shewhart: 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)

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.

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).

Author

tools:::Rd_package_author("dfphase1").

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 $E$.

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, tools:::Rd_expr_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, tools:::Rd_expr_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, tools:::Rd_expr_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, tools:::Rd_expr_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

Run this code

# 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)

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