scipy.stats.zipfian#

scipy.stats.zipfian = <scipy.stats._discrete_distns.zipfian_gen object>[source]#

A Zipfian discrete random variable.

As an instance of the rv_discrete class, zipfian object inherits from it a collection of generic methods (see below for the full list), and completes them with details specific for this particular distribution.

Methods

rvs(a, n, loc=0, size=1, random_state=None)

Random variates.

pmf(k, a, n, loc=0)

Probability mass function.

logpmf(k, a, n, loc=0)

Log of the probability mass function.

cdf(k, a, n, loc=0)

Cumulative distribution function.

logcdf(k, a, n, loc=0)

Log of the cumulative distribution function.

sf(k, a, n, loc=0)

Survival function (also defined as 1 - cdf, but sf is sometimes more accurate).

logsf(k, a, n, loc=0)

Log of the survival function.

ppf(q, a, n, loc=0)

Percent point function (inverse of cdf — percentiles).

isf(q, a, n, loc=0)

Inverse survival function (inverse of sf).

stats(a, n, loc=0, moments=’mv’)

Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’).

entropy(a, n, loc=0)

(Differential) entropy of the RV.

expect(func, args=(a, n), loc=0, lb=None, ub=None, conditional=False)

Expected value of a function (of one argument) with respect to the distribution.

median(a, n, loc=0)

Median of the distribution.

mean(a, n, loc=0)

Mean of the distribution.

var(a, n, loc=0)

Variance of the distribution.

std(a, n, loc=0)

Standard deviation of the distribution.

interval(confidence, a, n, loc=0)

Confidence interval with equal areas around the median.

See also

zipf

Notes

The probability mass function for zipfian is:

for k \in \{1, 2, \dots, n-1, n\}, a \ge 0, n \in \{1, 2, 3, \dots\}.

zipfian takes a and n as shape parameters. H_{n,a} is the nth generalized harmonic number of order a.

The Zipfian distribution reduces to the Zipf (zeta) distribution as n \rightarrow \infty.

The probability mass function above is defined in the “standardized” form. To shift distribution use the loc parameter. Specifically, zipfian.pmf(k, a, n, loc) is identically equivalent to zipfian.pmf(k - loc, a, n).

References

[1]

“Zipf’s Law”, Wikipedia, https://en.wikipedia.org/wiki/Zipf’s_law

[2]

Larry Leemis, “Zipf Distribution”, Univariate Distribution Relationships. http://www.math.wm.edu/~leemis/chart/UDR/PDFs/Zipf.pdf

Examples

>>> import numpy as np
>>> from scipy.stats import zipfian
>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots(1, 1)

Get the support:

>>> a, n = 1.25, 10
>>> lb, ub = zipfian.support(a, n)

Calculate the first four moments:

>>> mean, var, skew, kurt = zipfian.stats(a, n, moments='mvsk')

Display the probability mass function (pmf):

>>> x = np.arange(zipfian.ppf(0.01, a, n),
...               zipfian.ppf(0.99, a, n))
>>> ax.plot(x, zipfian.pmf(x, a, n), 'bo', ms=8, label='zipfian pmf')
>>> ax.vlines(x, 0, zipfian.pmf(x, a, n), colors='b', lw=5, alpha=0.5)

Alternatively, the distribution object can be called (as a function) to fix the shape and location. This returns a “frozen” RV object holding the given parameters fixed.

Freeze the distribution and display the frozen pmf:

>>> rv = zipfian(a, n)
>>> ax.vlines(x, 0, rv.pmf(x), colors='k', linestyles='-', lw=1,
...         label='frozen pmf')
>>> ax.legend(loc='best', frameon=False)
>>> plt.show()
../../_images/scipy-stats-zipfian-1_00_00.png

Check accuracy of cdf and ppf:

>>> prob = zipfian.cdf(x, a, n)
>>> np.allclose(x, zipfian.ppf(prob, a, n))
True

Generate random numbers:

>>> r = zipfian.rvs(a, n, size=1000)

Confirm that zipfian reduces to zipf for large n, a > 1.

>>> import numpy as np
>>> from scipy.stats import zipf, zipfian
>>> k = np.arange(11)
>>> np.allclose(zipfian.pmf(k, a=3.5, n=10000000), zipf.pmf(k, a=3.5))
True