scipy.stats.boxcox_llf(lmb, data)[source]

The boxcox log-likelihood function.

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.

llf : float or ndarray

Box-Cox log-likelihood of data given lmb. A float for 1-D data, an array otherwise.


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

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

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


>>> from scipy import stats
>>> import matplotlib.pyplot as plt
>>> from mpl_toolkits.axes_grid1.inset_locator import inset_axes
>>> np.random.seed(1245)

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

>>> x = stats.loggamma.rvs(5, loc=10, size=1000)
>>> 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('$\lambda=%1.2f$' % lmbda)

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