================
Two-sided Median
================
:doc:`/WorkProcessClassifiers/LocalAlgorithm/index` - :doc:`/WorkProcessClassifiers/OneDimensionalAlgorithm/index`

*Two-sided Median* algorithm computes the median

.. math::

    m_n^k = \text{Median}\{Y_{n-k}, \ldots, Y_{n-1}, Y_{n+1}, \ldots, Y_{n+k} \}.

:math:`Y_n` is the observed point and :math:`2k` is the size of the neighborhood window. If

.. math::

   |m_n^k - Y_n| \geq \tau \, \text{,}

:math:`Y_n` is treated as a value outside the region of interest.

.. rubric:: Input Parameters

+------------------------+------------------------------------------------+--------------------------------------------------------+------------------------------------------------+---------+
| Parameter              | Type                                           | Constraint                                             | Description                                    | Remarks |
+========================+================================================+========================================================+================================================+=========+
| :math:`Y`              | :math:`Y \in \mathbb R^N`                      | :math:`N \in \mathbb{N}, N \geq 3`                     | Input data vector of length :math:`N`          |         |
+------------------------+------------------------------------------------+--------------------------------------------------------+------------------------------------------------+---------+
| :math:`\tau`           | :math:`\tau \in \mathbb R^+`                   |                                                        | User-specified threshold                       |         |
+------------------------+------------------------------------------------+--------------------------------------------------------+------------------------------------------------+---------+

.. 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:: Single Steps using the Algorithm

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

.. rubric:: References

- \  S.\  Basu, M. Meckesheimer, Automatic outlier detection for time series: an application to sensor data, Knowledge and Information Systems, vol. 11, Issue 2, pp 137-154, 2007.