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 x^b 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, …,y_n} (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"

Automatic

prescribe the exponent

"Prefactor"

Automatic

prescribe 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]:=

Out[2]=

In[3]:=

Out[3]=

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

In[4]:=

Out[4]=

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

In[5]:=

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]:=

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