xDcusum.arl {spc} | R Documentation |
Compute ARLs of CUSUM control charts under drift
Description
Computation of the (zero-state and other) Average Run Length (ARL) under drift for one-sided CUSUM control charts monitoring normal mean.
Usage
xDcusum.arl(k, h, delta, hs = 0, sided = "one",
mode = "Gan", m = NULL, q = 1, r = 30, with0 = FALSE)
Arguments
k |
reference value of the CUSUM control chart. |
h |
decision interval (alarm limit, threshold) of the CUSUM control chart. |
delta |
true drift parameter. |
hs |
so-called headstart (enables fast initial response). |
sided |
distinguishes between one- and two-sided CUSUM control chart
by choosing |
mode |
decide whether Gan's or Knoth's approach is used. Use
|
m |
parameter used if |
q |
change point position. For |
r |
number of quadrature nodes, dimension of the resulting linear
equation system is equal to |
with0 |
defines whether the first observation used for the RL
calculation follows already 1*delta or still 0*delta.
With |
Details
Based on Gan (1991) or Knoth (2003), the ARL is calculated for one-sided CUSUM control charts under drift. In case of Gan's framework, the usual ARL function with mu=m*delta is determined and recursively via m-1, m-2, ... 1 (or 0) the drift ARL determined. The framework of Knoth allows to calculate ARLs for varying parameters, such as control limits and distributional parameters. For details see the cited papers. Note that two-sided CUSUM charts under drift are difficult to treat.
Value
Returns a single value which resembles the ARL.
Author(s)
Sven Knoth
References
F. F. Gan (1992), CUSUM control charts under linear drift, Statistician 41, 71-84.
F. F. Gan (1996), Average Run Lengths for Cumulative Sum control chart under linear trend, Applied Statistics 45, 505-512.
S. Knoth (2003), EWMA schemes with non-homogeneous transition kernels, Sequential Analysis 22, 241-255.
S. Knoth (2012), More on Control Charting under Drift, in: Frontiers in Statistical Quality Control 10, H.-J. Lenz, W. Schmid and P.-T. Wilrich (Eds.), Physica Verlag, Heidelberg, Germany, 53-68.
C. Zou, Y. Liu and Z. Wang (2009), Comparisons of control schemes for monitoring the means of processes subject to drifts, Metrika 70, 141-163.
See Also
xcusum.arl
and xcusum.ad
for zero-state and
steady-state ARL computation of CUSUM control charts
for the classical step change model.
Examples
## Gan (1992)
## Table 1
## original values are
# deltas arl
# 0.0001 475
# 0.0005 261
# 0.0010 187
# 0.0020 129
# 0.0050 76.3
# 0.0100 52.0
# 0.0200 35.2
# 0.0500 21.4
# 0.1000 15.0
# 0.5000 6.95
# 1.0000 5.16
# 3.0000 3.30
## Not run: k <- .25
h <- 8
r <- 50
DxDcusum.arl <- Vectorize(xDcusum.arl, "delta")
deltas <- c(0.0001, 0.0005, 0.001, 0.002, 0.005, 0.01, 0.02, 0.05, 0.1, 0.5, 1, 3)
arl.like.Gan <-
round(DxDcusum.arl(k, h, deltas, r=r, with0=TRUE), digits=2)
arl.like.Knoth <-
round(DxDcusum.arl(k, h, deltas, r=r, mode="Knoth", with0=TRUE), digits=2)
data.frame(deltas, arl.like.Gan, arl.like.Knoth)
## End(Not run)
## Zou et al. (2009)
## Table 1
## original values are
# delta arl1 arl2 arl3
# 0 ~ 1730
# 0.0005 345 412 470
# 0.001 231 275 317
# 0.005 86.6 98.6 112
# 0.01 56.9 61.8 69.3
# 0.05 22.6 21.6 22.7
# 0.1 15.4 14.7 14.2
# 0.5 6.60 5.54 5.17
# 1.0 4.63 3.80 3.45
# 2.0 3.17 2.67 2.32
# 3.0 2.79 2.04 1.96
# 4.0 2.10 1.98 1.74
## Not run:
k1 <- 0.25
k2 <- 0.5
k3 <- 0.75
h1 <- 9.660
h2 <- 5.620
h3 <- 3.904
deltas <- c(0.0005, 0.001, 0.005, 0.01, 0.05, 0.1, 0.5, 1:4)
arl1 <- c(round(xcusum.arl(k1, h1, 0, r=r), digits=2),
round(DxDcusum.arl(k1, h1, deltas, r=r), digits=2))
arl2 <- c(round(xcusum.arl(k2, h2, 0), digits=2),
round(DxDcusum.arl(k2, h2, deltas, r=r), digits=2))
arl3 <- c(round(xcusum.arl(k3, h3, 0, r=r), digits=2),
round(DxDcusum.arl(k3, h3, deltas, r=r), digits=2))
data.frame(delta=c(0, deltas), arl1, arl2, arl3)
## End(Not run)