Module « scipy.stats »

Fonction yulesimon - module scipy.stats

Signature de la fonction yulesimon

def yulesimon(*args, **kwds) 

Description

yulesimon.__doc__

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.
    
    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 fraction alpha [0, 1] of the
        distribution

    Notes
    -----

    The probability mass function for the `yulesimon` is:

    .. math::

        f(k) =  \alpha B(k, \alpha+1)

    for :math:`k=1,2,3,...`, where :math:`\alpha>0`.
    Here :math:`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 :math:`\alpha=a-1`.

    For details see the wikipedia entry [2]_.

    References
    ----------
    .. [1] Devroye, Luc. "Non-uniform Random Variate Generation",
         (1986) Springer, New York.

    .. [2] https://en.wikipedia.org/wiki/Yule-Simon_distribution

    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 to ``yulesimon.pmf(k - loc, alpha)``.

    Examples
    --------
    >>> from scipy.stats import yulesimon
    >>> import matplotlib.pyplot as plt
    >>> fig, ax = plt.subplots(1, 1)
    
    Calculate the first four 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`` and ``ppf``:
    
    >>> prob = yulesimon.cdf(x, alpha)
    >>> np.allclose(x, yulesimon.ppf(prob, alpha))
    True
    
    Generate random numbers:
    
    >>> r = yulesimon.rvs(alpha, size=1000)