# Problem with NonlinearModelFit: “The function value… is not a list of real numbers…”

I'm trying to use NonlinearModelFit to fit the data:

Data = {{2.046204620462046, 1.274347668}, {2.012987012987013,
1.131369542}, {1.984, 0.939944276}, {1.952755905511811,
0.721156437}, {1.9224806201550388,
0.614591638}, {1.8931297709923665,
0.572324748}, {1.8674698795180722,
0.536605273}, {1.8397626112759644,
0.49948005}, {1.8128654970760234,
0.478452409}, {1.7867435158501441,
0.455583518}, {1.7613636363636365,
0.439702881}, {1.7342657342657342,
0.411870668}, {1.712707182320442,
0.365204419}, {1.6870748299319729,
0.303633546}, {1.6666666666666667,
0.232056102}, {1.6445623342175066,
0.140710942}, {1.6230366492146597,
0.070744185}, {1.6020671834625324,
0.035633753}, {1.597938144329897, 0.044761696}};

fit = NonlinearModelFit[Data, B/(
1 + g (x - (1.42 + Ea))^(-3/2)), {B, g, Ea}, x]

which gives:

NonlinearModelFit::nrlnum: "The function value {-1.22471-0.217191\ I,-1.0682-0.243262\ I,-0.863406-0.265857\ I,-0.628591-0.289822\ I,<<12>>,0.26533 -0.472365\ I,0.318041 -0.47811\ I,0.312374 -0.479155\ I}\n is not a list of real numbers with dimensions {19} at {B,g,Ea} = {1.,1.,1.}."

The function I am using to fit my data gets imaginary values for some values of the parameters. But how can I force Mathematica to avoid these values?

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The reason is because it hit a negative value under the root, which ended up generating imaginary numbers. You can try using either assumptions, or use Abs, like this: !Mathematica graphics now it worked. If this meets your model or not, I do not know. I do not do stats. But that is the reason why it failed. –  Nasser Aug 25 '13 at 5:05
another option is to add a constraint, like this !Mathematica graphics again, I do not know if this meets your model needs or not. –  Nasser Aug 25 '13 at 5:16
When I try to evaluate the code, Mathematica 9.0.1 neither evaluates the code nor returns an error !Mathematica graphics. However following what @Nasser has suggested fixes the problem. –  M6299 Aug 25 '13 at 5:31
@Nasser would you consider an answer? You solved the problem, so why not document it? –  Yves Klett Aug 25 '13 at 6:42
This is a common problem. I've addressed it in my preferred fashion in this answer. –  Oleksandr R. Aug 25 '13 at 15:37

I run a trace on the command, Trace[NonlinearModelFit[Data, B/(1 + g (x - (1.42 + Ea))^(-3/2)), {B, g, Ea}, x]] and looking at the result, I noticed a check is being done using FreeQ[...,Complex] and right after that, saw lots of \$Failed messages (hard to read trace messages).

So when I Saw that I realized it was the use of the root in the model. Internally, it was generating a numerical data to fit, and used values that caused a negative value to appear, for example

B/(1 + g (x - (1.42 + Ea))^(-3/2)) /. {B -> 1, g -> 1, Ea -> 1,   x -> 1}
(*  0.7411531317364896 - 0.4380013322510369 I  *)

And NonlinearModelFit does not like to work with complex numbers to make the fit.

So, what to do? Add a constraints, or use Abs value to make sure the value under the root remain non-negative. I also tried Assumptions, but these did not help.

The constraint is added after form. For example

NonlinearModelFit[Data, {B/(1 + g (x - (1.42 + Ea))^(-3/2)), Ea <= 0}, {B, g, Ea}, x]

It seems your model did not consider that complex numbers can be generated for some values, and the correct constraint is needed to insure this does not happen.

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Important to note is that specifying constraints also changes the method from Levenberg-Marquardt to NLIP. Although the latter will certainly obey any constraints given, its convergence properties do not seem to be as good. –  Oleksandr R. Aug 25 '13 at 15:43