Module « scipy.stats »

Fonction crystalball - module scipy.stats

Signature de la fonction crystalball

def crystalball(*args, **kwds) 

Description

crystalball.__doc__

    Crystalball distribution

    As an instance of the `rv_continuous` class, `crystalball` 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(beta, m, loc=0, scale=1, size=1, random_state=None)
        Random variates.
    pdf(x, beta, m, loc=0, scale=1)
        Probability density function.
    logpdf(x, beta, m, loc=0, scale=1)
        Log of the probability density function.
    cdf(x, beta, m, loc=0, scale=1)
        Cumulative distribution function.
    logcdf(x, beta, m, loc=0, scale=1)
        Log of the cumulative distribution function.
    sf(x, beta, m, loc=0, scale=1)
        Survival function  (also defined as ``1 - cdf``, but `sf` is sometimes more accurate).
    logsf(x, beta, m, loc=0, scale=1)
        Log of the survival function.
    ppf(q, beta, m, loc=0, scale=1)
        Percent point function (inverse of ``cdf`` --- percentiles).
    isf(q, beta, m, loc=0, scale=1)
        Inverse survival function (inverse of ``sf``).
    moment(n, beta, m, loc=0, scale=1)
        Non-central moment of order n
    stats(beta, m, loc=0, scale=1, moments='mv')
        Mean('m'), variance('v'), skew('s'), and/or kurtosis('k').
    entropy(beta, m, loc=0, scale=1)
        (Differential) entropy of the RV.
    fit(data)
        Parameter estimates for generic data.
        See `scipy.stats.rv_continuous.fit <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.rv_continuous.fit.html#scipy.stats.rv_continuous.fit>`__ for detailed documentation of the
        keyword arguments.
    expect(func, args=(beta, m), loc=0, scale=1, lb=None, ub=None, conditional=False, **kwds)
        Expected value of a function (of one argument) with respect to the distribution.
    median(beta, m, loc=0, scale=1)
        Median of the distribution.
    mean(beta, m, loc=0, scale=1)
        Mean of the distribution.
    var(beta, m, loc=0, scale=1)
        Variance of the distribution.
    std(beta, m, loc=0, scale=1)
        Standard deviation of the distribution.
    interval(alpha, beta, m, loc=0, scale=1)
        Endpoints of the range that contains fraction alpha [0, 1] of the
        distribution

    Notes
    -----
    The probability density function for `crystalball` is:

    .. math::

        f(x, \beta, m) =  \begin{cases}
                            N \exp(-x^2 / 2),  &\text{for } x > -\beta\\
                            N A (B - x)^{-m}  &\text{for } x \le -\beta
                          \end{cases}

    where :math:`A = (m / |\beta|)^m  \exp(-\beta^2 / 2)`,
    :math:`B = m/|\beta| - |\beta|` and :math:`N` is a normalisation constant.

    `crystalball` takes :math:`\beta > 0` and :math:`m > 1` as shape
    parameters.  :math:`\beta` defines the point where the pdf changes
    from a power-law to a Gaussian distribution.  :math:`m` is the power
    of the power-law tail.

    References
    ----------
    .. [1] "Crystal Ball Function",
           https://en.wikipedia.org/wiki/Crystal_Ball_function

    The probability density above is defined in the "standardized" form. To shift
    and/or scale the distribution use the ``loc`` and ``scale`` parameters.
    Specifically, ``crystalball.pdf(x, beta, m, loc, scale)`` is identically
    equivalent to ``crystalball.pdf(y, beta, m) / scale`` with
    ``y = (x - loc) / scale``. Note that shifting the location of a distribution
    does not make it a "noncentral" distribution; noncentral generalizations of
    some distributions are available in separate classes.

    .. versionadded:: 0.19.0

    Examples
    --------
    >>> from scipy.stats import crystalball
    >>> import matplotlib.pyplot as plt
    >>> fig, ax = plt.subplots(1, 1)
    
    Calculate the first four moments:
    
    >>> beta, m = 2, 3
    >>> mean, var, skew, kurt = crystalball.stats(beta, m, moments='mvsk')
    
    Display the probability density function (``pdf``):
    
    >>> x = np.linspace(crystalball.ppf(0.01, beta, m),
    ...                 crystalball.ppf(0.99, beta, m), 100)
    >>> ax.plot(x, crystalball.pdf(x, beta, m),
    ...        'r-', lw=5, alpha=0.6, label='crystalball pdf')
    
    Alternatively, the distribution object can be called (as a function)
    to fix the shape, location and scale parameters. This returns a "frozen"
    RV object holding the given parameters fixed.
    
    Freeze the distribution and display the frozen ``pdf``:
    
    >>> rv = crystalball(beta, m)
    >>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')
    
    Check accuracy of ``cdf`` and ``ppf``:
    
    >>> vals = crystalball.ppf([0.001, 0.5, 0.999], beta, m)
    >>> np.allclose([0.001, 0.5, 0.999], crystalball.cdf(vals, beta, m))
    True
    
    Generate random numbers:
    
    >>> r = crystalball.rvs(beta, m, size=1000)
    
    And compare the histogram:
    
    >>> ax.hist(r, density=True, histtype='stepfilled', alpha=0.2)
    >>> ax.legend(loc='best', frameon=False)
    >>> plt.show()