# CUSUM chart with estimated in-control state

## Using normality assumptions

The following is an example of an application to a basic CUSUM chart, assuming that all observations are normally distributed.

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

set.seed(12381900)


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

X <-  rnorm(250)

par(mar=c(4,5,0,0))
plot(-(250:1),X,xlab="t",ylab=expression(X[t]))


Next, we initialise the chart and compute the estimates needed 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)


## 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


The number of bootstrap replications (the argument nrep) shoud be increased 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 are in-control.

newX <- rnorm(100)
S <- runchart(chart, newdata=newX,xi=xihat)


Then we plot the data and the chart.

par(mfrow=c(1,2),mar=c(4,5,0.1,0.1))
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")


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

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

par(mfrow=c(1,2),mar=c(4,5,0.1,0.1))
plot(newX,xlab="t")
plot(S,ylab=expression(S[t]),xlab="t",type="b",ylim=pmin(range(S,cal@res,cal@raw),15))
lines(c(0,100),rep(cal@res,2),col="red")
lines(c(0,100),rep(cal@raw,2),col="blue")