StatTools : CUSUM for Exponentially Distributed Values Explained

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CUSUM for Exponentially Distributed Measurements Program Page

CUSUM Exponential Dist Notation & Data Example 1 Example 2 References
The CUSUM chart is used to detect small and persistent change, and is based on the cumulative sum of differences between sampling measurements and the mean, (thus CUSUM). This assumes that, in the "in control" state the CUSUM would hover around the expected mean level, as deviation around the mean would cancel each other out. In the "out of control" state, there will be a bias away from the expected mean, and the CUSUM will drift away from the expected mean level.

The CUSUM programs on this site follow the approach outlined in the text book by Hawkins and Olwell (see references), summarized as follows.

  • The user defines the "in control". The central tendency and variance is defined, according to the nature of the data. In the normally distributed measurements, these are the mean and the Standard Deviation.
  • Using specific algorithms, the level of departure (h) from the central tendency to decide that the "out of control" alarm should be triggered is calculated. This level is abbreviated as h
  • h is calculated, depending on the amount of departure (k) the system is designed to detect, and the sensitivity of the detection, in terms of the averaged run length (ARL). The ARL is the expected number of observations between false alarms when the situation is "in control". Conceptually this represents the probability of Type I Error (α). An average run length of 20 is equivalent to α=0.05, ARL of 100, α=0.01.
  • Once the chart and the ARL are defined, sampling takes place at regular intervals. The departure from the expected is corrected by k, then added to CUSUM. If the CUSUM regressed to 0 or beyond, as it often does when the situation is "in control", the CUSUM recommences at 0.
  • In most cases therefore, two CUSUMs can be plotted, one for excessive increase in value, and one for excessive decrease in value (two tails). In most quality control situation however, only one of the tails is of interest.
  • In the programs of this site, 3 levels of h are offered in default, for ARLs of 20 (α=0.05) for yellow alert, 50 (α=0.02) for orange alert, and 100 (α=0.01) for red alert. The idea is that a yellow alert should trigger a heightened expectancy, orange alert triggers an investigation, and red alert triggers immediate response. However, these are merely recommendations, and users should define their own levels of sensitivity.