# `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` | 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_t2_select`.
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_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);
```
:::
:::{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} 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 ≤ 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`](../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 · ✗ 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 ≤ 1e+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
| Backend | Parity | 50×250 (ms) | 100×50 (ms) | 100×500 (ms) | 100×2500 (ms) | 200×40 (ms) | 250×50 (ms) | 500×50 (ms) | 500×500 (ms) | 500×2500 (ms) | 2500×50 (ms) | 2500×500 (ms) | 2500×2500 (ms) | 10000×50 (ms) | 10000×500 (ms) |
|---|
| C++ native · libn4m |
|---|
pls4all.cpp.blas | ≈ +4e-01 | 2.91 ms | 8.20 ms | 42.2 ms | 183.5 ms🏆 | 1.71 ms | 3.08 ms | 25.4 ms | 126.7 ms🏆 | 986.3 ms | 118.5 ms | 1.1 s | 5.9 s🏆 | 499.9 ms🏆 | 4.2 s🏆 |
pls4all.cpp.blas+omp | ≈ +4e-01 | 2.78 ms | 8.15 ms🏆 | 39.5 ms | 219.4 ms | 1.68 ms🏆 | 2.81 ms | 24.6 ms🏆 | 200.8 ms | 959.2 ms🏆 | 100.7 ms🏆 | 1.1 s | 7.0 s | 503.5 ms | 4.7 s |
pls4all.cpp.omp | ≈ +4e-01 | 2.55 ms🏆 | 8.76 ms | 37.5 ms🏆 | 247.2 ms | 1.70 ms | 2.88 ms | 24.6 ms | 177.1 ms | 1.2 s | 123.9 ms | 996.1 ms🏆 | 6.2 s | 562.0 ms | 4.6 s |
pls4all.cpp.ref | ≈ +4e-01 | 2.67 ms | 8.27 ms | 41.2 ms | 202.6 ms | 1.77 ms | 2.99 ms | 24.8 ms | 181.5 ms | 982.8 ms | 120.1 ms | 1.0 s | 6.1 s | 536.9 ms | 4.4 s |
| Python · pls4all |
|---|
pls4all.python | ✓ bind | 2.79 ms | — | — | — | 1.75 ms | 2.77 ms🏆 | — | — | — | — | — | — | — | — |
pls4all.sklearn | ✓ bind | 3.10 ms | — | — | — | 2.31 ms | 2.96 ms | — | — | — | — | — | — | — | — |
| R · pls4all |
|---|
pls4all.R | ✓ bind | 12.0 ms | — | — | — | 5.63 ms | 12.4 ms | — | — | — | — | — | — | — | — |
pls4all.R.formula | ✓ bind | 20.1 ms | — | — | — | 8.64 ms | 10.3 ms | — | — | — | — | — | — | — | — |
pls4all.R.mdatools | ✓ bind | 19.8 ms | — | — | — | 7.14 ms | 10.6 ms | — | — | — | — | — | — | — | — |
pls4all.R.pls | ✓ bind | 21.3 ms | — | — | — | 7.30 ms | 9.58 ms | — | — | — | — | — | — | — | — |
| MATLAB · pls4all |
|---|
pls4all.matlab | ✗ +1e+00 | 4.24 ms | — | — | — | 2.93 ms | 4.27 ms | — | — | — | — | — | — | — | — |
pls4all.matlab.classdef | ✗ +1e+00 | 6.80 ms | — | — | — | 3.32 ms | 5.53 ms | — | — | — | — | — | — | — | — |
| R · external |
|---|
📐ref.r_plsvarsel | source | 102.7 ms | — | — | — | 44.6 ms | 42.2 ms | — | — | — | — | — | — | — | — |
:::
:::{tab-item} 3 threads
:sync: threads-3
| Backend | Parity | 50×250 (ms) | 100×50 (ms) | 100×500 (ms) | 100×2500 (ms) | 200×40 (ms) | 250×50 (ms) | 500×50 (ms) | 500×500 (ms) | 500×2500 (ms) | 2500×50 (ms) | 2500×500 (ms) | 2500×2500 (ms) | 10000×50 (ms) | 10000×500 (ms) |
|---|
| C++ native · libn4m |
|---|
