I am trying to find the coefficients for following equation:

$\quad \quad f(x,\,y,\,z) = a (x^2)\, exp(b x + y^2) + c z^2 y$

using NonlinearModelFit. But I can't find an example of how to formulate the data matrix and equation for NonlinearModelFit when there is more than one independent variable. Can someone provide an example?


closed as off-topic by Sjoerd C. de Vries, Oleksandr R., bbgodfrey, Dr. belisarius, Daniel Lichtblau May 1 '15 at 14:20

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  • $\begingroup$ If you mean to generate a matrix with this formula, you can try Table. $\endgroup$ – happy fish May 1 '15 at 4:29
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  • $\begingroup$ The first example in the "Scope" block of the NonlinearModelFit help page precisely shows the necessary steps. Took me less than 15 secs to find. $\endgroup$ – Sjoerd C. de Vries May 1 '15 at 6:42

This is what I understand you're asking:

For your problem the data matrix has to be of the form {{x,y,z,f},{...},...}. I will create some data.

data = MapThread[{#1[[1]], #1[[2]], #1[[3]], 
          0.34*(#1[[1]]^2)*Exp[-1.82*#1[[1]] + #1[[2]]^2] + 
            0.94*(#1[[3]]^2)*#1[[2]] + #2} &,
  {RandomReal[1, {100, 3}], RandomReal[{-0.01, .01}, 100]}];


model = a*(x^2)*Exp[b*x + y^2] + c*(z^2)*y   
NonlinearModelFit[data, model, {a, b, c}, {x, y, z}]

(*FittedModel[0.385936 E^(-1.89747 x + y^2) x^2 + 0.943513 y z^2]*)

To generate data using the given equation, you can use different values for {a,b,c} to generate data. Here's a method using With where {a,b,c} are local variables:

f[x_, y_, z_] := With[{a = 0.1, b = -3., c = 1.2}, a *x^2 *Exp[b *x + y^2] + c* z^2* y];

Then you need to build you dataset; You can use Table and Flatten to have your data in suitable format ({x,y,z,f[x,y,z]}):

data = Flatten[
   Table[{x, y, z, f[x, y, z]}, {x, -3, 3}, {y, -3, 3}, {z, -3, 3}], 

then using NonLinearModelFit is simple:

 a *x^2 *Exp[b *x + y^2] + c* z^2* y, {a, b, c}, {x, y, z}]
(* 0.1 *x^2 *Exp[-0.3 *x + y^2] + 1.2 * z^2 * y *)

therefore fitted values for {a,b,c} are the same as given in $f(x,y,z)$ definition.


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