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I have a set of data, how can I fit the best function to that? i use FindFit and then I guess the function myself, but I want Mathematica to find the best fit. How can it be done?

y1 = {48.999, 57.180, 63.8421, 68.81, 72.228, 74.428, 75.782, 76.60, 
   77.138, 77.531, 77.886, 78.264, 78.697, 79.201, 79.784, 80.444, 
   81.180, 81.986, 82.857, 83.786, 84.7678, 85.796, 86.866, 87.974`, 
   89.113, 90.281, 91.474, 92.687`, 93.920, 95.167, 96.428, 97.700, 
   98.981, 100.268, 101.562, 102.859, 104.158, 105.460, 106.761, 
   108.062, 109.361, 110.658, 111.951};

y2 = Table[y2, {y2, 50, 470, 10}];

data = Transpose[{y2, y1}];
x1 = ListLinePlot[data]

enter image description here

FindFit[data, d x^4 + a x^3 + b x^2 + c x + f, {d, a, b, c, f}, x]

Show[ListPlot[data, PlotStyle -> Red], 
 Plot[d x^4 + a x^3 + b x^2 + c x + 
    f /. {d -> -1.2911673126124723`*^-8, 
    a -> 0.000014771971357899056`, b -> -0.005794083194578543`, 
    c -> 0.9998897221123239`, f -> 16.20821389434905`}, {x, 50, 500}]]

enter image description here

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    $\begingroup$ Easiest would be to get some inspiration from theoretical insights. Where do these data come from? What do you know about the asymptotes? etc. $\endgroup$
    – Roman
    Commented Oct 12, 2023 at 19:42
  • $\begingroup$ Put in plots of data and fit. Revert if you don't want those shown. $\endgroup$
    – Hugh
    Commented Oct 12, 2023 at 20:53
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    $\begingroup$ If you wish to use the fitted function in further analysis then an interpolation function could be suitable. You can do everything with an interpolation function that you could do with a fitted function. Do you need an example of the approach. What are you going to do with your fitted function? $\endgroup$
    – Hugh
    Commented Oct 12, 2023 at 20:56
  • $\begingroup$ @Hugh i want to enter it in another formula. $\endgroup$
    – li xu
    Commented Oct 12, 2023 at 21:12
  • $\begingroup$ You can put an interpolation function in a formula. It will work fine. $\endgroup$
    – Hugh
    Commented Oct 12, 2023 at 21:15

2 Answers 2

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I couldn't resist devoting 30 seconds of DataModeler at this. FindFormula[ ] was not all that good but evolution did a pretty decent job:

Table of developed models and fit to data

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  • $\begingroup$ hi, thanks, can you elaborate on that? Is it a package? I used FindFormula and it woorked quite good, but this one is much better. $\endgroup$
    – li xu
    Commented Oct 13, 2023 at 6:48
  • $\begingroup$ DataModeler is a commercial product which implements symbolic regression as well as the surrounding infrastructure you need for multivariate feature selection and model development. You can get a trial license from www.evolved-analytics.com. $\endgroup$
    – Mark Kotanchek
    Commented Oct 13, 2023 at 15:38
  • $\begingroup$ Thanks for the explanation $\endgroup$
    – li xu
    Commented Oct 13, 2023 at 16:40
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Many of the other answers are very good but I do think a simple interpolation function could be the simplest. Particularly as your data does not seem to include and noise. Below I make an interpolation function then invent a function to show how easy it is to use in other functions.

    y1 = {48.999, 57.180, 63.8421, 68.81, 72.228, 74.428, 75.782, 76.60, 
       77.138, 77.531, 77.886, 78.264, 78.697, 79.201, 79.784, 80.444, 
       81.180, 81.986, 82.857, 83.786, 84.7678, 85.796, 86.866, 87.974`, 
       89.113, 90.281, 91.474, 92.687`, 93.920, 95.167, 96.428, 97.700, 
       98.981, 100.268, 101.562, 102.859, 104.158, 105.460, 106.761, 
       108.062, 109.361, 110.658, 111.951};
    
    
    y2 = Table[y2, {y2, 50, 470, 10}];
    
    
    data = Transpose[{y2, y1}];
    {x1, x2} = {y2[[1]], y2[[-1]]};
    
    ClearAll[f];
    f[x_] := Evaluate[ Interpolation[data, Method -> "Spline"][x]]
Plot[f[x], {x, x1, x2}]

enter image description here

Now lets put the new function into an invented function and plot that

  Plot[f[x]^2/(90 - f[x] + 0.5 x), {x, 50, 470}, PlotRange -> All]

enter image description here

You can take the derivative of a function containing the interpolation function.

g[x_] := f[x]^2/(90 - f[x] + 0.5 x)

dg = D[g[x], x];

Plot[dg, {x, x1, x2}]

enter image description here

The only tricky issue is to integrate. You need a starting value but then you can use NDSolve to give you a new integrated interpolation function.

int = y /. First@NDSolve[{y'[x] == g[x], y[0] == 0}, y, {x, x1, x2}];

Plot[int[x], {x, x1, x2}]

enter image description here

Hope that helps. I think an interpolation function is simplest if you do not have noise in your data and you have no need to smooth.

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  • $\begingroup$ thanks, I should work on it, but I need something like y=F(x) can you explain this part, I did not get it: {x1, x2} = {y2[[1]], y2[[-1]]}; $\endgroup$
    – li xu
    Commented Oct 13, 2023 at 9:29
  • $\begingroup$ The x1 and x2 are the first and last points of the x values. You called your abscissi y1 and I was just trying to make things easier by having a symbol rather than typing the numbers into plot etc. [[1]] is the first point and [[-1]] the last point of a list. $\endgroup$
    – Hugh
    Commented Oct 13, 2023 at 11:36
  • $\begingroup$ thanks for the explanation, I got it. $\endgroup$
    – li xu
    Commented Oct 13, 2023 at 12:56

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