Problem with parameters from fitting

Question

I am trying to fit the following data (called TflogqIND):

TflogqIND={{0., 36.4886}, {Log[2]/Log[10], 37.1485}, {Log[5]/Log[10], 
  38.3859}, {1, 39.2263}, {Log[20]/Log[10], 39.8772}, {Log[30]/
  Log[10], 40.0107}}

to the following equation:

 eqn = ((log10q - Log10[qref]) == 
       c1*(Tfp - Tfpref)/(c2 + (Tfp - Tfpref))); (*WLF equation*)
    model = Tfp /. Solve[eqn, Tfp][[1]] // FullSimplify;
    constIND = {Tfpref -> 39.2263, 
       qref -> 10};
    modelIND = model /. (constIND // Rationalize) // FullSimplify;
    
    nlmIND = NonlinearModelFit[
       TflogqIND, {modelIND, c1 > 5, c2 > 5}, {c1, c2}, log10q]; 

As you can see from plotting:

Show[ ListPlot[TflogqIND, PlotMarkers -> Style[\[FilledSquare], 18, Red], Frame -> True, Axes -> False, FrameStyle -> 16, LabelStyle -> {Black, Bold, 10}, ImageSize -> Large, GridLines -> Automatic, GridLinesStyle -> Lighter[Gray, .8], PlotRange -> All], Plot[nlmIND[log10q], {log10q, 0, 1.4}, PlotStyle -> {Red, Dashed}]]

The model seems to describe the data well. However, the parameters (obtained from nlmIND["BestFitParameters"] with {c1 -> 1331.87, c2 -> 3520.53}) don't make any sense. Usually c1 and c2 parameters are from 1 to around 100 or 200 at the most.

I think there must be a mistake somewhere but I cannot see it. How can I fix the model to give better c1 and c2 values and better describe the data using that equation?

Answer 1

The problem is that with the available data and proposed model there are no finite values of the two parameters ($c_1$ and $c_2$) that can minimize the sum of squares (which is essentially what NonlinearModelFit uses in this case). In this case lack of numerical precision is not the culprit (although it is one of the usual suspects in many cases).

The "good" news is that an infinite number of solutions will give almost the same set of predictions. The "bad" news is that one can't place any interpretation on any particular chosen set of parameters.

Here's why. To avoid confusion with the lack of numerical precision culprit Rationalize the data:

TflogqIND = {{0., 36.4886}, {Log[2]/Log[10], 37.1485}, {Log[5]/Log[10], 38.3859},
  {1, 39.2263}, {Log[20]/Log[10], 39.8772}, {Log[30]/Log[10], 40.0107}};
TflogqIND = Rationalize[TflogqIND, 0];

The resulting model ends up being the following:

modelIND = 392263/10000 + (c2 (-1 + log10q))/(1 + c1 - log10q);

The sum of squares to be minimized is the following:

sumOfSquares = Total[(#[[2]] - modelIND /. log10q -> #[[1]])^2 & /@ TflogqIND];

The value of c2 that minimizes the sum of squares (assuming that there is a value that does so) is found by finding where the derivative of the sumOfSquares with respect to c2:

solc2 = Solve[D[sumOfSquares, c2] == 0, c2][[1]];

So we now have a solution for the minimum of the sum of squares given a value of c1. Plotting that minimum function against values of c1 shows that the absolute minimum is never realized for any finite value of c1:

Plot[sumOfSquares /. solc2, {c1, 0, 50000}]

The limit of the sum of squares as c1 -> ∞ is

Limit[sumOfSquares /. solc2, c1 -> ∞] // N
(* 0.312188 *)

So any set of values with c2/.solc2 that has a sum of squares close to 0.312188 will give an almost identical fit.

Consider c1 = 4000. Then the values of c2 and the sum of squares are

solc2 /. c1 -> 4000 // N
(* {c2 -> 10569.9} *)

sumOfSquares /. solc2 /. c1 -> 4000 // N
(* 0.31263 *)

With c1 = 40000 we have

solc2 /. c1 -> 40000 // N
(* {c2 -> 105685.} *)

sumOfSquares /. solc2 /. c1 -> 40000 // N
(* 0.312232 *)

Finally: given the data that you have, just fitting a + b*log10q might be a better approach.