CUSUM

As its name implies, CUSUM involves the calculation of a cumulative sum (which is what makes it "sequential"). Samples from a process ${\displaystyle x_{n}}$ are assigned weights ${\displaystyle \omega _{n}}$, and summed as follows:

${\displaystyle S_{0}=0}$

${\displaystyle S_{n+1}=\max(0,S_{n}+x_{n+1}-\omega _{n})}$

When the value of S exceeds a certain threshold value, a change in value has been found. The above formula only detects changes in the positive direction. When negative changes need to be found as well, the min operation should be used instead of the max operation, and this time a change has been found when the value of S is below the (negative) value of the threshold value.

Page did not explicitly say that ${\displaystyle \omega }$ represents the likelihood function, but this is common usage.

Note that this differs from SPRT by always using zero function as the lower "holding barrier" rather than a lower "holding barrier".[1] Also, CUSUM does not require the use of the likelihood function.

As a means of assessing CUSUM's performance, Page defined the average run length (A.R.L.) metric; "the expected number of articles sampled before action is taken." He further wrote:[2]

When the quality of the output is satisfactory the A.R.L. is a measure of the expense incurred by the scheme when it gives false alarms, i.e., Type I errors (Neyman & Pearson, 1936[4]). On the other hand, for constant poor quality the A.R.L. measures the delay and thus the amount of scrap produced before the rectifying action is taken, i.e., Type II errors.

The following example shows 20 observations ${\displaystyle X}$ of a process with a mean of 0 and a standard deviation of 0.5.

From the ${\displaystyle Z}$ column, it can be seen that ${\displaystyle X}$ never deviates by 3 standard deviations (${\displaystyle 3\sigma }$), so simply alerting on a high deviation will not detect a failure, whereas CUSUM shows that the ${\displaystyle S_{H}}$ value exceeds 4 at the 17th observation.

Column	Description
${\displaystyle X}$	The observations of the process with an expected mean ${\displaystyle {\bar {x}}}$ of 0 and an expected standard deviation ${\displaystyle \sigma _{X}}$ of 0.5
${\displaystyle Z}$	The normalized observations, i.e. centered around the mean and scaled by the standard deviation ${\displaystyle Z_{n}={\frac {X_{n}-{\bar {x}}}{\sigma _{X}}}}$
${\displaystyle S_{H}}$	The high CUSUM value, detecting a positive anomaly, ${\displaystyle {S_{H}}_{n+1}=\max(0,{S_{H}}_{n}+Z_{n+1}-\omega )}$
${\displaystyle S_{L}}$	The low CUSUM value, detecting a negative anomaly, ${\displaystyle {S_{L}}_{n+1}=\max(0,{S_{L}}_{n}-Z_{n+1}-\omega )}$

where ${\displaystyle \omega }$ is a critical level parameter (tunable, same as threshold T) that's used to adjust the sensitivity of change detection: larger ${\displaystyle \omega }$ makes CUSUM less sensitive to the change and vice versa.

Cumulative observed-minus-expected plots[1] are a related method.

• Michèle Basseville and Igor V. Nikiforov (April 1993). Detection of Abrupt Changes: Theory and Application. Englewood Cliffs, NJ: Prentice-Hall. ISBN 0-13-126780-9.
• Mishra, S., Vanli, O. A., & Park, C (2015). "A Multivariate Cumulative Sum Method for Continuous Damage Monitoring with Lamb-wave Sensors", International Journal of Prognostics and Health Management, ISSN 2153-2648

• "Engineering Statistics Handbook - Cusum Control Charts"