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 a routine from the Boost Math C++ library [1]. The F distribution is also available as scipy.stats.f. Calling fdtrc directly can improve performance compared to the sf method of scipy.stats.f (see last example below).

fdtrc has experimental support for Python Array API Standard compatible backends in addition to NumPy. Please consider testing these features by setting an environment variable SCIPY_ARRAY_API=1 and providing CuPy, PyTorch, JAX, or Dask arrays as array arguments. The following combinations of backend and device (or other capability) are supported.

Library	CPU	GPU
NumPy	✅	n/a
CuPy	n/a	✅
PyTorch	✅	⛔
JAX	⚠️ no JIT	⛔
Dask	✅	n/a

See Support for the array API standard for more information.

References

[1]

The Boost Developers. “Boost C++ Libraries”. https://www.boost.org/.

Examples

Calculate the function for dfn=1 and dfd=2 at x=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()

../../_images/scipy-special-fdtrc-1_00_00.png

The F distribution is also available as scipy.stats.f. Using fdtrc directly can be much faster than calling the sf method of scipy.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