scipy.special.gdtrc#
- scipy.special.gdtrc(a, b, x, out=None) = <ufunc 'gdtrc'>#
Gamma distribution survival function.
Integral from x to infinity of the gamma probability density function,
\[F = \int_x^\infty \frac{a^b}{\Gamma(b)} t^{b-1} e^{-at}\,dt,\]where \(\Gamma\) is the gamma function.
- Parameters:
- aarray_like
The rate parameter of the gamma distribution, sometimes denoted \(\beta\) (float). It is also the reciprocal of the scale parameter \(\theta\).
- barray_like
The shape parameter of the gamma distribution, sometimes denoted \(\alpha\) (float).
- xarray_like
The quantile (lower limit of integration; float).
- outndarray, optional
Optional output array for the function values
- Returns:
- Fscalar or ndarray
The survival function of the gamma distribution with parameters a and b evaluated at x.
See also
gdtr
Gamma distribution cumulative distribution function
scipy.stats.gamma
Gamma distribution
gdtrix
Notes
The evaluation is carried out using the relation to the incomplete gamma integral (regularized gamma function).
Wrapper for the Cephes [1] routine
gdtrc
. Callinggdtrc
directly can improve performance compared to thesf
method ofscipy.stats.gamma
(see last example below).References
[1]Cephes Mathematical Functions Library, http://www.netlib.org/cephes/
Examples
Compute the function for
a=1
andb=2
atx=5
.>>> import numpy as np >>> from scipy.special import gdtrc >>> import matplotlib.pyplot as plt >>> gdtrc(1., 2., 5.) 0.04042768199451279
Compute the function for
a=1
,b=2
at several points by providing a NumPy array for x.>>> xvalues = np.array([1., 2., 3., 4]) >>> gdtrc(1., 1., xvalues) array([0.36787944, 0.13533528, 0.04978707, 0.01831564])
gdtrc
can evaluate different parameter sets by providing arrays with broadcasting compatible shapes for a, b and x. Here we compute the function for three different a at four positions x andb=3
, resulting in a 3x4 array.>>> a = np.array([[0.5], [1.5], [2.5]]) >>> x = np.array([1., 2., 3., 4]) >>> a.shape, x.shape ((3, 1), (4,))
>>> gdtrc(a, 3., x) array([[0.98561232, 0.9196986 , 0.80884683, 0.67667642], [0.80884683, 0.42319008, 0.17357807, 0.0619688 ], [0.54381312, 0.12465202, 0.02025672, 0.0027694 ]])
Plot the function for four different parameter sets.
>>> a_parameters = [0.3, 1, 2, 6] >>> b_parameters = [2, 10, 15, 20] >>> linestyles = ['solid', 'dashed', 'dotted', 'dashdot'] >>> parameters_list = list(zip(a_parameters, b_parameters, linestyles)) >>> x = np.linspace(0, 30, 1000) >>> fig, ax = plt.subplots() >>> for parameter_set in parameters_list: ... a, b, style = parameter_set ... gdtrc_vals = gdtrc(a, b, x) ... ax.plot(x, gdtrc_vals, label=fr"$a= {a},\, b={b}$", ls=style) >>> ax.legend() >>> ax.set_xlabel("$x$") >>> ax.set_title("Gamma distribution survival function") >>> plt.show()
The gamma distribution is also available as
scipy.stats.gamma
. Usinggdtrc
directly can be much faster than calling thesf
method ofscipy.stats.gamma
, especially for small arrays or individual values. To get the same results one must use the following parametrization:stats.gamma(b, scale=1/a).sf(x)=gdtrc(a, b, x)
.>>> from scipy.stats import gamma >>> a = 2 >>> b = 3 >>> x = 1. >>> gdtrc_result = gdtrc(a, b, x) # this will often be faster than below >>> gamma_dist_result = gamma(b, scale=1/a).sf(x) >>> gdtrc_result == gamma_dist_result # test that results are equal True