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Fonction zipf - module scipy.stats

Signature de la fonction zipf

def zipf(*args, **kwds) 

Description

zipf.__doc__

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(alpha, a, loc=0)
        Endpoints of the range that contains fraction alpha [0, 1] of the
        distribution

    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
    --------
    >>> 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`` 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`.

    >>> from scipy.stats import zipfian
    >>> k = np.arange(11)
    >>> np.allclose(zipf.pmf(k, a), zipfian.pmf(k, a, n=10000000))
    True