The boxcox log-likelihood function.
Parameters: | lmb : scalar
data : array_like
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Returns: | llf : float or ndarray
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See also
Notes
The Box-Cox log-likelihood function is defined here as
where y is the Box-Cox transformed input data x.
Examples
>>> 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=np.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)
>>> plt.show()