FindFit “is not a real number” error when fitting a parametric function to the data

Question

I am using Mathematica's FindFit function to fit a parametric solution to my data and I am getting the error

FindFit::nrnum: The function value [complicated expression] is not a real number at {ka,kd,beta} = {0.7,0.1,0.1}

he initial guesses of the parameters {ka=0.7,kd=0.1,beta=0.1} are very close to the values I used to mock up the data, and I can't figure out where I am going wrong. I posted a similar version of this question on stack exchange but didn't get any response.

First I use the ParametricNDSolveValue function to parametrically solve two coupled first order differential equations for the functions 'a[t]' and 'b[t]' with parameters ka, kd, and beta:

sol = ParametricNDSolveValue[{D[a[t], t] == -kd*a[t] + ka*(1 - a[t] - b[t]), a[0] == ka/(ka + kd), 
  D[b[t], t] == -kd*b[t] + ka*(1 - Exp[-beta*t]) (1 - a[t] - b[t]), b[0] == 0 }, {a, b}, 
  {t, 0, 20}, {ka, kd, beta}]

I can extract the solution for a[t] for ka=0.7, kd=0.1, beta=0.1 by using:

y1 = sol[0.7, 0.1, 0.1][[1]]

Plotting y1 as a function of t gives the expected curve:

Plot[y1[t], {t, 0, 20}, PlotRange -> All]

Then I create a data table eyeballing the values from the above plot:

dataA = {{0, 0.87}, {5, 0.84}, {10, 0.75}, {15, 0.68}, {20, 0.62}}

Finally to test FindFit, I use FindFit with sol[ka, kd, beta][[1]][t] to fit the data giving the same initial values for ka, kd, and beta that I used to generate the above data:

fit = FindFit[dataA, sol[ka, kd, beta][[1]][t], {{ka, 0.7}, {kd, 0.1}, {beta, 0.1}}, t]

At which point I get the error:

FindFit::nrnum: "The function value 1/2\ ((-0.87+Abs[0.7[0.]])^2+(-0.84+Abs[0.7[5.]])^2+(-0.75+Abs[0.7[10.]])^2+(-0.68+Abs[0.7[15.]])^2+(-0.62+Abs[0.7[20.]])^2) is not a real number at {ka,kd,beta} = {0.7,0.1,0.1}."

I tried using Abs[sol[ka, kd, beta][[1]][t]] in FindFit, that didn't work. Then I tried adding constraints:

fit = FindFit[dataA, {Abs[sol[ka, kd, beta][[1]][t]], {ka > 0, kd > 0, 
  beta > 0}}, {{ka, 0.7}, {kd, 0.1}, {beta, 0.1}}, t]

And that failed too.

Any clues what is wrong with my code? Thanks a lot!

Answer 1

sol = ParametricNDSolve[{D[a[t], t] == -kd*a[t] + ka*(1 - a[t] - b[t]), a[0] == ka/(ka + kd), 
                        D[b[t], t] == -kd*b[t] + ka*(1 - Exp[-beta*t]) (1 - a[t] - b[t]), 
                        b[0] == 0}, {a, b}, {t, 0, 20}, {ka, kd, beta}];

dataA = {{0, 0.87}, {5, 0.84}, {10, 0.75}, {15, 0.68}, {20, 0.62}};

fit = FindFit[dataA, a[ka, kd, beta][t] /. sol, {{ka, 0.7}, {kd, 0.1}, {beta, 0.1}}, t]

 (* {ka -> 0.932926, kd -> 0.136649, beta -> 0.0603393} *)

Plot[{a[.7, .1, .1][t] /. sol, a[ka, kd, beta][t] /. fit /. sol}, {t, 0, 20}, PlotRange -> All]

Answer 2

The problem is that sol[ka, kd, beta][[1]][t] evaluates to ka[t] before numeric values are substituted for the parameters. You need to prevent evaluation of [[1]] until after sol has evaluated to a list of solutions.

Thanks to Mr.Wizard for pointing out Indexed, which will do what we're after.

fit = FindFit[dataA, Indexed[sol[ka, kd, beta], 1][t], {{ka, 0.7}, {kd, 0.1}, {beta, 0.1}}, t]
(*  {ka -> 0.932928, kd -> 0.136649, beta -> 0.0603392}  *)

In pre-V10, something like this will work (original answer):

firstIfList[l_List] := First[l]

fit = FindFit[dataA, firstIfList[sol[ka, kd, beta]][t], {{ka, 0.7}, {kd, 0.1}, {beta, 0.1}}, t]
(*  {ka -> 0.932928, kd -> 0.136649, beta -> 0.0603392}  *)

See Prevent Part[] from trying to extract parts of symbolic expressions for a similar issue with Part.