spc (version 0.3)

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", r = 30)

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.

r

number of quadrature nodes, dimension of the resulting linear equation system is equal to r+1 (one-, two-sided) or 2r+1 (Crosier).

Value

• Returns a single value which resembles the ARL.

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.

See Also

xewma.arl for zero-state ARL computation of EWMA control charts and xcusum.ad for the steady-state ARL.

Examples

## 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.