R: Univariate Shewhart-type control charts

shewhart {dfphase1}

R Documentation

Univariate Shewhart-type control charts

Description

shewhart computes, and, optionally, plots, Shewhart-type Phase I control charts for detecting changes in location and scale of univariate subgrouped data.

shewhart.normal.limits pre-computes the corresponding control limits when the in-control distribution is normal.

Usage

shewhart(x, subset, 
         stat = c("XbarS", "Xbar", "S", 
                  "Rank", "lRank", "sRank",
                  "Lepage", "Cucconi"),
         aggregation = c("mean", "median"), 
         plot = TRUE, 
         FAP = 0.05,
         seed = 11642257, 
         L = 1000, 
         limits = NA)

shewhart.normal.limits(n, m, 
                       stat = c("XbarS", "Xbar", "S", 
                                "Rank", "lRank", "sRank", 
                                "Lepage", "Cucconi"),
                       aggregation = c("mean", "median"), 
                       FAP = 0.05,
                       seed = 11642257, 
                       L = 100000)

Arguments

`x`	a nxm data numeric matrix (n observations gathered at m time points).
`subset`	an optional vector specifying a subset of subgroups/time points to be used
`stat`	character: the control statistic[s] to use; see Details.
`aggregation`	character: it specify how to aggregate the subgroup means and standard deviations. Used only when `stat` is `XbarS`, `Xbar` or `S`.
`plot`	logical; if `TRUE`, control statistic[s] is[are] displayed.
`FAP`	numeric (between 0 and 1): desired false alarm probability. Unused by `shewhart` when `limits` is not `NA`.
`seed`	positive integer; if not `NA`, the RNG's state is resetted using `seed`. The current `.Random.seed` will be preserved. Unused by `shewhart` when `limits` is not `NA`.
`L`	positive integer: number of random permutations used to compute the control limits. Unused by `shewhart` when `limits` is not `NA`.
`limits`	numeric: a precomputed vector of control limits. The vector should contain `(A,B_1,B_2)` when `stat=XbarS`, `(A)` when `stat=Xbar`, `(B_1,B_2)` when `stat=S`, `(C,D)` when `stat=Rank`, `(C)` when `stat=lRank`, `(D)` when `stat=sRank`, and `(E)` when `stat=Lepage` or `stat=Cucconi`. See Details for the definition of the critical values `A`, `B_1`, `B_2`, `C`, `D` and `E`.
`n`	integer: size of each subgroup (number of observations gathered at each time point).
`m`	integer: number of subgroups (time points).

Details

The implemented control charts are:

XbarS: combination of the Xbar and S control charts described in the following.
Xbar: chart based on plotting the subgroup means with control limits

\hat{\mu}\pm A\frac{\hat{\sigma}}{\sqrt{n}}

where \hat{\mu} (\hat{\sigma}) denotes the estimate of the in-control mean (standard deviation) computed as the mean or median of the subgroup means (standard deviations).
S: chart based on plotting the (unbiased) subgroup standard deviations with lower control limit B_1\hat{\sigma} and upper control limit B_2\hat{\sigma}.
Rank: combination of the lRank and sRank control charts described in the following.
lRank: control chart based on the standardized rank-sum control statistic suggested by Jones-Farmer et al. (2009) for detecting changes in the location parameter. Control limits are of the type \pm C.
sRank: chart based on the standardized rank-sum control statistic suggested by Jones-Farmer and Champ (2010) for detecting changes in the scale parameter. Control limits are of the type \pm D.
Lepage: chart based on the Lepage control statistic suggested by Li et al. (2019) for detecting changes in location and/or scale. There is only a upper control limit equal to E.
Cucconi: chart based on the Cucconi control statistic suggested by Li et al. (2020) for detecting changes in location and/or scale. There is only a upper control limit equal to E.

