============================
Single Exponential Smoothing
============================
:doc:`/WorkProcessClassifiers/LocalAlgorithm/index` - :doc:`/WorkProcessClassifiers/OneDimensionalAlgorithm/index`

*Single Exponential Smoothing* algorithm uses a parameter :math:`\alpha` to establish an exponentially decreasing weighting factor for time series data. When :math:`\alpha` is equal to one, there is no smoothing, and when :math:`\alpha` is equal to zero, a constant time series is returned. The basic formulas are stated as follows:

.. math::

   (1) \quad s_1 = Y_0 \, \text{,}



.. math::

   (2) \quad s_t = \alpha(Y_{t-1} - s_{t-1}) + s_{t-1}, \quad t > 1 \, \text{,}



.. math::

   (3) \quad \hat{Y}_t = s_t \, \text{,}

where :math:`Y` is the data sequence beginning at time :math:`t = 0` and :math:`\hat{Y}_{t}` is the smoothed forecast for time :math:`t`\ . This function is exponential through the nesting of the function at each subsequent data value.

.. 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:`\alpha`             | :math:`\alpha \in \mathbb R`                   | :math:`0 \leq \alpha \leq 1`                   |                                                  |         |
+----------------------------+------------------------------------------------+------------------------------------------------+--------------------------------------------------+---------+

.. rubric:: Output Parameters

+----------------------------+----------------------------------------------------+------------+-------------+---------+
| Parameter                  | Type                                               | Constraint | Description | Remarks |
+============================+====================================================+============+=============+=========+
| :math:`\hat{Y}`            | :math:`\hat{Y} \in \mathbb R^N`                    |            |             |         |
+----------------------------+----------------------------------------------------+------------+-------------+---------+

.. rubric:: Tool Support

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

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

.. rubric:: Single Steps using the Algorithm

* :doc:`/DataPreprocessing/DataCleaning/DataDenoising/DataDenoisingWithSingleExponentialSmoothing/index`

.. rubric:: References

- K.D.\  Kammeyer and K. Kroschel, Digitale Signalverarbeitung, 5th ed. Stuttgart: Teubner, 2002.