Nonlinear least squares

Question

I have a typical nonlinear least square problem and it seems very similar to the example provided in the section on parameter fitting at the Wolfram website. Unfortunately I always get some error messages.

My code is as follows - could anyone help get rid of the error messages?

fitdata = {{1962, 0}, {1963, 0.9}, {1964, 2.4}, {1965, 6.2}, {1966, 11.3}, 
           {1967, 17.1}, {1968, 22.95}, {1969, 28.85}, {1970, 33.55}};

pfun = ParametricNDSolveValue[
           { Derivative[1][x][t] == m (a (1 - x[t]) + b (1 - x[t]) x[t]^h), 
             x[0] == 0}, 
            x, {t, 0, 10}, {a , b, h, m}];

fit = FindFit[ fitdata, 
               pfun[a, b, h, m][t], 
               {{a, 0.02}, {b, 0.5}, {h, 0.8}, {m, 42}}, t]

Answer 1

This is an old (and possibly abandoned) question; however, its solution involves a number of issues related to NonlinearModelFit and ParametricNDSolveValue that show up on this site and it's perhaps constructive to look at this problem as a case study.

Rescaling the fit data

We wish to fit the following data:

fitdata = {{1962, 0}, {1963, 0.9}, {1964, 2.4}, {1965, 6.2}, {1966, 
    11.3}, {1967, 17.1}, {1968, 22.95}, {1969, 28.85}, {1970, 33.55}};

The first issue is one of scaling, since the model we want to use to fit the data has bounds (pardon me, Mathematicians) on y from 0 to 1. Additionally, it is practical to shift the x-data such that the first point is 0. This latter issue is easily addressed, but since we do not know the upper limit of fitdata[[All,2]] we run in to a problem. I attempt to solve this problem by introducing another adjustable parameter, ymax:

rsfitdata = fitdata;
rsfitdata[[All, 1]] = fitdata[[All, 1]] - 1962;
rsfitdata[[All, 2]] = Rescale[fitdata[[All, 2]], {0, ymax}, {0, 1}]

The re-scaled fitdata (rsfitdata) is now ready for the linear regression, so long as we have a good approximate value for ymax.

Model shape

I find it instructive to look at the model to make sure there is some hope of fitting the data to this model. We do this with a manual approach using Manipulate:

pfun = ParametricNDSolveValue[{Derivative[1][x][
      t] == (a (1 - x[t]) + b (1 - x[t]) x[t]^h), x[0] == 0}, 
   x, {t, 0, 10}, {a, b, h}];

Manipulate[
 Plot[pfun[a, b, h][t], {t, 0, 10}, 
  Epilog -> {Red, Point@(rsfitdata /. ymax -> ymax1)}, 
  PlotRange -> {0, 1}], {{a, 0.02}, 0, 1}, {{b, 0.544}, 0, 
  1}, {{h, 0.93}, 0, 1}, {{ymax1, 40.8}, 1, 100}]

The results here are promising, and a manual approach, using the approximate values provided by the OP, yields a model that looks like it could fit the data satisfactorily.

Using NonlinearModelFit with ParametricNDSolveValue

First, it is beneficial to ensure that the model is using numerical values. One approach is to create a model wrapper such as:

model[a_, b_, h_, y_][t_] := 
 y pfun[a, b, h][t] /; And @@ NumericQ /@ {a, b, h, y, t}

Here, I will move ymax from the data and use it as another adjustable parameter in the model. Finally, when we execute the nonlinear regression, we need to make sure that the appropriate constraints are applied to the parameters; in this case, a, b, and h must all be > 0 and y must be > 1. (Technically >=, but this works.)

nlm = NonlinearModelFit[
  rsfitdata /. ymax -> 1, {model[a, b, h, y][t], a > 0, b > 0, h > 0, 
   y > 1}, {{a, 0.02}, {b, 0.544}, {h, 0.93}, {y, 40.8}}, t]

This command takes a few minutes on my machine, and the algorithm does not converge in 500 iterations; however we do obtain results which are reasonable:

nlm["BestFitParameters"]
(* {a -> 0.0067278, b -> 0.458518, h -> 0.7424, y -> 43.6642} *)
Plot[model[a, b, h, y][t - 1962] /. nlm["BestFitParameters"], {t, 
  1962, 1972}, Epilog -> {Red, Point@fitdata}]

Note in the last plot I am overlaying the original fitdata.

Summary

When trying to perform sophisticated fits of this nature, I suggest the following general approach:

• Ensure that your data are scaled appropriately
• Spend time manually finding suitable starting values
• Use some technique to ensure numeric values will be passed to NonlinearModelFit
• Verify the results of your fit with the original data

Appendix

Thanks to Oleksandr R. for pointing this out: using Method -> {"NMinimize", Method->"NelderMead"} results in convergence of the model fit providing parameters very close to what is obtained above:

nlm["BestFitParameters"] (* With NMinimize/NelderMead method *)
(* {a -> 0.00670292, b -> 0.440683, h -> 0.731268, y -> 44.5793} *)