Monitoring multivariate processes with IQCC • IQCC

Scope

Multivariate process monitoring separates two questions that are often mixed in practice. Hotelling T-squared charts monitor shifts in a process mean vector, while generalized-variance and trace charts monitor changes in the covariance structure. IQCC contains all three workflows:

Question	Statistic	Main functions
Has the mean vector shifted?	Hotelling T-squared	`T2.1()`, `T2.2()`, `cchart.T2.1()`, `cchart.T2.2()`
Has total generalized variance changed?	`\|S\|`	`gv_stat()`, `gv_limits()`, `gv_alpha_risk()`, `cchart.GV()`
Has standardized covariance structure changed?	`tr(V)`	`trv_stat()`, `trv_limits()`, `trv_alpha_risk()`, `cchart.trV()`

The covariance charts are complementary. The determinant |S| summarizes the volume of the subgroup covariance matrix. The trace statistic (n - 1) tr(Sigma0^-1 S) can change even when the determinant is nearly unchanged.

Input formats

The generalized-variance and trace-statistic functions use the multivariate subgroup formats accepted by gv_stat() and cchart.GV(): a list of subgroup matrices, a subgroup-by-observation-by-variable array, or a stacked matrix with the subgroup size supplied separately. The examples below use a list because it makes the subgroup boundaries explicit.

g1 <- cbind(
  x = c(-1, 0, 1, -1, 1),
  y = c(-1, 1, 0, 1, -1)
)
g2 <- sweep(g1, 2, c(0.2, -0.1), "+")
g3 <- g1 %*% matrix(c(1.1, 0.1, 0, 0.9), nrow = 2)
groups <- list(g1, g2, g3)

data.frame(
  subgroup = seq_along(groups),
  generalized_variance = gv_stat(groups),
  trace_statistic = trv_stat(groups, Sigma0 = diag(2))
)
#>   subgroup generalized_variance trace_statistic
#> 1        1            0.9375000             8.0
#> 2        2            0.9375000             8.0
#> 3        3            0.9188438             7.9

The legacy Hotelling helpers use a different array convention: observation-by-variable-by-subgroup. Check dim() before moving data between the Hotelling helpers and the generalized-variance or trace workflows.

Hotelling T-squared

The Hotelling workflow estimates Phase I location and covariance, computes Phase I T-squared statistics, and then evaluates Phase II observations against the Phase II reference distribution.

set.seed(2026)
phase1 <- data.1(m = 8, n = 5, mu = c(0, 0), Sigma = diag(2))
reference <- IQCC::stats(phase1, m = 8, n = 5, p = 2)
t2_phase1 <- T2.1(reference, m = 8, n = 5)

phase2 <- data.2(reference, n = 5, p = 2)
t2_phase2 <- T2.2(phase2, reference, n = 5)

data.frame(
  phase = c(rep("Phase I", length(t2_phase1)), "Phase II"),
  statistic = c(t2_phase1, t2_phase2)
)
#>      phase statistic
#> 1  Phase I 6.5124530
#> 2  Phase I 2.8083634
#> 3  Phase I 8.1145908
#> 4  Phase I 0.5432524
#> 5  Phase I 1.5197014
#> 6  Phase I 4.1999052
#> 7  Phase I 1.6098032
#> 8  Phase I 3.6227786
#> 9 Phase II 3.8565799

cchart.T2.1() and cchart.T2.2() draw the two chart types. Their control limits are not interchangeable because Phase I and Phase II use different reference distributions.

Generalized variance

For each subgroup, gv_stat() computes the determinant of the sample covariance matrix. gv_limits() exposes normal, Cornish-Fisher, selected exact, and simulation-based limits. The exact implementation is deliberately scoped; generic Meijer-G quantiles are not claimed by the package.

gv_methods <- do.call(
  rbind,
  lapply(c("normal", "cf", "exact"), function(method) {
    limits <- gv_limits(n = 5, p = 2, det_sigma = 1, type = method)
    data.frame(
      method = method,
      center = limits$center,
      lcl = limits$lcl,
      ucl = limits$ucl
    )
  })
)

gv_sim <- gv_limits(
  n = 5,
  p = 2,
  det_sigma = 1,
  type = "simulation",
  nsim = 2000,
  seed = 2026
)

gv_methods
#>   method center lcl      ucl
#> 1 normal   0.75   0 3.305568
#> 2     cf   0.75   0 6.769365
#> 3  exact   0.75   0 6.288749
gv_sim[c("type", "ucl", "nsim", "seed")]
#> $type
#> [1] "simulation"
#> 
#> $ucl
#> [1] 5.379544
#> 
#> $nsim
#> [1] 2000
#> 
#> $seed
#> [1] 2026

