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[?] Fit noisy data as a Least squares problem?

Lorenzo Fuggiano, University
Posted 4 years ago

I want to fit a set of noisy data, obtained from the equation fm with added noise, to the same equation (with y=0; t=1).

 Dif = 0.0000001; K0 = 0.5; z = 0; Intens = 25000;
    angulo1 = 45*Pi/180;
    lsup1 = Cos[angulo1]; linf\[Alpha] = 0.0;
    m1 = -Tan[angulo1];

fm = T145[x_, y_, t_, b_, l_, d_] := 
  Intens/(2*\[Pi]*K0)*
   NIntegrate[
    Sqrt[1 + m1^2]*
     Erfc[Sqrt[(x - \[Alpha])^2 + (y - \[Beta])^2 + (z - (m1*\[Alpha] \
- d))^2]/Sqrt[
        Dif*4*t]]/(Sqrt[(x - \[Alpha])^2 + (y - \[Beta])^2 + (z - \
(m1*\[Alpha] - d))^2]), {\[Beta], -b/2, b/2}, {\[Alpha], linf\[Alpha],
      l*lsup1}]

Here data with added noise and then fitting:

Block[{y = 0, t = 1, b = 0.001, l = 0.001, d = 0.0001}, 
  data = Table[{x, 
     fm[x, y, t, b, l, d] + 
      Random[NormalDistribution[0, 0.1]]}, {x, -0.0015, 0.002, 
     0.000015}]];

model = Intens/(2*\[Pi]*K0)*
   NIntegrate[
    Sqrt[1 + m1^2]*
     Erfc[Sqrt[(x - \[Alpha])^2 + (-\[Beta])^2 + (z - (m1*\[Alpha] - 
             d))^2]/Sqrt[
        Dif*4*1]]/(Sqrt[(x - \[Alpha])^2 + (-\[Beta])^2 + (z - (m1*\
\[Alpha] - d))^2]), {\[Beta], -b/2, b/2}, {\[Alpha], linf\[Alpha], 
     l*lsup1}];

fit = FindFit[data, model, {b, l, d}, x, 
  Method -> "LevenbergMarquardt"]

But the results are {b->1.,l->1.,d->1.}, that are patently wrong. Where is the problem? Thanks to all.
POSTED BY: Lorenzo Fuggiano
4 Replies
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Well I think you only have to find the correct order of magnitude of the fit parameters. Which I think you should know beforehand.

If not you could do just a brute force evaluation of different orders of magnitude of the initial parameters and select the best values for your initial guess.

Block[{y = 0, t = 1, b = 0.0015, l = 0.0011, d = 0.00013}, 
  data = Table[{x, 
     fm[x, y, t, b, l, d] + 
      Random[NormalDistribution[0, 0.1]]}, {x, -0.0015, 0.002, 
     0.000015}]];

bi = li = di = {10.^-2, 10.^-3, 10.^-4, 10.^-5}
Block[{y = 0, t = 1},
  all = SortBy[Flatten[Table[
      datai = 
       Table[{x, 
         fm[x, y, t, b, l, d] + 
          Random[NormalDistribution[0, 0.1]]}, {x, -0.0015, 0.002, 
         0.000015}];
      {RootMeanSquare[data[[All, 2]] - datai[[All, 2]]], 
       datai, {b, l, d}},
      {b, bi}, {l, li}, {d, di}], 2], First]
  ];
{lsq, datai, {bi, li, di}} = First[all];

{lsq, {bi, li, di}}
ListLinePlot[all[[All, 2]], PlotRange -> All]
ListLinePlot[{data, datai}, PlotRange -> All]

enter image description here
POSTED BY: Martijn Froeling
Updating Name
Updating Name
Posted 4 years ago

Do you think it would be possible to obtain a more robust system to the variations of the initial parameters? I mean that I would like the fitting results (b, l, d) to be as similar as possible even by varying the initial search values. Thanks and sorry for the further inconvenience.
POSTED BY: Updating Name

Thank you very much. It solved my problem. So adding the command I'm saying that the function must be evaluated only if the input parameters are numeric. Thanks!
POSTED BY: Lorenzo Fuggiano
POSTED BY: Martijn Froeling
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