Approximating data with weighted smoothing splines

Abstract

Given a data set (t_i, y_i), i = 1,... ,n with the t_i ∈ [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. A common desideratum is that the function fn be smooth but the path towards this goal is often the indirect one of assuming a “true” data generating function f and then measuring performance by the expected mean square. The approach taken in this paper is a different one. We specify precisely what we mean by a function fn being an adequate approximation to the data and then, using weighted splines, we try to maximize the smoothness given the approximation constraints.