objective error function. Curvature information is used to narrow the searching area down to a data subset where the local error function becomes one-dimensional, convex, and bounded, thus guaranteeing a fast numerical solution. Unlike standard curvature-guided methods, typically relying on heuristic rules, the novel method here presented is based on a phenomenological approach as the error function to be minimized represents geometrical properties of the curve to be fitted, consequently reducing case-sensitivity issues and the possibility of defining spurious knots. A knot-readjustment procedure performed in the vicinity of a newly created knot has the ability of dispersing knots from otherwise highly knot-populated regions, a feature known to generate undesired local oscillations. The performance of the introduced method is tested against three other methods described in the literature, each handling the knot-placement problem via a different paradigm. The quality of the fitted splines for several datasets is compared in terms of the overall accuracy, the number of knots, and the computing efficiency. It is demonstrated that the novel algorithm leads to a significantly smaller knot vector and a much lower computing time, while preserving or improving the overall accuracy.