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\name{xshewhartrunsrules.arl}
\alias{xshewhartrunsrules.arl}
\alias{xshewhartrunsrules.crit}
\alias{xshewhartrunsrules.ad}
\alias{xshewhartrunsrules.matrix}
\title{Compute ARLs of Shewhart control charts with and without runs rules}
\description{Computation of the (zero-state and steady-state) Average Run Length (ARL)
for Shewhart control charts with and without runs rules
monitoring normal mean.}
\usage{xshewhartrunsrules.arl(mu, c = 1, type = "12")
xshewhartrunsrules.crit(L0, mu = 0, type = "12")
xshewhartrunsrules.ad(mu1, mu0 = 0, c = 1, type = "12")
xshewhartrunsrules.matrix(mu, c = 1, type = "12")}
\arguments{
\item{mu}{true mean.}
\item{L0}{pre-defined in-control ARL, that is, determine \code{c} so that the mean
number of observations until a false alarm is \code{L0}.}
\item{mu1, mu0}{for the steady-state ARL two means are specified, mu0 is the in-control one
and usually equal to 0 , and mu1 must be given.}
\item{c}{normalizing constant to ensure specific alarming behavior.}
\item{type}{controls the type of Shewhart chart used, seed details section.}
}
\details{
\code{xshewhartrunsrules.arl} determines the zero-state Average Run Length (ARL)
based on the Markov chain approach given in Champ and Woodall (1987).
\code{xshewhartrunsrules.matrix} provides the corresponding
transition matrix that is also used in \code{xDshewhartrunsrules.arl} (ARL for control charting drift).
\code{xshewhartrunsrules.crit} allows to find the normalization constant \code{c} to
attain a fixed in-control ARL. Typically this is needed to calibrate the chart.
With \code{xshewhartrunsrules.ad} the steady-state ARL is calculated.
With the argument \code{type} certain runs rules could be set. The following list gives an overview.
\itemize{
\item{"1"}{ The classical Shewhart chart with \code{+/- 3 c sigma} control limits (\code{c} is typically
equal to 1 and can be changed by the argument \code{c}).}
\item{"12"}{ The classic and the following runs rule: 2 of 3 are beyond \code{+/- 2 c sigma} on the same
side of the chart.}
\item{"13"}{ The classic and the following runs rule: 4 of 5 are beyond \code{+/- 1 c sigma} on the same
side of the chart.}
\item{"14"}{ The classic and the following runs rule: 8 of 8 are on the same side of the chart
(referring to the center line).}}
}
\value{Returns a single value which resembles the zero-state or steady-state ARL.
\code{xshewhartrunsrules.matrix} returns a matrix.}
\references{
C. W. Champ and W. H. Woodall (1987),
Exact results for Shewhart control charts with supplementary runs rules,
\emph{Technometrics 29}, 393-399.
}
\author{Sven Knoth}
\seealso{
\code{xDshewhartrunsrules.arl} for zero-state ARL of Shewhart control charts
with or without runs rules under drift.
}
\examples{
## Champ/Woodall (1987)
## Table 1
mus <- (0:15)/5
Mxshewhartrunsrules.arl <- Vectorize(xshewhartrunsrules.arl, "mu")
# standard (1 of 1 beyond 3 sigma) Shewhart chart without runs rules
C1 <- round(Mxshewhartrunsrules.arl(mus, type="1"), digits=2)
# standard + runs rule: 2 of 3 beyond 2 sigma on the same side
C12 <- round(Mxshewhartrunsrules.arl(mus, type="12"), digits=2)
# standard + runs rule: 4 of 5 beyond 1 sigma on the same side
C13 <- round(Mxshewhartrunsrules.arl(mus, type="13"), digits=2)
# standard + runs rule: 8 of 8 on the same side of the center line
C14 <- round(Mxshewhartrunsrules.arl(mus, type="14"), digits=2)
## original results are
# mus C1 C12 C13 C14
# 0.0 370.40 225.44 166.05 152.73
# 0.2 308.43 177.56 120.70 110.52
# 0.4 200.08 104.46 63.88 59.76
# 0.6 119.67 57.92 33.99 33.64
# 0.8 71.55 33.12 19.78 21.07
# 1.0 43.89 20.01 12.66 14.58
# 1.2 27.82 12.81 8.84 10.90
# 1.4 18.25 8.69 6.62 8.60
# 1.6 12.38 6.21 5.24 7.03
# 1.8 8.69 4.66 4.33 5.85
# 2.0 6.30 3.65 3.68 4.89
# 2.2 4.72 2.96 3.18 4.08
# 2.4 3.65 2.48 2.78 3.38
# 2.6 2.90 2.13 2.43 2.81
# 2.8 2.38 1.87 2.14 2.35
# 3.0 2.00 1.68 1.89 1.99
data.frame(mus, C1, C12, C13, C14)
## plus calibration, i. e. L0=250 (the maximal value for "14" is 255!
L0 <- 250
c1 <- xshewhartrunsrules.crit(L0, type = "1")
c12 <- xshewhartrunsrules.crit(L0, type = "12")
c13 <- xshewhartrunsrules.crit(L0, type = "13")
c14 <- xshewhartrunsrules.crit(L0, type = "14")
C1 <- round(Mxshewhartrunsrules.arl(mus, c=c1, type="1"), digits=2)
C12 <- round(Mxshewhartrunsrules.arl(mus, c=c12, type="12"), digits=2)
C13 <- round(Mxshewhartrunsrules.arl(mus, c=c13, type="13"), digits=2)
C14 <- round(Mxshewhartrunsrules.arl(mus, c=c14, type="14"), digits=2)
data.frame(mus, C1, C12, C13, C14)
## and the steady-state ARL
Mxshewhartrunsrules.ad <- Vectorize(xshewhartrunsrules.ad, "mu1")
C1 <- round(Mxshewhartrunsrules.ad(mus, c=c1, type="1"), digits=2)
C12 <- round(Mxshewhartrunsrules.ad(mus, c=c12, type="12"), digits=2)
C13 <- round(Mxshewhartrunsrules.ad(mus, c=c13, type="13"), digits=2)
C14 <- round(Mxshewhartrunsrules.ad(mus, c=c14, type="14"), digits=2)
data.frame(mus, C1, C12, C13, C14)
}
\keyword{ts}
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