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Module « scipy.stats »

Fonction crystalball - module scipy.stats

Signature de la fonction crystalball

def crystalball(*args, **kwds) 

Description

help(scipy.stats.crystalball)

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(order, beta, m, loc=0, scale=1)
    Non-central moment of the specified order.
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(confidence, beta, m, loc=0, scale=1)
    Confidence interval with equal areas around the median.

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.

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

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

Examples
--------
>>> import numpy as np
>>> 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, bins='auto', histtype='stepfilled', alpha=0.2)
>>> ax.set_xlim([x[0], x[-1]])
>>> ax.legend(loc='best', frameon=False)
>>> plt.show()

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