======================
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.