CUSUM

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For the Roman town of Cusum, see Petrovaradin.

CUSUM is a sequential analysis technique due to E. S. Page of the University of Cambridge. It is typically used for monitoring change detection[1]. CUSUM was announced in Biometrika a few years after the publication of Wald's SPRT algorithm[2].

Page referred to a "quality number" θ, 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.

A few years later, Barnard developed a visualization method, the V-mask chart, to detect both increases and decreases in θ[3].

[edit] Method

As its name implies, CUSUM involves the calculation of a cumulative sum (which is what makes it "sequential"). Samples from a process xn are assigned weights wn, and summed as follows:

Sn = max(0,Sn − 1 + Wn)

Page did not explicitly say that W represents the likelihood function, but this is common usage. Monitoring stops and action is taken when the sum exceeds a certain threshold, h. 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 assess 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.

[edit] Example

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[edit] References

  1. ^ a b Grigg et al (2003). "The Use of Risk-Adjusted CUSUM and RSPRT Charts for Monitoring in Medical Contexts". Statistical Methods in Medical Research 12: 147–170. doi:10.1177/096228020301200205. 
  2. ^ a b Page, E. S. (June, 1954). "Continuous Inspection Scheme". Biometrika 41 (1/2): 100–115. 
  3. ^ Barnard, G.A. (1959). "Control charts and stochastic processes". Journal of the Royal Statistical Society B (Methodological) (21): 239–71. 
  4. ^ "Sufficient statistics and uniformly most powerful tests of statistical hypotheses" . Statistical Research Memoirs I: 113–137.