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Module « scipy.stats »

Fonction boxcox_llf - module scipy.stats

Signature de la fonction boxcox_llf

def boxcox_llf(lmb, data) 

Description

help(scipy.stats.boxcox_llf)

The boxcox log-likelihood function.

Parameters
----------
lmb : scalar
    Parameter for Box-Cox transformation.  See `boxcox` for details.
data : array_like
    Data to calculate Box-Cox log-likelihood for.  If `data` is
    multi-dimensional, the log-likelihood is calculated along the first
    axis.

Returns
-------
llf : float or ndarray
    Box-Cox log-likelihood of `data` given `lmb`.  A float for 1-D `data`,
    an array otherwise.

See Also
--------
boxcox, probplot, boxcox_normplot, boxcox_normmax

Notes
-----
The Box-Cox log-likelihood function is defined here as

.. math::

    llf = (\lambda - 1) \sum_i(\log(x_i)) -
          N/2 \log(\sum_i (y_i - \bar{y})^2 / N),

where ``y`` is the Box-Cox transformed input data ``x``.

Examples
--------
>>> import numpy as np
>>> from scipy import stats
>>> import matplotlib.pyplot as plt
>>> from mpl_toolkits.axes_grid1.inset_locator import inset_axes

Generate some random variates and calculate Box-Cox log-likelihood values
for them for a range of ``lmbda`` values:

>>> rng = np.random.default_rng()
>>> x = stats.loggamma.rvs(5, loc=10, size=1000, random_state=rng)
>>> lmbdas = np.linspace(-2, 10)
>>> llf = np.zeros(lmbdas.shape, dtype=float)
>>> for ii, lmbda in enumerate(lmbdas):
...     llf[ii] = stats.boxcox_llf(lmbda, x)

Also find the optimal lmbda value with `boxcox`:

>>> x_most_normal, lmbda_optimal = stats.boxcox(x)

Plot the log-likelihood as function of lmbda.  Add the optimal lmbda as a
horizontal line to check that that's really the optimum:

>>> fig = plt.figure()
>>> ax = fig.add_subplot(111)
>>> ax.plot(lmbdas, llf, 'b.-')
>>> ax.axhline(stats.boxcox_llf(lmbda_optimal, x), color='r')
>>> ax.set_xlabel('lmbda parameter')
>>> ax.set_ylabel('Box-Cox log-likelihood')

Now add some probability plots to show that where the log-likelihood is
maximized the data transformed with `boxcox` looks closest to normal:

>>> locs = [3, 10, 4]  # 'lower left', 'center', 'lower right'
>>> for lmbda, loc in zip([-1, lmbda_optimal, 9], locs):
...     xt = stats.boxcox(x, lmbda=lmbda)
...     (osm, osr), (slope, intercept, r_sq) = stats.probplot(xt)
...     ax_inset = inset_axes(ax, width="20%", height="20%", loc=loc)
...     ax_inset.plot(osm, osr, 'c.', osm, slope*osm + intercept, 'k-')
...     ax_inset.set_xticklabels([])
...     ax_inset.set_yticklabels([])
...     ax_inset.set_title(r'$\lambda=%1.2f$' % lmbda)

>>> plt.show()

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