Module control 

Control limits for SPM monitoring statistics.

Provides upper control limits (UCL) for T-squared and SPE statistics:

• T-squared: chi-squared distribution quantile. For finite calibration samples of size n, T² follows (n·ncomp/(n−ncomp))·F(ncomp, n−ncomp) rather than χ²(ncomp). The chi-squared limit is the large-sample (n → ∞) limit.
• SPE: moment-matched chi-squared approximation (Box, 1954, Theorem 1, pp. 292–295). The derivation matches the first two moments of the SPE distribution to a scaled chi-squared: E[a·χ²(b)] = a·b = mean, Var[a·χ²(b)] = 2a²·b = var, giving a = var/(2·mean), b = 2·mean²/var.

Accuracy

The moment-matching approximation is exact when SPE follows a scaled chi-squared distribution (holds under Gaussian scores). For non-Gaussian data, the approximation error is O(κ₄) where κ₄ is the excess kurtosis of the SPE distribution. Use spe_moment_match_diagnostic to assess adequacy.

References

• Box, G.E.P. (1954). Some theorems on quadratic forms applied in the study of analysis of variance problems, I. Annals of Mathematical Statistics, 25(2), 290–302. Theorem 1, pp. 292–295.
• Woodall, W.H. & Ncube, M.M. (1985). Multivariate CUSUM quality-control procedures. Technometrics, 27(3), 285–292. §2, pp. 286–288.

Structs

ControlLimit

A control limit for a monitoring statistic.

Functions

spe_control_limit

Compute the SPE control limit using moment-matched chi-squared approximation.

spe_moment_match_diagnostic

Diagnostic for the SPE moment-match chi-squared approximation.

t2_control_limit

Compute the T-squared control limit based on the chi-squared distribution.