pois.cusum.crit {spc}

Compute alarm thresholds and randomization constants of Poisson CUSUM control charts

Description

Computation of the CUSUM upper limit and, if needed, of the randomization probability, given mean mu0.

Usage

pois.cusum.crit(mu0, km, A, m, i0=0, sided="upper", rando=FALSE)

Arguments

mu0

actual in-control mean.

km

enumerator of rational approximation of reference value k.

A

target in-control ARL (average run length).

m

denominator of rational approximation of reference value.

i0

head start value as integer multiple of 1/m; should be an element of 0:100 (a more reasonable upper limit will be established soon). It is planned, to set i0 as a fraction of the final threshold.

sided

distinguishes between different one- and two-sided CUSUM control chart by choosing "upper", "lower" and "two", respectively.

rando

Switch for activating randomization in order to allow continuous ARL control.

Details

The monitored data follow a Poisson distribution with mu (here the in-control level mu0). The ARL values of the resulting EWMA control chart are determined via Markov chain calculations. With some grid search, we obtain the smallest value for the integer threshold component hm so that the resulting ARL is not smaller than A. If equality is needed, then activating rando=TRUE yields the corresponding randomization probability gamma. More details will follow in a paper that will be submitted in 2020.

Value

Returns two single values, integer threshold hm resulting in the final alarm threshold h=hm/m, and the randomization probability.

Author(s)

Sven Knoth

References

J. M. Lucas (1985) Counted data CUSUM's, Technometrics 27(2), 129-144.

C. H. White and J. B. Keats (1996) ARLs and Higher-Order Run-Length Moments for the Poisson CUSUM, Journal of Quality Technology 28(3), 363-369.

C. H. White, J. B. Keats and J. Stanley (1997) Poisson CUSUM versus c chart for defect data, Quality Engineering 9(4), 673-679.

G. Rossi and L. Lampugnani and M. Marchi (1999), An approximate CUSUM procedure for surveillance of health events, Statistics in Medicine 18(16), 2111-2122.

S. W. Han, K.-L. Tsui, B. Ariyajunya, and S. B. Kim (2010), A comparison of CUSUM, EWMA, and temporal scan statistics for detection of increases in poisson rates, Quality and Reliability Engineering International 26(3), 279-289.

M. B. Perry and J. J. Pignatiello Jr. (2011) Estimating the time of step change with Poisson CUSUM and EWMA control charts, International Journal of Production Research 49(10), 2857-2871.

See Also

later.

Examples

## Lucas 1985
mu0 <- 0.25
km <- 1
A <- 430
m  <- 4
#cv <- pois.cusum.crit(mu0, km, A, m)
cv <- c(40, 0)
# Lucas reported h = 10 alias hm = 40 (in Table 2, first block, row 10.0 .25 .0 ..., column 1.0
# Recall that Lucas and other trigger an alarm, if the CUSUM statistic is greater than
# or equal to the alarm threshold h
print(cv)