Skip to main content
`Ctrl`+`K`
scipy.stats.

# boxcox_llf

scipy.stats.boxcox_llf(lmb, data, *, axis=0, keepdims=False, nan_policy='propagate')

  The boxcox log-likelihood function.

  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.

    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.

    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.

    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.

  Returns:
    llf float or ndarray

      Box-Cox log-likelihood of data given lmb. A float for 1-D data, an array otherwise.

  See also

  `boxcox`, `probplot`, `boxcox_normplot`, `boxcox_normmax`

  Notes

  The Box-Cox log-likelihood function \(l\) is defined here as

  \[l = (\lambda - 1) \sum_i^N \log(x_i) - \frac{N}{2} \log\left(\sum_i^N (y_i - \bar{y})^2 / N\right),\]

  where \(N\) is the number of data points `data` and \(y\) is the Box-Cox transformed input data. This corresponds to the profile log-likelihood of the original data \(x\) with some constant terms dropped.

  Array API Standard Support

  `boxcox_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 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()