Smoothing vs. Fitting Splines

*To*: mathgroup at smc.vnet.net*Subject*: [mg7171] Smoothing vs. Fitting Splines*From*: Peter Buttgereit <Buttgereit at compuserve.com>*Date*: Tue, 13 May 1997 01:58:10 -0400 (EDT)*Sender*: owner-wri-mathgroup at wolfram.com

Dear Mathgroup, in an earlier message I have asked about experiences with the implementation of Smoothing Splines. I think I had better defined what I meant with smoothing splines... Smoothing Splines - contrary to Fitting Splines (SplineFit.m) - do not interpolate data points but represent an estimate of the smooth part of a data series. The basic idea is that the more often a function g is differenciable and the smaller the value g^(k) of the low differenciation order k of that function, the smoother it will look like. Usually you take k = 2. On the other hand, you want g to have something to do with your data. So you have to compromise between these two demands: Smoothness: minimise Intergrate [ d^2 g(t) / dt^2 ]^2 dt Data representation: minimise Sum [ (data(ti) - g(ti) )^2 ] the latter being the least squares approach. All in all, the task is then to minimise beta^2 Sum [ g(ti) - 2 g(ti-1) + g(ti-2)]^2 + Sum [data(ti) - g(ti)]^2 with beta^2 as a Lagrange parameter for smoothness. Does someone know of a good implementation (in Mathematica)? I have scanned MathSource; Prest et al. "Numerical recipes..." don't help, either... Thanks in advance, Peter