# scipy.stats.yeojohnson_llf

scipy.stats.yeojohnson_llf(lmb, data)

  The yeojohnson 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. If data is multi-dimensional, the log-likelihood is calculated along the first axis.

  Returns:
    llf float

      Yeo-Johnson log-likelihood of data given lmb.

  See also

  `yeojohnson`, `probplot`, `yeojohnson_normplot`, `yeojohnson_normmax`

  Notes

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

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

  where \(\hat{\sigma}^2\) is estimated variance of the Yeo-Johnson transformed input data `x`.

  New in version 1.2.0.

  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()