CUSUM Chart with estimated In-Control State

Using Normality Assumptions

The following is a simple application for CUSUM charts, assuming that all observations are normally distributed.

Based on \(n\) past in-control observations \(X_{-n},\dots,X_{-1}\), the in-control mean is estimated by \(\hat \mu = \frac{1}{n}\sum_{i=-n}^{-1} X_i\) and the in-control variance by \(\hat \sigma^2=\frac{1}{n-1}\sum_{i=-n}^{-1} (X_i-\hat \mu)^2\). Based on new observations \(X_1,X_2,\dots\), the CUSUM chart is then defined by \[ S_0=0, \quad S_t=\max\left(0,\frac{S_{t-1}+X_t-\hat \mu-\Delta/2}{\hat \sigma}\right). \]

The following generates a data set of past observations (replace this with your observed past data).

X <-  rnorm(250)

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Next, we initialise and compute the resulting estimate for running the chart - in this case \(\hat \mu\) and \(\hat \sigma\).

library(spcadjust)
chart <- new("SPCCUSUM",model=SPCModelNormal(Delta=1));
xihat <- xiofdata(chart,X)
str(xihat)
## List of 3
##  $ mu: num 0.0251
##  $ sd: num 1.05
##  $ m : int 250

Calibrating the Chart to a Given Average Run Length (ARL)

We now compute a threshold that with roughly 90\% probability results in an average run length of at least 100 in control. This is based on parametric resampling assuming normality of the observations.

cal <- SPCproperty(data=X,nrep=50,
            property="calARL",chart=chart,params=list(target=100),quiet=TRUE)
cal
## 90 % CI: A threshold of 3.328 gives an in-control ARL of at least
##   100. 
## Unadjusted result:  2.949 
## Based on  50 bootstrap repetitions.

You should increase the number of bootstrap replications (the argument nrep) for real applications. Use the parallell option to speed up the bootstrap by parallel processing.

Run the chart

Next, we run the chart with new observations (that happen to be in-control).

newX <- rnorm(100)
S <- runchart(chart, newdata=newX,xi=xihat)
par(mfrow=c(1,2),mar=c(4,5,0,0))
plot(newX,xlab="t")
plot(S,ylab=expression(S[t]),xlab="t",type="b",ylim=range(S,cal@res+1,cal@raw))
lines(c(0,100),rep(cal@res,2),col="red")
lines(c(0,100),rep(cal@raw,2),col="blue")
legend("topleft",c("Adjusted Threshold","Unadjusted Threshold"),col=c("red","blue"),lty=1)

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In the next example, the chart is run with data that is out-of-control from time 51 onwards.

newX <- rnorm(100,mean=c(rep(0,50),rep(1,50)))
S <- runchart(chart, newdata=newX,xi=xihat)

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