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%\VignetteIndexEntry{EWMA Chart with estimated In-Control State}
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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
$$
M_0=0, \quad  M_t=\lambda \frac{X_t-\hat \mu}{\hat \sigma}+(1-\lambda) M_{t-1}
$$


```{r,echo=FALSE}
set.seed(12381900)
```

The following generates a data set of past observations (replace this with your observed past data).
```{r}
X <-  rnorm(250)
```


Next, we initialise the chart and compute the estimates needed for running the
chart - in this case $\hat \mu$ and $\hat \sigma$. 
```{r}
library(spcadjust)
chart <- new("SPCEWMA",model=SPCModelNormal(Delta=0),lambda=0.1);
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.

```{r}
cal <- SPCproperty(data=X,nrep=50,
            property="calARL",chart=chart,params=list(target=100),quiet=TRUE)
cal
``` 
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.
```{r}
newX <- rnorm(100)
S <- runchart(chart, newdata=newX,xi=xihat)
```

Then we plot the data and the chart. 
```{r,fig=TRUE,fig.width=10,fig.height=4}
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)
```

```{r,echo=FALSE}
set.seed(123819123)
```

In the next example, the  chart is run with data that are
out-of-control from time 51 and onwards.
```{r}
newX <- rnorm(100,mean=c(rep(0,50),rep(-1,50)))
S <- runchart(chart, newdata=newX,xi=xihat)
```

```{r,fig=TRUE,fig.width=10,fig.height=4,echo=FALSE}
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+0.5,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)
```