xewma.arl {spc} | R Documentation |
Computation of the (zero-state) Average Run Length (ARL) for different types of EWMA control charts monitoring normal mean.
xewma.arl(l,c,mu,zr=0,hs=0,sided="one",limits="fix",r=40)
l |
smoothing parameter lambda of the EWMA control chart. |
c |
critical value (similar to alarm limit) of the EWMA control chart. |
mu |
true mean. |
zr |
reflection border for the one-sided chart. |
hs |
so-called headstart (give fast initial response). |
sided |
distinguish between one- and two-sided
two-sided EWMA control chart by choosing "one" and "two" ,
respectively. |
limits |
distinguish between different control limits behavior. |
r |
number of quadrature nodes, dimension of the resulting linear
equation system is equal to r+1 (one-sided) or r
(two-sided). |
In case of the two-sided chart with fixed control limits and
of the one-sided chart,
xewma.arl
determines the Average Run Length (ARL) by numerically
solving the related ARL integral equation by means of the Nystroem method
based on Gauss-Legendre quadrature.
Otherwise, for the two-sided chart with varying limits the method
presented in Knoth (2003) is utilized.
Returns a single value which resembles the ARL.
Sven Knoth
K.-H. Waldmann (1986), Bounds for the distribution of the run length of geometric moving average charts, Appl. Statist. 35, 151-158.
S. V. Crowder (1987), A simple method for studying run-length distributions of exponentially weighted moving average charts, Technometrics 29, 401-407.
J. M. Lucas, M. S. Saccucci (1990), Exponentially weighted moving average control schemes: Properties and enhancements, Technometrics 32, 1-12.
S. Chandrasekaran, J. R. English, R. L. Disney (1995), Modeling and analysis of EWMA control schemes with variance-adjusted control limits, IIE Transactions 277, 282-290.
T. R. Rhoads, D. C. Montgomery, C. M. Mastrangelo (1996), Fast initial response scheme for exponentially weighted moving average control chart, Quality Engineering 9, 317-327.
S. H. Steiner (1999), EWMA control charts with time-varying control limits and fast initial response, Journal of Quality Technology 31, 75-86.
S. Knoth (2003), EWMA schemes with non-homogeneous transition kernels, Sequential Analysis 22, 241-255.
S. Knoth (2004), Fast initial response features for EWMA Control Charts, Statistical Papers 46, 47-64.
xcusum.arl
for zero-state ARL computation of CUSUM control charts
and xewma.ad
for the steady-state ARL.
