Wolfram Research

Function Repository Resource:

FitPowerLaw

Source Notebook

Fit a power law to data

Contributed by: Sander Huisman

ResourceFunction["FitPowerLaw"][data]

fits data with a power law by fitting the log of data with a linear function.

ResourceFunction["FitPowerLaw"][data,x]

fits data and uses the variable x for the resulting fits.

Details and Options

ResourceFunction["FitPowerLaw"] effectively fits the function a xb to the data, but does so by fitting the linear function c x+d to the log of the data.
ResourceFunction["FitPowerLaw"] fits in 'log space' such that fits that span multiple decades of data are not dominated by a few of the largest values and the smallest values are basically neglegible. The errors are therefore not in the sense of a classical least-square but rather the square of the log of the ratio of the fitted value and the original data. Visually this then results in a 'good fit' when the data is visualized in a log-log plot of the data.
The data data should either be a list of points {y1, y2, y3, …,yn} (to which the x coordinates 1 through n are assigned), or the data is {x,y} pairs: {{x1,y1},{x2,y2},…}, for a list of lists of data ResourceFunction["FitPowerLaw"] will fit each sublist, returning in that case a list of associations.
Only positive values are fitted. Negative points are ignored.
ResourceFunction["FitPowerLaw"] has the following options:
"Exponent"Automaticprescribe the exponent
"Prefactor"Automaticprescribe the prefactor
If both options are given, only the "Prefactor" option is used.
ResourceFunction["FitPowerLaw"] returns an Association with fit and fit parameters.

Examples

Basic Examples (1) 

Create some data and fit it:

In[1]:=
ResourceFunction["FitPowerLaw"][
 Table[{i, 8.9 i^0.34}, {i, 10^Range[1, 4, 0.1]}]]
Out[1]=

Scope (3) 

For a single list of values the x coordinates are assumed to be 1,2,3,4…:

In[2]:=
ResourceFunction["FitPowerLaw"][{1, 4, 9, 15, 25, 37}]
Out[2]=
In[3]:=
ResourceFunction[
 "FitPowerLaw"][{{1, 1}, {2, 4}, {3, 9}, {4, 15}, {5, 25}, {6, 37}}]
Out[3]=

For a list of lists of data each list will be fitted separately:

In[4]:=
ResourceFunction[
 "FitPowerLaw"][{{1, 4, 9, 15, 25, 37}, {1, 6, 7, 8, 3}}]
Out[4]=

For a list of pairs the data is assumed to be in {x,y} pairs:

In[5]:=
ResourceFunction["FitPowerLaw"][{{1, 2}, {3, 4}, {5, 6}}]
Out[5]=

Options (2) 

Prescribe the exponent:

In[6]:=
SeedRandom[1234];
d = Table[{i, 14 (i RandomVariate[NormalDistribution[1, 0.1]])^0.5}, {i, 10^Range[1, 4, 0.01]}];
ResourceFunction["FitPowerLaw"][d, "Exponent" -> 1/2]
Out[8]=

Prescribe the prefactor:

In[9]:=
SeedRandom[1234];
d = Table[{i, 8 (i RandomVariate[NormalDistribution[1, 0.1]])^0.312}, {i, 10^Range[1, 4, 0.01]}];
ResourceFunction["FitPowerLaw"][d, "Prefactor" -> 8]
Out[10]=

Applications (1) 

Create some data, fit it, and then plot both:

In[11]:=
SeedRandom[1234];
d = Table[{i, 8 (i RandomVariate[NormalDistribution[1, 0.2]])^0.312}, {i, 10^Range[1, 4, 0.01]}];
fit = ResourceFunction["FitPowerLaw"][d, x];
Show[{ListLogLogPlot[d], LogLogPlot[fit["BestFit"], {x, 10, 10000}, PlotStyle -> Red]}]
Out[12]=

Possible Issues (3) 

The difference between a classical least-squares fit or the fitting of the least-squares in log-space becomes very apparent when fitting over several decades. Because the error is much larger for the higher decades the error is completely dominated by those larger points, resulting in a strange fit when viewed in log-space:

In[13]:=
d = CompressedData["
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"];
fit = ResourceFunction["FitPowerLaw"][d, x];
fm = NonlinearModelFit[d, a x^b, {a, b}, x];
Show[{ListLogLogPlot[d, PlotRange -> All], LogLogPlot[{fit["BestFit"], fm["BestFit"]}, {x, 10, 10000}, Sequence[PlotStyle -> {
Darker[Green], Red}, PlotLegends -> {"This method", "Classical least squares fit"}]]}]
Out[16]=

The error for the classical method is lower:

In[17]:=
err1 = (fit["BestFit"] /. x -> d[[All, 1]]) - d[[All, 2]];
err2 = (fm["BestFit"] /. x -> d[[All, 1]]) - d[[All, 2]];
{err1 . err1, err2 . err2}
Out[19]=

But not in log space, here the current method is lower:

In[20]:=
err1 = Log[fit["BestFit"] /. x -> d[[All, 1]]] - Log[d[[All, 2]]];
err2 = Log[fm["BestFit"] /. x -> d[[All, 1]]] - Log[d[[All, 2]]];
{err1 . err1, err2 . err2}
Out[22]=

Publisher

SHuisman

Version History

  • 1.0.0 – 07 October 2022

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