p.ewma.arl: Compute ARLs of binomial EWMA p control charts

In spc: Statistical Process Control -- Calculation of ARL and Other Control Chart Performance Measures

Compute ARLs of binomial EWMA p control charts

Description

Computation of the (zero-state) Average Run Length (ARL) at given rate p.

Usage

p.ewma.arl(lambda, ucl, n, p, z0, sided="upper", lcl=NULL, d.res=1,
r.mode="ieee.round", i.mode="integer")

Arguments

lambda	smoothing parameter of the EWMA p control chart.
ucl	upper control limit of the EWMA p control chart.
n	subgroup size.
p	(failure/success) rate.
z0	so-called headstart (give fast initial response).
sided	distinguishes between one- and two-sided EWMA control chart by choosing "upper", "lower", and "two", respectively.
lcl	lower control limit of the EWMA p control chart; needed for two-sided design.
d.res	resolution (see details).
r.mode	round mode – allowed modes are "gan.floor", "floor", "ceil", "ieee.round", "round", "mix".
i.mode	type of interval center – "integer" or "half" integer.

Details

The monitored data follow a binomial distribution with size n and failure/success probability p. The ARL values of the resulting EWMA control chart are determined by Markov chain approximation. Here, the original EWMA values are approximated by multiples of one over d.res. Different ways of rounding (see r.mode) to the next multiple are implemented. Besides Gan's paper nothing is published about the numerical subtleties.

Value

Return single value which resemble the ARL.

Author(s)

Sven Knoth

References

F. F. Gan (1990), Monitoring observations generated from a binomial distribution using modified exponentially weighted moving average control chart, J. Stat. Comput. Simulation 37, 45-60.

S. Knoth and S. Steinmetz (2013), EWMA p charts under sampling by variables, International Journal of Production Research 51, 3795-3807.

See Also

later.

Examples

## Gan (1990)

# Table 1

n <- 150
p0 <- .1
z0 <- n*p0

lambda <- c(1, .51, .165)
hu <- c(27, 22, 18)

p.value <- .1 + (0:20)/200

p.EWMA.arl <- Vectorize(p.ewma.arl, "p")

arl1.value <- round(p.EWMA.arl(lambda[1], hu[1], n, p.value, z0, r.mode="round"), digits=2)
arl2.value <- round(p.EWMA.arl(lambda[2], hu[2], n, p.value, z0, r.mode="round"), digits=2)
arl3.value <- round(p.EWMA.arl(lambda[3], hu[3], n, p.value, z0, r.mode="round"), digits=2)

arls <- matrix(c(arl1.value, arl2.value, arl3.value), ncol=length(lambda))
rownames(arls) <- p.value
colnames(arls) <- paste("lambda =", lambda)
arls

## Knoth/Steinmetz (2013)

n <- 5
p0 <- 0.02
z0 <- n*p0
lambda <- 0.3
ucl <- 0.649169922 ## in-control ARL 370.4 (determined with d.res = 2^14 = 16384)

res.list <- 2^(1:11)
arl.list <- NULL
for ( res in res.list ) {
  arl <- p.ewma.arl(lambda, ucl, n, p0, z0, d.res=res)
  arl.list <- c(arl.list, arl)
}
cbind(res.list, arl.list)

spc documentation built on Oct. 24, 2022, 5:07 p.m.