spc (version 0.3)

sewma.crit: Compute critical values of EWMA control charts (variance charts)

Description

Computation of the critical values (similar to alarm limits) for different types of EWMA control charts (based on the sample variance $S^2$) monitoring normal variance.

Usage

sewma.crit(l,L0,df,sigma0=1,cl=NULL,cu=NULL,hs=1,s2.on=TRUE,sided="upper",mode="fixed",r=40,qm=30)

Arguments

l

smoothing parameter lambda of the EWMA control chart.

L0

in-control ARL.

df

actual degrees of freedom, corresponds to batch size.

sigma0

in-control standard deviation.

cl

deployed for sided="Rupper", that is, upper variance control chart with lower reflecting barrier cl.

cu

for two-sided (sided="two") and fixed upper control limit (mode="fixed") a value larger than sigma0 has to been given, for all other cases cu is ignored.

hs

so-called headstart (give fast initial response).

s2.on

distinguish between $S^2$ and $S$ chart.

sided

distinguish between one- and two-sided two-sided EWMA-$S^2$ control charts by choosing "upper" (upper chart without reflection at cl -- the actual value of cl is not used), "Rupper" (upper chart with ref

mode

only deployed for sided="two" -- with "fixed" an upper control limit (see cu) is set and only the lower is calculated to obtain the in-control ARL L0, while with "unbiased" a ce

r

dimension of the resulting linear equation system.

qm

number of quadrature nodes.

Value

• Returns the lower and upper control limit cl and cu.

Details

sewma.crit determines the critical values (similar to alarm limits) for given in-control ARL L0 by applying secant rule and using sewma.arl(). In case of sided="two" and mode="unbiased" a two-dimensional secant rule is applied that also ensures that the maximum of the ARL function for given standard deviation is attained at sigma0. See ? and the related example.

References

H.-J. Mittag and D. Stemann and B. Tewes (1998), EWMA-Karten zur "Uberwachung der Streuung von Qualit"atsmerkmalen, Allgemeines Statistisches Archiv 82, 327-338,

C. A. Acosta-Mej'ia and J. J. Pignatiello Jr. and B. V. Rao (1999), A comparison of control charting procedures for monitoring process dispersion, IIE Transactions 31, 569-579.

S. Knoth (2005), Accurate ARL computation for EWMA-$S^2$ control charts, Statistics and Computing 15, 341-352.

S. Knoth (2006), Computation of the ARL for CUSUM-$S^2$ schemes, Computational Statistics & Data Analysis 51, 499-512.

See Also

sewma.arl for calculation of ARL of variance charts.

Examples

## Mittag et al. (1998)
## compare their upper critical value 2.91 that
## leads to the upper control limit via the formula shown below
## (for the usual upper EWMA \eqn{S^2}{S^2})

l  <- 0.18
L0 <- 250
df <- 4
limits <- sewma.crit(l, L0, df)
limits["cu"]

limits.cu.mittag_et_al <- 1 + sqrt(l/(2-l))*sqrt(2/df)*2.91
limits.cu.mittag_et_al

## Knoth{2005}
## reproduce the critical value given in Figure 2 (c=1.661865) for
## upper EWMA \eqn{S^2}{S^2} with df=1

l  <- 0.025
L0 <- 250
df <- 1
limits <- sewma.crit(l, L0, df)
cv.Fig2 <- (limits["cu"]-1)/( sqrt(l/(2-l))*sqrt(2/df) )
cv.Fig2

## the small difference (sixth digit after decimal point) stems from
## tighter criterion in the secant rule implemented in the R package.

## demo of unbiased ARL curves
## Deploy, please, not matrix dimensions smaller than 50 -- for the
## sake of accuracy, the value 80 was used.
## Additionally, this example needs between 1 and 2 minutes on a 1.6 Ghz box. 

l  <- 0.1
L0 <- 500
df <- 4
limits <- sewma.crit(l, L0, df, sided="two", mode="unbiased", r=80)
SEWMA.arl <- Vectorize(sewma.arl, "sigma")
SEWMA.ARL <- function(sigma) 
  SEWMA.arl(l, limits[1], limits[2], sigma, df, sided="two", r=80)
layout(matrix(1:2, nrow=1))
curve(SEWMA.ARL, .75, 1.25, log="y")
curve(SEWMA.ARL, .95, 1.05, log="y")

## control limits for upper and lower EWMA charts with reflecting barriers
## (reflection at in-control level sigma0 = 1)
## examples from Knoth (2006), Tables 4 and 5

## upper chart with reflection at sigma0=1 in Table 4: c = 2.4831
l <- 0.15
L0 <- 100
df <- 4
limits <- sewma.crit(l, L0, df, cl=1, sided="Rupper", r=100)
cv.Tab4 <- (limits["cu"]-1)/( sqrt(l/(2-l))*sqrt(2/df) )
cv.Tab4

## lower chart with reflection at sigma0=1 in Table 5: c = 2.0613
l <- 0.115
L0 <- 200
df <- 5
limits <- sewma.crit(l, L0, df, cu=1, sided="Rlower", r=100)
cv.Tab5 <- -(limits["cl"]-1)/( sqrt(l/(2-l))*sqrt(2/df) )
cv.Tab5