Mathematica "FindFit" starting value

Question

I believe I have encountered a case in Mathematica where the FindFit function is not able to capture the fitting parameter value even when the starting value is approximately equal to the best-fit paremeter value.

I have data I am trying to fit:

data = {{0., 1.}, {0.0001, 0.0946391}, {0.0002, 0.050354}, {0.0003, 
  0.034469}, {0.0004, 0.0262545}, {0.0005, 0.0212236}, {0.0006, 
  0.0178217}, {0.0007, 0.0153659}, {0.0008, 0.0135088}, {0.0009, 
  0.0120546}, {0.001, 0.0108848}, {0.0011, 0.00992321}, {0.0012, 
  0.00911858}, {0.0013, 0.00843532}, {0.0014, 0.00784783}, {0.0015, 
  0.00733724}, {0.0016, 0.00688935}, {0.0017, 0.00649325}, {0.0018, 
  0.00614043}, {0.0019, 0.00582415}, {0.002, 0.005539}, {0.0021, 
  0.00528059}, {0.0022, 0.00504532}, {0.0023, 0.00483021}, {0.0024, 
  0.00463277}, {0.0025, 0.0044509}, {0.0026, 0.00428282}, {0.0027, 
  0.00412703}, {0.0028, 0.00398222}, {0.0029, 0.00384727}, {0.003, 
  0.00372119}, {0.0031, 0.00360315}, {0.0032, 0.0034924}, {0.0033, 
  0.00338827}, {0.0034, 0.0032902}, {0.0035, 0.00319767}, {0.0036, 
  0.00311021}, {0.0037, 0.00302743}, {0.0038, 0.00294896}, {0.0039, 
  0.00287446}, {0.004, 0.00280365}, {0.0041, 0.00273626}, {0.0042, 
  0.00267204}, {0.0043, 0.00261077}, {0.0044, 0.00255226}, {0.0045, 
  0.00249633}, {0.0046, 0.0024428}, {0.0047, 0.00239152}, {0.0048, 
  0.00234236}, {0.0049, 0.00229519}, {0.005, 0.00224988}, {0.0051, 
  0.00220634}, {0.0052, 0.00216445}, {0.0053, 0.00212413}, {0.0054, 
  0.00208529}, {0.0055, 0.00204785}, {0.0056, 0.00201173}, {0.0057, 
  0.00197686}, {0.0058, 0.00194319}, {0.0059, 0.00191065}, {0.006, 
  0.00187919}, {0.0061, 0.00184874}, {0.0062, 0.00181927}, {0.0063, 
  0.00179073}, {0.0064, 0.00176307}, {0.0065, 0.00173626}, {0.0066, 
  0.00171025}, {0.0067, 0.00168501}, {0.0068, 0.00166051}, {0.0069, 
  0.00163671}, {0.007, 0.00161359}, {0.0071, 0.00159111}, {0.0072, 
  0.00156925}, {0.0073, 0.00154798}, {0.0074, 0.00152729}, {0.0075, 
  0.00150714}, {0.0076, 0.00148752}, {0.0077, 0.0014684}, {0.0078, 
  0.00144977}, {0.0079, 0.00143161}, {0.008, 0.0014139}, {0.0081, 
  0.00139662}, {0.0082, 0.00137976}, {0.0083, 0.00136331}, {0.0084, 
  0.00134724}, {0.0085, 0.00133155}, {0.0086, 0.00131622}, {0.0087, 
  0.00130124}, {0.0088, 0.0012866}, {0.0089, 0.00127228}, {0.009, 
  0.00125828}, {0.0091, 0.00124459}, {0.0092, 0.00123119}, {0.0093, 
  0.00121808}, {0.0094, 0.00120524}, {0.0095, 0.00119267}, {0.0096, 
  0.00118037}, {0.0097, 0.00116831}, {0.0098, 0.0011565}, {0.0099, 
  0.00114493}, {0.01, 0.00113358}}

This data, above, is basically a type of exponential decay with y-intercept at 1.

The fitting function I am using is:

fit[a_,T_] = 1/(1 + 9.58814*10^7 T^(3/2))

A quick plot of the data and the fitting function shows that the fitting parameter, a, should be relatively close to 2*10^-2.

Show[
     data,
     Plot[
          fit[0.02,T], {T, 0, 0.01}, 
          PlotStyle -> Red
          ]
 ]

which gives

However, even when providing a starting value of 0.02 in the FindFit function, Mathematica is still not able to find a best-fit value...

FindFit[
        data,
        fit[a,T],
        {
        {a, 0.02}
        },
        T
        ][[1]]

which yields the error

FindFit::fmgz: Encountered a gradient that is effectively zero. The result returned may not be a minimum; it may be a maximum or a saddle point.
a -> 0.02

Is this a fundamental flaw in the fitting procedure used by FindFit, or is there something I'm neglecting to incorporate into the function FindFit?

Answer 1

For the data, f = 1/(a + b*T) fits well.

Clear[f,a,b,sol];
f = 1/(a + b*T);
sol = FindFit[data, f, {a, b}, T];
Show[ListPlot[data], Plot[f /. sol, {T, 0, .01}, PlotStyle -> Red]]

If we remove the first of the data,then we can use f = 1/(a + b*T^c)

Clear[f, fit];
f = 1/(a + b*T^c);
fit = NonlinearModelFit[Rest@data, f, {a, b, {c, .5}}, T]
Show[ListPlot[data], Plot[Normal@fit, {T, 0, 0.01}, PlotStyle -> Red]]

Answer 2

In short: the model is completely inadequate for the data and FindFit worked just fine.

It appears that your model is

fit[a_, T_] := 1/(1 + a 9.58814*10^7 T^(3/2))

When you plot the data against fit[0.2, T] (where you have chosen 0.2 for the parameter I assume for the reason because it overestimates roughly equally to the underestimates to the left and right, respectively, of the middle of the predictor values), you get a very wrong picture of that fit because you don't show the first 8 data points. (The first point {0, 1} isn't really a data point but something the model demands.) Here's what that figure should look like with the appropriate result from FindFit given the model you chose:

fit[a_, T_] := 1/(1 + a 9.58814*10^7 T^(3/2))

results = FindFit[data[[2 ;;]], fit[a, T], a, T]
(* {a -> 0.0858187} *)

Show[ListPlot[data[[2 ;;]], PlotRange -> {All, {0, 0.1}}, ImageSize -> Large],
 Plot[{fit[0.02, t], fit[a /. results, t]}, {t, 0, 0.01},
  PlotStyle -> {Red, Green}, PlotRange -> All,
  PlotLegends -> {"Wrong fit", "Correct fit"}]]

When finding the fit that minimizes the sum of squares, the points with the smallest T values have much more influence on the curve chosen. The root mean squares of the two fits are as follows:

(* a = 0.02  *)
Sqrt[((data[[2 ;;, 2]] - fit[0.02, data[[2 ;;, 1]]])^2 // Total)/(Length[data] - 2)]
(* 0.0281204 *)

(* a = result from `FindFit` *)
Sqrt[((data[[2 ;;, 2]] - fit[a /. results, data[[2 ;;, 1]]])^2 // Total)/(Length[data] - 2)]
(* 0.00401312 *)

The guess of 0.02 for a results in a root mean square error of about 7 times that of the result from FindFit.

In short: the model is completely inadequate for the data and FindFit worked just fine.

To describe the data in an adequate compact form, use @cvgmt 's answer.