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mshewhart computes, and, optionally, plots,
several Shewhart-type Phase I control charts for detecting
location and scale changes in multivariate subgrouped data.
mshewhart.normal.limits pre-computes
the corresponding control limits when the in-control distribution is
multivariate normal.
mshewhart(x, subset, stat = c("T2Var", "T2", "Var", "Depth Ranks"), score = c("Identity",
"Signed Ranks", "Spatial Signs", "Spatial Ranks", "Marginal Ranks"),
loc.scatter = c("Classic", "Robust"), plot = TRUE, FAP = 0.05,
seed = 11642257, L = 1000, limits = NA)mshewhart.normal.limits(p, n, m, stat = c("T2Var", "T2", "Var", "Depth Ranks"),
score = c("Identity", "Signed Ranks", "Spatial Signs", "Spatial Ranks",
"Marginal Ranks"), loc.scatter = c("Classic", "Robust"),
FAP = 0.05, seed = 11642257, L = 100000)
mshewhart returns an invisible list with elements:
T2 stat is T2Var or T2.
Var stat is T2Var or Var.
DepthRanks control statistic based on the rank of
the Mahalanobis depths; this element is present only if
stat is Depth Ranks.
center, scatter estimates of the multivariate location and scatter used to standardized the observations.
limits control limits.
stat, score, loc.scatter, FAP,
L, seed input arguments.
mshewhart.normal.limits returns a numeric vector
containing the control limits.
a pxnxm data numeric array (n observations gathered at m time points on p variables).
integer: number of monitored variables.
integer: size of each subgroup (number of observations gathered at each time point).
integer: number of subgroups (time points).
an optional vector specifying a subset of subgroups/time points to be used
character: control statistic[s] to use; see Details.
character: transformation to use; unused when
stat=Depth Ranks; see Details.
character: estimates of the multivariate location and scatter
to use when no preliminary rank transformation is applied.
Unused when stat is equal to Depth Ranks or score is
Marginal Ranks. See Details.
logical; if TRUE, control statistic[s] is[are] displayed.
numeric (between 0 and 1): desired false alarm probability.
positive integer; if not NA, the RNG's state is resetted
using seed. The current .Random.seed will be
preserved. Unused by mshewhart when limits is not NA.
positive integer: number of Monte Carlo replications used to
compute the control limits. Unused by mshewhart when limits is not NA.
numeric: pre-computed vector of control limits.
This vector should contain stat=T2Var, stat=T2,
stat=Var and
stat=Depth Ranks.
See Details for the definition of the critical values
tools:::Rd_package_author("dfphase1").
The implemented control statistics are
T2Var: combination of the T2 and
Var statistics described below.
T2: Hotelling's
Var: normal likelihood ratio control statistics
for detecting changes in the multivariate dispersion
(see Montgomery, 2009, equation 11.34), with control limit
equal to
Depth Ranks:
control statistics based on the rank of the Mahalanobis
depths, proposed by Bell et. al.. As suggested Bell et
al., the Mahalanobis depths are computed using the BACON estimates
of the multivariate mean vector and the mean of the subgroups sample
covariance matrices.
An alarm is signalled if any of the statistics is greater
than a positive control limit
The T2 and Var control statistics are computed
score=Identical:
from the original data standardized
using either the classical pooled estimates of the mean vector
and dispersion matrix (Montgomery, 2009, equations 11.14--11.18;
Qiu, 2013, equations at page 269) or the highly robust minimum covariance determinant (MCD)
estimate when argument loc.scatter is equal to
Classic or Robust, respectively.
score=Signed Ranks, Spatial Signs, Spatial Ranks,
Marginal Ranks: from a “rank” transformation of the original
data. In particular, see Hallin and Paindaveine (2005) for the
definition of the multivariate signed ranks and Oja (2010) for those of the
spatial signs, spatial ranks, and marginal ranks. Multivariate signed
ranks, spatial signs and
ranks are “inner” standardized while marginal ranks are
“outer” standardized (see Oja (2010) for the definition of
“inner” and “outer” standardization).
When loc.scatter is equal to Classic,
inner standardization takes into account the subgroup structure of the
data imposing that the average of the within-group covariances of the
transformed data is proportional to the identity matrix.
Otherwise, i.e., when
loc.scatter is equal to Robust, it is based on
a standard Hettmansperger-Randles-like scatter estimate.
Note that the loc.scatter is equal to
Robust.
R. C. Bell, L. A. Jones-Farmer, N. Billor (2014) “A distribution-free multivariate Phase I location control chart for subgrouped data from elliptical distributions”. Technometrics, 56, pp. 528--538, tools:::Rd_expr_doi("10.1080/00401706.2013.879264").
C. R. Cheng, J. J. H. Shiau JJH (2015) “A distribution-free multivariate control chart for Phase I applications”. Quality and Reliability Engineering International, 31, pp. 97--111, tools:::Rd_expr_doi("10.1002/qre.1751").
M. Hallin and D. Paindaveine (2005) “Affine-Invariant Aligned Rank Tests for the Multivariate General Linear Model with VARMA Errors”. Journal of Multivariate Analysis, 93, pp. 122--163, tools:::Rd_expr_doi("10.1016/j.jmva.2004.01.005").
D. C. Montgomery (2009) Introduction to Statistical Quality Control, 6th edn. Wiley.
H. Oja (2010) Multivariate Nonparametric Methods with R. An Approach Based on Spatial Signs and Ranks. Springer.
P. Qiu (2013) Introduction to Statistical Process Control. Chapman & Hall/CRC Press.
data(ryan)
mshewhart(ryan)
mshewhart(ryan,subset=-10)
mshewhart(ryan,subset=-c(10,20))
mshewhart(ryan,score="Signed Ranks")
mshewhart(ryan,subset=-10,score="Signed Ranks")
mshewhart(ryan,subset=-c(10,20),score="Signed Ranks")
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