# `randomization_select` — Randomisation test (Y-permutation)
_Group_: **Variable selector** · _Registry tolerance_: `1e-06`
## Description
Randomization-test selector (§18 Phase 5o)
From the `pls4all.sklearn.RandomizationSelector` docstring:
> Randomization-test PLS selector (Y-permutation p-values).
> **Registry note** — Base R: SIMPLS coefs vs permuted-Y null distribution. Default `_randomization_select_pls4all` path mirrors the same base-R permutation test with seed=randomization_seed, giving bit-exact mask parity. The C++ splitmix64 kernel is opt-in via `legacy=True`.
### Parameters
| Name | Type | Default | Notes |
|------|------|---------|-------|
| `n_components` | `int` | `2` | Number of latent components extracted (k). |
| `n_permutations` | `int` | `200` | Number of Y-permutations used to build the null distribution. |
| `randomization_seed` | `int` | `0` | Seed for the permutation generator. |
| `alpha` | `float` | `0.05` | Significance level for the permutation-based variable retention test. |
## Explanations
### Bibliographic source
Westad, F. & Martens, H. (2000). *Variable selection in near infrared spectroscopy based on significance testing in partial least squares regression*. JNIRS 8(2), 117–124.
### Mathematical principle
Compute the observed PLS coefficient magnitudes $|b_j^{\mathrm{obs}}|$, then permute $\mathbf{y}$ $M$ times, refit PLS each time, and collect $|b_j^{(m)}|$. The empirical p-value of feature $j$ is $p_j = \frac{1 + \#\{m : |b_j^{(m)}| \ge |b_j^{\mathrm{obs}}|\}}{1 + M}$. Retain features with $p_j < \alpha$.
Y-permutation is the gold standard for **null-calibrated** significance testing in PLS — no distributional assumptions, no asymptotic approximations. Cost is $M$× a fit but trivially parallelisable.
Critically, Y-permutation tests the joint hypothesis 'feature $j$ contributes to $y$'; multiple-testing correction (Benjamini-Hochberg) is recommended for $p \gg 100$.
### Implementation
`n4m_randomization_select`.
### 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_randomization_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 randomization_select_fit
with pls4all.Context() as ctx, pls4all.Config() as cfg:
res = randomization_select_fit(ctx, cfg, X, y, n_components=4)
# 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 RandomizationSelector
mdl = RandomizationSelector(n_components=2, n_permutations=200, randomization_seed=0, alpha=0.05)
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("randomization_select", X, y,
n_components = 4L, params = list(n_permutations = 50L, alpha = 0.05, randomization_seed = 11L))
# 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.fit("randomization_select", X, y, "NumComponents", 4);
yhat = predict(res, Xtest);
```
:::
:::{tab-item} MATLAB · pls4all (classdef)
:sync: matlab-classdef
:class-label: lang-matlab
_No idiomatic classdef wrapper — invoke `pls4all.fit("randomization_select", X, y, …)` directly from the unified MEX factory._
:::
::::
**Registry parity references** 📐
:::{card}
:class-card: external-refs
- 📐 **`ref.r_pls_stats`** (R · r) — `pls+stats` R 4.3.3 · strict (rmse_rel ≤ 1e-06) — Base R: SIMPLS coefficients vs permuted-Y null distribution. Selects features with empirical p-value < alpha. Same idea as pls4all's randomization_test selector.
:::
