Spatially adaptive smoothing splines

Abstract

SUMMARY We use a reproducing kernel Hilbert space representation to derive the smoothing spline solution when the smoothness penalty is a function A(t) of the design space t, thereby allowing the model to adapt to various degrees of smoothness in the structure of the data. We propose a convenient form for the smoothness penalty function and discuss computational algorithms for automatic curve fitting using a generalised crossvalidation measure. of order m, where m is typically taken to be two. The smoothing spline solution uses a global smoothing parameter il which implies that the true underlying mean process has a constant degree of smoothness. In this paper we suggest a more general, 'spatially adaptive', framework that accommodates varying degrees of roughness by seeking solutions where the smoothness penalty A(t) is now a function of the design t. We derive the solution within a reproducing kernel Hilbert space (Gu, 2002, Ch. 2; Wahba, 1990, Ch. 1). When applying the method to data we propose a piecewise-constant model for the smoothing function A(t). This provides a convenient computational framework with closed-form solutions for the corresponding reproducing kernels of the Hilbert space. These kernels show similarities to regression splines in that the use of a piecewise-constant i~(t) induces breaks in the mth, ... ., (2m -l1)th derivatives at the changepoints in L(t).