## Waldmann (1986), one-sided EWMA l <- .75 round(xewma.arl(l,2*sqrt((2-l)/l),0,zr=-4*sqrt((2-l)/l)),digits=1) l <- .5 round(xewma.arl(l,2*sqrt((2-l)/l),0,zr=-4*sqrt((2-l)/l)),digits=1) ## original values are 209.3 and 3907.5 (in Table 2). ## Waldmann (1986), two-sided EWMA with fixed control limits l <- .75 round(xewma.arl(l,2*sqrt((2-l)/l),0,sided="two"),digits=1) l <- .5 round(xewma.arl(l,2*sqrt((2-l)/l),0,sided="two"),digits=1) ## original values are 104.0 and 1952 (in Table 1). ## Crowder (1987), two-sided EWMA with fixed control limits l1 <- .5 l2 <- .05 c <- 2 mu <- (0:16)/4 arl1 <- sapply(mu,l=l1,c=c,sided="two",xewma.arl) arl2 <- sapply(mu,l=l2,c=c,sided="two",xewma.arl) round(cbind(mu,arl1,arl2),digits=2) ## original results are (in Table 1) ## 0.00 26.45 127.53 ## 0.25 20.12 43.94 ## 0.50 11.89 18.97 ## 0.75 7.29 11.64 ## 1.00 4.91 8.38 ## 1.25 3.95* 6.56 ## 1.50 2.80 5.41 ## 1.75 2.29 4.62 ## 2.00 1.94 4.04 ## 2.25 1.70 3.61 ## 2.50 1.51 3.26 ## 2.75 1.37 2.99 ## 3.00 1.26 2.76 ## 3.25 1.18 2.56 ## 3.50 1.12 2.39 ## 3.75 1.08 2.26 ## 4.00 1.05 2.15 (* -- in Crowder (1987) typo!?). ## Lucas/Saccucci (1990) ## two-sided EWMA ## with fixed limits l1 <- .5 l2 <- .03 c1 <- 3.071 c2 <- 2.437 mu <- c(0,.25,.5,.75,1,1.5,2,2.5,3,3.5,4,5) arl1 <- sapply(mu,l=l1,c=c1,sided="two",xewma.arl) arl2 <- sapply(mu,l=l2,c=c2,sided="two",xewma.arl) round(cbind(mu,arl1,arl2),digits=2) ## original results are (in Table 3) ## 0.00 500. 500. ## 0.25 255. 76.7 ## 0.50 88.8 29.3 ## 0.75 35.9 17.6 ## 1.00 17.5 12.6 ## 1.50 6.53 8.07 ## 2.00 3.63 5.99 ## 2.50 2.50 4.80 ## 3.00 1.93 4.03 ## 3.50 1.58 3.49 ## 4.00 1.34 3.11 ## 5.00 1.07 2.55. ## with fir feature l1 <- .5 l2 <- .03 c1 <- 3.071 c2 <- 2.437 hs1 <- c1/2 hs2 <- c2/2 mu <- c(0,.5,1,2,3,5) arl1 <- sapply(mu,l=l1,c=c1,hs=hs1,sided="two",limits="fir",xewma.arl) arl2 <- sapply(mu,l=l2,c=c2,hs=hs2,sided="two",limits="fir",xewma.arl) round(cbind(mu,arl1,arl2),digits=2) ## original results are (in Table 5) ## 0.0 487. 406. ## 0.5 86.1 18.4 ## 1.0 15.9 7.36 ## 2.0 2.87 3.43 ## 3.0 1.45 2.34 ## 5.0 1.01 1.57. ## Chandrasekaran, English, Disney (1995) ## two-sided EWMA with fixed and variance adjusted limits (vacl) l1 <- .25 l2 <- .1 c1s <- 2.9985 c1n <- 3.0042 c2s <- 2.8159 c2n <- 2.8452 mu <- c(0,.25,.5,.75,1,2) arl1s <- sapply(mu,l=l1,c=c1s,sided="two",xewma.arl) arl1n <- sapply(mu,l=l1,c=c1n,sided="two",limits="vacl",xewma.arl) arl2s <- sapply(mu,l=l2,c=c2s,sided="two",xewma.arl) arl2n <- sapply(mu,l=l2,c=c2n,sided="two",limits="vacl",xewma.arl) round(cbind(mu,arl1s,arl1n,arl2s,arl2n),digits=2) ## original results are (in Table 2) ## 0.00 500. 500. 500. 