# intended to implement a power-law fitting routine as specified in.....
# http://www.santafe.edu/~aaronc/powerlaws/
#
# The MLE for the power-law alpha is very easy to derive given knowledge
# of the lowest value at which a power law holds, but that point is
# difficult to derive and must be acquired iteratively.
"""
Pure-Python version of plfit.py
===============================
A *pure* python power-law distribution fitter based on code by Aaron Clauset.
This is the slowest implementation, but has no dependencies.
Example very simple use::
from plfit_py import plfit
MyPL = plfit(mydata)
MyPL.plotpdf(log=True)
"""
import time
import random
import math
[docs]class plfit:
"""
A Python implementation of the Matlab code http://www.santafe.edu/~aaronc/powerlaws/plfit.m
from http://www.santafe.edu/~aaronc/powerlaws/
See A. Clauset, C.R. Shalizi, and M.E.J. Newman, "Power-law distributions
in empirical data" SIAM Review, 51, 661-703 (2009). (arXiv:0706.1062)
http://arxiv.org/abs/0706.1062
The output "alpha" is defined such that :math:`p(x) \sim (x/xmin)^{-alpha}`
"""
def __init__(self,x,**kwargs):
"""
Initializes and fits the power law. Can pass "quiet" to turn off
output (except for warnings; "silent" turns off warnings)
"""
neg = [i<0 for i in x]
if any(neg) > 0:
print "Removed %i negative points" % (sum(neg))
x = [i for i in x if i > 0]
self.data = x
self.plfit(**kwargs)
[docs] def alpha_(self,x):
""" Create a mappable function alpha to apply to each xmin in a list of xmins.
This is essentially the slow version of fplfit/cplfit, though I bet it could
be speeded up with a clever use of parellel_map. Not intended to be used by users."""
def alpha(xmin,x=x):
"""
given a sorted data set and a minimum, returns power law MLE fit
data is passed as a keyword parameter so that it can be vectorized
"""
x = [i for i in x if i>=xmin]
n = sum(x)
divsum = sum([math.log(i/xmin) for i in x])
if divsum == 0:
return float('inf')
# the "1+" here is unimportant because alpha_ is only used for minimization
a = 1 + float(n) / divsum
return a
return alpha
[docs] def kstest_(self,x):
def kstest(xmin,x=x):
"""
given a sorted data set and a minimum, returns power law MLE ks-test w/data
data is passed as a keyword parameter so that it can be vectorized
The returned value is the "D" parameter in the ks test...
"""
x = [i for i in x if i>=xmin]
n = len(x)
if n == 0: return float('inf')
divsum = sum([math.log(i/xmin) for i in x])
if divsum == 0: return float('inf')
a = float(n) / divsum
cx = [float(i)/float(n) for i in xrange(int(n))]
cf = [1-(xmin/i)**a for i in x]
ks = max([abs(a-b) for a,b in zip(cf,cx)])
return ks
return kstest
[docs] def plfit(self,nosmall=True,finite=False,quiet=False,silent=False,
xmin=None, verbose=False):
"""
A pure-Python implementation of the Matlab code http://www.santafe.edu/~aaronc/powerlaws/plfit.m
from http://www.santafe.edu/~aaronc/powerlaws/
See A. Clauset, C.R. Shalizi, and M.E.J. Newman, "Power-law distributions
in empirical data" SIAM Review, 51, 661-703 (2009). (arXiv:0706.1062)
http://arxiv.org/abs/0706.1062
nosmall is on by default; it rejects low s/n points
can specify xmin to skip xmin estimation
This is only for continuous distributions; I have not implemented a
pure-python discrete distribution fitter
"""
x = self.data
z = sorted(x)
t = time.time()
possible_xmins = sorted(set(z))
argxmins = [z.index(i) for i in possible_xmins]
self._nunique = len(possible_xmins)
if xmin is None:
av = map(self.alpha_(z),possible_xmins)
dat = map(self.kstest_(z),possible_xmins)
sigma = [(a-1)/math.sqrt(len(z)-i+1) for a,i in zip(av,argxmins)]
if nosmall:
# test to make sure the number of data points is high enough
# to provide a reasonable s/n on the computed alpha
goodvals = [s<0.1 for s in sigma]
if False in goodvals:
nmax = goodvals.index(False)
dat = dat[:nmax]
possible_xmins = possible_xmins[:nmax]
av = av[:nmax]
else:
print "Not enough data left after flagging - using all positive data."
if not quiet: print "PYTHON plfit executed in %f seconds" % (time.time()-t)
self._av = av
self._xmin_kstest = dat
self._sigma = sigma
# [:-1] to weed out the very last data point; it cannot be correct
# (can't have a power law with 1 data point).
# However, this should only be done if the ends have not previously
# been excluded with nosmall
if nosmall:
xmin = possible_xmins[dat.index(min(dat))]
else:
xmin = possible_xmins[dat.index(min(dat[:-1]))]
z = [i for i in z if i >= xmin]
n = len(z)
alpha = 1 + n / sum([math.log(a/xmin) for a in z])
if finite:
alpha = alpha*(n-1.)/n+1./n
if n == 1 and not silent:
print "Failure: only 1 point kept. Probably not a power-law distribution."
