fitting - Using Manipulate to find best starting parameters to fit data

Often, curve fitting is very sensitive to starting values of parameters. It would be great, if I could to find such starting values of parameters "by hand" using Manipulate.

So, I plot in Manipulate my experimental data points and theoretical curve. I can change the shape of this curve by changing all parameters in manipulate in such way that they approximate my experimental data relatively well. Now I would like to run fitting using current parameters in Manipulate as starting values. Finally I would like to insert the parameters found by fitting back into Manipulate.

Here is example code for simple function. My data is more complex.

data = Table[{x, 8 x^3 - 7 x^2 - 10 x + 1 + RandomReal[{-5, 5}]}, {x, -2, 2, 0.1}];
Manipulate[
 Show[
  Plot[a x^3 + b x^2 + c x + d, {x, -2, 2}],
  ListPlot[data]
 ],
 {a, -10, 10},

 {b, -10, 10},
 {c, -10, 10},
 {d, -10, 10}
]

Answer

As I understand the question a curve fitting procedure that has the following properties is sought:

1. Manually adjust the parameters to get an approximate fit.


2. Use these parameters as the starting values for FindFit.


3. Propagate the solution from FindFit back to the Manipulate parameters.



4. Subsequently enable further editing of the Manipulate parameters and repeat the cycle.

The following code satisfies this criteria by wrapping Manipulate inside a DynamicModule and the use of a Button to indicate when FindFit should be run.

data = Table[{x, 8 x^3 - 7 x^2 - 10 x + 1 + RandomReal[{-5, 5}]}, {x, -2, 2, 0.1}];

DynamicModule[
 {
  sol
  },


 Manipulate[

  If[computeFlag == True,
   sol = FindFit[data, 
     aa x^3 + bb x^2 + cc x + 
      dd, {{aa, a}, {bb, b}, {cc, c}, {dd, d}}, x];
   {a, b, c, d} = {aa, bb, cc, dd} /. sol;
   computeFlag = False;
   ];


  Column[{
    Dynamic[
     Button["Compute",
      computeFlag = True
      ]
     ],
    Show[
     Plot[a x^3 + b x^2 + c x + d, {x, -2, 2},
      PlotStyle -> Black],
     ListPlot[data, PlotStyle -> Red],

     ImageSize -> 300,
     PlotRange -> {{-2.05, 2.05}, All}
     ]
    }],

  (*Manipulate variables*)
  {{computeFlag, False}, ControlType -> None},

  {{a, 1}, -10, 10, Appearance -> "Open"},
  {{b, 1}, -10, 10, Appearance -> "Open"},

  {{c, 1}, -10, 10, Appearance -> "Open"},
  {{d, 1}, -10, 10, Appearance -> "Open"}

  ] (* end of Manipulate *)
 ] (* end of DynamicModule *)

Below is a figure where the parameters have been manually adjusted.

After clicking the Compute button FindFit propagates the solution back to the Manipulate parameters.

The user is free to re-edit the Manipulate parameters and repeat the cycle.