500. ## 0.25 170.09 167.54 105.90 96.6 ## 0.50 48.14 45.65 31.08 24.35 ## 0.75 20.02 19.72 15.71 10.74 ## 1.00 11.07 9.37 10.23 6.35 ## 2.00 3.59 2.64 4.32 2.73. ## The results in Chandrasekaran, English, Disney (1995) are not ## that accurate. Let us consider the more appropriate comparison c1s <- xewma.crit(l1,500,sided="two") c1n <- xewma.crit(l1,500,sided="two",limits="vacl") c2s <- xewma.crit(l2,500,sided="two") c2n <- xewma.crit(l2,500,sided="two",limits="vacl") mu <- c(0,.25,.5,.75,1,2) arl1s <- sapply(mu,l=l1,c=c1s,sided="two",xewma.arl) arl1n <- sapply(mu,l=l1,c=c1n,sided="two",limits="vacl",xewma.arl) arl2s <- sapply(mu,l=l2,c=c2s,sided="two",xewma.arl) arl2n <- sapply(mu,l=l2,c=c2n,sided="two",limits="vacl",xewma.arl) round(cbind(mu,arl1s,arl1n,arl2s,arl2n),digits=2) ## which demonstrate the abilities of the variance-adjusted limits ## scheme more explicitely. ## Rhoads, Montgomery, Mastrangelo (1996) ## two-sided EWMA with fixed and variance adjusted limits (vacl), ## with fir and both features l <- .03 c <- 2.437 mu <- c(0,.5,1,1.5,2,3,4) sl <- sqrt(l*(2-l)) arlfix <- sapply(mu,l=l,c=c,sided="two",xewma.arl) arlvacl <- sapply(mu,l=l,c=c,sided="two",limits="vacl",xewma.arl) arlfir <- sapply(mu,l=l,c=c,hs=c/2,sided="two",limits="fir",xewma.arl) arlboth <- sapply(mu,l=l,c=c,hs=c/2*sl,sided="two",limits="both",xewma.arl) round(cbind(mu,arlfix,arlvacl,arlfir,arlboth),digits=1) ## original results are (in Table 1) ## 0.0 477.3* 427.9* 383.4* 286.2* ## 0.5 29.7 20.0 18.6 12.8 ## 1.0 12.5 6.5 7.4 3.6 ## 1.5 8.1 3.3 4.6 1.9 ## 2.0 6.0 2.2 3.4 1.4 ## 3.0 4.0 1.3 2.4 1.0 ## 4.0 3.1 1.1 1.9 1.0 ## * -- the in-control values differ sustainably from the true values! ## Steiner (1999) ## two-sided EWMA control charts with various modifications ## fixed vs. variance adjusted limits l <- .05 c <- 3 mu <- c(0,.25,.5,.75,1,1.5,2,2.5,3,3.5,4) arlfix <- sapply(mu,l=l,c=c,sided="two",xewma.arl) arlvacl <- sapply(mu,l=l,c=c,sided="two",limits="vacl",xewma.arl) round(cbind(mu,arlfix,arlvacl),digits=1) ## original results are (in Table 2) ## 0.00 1379.0 1353.0 ## 0.25 135.0 127.0 ## 0.50 37.4 32.5 ## 0.75 20.0 15.6 ## 1.00 13.5 9.0 ## 1.50 8.3 4.5 ## 2.00 6.0 2.8 ## 2.50 4.8 2.0 ## 3.00 4.0 1.6 ## 3.50 3.4 1.3 ## 4.00 3.0 1.1. ## fir, both, and Steiner's modification l <- .03 cfir <- 2.44 cboth <- 2.54 cstein <- 2.55 hsfir <- cfir/2 hsboth <- cboth/2*sqrt(l*(2-l)) mu <- c(0,.5,1,1.5,2,3,4) arlfir <- sapply(mu,l=l,c=cfir,hs=hsfir,sided="two",limits="fir",xewma.arl) arlboth <- sapply(mu,l=l,c=cboth,hs=hsboth,sided="two",limits="both",xewma.arl) arlstein <- sapply(mu,l=l,c=cstein,sided="two",limits="Steiner",xewma.arl) round(cbind(mu,arlfir,arlboth,arlstein),digits=1) ## original values are (in Table 5) ## 0.0 383.0 384.0 391.0 ## 0.5 18.6 14.9 13.8 ## 1.0 7.4 3.9 3.6 ## 1.5 4.6 2.0 1.8 ## 2.0 3.4 1.4 1.3 ## 3.0 2.4 1.1 1.0 ## 4.0 1.9 1.0 1.0. ## SAS/QC manual 1999 ## two-sided EWMA control charts with fixed limits l <- .25 c <- 3 mu <- 1 print(xewma.arl(l,c,mu,sided="two"),digits=11) # original value is 11.154267016.