# Cumulative Sum (CUSUM)¶

Cumulative Sum algorithm calculates the cumulative sum of an input data sequence. The basic formula is stated as follows:

$S_0 = 0 \, \text{,}$
$S_n = max(0, S_{n-1} + Y_n - k) \, \text{,}$

where $$Y$$ is the input data sequence and $$S$$ represents the cumulative sums. $$k$$ is a user-given reference value.

Input Parameters

Parameter Type Constraint Description Remarks
$$Y$$ $$Y \in \mathbb R^N$$ $$N \in \mathbb{N}$$ Input data sequence of length $$N$$ None
$$k$$ $$k \in \mathbb R$$ None None None

Output Parameters

Parameter Type Constraint Description Remarks
$$S$$ $$S \in \mathbb R^N$$ $$N \in \mathbb{N}$$ None None

Single Steps using the Algorithm

References

• E.S. Page, Continuous Inspection Scheme, Biometrika, vol. 41(1/2), pp. 100-115, 1954.