multivariate op• Data kinds: matrix → table
• Call: import fullseye as fs; fs.ledger.spc_hotelling_t2(data, alpha=0.0027, mean=None, cov=None) (to call the implementation directly, import spc; spc.spc_hotelling_t2(data, alpha=0.0027, mean=None, cov=None); from the registry, opsspc.get("spc_hotelling_t2"))
• GPU: none (CPU reference implementation only)
Multivariate SPC by Hotelling's T² with the control limit of the phase being charted.
data is a 2-D array of shape (m, p) — m observations of p correlated features. Each row's statistic is::
T^2_i = (x_i - mu)' S^-1 (x_i - mu)
The control limit depends on where mu and S come from, and the op picks it from what you pass:
• Phase I (default, neither mean nor cov) — mu and S are the sample mean and ddof=1 covariance of the *same* rows. Then m T^2_i / (m-1)^2 follows Beta(p/2, (m-p-1)/2) exactly (Gnanadesikan & Kettenring 1972; Tracy, Young & Mason 1992), so::
UCL = (m-1)^2 / m * Beta_{1-alpha}(p/2, (m-p-1)/2)
In phase I T^2_i <= (m-1)^2/m holds for every row whatever the data, so
the phase-II F limit (which exceeds that bound for small m) can never
alarm there. Needs m >= p + 2.
• Known parameters (both mean and cov) — the in-control mean and covariance are taken as known (a specification, or estimates from a very large reference). Then T^2 ~ chi^2_p and UCL = chi^2_{p, 1-alpha}, independent of how many rows are charted (one row is enough).
• Phase II with estimated parameters — if mean / cov were estimated from a separate in-control reference of n rows and new rows are charted against them, the exact limit is p (n+1)(n-1) / (n (n-p)) * F_{1-alpha}(p, n-p) with n the size of the *reference*, not of the charted data. This op treats passed parameters as known (chi-square limit, the n -> inf limit of that F form); for a small reference compute the F limit with your n and compare t2 against it.
Passing only one of mean / cov is refused: no standard control limit exists for a half-known parameter set.
At false-alarm rate alpha (default 0.0027, the 3-sigma-equivalent). Returns a dict with the per-row t2 array, the ucl, the phase used ("phase_I" / "known"), the integer indices out_of_control, and boolean in_control. The mean row (x == mu) has T^2 == 0 (pinned in the tests).
Raises ValueError: a non-2-D / empty *data*, fewer than p+2 observations in phase I, only one of mean / cov, alpha outside (0, 1), a singular covariance, or non-finite / mislabelled input.
• Sample-data catalog (download URLs / licences) — 2-D uses skimage.data (BSD/public domain) plus synthetic images; 3-D lists download URLs for real data sources (Stanford, PDS, …).
• Operator provenance and references — the sources of the research/methods this op family came from.
• The canonical algorithm (author, year) and its uses are named in the family usage guide above.
• poc_spc — py -3.11 examples/poc_spc.py
table as input)msa_anova_table · msa_gauge_rr · msa_bias_linearity · msa_attribute_agreement · gum_standard_uncertainty · gum_propagate · gum_expanded · gum_monte_carlo
multivariate)—
*Provenance: spc.py — SPC operator registry. This per-op note is generated by tools/opdocs.py md (do not hand-edit).*
© 2026 Kazufumi Furuse — Fullseye operator documentation. Licensed under Apache-2.0.