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False-Alarm Risk for Generalized Variance Charts

Source: R/gv_alpha_risk.R
gv_alpha_risk.Rd

Evaluate the actual in-control probability that the generalized variance statistic crosses limits produced by gv_limits(), and report the corresponding zero-state in-control average run length.

Usage

gv_alpha_risk(
  n,
  p,
  det_sigma = 1,
  alpha = 0.0027,
  type = c("normal", "cf", "exact", "simulation"),
  side = c("upper", "two.sided"),
  cf_order = 1,
  nsim = 2e+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:

alpha

Actual or Monte Carlo estimated false-alarm probability.

nominal_alpha

The requested nominal value.

arl0

Zero-state in-control average run length, 1 / alpha; Inf if no simulated crossing is observed.

limits

The complete object returned by gv_limits().

method

Either "exact evaluation" for \(p=2\) or "simulation" for higher dimensions.

Details

The diagnostic is useful because Gaussian and Cornish-Fisher limits are approximations and therefore need not attain the requested nominal risk exactly. For dimension \(p=2\), the crossing probability is evaluated from the exact chi-square representation of \(|S|\). For \(p>2\), it is estimated independently by Monte Carlo simulation from Bartlett's product-of-chi-squares representation.

The function first calls gv_limits() using the requested type. It then evaluates $$P(|S| > UCL)$$ for an upper chart, or $$P(|S| < LCL) + P(|S| > UCL)$$ for a two-sided chart. The inequalities are strict, matching the signaling rule used by cchart.GV().

For \(p>2\), nsim controls both any simulation used to construct the limits and the independent simulation used to evaluate their risk. If seed is supplied, the limits use that seed and the risk evaluation uses seed + 1; each seeded simulation restores the caller's previous random-number state on exit. With seed = NULL, the global RNG state advances normally.

Monte Carlo uncertainty

For \(p>2\), the reported risk is a binomial proportion based on nsim draws. Its approximate Monte Carlo standard error is \(\sqrt{\hat\alpha(1-\hat\alpha)/nsim}\). Very small false-alarm risks therefore require substantially more simulations than routine examples.

Errors

Input and exact-method restrictions are inherited from gv_limits(). In addition, simulation requires an integer nsim of at least 1000 and a valid optional 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.

See also

gv_limits, gv_stat, cchart.GV

Examples

# Exact evaluation shows that exact p = 2 limits attain the nominal risk.
exact_risk <- gv_alpha_risk(10, 2, type = "exact")
exact_risk$alpha
#> [1] 0.0027
exact_risk$arl0
#> [1] 370.3704

# Diagnose the false-alarm inflation of moment-matched normal limits.
gv_alpha_risk(10, 2, type = "normal")
#> $alpha
#> [1] 0.02078953
#> 
#> $nominal_alpha
#> [1] 0.0027
#> 
#> $arl0
#> [1] 48.10115
#> 
#> $limits
#> $limits$lcl
#> [1] 0
#> 
#> $limits$ucl
#> [1] 2.685498
#> 
#> $limits$center
#> [1] 0.8888889
#> 
#> $limits$type
#> [1] "normal"
#> 
#> $limits$side
#> [1] "upper"
#> 
#> $limits$alpha
#> [1] 0.0027
#> 
#> $limits$n
#> [1] 10
#> 
#> $limits$p
#> [1] 2
#> 
#> $limits$det_sigma
#> [1] 1
#> 
#> $limits$moments
#> $limits$moments$ordinary
#> [1] 0.8888889 1.2071331 2.3248489 6.0273859
#> 
#> $limits$moments$mean
#> [1] 0.8888889
#> 
#> $limits$moments$variance
#> [1] 0.4170096
#> 
#> $limits$moments$sd
#> [1] 0.6457628
#> 
#> $limits$moments$skewness
#> [1] 1.895698
#> 
#> $limits$moments$excess_kurtosis
#> [1] 6.264543
#> 
#> $limits$moments$b1
#> [1] 0.8888889
#> 
#> $limits$moments$b2
#> [1] 0.4170096
#> 
#> $limits$moments$n
#> [1] 10
#> 
#> $limits$moments$p
#> [1] 2
#> 
#> $limits$moments$det_sigma
#> [1] 1
#> 
#> 
#> $limits$cf_order
#> [1] 1
#> 
#> $limits$nsim
#> [1] 2e+05
#> 
#> $limits$seed
#> NULL
#> 
#> 
#> $method
#> [1] "exact evaluation"
#> 

# \donttest{
# Reproducible risk evaluation for a higher-dimensional chart.
gv_alpha_risk(8, 3, type = "cf", nsim = 10000, seed = 2026)
#> $alpha
#> [1] 0.0017
#> 
#> $nominal_alpha
#> [1] 0.0027
#> 
#> $arl0
#> [1] 588.2353
#> 
#> $limits
#> $limits$lcl
#> [1] 0
#> 
#> $limits$ucl
#> [1] 5.631511
#> 
#> $limits$center
#> [1] 0.6122449
#> 
#> $limits$type
#> [1] "cf"
#> 
#> $limits$side
#> [1] "upper"
#> 
#> $limits$alpha
#> [1] 0.0027
#> 
#> $limits$n
#> [1] 8
#> 
#> $limits$p
#> [1] 3
#> 
#> $limits$det_sigma
#> [1] 1
#> 
#> $limits$moments
#> $limits$moments$ordinary
#> [1]  0.6122449  0.8996252  2.5965857 12.9904989
#> 
#> $limits$moments$mean
#> [1] 0.6122449
#> 
#> $limits$moments$variance
#> [1] 0.5247813
#> 
#> $limits$moments$sd
#> [1] 0.7244179
#> 
#> $limits$moments$skewness
#> [1] 3.691082
#> 
#> $limits$moments$excess_kurtosis
#> [1] 26.89629
#> 
#> $limits$moments$b1
#> [1] 0.6122449
#> 
#> $limits$moments$b2
#> [1] 0.5247813
#> 
#> $limits$moments$n
#> [1] 8
#> 
#> $limits$moments$p
#> [1] 3
#> 
#> $limits$moments$det_sigma
#> [1] 1
#> 
#> 
#> $limits$cf_order
#> [1] 1
#> 
#> $limits$nsim
#> [1] 10000
#> 
#> $limits$seed
#> [1] 2026
#> 
#> 
#> $method
#> [1] "simulation"
#> 
# }