pls4all.cpp.blas | ~ shape 4e-01 | — | — | — | — | 2.79 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.blas+omp | ~ shape 4e-01 | — | — | — | — | 1.89 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.omp | ~ shape 4e-01 | — | — | — | — | 1.72 ms🏆 | — | — | — | — | — | — | — | — | — |
pls4all.cpp.ref | ~ shape 4e-01 | — | — | — | — | 2.66 ms | — | — | — | — | — | — | — | — | — |
| Python · pls4all |
|---|
pls4all.python | ✓ bind | — | — | — | — | 2.64 ms | — | — | — | — | — | — | — | — | — |
pls4all.sklearn | ✓ bind | — | — | — | — | 2.35 ms | — | — | — | — | — | — | — | — | — |
| R · pls4all |
|---|
pls4all.R | ✓ bind | — | — | — | — | 4.94 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.formula | ✓ bind | — | — | — | — | 6.84 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.mdatools | ✓ bind | — | — | — | — | 6.29 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.pls | ✓ bind | — | — | — | — | 6.48 ms | — | — | — | — | — | — | — | — | — |
| MATLAB · pls4all |
|---|
pls4all.matlab | ✗ +1e+00 | — | — | — | — | 2.68 ms | — | — | — | — | — | — | — | — | — |
pls4all.matlab.classdef | ✗ +1e+00 | — | — | — | — | 3.08 ms | — | — | — | — | — | — | — | — | — |
| R · external |
|---|
📐ref.r_plsvarsel | source | — | — | — | — | 40.6 ms | — | — | — | — | — | — | — | — | — |
:::
:::{tab-item} 10 threads
:sync: threads-10
| Backend | Parity | 50×250 (ms) | 100×50 (ms) | 100×500 (ms) | 100×2500 (ms) | 200×40 (ms) | 250×50 (ms) | 500×50 (ms) | 500×500 (ms) | 500×2500 (ms) | 2500×50 (ms) | 2500×500 (ms) | 2500×2500 (ms) | 10000×50 (ms) | 10000×500 (ms) |
|---|
| C++ native · libn4m |
|---|
pls4all.cpp.blas | ~ shape 4e-01 | — | — | — | — | 1.67 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.blas+omp | ~ shape 4e-01 | — | — | — | — | 1.62 ms🏆 | — | — | — | — | — | — | — | — | — |
pls4all.cpp.omp | ~ shape 4e-01 | — | — | — | — | 1.64 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.ref | ~ shape 4e-01 | — | — | — | — | 1.65 ms | — | — | — | — | — | — | — | — | — |
| Python · pls4all |
|---|
pls4all.python | ✓ bind | — | — | — | — | 1.65 ms | — | — | — | — | — | — | — | — | — |
pls4all.sklearn | ✓ bind | — | — | — | — | 1.82 ms | — | — | — | — | — | — | — | — | — |
| R · pls4all |
|---|
pls4all.R | ✓ bind | — | — | — | — | 4.08 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.formula | ✓ bind | — | — | — | — | 4.94 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.mdatools | ✓ bind | — | — | — | — | 5.22 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.pls | ✓ bind | — | — | — | — | 4.91 ms | — | — | — | — | — | — | — | — | — |
| MATLAB · pls4all |
|---|
pls4all.matlab | ✗ +1e+00 | — | — | — | — | 2.46 ms | — | — | — | — | — | — | — | — | — |
pls4all.matlab.classdef | ✗ +1e+00 | — | — | — | — | 2.77 ms | — | — | — | — | — | — | — | — | — |
| R · external |
|---|
📐ref.r_plsvarsel | source | — | — | — | — | 32.9 ms | — | — | — | — | — | — | — | — | — |
:::
::::
---
_See also_: [benchmark overview](../benchmarks/overview.md) · [methods index](index.md) · [interactive dashboard](../landing/dashboard.md)