Fitting Experimental data knowing the possible fitting formula

Question

Good Morning, I have a set of experimental data which i want to fit with a formula that I know it works. I tried to write a code in mathematica notebook that I put it here
data = {{1.45045*10^8, 1.}, {7.44768*10^8, 0.8787}, {1.81148*10^9,0.7013}, {3.34183*10^9, 0.5414}, {5.33711*10^9,0.3959}, {7.80159*10^9, 0.2703}, {1.07275*10^10,0.1857}, {1.41183*10^10, 0.1399}, {1.79806*10^10,0.1022}, {2.2302*10^10, 0.07084}, {2.70884*10^10,0.04772}, {3.23398*10^10,0.03562}, {3.80655*10^10,0.02567}, {4.42474*10^10, 0.01419}, {5.08944*10^10,0.01431}, {5.80178*10^10, 0.01108}};

model = 1/(1 + σ^2/DD*b)^(DD/σ)^2;
nml = NonlinearModelFit[data, model, {DD, σ}, b]

Show[
  Plot[nlm[b], {b, 0, 10^11}],
  ListPlot[data, PlotStyle -> {Darker@Green, PointSize[0.03]}]
]

It gives some errors.
I want to show the superposition of original data and the fitting function on semiLog (in the ordinate axis).
Can you help me Please? Thank you.
Have a nice day.

Anton Antonov · Answer

Consider not using a pre-conceived model:
Quantile Regression
qrFunc = ResourceFunction["QuantileRegression"][data, 6, 0.5][[1]];

Simplify[qrFunc[t]]

ListPlot[{data, {#, qrFunc[#]} & /@ data[[All, 1]]}, 
 Joined -> {False, True}, 
 PlotLegends -> {"Original", "QuantileRegression"}]

FindFormula
ffFunc = FindFormula[data];

(* ffFunc = Piecewise[{{1.3784372096038422 - 
       0.018976070116910493*Log[#1] - 1.504938184291008*^-10*#1, 
      Inequality[1.45045*^8, LessEqual, #1, Less, 
              2.664504676051075*^9]}, {9.462060930568198 - 
       0.4091705591204782*Log[#1] + 1.6776712526675216*^-11*#1, 
            
      Inequality[2.664504676051075*^9, LessEqual, #1, Less, 
       2.177107509602669*^10]}, {2.3369081717676714 - 
       0.09516385999471486*Log[#1], 
            
      Inequality[2.177107509602669*^10, LessEqual, #1, Less, 
       3.5441247654932556*^10]}, {1.5828759881621233 - 
       0.0634289715131976*Log[#1], 
            
      Inequality[3.5441247654932556*^10, LessEqual, #1, Less, 
       4.756828177139195*^10]}}, 0] & ; *)

ListPlot[{data, {#, ffFunc[#]} & /@ data[[All, 1]]}, 
 PlotLegends -> {"Original", "FindFormula"}]

JimB · Answer

Changing the model definition from
model = 1/(1 + σ^2/DD*b)^(DD/σ)^2;

to
model = 1/(1 + σ^2/DD*(b/10^10))^(DD/σ)^2;

will fix the problem.
data = {{1.45045*10^8, 1.}, {7.44768*10^8, 0.8787}, {1.81148*10^9, 0.7013},
  {3.34183*10^9, 0.5414}, {5.33711*10^9, 0.3959}, {7.80159*10^9, 0.2703},
  {1.07275*10^10, 0.1857}, {1.41183*10^10, 0.1399}, {1.79806*10^10, 0.1022}, 
  {2.2302*10^10, 0.07084}, {2.70884*10^10, 0.04772}, {3.23398*10^10, 0.03562},
  {3.80655*10^10, 0.02567}, {4.42474*10^10, 0.01419}, {5.08944*10^10, 0.01431},
  {5.80178*10^10, 0.01108}};
model = 1/(1 + σ^2/DD*(b/10^10))^(DD/σ)^2;
nlm = NonlinearModelFit[data, model, {DD, σ}, b];
nlm["BestFitParameters"]
(* {DD -> 2.05829, σ -> 1.22768} *)
Show[ListPlot[data], Plot[nlm[x], {x, 0, 6 10^10}, PlotRange -> All]]

Fitting Experimental data knowing the possible fitting formula

2 Answers

Quantile Regression

`FindFormula`

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