scipy.special.

gdtrc#

scipy.special.gdtrc(a, b, x, out=None)#

Gamma distribution survival function.

Integral from x to infinity of the gamma probability density function,

\[F = \int_x^\infty \frac{a^b}{\Gamma(b)} t^{b-1} e^{-at},円dt,\]

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. Calling gdtrc directly can improve performance compared to the sf method of scipy.stats.gamma (see last example below).

Array API Standard Support

gdtrc 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 ]

Cephes Mathematical Functions Library, http://www.netlib.org/cephes/

Examples

Compute the function for a=1 and b=2 at x=5.

>>> importnumpyasnp
>>> fromscipy.specialimport gdtrc
>>> importmatplotlib.pyplotasplt
>>> 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 and b=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"$a= {a},,円 b={b}$", ls=style)
>>> ax.legend()
>>> ax.set_xlabel("$x$")
>>> ax.set_title("Gamma distribution survival function")
>>> plt.show()

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

The gamma distribution is also available as scipy.stats.gamma. Using gdtrc directly can be much faster than calling the sf method of scipy.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).

>>> fromscipy.statsimport 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