Trouble fitting an explicitly complex model with the TransformedFit package

Question

This is follow up from a previous question in which @Oleksandr R. suggested the use of a package he developed for splitting a model and data into real and imaginary parts to avoid a problem with FindFit when it encounters complex solutions. I eventually had success with fitting the model in the previous question but now I have moved on to a more complex model and I'm getting an error I don't know how to interpret.

The model is called qres and is defined as:

Clear[ns, ns\[Infinity], m, A, \[CapitalDelta]E, T, a, b, p, qst, \
qobs, R]
\[Theta] = ns/ns\[Infinity];
b = A*Exp[\[CapitalDelta]E/(R*T)];
p = (b/(\[Theta]^-m - 1))^(1/m);

qst = R*T*
Log[Psat[T]/b^(1/m)*(\[Theta]^m/(1 - \[Theta]^m))^((m - 1)/m)] + 
R*T*Z[T] + \[Lambda]p;

qobs = FullSimplify[Integrate[qst, ns]]*.039;

qres = qobs - (513.4933382*ns - 98.64056065*ns^2);

Several variables are dependent functions of ns and T:

R = 8.314;
Psat[85.] = 78896.59231;
Psat[100.] = 323767.1859;
Psat[120.] = 1213037.739;
Psat[140.] = 3168227.12;
Z[85.] = -1.584*10^-13*p^2 - 3.494*10^-7*p + 1;
Z[100.] = -6.147*10^-14*p^2 - 2.185*10^-7*p + 1;
Z[120.] = -2.519*10^-14*p^2 - 1.264*10^-7*p + .9997;
Z[140.] = -4.203*10^-21*p^3 + 3.235*10^-15*p^2 - 8.985*10^-8*p + 1;
\[Lambda]p = 
Piecewise[{{6540.2, Re[p] <= 68890.88832}}, 
3.74853*10^-35*p^6 - 3.91100*10^-28*p^5 + 1.60666*10^-21*p^4 - 
3.33138*10^-15*p^3 + 3.74416*10^-9*p^2 - 3.12419*10^-3*p + 
6723.71];

By choosing values for T and ns as well as guessing at the model parameters ns[Infinity], m, A, [CapitalDelta]E I can see that various parts of the model return the expected numerical results:

T = 140; ns = .5; ns\[Infinity] = 1.3052144404401276; m = \
0.33692827685320603; A = 2907.997380634411; \[CapitalDelta]E = \
-4579.218711936131;
Psat[T]
p
Z[T]
\[Lambda]p
qobs
qres
Clear[T, ns, ns\[Infinity], m, A, \[CapitalDelta]E]

The data to fit the model to are:

Data = {{0, 85, 0}, {0.131592244`, 85, 3.209459459`}, {0.230286428`, 
85, 2.432432432`}, {0.255592629`, 85, 
2.027027027`}, {0.382123633`, 85, 0.033783784`}, {0.426409484`, 
85, -0.743243243`}, {0.516246497`, 
85, -1.047297297`}, {0.60102227`, 
85, -1.689189189`}, {0.642777502`, 
85, -0.540540541`}, {0.749063545`, 85, 
0.168918919`}, {0.760451336`, 85, -0.472972973`}, {0.762981956`, 
85, 0.27027027`}, {0, 100, 0}, {0.074653293`, 100, 
1.689189189`}, {0.111347284`, 100, 1.689189189`}, {0.127796314`, 
100, 1.689189189`}, {0.212572087`, 100, 
0.033783784`}, {0.269511039`, 100, -1.081081081`}, {0.29481724`, 
100, -1.790540541`}, {0.408695144`, 
100, -4.189189189`}, {0.523838358`, 
100, -5.810810811`}, {0.536491458`, 
100, -6.013513514`}, {0.551675179`, 
100, -6.081081081`}, {0.58710386`, 
100, -6.317567568`}, {0.632655021`, 100, -6.621621622`}, {0, 120, 
0}, {0.072122672`, 120, 0.033783784`}, {0.127796314`, 
120, -0.878378378`}, {0.258123249`, 
120, -1.891891892`}, {0.260653869`, 
120, -5.304054054`}, {0.29481724`, 
120, -5.574324324`}, {0.349225572`, 
120, -6.722972973`}, {0.356817432`, 
120, -8.378378378`}, {0.361878672`, 
120, -8.344594595`}, {0.372001153`, 
120, -8.986486486`}, {0.415021694`, 
120, -10.91216216`}, {0.430205415`, 
120, -10.70945946`}, {0.474491266`, 
120, -12.16216216`}, {0.526368978`, 120, -13.51351351`}, {0, 140, 
0}, {0.051877712`, 140, -0.912162162`}, {0.080979843`, 
140, -1.385135135`}, {0.111347284`, 
140, -2.668918919`}, {0.116408524`, 
140, -2.804054054`}, {0.158163755`, 
140, -4.391891892`}, {0.169551546`, 140, -5}, {0.187265886`, 
140, -5.878378378`}, {0.248000768`, 140, -8.918918919`}};

I have installed Oleksandr R.'s "TransformedFit" package and tried to use it to fit the function qres to the data:

ComplexFit[Data, qres, {{ns\[Infinity], 1.3052144404401276}, {m, 
0.33692827685320603}, {A, 
2907.997380634411}, {\[CapitalDelta]E, -4579.218711936131}} , {ns, 
T}]

I get the error message:

FindFit::nrlnum: "The function value {0.,-69.0731+0.0051321\ (12314.5 +706.69\ (1. +Re  
[Plus[<<2>>]])),-115.452+0.00898117\ (11880.9 +706.69\ (1. +Re[Plus
[<<2>>]])),<<45>>,0.,0. +0.0051321\ (0. +706.69\ Im[Times[<<2>>]+Times[<<2>>]]),<<46>>} 
is not a list of real numbers with dimensions {96} at {TransformedParameter
$11,TransformedParameter$15,TransformedParameter$12,TransformedParameter
$16,TransformedParameter$13,TransformedParameter$17,TransformedParameter
$14,TransformedParameter$18} = {1.30521,0.,0.336928,0.,2908.,0.,-4579.22,0.}."

