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Concept# CUSUM

Summary

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).
E. S. Page referred to a "quality number" , 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 .
As its name implies, CUSUM involves the calculation of a cumulative sum (which is what makes it "sequential"). Samples from a process are assigned weights , and summed as follows:
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 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". 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:
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).

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Change detection

In statistical analysis, change detection or change point detection tries to identify times when the probability distribution of a stochastic process or time series changes. In general the problem concerns both detecting whether or not a change has occurred, or whether several changes might have occurred, and identifying the times of any such changes. Specific applications, like step detection and edge detection, may be concerned with changes in the mean, variance, correlation, or spectral density of the process.

Sequential analysis

In statistics, sequential analysis or sequential hypothesis testing is statistical analysis where the sample size is not fixed in advance. Instead data are evaluated as they are collected, and further sampling is stopped in accordance with a pre-defined stopping rule as soon as significant results are observed. Thus a conclusion may sometimes be reached at a much earlier stage than would be possible with more classical hypothesis testing or estimation, at consequently lower financial and/or human cost.

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