# 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
\[
S_0=0, \quad S_t=\max\left(0,\frac{S_{t-1}+X_t-\hat \mu-\Delta/2}{\hat \sigma}\right).
\]

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

```
X <- rnorm(250)
```

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)
```

```
## 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.
```

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")
legend("topleft",c("Adjusted Threshold","Unadjusted Threshold"),col=c("red","blue"),lty=1)
```

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)
```