Approximating data with weighted smoothing splines∗

Given a data set (ti, yi), i = 1, . . . , n with the ti ∈ [0, 1] non-parametric regression is concerned with the problem of specifying a suitable function fn : [0, 1] → R such that the data can be reasonably approximated by the points (ti, fn(ti)), 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. AMS 2000 Subject classifications: Primary 62G08, secondary 62G15, 62G20