# [Numpy-discussion] Power distribution

alan@ajackso... alan@ajackso...
Fri Aug 7 21:17:38 CDT 2009

```Thanks! That helps a lot.

>On Fri, Aug 7, 2009 at 8:54 PM, <josef.pktd@gmail.com> wrote:
>> On Fri, Aug 7, 2009 at 6:57 PM, <josef.pktd@gmail.com> wrote:
>>> On Fri, Aug 7, 2009 at 6:13 PM, <josef.pktd@gmail.com> wrote:
>>>> On Fri, Aug 7, 2009 at 5:42 PM, <josef.pktd@gmail.com> wrote:
>>>>> On Fri, Aug 7, 2009 at 5:25 PM, Andrew Hawryluk<HAWRYLA@novachem.com> wrote:
>>>>>> Hmm ... good point.
>>>>>> It appears to give a probability distribution proportional to x**(a-1),
>>>>>> but I see no good reason why the domain should be limited to [0,1].
>>>>>>
>>>>>> def test(a):
>>>>>>    nums =
>>>>>> plt.hist(np.random.power(a,100000),bins=100,ec='none',fc='#dddddd')
>>>>>>    x = np.linspace(0,1,200)
>>>>>>    plt.plot(x,nums[0][-1]*x**(a-1))
>>>>>>
>>>>>> Andrew
>>>>>>
>>>>>>
>>>>>>
>>>>>>> -----Original Message-----
>>>>>>> From: numpy-discussion-bounces@scipy.org [mailto:numpy-discussion-
>>>>>>> bounces@scipy.org] On Behalf Of alan@ajackson.org
>>>>>>> Sent: 7 Aug 2009 2:49 PM
>>>>>>> To: Discussion of Numerical Python
>>>>>>> Subject: Re: [Numpy-discussion] Power distribution
>>>>>>>
>>>>>>> I don't think that is it, since the one in numpy has a range
>>>>>> restricted
>>>>>>> to the interval 0-1.
>>>>>>>
>>>>>>> Try out hist(np.random.power(5, 1000000), bins=100)
>>>>>>>
>>>>>>> >You might get better results for 'power-law distribution'
>>>>>>> >http://en.wikipedia.org/wiki/Power_law
>>>>>>> >
>>>>>>> >Andrew
>>>>>>> >
>>>>>>> >> -----Original Message-----
>>>>>>> >> From: numpy-discussion-bounces@scipy.org [mailto:numpy-discussion-
>>>>>>> >> bounces@scipy.org] On Behalf Of alan@ajackson.org
>>>>>>> >> Sent: 7 Aug 2009 11:45 AM
>>>>>>> >> To: Discussion of Numerical Python
>>>>>>> >> Subject: [Numpy-discussion] Power distribution
>>>>>>> >>
>>>>>>> >> Documenting my way through the statistics modules in numpy, I ran
>>>>>>> >> into the Power Distribution.
>>>>>>> >>
>>>>>>> >> Anyone know what that is? I Googled for it, and found a lot of
>>>>>> stuff
>>>>>>> >on
>>>>>>> >> electricity, but no reference for a statistical distribution of
>>>>>> that
>>>>>>> >> name. Does it have a common alias?
>>>>>>> >>
>>>>>>> >> --
>>>>>
>>>>>
>>>>> same is in Travis' notes on the distribution and scipy.stats.distributions
>>>>> domain in [0,1], but I don't know anything about it either
>>>>>
>>>>> ## Power-function distribution
>>>>> ##   Special case of beta dist. with d =1.0
>>>>>
>>>>> class powerlaw_gen(rv_continuous):
>>>>>    def _pdf(self, x, a):
>>>>>        return a*x**(a-1.0)
>>>>>    def _cdf(self, x, a):
>>>>>        return x**(a*1.0)
>>>>>    def _ppf(self, q, a):
>>>>>        return pow(q, 1.0/a)
>>>>>    def _stats(self, a):
>>>>>        return a/(a+1.0), a*(a+2.0)/(a+1.0)**2, \
>>>>>               2*(1.0-a)*sqrt((a+2.0)/(a*(a+3.0))), \
>>>>>               6*polyval([1,-1,-6,2],a)/(a*(a+3.0)*(a+4))
>>>>>    def _entropy(self, a):
>>>>>        return 1 - 1.0/a - log(a)
>>>>> powerlaw = powerlaw_gen(a=0.0, b=1.0, name="powerlaw",
>>>>>                        longname="A power-function",
>>>>>
>>>>> Power-function distribution
>>>>>
>>>>> powerlaw.pdf(x,a) = a*x**(a-1)
>>>>> for 0 <= x <= 1, a > 0.
>>>>> """
>>>>>                        )
>>>>>
>>>>
>>>>
>>>> it looks like it's the same distribution, even though it doesn't use
>>>> the random numbers from the numpy function
>>>>
>>>> high p-values with Kolmogorov-Smirnov, see below
>>>>
>>>> I assume it is a truncated version of *a* powerlaw distribution, so
>>>> that a can be large, which would be impossible in the open domain
>>>> case. But a quick search, I only found powerlaw applications that
>>>> refer to the tail behavior.
