======================
Box-Cox Transformation
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:doc:`/WorkProcessClassifiers/GlobalAlgorithm/index` - :doc:`/WorkProcessClassifiers/OneDimensionalAlgorithm/index`

*Box-Cox Transformation* algorithm is a useful data pre-processing technique. It can be used to make the non-normally distributed data normal and stabilise variance of the time series data that is non-stationary in variance. 

.. rubric:: Input Parameters

+--------------------+--------------------------------------------+----------------------------------------------------+------------------------------------------------+---------+
| Parameter          | Type                                       | Constraint                                         | Description                                    | Remarks |
+====================+============================================+====================================================+================================================+=========+
| :math:`Y`          | :math:`Y \in \mathbb R^N`                  | :math:`N \in \mathbb{N}, Y > 0`                    | Input data vector of length :math:`N`          |         |
+--------------------+--------------------------------------------+----------------------------------------------------+------------------------------------------------+---------+

.. rubric:: Output Parameters

+----------------------------+----------------------------------------------------+----------------------------------------+--------------------------------------------------------------------------------------+------------------------------------------------------------------+
| Parameter                  | Type                                               | Constraint                             | Description                                                                          | Remarks                                                          |
+============================+====================================================+========================================+======================================================================================+==================================================================+
| :math:`\hat{Y}`            | :math:`\hat{Y} \in \mathbb R^N`                    | :math:`N \in \mathbb{N}`               | Data vector of length :math:`N`          that has approximately normal distribution  |                                                                  |
+----------------------------+----------------------------------------------------+----------------------------------------+--------------------------------------------------------------------------------------+------------------------------------------------------------------+
| :math:`\lambda`            | :math:`\lambda \in \mathbb R`                      |                                        | Estimated transformation parameter                                                   | In some cases, it may be given by the user as an input parameter |
+----------------------------+----------------------------------------------------+----------------------------------------+--------------------------------------------------------------------------------------+------------------------------------------------------------------+

.. rubric:: Tool Support

* :doc:`/Tools/MatlabTool/index`

  For details refer to the online documentation of the function `'boxcox' <http://www.mathworks.de/help/toolbox/finance/boxcox.html>`__.

.. rubric:: Single Steps using the Algorithm

* :doc:`/DataPreprocessing/MathematicalComputation/HandlingDataNonNormality/ReducingDataNonNormalityWithBoxCoxTransformation/index`

* :doc:`/DataPreprocessing/MathematicalComputation/HandlingDataStationarity/StabilizingVarianceWithBoxCoxTransformation/index`

.. rubric:: References

- R.M.\  Sakia, The Box-Cox transformation technique: a review, The Statistician, vol. 41, pp. 169-178, 1992.

  `http://staff.ustc.edu.cn/~zwp/teach/Reg/Boxcox.pdf <http://staff.ustc.edu.cn/~zwp/teach/Reg/Boxcox.pdf>`__