Methods¶
CUSUM parameters¶
Parameter |
Description |
|---|---|
μ_in |
The mean of the performance metric when the process is in-control, i.e., when there is no performance drift |
ARL_0 |
Number of observations before the control chart signals a false detection |
σ_in |
The in-control standard deviation of the metric |
ARL_1 |
Number of observations before the control chart signals a true detection |
k |
The normalized reference value, which is related to the magnitude of change that one is interested in detecting. k = 0.5 is the default choice for detecting a unit standard deviation change |
S_hi |
Cumulative sum of positive changes in the metric |
h |
The normalized threshold or control limit (default =4). This threshold determines when the control chart signals a detection |
S_lo |
Cumulative sum of negative changes in the metric |
CUSUM chart¶
A two-sided CUSUM control chart computes the cumulative differences or deviations of individual observations from the target mean (or in-control mean, \(\mu_{in}\)). The positive and negative cumulative sums are calculated:
where d denotes a unit of time, \(x_d\) is the value of quantity being monitored at time \(d\), \(\hat{\mu}_{in}\) is the in-control mean of \(x_d\), and \(K\) is a “reference value” related to the magnitude of change that one is interested in detecting. \(S_{hi}\) and \(S_{lo}\) are the cumulative sum of positive and negative changes. To detect a change in the observed values from the in-control mean, the CUSUM scheme accumulates deviations that are \(K\) units away from the in-control mean. Let \(\sigma_{in}\) denote the in-control standard deviation of \(x_d\).