Hybrid Adaptive Splines

Abstract

An adaptive spline method for smoothing is proposed that combines features from both regression spline and smoothing spline approaches. One of its advantages is the ability to vary the amount of smoothing in response to the inhomogeneous “curvature” of true functions at different locations. This method can be applied to many multivariate function estimation problems, which is illustrated by an application to smoothing temperature data on the globe. The method's performance in a simulation study is found to be comparable to the wavelet shrinkage methods proposed by Donoho and Johnstone. The problem of how to count the degrees of freedom for an adaptively chosen set of basis functions is addressed. This issue arises also in the MARS procedure proposed by Friedman and other adaptive regression spline procedures.

Key Words:

• Inflated degrees of freedom
• Regression spline
• Smoothing on the sphere
• Smoothing spline
• Spatial adaptability
• Stepwise regression