scipy.stats.yulesimon#

scipy.stats.yulesimon = <scipy.stats._discrete_distns.yulesimon_gen object>[source]#

A Yule-Simon discrete random variable.

As an instance of the rv_discrete class, yulesimon 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(alpha, loc=0, size=1, random_state=None)	Random variates.
pmf(k, alpha, loc=0)	Probability mass function.
logpmf(k, alpha, loc=0)	Log of the probability mass function.
cdf(k, alpha, loc=0)	Cumulative distribution function.
logcdf(k, alpha, loc=0)	Log of the cumulative distribution function.
sf(k, alpha, loc=0)	Survival function (also defined as `1 - cdf`, but sf is sometimes more accurate).
logsf(k, alpha, loc=0)	Log of the survival function.
ppf(q, alpha, loc=0)	Percent point function (inverse of `cdf` — percentiles).
isf(q, alpha, loc=0)	Inverse survival function (inverse of `sf`).
stats(alpha, loc=0, moments=’mv’)	Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’).
entropy(alpha, loc=0)	(Differential) entropy of the RV.
expect(func, args=(alpha,), loc=0, lb=None, ub=None, conditional=False)	Expected value of a function (of one argument) with respect to the distribution.
median(alpha, loc=0)	Median of the distribution.
mean(alpha, loc=0)	Mean of the distribution.
var(alpha, loc=0)	Variance of the distribution.
std(alpha, loc=0)	Standard deviation of the distribution.
interval(confidence, alpha, loc=0)	Confidence interval with equal areas around the median.

Notes

The probability mass function for the yulesimon is:

\[f(k) = \alpha B(k, \alpha+1)\]

for \(k=1,2,3,...\), where \(\alpha>0\). Here \(B\) refers to the scipy.special.beta function.

The sampling of random variates is based on pg 553, Section 6.3 of [1]. Our notation maps to the referenced logic via \(\alpha=a-1\).

For details see the wikipedia entry [2].

References

[1]

Devroye, Luc. “Non-uniform Random Variate Generation”, (1986) Springer, New York.

[2]

https://en.wikipedia.org/wiki/Yule-Simon_distribution

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

Examples

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

Calculate the first four moments:

>>> alpha = 11
>>> mean, var, skew, kurt = yulesimon.stats(alpha, moments='mvsk')

Display the probability mass function (pmf):

>>> x = np.arange(yulesimon.ppf(0.01, alpha),
...               yulesimon.ppf(0.99, alpha))
>>> ax.plot(x, yulesimon.pmf(x, alpha), 'bo', ms=8, label='yulesimon pmf')
>>> ax.vlines(x, 0, yulesimon.pmf(x, alpha), 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 = yulesimon(alpha)
>>> 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-yulesimon-1_00_00.png

Check accuracy of cdf and ppf:

>>> prob = yulesimon.cdf(x, alpha)
>>> np.allclose(x, yulesimon.ppf(prob, alpha))
True

Generate random numbers:

>>> r = yulesimon.rvs(alpha, size=1000)