MaplePrimes - answers and comments on Question, Can we get an exact curve fitting line?

Hermite interpolation and Hermite spline

Alec Mihailovs — Fri, 20 May 2011 05:11:34 Z

You could try a Hermite interpolation and a proper Hermite spline (which, probably, would require more data entered).

It's easier in Mathematica,

L = {3, 4, 6, 7, 2, 3, 5, 4, 6, 8, 20, 4, 5, 12, 0, 5, 5, 5, 3};

Show[Plot[ListInterpolation[L][x], {x, 1, 19}], ListPlot[L]]

Alec

Curve Fitting Options

Doug Meade — Fri, 20 May 2011 09:13:09 Z

@Christopher2222 The oscillations and overshoots become more likely as the degree of the interpolating polynomial increases. One possibility to combat this effect is to use interpolating piecewise polynomials (often b-splines). Maple supports this.

Another option is to relax your search and not insist that the approximation interpolate the data. If you look for a best fitting curve it won't pass through most points, but overall it will reflect various general trends within the data set. You do appear to want quite a bit of turning throughout the interval, so this might not be so appropriate.

If you really know where your curve will have critical points and inflection points, these become additional constraints. While, in principle, these require higher-order polynomials, they often have other oscillations in addition to the ones prescribed and so might not be any better than what you are seing at present. I think I might be tempted to utilize piecewise cubic (or quartic) with additional constraints at the transition points between cubics to ensure that the overall curve is continuous and that an appropriate number of derivatives agree at the transition points.

I hope at least one of these ideas makes some sense to you, and that you are able to put these ideas to good use. Post again if you have more questions or encounter any problems.

Doug

Hermite splines

Alec Mihailovs — Sat, 21 May 2011 06:55:42 Z

I've just produced the following picture in Maple using cubic Hermite splines,

Is it what you wanted?

_______________
Alec Mihailovs, PhD

PS I put the procedures in a separate post. -Alec

Smooth...

longrob — Fri, 20 May 2011 13:56:31 Z

Alec, is it possible to use that method while maintainng smothness ?

cspline

Christopher2222 — Fri, 20 May 2011 17:33:36 Z

So maybe the Cubic Hermite Spline (cspline) is what I need. From the wikipedia page link Alec provided the finite difference method or finite difference tangents looks promising. I just need to interperet all that algebra (makes my head spin) into maple code.

I would need some help creating this cspline procedure.

smoothness

Alec Mihailovs — Fri, 20 May 2011 18:22:29 Z

@longrob

Sure, just requires some manual adjusting, adding derivative values, for instance, and second derivatives may be added etc. - there are many examples in the Mathematica help - it has options Method->Spline, which produces a more smooth looking curve, Method->Hermite etc.

On the other hand, for such irregularly oscillating data as in this particular example, a simple line graph (not smooth) - i.e. just connecting the points with line segments, gives, maybe, even better idea about what is happening - and it can be easily done in Maple using plot; maybe with adding moving average, which is available in the Statistics package in Maple if I recall correctly - I'll check it later.

Alec