xcusum.arl: Compute ARLs of CUSUM control charts

Description

Computation of the (zero-state) Average Run Length (ARL) for different types of CUSUM control charts monitoring normal mean.

Usage

xcusum.arl(k, h, mu, hs = 0, sided = "one", method = "igl", q = 1, r = 30)

Value

Returns a vector of length q which resembles the ARL and the sequence of conditional expected delays for q=1 and q>1, respectively.

Arguments

k: reference value of the CUSUM control chart.
h: decision interval (alarm limit, threshold) of the CUSUM control chart.
mu: true mean.
hs: so-called headstart (give fast initial response).
sided: distinguish between one-, two-sided and Crosier's modified two-sided CUSUM scheme by choosing "one", "two", and "Crosier", respectively.
method: deploy the integral equation ("igl") or Markov chain approximation ("mc") method to calculate the ARL (currently only for two-sided CUSUM implemented).
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, \(E_q(L-q+1|L\ge q)\), will be determined. Note that mu0=0 is implicitely fixed.
r: number of quadrature nodes, dimension of the resulting linear equation system is equal to r+1 (one-, two-sided) or 2r+1 (Crosier).

Author

Sven Knoth

Details

xcusum.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.

References

A. L. Goel, S. M. Wu (1971), Determination of A.R.L. and a contour nomogram for CUSUM charts to control normal mean, Technometrics 13, 221-230.

D. Brook, D. A. Evans (1972), An approach to the probability distribution of cusum run length, Biometrika 59, 539-548.

J. M. Lucas, R. B. Crosier (1982), Fast initial response for cusum quality-control schemes: Give your cusum a headstart, Technometrics 24, 199-205.

L. C. Vance (1986), Average run lengths of cumulative sum control charts for controlling normal means, Journal of Quality Technology 18, 189-193.

K.-H. Waldmann (1986), Bounds for the distribution of the run length of one-sided and two-sided CUSUM quality control schemes, Technometrics 28, 61-67.

R. B. Crosier (1986), A new two-sided cumulative quality control scheme, Technometrics 28, 187-194.

Examples

Run this code

## Brook/Evans (1972), one-sided CUSUM
## Their results are based on the less accurate Markov chain approach.

k <- .5
h <- 3
round(c( xcusum.arl(k,h,0), xcusum.arl(k,h,1.5) ),digits=2)

## results in the original paper are L0 = 117.59, L1 = 3.75 (in Subsection 4.3).

## Lucas, Crosier (1982)
## (one- and) two-sided CUSUM with possible headstarts

k <- .5
h <- 4
mu <- c(0,.25,.5,.75,1,1.5,2,2.5,3,4,5)
arl1 <- sapply(mu,k=k,h=h,sided="two",xcusum.arl)
arl2 <- sapply(mu,k=k,h=h,hs=h/2,sided="two",xcusum.arl)
round(cbind(mu,arl1,arl2),digits=2)

## results in the original paper are (in Table 1)
## 0.00 168.   149.
## 0.25  74.2   62.7
## 0.50  26.6   20.1
## 0.75  13.3    8.97
## 1.00   8.38   5.29
## 1.50   4.75   2.86
## 2.00   3.34   2.01
## 2.50   2.62   1.59
## 3.00   2.19   1.32
## 4.00   1.71   1.07
## 5.00   1.31   1.01

## Vance (1986), one-sided CUSUM
## The first paper on using Nystroem method and Gauss-Legendre quadrature
## for solving the ARL integral equation (see as well Goel/Wu, 1971)

k <- 0
h <- 10
mu <- c(-.25,-.125,0,.125,.25,.5,.75,1)
round(cbind(mu,sapply(mu,k=k,h=h,xcusum.arl)),digits=2)

## results in the original paper are (in Table 1 incl. Goel/Wu (1971) results)
##  -0.25  2071.51
##  -0.125  400.28
##   0.0    124.66
##   0.125   59.30
##   0.25    36.71
##   0.50    20.37
##   0.75    14.06
##   1.00    10.75

## Waldmann (1986),
## one- and two-sided CUSUM

## one-sided case

k <- .5
h <- 3
mu <- c(-.5,0,.5)
round(sapply(mu,k=k,h=h,xcusum.arl),digits=2)

## results in the original paper are 1963, 117.4, and 17.35, resp.
## (in Tables 3, 1, and 5, resp.).

## two-sided case

k <- .6
h <- 3
round(xcusum.arl(k,h,-.2,sided="two"),digits=1)  # fits to Waldmann's setup

## result in the original paper is 65.4 (in Table 6).

## Crosier (1986), Crosier's modified two-sided CUSUM
## He introduced the modification and evaluated it by means of
## Markov chain approximation

k <- .5
h <- 3.73
mu <- c(0,.25,.5,.75,1,1.5,2,2.5,3,4,5)
round(cbind(mu,sapply(mu,k=k,h=h,sided="Crosier",xcusum.arl)),digits=2)

## results in the original paper are (in Table 3)
## 0.00 168.
## 0.25  70.7
## 0.50  25.1
## 0.75  12.5
## 1.00   7.92
## 1.50   4.49
## 2.00   3.17
## 2.50   2.49
## 3.00   2.09
## 4.00   1.60
## 5.00   1.22

## SAS/QC manual 1999
## one- and two-sided CUSUM schemes

## one-sided

k <- .25
h <- 8
mu <- 2.5
print(xcusum.arl(k,h,mu),digits=12)
print(xcusum.arl(k,h,mu,hs=.1),digits=12)

## original results are 4.1500836225 and 4.1061588131.

## two-sided

print(xcusum.arl(k,h,mu,sided="two"),digits=12)

## original result is 4.1500826715.

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