The chart wrapper records the phase split, limits, covariance source, and signaled subgroups. Use summary() to inspect the fitted object instead of reading values from a plot.

gv_chart <- cchart.GV(
  x1 = groups[1:2],
  x2 = groups[3],
  type = "cf",
  plot = FALSE
)
summary(gv_chart)
#> Generalized Variance Control Chart
#>   Dimension: p = 2 ; subgroup size n = 5 
#>   Subgroups: 3 (Phase I: 2 ; Phase II: 1 )
#>   Limits: cf / upper ; nominal alpha = 0.0027 
#>   Covariance: estimated from Phase I subgroups 
#>   LCL = 0 ; center = 0.8036 ; UCL = 7.253 
#>   Signals: 0

If Sigma is omitted, IQCC uses Phase I covariance information as a plug-in reference. The nominal known-parameter calibration does not include the extra uncertainty introduced by estimating the reference from a finite Phase I sample.

Auxiliary trace chart

The trace statistic is

$T = (n - 1)\operatorname{tr}(\Sigma_0^{-1} S).$

Under multivariate normality and known in-control covariance Sigma0, T has a chi-square distribution with p * (n - 1) degrees of freedom.

trv_limits(n = 5, p = 2, alpha = 0.0027)
#> $lcl
#> [1] 0
#> 
#> $ucl
#> [1] 23.57439
#> 
#> $center
#> [1] 8
#> 
#> $type
#> [1] "chisq"
#> 
#> $alpha
#> [1] 0.0027
#> 
#> $n
#> [1] 5
#> 
#> $p
#> [1] 2
#> 
#> $df
#> [1] 8
#> 
#> $nsim
#> [1] 1e+05
#> 
#> $seed
#> NULL
trv_alpha_risk(n = 5, p = 2, ucl = trv_limits(5, 2)$ucl)
#> $alpha
#> [1] 0.0027
#> 
#> $arl0
#> [1] 370.3704
#> 
#> $ucl
#> [1] 23.57439
#> 
#> $n
#> [1] 5
#> 
#> $p
#> [1] 2
#> 
#> $df
#> [1] 8
#> 
#> $method
#> [1] "exact chi-square"

The trace chart is most useful beside |S|, not as a replacement for it. The following deterministic example has two covariance matrices with the same determinant but different standardized trace.

base <- sqrt(3 / 4) * rbind(
  c(1, 1), c(1, -1), c(-1, 1), c(-1, -1)
)
same_volume <- list(
  base %*% chol(diag(2)),
  base %*% chol(diag(c(4, 0.25)))
)

data.frame(
  generalized_variance = gv_stat(same_volume),
  trace_statistic = trv_stat(same_volume, Sigma0 = diag(2))
)
#>   generalized_variance trace_statistic
#> 1                    1            6.00
#> 2                    1           12.75

trv_chart <- cchart.trV(
  x = same_volume[1],
  newdata = same_volume[2],
  Sigma0 = diag(2),
  alpha = 0.05,
  type = "chisq",
  plot = FALSE
)
summary(trv_chart)
#> Trace-Statistic Control Chart
#>   Dimension: p = 2 ; subgroup size n = 4 
#>   Subgroups: 2 (Phase I: 1 ; Phase II: 1 )
#>   Limits: chisq ; nominal alpha = 0.05 ; df = 6 
#>   Covariance: supplied Sigma0 
#>   Center = 6 ; UCL = 12.59 
#>   Signals: 1 
#>  index phase statistic
#>      2    II     12.75

The published generalized-variance report contains a Case B where the printed summary matrices keep determinants close while the trace chart signals in Phase II. IQCC reproduces the printed determinants and the published trace UCL, but the full signal sequence cannot be recomputed because the report does not print all simulated subgroups or the random seed.

Practical sequence

Use Hotelling T-squared when the primary question is a shift in the mean vector.
Use |S| when the primary question is a change in generalized variance or covariance volume.
Add tr(V) when covariance structure may change without a large determinant change.
State whether covariance references are known parameters or Phase I plug-in estimates.
Use simulated limits as Monte Carlo approximations and report nsim, seed, and simulation error when they support a scientific claim.

cat("<!-- IQCC_EXECUTED_MULTIVARIATE_MONITORING -->\n")