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
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
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,
Thanks in advance,
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