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I asked a question in a previous post that was closed because "The title of the question is not correct and the issue here is too trivial to help anyone later".

https://mathematica.stackexchange.com/questions/19297/nonlinearmodelfit

It's possible because I'm not so smart in Mathematica. But in this period I've tried a lot of alternative solutions.

Recall my problem

 data={{220., 542.821}, {225., 531.898}, {230., 521.391}, {235., 
 510.596}, {240., 499.603}, {245., 488.585}, {250., 479.002}, {255., 
 468.574}}
 func= a0*Exp[a1 t + a2 t^2]
 nlm = NonlinearModelFit[data, func , {a0, a1, a2}, t]

which gives the result: 

FittelModel = 0.

Some answers told me that it is not an exponential, but linear problem. But this problem is a physical problem that comes out from a lot of papers that describes this phenomenum as having exponential behavior. So also if my question was trivial, I've continued to solve it.

I read a lot of posts, and I tried this solution in Mathematica, using NMinimize and not NonLiNearModelFit.

f[a0_, a1_, a2_][t_] := a0*Exp[a1*t + a2*t^2 ];
error[a0_, a1_, a2_][x_, y_] := (f[a0, a1, a2]@x - y);
errorSum[a0_, a1_, a2_] = Total[(error[a0, a1, a2] @@@ data)];
bestFitVals = 
NMinimize[{errorSum[a0, a1, 
a2], {600 < a0 < 1500&& -0.01 < a1 < 0 && -0.0000001 < a2 < 0}}(*,
Thread[0<{a0,a1,a2}]*), {a0, a1, a2}, Method -> "SimulatedAnnealing"]
soln[x_] = f[a0, a1, a2]@x /. bestFitVals[[2]]
limits = Flatten@{x, Through[{Min, Max}@data[[All, 1]]]};
Show@{Plot[soln@x, Evaluate@limits, PlotRange -> All], 
ListPlot[data, PlotStyle -> Red]}

I tried all the methods available to NMinimize, changing starting values or removing them, but no choice works. So finally a collegue suggested that I try another software package: SigmaPlot. I said him that Mathematica is superior, but Sigmaplot gives me an exponential solution, with these coefficients

a0 = 1072.637060511283
a1 = -2.128998307513597e-3
a2 = -4.390602833555603e-6.

and with residuals that are very low.

I tried with another set of data, but the solution is the same: Mathematica is not be able to regress or minimize and Sigmaplot is able to.

I hope my question is not trivial, but I hope more that someone can solve my question: is Sigmaplot superior of Mathematica, or is it the case that I'm not applying Mathematica properly to my problem?

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    $\begingroup$ So, it would appear this is an issue of giving reasonable starting values. Obviously, the "preferred" value for a0 is large, while a1 and a2 are close to zero. If you don't specify initial values, Mathematica just chooses 1 by default, hence the difficulty you encountered. Try nlm = NonlinearModelFit[data, func, {{a0, 1000}, {a1, 0}, {a2, 0}}, t] instead and you will get the same result as from SigmaPlot. $\endgroup$
    – Oleksandr R.
    Commented Feb 26, 2013 at 19:16
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    $\begingroup$ @s0rce: the data are the same I put inside Sigmaplot and the residuals of exponential fitting is very low compared with linear fit. $\endgroup$
    – Mary
    Commented Feb 26, 2013 at 19:19
  • $\begingroup$ @Oleksandr R: Very good idea. $\endgroup$
    – Mary
    Commented Feb 26, 2013 at 19:22
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    $\begingroup$ Unless you are confident the error variance is constant across different values of $x$, you should instead be fitting the model $\log(y) = \log(a_0) + a_1 t + a_2 t^2$. Use LinearModelFit; it's fast, easy, and can output a huge amount of additional information. $\endgroup$
    – whuber
    Commented Feb 27, 2013 at 0:33
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    $\begingroup$ @rc Constant error variance is assumed by (unweighted) least-squares fitting. When that variance changes, intuitively it makes sense to put less weight on the points with larger variance--and in fact using appropriately weighted least squares is one way to fix up the procedure. But there are great advantages to re-expressing the data to produce constant error variances: interpretation is simpler, prediction is simpler and more reliable, and in physical applications often the re-expressions have meaning. These issues are extensively discussed on stats.stackexchange.com, Mary. $\endgroup$
    – whuber
    Commented Feb 27, 2013 at 14:07

1 Answer 1

Reset to default
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As mentioned in the comments, the default initial parameter (1) for NonlinearModelFit is not appropriate for this question, and initial parameters need to be supplied:

nlm = NonlinearModelFit[data, func, {{a0, 1000}, {a1, 0}, {a2, 0}}, t]
nlm["BestFitParameters"]
(* {a0 -> 1072.64, a1 -> -0.002129, a2 -> -4.3906*10^-6} *)

Apparently SigmaPlot has some mechanism for automatically determining initial parameters but those are contained in libraries that are not immediately accessible to those who don't own a copy of SigmaPlot.

I can only speculate on the rules that SigmaPlot uses to determine initial parameters, and since it uses a library of functions (see page 867 of this long PDF), there is likely no one-shoe-fits-all approach.

Instead of initializing all parameters to 1, another trivial approach would be to initialize all parameters to 0. Oddly enough, this approach works in this instance:

nlm = NonlinearModelFit[data, func, {{a0, 0}, {a1, 0}, {a2, 0}}, t]
nlm["BestFitParameters"] 
{a0 -> 1072.64, a1 -> -0.002129, a2 -> -4.3906*10^-6}
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