scipy.special.fdtrc#
- scipy.special.fdtrc(dfn, dfd, x, out=None) = <ufunc 'fdtrc'>#
F survival function.
Returns the complemented F-distribution function (the integral of the density from x to infinity).
- Parameters:
- dfnarray_like
First parameter (positive float).
- dfdarray_like
Second parameter (positive float).
- xarray_like
Argument (nonnegative float).
- outndarray, optional
Optional output array for the function values
- Returns:
- yscalar or ndarray
The complemented F-distribution function with parameters dfn and dfd at x.
See also
fdtr
F distribution cumulative distribution function
fdtri
F distribution inverse cumulative distribution function
scipy.stats.f
F distribution
Notes
The regularized incomplete beta function is used, according to the formula,
\[F(d_n, d_d; x) = I_{d_d/(d_d + xd_n)}(d_d/2, d_n/2).\]Wrapper for the Cephes [1] routine
fdtrc
. The F distribution is also available asscipy.stats.f
. Callingfdtrc
directly can improve performance compared to thesf
method ofscipy.stats.f
(see last example below).References
[1]Cephes Mathematical Functions Library, http://www.netlib.org/cephes/
Examples
Calculate the function for
dfn=1
anddfd=2
atx=1
.>>> import numpy as np >>> from scipy.special import fdtrc >>> fdtrc(1, 2, 1) 0.42264973081037427
Calculate the function at several points by providing a NumPy array for x.
>>> x = np.array([0.5, 2., 3.]) >>> fdtrc(1, 2, x) array([0.5527864 , 0.29289322, 0.22540333])
Plot the function for several parameter sets.
>>> import matplotlib.pyplot as plt >>> dfn_parameters = [1, 5, 10, 50] >>> dfd_parameters = [1, 1, 2, 3] >>> linestyles = ['solid', 'dashed', 'dotted', 'dashdot'] >>> parameters_list = list(zip(dfn_parameters, dfd_parameters, ... linestyles)) >>> x = np.linspace(0, 30, 1000) >>> fig, ax = plt.subplots() >>> for parameter_set in parameters_list: ... dfn, dfd, style = parameter_set ... fdtrc_vals = fdtrc(dfn, dfd, x) ... ax.plot(x, fdtrc_vals, label=rf"$d_n={dfn},\, d_d={dfd}$", ... ls=style) >>> ax.legend() >>> ax.set_xlabel("$x$") >>> ax.set_title("F distribution survival function") >>> plt.show()
The F distribution is also available as
scipy.stats.f
. Usingfdtrc
directly can be much faster than calling thesf
method ofscipy.stats.f
, especially for small arrays or individual values. To get the same results one must use the following parametrization:stats.f(dfn, dfd).sf(x)=fdtrc(dfn, dfd, x)
.>>> from scipy.stats import f >>> dfn, dfd = 1, 2 >>> x = 1 >>> fdtrc_res = fdtrc(dfn, dfd, x) # this will often be faster than below >>> f_dist_res = f(dfn, dfd).sf(x) >>> f_dist_res == fdtrc_res # test that results are equal True