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EWMA chart with estimated in-control state

Using normality assumptions

The following is an illustration for EWMA charts, assuming that all observations are normally distributed.

Based on \(n\) past in-control observations \(X_{-n},\dots,X_{-1}\), the in-control mean and standard deviation can be estimated by the sample mean, \(\hat \mu\), and sample standard deviation, \(\hat \sigma\). For new observations \(X_1,X_2,\dots\), an EWMA chart based on these estimated parameters is defined by \[ M_0=0, \quad M_t=\lambda \frac{X_t-\hat \mu}{\hat \sigma}+(1-\lambda) M_{t-1} \]

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("SPCEWMA",model=SPCModelNormal(Delta=0),lambda=0.1);
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 +/- 0.5342 gives an in-control ARL of at
##   least 100. 
## Unadjusted result:  0.4928 
## Based on  50 bootstrap repetitions.

You should increase the number of bootstrap replications (the argument nrep) for real applications.

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(-cal@res,S,cal@res+0.3,cal@raw))
lines(c(0,100),rep(cal@res,2),col="red")
lines(c(0,100),rep(cal@raw,2),col="blue")
abline(0,0,lty=3)
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 that 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)

These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.