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Module « scipy.stats »

Fonction ncf - module scipy.stats

Signature de la fonction ncf

def ncf(*args, **kwds) 

Description

help(scipy.stats.ncf)

A non-central F distribution continuous random variable.

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

See Also
--------
scipy.stats.f : Fisher distribution

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

.. math::

    f(x, n_1, n_2, \lambda) =
        \exp\left(\frac{\lambda}{2} +
                  \lambda n_1 \frac{x}{2(n_1 x + n_2)}
            \right)
        n_1^{n_1/2} n_2^{n_2/2} x^{n_1/2 - 1} \\
        (n_2 + n_1 x)^{-(n_1 + n_2)/2}
        \gamma(n_1/2) \gamma(1 + n_2/2) \\
        \frac{L^{\frac{n_1}{2}-1}_{n_2/2}
            \left(-\lambda n_1 \frac{x}{2(n_1 x + n_2)}\right)}
        {B(n_1/2, n_2/2)
            \gamma\left(\frac{n_1 + n_2}{2}\right)}

for :math:`n_1, n_2 > 0`, :math:`\lambda \ge 0`.  Here :math:`n_1` is the
degrees of freedom in the numerator, :math:`n_2` the degrees of freedom in
the denominator, :math:`\lambda` the non-centrality parameter,
:math:`\gamma` is the logarithm of the Gamma function, :math:`L_n^k` is a
generalized Laguerre polynomial and :math:`B` is the beta function.

`ncf` takes ``dfn``, ``dfd`` and ``nc`` as shape parameters. If ``nc=0``,
the distribution becomes equivalent to the Fisher distribution.

This distribution uses routines from the Boost Math C++ library for
the computation of the ``pdf``, ``cdf``, ``ppf``, ``stats``, ``sf`` and
``isf`` methods. [1]_

The probability density above is defined in the "standardized" form. To shift
and/or scale the distribution use the ``loc`` and ``scale`` parameters.
Specifically, ``ncf.pdf(x, dfn, dfd, nc, loc, scale)`` is identically
equivalent to ``ncf.pdf(y, dfn, dfd, nc) / 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] The Boost Developers. "Boost C++ Libraries". https://www.boost.org/.

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

Calculate the first four moments:

>>> dfn, dfd, nc = 27, 27, 0.416
>>> mean, var, skew, kurt = ncf.stats(dfn, dfd, nc, moments='mvsk')

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

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

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

>>> vals = ncf.ppf([0.001, 0.5, 0.999], dfn, dfd, nc)
>>> np.allclose([0.001, 0.5, 0.999], ncf.cdf(vals, dfn, dfd, nc))
True

Generate random numbers:

>>> r = ncf.rvs(dfn, dfd, nc, 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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