pls_diagnostic_t2 — Hotelling T² score

Group: Diagnostic · Registry tolerance: 1e-08

Description

PLS Hotelling T² (§9)

Registry note — R mdatools::pls is the only widely installable external reference. Both use SIMPLS-style but differ on score normalization conventions — tolerance is wide enough to flag the R ref’s presence.

Parameters

Name	Type	Default	Notes
n_components	int	4	registry benchmark cell value

Explanations

Bibliographic source

Hotelling, H. (1931). The generalization of Student’s ratio. Annals of Mathematical Statistics 2(3), 360–378. DOI 10.1214/aoms/1177732979.

Mathematical principle

Hotelling T² measures how unusual a sample is within the latent score space: \(T_i^2 = \mathbf{t}_i^{\top} \boldsymbol{\Lambda}^{-1} \mathbf{t}_i\) where \(\boldsymbol{\Lambda}\) is the diagonal matrix of score variances. Under multivariate normality of the scores, \(\frac{n(n-k)}{k(n^2-1)} T^2 \sim F_{k, n-k}\), giving an exact upper control limit at any \(\alpha\).

T² complements the Q residual (next entry): Q measures the distance to the model (variation in \(\mathbf{X}\) that the latent space does not explain), while T² measures the distance within the model (unusual score combination on otherwise well-explained samples). Joint T²/Q monitoring catches both kinds of out-of-control points.

Reported per-sample as a 1-D vector aligned with the rows of the input.

Appropriate uses

Flagging observations with unusual leverage in the fitted PLS score space.

Limits and validation

T-squared detects score-space leverage rather than large residuals; thresholds require an appropriate reference population.

Implementation

n4m_metrics_pls_diagnostics_compute with stat=’t2’. Reference: R mdatools 0.15.0 (Kucheryavskiy).

Sources and provenance

Current implementation: cpp/src/core/pls_diagnostics.cpp.

API and bindings

C ABI (ABI 2): n4m_metrics_pls_diagnostics_compute. Use the linked public header for the exact signature, configuration, and result handles.

Python: no current AST-verified public n4m re-export was found for this method. The linked C ABI above is the documented surface in this checkout.

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.

Registry parity references 📐

• 📐 ref.r_mdatools (R · r) — mdatools 0.15.0 · strict (rmse_rel ≤ 1e-08) — R mdatools::pls with predict()$xdecomp$T2 / $Q. DModX is derived locally from $Q + DOF. mdatools uses different SIMPLS deflation / normalization conventions than pls4all, so cross-implementation parity is qualitative.

Benchmarks

Archived measurement identity. Backend labels in this table are the raw IDs recorded when the benchmark ran (including historical pls4all.* IDs). They preserve measurement provenance and do not describe a current public Python, R, or MATLAB binding; use the source-verified API and bindings section above for current entry points.

Adaptive wall-clock per cell measured against full_matrix.csv. Only backends that implement this method are listed; libraries without the method are omitted.

Verdict  ·  ✓ ref / ≈ ref / ~ shape mark a reference-gate pass at strict / relaxed / qualitative tolerance  ·  ✓ bind = archived binding-harness result agrees with the C++ baseline  ·  ⇄ cross-check = documented by-design selector/RNG/model, noncanonical API/facade convention, or secondary oracle  ·  ✗ divergent  ·  ⚠ error  ·  — not run. The fastest backend per column is marked 🏆.

Reference gate: strict — numeric equivalence (rmse_rel_tol ≤ 1e-08).

Rows tagged with 📐 are the canonical parity references for this method (declared in parity_timing.registry). C++ and external rows show reference parity; archived language-harness rows show binding parity against the C++ backend. Hover the icon for role and tolerance band.

1 thread

Backend	Parity	200×30 (ms)
C++ native · libn4m
pls4all.cpp.blas+omp	✓ ref 3e-15	1.19 ms🏆
Python · archived pls4all benchmark
pls4all.python	✓ bind	1.22 ms
pls4all.sklearn	✓ bind	1.31 ms
R · archived pls4all benchmark
pls4all.R	✓ 1e-13	2.67 ms
pls4all.R.formula	✓ 1e-13	3.38 ms
pls4all.R.mdatools	✓ 1e-13	3.70 ms
pls4all.R.pls	✓ 1e-13	4.11 ms
R · external
📐ref.r_mdatools	source	14.2 ms

3 threads

Backend	Parity	200×30 (ms)
C++ native · libn4m
pls4all.cpp.blas+omp	✓ ref 3e-15	1.19 ms
Python · archived pls4all benchmark
pls4all.python	✓ bind	1.19 ms🏆
pls4all.sklearn	✓ bind	1.37 ms
R · archived pls4all benchmark
pls4all.R	✓ 1e-13	2.74 ms
pls4all.R.formula	✓ 1e-13	3.29 ms
pls4all.R.mdatools	✓ 1e-13	3.43 ms
pls4all.R.pls	✓ 1e-13	3.70 ms
R · external
📐ref.r_mdatools	source	13.5 ms

10 threads

Backend	Parity	200×30 (ms)
C++ native · libn4m
pls4all.cpp.blas+omp	✓ ref 3e-15	1.21 ms🏆
Python · archived pls4all benchmark
pls4all.python	✓ bind	2.06 ms
pls4all.sklearn	✓ bind	1.34 ms
R · archived pls4all benchmark
pls4all.R	✓ 1e-13	2.78 ms
pls4all.R.formula	✓ 1e-13	3.45 ms
pls4all.R.mdatools	✓ 1e-13	3.65 ms
pls4all.R.pls	✓ 1e-13	3.45 ms
R · external
📐ref.r_mdatools	source	13.5 ms

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See also: benchmark overview · methods index · interactive dashboard