scipy.special.tklmbda#

scipy.special.tklmbda(x, lmbda, out=None) = <wrapped_ufunc 'tklmbda'>#

Cumulative distribution function of the Tukey lambda distribution.

Parameters:

x, lmbdaarray_like: Parameters
outndarray, optional: Optional output array for the function results

Returns:

cdfscalar or ndarray: Value of the Tukey lambda CDF

See also

scipy.stats.tukeylambda: Tukey lambda distribution

Notes

Array API Standard Support

tklmbda 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 typically 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. tklmbda does not currently support out for the PyTorch backend.

See Support for the array API standard for more information.

Examples

>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from scipy.special import tklmbda, expit

Compute the cumulative distribution function (CDF) of the Tukey lambda distribution at several x values for lmbda = -1.5.

>>> x = np.linspace(-2, 2, 9)
>>> x
array([-2. , -1.5, -1. , -0.5,  0. ,  0.5,  1. ,  1.5,  2. ])
>>> tklmbda(x, -1.5)
array([0.34688734, 0.3786554 , 0.41528805, 0.45629737, 0.5       ,
       0.54370263, 0.58471195, 0.6213446 , 0.65311266])

When lmbda is 0, the function is the logistic sigmoid function, which is implemented in scipy.special as expit.

>>> tklmbda(x, 0)
array([0.11920292, 0.18242552, 0.26894142, 0.37754067, 0.5       ,
       0.62245933, 0.73105858, 0.81757448, 0.88079708])
>>> expit(x)
array([0.11920292, 0.18242552, 0.26894142, 0.37754067, 0.5       ,
       0.62245933, 0.73105858, 0.81757448, 0.88079708])

When lmbda is 1, the Tukey lambda distribution is uniform on the interval [-1, 1], so the CDF increases linearly.

>>> t = np.linspace(-1, 1, 9)
>>> tklmbda(t, 1)
array([0.   , 0.125, 0.25 , 0.375, 0.5  , 0.625, 0.75 , 0.875, 1.   ])

In the following, we generate plots for several values of lmbda.

The first figure shows graphs for lmbda <= 0.

>>> styles = ['-', '-.', '--', ':']
>>> fig, ax = plt.subplots()
>>> x = np.linspace(-12, 12, 500)
>>> for k, lmbda in enumerate([-1.0, -0.5, 0.0]):
...     y = tklmbda(x, lmbda)
...     ax.plot(x, y, styles[k], label=rf'$\lambda$ = {lmbda:-4.1f}')

>>> ax.set_title(r'tklmbda(x, $\lambda$)')
>>> ax.set_label('x')
>>> ax.legend(framealpha=1, shadow=True)
>>> ax.grid(True)

The second figure shows graphs for lmbda > 0. The dots in the graphs show the bounds of the support of the distribution.

>>> fig, ax = plt.subplots()
>>> x = np.linspace(-4.2, 4.2, 500)
>>> lmbdas = [0.25, 0.5, 1.0, 1.5]
>>> for k, lmbda in enumerate(lmbdas):
...     y = tklmbda(x, lmbda)
...     ax.plot(x, y, styles[k], label=fr'$\lambda$ = {lmbda}')

>>> ax.set_prop_cycle(None)
>>> for lmbda in lmbdas:
...     ax.plot([-1/lmbda, 1/lmbda], [0, 1], '.', ms=8)

>>> ax.set_title(r'tklmbda(x, $\lambda$)')
>>> ax.set_xlabel('x')
>>> ax.legend(framealpha=1, shadow=True)
>>> ax.grid(True)

>>> plt.tight_layout()
>>> plt.show()

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

../../_images/scipy-special-tklmbda-1_00_01.png

The CDF of the Tukey lambda distribution is also implemented as the cdf method of scipy.stats.tukeylambda. In the following, tukeylambda.cdf(x, -0.5) and tklmbda(x, -0.5) compute the same values:

>>> from scipy.stats import tukeylambda
>>> x = np.linspace(-2, 2, 9)

>>> tukeylambda.cdf(x, -0.5)
array([0.21995157, 0.27093858, 0.33541677, 0.41328161, 0.5       ,
       0.58671839, 0.66458323, 0.72906142, 0.78004843])

>>> tklmbda(x, -0.5)
array([0.21995157, 0.27093858, 0.33541677, 0.41328161, 0.5       ,
       0.58671839, 0.66458323, 0.72906142, 0.78004843])

The implementation in tukeylambda also provides location and scale parameters, and other methods such as pdf() (the probability density function) and ppf() (the inverse of the CDF), so for working with the Tukey lambda distribution, tukeylambda is more generally useful. The primary advantage of tklmbda is that it is significantly faster than tukeylambda.cdf.