imr.RuRl_alone {spc}

Compute control limits of MR charts for normal data

Description

Computation of control limits of standalone MR charts.

Usage

imr.RuRl_alone(L0, N=30, qm=30, M0=12, eps=1e-3)

imr.RuRl_alone_s3(L0, N=30, qm=30, M0=12)

imr.RuRl_alone_tail(L0, N=30, qm=30, M0=12)

imr.Ru_Rlgiven(Rl, L0, N=30, qm=30, M0=12)

imr.Rl_Rugiven(Ru, L0, N=30, qm=30, M0=12)

Arguments

L0

pre-defined in-control ARL, that is, determine Ru and Rl so that the mean number of observations until a false alarm is L0.

N

controls the dimension of the linear equation system and consequently the accuracy of the result. See details.

qm

number of quadrature nodes (and weights) to determine the definite collocation integrals.

M0

mimics Inf — by setting M0 to some large value (having a standard normal distribution in mind), the algorithm for IMR charts could be used as well for the standalone MR chart.

eps

resolution parameter, which controls the approximation of the ARL slope at the in-control level of the monitored standard deviation. It ensures the pattern that is called ARL unbiasedness. A small value is recommended.

Rl

lower control limit multiple for moving range chart.

Ru

upper control limit multiple for moving range chart.

Details

Crowder (1987a) provided some math to determine the ARL of the so-called individual moving range (IMR) chart, which consists of the mean X chart and the standard deviation MR chart. Making the alarm threshold, M0, huge (default value here is 12) for the X chart allows us to utilize Crowder's setup for standalone MR charts. For details about the IMR numerics see imr.arl. The three different versions of imr.RuRl_alone determine limits that form an ARL unbiased design, follow the restriction Rl = 1/Ru^3 and feature equal probability tails for the MR's half-normal distribution, respectively in the order given above). The other two functions are helper routines for imr.RuRl_alone. Note that the elegant approach given in Acosta-Mejia/Pignatiello (2000) is only an approximation, because the MR series is not Markovian.

Value

Returns control limit factors (alias multiples).

Author(s)

Sven Knoth

References

S. V. Crowder (1987a) Computation of ARL for Combined Individual Measurement and Moving Range Charts, Journal of Quality Technology 19(2), 98-102.

S. V. Crowder (1987b) A Program for the Computation of ARL for Combined Individual Measurement and Moving Range Charts, Journal of Quality Technology 19(2), 103-106.

D. Radson, L. C. Alwan (1995) Detecting Variance Reductions Using the Moving Range, Quality Engineering 8(1), 165-178.

C. A. Acosta-Mejia, J. J. Pignatiello (2000) Monitoring process dispersion without subgrouping, Journal of Quality Technology 32(2), 89-102.

See Also

later.

Examples

## Radson, Alwan (1995), Table 2 (Monte Carlo based), half-normal, known parameter case
## two-sided MR-alone chart, hence the ARL results has to be decreased by 1
## Here: a large M0=12 (default of the functions above) is deployed to mimic Inf
alpha <- 0.00915
Ru <- sqrt(2) * qnorm(1-alpha/4)
Rl <- sqrt(2) * qnorm(0.5+alpha/4)
M0 <- 12
## Not run: 
ARL0 <- imr.arl(M0, Ru, 0, 1, vsided="two", Rl=Rl)
RRR1995 <- imr.RuRl_alone_tail(ARL0)
RRRs <- imr.RuRl_alone_s3(ARL0)
RRR <- imr.RuRl_alone(ARL0)
results <- rbind(c(Rl, Ru), RRR1995, RRRs, RRR)
results
## End(Not run)