xewma.arl: Compute ARLs of EWMA control charts
In spc: Statistical Process Control -- Calculation of ARL and Other Control Chart Performance Measures
xewma.arl R Documentation
Compute ARLs of EWMA control charts
Description
Computation of the (zero-state) Average Run Length (ARL)
for different types of EWMA control charts monitoring normal mean.
Usage
xewma.arl(l,c,mu,zr=0,hs=0,sided="one",limits="fix",q=1,
steady.state.mode="conditional",r=40)
Arguments
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 (enables fast initial response).
sided
distinguishes between one- and two-sided EWMA control chart
by choosing "one" and "two", respectively.
limits
distinguishes between different control limits behavior.
q
change point position. For q=1 and
μ=μ_0 and μ=μ_1, the usual
zero-state ARLs for the in-control and out-of-control case, respectively,
are calculated. For q>1 and μ!=0 conditional delays, that is,
E_q(L-q+1|L≥ q), will be determined. Note that mu0=0 is implicitely fixed.
steady.state.mode
distinguishes between two steady-state modes – conditional and cyclical
(needed for q>1).
r
number of quadrature nodes, dimension of the resulting linear
equation system is equal to r+1 (one-sided) or r
(two-sided).
Details
In case of the EWMA chart with fixed control limits,
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.
If limits is not "fix", then the method presented in Knoth (2003) is utilized.
Note that for one-sided EWMA charts (sided="one"), only
"vacl" and "stat" are deployed, while for two-sided ones
(sided="two") also "fir", "both"
(combination of "fir" and "vacl"), and "Steiner" are implemented.
For details see Knoth (2004).
Value
Except for the fixed limits EWMA charts it returns a single value which resembles the ARL.
For fixed limits charts, it returns a vector of length q which resembles the ARL and the
sequence of conditional expected delays for
q=1 and q>1, respectively.
Author(s)
Sven Knoth
References
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 and M. S. Saccucci (1990),
Exponentially weighted moving average control schemes: Properties
and enhancements, Technometrics 32, 1-12.
S. Chandrasekaran, J. R. English and 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 and 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.
See Also
xcusum.arl for zero-state ARL computation of CUSUM control charts
and xewma.ad for the steady-state ARL.
Examples
## 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
## Not run:
## 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.
## Some recent examples for one-sided EWMA charts
## with varying limits and in the so-called stationary mode
# 1. varying limits = "vacl"
lambda <- .1
L0 <- 500
## Monte Carlo results (10^9 replicates)
# mu ARL s.e.
# 0 500.00 0.0160
# 0.5 21.637 0.0006
# 1 6.7596 0.0001
# 1.5 3.5398 0.0001
# 2 2.3038 0.0000
# 2.5 1.7004 0.0000
# 3 1.3675 0.0000
zr <- -6
r <- 50
c <- xewma.crit(lambda, L0, zr=zr, limits="vacl", r=r)
Mxewma.arl <- Vectorize(xewma.arl, "mu")
mus <- (0:6)/2
arls <- round(Mxewma.arl(lambda, c, mus, zr=zr, limits="vacl", r=r), digits=4)
data.frame(mus, arls)
# 2. stationary mode, i. e. limits = "stat"
## Monte Carlo results (10^9 replicates)
# mu ARL s.e.
# 0 500.00 0.0159
# 0.5 22.313 0.0006
# 1 7.2920 0.0001
# 1.5 3.9064 0.0001
# 2 2.5131 0.0000
# 2.5 1.7983 0.0000
# 3 1.4029 0.0000
c <- xewma.crit(lambda, L0, zr=zr, limits="stat", r=r)
arls <- round(Mxewma.arl(lambda, c, mus, zr=zr, limits="stat", r=r), digits=4)
data.frame(mus, arls)
## End(Not run)
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| xewma.arl | R Documentation |
Compute ARLs of EWMA control charts
Description
Computation of the (zero-state) Average Run Length (ARL) for different types of EWMA control charts monitoring normal mean.
Usage
xewma.arl(l,c,mu,zr=0,hs=0,sided="one",limits="fix",q=1, steady.state.mode="conditional",r=40)
Arguments
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 (enables fast initial response). |
sided |
distinguishes between one- and two-sided EWMA control chart
by choosing |
limits |
distinguishes between different control limits behavior. |
q |
change point position. For q=1 and μ=μ_0 and μ=μ_1, the usual zero-state ARLs for the in-control and out-of-control case, respectively, are calculated. For q>1 and μ!=0 conditional delays, that is, E_q(L-q+1|L≥ q), will be determined. Note that mu0=0 is implicitely fixed. |
steady.state.mode |
distinguishes between two steady-state modes – conditional and cyclical
(needed for |
r |
number of quadrature nodes, dimension of the resulting linear
equation system is equal to |
Details
In case of the EWMA chart with fixed control limits,
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.
If limits is not "fix", then the method presented in Knoth (2003) is utilized.
Note that for one-sided EWMA charts (sided="one"), only
"vacl" and "stat" are deployed, while for two-sided ones
(sided="two") also "fir", "both"
(combination of "fir" and "vacl"), and "Steiner" are implemented.
For details see Knoth (2004).
Value
Except for the fixed limits EWMA charts it returns a single value which resembles the ARL.
For fixed limits charts, it returns a vector of length q which resembles the ARL and the
sequence of conditional expected delays for
q=1 and q>1, respectively.
Author(s)
Sven Knoth
References
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 and M. S. Saccucci (1990), Exponentially weighted moving average control schemes: Properties and enhancements, Technometrics 32, 1-12.
S. Chandrasekaran, J. R. English and 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 and 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.
See Also
xcusum.arl for zero-state ARL computation of CUSUM control charts
and xewma.ad for the steady-state ARL.
Examples
## 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 ## Not run: ## 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. ## Some recent examples for one-sided EWMA charts ## with varying limits and in the so-called stationary mode # 1. varying limits = "vacl" lambda <- .1 L0 <- 500 ## Monte Carlo results (10^9 replicates) # mu ARL s.e. # 0 500.00 0.0160 # 0.5 21.637 0.0006 # 1 6.7596 0.0001 # 1.5 3.5398 0.0001 # 2 2.3038 0.0000 # 2.5 1.7004 0.0000 # 3 1.3675 0.0000 zr <- -6 r <- 50 c <- xewma.crit(lambda, L0, zr=zr, limits="vacl", r=r) Mxewma.arl <- Vectorize(xewma.arl, "mu") mus <- (0:6)/2 arls <- round(Mxewma.arl(lambda, c, mus, zr=zr, limits="vacl", r=r), digits=4) data.frame(mus, arls) # 2. stationary mode, i. e. limits = "stat" ## Monte Carlo results (10^9 replicates) # mu ARL s.e. # 0 500.00 0.0159 # 0.5 22.313 0.0006 # 1 7.2920 0.0001 # 1.5 3.9064 0.0001 # 2 2.5131 0.0000 # 2.5 1.7983 0.0000 # 3 1.4029 0.0000 c <- xewma.crit(lambda, L0, zr=zr, limits="stat", r=r) arls <- round(Mxewma.arl(lambda, c, mus, zr=zr, limits="stat", r=r), digits=4) data.frame(mus, arls) ## End(Not run)
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