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.

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

>>> 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)

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(alpha, alpha, loc=0)

Endpoints of the range that contains fraction alpha [0, 1] of the distribution