Value

shewhart returns an invisible list with elements

`Xbar`	subgroup means; this element is present only if `stat` is `XbarS` or `Xbar`.
`S`	subgroup standard deviation; this element is present only if `stat` is `XbarS` or `S`.
`lRank`	rank-based control statistics for detecting changes in location; this element is present only if `stat` is `Rank` or `lRank`.
`sRank`	rank-based control-statistics for detecting changes in scale; this element is present only if `stat` is `Rank` or `sRank`.
`Lepage`, `W2`, `AB2`	Lepage, squared Wilcoxon and squared Ansari-Bradley statistics; these elements are present only if `stat` is `Lepage`.
`Cucconi`, `lCucconi`, `sCucconi`	Cucconi control statistic and its location and scale components; these elements are present only if `stat` is `Cucconi`.
`limits`	control limits.
`center`, `scale`	estimates `\hat{\mu}` and `\hat{\sigma}` of the in-control mean and standard deviation; these elements are present only if `stat` is `XbarS`, `Xbar` and `S`.
`stat`, `L`, `aggregation`, `FAP`, `seed`	input arguments.

shewhart.normal.limits returns a numeric vector containing the limits.

Note

If argument limits is NA, shewhart computes the control limits by permutation. The resulting control chart are distribution-free.
Pre-computed limits, such as those computed using shewhart.normal.limits, are not recommended when stat is XbarS, Xbar or S. Indeed, the resulting control chart will not be distribution-free.
When stat is Rank, lRank, sRank, Lepage or Cucconi the control limits computed by shewhart.normal.limits are distribution-free in the class of all univariate continuous distributions. So, if user plan to apply rank-based control charts on a repeated number of samples of the same size, pre-computing the control limits using mshewhart.normal.limits can reduce the overall computing time.

Author(s)

Giovanna Capizzi and Guido Masarotto.

References

L. A. Jones-Farmer, V. Jordan, C. W. Champs (2009) “Distribution-free Phase I control charts for subgroup location”, Journal of Quality Technology, 41, pp. 304–316, doi:10.1080/00224065.2009.11917784.

L. A. Jones-Farmer, C. W. Champ (2010) “A distribution-free Phase I control chart for subgroup scale”. Journal of Quality Technology, 42, pp. 373–387, doi:10.1080/00224065.2010.11917834

C. Li, A. Mukherjee, Q. Su (2019) “A distribution-free Phase I monitoring scheme for subgroup location and scale based on the multi-sample Lepage statistic”, Computers & Industrial Engineering, 129, pp. 259–273, doi:10.1016/j.cie.2019.01.013

C. Li, A. Mukherjee, M. Marozzi (2020) “A new distribution-free Phase-I procedure for bi-aspect monitoring based on the multi-sample Cucconi statistic”, Computers & Industrial Engineering, 149, doi:10.1016/j.cie.2020.106760

D. C. Montgomery (2009) Introduction to Statistical Quality Control, 6th edn. Wiley.

P. Qiu (2013) Introduction to Statistical Process Control. Chapman & Hall/CRC Press.

Examples

# A simulated example
set.seed(12345)
y <- matrix(rt(100,3),5)
y[,20] <- y[,20]+3
shewhart(y)
shewhart(y, stat="Rank")
shewhart(y, stat="Lepage")
shewhart(y, stat="Cucconi")
# Reproduction of the control chart shown
# by Jones-Farmer et. al. (2009)
data(colonscopy)
u <- shewhart.normal.limits(NROW(colonscopy),NCOL(colonscopy), 
                            stat="lRank", FAP=0.1, L=10000)
# In Jones-Farmer et al. (2009) is estimated as 2.748
u
shewhart(colonscopy,stat="lRank",limits=u)
# Examples of control limits for comparisons
# with Li et al. (2019) and (2020) but
# using a limited number of Monte Carlo
# replications
# Lepage: in Li et al. (2019) is estimated as 11.539
shewhart.normal.limits(5, 25, stat="Lepage", L=10000)
# Cucconi: in Li et al. (2020) is estimated as 0.266
shewhart.normal.limits(5, 25, stat="Cucconi", L=10000)

[Package dfphase1 version 1.2.0 Index]