Re: Estimation of curvature based on noisy data
- From: Han de Bruijn <Han.deBruijn@xxxxxxxxxxxxxx>
- Date: Wed, 09 Jul 2008 09:29:03 +0200
Anders wrote:
On Jul 8, 4:22 pm, Han de Bruijn <Han.deBru...@xxxxxxxxxxxxxx> wrote:
Peter Spellucci wrote:
In article <9287e0fe-96e7-4379-ad63-279a04653...@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
Anders <Anders.Lyckega...@xxxxxxxxx> writes:
>Hi all,
>I'm working on a bit postprocessing of experimental data.I'm trying to
>find a robust algorithm for estimation of the curvature of a beam
>based on measurements of deflections and inclination of a beam.
>I have a few measurements of deflection and inclinations of a beam
>along the length of it. Now I would like to find the curvature and an
>estimate of the accuracy of my estimate.
>Furthermore, I could possibly define some bounds for the magnitude of
>curvature and a physical model for the beam, i.e. equilibrium.
>I have tried fitting different polynomials to the data points using
>regression analysis, but I get different results based on my choice of
>polynomial.
>Thus, I was hoping to find a method that defines the problem in a more
>general way, and can provide an accurate estimate and an estimate of
>the accuracy.
>I have a feeling that estimation theory might be helpful, but I'm not
>familiar with it and don't know were to start off.
>http://en.wikipedia.org/wiki/Estimation_theory
>Any suggestions are welcome,
>Best,
>Anders
using polynomials is not a good idea for this task since these will exhibit
a strong oscillatory behaviour especially for higher degrees.
for your task a smoothing spline with a variable number
of nodes (and using estimates of the variance of the noise) might be usable,
however, since you
also have inclination data you cannot use existing software off the shelf
which allows (x,y=deflection) data only. but the basic principle of
derivation of such a spline applies: minimizing a weighted sum
of the integral of the square of curvature and the least squares sum
of deviation of model data from measured data.
hth
peter
Splines, yes. The simplest ones are quadratic. Here are some basics:
http://hdebruijn.soo.dto.tudelft.nl/www/programs/delphi.htm#knuth
Han de Bruijn
Thank you for the suggestion.
We did try splines. The problem is that if we use interpolation based
on splines to fit the deflection and inclinations we an amplification
of the noise once we find the curvature. In fact, the curvature ends
up outside of the bounds that we think are physically reasonable.
Best,
Anders
I hope you didn't just fit the splines to your noisy data. The curvature
contains first and second derivatives; therefore an amplification of the
noise can be expected then. I think you should formulate kind of a least
squares approximation with splines in the first place, no interpolation,
as has been suggested by Peter Spellucci. Did you do that?
Han de Bruijn
.
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