multivariate op• 資料種類:matrix → table
• 呼叫: import fullseye as fs; fs.ledger.spc_hotelling_t2(data, alpha=0.0027, mean=None, cov=None)(要直接呼叫實作,import spc; spc.spc_hotelling_t2(data, alpha=0.0027, mean=None, cov=None);從台帳取用則 opsspc.get("spc_hotelling_t2"))
• GPU:無(僅 CPU 參考實作)
基於 Hotelling T² 的多變量 SPC，管制界限依所處階段選取(第 I 階段用 Beta 分配，參數已知時用 χ² 分配)。
> 以下的詳細說明為原文 —— 摘要與標題已翻譯。
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.
• 範例資料目錄(下載 URL / 授權) —— 2-D 用 skimage.data(BSD/公有領域)加合成圖,3-D 給出真實資料源(Stanford/PDS 等)的下載 URL。
• 運算子來歷與參考文獻 —— 該運算子族所依據的研究/方法出處。
• 演算法的正典(作者、年份)與用途見上面的族使用指南。
• poc_spc — py -3.11 examples/poc_spc.py
table 作為輸入)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 運算子登記表。本條目由 tools/opdocs.py md 自動產生(請勿手動編輯)。*
© 2026 Kazufumi Furuse — Fullseye operator documentation. Licensed under Apache-2.0.