Generalized Variance Control Limits

Compute one-sided upper or equal-tail two-sided control limits for the generalized variance statistic \(|S|\) under independent sampling from a \(p\)-variate normal process with in-control covariance matrix \(\Sigma\).

Usage

gv_limits(
  n,
  p,
  det_sigma = 1,
  alpha = 0.0027,
  type = c("normal", "cf", "exact", "simulation"),
  side = c("upper", "two.sided"),
  cf_order = 1,
  nsim = 1e+05,
  seed = NULL
)

Arguments

n

Integer subgroup sample size. It must satisfy \(n > p\).

p

Integer process dimension, at least 2.

det_sigma

Positive finite scalar equal to \(|\Sigma|\), the determinant of the in-control covariance matrix. Control limits scale linearly with this value.

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.

Value

A list with components:

lcl, ucl: Lower and upper control limits.
center: The exact in-control mean of \(|S|\); this is the center line used by cchart.GV().
type, side, alpha: The selected method, chart sidedness, and nominal false-alarm probability.
n, p, det_sigma: The validated design parameters.
moments: A list containing the first four ordinary moments, mean, variance, standard deviation, skewness, excess kurtosis, the constants b1 and b2, and the design parameters.
cf_order, nsim, seed: The requested numerical settings.

Details

If \(S\) is the usual covariance matrix from a subgroup of size \(n\), then \((n-1)S\) has a Wishart distribution and \(|S|\) can be represented as a scaled product of independent chi-square variables. The function uses that representation to obtain moments, approximations, exact cases, or simulated quantiles.

Exact-method scope

The \(p=2\) exact distribution is available for both upper and two-sided charts. The \(p=3\) implementation is deliberately limited to the published one-sided upper quantiles described above. No generic Meijer-G evaluator is claimed; use type = "simulation" outside those cases.

Decision convention

A plotted subgroup signals when \(|S| < LCL\) or \(|S| > UCL\). Equality to either limit is treated as in control.

Errors

Errors are raised for invalid dimensions, \(n <= p\), a nonpositive det_sigma, invalid alpha, unsupported exact cases, invalid Cornish-Fisher order, fewer than 1000 simulations, or an invalid seed.

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.

Anderson, T. W. (1984). An Introduction to Multivariate Statistical Analysis, 2nd ed. Wiley.

Cornish, E. A. and Fisher, R. A. (1960). The percentage points of distributions having known cumulants. Technometrics, 2, 209–225.

Examples

# Published dimension-two illustration: |Sigma| = 0.5320, n = 10.
gv_limits(10, 2, det_sigma = 0.5320, type = "normal")$ucl
#> [1] 1.428685
gv_limits(10, 2, det_sigma = 0.5320, type = "cf")$ucl
#> [1] 2.160305
gv_limits(10, 2, det_sigma = 0.5320, type = "exact")$ucl
#> [1] 2.153629

# Equal-tail exact limits are also available in dimension two.
gv_limits(10, 2, side = "two.sided", type = "exact")
#> $lcl
#> [1] 0.0527836
#> 
#> $ucl
#> [1] 4.538591
#> 
#> $center
#> [1] 0.8888889
#> 
#> $type
#> [1] "exact"
#> 
#> $side
#> [1] "two.sided"
#> 
#> $alpha
#> [1] 0.0027
#> 
#> $n
#> [1] 10
#> 
#> $p
#> [1] 2
#> 
#> $det_sigma
#> [1] 1
#> 
#> $moments
#> $moments$ordinary
#> [1] 0.8888889 1.2071331 2.3248489 6.0273859
#> 
#> $moments$mean
#> [1] 0.8888889
#> 
#> $moments$variance
#> [1] 0.4170096
#> 
#> $moments$sd
#> [1] 0.6457628
#> 
#> $moments$skewness
#> [1] 1.895698
#> 
#> $moments$excess_kurtosis
#> [1] 6.264543
#> 
#> $moments$b1
#> [1] 0.8888889
#> 
#> $moments$b2
#> [1] 0.4170096
#> 
#> $moments$n
#> [1] 10
#> 
#> $moments$p
#> [1] 2
#> 
#> $moments$det_sigma
#> [1] 1
#> 
#> 
#> $cf_order
#> [1] 1
#> 
#> $nsim
#> [1] 1e+05
#> 
#> $seed
#> NULL
#> 

# \donttest{
# A reproducible simulation-based limit for higher dimension.
gv_limits(8, 3, type = "simulation", nsim = 5000, seed = 2026)
#> $lcl
#> [1] 0
#> 
#> $ucl
#> [1] 5.781372
#> 
#> $center
#> [1] 0.6122449
#> 
#> $type
#> [1] "simulation"
#> 
#> $side
#> [1] "upper"
#> 
#> $alpha
#> [1] 0.0027
#> 
#> $n
#> [1] 8
#> 
#> $p
#> [1] 3
#> 
#> $det_sigma
#> [1] 1
#> 
#> $moments
#> $moments$ordinary
#> [1]  0.6122449  0.8996252  2.5965857 12.9904989
#> 
#> $moments$mean
#> [1] 0.6122449
#> 
#> $moments$variance
#> [1] 0.5247813
#> 
#> $moments$sd
#> [1] 0.7244179
#> 
#> $moments$skewness
#> [1] 3.691082
#> 
#> $moments$excess_kurtosis
#> [1] 26.89629
#> 
#> $moments$b1
#> [1] 0.6122449
#> 
#> $moments$b2
#> [1] 0.5247813
#> 
#> $moments$n
#> [1] 8
#> 
#> $moments$p
#> [1] 3
#> 
#> $moments$det_sigma
#> [1] 1
#> 
#> 
#> $cf_order
#> [1] 1
#> 
#> $nsim
#> [1] 5000
#> 
#> $seed
#> [1] 2026
#> 
# }