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Module « scipy.special »

Fonction ncfdtr - module scipy.special

Signature de la fonction ncfdtr

def ncfdtr(*args, **kwargs) 

Description

help(scipy.special.ncfdtr)

ncfdtr(x1, x2, x3, x4, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])

ncfdtr(dfn, dfd, nc, f, out=None)

Cumulative distribution function of the non-central F distribution.

The non-central F describes the distribution of,

.. math::
    Z = \frac{X/d_n}{Y/d_d}

where :math:`X` and :math:`Y` are independently distributed, with
:math:`X` distributed non-central :math:`\chi^2` with noncentrality
parameter `nc` and :math:`d_n` degrees of freedom, and :math:`Y`
distributed :math:`\chi^2` with :math:`d_d` degrees of freedom.

Parameters
----------
dfn : array_like
    Degrees of freedom of the numerator sum of squares.  Range (0, inf).
dfd : array_like
    Degrees of freedom of the denominator sum of squares.  Range (0, inf).
nc : array_like
    Noncentrality parameter.  Range [0, inf).
f : array_like
    Quantiles, i.e. the upper limit of integration.
out : ndarray, optional
    Optional output array for the function results

Returns
-------
cdf : scalar or ndarray
    The calculated CDF.  If all inputs are scalar, the return will be a
    float.  Otherwise it will be an array.

See Also
--------
ncfdtri : Quantile function; inverse of `ncfdtr` with respect to `f`.
ncfdtridfd : Inverse of `ncfdtr` with respect to `dfd`.
ncfdtridfn : Inverse of `ncfdtr` with respect to `dfn`.
ncfdtrinc : Inverse of `ncfdtr` with respect to `nc`.
scipy.stats.ncf : Non-central F distribution.

Notes
-----
This function calculates the CDF of the non-central f distribution using
the Boost Math C++ library [1]_.

The cumulative distribution function is computed using Formula 26.6.20 of
[2]_:

.. math::
    F(d_n, d_d, n_c, f) = \sum_{j=0}^\infty e^{-n_c/2}
    \frac{(n_c/2)^j}{j!} I_{x}(\frac{d_n}{2} + j, \frac{d_d}{2}),

where :math:`I` is the regularized incomplete beta function, and
:math:`x = f d_n/(f d_n + d_d)`.

Note that argument order of `ncfdtr` is different from that of the
similar ``cdf`` method of `scipy.stats.ncf`: `f` is the last
parameter of `ncfdtr` but the first parameter of ``scipy.stats.ncf.cdf``.

References
----------
.. [1] The Boost Developers. "Boost C++ Libraries". https://www.boost.org/.
.. [2] Milton Abramowitz and Irene A. Stegun, eds.
       Handbook of Mathematical Functions with Formulas,
       Graphs, and Mathematical Tables. New York: Dover, 1972.

Examples
--------
>>> import numpy as np
>>> from scipy import special
>>> from scipy import stats
>>> import matplotlib.pyplot as plt

Plot the CDF of the non-central F distribution, for nc=0.  Compare with the
F-distribution from scipy.stats:

>>> x = np.linspace(-1, 8, num=500)
>>> dfn = 3
>>> dfd = 2
>>> ncf_stats = stats.f.cdf(x, dfn, dfd)
>>> ncf_special = special.ncfdtr(dfn, dfd, 0, x)

>>> fig = plt.figure()
>>> ax = fig.add_subplot(111)
>>> ax.plot(x, ncf_stats, 'b-', lw=3)
>>> ax.plot(x, ncf_special, 'r-')
>>> plt.show()

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