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Re: Interpolation problems
*To*: mathgroup at smc.vnet.net
*Subject*: [mg120267] Re: Interpolation problems
*From*: Ray Koopman <koopman at sfu.ca>
*Date*: Fri, 15 Jul 2011 05:26:20 -0400 (EDT)
*References*: <ivmclp$fbj$1@smc.vnet.net>
On Jul 14, 2:25 am, James Womack <james.c.wom... at gmail.com> wrote:
> Hello all,
>
> I am having difficulty with interpolating a data set. My data is a
> function of one coordinate -- each data point consists of 2 numbers, one
> denoting position and one denoting the value of the function. If you are
> interested, the data represents a potential energy function.
>
> The best way to demonstrate my issue is to link to some images of plots.
>
> First, here is the result of:
> ListPlot[data, PlotRange -> {{0.4, 0.6}, {-350, 350}}, Joined -> True]http://jcwx86.dyndns.org/public/listplot.jpeg
>
> Second, here is the result of:
> Plot[Interpolation[data][x], {x, 0.4, 0.6}, PlotRange -> {-350, 350}]http://jcwx86.dyndns.org/public/interpolationfunction.jpeg
>
> These plots focus only on the relevant region of the data. The data
> spans x = 0.000857412 to x = 25.6297. The peak you see in the plots is
> cut off with a maximum value of the function at about y = 4000.
>
> As you can see, the interpolation results in two small additional minima
> at the base of the peak. This is problematic, since the function is a
> potential energy function and minima represent stable-states. Additional
> minima present in the data therefore result in problems with subsequent
> calculations using the InterpolationFunction of the data.
>
> I suspect that the issue is to do with the Interpolation command trying
> to fit the function to every datapoint. The rapid change in gradient at
> the base of the peak means that in order to fit splines to the function,
> these minima are necessary.
>
> Ideally, I would like an InterpolationFunction of the data that is (i)
> smooth and (ii) lacks these spurious minima. I have considered deleting
> data points from the base of the peak, as this might enable the splines
> to better fit the peak. Does anyone know of a way that I can modify the
> Interpolation command that enables a fit which better represents the
> data? Or perhaps you know of an alternative method of creating
> interpolated functions which might suit my needs better? I have already
> tried increasing the interpolation order -- this seems only to result in
> more "wobbles" at the base of the peak. Decreasing the interpolation
> order to 1 removes the minima, but also means the function is no longer
> "smooth".
>
> If you want to play around with the data yourself,
> I will post it below in full.
>
> Thanks in advance for your thoughts,
>
> James
>
> data = << ...deleted... >>
Except for the glitch near x = .53, a plot of the data resembles a
left-truncated Cauchy probability density. Using only the first 200
points (x < .48):
Clear[b,c,d,x];
FindFit[Take[data,200], b/(c^2 + (x-d)^2), {b,{c,.01},{d,.002}}, x]
{b -> 0.951445, c -> 0.00793639, d -> 0.00200581}
Using all but points 201...216 (.48 < x < .60) gives the same
coefficients to 6 significant digits.
Try fitting something to the residuals, y - f[x]. I tried another
Cauchy, but it didn't work. I suspect there is simply not enough
data to support blind curve-fitting. You would need more points in
(.48, .60). Do you have substantive information that might suggest
the form of the second function?
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