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Mean Absolute Deviation
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:doc:`/WorkProcessClassifiers/GlobalAlgorithm/index` - :doc:`/WorkProcessClassifiers/OneDimensionalAlgorithm/index`

*Mean Absolute Deviation*\algorithm is commonly used as a robust measure of the variability of data. The basic formulat is stated as follows:

.. math::

    \text{MAD} = \text{Median}[(|Y_1-\text{Median}(Y)|, |Y_2-\text{Median}(Y)|, \ldots, |Y_N-\text{Median}(Y)|)^T \, \text{,} 

where :math:`Y_i` denotes the :math:`i`\ th element of the input data vector :math:`Y`\ . :math:`|\cdot|` returns the absolute value and :math:`T` denotes the transpose of the vector. *Median* algorithm returns the Median value of :math:`Y`\ .

.. rubric:: Input Parameters

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

.. rubric:: Output Parameters

+--------------------------------+----------------------------------------------------+------------+-------------+---------+
| Parameter                      | Type                                               | Constraint | Description | Remarks |
+================================+====================================================+============+=============+=========+
| :math:`\text{MAD}`             | :math:`\text{MAD} \in \mathbb R`                   |            |             |         |
+--------------------------------+----------------------------------------------------+------------+-------------+---------+

.. rubric:: Tool Support

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

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

.. rubric:: Single Steps using the Algorithm

* :doc:`/DataPreprocessing/MathematicalComputation/VarianceEstimation/VarianceEstimationWithMeanAbsoluteDeviation/index`

.. rubric:: References

- F.\  Mosteller, J. Tukey, Data Analysis and Regression, Upper Saddle River, NJ: Addison-Wesley, 1977.

- L.\  Sachs, Applied Statistics: A Handbook of Techniques, New York: Springer-Verlag, 1984.