hotelling_t2 — n4m.outlier_detection.hotelling_t2¶
Namespace: n4m.outlier_detection · Fully-qualified: n4m.outlier_detection.hotelling_t2 · Catalog id: utilities.hotelling_t2
API surface¶
C ABI (ABI 2): n4m_outlier_detection_hotelling_t2 · n4m_outlier_detection_hotelling_t2_resolved. Use the linked public header for the exact signature, configuration, and result handles.
Python (verified public re-export): from n4m.outlier_detection import hotelling_t2
Signature: hotelling_t2(X, n_components: int = 5, alpha: float = 0.05)
R: no current source-verified entry point was found for this catalog method.
MATLAB / Octave: no current source-verified entry point was found for this catalog method.
Parameters¶
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Explanations¶
Bibliographic source¶
Hotelling (1931), The Generalization of Student’s Ratio, Annals of Mathematical Statistics 2, 360–378, https://doi.org/10.1214/aoms/1177732979.
Mathematical principle¶
The input matrix is column-centered with a corrected two-pass mean (constant columns centre to exact zeros) and decomposed by compact SVD. Only resolved directions enter, \(\sigma_j > 16\max(n,p)\,\varepsilon\,\sigma_1\); with \(r\le k\) of them, \(T_i^2=\sum_{j=1}^r t_{ij}^2/\lambda_j=(n-1)\sum_{j=1}^r u_{ij}^2\), evaluated in the second, scale-invariant form. The upper control limit is \(r(n-1)(n+1)/[n(n-r)]\,F_{1-\alpha}(r,n-r)\), solved directly in the UCL \(G\): with \(\kappa=(n^2-1)/n\), \(G/(\kappa+G)\) follows a Beta\((r/2,(n-r)/2)\) law whose small tail (upper for \(\alpha\le 0.5\), lower otherwise) is evaluated in logarithms from \(G/\kappa\) and inverted by bisection on the normal doubles; \(r=0\) gives \(T^2=0\) and UCL \(0\).
Appropriate uses¶
Flagging multivariate score-space extremes directly from a calibration matrix when no PLS model object is involved.
Limits and validation¶
The implementation requires rows >= cols, \(k< n\), finite dense F64 data, and treats the same input as both PCA fit and scored set. The finite-sample F limit assumes the classical approximately normal model. The measured relative error of the UCL is below \(10^{-12}\) against mpmath references (\(n\) up to \(10^6\), \(\alpha\) from \(10^{-300}\) to \(1-2^{-53}\)), not a proven bound and not uniform in the rank: columns 1–127 of \(H_{128}\) with \(k=127\) and \(\alpha=10^{-100}\) give \(7.2\times10^{-12}\). A UCL that is not a finite normal double (only for extreme \(\alpha\), e.g. \(n=3\), \(r=1\), \(\alpha<7.4\times10^{-309}\)) returns N4M_ERR_NUMERICAL_FAILURE.
Implementation¶
n4m.outlier_detection.hotelling_t2 wraps the stateless ABI-2 symbol n4m_outlier_detection_hotelling_t2; SVD, statistic, and UCL are in cpp/src/core/utilities/hotelling_t2.c.
Sources and provenance¶
https://doi.org/10.1214/aoms/1177732979; https://github.com/GBeurier/nirs4all-methods/blob/main/cpp/src/core/utilities/hotelling_t2.c
See also: methods index.