>>>>
>>>> Josef
>>>>
>>>>>>> rvs = np.random.power(5, 100000)
>>>>>>> stats.kstest(rvs,'powerlaw',(5,))
>>>> (0.0021079715221341555, 0.76587118275752697)
>>>>>>> rvs = np.random.power(5, 1000000)
>>>>>>> stats.kstest(rvs,'powerlaw',(5,))
>>>> (0.00063983013407076239, 0.80757958281509501)
>>>>>>> rvs = np.random.power(0.5, 1000000)
>>>>>>> stats.kstest(rvs,'powerlaw',(0.5,))
>>>> (0.00081823148457027539, 0.51478478398950211)
>>>>
>>>
>>> I found a short reference in Johnson, Kotz, Balakrishnan vol. 1 where
>>> it is refered to as the "power-function" distribution.
>>> roughly: if X is pareto (which kind) distributed, then Y=X**(-1) is
>>> distributed according to the power-function distribution. JKB have an
>>> extra parameter in there and is a bit more general then the scipy
>>> version, or maybe it is just the scale parameter included in the
>>> density function.
>>>
>>> It is also in NIST data plot, but I didn't find the html reference
>>> page, but only the pdf
>>>
>>>
>>> the pdf-files for powpdf and powcdf  are here
>>> http://www.itl.nist.gov/div898/software/dataplot/refman2/auxillar/homepage.htm
>>>
>>>
>>> I can look some more a bit later tonight.
>>>
>>> Josef
>>>
>>
>>
>> for the relationship to pareto, below are some kstests and graphs
>> a reminder that numpy.random.pareto uses a non-standard 0 bound, instead of 1
>> ks tests don't show good numbers every once in a while, since they are random
>>
>> I checked the definitions in JKB (page 607) and my previous
>> interpretation was correct.
>> if X has a pareto distribution with lower bound at 1 and shape
>> parameter a>0, then 1/X has a density function
>> p(y) = a*y**(a-1),  (0<y<1)
>> weak inequality in JKB instead of strict as in scipy.stats.powerlaw docstring
>> (the actual scipy.stats.powerlaw docstring  has a typo, a**x**(a-1),
>> which I will correct)
>>
>> Josef
>>
>>
>> import numpy as np
>> from scipy import stats
>> import matplotlib.pyplot as plt
>>
>>
>> rvs = np.random.power(5, 1000000)
>> rvsp = np.random.pareto(5, 1000000)
>> rvsps = stats.pareto.rvs(5, size=100)
>>
>> print "stats.kstest(1./rvsps,'powerlaw',(5,))"
>> print stats.kstest(1./rvsps,'powerlaw',(5,))
>>
>> print "stats.kstest(1./(1+rvsp),'powerlaw',(5,))"
>> print stats.kstest(1./(1+rvsp),'powerlaw',(5,))
>>
>> print "stats.kstest(rvs,'powerlaw',(5,))"
>> print stats.kstest(rvs,'powerlaw',(5,))
>>
>> print "stats.ks_2samp(rvs,1./(rvsp+1))"
>> print stats.ks_2samp(rvs,1./(rvsp+1))
>> print "stats.ks_2samp(rvs,1./rvsps)"
>> print stats.ks_2samp(rvs,1./rvsps)
>> print "stats.ks_2samp(1+rvsp, rvsps)"
>> print stats.ks_2samp(1+rvsp, rvsps)
>
>
>Improvements to graphs, compare with theoretical pdf
>
>Josef
>
>xx = np.linspace(0,1,100)
>powpdf = stats.powerlaw.pdf(xx,5)
>
>plt.figure()
>plt.hist(rvs, bins=50, normed=True)
>plt.plot(xx,powpdf,'r-')
>plt.title('np.random.power(5)')
>plt.figure()
>plt.hist(1./(1.+rvsp), bins=50, normed=True)
>plt.plot(xx,powpdf,'r-')
>plt.title('inverse of 1 + np.random.pareto(5)')
>plt.figure()
>plt.hist(1./(1.+rvsp), bins=50, normed=True)
>plt.plot(xx,powpdf,'r-')
>plt.title('inverse of stats.pareto(5)')
>
>
>
>>
>> plt.figure()
>> plt.hist(rvs, bins=50)
>> plt.title('np.random.power(5)')
>> plt.figure()
>> plt.hist(1./(1.+rvsp), bins=50)
>> plt.title('inverse of 1 + np.random.pareto(5)')
>> plt.figure()
>> plt.hist(1./(1.+rvsp), bins=50)
>> plt.title('inverse of stats.pareto(5)')
>> #plt.show()
>>

--
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| Alan K. Jackson            | To see a World in a Grain of Sand      |
| alan@ajackson.org          | And a Heaven in a Wild Flower,         |
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