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boxcox_llf(lmb, data)

Notes

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.

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.

The boxcox log-likelihood function.

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()

See :

Back References

The following pages refer to to this document either explicitly or contain code examples using this.

scipy.stats._morestats:boxcox scipy.stats._morestats:boxcox_normmax scipy.stats._morestats:boxcox_normplot

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