xewma.ad {spc}

Compute steady-state ARLs of EWMA control charts

Description

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

Usage

xewma.ad(l, c, mu1, mu0=0, zr=0, z0=0, sided="one", limits="fix",
steady.state.mode="conditional", r=40)

Arguments

l

smoothing parameter lambda of the EWMA control chart.

c

critical value (similar to alarm limit) of the EWMA control chart.

mu1

out-of-control mean.

mu0

in-control mean.

zr

reflection border for the one-sided chart.

z0

restarting value of the EWMA sequence in case of a false alarm in steady.state.mode="cyclical".

sided

distinguishes between one- and two-sided two-sided EWMA control chart by choosing "one" and "two", respectively.

limits

distinguishes between different control limits behavior.

steady.state.mode

distinguishes between two steady-state modes – conditional and cyclical.

r

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

Details

xewma.ad determines the steady-state Average Run Length (ARL) by numerically solving the related ARL integral equation by means of the Nystroem method based on Gauss-Legendre quadrature and using the power method for deriving the largest in magnitude eigenvalue and the related left eigenfunction.

Value

Returns a single value which resembles the steady-state ARL.

Author(s)

Sven Knoth

References

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

S. V. Crowder (1987), A simple method for studying run-length distributions of exponentially weighted moving average charts, Technometrics 29, 401-407.

J. M. Lucas and M. S. Saccucci (1990), Exponentially weighted moving average control schemes: Properties and enhancements, Technometrics 32, 1-12.

See Also

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

Examples

## comparison of zero-state (= worst case ) and steady-state performance
## for two-sided EWMA control charts

l <- .1
c <- xewma.crit(l,500,sided="two")
mu <- c(0,.5,1,1.5,2)
arl <- sapply(mu,l=l,c=c,sided="two",xewma.arl)
ad <- sapply(mu,l=l,c=c,sided="two",xewma.ad)
round(cbind(mu,arl,ad),digits=2)

## Lucas/Saccucci (1990)
## two-sided EWMA

## with fixed limits
l1 <- .5
l2 <- .03
c1 <- 3.071
c2 <- 2.437
mu <- c(0,.25,.5,.75,1,1.5,2,2.5,3,3.5,4,5)
ad1 <- sapply(mu,l=l1,c=c1,sided="two",xewma.ad)
ad2 <- sapply(mu,l=l2,c=c2,sided="two",xewma.ad)
round(cbind(mu,ad1,ad2),digits=2)

## original results are (in Table 3)
## 0.00 499.   480.  
## 0.25 254.    74.1
## 0.50  88.4   28.6
## 0.75  35.7   17.3
## 1.00  17.3   12.5
## 1.50   6.44   8.00
## 2.00   3.58   5.95
## 2.50   2.47   4.78
## 3.00   1.91   4.02
## 3.50   1.58   3.49
## 4.00   1.36   3.09
## 5.00   1.10   2.55