| x.res.ewma.arl {spc} | R Documentation |
Compute ARLs of EWMA residual control charts
Description
Computation of the (zero-state) Average Run Length (ARL) for EWMA residual control charts monitoring normal mean, variance, or mean and variance simultaneously. Additionally, the probability of misleading signals (PMS) is calculated.
Usage
x.res.ewma.arl(l, c, mu, alpha=0, n=5, hs=0, r=40)
s.res.ewma.arl(l, cu, sigma, mu=0, alpha=0, n=5, hs=1, r=40, qm=30)
xs.res.ewma.arl(lx, cx, ls, csu, mu, sigma, alpha=0,
n=5, hsx=0, rx=40, hss=1, rs=40, qm=30)
xs.res.ewma.pms(lx, cx, ls, csu, mu, sigma, type="3",
alpha=0, n=5, hsx=0, rx=40, hss=1, rs=40, qm=30)
Arguments
l, lx, ls |
smoothing parameter(s) lambda of the EWMA control chart. |
c, cu, cx, csu |
critical value (similar to alarm limit) of the EWMA control charts. |
mu |
true mean. |
sigma |
true standard deviation. |
alpha |
the AR(1) coefficient – first order autocorrelation of the original data. |
n |
batch size. |
hs, hsx, hss |
so-called headstart (enables fast initial response). |
r, rx, rs |
number of quadrature nodes or size of collocation base,
dimension of the resulting linear
equation system is equal to |
qm |
number of nodes for collocation quadratures. |
type |
PMS type, for |
Details
The above list of functions provides the application of
algorithms developed for iid data to
the residual case. To be more precise, the underlying model is a sequence of normally
distributed batches with size n with autocorrelation within
the batch and independence between the batches
(see also the references below). It is restricted to the
classical EWMA chart types, that
is two-sided for the mean, upper charts for the variance,
and all equipped with fixed limits.
The autocorrelation is modeled by an AR(1) process with parameter
alpha. Additionally,
with xs.res.ewma.pms the probability of misleading signals
(PMS) of type is
calculated. This is offered exclusively in this small
collection so that for iid data
this function has to be used too (with alpha=0).
Value
Return single values which resemble the ARL and the PMS, respectively.
Author(s)
Sven Knoth
References
S. Knoth, M. C. Morais, A. Pacheco, W. Schmid (2009), Misleading Signals in Simultaneous Residual Schemes for the Mean and Variance of a Stationary Process, Commun. Stat., Theory Methods 38, 2923-2943.
S. Knoth, W. Schmid, A. Schoene (2001), Simultaneous Shewhart-Type Charts for the Mean and the Variance of a Time Series, Frontiers of Statistical Quality Control 6, A. Lenz, H.-J. & Wilrich, P.-T. (Eds.), 6, 61-79.
S. Knoth, W. Schmid (2002) Monitoring the mean and the variance of a stationary process, Statistica Neerlandica 56, 77-100.
See Also
xewma.arl, sewma.arl, and xsewma.arl as more
elaborated functions in the iid case.
Examples
## Not run:
## S. Knoth, W. Schmid (2002)
cat("\nFragments of Table 2 (n=5, lambda.1=lambda.2)\n")
lambdas <- c(.5, .25, .1, .05)
L0 <- 500
n <- 5
crit <- NULL
for ( lambda in lambdas ) {
cs <- xsewma.crit(lambda, lambda, L0, n-1)
x.e <- round(cs[1], digits=4)
names(x.e) <- NULL
s.e <- round((cs[3]-1) * sqrt((2-lambda)/lambda)*sqrt((n-1)/2), digits=4)
names(s.e) <- NULL
crit <- rbind(crit, data.frame(lambda, x.e, s.e))
}
## orinal values are (Markov chain approximation with 50 states)
# lambda x.e s.e
# 0.50 3.2765 4.6439
# 0.25 3.2168 4.0149
# 0.10 3.0578 3.3376
# 0.05 2.8817 2.9103
print(crit)
