# `one_se_rule` — One-SE rule for component selection
_Group_: **Diagnostic** · _Registry tolerance_: `1e-06`
## Description
One-SE component selection rule (§10)
> **Registry note** — R `pls::plsr(validation='CV', segment.type='consecutive', method='simpls', scale=FALSE)` + onesigma rule. pls4all's wrapper runs the same consecutive-fold CV with a SIMPLS kernel matching `pls::simpls.fit` bit-for-bit, then feeds the pooled per-component RMSEP into the C-side `n4m_one_se_rule_compute`. Per-component CV-RMSEP vectors agree to ~1e-12.
### Parameters
| Name | Type | Default | Notes |
|------|------|---------|-------|
| `max_components` | `int` | `8` | registry benchmark cell value |
| `n_folds` | `int` | `5` | registry benchmark cell value |
## Explanations
### Bibliographic source
Hastie, T., Tibshirani, R. & Friedman, J. (2009). *The Elements of Statistical Learning*, 2nd ed., Springer, §7.10.
### Mathematical principle
Cross-validated RMSE as a function of $k$ is typically U-shaped with a relatively flat minimum. Picking the absolute minimum $k^{\star}$ can over-fit because it exploits sampling noise. The one-SE rule instead picks the **smallest** $k$ whose CV-RMSE is within one standard error of $\mathrm{RMSE}(k^{\star})$.
This yields a more parsimonious model with negligible predictive cost — the smaller-$k$ alternative is indistinguishable from the optimum within the noise of the CV estimate. The rule is non-parametric (no assumption about the CV-RMSE distribution) and is the standard practice in regularised regression (`glmnet`, `pls::pls`).
Inputs: a fold × component RMSE matrix from cross-validation. Output: an integer component count.
### Implementation
`n4m_one_se_rule_compute`. Returns an integer.
MATLAB header (`bindings/matlab/+pls4all/one_se_rule.m`):
```text
pls4all.one_se_rule One-SE component selection from a fold RMSE matrix.
fold_rmse_matrix: (max_components × n_folds) matrix of fold RMSE values.
```
### 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_one_se_rule_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);
```
:::
:::{tab-item} Python · pls4all (raw)
:sync: python-raw
:class-label: lang-python
```python
import pls4all
from pls4all._methods import one_se_rule_compute
with pls4all.Context() as ctx, pls4all.Config() as cfg:
res = one_se_rule_compute(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 one_se_rule
result = one_se_rule(X, y, n_components=2)
```
:::
:::{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("one_se_rule", X, y,
n_components = 2L, params = list(max_components = 8L, n_folds = 5L))
# 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.one_se_rule(X, y, 2);
% see header of bindings/matlab/+pls4all/one_se_rule.m for full
% parameter surface:
% res = one_se_rule(fold_rmse_matrix)
yhat = predict(res, Xtest);
```
:::
:::{tab-item} MATLAB · pls4all (classdef)
:sync: matlab-classdef
:class-label: lang-matlab
_No idiomatic classdef wrapper — invoke `pls4all.fit("one_se_rule", X, y, …)` directly from the unified MEX factory._
:::
::::
**Registry parity references** 📐
:::{card}
:class-card: external-refs
- 📐 **`ref.r_pls`** (R · r) — `pls` 2.8.5 · strict (rmse_rel ≤ 1e-06) — R `pls::plsr(validation='CV', segment.type='consecutive', method='simpls', scale=FALSE)` + `pls::selectNcomp(method='onesigma')`. The pls4all wrapper performs the same consecutive-fold CV with a SIMPLS kernel matching `pls::simpls.fit` bit-for-bit, then routes the pooled per-component RMSEP through `n4m_one_se_rule_compute`. We compare `mean_rmse_per_component` directly.
