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Module « scipy.stats »

Fonction alpha - module scipy.stats

Signature de la fonction alpha

def alpha(*args, **kwds) 

Description

help(scipy.stats.alpha)

An alpha continuous random variable.

As an instance of the `rv_continuous` class, `alpha` 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, scale=1, size=1, random_state=None)
    Random variates.
pdf(x, a, loc=0, scale=1)
    Probability density function.
logpdf(x, a, loc=0, scale=1)
    Log of the probability density function.
cdf(x, a, loc=0, scale=1)
    Cumulative distribution function.
logcdf(x, a, loc=0, scale=1)
    Log of the cumulative distribution function.
sf(x, a, loc=0, scale=1)
    Survival function  (also defined as ``1 - cdf``, but `sf` is sometimes more accurate).
logsf(x, a, loc=0, scale=1)
    Log of the survival function.
ppf(q, a, loc=0, scale=1)
    Percent point function (inverse of ``cdf`` --- percentiles).
isf(q, a, loc=0, scale=1)
    Inverse survival function (inverse of ``sf``).
moment(order, a, loc=0, scale=1)
    Non-central moment of the specified order.
stats(a, loc=0, scale=1, moments='mv')
    Mean('m'), variance('v'), skew('s'), and/or kurtosis('k').
entropy(a, 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=(a,), 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(a, loc=0, scale=1)
    Median of the distribution.
mean(a, loc=0, scale=1)
    Mean of the distribution.
var(a, loc=0, scale=1)
    Variance of the distribution.
std(a, loc=0, scale=1)
    Standard deviation of the distribution.
interval(confidence, a, loc=0, scale=1)
    Confidence interval with equal areas around the median.

Notes
-----
The probability density function for `alpha` ([1]_, [2]_) is:

.. math::

    f(x, a) = \frac{1}{x^2 \Phi(a) \sqrt{2\pi}} *
              \exp(-\frac{1}{2} (a-1/x)^2)

where :math:`\Phi` is the normal CDF, :math:`x > 0`, and :math:`a > 0`.

`alpha` takes ``a`` as a shape parameter.

The probability density above is defined in the "standardized" form. To shift
and/or scale the distribution use the ``loc`` and ``scale`` parameters.
Specifically, ``alpha.pdf(x, a, loc, scale)`` is identically
equivalent to ``alpha.pdf(y, a) / 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.

References
----------
.. [1] Johnson, Kotz, and Balakrishnan, "Continuous Univariate
       Distributions, Volume 1", Second Edition, John Wiley and Sons,
       p. 173 (1994).
.. [2] Anthony A. Salvia, "Reliability applications of the Alpha
       Distribution", IEEE Transactions on Reliability, Vol. R-34,
       No. 3, pp. 251-252 (1985).

Examples
--------
>>> import numpy as np
>>> from scipy.stats import alpha
>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots(1, 1)

Calculate the first four moments:

>>> a = 3.57
>>> mean, var, skew, kurt = alpha.stats(a, moments='mvsk')

Display the probability density function (``pdf``):

>>> x = np.linspace(alpha.ppf(0.01, a),
...                 alpha.ppf(0.99, a), 100)
>>> ax.plot(x, alpha.pdf(x, a),
...        'r-', lw=5, alpha=0.6, label='alpha 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 = alpha(a)
>>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')

Check accuracy of ``cdf`` and ``ppf``:

>>> vals = alpha.ppf([0.001, 0.5, 0.999], a)
>>> np.allclose([0.001, 0.5, 0.999], alpha.cdf(vals, a))
True

Generate random numbers:

>>> r = alpha.rvs(a, 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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