pls_diagnostic_t2 — Hotelling T² score

Group: Diagnostic · Registry tolerance: 10.0

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. — applied to PLS scores by MacGregor & Kourti 1995.

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.

Implementation

n4m_pls_diagnostics_compute with stat=’t2’. Reference: R mdatools 0.15.0 (Kucheryavskiy).

MATLAB header (bindings/matlab/+pls4all/pls_diagnostics.m):

pls4all.pls_diagnostics  Hotelling T2, Q residuals, DModX from a SIMPLS fit.

   res = pls4all.pls_diagnostics(X, Y, n_components)
   res = pls4all.pls_diagnostics(X, Y, n_components, X_reference)

 Fits an internal SIMPLS model (store_scores=1) and evaluates row-wise

Usage

Every pls4all binding tab dispatches into the same C kernel; the external libraries listed at the bottom of the page are the parity references registered in benchmarks.parity_timing.registry. Switch tabs to read the same fit in your language. The R package now ships drop-in-compatible facades for the CRAN pls package (plsr, pcr, mvr) and for the mdatools::pls(x, y, ...) matrix idiom — those tabs appear only on the methods that have a meaningful equivalence.

pls4all bindings

/* C ABI — libn4m */
n4m_context_t* ctx = n4m_context_create();
n4m_config_t*  cfg = n4m_config_create();
n4m_method_result_t* res = NULL;
n4m_pls_diagnostics_compute(ctx, cfg, &x_view, &y_view, /* hyperparams */, &res);
/* … read coefficients / mask / scores via */
/* n4m_method_result_get_double_matrix / vector / scalar … */
n4m_method_result_destroy(res);
n4m_config_destroy(cfg);
n4m_context_destroy(ctx);

import pls4all
from pls4all._methods import pls_diagnostics_compute
with pls4all.Context() as ctx, pls4all.Config() as cfg:
    res = pls_diagnostics_compute(ctx, cfg, X, y, n_components=4)
# then: res.matrix("predictions"), res.matrix("coefficients"),
# res.vector("mask"), res.scalar("intercept"), …

from pls4all.sklearn import t2_score
result = t2_score(X, y, n_components=4)

library(pls4all)
# Unified low-level dispatcher (May 2026 R cleanup):
res <- pls4all_method("pls_diagnostic_t2", X, y,
                      n_components = 4L)
# res is a named list with MethodResult arrays/scalars.
# selected_indices / top_k_intervals are 1-based.

res = pls4all.pls_diagnostics(X, y, 4);
% see header of bindings/matlab/+pls4all/pls_diagnostics.m for full
% parameter surface:
%   res = pls_diagnostics(X, Y, n_components, X_reference)
yhat = predict(res, Xtest);

No idiomatic classdef wrapper — invoke pls4all.fit("pls_diagnostic_t2", X, y, …) directly from the unified MEX factory.

Registry parity references 📐

  • 📐 ref.r_mdatools (R · r) — mdatools 0.15.0 · qualitative (rmse_rel ≤ 1e+01) — 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

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 = pls4all binding 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; pls4all language bindings show binding parity against the C++ backend. Hover the icon for role and tolerance band.

BackendParity200×30 (ms)
C++ native · libn4m
pls4all.cpp.blas+omp✓ ref 3e-151.19 ms🏆
Python · pls4all
pls4all.python✓ bind1.22 ms
pls4all.sklearn✓ bind1.31 ms
R · pls4all
pls4all.R✓ 1e-132.67 ms
pls4all.R.formula✓ 1e-133.38 ms
pls4all.R.mdatools✓ 1e-133.70 ms
pls4all.R.pls✓ 1e-134.11 ms
R · external
📐ref.r_mdatoolssource14.2 ms
BackendParity200×30 (ms)
C++ native · libn4m
pls4all.cpp.blas+omp✓ ref 3e-151.19 ms
Python · pls4all
pls4all.python✓ bind1.19 ms🏆
pls4all.sklearn✓ bind1.37 ms
R · pls4all
pls4all.R✓ 1e-132.74 ms
pls4all.R.formula✓ 1e-133.29 ms
pls4all.R.mdatools✓ 1e-133.43 ms
pls4all.R.pls✓ 1e-133.70 ms
R · external
📐ref.r_mdatoolssource13.5 ms
BackendParity200×30 (ms)
C++ native · libn4m
pls4all.cpp.blas+omp✓ ref 3e-151.21 ms🏆
Python · pls4all
pls4all.python✓ bind2.06 ms
pls4all.sklearn✓ bind1.34 ms
R · pls4all
pls4all.R✓ 1e-132.78 ms
pls4all.R.formula✓ 1e-133.45 ms
pls4all.R.mdatools✓ 1e-133.65 ms
pls4all.R.pls✓ 1e-133.45 ms
R · external
📐ref.r_mdatoolssource13.5 ms

See also: benchmark overview · methods index · interactive dashboard