spc (version 0.3)

xewma.arl: 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,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 (give fast initial response).

sided

distinguish between one- and two-sided EWMA control chart by choosing "one" and "two", respectively.

limits

distinguish between different control limits behavior.

q

change point position. For $q=1$ and $\mu=\mu_0$ and $\mu=\mu_1$, the usual zero-state ARLs for the in-control and out-of-control case, respectively, are calculated. For $q>1$ and $\mu!=0$ conditional delays, that is, $E_q(L-q+1|L\geq)$, will be determine

r

number of quadrature nodes, dimension of the resulting linear equation system is equal to r+1 (one-sided) or r (two-sided).

Value

• Returns a single value which resembles the ARL.

Details

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. 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).

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, 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.

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.

## 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)