Spatially Adaptive Regression Splines and Accurate Knot Selection Schemes

Abstract

Spline procedures have proven effective in estimating smooth functions. However, spline procedures based on stepwise addition and/or deletion have some drawbacks. They suffer from the knot compounding problem, making their performance suboptimal. Furthermore, due to computational complexity, spline procedures may not achieve their full potential. In this article, we propose a novel knot selection algorithm for regression spline estimation in nonparametric regression. The algorithm includes three new components: knot relocation, guided search, and local fitting. The local properties of the spline functions are used to efficiently implement the algorithm. Extensive simulation studies are performed to demonstrate the improvement of the new knot selection algorithm over the stepwise addition and deletion scheme, and the advantages of the spline procedure with the new knot selection scheme over alternative adaptive methods. In the simulations, our procedure achieves very competitive performance with alternative methods and has substantial advantage in nonsmooth functions. Finally, the usefulness of the proposed method is illustrated by an application to signal recovery in speech signal processing.

KEY WORDS:

• Knot relocation
• Regression spline
• Spatial adaptation
• Stepwise Knot selection