Module « scipy.stats »

Fonction zipfian - module scipy.stats

Signature de la fonction zipfian

def zipfian(*args, **kwds) 

Description

zipfian.__doc__

A Zipfian discrete random variable.

    As an instance of the `rv_discrete` class, `zipfian` 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, n, loc=0, size=1, random_state=None)
        Random variates.
    pmf(k, a, n, loc=0)
        Probability mass function.
    logpmf(k, a, n, loc=0)
        Log of the probability mass function.
    cdf(k, a, n, loc=0)
        Cumulative distribution function.
    logcdf(k, a, n, loc=0)
        Log of the cumulative distribution function.
    sf(k, a, n, loc=0)
        Survival function  (also defined as ``1 - cdf``, but `sf` is sometimes more accurate).
    logsf(k, a, n, loc=0)
        Log of the survival function.
    ppf(q, a, n, loc=0)
        Percent point function (inverse of ``cdf`` --- percentiles).
    isf(q, a, n, loc=0)
        Inverse survival function (inverse of ``sf``).
    stats(a, n, loc=0, moments='mv')
        Mean('m'), variance('v'), skew('s'), and/or kurtosis('k').
    entropy(a, n, loc=0)
        (Differential) entropy of the RV.
    expect(func, args=(a, n), loc=0, lb=None, ub=None, conditional=False)
        Expected value of a function (of one argument) with respect to the distribution.
    median(a, n, loc=0)
        Median of the distribution.
    mean(a, n, loc=0)
        Mean of the distribution.
    var(a, n, loc=0)
        Variance of the distribution.
    std(a, n, loc=0)
        Standard deviation of the distribution.
    interval(alpha, a, n, loc=0)
        Endpoints of the range that contains fraction alpha [0, 1] of the
        distribution

    See Also
    --------
    zipf

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

    .. math::

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

    for :math:`k \in \{1, 2, \dots, n-1, n\}`, :math:`a \ge 0`,
    :math:`n \in \{1, 2, 3, \dots\}`.

    `zipfian` takes :math:`a` and :math:`n` as shape parameters.
    :math:`H_{n,a}` is the :math:`n`:sup:`th` generalized harmonic
    number of order :math:`a`.

    The Zipfian distribution reduces to the Zipf (zeta) distribution as
    :math:`n \rightarrow \infty`.

    The probability mass function above is defined in the "standardized" form.
    To shift distribution use the ``loc`` parameter.
    Specifically, ``zipfian.pmf(k, a, n, loc)`` is identically
    equivalent to ``zipfian.pmf(k - loc, a, n)``.

    References
    ----------
    .. [1] "Zipf's Law", Wikipedia, https://en.wikipedia.org/wiki/Zipf's_law
    .. [2] Larry Leemis, "Zipf Distribution", Univariate Distribution
           Relationships. http://www.math.wm.edu/~leemis/chart/UDR/PDFs/Zipf.pdf

    Examples
    --------
    >>> from scipy.stats import zipfian
    >>> import matplotlib.pyplot as plt
    >>> fig, ax = plt.subplots(1, 1)
    
    Calculate the first four moments:
    
    >>> a, n = 1.25, 10
    >>> mean, var, skew, kurt = zipfian.stats(a, n, moments='mvsk')
    
    Display the probability mass function (``pmf``):
    
    >>> x = np.arange(zipfian.ppf(0.01, a, n),
    ...               zipfian.ppf(0.99, a, n))
    >>> ax.plot(x, zipfian.pmf(x, a, n), 'bo', ms=8, label='zipfian pmf')
    >>> ax.vlines(x, 0, zipfian.pmf(x, a, n), 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 = zipfian(a, n)
    >>> 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 = zipfian.cdf(x, a, n)
    >>> np.allclose(x, zipfian.ppf(prob, a, n))
    True
    
    Generate random numbers:
    
    >>> r = zipfian.rvs(a, n, size=1000)

    Confirm that `zipfian` reduces to `zipf` for large `n`, `a > 1`.

    >>> from scipy.stats import zipf
    >>> k = np.arange(11)
    >>> np.allclose(zipfian.pmf(k, a=3.5, n=10000000), zipf.pmf(k, a=3.5))
    True