scipy.stats.

yeojohnson_llf#

scipy.stats.yeojohnson_llf(lmb, data, *, axis=0, nan_policy='propagate', keepdims=False)[source]#

The Yeo-Johnson log-likelihood function.

Parameters:

lmbscalar

Parameter for Yeo-Johnson transformation. See yeojohnson for details.

dataarray_like

Data to calculate Yeo-Johnson log-likelihood for.

axisint, default: 0

If an int, the axis of the input along which to compute the statistic. The statistic of each axis-slice (e.g. row) of the input will appear in a corresponding element of the output. If None, the input will be raveled before computing the statistic.

nan_policy{‘propagate’, ‘omit’, ‘raise’

Defines how to handle input NaNs.

propagate: if a NaN is present in the axis slice (e.g. row) along which the statistic is computed, the corresponding entry of the output will be NaN.
omit: NaNs will be omitted when performing the calculation. If insufficient data remains in the axis slice along which the statistic is computed, the corresponding entry of the output will be NaN.
raise: if a NaN is present, a ValueError will be raised.

keepdimsbool, default: False

If this is set to True, the axes which are reduced are left in the result as dimensions with size one. With this option, the result will broadcast correctly against the input array.

Returns:

llffloat: Yeo-Johnson log-likelihood of data given lmb.

See also

yeojohnson, probplot, yeojohnson_normplot, yeojohnson_normmax

Notes

The Yeo-Johnson log-likelihood function \(l\) is defined here as

\[l = -\frac{N}{2} \log(\hat{\sigma}^2) + (\lambda - 1) \sum_i^N \text{sign}(x_i) \log(|x_i| + 1)\]

where \(N\) is the number of data points \(x`=``data`\) and \(\hat{\sigma}^2\) is the estimated variance of the Yeo-Johnson transformed input data \(x\). This corresponds to the profile log-likelihood of the original data \(x\) with some constant terms dropped.

Added in version 1.2.0.

Array API Standard Support

yeojohnson_llf has experimental support for Python Array API Standard compatible backends in addition to NumPy. Please consider testing these features by setting an environment variable SCIPY_ARRAY_API=1 and providing CuPy, PyTorch, JAX, or Dask arrays as array arguments. The following combinations of backend and device (or other capability) are supported.

Library	CPU	GPU
NumPy	✅	n/a
CuPy	n/a	✅
PyTorch	✅	✅
JAX	✅	✅
Dask	⛔	n/a

See Support for the array API standard for more information.

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 Yeo-Johnson 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.yeojohnson_llf(lmbda, x)

Also find the optimal lmbda value with yeojohnson:

>>> x_most_normal, lmbda_optimal = stats.yeojohnson(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.yeojohnson_llf(lmbda_optimal, x), color='r')
>>> ax.set_xlabel('lmbda parameter')
>>> ax.set_ylabel('Yeo-Johnson log-likelihood')

Now add some probability plots to show that where the log-likelihood is maximized the data transformed with yeojohnson 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.yeojohnson(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()

../../_images/scipy-stats-yeojohnson_llf-1.png