==================
Maximum Likelihood
==================
:doc:`/WorkProcessClassifiers/GlobalAlgorithm/index` - :doc:`/WorkProcessClassifiers/OneDimensionalAlgorithm/index`

*Maximum Likelihood* algorithm decides whether a value is a value outside the region of interest based on that value's distance from the estimated mean distribution. Generally, if

.. math::

   |Y_i - \mu| > 3 \cdot \sigma \, \text{,}

the data point is considered outside the region of interest. :math:`\mu` is the mean distribution, :math:`Y_i` is :math:`i`\ th element of :math:`Y` and :math:`\sigma` denotes the standard deviation. Note that :math:`\mu \pm 3\sigma` contains :math:`99.7%` data under the assumption of normal distribution.

.. rubric:: Input Parameters

+----------------------------+------------------------------------------------+----------------------------------------+--------------------------------------------------+---------+
| Parameter                  | Type                                           | Constraint                             | Description                                      | Remarks |
+============================+================================================+========================================+==================================================+=========+
| :math:`Y`                  | :math:`Y \in \mathbb R^N`                      | :math:`N \in \mathbb{N}`               | Input data sequence of length :math:`N`          |         |
+----------------------------+------------------------------------------------+----------------------------------------+--------------------------------------------------+---------+
| :math:`\mu`                | :math:`\mu \in \mathbb{R}`                     |                                        | Mean distribution of :math:`Y`                   |         |
+----------------------------+------------------------------------------------+----------------------------------------+--------------------------------------------------+---------+
| :math:`\sigma`             | :math:`\sigma\in \mathbb{R}`                   |                                        | Standard deviation of :math:`Y`                  |         |
+----------------------------+------------------------------------------------+----------------------------------------+--------------------------------------------------+---------+

.. rubric:: Output Parameters

+----------------------------+----------------------------------------------------+------------+-------------------------------------------------------------------------------------------+---------+
| Parameter                  | Type                                               | Constraint | Description                                                                               | Remarks |
+============================+====================================================+============+===========================================================================================+=========+
| :math:`\hat{Y}`            | :math:`\hat{Y} \in \mathbb R^N`                    |            | Values in the :math:`Y`          list which are outside the region of interest are marked |         |
+----------------------------+----------------------------------------------------+------------+-------------------------------------------------------------------------------------------+---------+

.. rubric:: Tool Support

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

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

.. rubric:: Single Steps using the Algorithm

* :doc:`/DataPreprocessing/DataCleaning/OutlierDetection/OutlierDetectionWithMaximumLikelihood/index`

.. rubric:: References

- J.\  Han, M. Kamber and J. Pei, Data Mining - Concepts and Techniques, 3rd ed., Amsterdam: Morgan Kaufmann Publishers, 2012.