Approximating Data with weighted smoothing Splines

TL;DR

This paper introduces a method using weighted smoothing splines to adaptively approximate data, ensuring local variations and derivatives are accurately captured while maintaining overall smoothness.

Contribution

It presents a simple approach for achieving local adaptivity in both the regression function and its derivatives using residual-based weighted smoothing splines.

Findings

Effective local adaptivity in function and derivatives

Balances local fit with global smoothness

Applicable to data with large local variations

Abstract

Given a data set (t_i, y_i), i=1,..., n with the t_i in [0,1] non-parametric regression is concerned with the problem of specifying a suitable function f_n:[0,1] -> R such that the data can be reasonably approximated by the points (t_i, f_n(t_i)), i=1,..., n. If a data set exhibits large variations in local behaviour, for example large peaks as in spectroscopy data, then the method must be able to adapt to the local changes in smoothness. Whilst many methods are able to accomplish this they are less successful at adapting derivatives. In this paper we show how the goal of local adaptivity of the function and its first and second derivatives can be attained in a simple manner using weighted smoothing splines. A residual based concept of approximation is used which forces local adaptivity of the regression function together with a global regularization which makes the function as smooth as possible subject to the approximation constraints.