I would be grateful for any suggestions.

I tested the package on all of the models discussed on this site that led to problems, but I never tried it with a model as complicated as yours. Let me look into it and I'll get back to you later. — Oleksandr R., Apr 28 '14 at 09:59
Thank you very much @Oleksandr R.. I've tried doing the transformation of the model and data "manually" as in the previous question I asked and got a similar error message: — Pete in Perth, Apr 29 '14 at 02:08
FindFit::nrlnum: "The function value {0.,-69.0731+0.0051321\ (12314.5 +706.69\ (1. +Re[Plus[<<2>>]])),<<47>>,0. +3.6268\ Im[-6656.38\ Times[<<2>>]^2.96799-5.7489*10^7\ Times[<<2>>]^5.93598],<<46>>} is not a list of real numbers with dimensions {96} at {ns\[Infinity],m,A,\[CapitalDelta]E} = {1.30521,0.336928,2908.,-4579.22}." — Pete in Perth, Apr 29 '14 at 02:16

score 0 · Accepted Answer · answered May 01 '14 at 07:47

Apologies for the delay in replying. I didn't need to look too deeply into the cause of the issue you encountered, because I didn't like the way your model was written, so the first thing I did was to change

Psat[85.] = 78896.59231;
Psat[100.] = 323767.1859;
Psat[120.] = 1213037.739;
Psat[140.] = 3168227.12;
Z[85.] = -1.584*10^-13*p^2 - 3.494*10^-7*p + 1;
Z[100.] = -6.147*10^-14*p^2 - 2.185*10^-7*p + 1;
Z[120.] = -2.519*10^-14*p^2 - 1.264*10^-7*p + .9997;
Z[140.] = -4.203*10^-21*p^3 + 3.235*10^-15*p^2 - 8.985*10^-8*p + 1;

into

Psat[T_] := Piecewise[{
    {78896.59231, T == 85}, {323767.1859, T == 100},
    {1213037.739, T == 120}, {3168227.12, T == 140},
    {Indeterminate, True}
   }];

Z[T_] := Piecewise[{
    {-1.584*10^-13*p^2 - 3.494*10^-7*p + 1, T == 85},
    {-6.147*10^-14*p^2 - 2.185*10^-7*p + 1, T == 100},
    {-2.519*10^-14*p^2 - 1.264*10^-7*p + .9997, T == 120},
    {-4.203*10^-21*p^3 + 3.235*10^-15*p^2 - 8.985*10^-8*p + 1, T == 140},
    {Indeterminate, True}
   }];

after which the problem immediately disappeared, giving what seems to be a sensible result:

ComplexFit[Data, qres, {
  {ns\[Infinity], 1.3052144404401276},
  {m, 0.33692827685320603},
  {A, 2907.997380634411},
  {\[CapitalDelta]E, -4579.218711936131}
 }, {ns, T}, "CoordinateSystem" -> "Real"
]

(* emits FindFit::sszero message *)
(* -> { ns\[Infinity] -> 2.74479, m -> 0.310828, 
        A -> 1586.86, \[CapitalDelta]E -> -4028.45 } *)

One can easily imagine why the former approach might give rise to problems: on the LHS is a pattern that is compared in a MatchQ sense with the argument, so for example Psat[85.] is defined while Psat[85] isn't. Also, supplying an argument corresponding to a temperature for which values are not known does not evaluate, whereas in fact it ought to evaluate to an indeterminate result.

Note that "CoordinateSystem" -> "Real" only forces the values of the parameters to be real; it does not matter for this whether the model contains complex numbers or if it evaluates to a complex result. Since you state in the title that the model is explicitly complex, yet I don't see any complex numbers in its definition, perhaps you wanted complex values for the parameters. I assumed not in the above because I can't see how complex Arrhenius parameters would be meaningful, but maybe I misunderstood something. If so, you can do as you did in the question and specify "CoordinateSystem" -> "Cartesian" (or specify it implicitly by omitting that option entirely), and you will get the best-fitting complex values for the parameters.

Thank you very much Oleksandr R. I suspected that there might be a problem with how I had formatted the model but I didn't know how to do it better. I work as an engineer but as you might guess, programing and working in the Mathematica language are a little outside my training - I'm working on it but I have a long ways to go. Anyway, I really appreciate your help, in just a couple of post you've taught me some useful things. You're quite right about my only wanting the real part of the solution. — Pete in Perth, May 02 '14 at 10:06
@PeteinPerth you're very welcome. Note though that this is not the same thing as just taking the "real part of the solution"--here we additionally force the imaginary part to be zero, so that the solution is purely real. If you allow for a complex result and take its real part, the results will be substantially different (not to mention, not correct). — Oleksandr R., May 02 '14 at 11:30

Trouble fitting an explicitly complex model with the TransformedFit package

1 Answers1