python scipy.stats.powerlaw negative exponent

Question

I want to supply a negative exponent for the scipy.stats.powerlaw routine, e.g. a=-1.5, in order to draw random samples:

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powerlaw.pdf(x, a) = a * x**(         


        

Answer 1

The Python package powerlaw can do this. Consider for a>1 a power law distribution with probability density function

f(x) = c * x^(-a) 

for x > x_min and f(x) = 0 otherwise. Here c is a normalization factor and is determined as

c = (a-1) * x_min^(a-1).

In the example below it is a = 1.5 and x_min = 1.0 and comparing the probability density function estimated from the random sample with the PDF from the expression above gives the expected result.

import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as pl

import numpy as np
import powerlaw

a, xmin = 1.5, 1.0
N = 10000

# generates random variates of power law distribution
vrs = powerlaw.Power_Law(xmin=xmin, parameters=[a]).generate_random(N)

# plotting the PDF estimated from variates
bin_min, bin_max = np.min(vrs), np.max(vrs)
bins = 10**(np.linspace(np.log10(bin_min), np.log10(bin_max), 100))
counts, edges = np.histogram(vrs, bins, density=True)
centers = (edges[1:] + edges[:-1])/2.

# plotting the expected PDF 
xs = np.linspace(bin_min, bin_max, 100000)
pl.plot(xs, [(a-1)*xmin**(a-1)*x**(-a) for x in xs], color='red')
pl.plot(centers, counts, '.')

pl.xscale('log')
pl.yscale('log')

pl.savefig('powerlaw_variates.png')

returns