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Re: Smoothing vs. Fitting Splines

• To: mathgroup at smc.vnet.net
• Subject: [mg7208] Re: [mg7171] Smoothing vs. Fitting Splines
• From: Edgar Camacho Cota <ecc at ds5000.super.unam.mx>
• Date: Fri, 16 May 1997 02:30:45 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

On Tue, 13 May 1997, Peter Buttgereit wrote:

> 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
> 
> 
In mathSource exist some implementation of Savinsky Golay Low Pass Filter for
Smoothing set of sampling data. this function like a filter for smoothing 
that show the book Numerical Recipes ...