======================= Backward Moving Average ======================= :doc:/WorkProcessClassifiers/LocalAlgorithm/index - :doc:/WorkProcessClassifiers/OneDimensionalAlgorithm/index *Backward Moving Average* algorithm replaces each original data value by the average over its neighbor values. The choice of filter width has a great impact on the final results. The basic formula (for filter width :math:L = M+1\ ) is stated as follows: .. math:: \hat{x}[n] = \frac{1}{M+1} \sum_{k=-M}^0 x[n+k] \, \text{,} where :math:x[n] and :math:\hat{x}[n] denote raw data and processed data, respectively. .. rubric:: Input Parameters +------------------------+------------------------------------------------+----------------------------------------------------------------+--------------------------------------------------+--------------------------------------------------------------------------------------------------------------+ | Parameter | Type | Constraint | Description | Remarks | +========================+================================================+================================================================+==================================================+==============================================================================================================+ | :math:x[n] | :math:x[n] \in \mathbb R^N | :math:N \in \mathbb{N} | Input data sequence of length :math:N | The algorithm assumes that input data contains no outliers and improper values such as 'nan', 'inf', 'null'. | +------------------------+------------------------------------------------+----------------------------------------------------------------+--------------------------------------------------+--------------------------------------------------------------------------------------------------------------+ | :math:L | :math:L \in \mathbb N | :math:L = M + 1, \quad M \in \mathbb{N} | | | +------------------------+------------------------------------------------+----------------------------------------------------------------+--------------------------------------------------+--------------------------------------------------------------------------------------------------------------+ .. rubric:: Output Parameters +--------------------------------+--------------------------------------------------------+----------------------------------------+---------------------------------------------------+---------+ | Parameter | Type | Constraint | Description | Remarks | +================================+========================================================+========================================+===================================================+=========+ | :math:\hat{x}[n] | :math:\hat{x}[n] \in \mathbb R^N | :math:N \in \mathbb{N} | Output data sequence of length :math:N | | +--------------------------------+--------------------------------------------------------+----------------------------------------+---------------------------------------------------+---------+ .. rubric:: Tool Support * :doc:/Tools/ExcelTool/index * :doc:/Tools/MatlabTool/index For details refer to the online documentation of the function 'filter' __. .. rubric:: Single Steps using the Algorithm * :doc:/DataPreprocessing/DataCleaning/DataDenoising/DataDenoisingWithBackwardMovingAverage/index .. rubric:: References - B.\ Jaehne, Digitale Bildverarbeitung, 5th ed. Berlin: Springer Verlag, 2002. - K.D.\ Kammeyer and K. Kroschel, Digitale Signalverarbeitung, 5th ed. Stuttgart: Teubner, 2002.