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Module « scipy.special »

Fonction tklmbda - module scipy.special

Signature de la fonction tklmbda

def tklmbda(*args, **kwargs) 

Description

help(scipy.special.tklmbda)

tklmbda(x1, x2, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])

tklmbda(x, lmbda, out=None)

Cumulative distribution function of the Tukey lambda distribution.

Parameters
----------
x, lmbda : array_like
    Parameters
out : ndarray, optional
    Optional output array for the function results

Returns
-------
cdf : scalar or ndarray
    Value of the Tukey lambda CDF

See Also
--------
scipy.stats.tukeylambda : Tukey lambda distribution

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()

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``.

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