Approximating data with weighted smoothing splines

Given a data set (t(i), y(i)), i = 1,...,n with t(i) epsilon [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 showed 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.