scipy.special.fdtri#
- scipy.special.fdtri(dfn, dfd, p, out=None) = <ufunc 'fdtri'>#
The p-th quantile of the F-distribution.
This function is the inverse of the F-distribution CDF,
fdtr
, returning the x such that fdtr(dfn, dfd, x) = p.- Parameters:
- dfnarray_like
First parameter (positive float).
- dfdarray_like
Second parameter (positive float).
- parray_like
Cumulative probability, in [0, 1].
- outndarray, optional
Optional output array for the function values
- Returns:
- xscalar or ndarray
The quantile corresponding to p.
See also
fdtr
F distribution cumulative distribution function
fdtrc
F distribution survival function
scipy.stats.f
F distribution
Notes
The computation is carried out using the relation to the inverse regularized beta function, \(I^{-1}_x(a, b)\). Let \(z = I^{-1}_p(d_d/2, d_n/2).\) Then,
\[x = \frac{d_d (1 - z)}{d_n z}.\]If p is such that \(x < 0.5\), the following relation is used instead for improved stability: let \(z' = I^{-1}_{1 - p}(d_n/2, d_d/2).\) Then,
\[x = \frac{d_d z'}{d_n (1 - z')}.\]Wrapper for the Cephes [1] routine
fdtri
.The F distribution is also available as
scipy.stats.f
. Callingfdtri
directly can improve performance compared to theppf
method ofscipy.stats.f
(see last example below).References
[1]Cephes Mathematical Functions Library, http://www.netlib.org/cephes/
Examples
fdtri
represents the inverse of the F distribution CDF which is available asfdtr
. Here, we calculate the CDF fordf1=1
,df2=2
atx=3
.fdtri
then returns3
given the same values for df1, df2 and the computed CDF value.>>> import numpy as np >>> from scipy.special import fdtri, fdtr >>> df1, df2 = 1, 2 >>> x = 3 >>> cdf_value = fdtr(df1, df2, x) >>> fdtri(df1, df2, cdf_value) 3.000000000000006
Calculate the function at several points by providing a NumPy array for x.
>>> x = np.array([0.1, 0.4, 0.7]) >>> fdtri(1, 2, x) array([0.02020202, 0.38095238, 1.92156863])
Plot the function for several parameter sets.
>>> import matplotlib.pyplot as plt >>> dfn_parameters = [50, 10, 1, 50] >>> dfd_parameters = [0.5, 1, 1, 5] >>> linestyles = ['solid', 'dashed', 'dotted', 'dashdot'] >>> parameters_list = list(zip(dfn_parameters, dfd_parameters, ... linestyles)) >>> x = np.linspace(0, 1, 1000) >>> fig, ax = plt.subplots() >>> for parameter_set in parameters_list: ... dfn, dfd, style = parameter_set ... fdtri_vals = fdtri(dfn, dfd, x) ... ax.plot(x, fdtri_vals, label=rf"$d_n={dfn},\, d_d={dfd}$", ... ls=style) >>> ax.legend() >>> ax.set_xlabel("$x$") >>> title = "F distribution inverse cumulative distribution function" >>> ax.set_title(title) >>> ax.set_ylim(0, 30) >>> plt.show()
The F distribution is also available as
scipy.stats.f
. Usingfdtri
directly can be much faster than calling theppf
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).ppf(x)=fdtri(dfn, dfd, x)
.>>> from scipy.stats import f >>> dfn, dfd = 1, 2 >>> x = 0.7 >>> fdtri_res = fdtri(dfn, dfd, x) # this will often be faster than below >>> f_dist_res = f(dfn, dfd).ppf(x) >>> f_dist_res == fdtri_res # test that results are equal True