scipy.special.nbdtri#

scipy.special.nbdtri(k, n, y, out=None) = <ufunc 'nbdtri'>#

Returns the inverse with respect to the parameter p of y = nbdtr(k, n, p), the negative binomial cumulative distribution function.

Parameters:

karray_like: The maximum number of allowed failures (nonnegative int).
narray_like: The target number of successes (positive int).
yarray_like: The probability of k or fewer failures before n successes (float).
outndarray, optional: Optional output array for the function results

Returns:

pscalar or ndarray: Probability of success in a single event (float) such that nbdtr(k, n, p) = y.

See also

nbdtr: Cumulative distribution function of the negative binomial.
nbdtrc: Negative binomial survival function.
scipy.stats.nbinom: negative binomial distribution.
nbdtrik: Inverse with respect to k of nbdtr(k, n, p).
nbdtrin: Inverse with respect to n of nbdtr(k, n, p).
scipy.stats.nbinom: Negative binomial distribution

Notes

Wrapper for the Cephes [1] routine nbdtri.

The negative binomial distribution is also available as scipy.stats.nbinom. Using nbdtri directly can improve performance compared to the ppf method of scipy.stats.nbinom.

Array API Standard Support

nbdtri has support for Python Array API Standard compatible backends in addition to NumPy. The following combinations of backend and device (or other capability) are supported.

Library	CPU	GPU
NumPy	✅	n/a
CuPy	n/a	✅
PyTorch	✅	⛔
JAX	✅	⛔
Dask	✅	n/a

For the NumPy backend, this function supports all NumPy ufunc keyword arguments. Other backends may support out, but none of the other ufunc kwargs. out is typically supported for CuPy and PyTorch, but not currently in cases where SciPy relies on a generic Array API implementation or, for PyTorch on CPU, falls back to the NumPy backend. out is never supported for JAX because JAX arrays are immutable. nbdtri does not currently support out for the PyTorch backend.

See Support for the array API standard for more information.

References

[1]

Cephes Mathematical Functions Library, https://netlib.org/cephes/

Examples

nbdtri is the inverse of nbdtr with respect to p. Up to floating point errors the following holds: nbdtri(k, n, nbdtr(k, n, p))=p.

>>> import numpy as np
>>> from scipy.special import nbdtri, nbdtr
>>> k, n, p = 5, 10, 0.2
>>> cdf_val = nbdtr(k, n, p)
>>> nbdtri(k, n, cdf_val)
0.20000000000000004

Compute the function for k=10 and n=5 at several points by providing a NumPy array or list for y.

>>> y = np.array([0.1, 0.4, 0.8])
>>> nbdtri(3, 5, y)
array([0.34462319, 0.51653095, 0.69677416])

Plot the function for three different parameter sets.

>>> import matplotlib.pyplot as plt
>>> n_parameters = [5, 20, 30, 30]
>>> k_parameters = [20, 20, 60, 80]
>>> linestyles = ['solid', 'dashed', 'dotted', 'dashdot']
>>> parameters_list = list(zip(n_parameters, k_parameters, linestyles))
>>> cdf_vals = np.linspace(0, 1, 1000)
>>> fig, ax = plt.subplots(figsize=(8, 8))
>>> for parameter_set in parameters_list:
...     n, k, style = parameter_set
...     nbdtri_vals = nbdtri(k, n, cdf_vals)
...     ax.plot(cdf_vals, nbdtri_vals, label=rf"$k={k},\ n={n}$",
...             ls=style)
>>> ax.legend()
>>> ax.set_ylabel("$p$")
>>> ax.set_xlabel("$CDF$")
>>> title = "nbdtri: inverse of negative binomial CDF with respect to $p$"
>>> ax.set_title(title)
>>> plt.show()

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

nbdtri can evaluate different parameter sets by providing arrays with shapes compatible for broadcasting for k, n and p. Here we compute the function for three different k at four locations p, resulting in a 3x4 array.

>>> k = np.array([[5], [10], [15]])
>>> y = np.array([0.3, 0.5, 0.7, 0.9])
>>> k.shape, y.shape
((3, 1), (4,))

>>> nbdtri(k, 5, y)
array([[0.37258157, 0.45169416, 0.53249956, 0.64578407],
       [0.24588501, 0.30451981, 0.36778453, 0.46397088],
       [0.18362101, 0.22966758, 0.28054743, 0.36066188]])