Generalized Variance Control Chart

Construct and optionally plot a Shewhart-type chart for multivariate process variability based on the generalized variance statistic $|S|$, the determinant of each subgroup sample covariance matrix.

Usage

cchart.GV(
  x1,
  size = NULL,
  x2 = NULL,
  Sigma = NULL,
  alpha = 0.0027,
  type = c("normal", "cf", "exact", "simulation"),
  side = c("upper", "two.sided"),
  cf_order = 1,
  nsim = 1e+05,
  seed = NULL,
  plot = TRUE,
  ...
)

# S3 method for class 'cchart.GV'
plot(x, ...)

Arguments

x1

Phase I subgroup data in any format accepted by gv_stat(): a list of equal-sized numeric matrices, a subgroup-by-observation-by-variable array, or a numeric matrix containing consecutive subgroups.

size

Positive integer subgroup size when x1 or x2 is a stacked matrix. It is ignored for list and array input.

x2

Optional Phase II subgroup data in a format accepted by gv_stat(). Phase II subgroups must have the same number of observations and variables as the Phase I subgroups.

Sigma

Optional finite symmetric positive-definite $p \times p$ in-control covariance matrix. If supplied, its determinant defines the chart limits. If omitted, the function averages the Phase I covariance matrices and estimates $$|\Sigma| = |\bar S| / b_3,$$ where $b_3$ is the finite-Phase-I determinant bias correction based on the number of Phase I subgroups, subgroup size, and dimension.

alpha

Nominal probability of a false alarm per subgroup, strictly between 0 and 1. For side = "upper", all probability is placed in the upper tail. For side = "two.sided", alpha / 2 is placed in each tail.

type

Character string selecting the limit calculation:

"normal": Moment-matched Gaussian quantiles using the exact mean and variance of $|S|$.
"cf": Cornish-Fisher corrected Gaussian quantiles using skewness, and optionally excess kurtosis.
"exact": Closed-form chi-square quantiles for $p=2$, or the published upper-limit table for $p=3$, $n=4,\ldots,15$, and alpha equal to 0.0020 or 0.0027.
"simulation": Monte Carlo quantiles from the Bartlett product-of-chi-squares representation.

Partial matching is not used.

side

Either "upper" or "two.sided". Since generalized variance is nonnegative, the lower limit for an upper chart is zero. Any negative approximated lower limit for a two-sided chart is also truncated to zero.

cf_order

Integer 1 or 2. Order 1 uses the skewness correction. Order 2 additionally uses excess kurtosis and the squared-skewness term. It is used only when type = "cf".

nsim

Integer number of Monte Carlo draws, at least 1000. It is used by type = "simulation" and stored in the returned object for all methods.

seed

NULL or a finite numeric scalar used as the Monte Carlo seed. A supplied seed makes simulated limits reproducible and the caller's existing .Random.seed is restored on exit.

plot

Logical scalar. If TRUE, draw the chart before returning the result. If FALSE, construct the object without plotting.

...

Additional graphical arguments passed to plot.cchart.GV() and then to graphics::plot().

x

An object of class "cchart.GV".

Value

An object of class "cchart.GV", a list with components:

statistics: Generalized variances for all Phase I followed by all Phase II subgroups.
phase1, phase2: Integer indices identifying the two phases in statistics.
limits: The complete list returned by gv_limits().
out.of.control: Indices for which statistics < lcl or statistics > ucl. Equality to a limit does not signal.
Sbar: The average Phase I covariance matrix when Sigma is estimated, otherwise NULL.
Sigma: The supplied in-control covariance matrix, or NULL when the determinant is estimated from Phase I.
b3: The determinant bias-correction factor when Phase I estimation is used, otherwise NA_real_.
n, p: Subgroup size and process dimension.
call: The matched function call.

For plot.cchart.GV(), x invisibly.

Details

Phase I subgroups define the chart design and, unless Sigma is supplied, estimate the in-control generalized variance. Optional Phase II subgroups are then evaluated using the same subgroup size, process dimension, and fixed control limits.

Each subgroup must contain finite observations on at least two variables, and the subgroup size must exceed the process dimension. When Sigma = NULL, the average Phase I covariance matrix must itself be positive definite. The control-line center is the exact in-control mean of $|S|$, not generally $|\Sigma|$.

plot.cchart.GV() displays subgroup generalized variances, the center line, lower and upper limits, a vertical separator before Phase II, and solid points at signaled subgroups. The plot method returns its input invisibly.

Phase I and Phase II convention

Phase I data are used both as historical plotted points and, when Sigma is absent, to estimate the in-control determinant. Phase II data do not alter the estimate or the limits. For a strict prospective chart with known parameters, supply Sigma.

Simulated limits

When type = "simulation", nsim and seed are passed to gv_limits(). A supplied seed makes the result reproducible and the caller's existing random-number state is restored.

Errors

Errors are raised for invalid subgroup representations, unequal subgroup dimensions, non-finite observations, $n <= p$, an invalid or non-positive-definite Sigma, a singular Phase I average covariance matrix, and any unsupported limit configuration documented in gv_limits().

References

Barbosa, E. P., Gneri, M. A., and Meneguetti, A. Improving Shewhart-type Generalized Variance Control Charts for Multivariate Process Variability Monitoring using Cornish-Fisher Quantile Correction, Meijer-G Function and Other Tools. Research report.

Alt, F. B. (1984). Multivariate quality control. In Johnson, N. L. and Kotz, S. (eds.), Encyclopedia of Statistical Sciences, Vol. 6, 110–122. Wiley.

Examples

set.seed(123)
phase1 <- array(rnorm(6 * 8 * 2), dim = c(6, 8, 2))

# Known in-control covariance, exact dimension-two limits.
chart <- cchart.GV(
    phase1,
    Sigma = diag(2),
    type = "exact",
    plot = FALSE
)
chart$limits[c("lcl", "center", "ucl")]
#> $lcl
#> [1] 0
#> 
#> $center
#> [1] 0.8571429
#> 
#> $ucl
#> [1] 4.621594
#> 
summary(chart)
#> Generalized Variance Control Chart
#>   Dimension: p = 2 ; subgroup size n = 8 
#>   Subgroups: 6 (Phase I: 6 ; Phase II: 0 )
#>   Limits: exact / upper ; nominal alpha = 0.0027 
#>   Covariance: supplied Sigma 
#>   LCL = 0 ; center = 0.8571 ; UCL = 4.622 
#>   Signals: 0 

# Estimate |Sigma| from Phase I and evaluate two Phase II subgroups.
phase2 <- array(rnorm(2 * 8 * 2), dim = c(2, 8, 2))
estimated <- cchart.GV(
    phase1,
    x2 = phase2,
    type = "cf",
    plot = FALSE
)
estimated$phase2
#> [1] 7 8