spc (version 0.7.1)

xDcusum.arl: 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)

Value

Returns a single value which resembles the ARL.

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 "one" and "two", respectively. Currentlly, the two-sided scheme is not implemented.

mode

decide whether Gan's or Knoth's approach is used. Use "Gan" and "Knoth", respectively.

m

parameter used if mode="Gan". m is design parameter of Gan's approach. If m=NULL, then m will increased until the resulting ARL does not change anymore.

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 determined. Note that mu0=0 is implicitely fixed. Deploy large q to mimic steady-state. It works only for mode="Knoth".

r

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

with0

defines whether the first observation used for the RL calculation follows already 1*delta or still 0*delta. With q additional flexibility is given.

Author

Sven Knoth

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.

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
if (FALSE) 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)

## 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
if (FALSE) {
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)}