CUSUM

In statistical quality control, the CUSUM (or cumulative sum control chart) is a sequential analysis technique developed by E. S. Page of the University of Cambridge. It is typically used for monitoring change detection. CUSUM was announced in Biometrika, in 1954, a few years after the publication of Wald's sequential probability ratio test (SPRT).

CUSUM chart
Originally proposed by	E. S. Page
Process observations
Rational subgroup size	n = 1
Measurement type	Cumulative sum of a quality characteristic
Quality characteristic type	Variables data
Underlying distribution	Normal distribution
Performance
Size of shift to detect	≤ 1.5σ
Process variation chart
Not applicable
Process mean chart
Center line	The target value, T, of the quality characteristic
Upper control limit	${\displaystyle C_{i}^{+}=\max \lbrack 0,x_{i}-\left(T+K\right)+C_{i-1}^{+}\rbrack }$
Lower control limit	${\displaystyle C_{i}^{-}=\max \lbrack 0,\left(T-K\right)-x_{i}+C_{i-1}^{-}\rbrack }$
Plotted statistic	${\displaystyle C_{i}=\sum _{j=1}^{i}{\bar {x}}_{j}-T}$

E. S. Page referred to a "quality number" ${\displaystyle \theta }$, by which he meant a parameter of the probability distribution; for example, the mean. He devised CUSUM as a method to determine changes in it, and proposed a criterion for deciding when to take corrective action. When the CUSUM method is applied to changes in mean, it can be used for step detection of a time series.

A few years later, George Alfred Barnard developed a visualization method, the V-mask chart, to detect both increases and decreases in ${\displaystyle \theta }$.