self._alpha = 0
self._alphaerr = 0
self._likelihood = 0
self._ks = 0
self._ks_prob = 0
self._xmin = xmin
return xmin,0
if n < 50 and not finite and not silent:
print '(PLFIT) Warning: finite-size bias may be present. n=%i' % n
# ks = max(abs( numpy.arange(n)/float(n) - (1-(xmin/z)**(alpha-1)) ))
ks = max( [abs( i/float(n) - (1-(xmin/b)**(alpha-1))) for i,b in zip(xrange(n),z)] )
# Parallels Eqn 3.5 in Clauset et al 2009, but zeta(alpha, xmin) = (alpha-1)/xmin. Really is Eqn B3 in paper.
#L = n*log((alpha-1)/xmin) - alpha*sum(log(z/xmin))
sl = sum([math.log(a/xmin) for a in z])
L = (n*math.log((alpha-1)/xmin) - alpha*sl)
#requires another map... Larr = arange(len(unique(x))) * log((av-1)/unique(x)) - av*sum
self._likelihood = L
self._xmin = xmin
self._xmins = possible_xmins
self._alpha= alpha
self._alphaerr = (alpha-1)/math.sqrt(n)
self._ks = ks # this ks statistic may not have the same value as min(dat) because of unique()
#if scipyOK: self._ks_prob = scipy.stats.kstwobign.sf(ks*numpy.sqrt(n))
self._ngtx = n
if math.isnan(L) or math.isnan(xmin) or math.isnan(alpha):
raise ValueError("plfit failed; returned a nan")
if not quiet:
if verbose: print "The lowest value included in the power-law fit, ",
print "xmin: %g" % xmin,
if verbose: print "\nThe number of values above xmin, ",
print "n(>xmin): %i" % n,
if verbose: print "\nThe derived power-law alpha (p(x)~x^-alpha) with MLE-derived error, ",
print "alpha: %g +/- %g " % (alpha,self._alphaerr),
if verbose: print "\nThe log of the Likelihood (the maximized parameter), ",
print "Log-Likelihood: %g " % L,
if verbose: print "\nThe KS-test statistic between the best-fit power-law and the data, ",
print "ks: %g" % (ks)
return xmin,alpha
[docs]def plexp(x,xm=1,a=2.5):
"""
CDF(x) for the piecewise distribution exponential x<xmin, powerlaw x>=xmin
This is the CDF version of the distributions drawn in fig 3.4a of Clauset et al.
"""
C = 1/(-xm/(1 - a) - xm/a + math.exp(a)*xm/a)
Ppl = lambda(X): 1+C*(xm/(1-a)*(X/xm)**(1-a))
Pexp = lambda(X): C*xm/a*math.exp(a)-C*(xm/a)*math.exp(-a*(X/xm-1))
d=Ppl(x)
d[x<xm]=Pexp(x)
return d
[docs]def plexp_inv(P,xm,a):
"""
Inverse CDF for a piecewise PDF as defined in eqn. 3.10
of Clauset et al.
"""
C = 1/(-xm/(1 - a) - xm/a + math.exp(a)*xm/a)
Pxm = 1+C*(xm/(1-a))
pp = P
x = xm*(pp-1)*(1-a)/(C*xm)**(1/(1-a)) if pp >= Pxm else (math.log( ((C*xm/a)*math.exp(a)-pp)/(C*xm/a)) - a) * (-xm/a)
#x[P>=Pxm] = xm*( (P[P>=Pxm]-1) * (1-a)/(C*xm) )**(1/(1-a)) # powerlaw
#x[P<Pxm] = (math.log( (C*xm/a*math.exp(a)-P[P<Pxm])/(C*xm/a) ) - a) * (-xm/a) # exp
return x
[docs]def pl_inv(P,xm,a):
"""
Inverse CDF for a pure power-law
"""
x = (1-P)**(1/(1-a)) * xm
return x
[docs]def test_fitter(xmin=1.0, alpha=2.5, niter=500, npts=1000, invcdf=plexp_inv,
quiet=True, silent=True):
"""
Tests the power-law fitter
Examples
========
Example (fig 3.4b in Clauset et al.)::
xminin=[0.25,0.5,0.75,1,1.5,2,5,10,50,100]
xmarr,af,ksv,nxarr = plfit.test_fitter(xmin=xminin,niter=1,npts=50000)
loglog(xminin,xmarr.squeeze(),'x')
Example 2::
xminin=[0.25,0.5,0.75,1,1.5,2,5,10,50,100]
xmarr,af,ksv,nxarr = plfit.test_fitter(xmin=xminin,niter=10,npts=1000)
loglog(xminin,xmarr.mean(axis=0),'x')
Example 3::
xmarr,af,ksv,nxarr = plfit.test_fitter(xmin=1.0,niter=1000,npts=1000)
hist(xmarr.squeeze());
# Test results:
# mean(xmarr) = 0.70, median(xmarr)=0.65 std(xmarr)=0.20
# mean(af) = 2.51 median(af) = 2.49 std(af)=0.14
# biased distribution; far from correct value of xmin but close to correct alpha
Example 4::
xmarr,af,ksv,nxarr = plfit.test_fitter(xmin=1.0,niter=1000,npts=1000,invcdf=pl_inv)
print("mean(xmarr): %0.2f median(xmarr): %0.2f std(xmarr): %0.2f" % (mean(xmarr),median(xmarr),std(xmarr)))
print("mean(af): %0.2f median(af): %0.2f std(af): %0.2f" % (mean(af),median(af),std(af)))
# mean(xmarr): 1.19 median(xmarr): 1.03 std(xmarr): 0.35
# mean(af): 2.51 median(af): 2.50 std(af): 0.07
"""
sz = niter
xmarr,alphaf_v,ksv,nxarr = ([0]*sz,)*4
for i in xrange(niter):
randarr = [random.random() for k in xrange(npts)]
fakedata = [invcdf(r,xmin,alpha) for r in randarr]
TEST = plfit(fakedata,quiet=quiet,silent=silent,nosmall=True)
alphaf_v[i] = TEST._alpha
ksv[i] = TEST._ks
nxarr[i] = TEST._ngtx
xmarr[i] = TEST._xmin
return xmarr,alphaf_v,ksv,nxarr