sewma.crit {spc}

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=NULL,s2.on=TRUE,
sided="upper",mode="fixed",ur=4,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 subgroup size (for known mean it is equal to the subgroup size, for unknown mean it is equal to subgroup size minus one.

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 (enables fast initial response); the default (NULL) yields the expected in-control value of S^2 (1) and S (c_4), respectively.

s2.on

distinguishes between S^2 and S chart.

sided

distinguishes 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 reflection at cl), "Rlower" (lower chart with reflection at cu), and "two" (two-sided chart), respectively.

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 certain unbiasedness of the ARL function is guaranteed (here, both the lower and the upper control limit are calculated). With "vanilla" limits symmetric around 1 (the in-control value of the variance) are determined, while for "eq.tails" the in-control ARL values of two single EWMA variance charts (decompose the two-sided scheme into one lower and one upper scheme) are matched.

ur

truncation of lower chart for eq.tails mode.

r

dimension of the resulting linear equation system (highest order of the collocation polynomials).

qm

number of quadrature nodes for calculating the collocation definite integrals.

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 Knoth (2010) and the related example.

Value

Returns the lower and upper control limit cl and cu.

Author(s)

Sven Knoth

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

S. Knoth (2006b), The art of evaluating monitoring schemes – how to measure the performance of control charts? in Frontiers in Statistical Quality Control 8, H.-J. Lenz and P.-T. Wilrich (Eds.), Physica Verlag, Heidelberg, Germany, 74-99.

S. Knoth (2010), Control Charting Normal Variance – Reflections, Curiosities, and Recommendations, in Frontiers in Statistical Quality Control 9, H.-J. Lenz and P.-T. Wilrich (Eds.), Physica Verlag, Heidelberg, Germany, 3-18.

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}).
## See Knoth (2006b) for a discussion of this EWMA setup and it's evaluation.

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. 

## Not run: 
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")
## End(Not run)
# the above stuff needs about 1 minute

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

## Not run: 
## 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
## End(Not run)