# `t2_select` — Hotelling T² loading selection _Group_: **Variable selector** · _Registry tolerance_: `2.0` ## 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` (Mehmood 2016) Hotelling-T² loading-weight selection, called with train=test so it matches pls4all's single-training-set selector. VERIFIED IDENTICAL where it matters: pls4all's per-feature T², the UCL = ((p-1)^2/p)*qbeta(1-alpha, a/2, (p-a-1)/2), the sample covariance over features (/(p-1)), the t2>ucl threshold, the top-k floor fallback, and the min-CV-error alpha pick are all bit-identical to plsVarSel t2_calc / T2_pls. The residual mask divergence is an UPSTREAM PLS loading-weight convention difference: plsVarSel computes T² from R `pls::plsr` `loading.weights` (== sklearn `x_weights_`), while pls4all computes T² from the fitted model's `weights_w`; on borderline features near the UCL the two select slightly different sets (Jaccard ~0.7–1.0; both capture the true signal). Both are valid T²-PLS selectors. Gated on Jaccard via the orchestrator SELECTION_DIVERGENCE_ALLOWLIST. ### 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` | Lower bound on the surviving feature count after thresholding. | ## 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_feature_selection_t2_select`. R roxygen note (`methods_extra.R::t2_select`): > T2-PLS - sweep over alpha thresholds. > @param n_components Integer. Number of latent components. > @param alpha_thresholds Method-specific parameter. See the underlying `*_fit()` function for the exact semantics. > @param min_selected Method-specific parameter. See the underlying `*_fit()` function for the exact semantics. > @param X Numeric matrix of predictors (rows = samples, cols = features). > @param Y Numeric matrix or vector of responses, with one row per sample. > @export MATLAB header (`bindings/matlab/+pls4all/t2_select.m`): ```text 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** ::::{tab-set} :class: pls4all-bindings :::{tab-item} C ABI · libn4m :sync: c :class-label: lang-c ```c /* C ABI — libn4m */ n4m_context_t* ctx = n4m_context_create(); n4m_config_t* cfg = n4m_config_create(); n4m_method_result_t* res = NULL; n4m_feature_selection_t2_select(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); ``` ::: :::{tab-item} Python · pls4all (raw) :sync: python-raw :class-label: lang-python ```python 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"), … ``` ::: :::{tab-item} Python · pls4all.sklearn :sync: python-sklearn :class-label: lang-python ```python 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) ``` ::: :::{tab-item} R · pls4all_method() :sync: r-dispatcher :class-label: lang-r ```r 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. ``` ::: :::{tab-item} R · pls4all (raw fn) :sync: r-raw :class-label: lang-r ```r library(pls4all) res <- t2_select(X, Y, n_components, alpha_thresholds, min_selected = NULL) yhat <- pls4all_predict(res, X_test) ``` ::: :::{tab-item} MATLAB · pls4all (MEX) :sync: matlab-mex :class-label: lang-matlab ```matlab 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); ``` ::: :::{tab-item} MATLAB · pls4all (classdef) :sync: matlab-classdef :class-label: lang-matlab _No idiomatic classdef wrapper — invoke `pls4all.fit("t2_select", X, y, …)` directly from the unified MEX factory._ ::: :::: **Registry parity references** 📐 :::{card} :class-card: external-refs - 📐 **`ref.r_plsvarsel`** (R · r) — `plsVarSel` 0.10.0 · qualitative (rmse_rel ≤ 2e+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`](../benchmarks/overview.md). 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`](../benchmarks/methodology.md)). 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. ::::{tab-set} :class: parity-tabs :::{tab-item} 1 thread :sync: threads-1
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
::: :::{tab-item} 3 threads :sync: threads-3
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
::: :::{tab-item} 10 threads :sync: threads-10
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](../benchmarks/overview.md) · [methods index](index.md) · [interactive dashboard](../landing/dashboard.md)