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Module « scipy.stats »

Fonction zipf - module scipy.stats

Signature de la fonction zipf

def zipf(*args, **kwds) 

Description

help(scipy.stats.zipf)

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.

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.

See Also
--------
zipfian

Notes
-----
The probability mass function for `zipf` is:

.. math::

    f(k, a) = \frac{1}{\zeta(a) k^a}

for :math:`k \ge 1`, :math:`a > 1`.

`zipf` takes :math:`a > 1` as shape parameter. :math:`\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 to ``zipf.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 = 6.6
>>> 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`` and ``ppf``:

>>> 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 of `zipfian`.

>>> 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

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