# CUSUM chart based on linear regression model

The following is an example of an application to a CUSUM chart based on a linear regression model.

Assume we have $n$ past in-control data $(Y_{-n},X_{-n}),\ldots,(Y_{-1},X_{-1})$, where $Y_i$ is a response variable and $X_i$ is a corresponding vector of covariates. The parameters $\beta$ of a linear model $\mbox{E} Y=X\beta$ are estimated using e.g. the lm function. The corresponding risk adjusted CUSUM chart to detect a shift of $\Delta>0$ in the mean of the response for new observations $(Y_{1},X_{1}),\ldots,(Y_{n},X_{n})$ is then defined by $$S_0=0, \quad S_t=\max\left(0,S_{t-1}+Y_t-X_t\hat\beta-\Delta/2 \right).$$

set.seed(12381900)


The following generates a data set of past observations (replace this with your observed past data) from the model $\mbox{E}Y=2+x_1+x_2+x_3$ with standard normal noise and distribution of the covariate values as specified below.

n <- 1000
Xlinreg <- data.frame(x1= rbinom(n,1,0.4), x2= runif(n,0,1), x3= rnorm(n))
Xlinreg$y <- 2 + Xlinreg$x1 + Xlinreg$x2 + Xlinreg$x3 + rnorm(n)


Next, we initialise the chart and compute the estimate needed for running the chart - in this case $\hat \beta$.

library(spcadjust)
chartlinreg <- new("SPCCUSUM",model=SPCModellm(Delta=1,formula="y~x1+x2+x3"))
xihat <- xiofdata(chartlinreg,Xlinreg)
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. In this regression model this is computed by non-parametric bootstrapping. The number of bootstrap replications (the argument nrep) shoud be increased for real applications.

cal <- SPCproperty(data=Xlinreg,
nrep=100,
property="calARL",chart=chartlinreg,params=list(target=100),quiet=TRUE)
cal

set.seed(12381903)


## Run the chart

Next, we run the chart with new observations that are in-control.

n <- 100
newXlinreg <- data.frame(x1= rbinom(n,1,0.4), x2= runif(n,0,1), x3=rnorm(n))
newXlinreg$y <- 2 + newXlinreg$x1 + newXlinreg$x2 + newXlinreg$x3 + rnorm(n)
S <- runchart(chartlinreg, newdata=newXlinreg,xi=xihat)

par(mfrow=c(1,1),mar=c(4,5,0,0))
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 that that are out-of-control from time 51 and onwards.

n <- 100
newXlinreg <- data.frame(x1= rbinom(n,1,0.4), x2= runif(n,0,1),x3=rnorm(n))
outind <- c(rep(0,50),rep(1,50))
newXlinreg$y <- 2 + newXlinreg$x1 + newXlinreg$x2 + newXlinreg$x3 + rnorm(n)+outind
S <- runchart(chartlinreg, newdata=newXlinreg,xi=xihat)

par(mfrow=c(1,1),mar=c(4,5,0,0))
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")