Toolbox for testing if a probability distribution fits a power law
powerlaw: A Python Package for Analysis of Heavy-Tailed Distributions
powerlaw is a toolbox using the statistical methods developed in
Clauset et al. 2007 <http://arxiv.org/abs/0706.1062>_ and
Klaus et al. 2011 <http://www.plosone.org/article/info%3Adoi%2F10.1371%2Fjournal.pone.0019779>_ to determine if a
probability distribution fits a power law. Academics, please cite as:
Jeff Alstott, Ed Bullmore, Dietmar Plenz. (2014). powerlaw: a Python package
for analysis of heavy-tailed distributions.
PLoS ONE 9(1): e85777 <http://www.plosone.org/article/info%3Adoi%2F10.1371%2Fjournal.pone.0085777>_
Also available at
arXiv:1305.0215 [physics.data-an] <http://arxiv.org/abs/1305.0215>_
For the simplest, typical use cases, this tells you everything you need to know.::
import powerlaw data = array([1.7, 3.2 ...]) # data can be list or numpy array results = powerlaw.Fit(data) print(results.power_law.alpha) print(results.power_law.xmin) R, p = results.distribution_compare('power_law', 'lognormal')
For more explanation, understanding, and figures, see the paper,
which illustrates all of
powerlaw's features. For details of the math,
see Clauset et al. 2007, which developed these methods.
Paper illustrating all of powerlaw's features, with figures <http://arxiv.org/abs/1305.0215>__
Code examples from manuscript, as an IPython Notebook <http://nbviewer.ipython.org/github/jeffalstott/powerlaw/blob/master/manuscript/Manuscript_Code.ipynb>__
Note: Some results involving lognormals will now be different from the
manuscript, as the lognormal fitting has been improved to allow for
greater numerical precision.
This code was developed and tested for Python 2.x with the
Enthought Python Distribution <http://www.enthought.com/products/epd.php>, and later amended to be
compatible with 3.x. The full version of Enthought is
available for free for academic use <http://www.enthought.com/products/edudownload.php>.
powerlaw is hosted on
PyPI <https://pypi.python.org/pypi/powerlaw>__, so installation is straightforward. The easiest way to install type this at the command line (Linux, Mac, or Windows)::
or, better yet::
pip install powerlaw
pip just need to be on your PATH, which for Linux or Mac is probably the case.
pip should install all dependencies automagically. These other dependencies are
matplotlib. These are all present in Enthought, Anaconda, and most other scientific Python stacks. To fit truncated power laws or gamma distributions,
mpmath is also required, which is less common and is installable with::
pip install mpmath
The requirement of
mpmath will be dropped if/when the
gammaincc are updated to have sufficient numerical accuracy for negative numbers.
You can also build from source from the code here on Github, though it may be a development version slightly ahead of the PyPI version.
Update Notifications and Mailing List
Get notified of updates by joining the Google Group
Questions/discussions/help go on the Google Group
here <https://groups.google.com/forum/?fromgroups#!forum/powerlaw-general>__. Also receives update info.
The original author of
powerlaw, Jeff Alstott, is now only writing minor tweaks, but
powerlaw remains open for further development by the community. If there's a feature you'd like to see in
powerlaw you can
submit an issue <https://github.com/jeffalstott/powerlaw/issues>_, but pull requests are even better. Offers for expansion or inclusion in other projects are welcomed and encouraged.
Many thanks to Andreas Klaus, Mika Rubinov and Shan Yu for helpful
discussions. Thanks also to
Andreas Klaus <http://neuroscience.nih.gov/Fellows/Fellow.asp?People_ID=2709>,
Aaron Clauset, Cosma Shalizi <http://tuvalu.santafe.edu/~aaronc/powerlaws/>,
Adam Ginsburg <http://code.google.com/p/agpy/wiki/PowerLaw>_ for making
their code available. Their implementations were a critical starting point for
Power Laws vs. Lognormals and powerlaw's 'lognormal_positive' option
When fitting a power law to a data set, one should compare the goodness of fit to that of a
lognormal distribution <https://en.wikipedia.org/wiki/Lognormal_distribution>. This is done because lognormal distributions are another heavy-tailed distribution, but they can be generated by a very simple process: multiplying random positive variables together. The lognormal is thus much like the normal distribution, which can be created by adding random variables together; in fact, the log of a lognormal distribution is a normal distribution (hence the name), and the exponential of a normal distribution is the lognormal (which maybe would be better called an expnormal). In contrast, creating a power law generally requires fancy or exotic generative mechanisms (this is probably why you're looking for a power law to begin with; they're sexy). So, even though the power law has only one parameter (
alpha: the slope) and the lognormal has two (
mu: the mean of the random variables in the underlying normal and
sigma: the standard deviation of the underlying normal distribution), we typically consider the lognormal to be a simpler explanation for observed data, as long as the distribution fits the data just as well. For most data sets, a power law is actually a worse fit than a lognormal distribution, or perhaps equally good, but rarely better. This fact was one of the central empirical results of the paper
Clauset et al. 2007 <http://arxiv.org/abs/0706.1062>, which developed the statistical methods that
However, for many data sets, the superior lognormal fit is only possible if one allows the fitted parameter
mu to go negative. Whether or not this is sensible depends on your theory of what's generating the data. If the data is thought to be generated by multiplying random positive variables,
mu is just the log of the distribution's median; a negative
mu just indicates those variables' products are typically below 1. However, if the data is thought to be generated by exponentiating a normal distribution, then
mu is interpreted as the median of the underlying normal data. In that case, the normal data is likely generated by summing random variables (positive and negative), and
mu is those sums' median (and mean). A negative
mu, then, indicates that the random variables are typically negative. For some physical systems, this is perfectly possible. For the data you're studying, though, it may be a weird assumption. For starters, all of the data points you're fitting to are positive by definition, since power laws must have positive values (indeed,
powerlaw throws out 0s or negative values). Why would those data be generated by a process that sums and exponentiates negative variables?
If you think that your physical system could be modeled by summing and exponentiating random variables, but you think that those random variables should be positive, one possible hacks is
lognormal_positive. This is just a regular lognormal distribution, except
mu must be positive. Note that this does not force the underlying normal distribution to be the sum of only positive variables; it only forces the sums' average to be positive, but it's a start. You can compare a power law to this distribution in the normal way shown above::
R, p = results.distribution_compare('power_law', 'lognormal_positive')
You may find that a lognormal where
mu must be positive gives a much worse fit to your data, and that leaves the power law looking like the best explanation of the data. Before concluding that the data is in fact power law distributed, consider carefully whether a more likely explanation is that the data was generated by multiplying positive random variables, or even by summing and exponentiating random variables; either one would allow for a lognormal with an intelligible negative value of
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