scipy.stats.yulesimon¶
-
scipy.stats.
yulesimon
(*args, **kwds) = <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
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 toyulesimon.pmf(k - loc, alpha)
.Examples
>>> from scipy.stats import yulesimon >>> import matplotlib.pyplot as plt >>> fig, ax = plt.subplots(1, 1)
Calculate a few first 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()
Check accuracy of
cdf
andppf
:>>> 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 alpha percent of the distribution