scipy.special.gdtr#

scipy.special.gdtr(a, b, x, out=None) = <ufunc 'gdtr'>#

Gamma distribution cumulative distribution function.

Returns the integral from zero to x of the gamma probability density function,

\[F = \int_0^x \frac{a^b}{\Gamma(b)} t^{b-1} e^{-at}\,dt,\]

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 (upper limit of integration; float).
outndarray, optional: Optional output array for the function values

Returns:

Fscalar or ndarray: The CDF of the gamma distribution with parameters a and b evaluated at x.

See also

gdtrc: 1 - CDF of the gamma distribution.
scipy.stats.gamma: Gamma distribution

Notes

The evaluation is carried out using the relation to the incomplete gamma integral (regularized gamma function).

Wrapper for the Cephes [1] routine gdtr. Calling gdtr directly can improve performance compared to the cdf method of scipy.stats.gamma (see last example below).

Array API Standard Support

gdtr 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, b=2 at x=5.

>>> import numpy as np
>>> from scipy.special import gdtr
>>> import matplotlib.pyplot as plt
>>> gdtr(1., 2., 5.)
0.9595723180054873

Compute the function for a=1 and b=2 at several points by providing a NumPy array for x.

>>> xvalues = np.array([1., 2., 3., 4])
>>> gdtr(1., 1., xvalues)
array([0.63212056, 0.86466472, 0.95021293, 0.98168436])

gdtr 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,))

>>> gdtr(a, 3., x)
array([[0.01438768, 0.0803014 , 0.19115317, 0.32332358],
       [0.19115317, 0.57680992, 0.82642193, 0.9380312 ],
       [0.45618688, 0.87534798, 0.97974328, 0.9972306 ]])

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
...     gdtr_vals = gdtr(a, b, x)
...     ax.plot(x, gdtr_vals, label=fr"$a= {a},\, b={b}$", ls=style)
>>> ax.legend()
>>> ax.set_xlabel("$x$")
>>> ax.set_title("Gamma distribution cumulative distribution function")
>>> plt.show()

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

The gamma distribution is also available as scipy.stats.gamma. Using gdtr directly can be much faster than calling the cdf 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).cdf(x)=gdtr(a, b, x).

>>> from scipy.stats import gamma
>>> a = 2.
>>> b = 3
>>> x = 1.
>>> gdtr_result = gdtr(a, b, x)  # this will often be faster than below
>>> gamma_dist_result = gamma(b, scale=1/a).cdf(x)
>>> gdtr_result == gamma_dist_result  # test that results are equal
True