:::
### 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) | 200×30 (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 | ✗ +7e-02 | 3.54 ms | 5.06 ms | 14.8 ms | 69.7 ms | 2.33 ms | 3.63 ms | 7.88 ms | 65.2 ms🏆 | 353.9 ms | 38.6 ms | 368.1 ms | 2.0 s | 131.7 ms🏆 | 1.4 s🏆 |
pls4all.cpp.blas+omp | ✗ +7e-02 | 3.48 ms🏆 | 3.35 ms | 15.4 ms | 70.9 ms | 2.22 ms | 3.62 ms | 9.38 ms | 66.9 ms | 332.0 ms🏆 | 33.5 ms🏆 | 355.9 ms🏆 | 2.0 s🏆 | 148.2 ms | 1.5 s |
pls4all.cpp.omp | ✗ +7e-02 | 3.70 ms | 4.34 ms | 12.9 ms🏆 | 64.5 ms🏆 | 2.23 ms | 3.60 ms🏆 | 7.08 ms🏆 | 72.1 ms | 343.7 ms | 39.8 ms | 363.5 ms | 2.0 s | 140.1 ms | 1.4 s |
pls4all.cpp.ref | ✗ +7e-02 | 3.49 ms | 2.48 ms | 15.5 ms | 65.3 ms | 2.21 ms🏆 | 4.19 ms | 8.58 ms | 67.2 ms | 341.4 ms | 38.0 ms | 357.3 ms | 2.1 s | 139.8 ms | 1.6 s |
| Python · pls4all |
|---|
pls4all.python | ✓ bind | 3.75 ms | — | — | — | 2.27 ms | 3.71 ms | — | — | — | — | — | — | — | — |
pls4all.sklearn | ✗ +1e+00 | 2.33 ms | — | — | — | 1.25 ms | 2.36 ms | — | — | — | — | — | — | — | — |
| R · pls4all |
|---|
pls4all.R | ✗ +1e+00 | 14.1 ms | — | — | — | 4.00 ms | 11.5 ms | — | — | — | — | — | — | — | — |
pls4all.R.formula | ✗ +1e+00 | 22.0 ms | — | — | — | 5.72 ms | 12.4 ms | — | — | — | — | — | — | — | — |
pls4all.R.mdatools | ✗ +1e+00 | 21.6 ms | — | — | — | 5.41 ms | 12.4 ms | — | — | — | — | — | — | — | — |
pls4all.R.pls | ✗ +1e+00 | 23.5 ms | — | — | — | 4.93 ms | 12.8 ms | — | — | — | — | — | — | — | — |
| MATLAB · pls4all |
|---|
pls4all.matlab | ✗ +1e+00 | 4.20 ms | — | — | — | 2.25 ms | 4.80 ms | — | — | — | — | — | — | — | — |
pls4all.matlab.classdef | ✗ +1e+00 | 5.44 ms | — | — | — | 2.51 ms | 4.97 ms | — | — | — | — | — | — | — | — |
| R · external |
|---|
📐ref.r_pls | source | 25.9 ms | — | — | — | 15.1 ms | 16.2 ms | — | — | — | — | — | — | — | — |
:::
:::{tab-item} 3 threads
:sync: threads-3
| Backend | Parity | 50×250 (ms) | 100×50 (ms) | 100×500 (ms) | 100×2500 (ms) | 200×30 (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 4e-14 | — | — | — | — | 2.43 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.blas+omp | ✓ ref 4e-14 | — | — | — | — | 2.34 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.omp | ✓ ref 4e-14 | — | — | — | — | 2.17 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.ref | ✓ ref 4e-14 | — | — | — | — | 2.83 ms | — | — | — | — | — | — | — | — | — |
| Python · pls4all |
|---|
pls4all.python | ✓ bind | — | — | — | — | 2.15 ms🏆 | — | — | — | — | — | — | — | — | — |
pls4all.sklearn | ✗ +9e-01 | — | — | — | — | 1.25 ms | — | — | — | — | — | — | — | — | — |
| R · pls4all |
|---|
pls4all.R | ✗ +9e-01 | — | — | — | — | 3.62 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.formula | ✗ +9e-01 | — | — | — | — | 5.25 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.mdatools | ✗ +9e-01 | — | — | — | — | 4.93 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.pls | ✗ +9e-01 | — | — | — | — | 5.07 ms | — | — | — | — | — | — | — | — | — |
| MATLAB · pls4all |
|---|
pls4all.matlab | ✗ +9e-01 | — | — | — | — | 2.21 ms | — | — | — | — | — | — | — | — | — |
pls4all.matlab.classdef | ✗ +9e-01 | — | — | — | — | 3.11 ms | — | — | — | — | — | — | — | — | — |
| R · external |
|---|
📐ref.r_pls | source | — | — | — | — | 13.7 ms | — | — | — | — | — | — | — | — | — |
:::
:::{tab-item} 10 threads
:sync: threads-10
| Backend | Parity | 50×250 (ms) | 100×50 (ms) | 100×500 (ms) | 100×2500 (ms) | 200×30 (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 4e-14 | — | — | — | — | 2.04 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.blas+omp | ✓ ref 4e-14 | — | — | — | — | 2.05 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.omp | ✓ ref 4e-14 | — | — | — | — | 2.03 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.ref | ✓ ref 4e-14 | — | — | — | — | 1.99 ms🏆 | — | — | — | — | — | — | — | — | — |
| Python · pls4all |
|---|
pls4all.python | ✓ bind | — | — | — | — | 2.08 ms | — | — | — | — | — | — | — | — | — |
pls4all.sklearn | ✗ +9e-01 | — | — | — | — | 1.09 ms | — | — | — | — | — | — | — | — | — |
| R · pls4all |
|---|
pls4all.R | ✗ +9e-01 | — | — | — | — | 3.10 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.formula | ✗ +9e-01 | — | — | — | — | 3.67 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.mdatools | ✗ +9e-01 | — | — | — | — | 3.53 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.pls | ✗ +9e-01 | — | — | — | — | 3.67 ms | — | — | — | — | — | — | — | — | — |
| MATLAB · pls4all |
|---|
pls4all.matlab | ✗ +9e-01 | — | — | — | — | 1.87 ms | — | — | — | — | — | — | — | — | — |
pls4all.matlab.classdef | ✗ +9e-01 | — | — | — | — | 2.15 ms | — | — | — | — | — | — | — | — | — |
| R · external |
|---|
📐ref.r_pls | source | — | — | — | — | 12.4 ms | — | — | — | — | — | — | — | — | — |
:::
::::
---
_See also_: [benchmark overview](../benchmarks/overview.md) · [methods index](index.md) · [interactive dashboard](../landing/dashboard.md)