fdtrc#

scipy.special.fdtrc(dfn,dfd,x,out=None)#

F survival function.

Returns the complemented F-distribution function (the integral of thedensity fromx 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 parametersdfn anddfd atx.

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 theformula,

\[F(d_n, d_d; x) = I_{d_d/(d_d + xd_n)}(d_d/2, d_n/2).\]

Wrapper for a routine from the Boost Math C++ library[1]. TheF distribution is also available asscipy.stats.f. Callingfdtrc directly can improve performance compared to thesfmethod ofscipy.stats.f (see last example below).

Array API Standard Support

fdtrc has experimental support for Python Array API Standard compatiblebackends in addition to NumPy. Please consider testing these featuresby setting an environment variableSCIPY_ARRAY_API=1 and providingCuPy, PyTorch, JAX, or Dask arrays as array arguments. The followingcombinations 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

SeeSupport for the array API standard for more information.

References

[1]

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

Examples

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Calculate the function fordfn=1 anddfd=2 atx=1.

>>>importnumpyasnp>>>fromscipy.specialimportfdtrc>>>fdtrc(1,2,1)0.42264973081037427

Calculate the function at several points by providing a NumPy array forx.

>>>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.

>>>importmatplotlib.pyplotasplt>>>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()>>>forparameter_setinparameters_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 asscipy.stats.f. Usingfdtrcdirectly 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).

>>>fromscipy.statsimportf>>>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 equalTrue

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