sewma.crit | R Documentation |
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.
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)
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 |
cu |
for two-sided ( |
hs |
so-called headstart (enables fast initial response); the default ( |
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 |
mode |
only deployed for |
ur |
truncation of lower chart for |
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. |
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.
Returns the lower and upper control limit cl
and cu
.
Sven Knoth
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.
sewma.arl
for calculation of ARL of variance charts.
## 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)
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