cat("\nFragments of Table 4 (n=5, lambda.1=lambda.2=0.1)\n\n")
lambda <- .1
# the algorithm used in Knoth/Schmid is less accurate -- proceed with their values
cx <- x.e <- 3.0578
s.e <- 3.3376
csu <- 1 + s.e * sqrt(lambda/(2-lambda))*sqrt(2/(n-1))
alpha <- .3
a.values <- c((0:6)/4, 2)
d.values <- c(1 + (0:5)/10, 1.75 , 2)
arls <- NULL
for ( delta in d.values ) {
row <- NULL
for ( mu in a.values ) {
arl <- round(xs.res.ewma.arl(lambda, cx, lambda, csu, mu*sqrt(n), delta, alpha=alpha, n=n),
digits=2)
names(arl) <- NULL
row <- c(row, arl)
}
arls <- rbind(arls, data.frame(t(row)))
}
names(arls) <- a.values
rownames(arls) <- d.values
## orinal values are (now Monte-Carlo with 10^6 replicates)
# 0 0.25 0.5 0.75 1 1.25 1.5 2
#1 502.44 49.50 14.21 7.93 5.53 4.28 3.53 2.65
#1.1 73.19 32.91 13.33 7.82 5.52 4.29 3.54 2.66
#1.2 24.42 18.88 11.37 7.44 5.42 4.27 3.54 2.67
#1.3 13.11 11.83 9.09 6.74 5.18 4.17 3.50 2.66
#1.4 8.74 8.31 7.19 5.89 4.81 4.00 3.41 2.64
#1.5 6.50 6.31 5.80 5.08 4.37 3.76 3.28 2.59
#1.75 3.94 3.90 3.78 3.59 3.35 3.09 2.83 2.40
#2 2.85 2.84 2.80 2.73 2.63 2.51 2.39 2.14
print(arls)
## S. Knoth, M. C. Morais, A. Pacheco, W. Schmid (2009)
cat("\nFragments of Table 5 (n=5, lambda=0.1)\n\n")
d.values <- c(1.02, 1 + (1:5)/10, 1.75 , 2)
arl.x <- arl.s <- arl.xs <- PMS.3 <- NULL
for ( delta in d.values ) {
arl.x <- c(arl.x, round(x.res.ewma.arl(lambda, cx/delta, 0, n=n),
digits=3))
arl.s <- c(arl.s, round(s.res.ewma.arl(lambda, csu, delta, n=n),
digits=3))
arl.xs <- c(arl.xs, round(xs.res.ewma.arl(lambda, cx, lambda, csu, 0, delta, n=n),
digits=3))
PMS.3 <- c(PMS.3, round(xs.res.ewma.pms(lambda, cx, lambda, csu, 0, delta, n=n),
digits=6))
}
## orinal values are (Markov chain approximation)
# delta arl.x arl.s arl.xs PMS.3
# 1.02 833.086 518.935 323.324 0.381118
# 1.10 454.101 84.208 73.029 0.145005
# 1.20 250.665 25.871 24.432 0.071024
# 1.30 157.343 13.567 13.125 0.047193
# 1.40 108.112 8.941 8.734 0.035945
# 1.50 79.308 6.614 6.493 0.029499
# 1.75 44.128 3.995 3.942 0.021579
# 2.00 28.974 2.887 2.853 0.018220
print(cbind(delta=d.values, arl.x, arl.s, arl.xs, PMS.3))
cat("\nFragments of Table 6 (n=5, lambda=0.1)\n\n")
alphas <- c(-0.9, -0.5, -0.3, 0, 0.3, 0.5, 0.9)
deltas <- c(0.05, 0.25, 0.5, 0.75, 1, 1.25, 1.5, 2)
PMS.4 <- NULL
for ( ir in 1:length(deltas) ) {
mu <- deltas[ir]*sqrt(n)
pms <- NULL
for ( alpha in alphas ) {
pms <- c(pms, round(xs.res.ewma.pms(lambda, cx, lambda, csu, mu, 1, type="4", alpha=alpha, n=n),
digits=6))
}
PMS.4 <- rbind(PMS.4, data.frame(delta=deltas[ir], t(pms)))
}
names(PMS.4) <- c("delta", alphas)
rownames(PMS.4) <- NULL
## orinal values are (Markov chain approximation)
# delta -0.9 -0.5 -0.3 0 0.3 0.5 0.9
# 0.05 0.055789 0.224521 0.279842 0.342805 0.391299 0.418915 0.471386
# 0.25 0.003566 0.009522 0.014580 0.025786 0.044892 0.066584 0.192023
# 0.50 0.002994 0.001816 0.002596 0.004774 0.009259 0.015303 0.072945
# 0.75 0.006967 0.000703 0.000837 0.001529 0.003400 0.006424 0.046602
# 1.00 0.005098 0.000402 0.000370 0.000625 0.001589 0.003490 0.039978
# 1.25 0.000084 0.000266 0.000202 0.000300 0.000867 0.002220 0.039773
# 1.50 0.000000 0.000256 0.000120 0.000163 0.000531 0.001584 0.042734
# 2.00 0.000000 0.000311 0.000091 0.000056 0.000259 0.001029 0.054543
print(PMS.4)
## End(Not run)