scipy.stats.zipf#
- scipy.stats.zipf = <scipy.stats._discrete_distns.zipf_gen object>[source]#
A Zipf (Zeta) discrete random variable.
As an instance of the
rv_discrete
class,zipf
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.See also
Notes
The probability mass function for
zipf
is:\[f(k, a) = \frac{1}{\zeta(a) k^a}\]for \(k \ge 1\), \(a > 1\).
zipf
takes \(a > 1\) as shape parameter. \(\zeta\) is the Riemann zeta function (scipy.special.zeta
)The Zipf distribution is also known as the zeta distribution, which is a special case of the Zipfian distribution (
zipfian
).The probability mass function above is defined in the “standardized” form. To shift distribution use the
loc
parameter. Specifically,zipf.pmf(k, a, loc)
is identically equivalent tozipf.pmf(k - loc, a)
.References
[1]“Zeta Distribution”, Wikipedia, https://en.wikipedia.org/wiki/Zeta_distribution
Examples
>>> import numpy as np >>> from scipy.stats import zipf >>> import matplotlib.pyplot as plt >>> fig, ax = plt.subplots(1, 1)
Calculate the first four moments:
>>> a = 6.5 >>> mean, var, skew, kurt = zipf.stats(a, moments='mvsk')
Display the probability mass function (
pmf
):>>> x = np.arange(zipf.ppf(0.01, a), ... zipf.ppf(0.99, a)) >>> ax.plot(x, zipf.pmf(x, a), 'bo', ms=8, label='zipf pmf') >>> ax.vlines(x, 0, zipf.pmf(x, a), 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 = zipf(a) >>> 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 = zipf.cdf(x, a) >>> np.allclose(x, zipf.ppf(prob, a)) True
Generate random numbers:
>>> r = zipf.rvs(a, size=1000)
Confirm that
zipf
is the large n limit ofzipfian
.>>> import numpy as np >>> from scipy.stats import zipf, zipfian >>> k = np.arange(11) >>> np.allclose(zipf.pmf(k, a), zipfian.pmf(k, a, n=10000000)) True
Methods
rvs(a, loc=0, size=1, random_state=None)
Random variates.
pmf(k, a, loc=0)
Probability mass function.
logpmf(k, a, loc=0)
Log of the probability mass function.
cdf(k, a, loc=0)
Cumulative distribution function.
logcdf(k, a, loc=0)
Log of the cumulative distribution function.
sf(k, a, loc=0)
Survival function (also defined as
1 - cdf
, but sf is sometimes more accurate).logsf(k, a, loc=0)
Log of the survival function.
ppf(q, a, loc=0)
Percent point function (inverse of
cdf
— percentiles).isf(q, a, loc=0)
Inverse survival function (inverse of
sf
).stats(a, loc=0, moments=’mv’)
Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’).
entropy(a, loc=0)
(Differential) entropy of the RV.
expect(func, args=(a,), loc=0, lb=None, ub=None, conditional=False)
Expected value of a function (of one argument) with respect to the distribution.
median(a, loc=0)
Median of the distribution.
mean(a, loc=0)
Mean of the distribution.
var(a, loc=0)
Variance of the distribution.
std(a, loc=0)
Standard deviation of the distribution.
interval(confidence, a, loc=0)
Confidence interval with equal areas around the median.