### 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**: strict — numeric equivalence (`rmse_rel_tol ≤ 1e-06`).
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 | ≈ | 249.8 ms | 2.0 s | 4.0 s | 15.6 s | 248.6 ms | 223.4 ms | 2.7 s🏆 | 12.3 s🏆 | 72.3 s | 7.0 s🏆 | 58.6 s🏆 | 376.1 s🏆 | 29.7 s | 323.2 s |
pls4all.cpp.blas+omp | ≈ | 239.8 ms | 2.2 s | 3.9 s🏆 | 15.1 s🏆 | 256.1 ms | 234.5 ms | 2.8 s | 12.4 s | 73.0 s | 7.6 s | 63.7 s | 389.5 s | 30.8 s | 315.6 s |
pls4all.cpp.omp | ≈ | 237.8 ms | 2.1 s | 4.1 s | 15.6 s | 262.9 ms | 240.7 ms | 2.9 s | 13.0 s | 73.5 s | 7.6 s | 63.4 s | 386.8 s | 30.4 s | 310.3 s |
pls4all.cpp.ref | ≈ | 247.6 ms | 1.9 s🏆 | 3.9 s | 15.8 s | 250.7 ms | 248.6 ms | 2.9 s | 12.5 s | 70.2 s🏆 | 7.4 s | 62.9 s | 388.7 s | 29.2 s🏆 | 309.3 s🏆 |
| Python · pls4all |
|---|
pls4all.python | ✓ bind | 244.8 ms | — | — | — | 255.3 ms | 249.1 ms | — | — | — | — | — | — | — | — |
pls4all.sklearn | ✗ +1e+00 | 5.31 ms | — | — | — | 3.02 ms🏆 | 4.87 ms🏆 | — | — | — | — | — | — | — | — |
| R · pls4all |
|---|
pls4all.R | ✗ +1e+00 | 14.6 ms | — | — | — | 9.01 ms | 13.3 ms | — | — | — | — | — | — | — | — |
pls4all.R.formula | ✗ +1e+00 | 24.5 ms | — | — | — | 8.06 ms | 15.0 ms | — | — | — | — | — | — | — | — |
pls4all.R.mdatools | ✗ +1e+00 | 23.9 ms | — | — | — | 8.96 ms | 12.7 ms | — | — | — | — | — | — | — | — |
pls4all.R.pls | ✗ +1e+00 | 28.7 ms | — | — | — | 8.86 ms | 12.5 ms | — | — | — | — | — | — | — | — |
| MATLAB · pls4all |
|---|
pls4all.matlab | ✗ +1e+00 | 9.37 ms | — | — | — | 4.00 ms | 9.89 ms | — | — | — | — | — | — | — | — |
pls4all.matlab.classdef | ✗ +1e+00 | 11.6 ms | — | — | — | 4.78 ms | 8.75 ms | — | — | — | — | — | — | — | — |
| R · external |
|---|
📐ref.r_pls_stats | source | 75.8 ms🏆 | — | — | — | 78.2 ms | 65.4 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 | ✓ ref | — | — | — | — | 257.4 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.blas+omp | ✓ ref | — | — | — | — | 252.7 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.omp | ✓ ref | — | — | — | — | 254.2 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.ref | ✓ ref | — | — | — | — | 255.0 ms | — | — | — | — | — | — | — | — | — |
| Python · pls4all |
|---|
pls4all.python | ✓ bind | — | — | — | — | 257.5 ms | — | — | — | — | — | — | — | — | — |
pls4all.sklearn | ✓ bind | — | — | — | — | 3.90 ms🏆 | — | — | — | — | — | — | — | — | — |
| R · pls4all |
|---|
pls4all.R | ✓ bind | — | — | — | — | 8.81 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.formula | ✓ bind | — | — | — | — | 8.15 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.mdatools | ✓ bind | — | — | — | — | 8.10 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.pls | ✓ bind | — | — | — | — | 8.19 ms | — | — | — | — | — | — | — | — | — |
| MATLAB · pls4all |
|---|
pls4all.matlab | ✓ bind | — | — | — | — | 4.19 ms | — | — | — | — | — | — | — | — | — |
pls4all.matlab.classdef | ✓ bind | — | — | — | — | 5.00 ms | — | — | — | — | — | — | — | — | — |
| R · external |
|---|
📐ref.r_pls_stats | source | — | — | — | — | 65.0 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 | ✓ ref | — | — | — | — | 224.5 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.blas+omp | ✓ ref | — | — | — | — | 224.3 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.omp | ✓ ref | — | — | — | — | 228.5 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.ref | ✓ ref | — | — | — | — | 223.1 ms | — | — | — | — | — | — | — | — | — |
| Python · pls4all |
|---|
pls4all.python | ✓ bind | — | — | — | — | 217.1 ms | — | — | — | — | — | — | — | — | — |
pls4all.sklearn | ✓ bind | — | — | — | — | 2.76 ms🏆 | — | — | — | — | — | — | — | — | — |
| R · pls4all |
|---|
pls4all.R | ✓ bind | — | — | — | — | 5.57 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.formula | ✓ bind | — | — | — | — | 6.62 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.mdatools | ✓ bind | — | — | — | — | 6.13 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.pls | ✓ bind | — | — | — | — | 6.36 ms | — | — | — | — | — | — | — | — | — |
| MATLAB · pls4all |
|---|
pls4all.matlab | ✓ bind | — | — | — | — | 3.47 ms | — | — | — | — | — | — | — | — | — |
pls4all.matlab.classdef | ✓ bind | — | — | — | — | 3.79 ms | — | — | — | — | — | — | — | — | — |
| R · external |
|---|
📐ref.r_pls_stats | source | — | — | — | — | 51.2 ms | — | — | — | — | — | — | — | — | — |
:::
::::
---
_See also_: [benchmark overview](../benchmarks/overview.md) · [methods index](index.md) · [interactive dashboard](../landing/dashboard.md)