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

*One-sided Median* algorithm computes the median

.. math::

   m_n^Y = \text{Median}\{Y_{n-2k}, \ldots, Y_{n-1}\}

and the median

.. math::

   m_n^Z = \text{Median}\{Z_{n-2k}, \ldots, Z_{n-1}\}

with

.. math::

   Z_n = Y_n - Y_{n-1}.

:math:`2k` is the size of the neighborhood window. Defining

.. math::

   \hat{m}_n = m_n^Y + k m_n^Z \, \text{,}

:math:`Y_n` is treated as a value outside the region of interest if :math:`|Y_n - \hat{m}_n| \geq \tau`\ .

.. 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:`\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/OutlierDetectionWithOneSidedMedian/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.