====================== Cumulative Sum (CUSUM) ====================== :doc:/WorkProcessClassifiers/LocalAlgorithm/index - :doc:/WorkProcessClassifiers/OneDimensionalAlgorithm/index *Cumulative Sum* algorithm calculates the cumulative sum of an input data sequence. The basic formula is stated as follows: .. math:: S_0 = 0 \, \text{,} .. math:: S_n = max(0, S_{n-1} + Y_n - k) \, \text{,} where :math:Y is the input data sequence and :math:S represents the cumulative sums. :math:k is a user-given reference 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:k | :math:k \in \mathbb R | | | | +--------------------+--------------------------------------------+----------------------------------------+--------------------------------------------------+---------+ .. rubric:: Output Parameters +--------------------+--------------------------------------------+----------------------------------------+-------------+---------+ | Parameter | Type | Constraint | Description | Remarks | +====================+============================================+========================================+=============+=========+ | :math:S | :math:S \in \mathbb R^N | :math:N \in \mathbb{N} | | | +--------------------+--------------------------------------------+----------------------------------------+-------------+---------+ .. rubric:: Tool Support * :doc:/Tools/MatlabTool/index .. rubric:: Single Steps using the Algorithm * :doc:/DataPreprocessing/MathematicalComputation/ComputingCUSUMValuesWithCUSUM/index .. rubric:: References - E.S.\ Page, Continuous Inspection Scheme, Biometrika, vol. 41(1/2), pp. 100-115, 1954.