bundles / scipy 2.0.0.dev0+git20260907.41eeb59 / scipy / stats / _morestats / yeojohnson_llf

function

`scipy.stats._morestats:yeojohnson_llf`

source: build-install/usr/lib/python3.14/site-packages/scipy/stats/_morestats.py:1759

Signature

def yeojohnson_llf(lmb, data, *, axis = 0, nan_policy = propagate, keepdims = False)

Summary

The Yeo-Johnson log-likelihood function.

Parameters

lmb : scalar

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

data : array_like

Data to calculate Yeo-Johnson log-likelihood for.

axis : int, 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.

keepdims : bool, 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

llf : float: Yeo-Johnson log-likelihood of data given lmb.

Notes

The Yeo-Johnson log-likelihood function $l$ is defined here as

l = - \frac{N}{2} lo g (\overset{σ}{^}^{2}) + (λ - 1) i \sum N sign (x_{i}) lo g (∣ x_{i} ∣ + 1)

where $N$ is the number of data points $x$ = data and $\overset{σ}{^}^{2}$ 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.

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')

✗

Text(0, 0.5, '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)

✗

[<matplotlib.lines.Line2D object at 0x7f3f68fd8440>, <matplotlib.lines.Line2D object at 0x7f3f68fd8d70>]
[Text(-4.0, 0, ''), Text(-3.0, 0, ''), Text(-2.0, 0, ''), Text(-1.0, 0, ''), Text(0.0, 0, ''), Text(1.0, 0, ''), Text(2.0, 0, ''), Text(3.0, 0, ''), Text(4.0, 0, '')]
[Text(0, 0.905, ''), Text(0, 0.91, ''), Text(0, 0.915, ''), Text(0, 0.92, ''), Text(0, 0.925, ''), Text(0, 0.93, ''), Text(0, 0.935, '')]
Text(0.5, 1.0, '$\\lambda=-1.00$')
[<matplotlib.lines.Line2D object at 0x7f3f607946e0>, <matplotlib.lines.Line2D object at 0x7f3f60795010>]
[Text(-4.0, 0, ''), Text(-3.0, 0, ''), Text(-2.0, 0, ''), Text(-1.0, 0, ''), Text(0.0, 0, ''), Text(1.0, 0, ''), Text(2.0, 0, ''), Text(3.0, 0, ''), Text(4.0, 0, '')]
[Text(0, 6000.0, ''), Text(0, 8000.0, ''), Text(0, 10000.0, ''), Text(0, 12000.0, ''), Text(0, 14000.0, ''), Text(0, 16000.0, ''), Text(0, 18000.0, ''), Text(0, 20000.0, ''), Text(0, 22000.0, '')]
Text(0.5, 1.0, '$\\lambda=4.36$')
[<matplotlib.lines.Line2D object at 0x7f3f58a881a0>, <matplotlib.lines.Line2D object at 0x7f3f58a88440>]
[Text(-4.0, 0, ''), Text(-3.0, 0, ''), Text(-2.0, 0, ''), Text(-1.0, 0, ''), Text(0.0, 0, ''), Text(1.0, 0, ''), Text(2.0, 0, ''), Text(3.0, 0, ''), Text(4.0, 0, '')]
[Text(0, -250000000.0, ''), Text(0, 0.0, ''), Text(0, 250000000.0, ''), Text(0, 500000000.0, ''), Text(0, 750000000.0, ''), Text(0, 1000000000.0, ''), Text(0, 1250000000.0, ''), Text(0, 1500000000.0, ''), Text(0, 1750000000.0, ''), Text(0, 2000000000.0, '')]
Text(0.5, 1.0, '$\\lambda=9.00$')

plt.show()

✓

Aliases

scipy.stats.yeojohnson_llf