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Module « scipy.stats »

Fonction norminvgauss - module scipy.stats

Signature de la fonction norminvgauss

def norminvgauss(*args, **kwds) 

Description

help(scipy.stats.norminvgauss)

A Normal Inverse Gaussian continuous random variable.

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

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

.. math::

    f(x, a, b) = \frac{a \, K_1(a \sqrt{1 + x^2})}{\pi \sqrt{1 + x^2}} \,
                 \exp(\sqrt{a^2 - b^2} + b x)

where :math:`x` is a real number, the parameter :math:`a` is the tail
heaviness and :math:`b` is the asymmetry parameter satisfying
:math:`a > 0` and :math:`|b| <= a`.
:math:`K_1` is the modified Bessel function of second kind
(`scipy.special.k1`).

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

A normal inverse Gaussian random variable `Y` with parameters `a` and `b`
can be expressed as a normal mean-variance mixture:
``Y = b * V + sqrt(V) * X`` where `X` is ``norm(0,1)`` and `V` is
``invgauss(mu=1/sqrt(a**2 - b**2))``. This representation is used
to generate random variates.

Another common parametrization of the distribution (see Equation 2.1 in
[2]_) is given by the following expression of the pdf:

.. math::

    g(x, \alpha, \beta, \delta, \mu) =
    \frac{\alpha\delta K_1\left(\alpha\sqrt{\delta^2 + (x - \mu)^2}\right)}
    {\pi \sqrt{\delta^2 + (x - \mu)^2}} \,
    e^{\delta \sqrt{\alpha^2 - \beta^2} + \beta (x - \mu)}

In SciPy, this corresponds to
`a = alpha * delta, b = beta * delta, loc = mu, scale=delta`.

References
----------
.. [1] O. Barndorff-Nielsen, "Hyperbolic Distributions and Distributions on
       Hyperbolae", Scandinavian Journal of Statistics, Vol. 5(3),
       pp. 151-157, 1978.

.. [2] O. Barndorff-Nielsen, "Normal Inverse Gaussian Distributions and
       Stochastic Volatility Modelling", Scandinavian Journal of
       Statistics, Vol. 24, pp. 1-13, 1997.

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

Calculate the first four moments:

>>> a, b = 1.25, 0.5
>>> mean, var, skew, kurt = norminvgauss.stats(a, b, moments='mvsk')

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

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

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

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

Generate random numbers:

>>> r = norminvgauss.rvs(a, b, 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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