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# scipy.stats.boxcox

scipy.stats.boxcox(x, lmbda=None, alpha=None, optimizer=None)

  Return a dataset transformed by a Box-Cox power transformation.

  Parameters:
    x ndarray

      Input array to be transformed.

      If lmbda is not None, this is an alias of `scipy.special.boxcox`. Returns nan if `x < 0`; returns -inf if `x == 0 and lmbda < 0`.

      If lmbda is None, array must be positive, 1-dimensional, and non-constant.

    lmbda scalar, optional

      If lmbda is None (default), find the value of lmbda that maximizes the log-likelihood function and return it as the second output argument.

      If lmbda is not None, do the transformation for that value.

    alpha float, optional

      If lmbda is None and `alpha` is not None (default), return the `100 * (1-alpha)%` confidence interval for lmbda as the third output argument. Must be between 0.0 and 1.0.

      If lmbda is not None, `alpha` is ignored.

    optimizer callable, optional

      If lmbda is None, optimizer is the scalar optimizer used to find the value of lmbda that minimizes the negative log-likelihood function. optimizer is a callable that accepts one argument:

      funcallable

        The objective function, which evaluates the negative log-likelihood function at a provided value of lmbda

      and returns an object, such as an instance of `scipy.optimize.OptimizeResult`, which holds the optimal value of lmbda in an attribute x.

      See the example in `boxcox_normmax` or the documentation of `scipy.optimize.minimize_scalar` for more information.

      If lmbda is not None, optimizer is ignored.

  Returns:
    boxcox ndarray

      Box-Cox power transformed array.

    maxlog float, optional

      If the lmbda parameter is None, the second returned argument is the lmbda that maximizes the log-likelihood function.

    (min_ci, max_ci)tuple of float, optional

      If lmbda parameter is None and `alpha` is not None, this returned tuple of floats represents the minimum and maximum confidence limits given `alpha`.

  See also

  `probplot`, `boxcox_normplot`, `boxcox_normmax`, `boxcox_llf`

  Notes

  The Box-Cox transform is given by:

      y = (x**lmbda - 1) / lmbda,  for lmbda != 0
          log(x),                  for lmbda = 0

  `boxcox` requires the input data to be positive. Sometimes a Box-Cox transformation provides a shift parameter to achieve this; `boxcox` does not. Such a shift parameter is equivalent to adding a positive constant to x before calling `boxcox`.

  The confidence limits returned when `alpha` is provided give the interval where:

  \[llf(\hat{\lambda}) - llf(\lambda) < \frac{1}{2}\chi^2(1 - \alpha, 1),\]

  with `llf` the log-likelihood function and \(\chi^2\) the chi-squared function.

  References

  G.E.P. Box and D.R. Cox, “An Analysis of Transformations”, Journal of the Royal Statistical Society B, 26, 211-252 (1964).

  Examples

      >>> from scipy import stats
      >>> import matplotlib.pyplot as plt

  We generate some random variates from a non-normal distribution and make a probability plot for it, to show it is non-normal in the tails:

      >>> fig = plt.figure()
      >>> ax1 = fig.add_subplot(211)
      >>> x = stats.loggamma.rvs(5, size=500) + 5
      >>> prob = stats.probplot(x, dist=stats.norm, plot=ax1)
      >>> ax1.set_xlabel('')
      >>> ax1.set_title('Probplot against normal distribution')

  We now use `boxcox` to transform the data so it’s closest to normal:

      >>> ax2 = fig.add_subplot(212)
      >>> xt, _ = stats.boxcox(x)
      >>> prob = stats.probplot(xt, dist=stats.norm, plot=ax2)
      >>> ax2.set_title('Probplot after Box-Cox transformation')

      >>> plt.show()