t2_select — Hotelling T² loading selection

Group: Variable selector · Registry tolerance: 1.2

Description

T²-PLS loading-weight selection (§18 Phase 5l)

From the pls4all.sklearn.T2Selector docstring:

T²-PLS loading-weight selection (plsVarSel::T2_pls style).

Registry note — R plsVarSel::T2_pls Hotelling T² loading selection with train=test to match pls4all’s single-training-set selector. The remaining divergence is threshold semantics: plsVarSel chooses the $mv$ min-error set across alpha levels, while pls4all thresholds training-score T² directly. Mask RMSE-rel ~0=perfect, ~1=half disagree, ~1.41=disjoint.

Parameters

Name

Type

Default

Notes

alpha_thresholds

—

None

Sequence of significance levels (α) defining the T² acceptance regions to sweep.

n_components

int

2

Number of latent components extracted (k).

min_selected

`int

None`

None

Explanations

Bibliographic source

Mehmood, T. (2016). Hotelling T² based variable selection in partial least squares regression. Chemometrics and Intelligent Laboratory Systems 154, 23–28. https://doi.org/10.1016/j.chemolab.2016.03.020 — proposes T²-PLS, the loading-weights-level Hotelling T² selector. See also Wold, Sjöström & Eriksson (2001), Chemometrics and Intelligent Laboratory Systems 58(2), 109–130 §6.2 for the original T²-vs-VIP discussion in PLS.

Mathematical principle

Apply Hotelling T² to the loading weights rather than the scores: features with loading vectors of large T² are deemed important. Threshold via the F-distribution upper control limit at a user-specified \(\alpha\), with a top-\(k\) fallback to avoid empty selections.

Distinct from sample-level T² monitoring (see pls_diagnostic_t2) — here T² acts as a multivariate feature ranker that respects correlation structure across components. Useful when the loadings have strong between-component structure and per-component VIP under-counts contributions spread across multiple components.

Implementation

n4m_t2_select.

MATLAB header (bindings/matlab/+pls4all/t2_select.m):

pls4all.t2_select  T²-based selector (sweep over alpha thresholds).

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_t2_select_fit(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 t2_select_fit
with pls4all.Context() as ctx, pls4all.Config() as cfg:
    res = t2_select_fit(ctx, cfg, X, y)
# then: res.matrix("predictions"), res.matrix("coefficients"),
# res.vector("mask"), res.scalar("intercept"), …

from pls4all.sklearn import T2Selector
mdl = T2Selector(alpha_thresholds, n_components=2, min_selected=None)
mdl.fit(X, y)
y_hat = mdl.predict(X_test)

library(pls4all)
# Unified low-level dispatcher (May 2026 R cleanup):
res <- pls4all_method("t2_select", X, y,
                      n_components = 2L)
# res is a named list with MethodResult arrays/scalars.
# selected_indices / top_k_intervals are 1-based.

res = pls4all.t2_select(X, y, 2);
% see header of bindings/matlab/+pls4all/t2_select.m for full
% parameter surface:
%   res = t2_select(X, Y, n_components, alpha_thresholds, min_selected)
yhat = predict(res, Xtest);

No idiomatic classdef wrapper — invoke pls4all.fit("t2_select", X, y, …) directly from the unified MEX factory.

Registry parity references 📐

  • 📐 ref.r_plsvarsel (R · r) — plsVarSel 0.10.0 · qualitative (rmse_rel ≤ 1e+00) — R plsVarSel::T2_pls — Hotelling T² loading-weight selection. Same idea as pls4all’s T2_select.

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: qualitative — shape/smoke comparison only. The external library and pls4all do not produce numerically equivalent output for this method (see the MethodSpec notes); the rmse_rel_tol ≤ 2e+00 budget is set wide on purpose. Treat ~ shape as “we ran both, both finished”, not as numerical agreement.

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.

BackendParity200×40 (ms)
C++ native · libn4m
pls4all.cpp.blas+omp⇄ J 0.801.67 ms🏆
Python · pls4all
pls4all.python✓ J 0.801.69 ms
pls4all.sklearn✓ J 0.801.83 ms
R · pls4all
pls4all.R✓ J 0.804.21 ms
pls4all.R.formula✓ J 0.806.34 ms
pls4all.R.mdatools✓ J 0.806.75 ms
pls4all.R.pls✓ J 0.805.80 ms
R · external
📐ref.r_plsvarselsource37.0 ms
BackendParity200×40 (ms)
C++ native · libn4m
pls4all.cpp.blas+omp⇄ J 0.801.67 ms🏆
Python · pls4all
pls4all.python✓ J 0.801.69 ms
pls4all.sklearn✓ J 0.801.89 ms
R · pls4all
pls4all.R✓ J 0.805.03 ms
pls4all.R.formula✓ J 0.805.50 ms
pls4all.R.mdatools✓ J 0.805.35 ms
pls4all.R.pls✓ J 0.805.34 ms
R · external
📐ref.r_plsvarselsource35.9 ms
BackendParity200×40 (ms)
C++ native · libn4m
pls4all.cpp.blas+omp⇄ J 0.801.66 ms🏆
Python · pls4all
pls4all.python✓ J 0.801.76 ms
pls4all.sklearn✓ J 0.801.83 ms
R · pls4all
pls4all.R✓ J 0.804.63 ms
pls4all.R.formula✓ J 0.805.24 ms
pls4all.R.mdatools✓ J 0.805.78 ms
pls4all.R.pls✓ J 0.805.58 ms
R · external
📐ref.r_plsvarselsource35.4 ms

See also: benchmark overview · methods index · interactive dashboard