The following is an example of an application to a CUSUM charts 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 \(\mbox{E} Y=X\beta\) is estimated using 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 \[ S_0=0, \quad S_t=\max\left(0,S_{t-1}+Y_t-X_t\hat\beta-\Delta/2 \right). \]
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 and compute the resulting estimate 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
##
## Call:
## lm(formula = formula, data = P)
##
## Coefficients:
## (Intercept) x1 x2 x3
## 2.003 1.091 1.003 1.016
Next find the threshold that with roughly 90\% probability results in an average run length of at least 100 in control. You should increase the number of bootstrap replications (the argument nrep) for real applications.
cal <- SPCproperty(data=Xlinreg,
nrep=100,
property="calARL",chart=chartlinreg,params=list(target=100),quiet=TRUE)
cal
## 90 % CI: A threshold of 3.06 gives an in-control ARL of at least
## 100.
## Unadjusted result: 2.817
## Based on 100 bootstrap repetitions.
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)
In the next example, the chart is run with data that is 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)