==================================== Brown's Double Exponential Smoothing ==================================== :doc:`/WorkProcessClassifiers/LocalAlgorithm/index` - :doc:`/WorkProcessClassifiers/OneDimensionalAlgorithm/index` The basic formulas of the *Brown's Double Exponential Smoothing* algorithm are stated as follows: .. math:: (1) \quad s^{'}_0 = Y_0 \, \text{,} .. math:: (2) \quad s^{''}_0 = Y_0 \, \text{,} .. math:: (3) \quad s^{'}_t = \alpha Y_t + (1-\alpha) s^{'}_{t-1} \, \text{,} .. math:: (4) \quad s^{''}_t = \alpha s^{'}_t + (1-\alpha) s^{''}_{t-1} \, \text{,} .. math:: (5) \quad a_t = 2 s^{'}_t - s^{''}_t \, \text{,} .. math:: (6) \quad b_t = \frac{\alpha}{1-\alpha} (s^{'}_t - s^{''}_t) \, \text{,} .. math:: (7) \quad \hat{Y}_{t+m} = a_t + m b_t \, \text{,} where :math:`Y` is the data sequence beginning at time :math:`t = 0` and :math:`\hat{Y}_{t+m}` is the smoothed forecast for time :math:`t + m`\ . .. 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/ExcelTool/index` .. rubric:: Single Steps using the Algorithm * :doc:`/DataPreprocessing/DataCleaning/DataDenoising/DataDenoisingWithBrownsDoubleExponentialSmoothing/index` .. rubric:: References - NIST/SEMATECH e-Handbook of Statistical Methods `http://www.itl.nist.gov/div898/handbook `__