# `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_metrics_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_metrics_one_se_rule_compute`. Returns an integer.
R roxygen note (`methods_extra.R::one_se_rule`):
> One-SE rule from a (max_components × n_folds) fold RMSE matrix.
> @param fold_rmse_matrix Method-specific parameter. See the underlying `*_fit()` function for the exact semantics.
> @export
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_metrics_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} R · pls4all (raw fn)
:sync: r-raw
:class-label: lang-r
```r
library(pls4all)
res <- one_se_rule(fold_rmse_matrix)
yhat <- pls4all_predict(res, X_test)
```
:::
:::{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_metrics_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 · ⇄ 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**: 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 | 200×30 (ms) |
|---|
| C++ native · libn4m |
|---|
pls4all.cpp.blas+omp | ✓ ref 4e-14 | 2.10 ms |
| Python · pls4all |
|---|
pls4all.python | ✓ bind | 2.08 ms |
pls4all.sklearn | ⇄ +9e-01 | 1.13 ms🏆 |
| R · pls4all |
|---|
pls4all.R | ⇄ +9e-01 | 3.25 ms |
pls4all.R.formula | ⇄ +9e-01 | 3.87 ms |
pls4all.R.mdatools | ⇄ +9e-01 | 3.96 ms |
pls4all.R.pls | ⇄ +9e-01 | 4.01 ms |
| R · external |
|---|
📐ref.r_pls | source | 11.6 ms |
:::
:::{tab-item} 3 threads
:sync: threads-3
| Backend | Parity | 200×30 (ms) |
|---|
| C++ native · libn4m |
|---|
pls4all.cpp.blas+omp | ✓ ref 4e-14 | 7.61 ms |
| Python · pls4all |
|---|
pls4all.python | ✓ bind | 2.19 ms |
pls4all.sklearn | ⇄ +9e-01 | 1.23 ms🏆 |
| R · pls4all |
|---|
pls4all.R | ⇄ +9e-01 | 3.28 ms |
pls4all.R.formula | ⇄ +9e-01 | 4.07 ms |
pls4all.R.mdatools | ⇄ +9e-01 | 4.07 ms |
pls4all.R.pls | ⇄ +9e-01 | 4.00 ms |
| R · external |
|---|
📐ref.r_pls | source | 11.6 ms |
:::
:::{tab-item} 10 threads
:sync: threads-10
| Backend | Parity | 200×30 (ms) |
|---|
| C++ native · libn4m |
|---|
pls4all.cpp.blas+omp | ✓ ref 4e-14 | 2.06 ms |
| Python · pls4all |
|---|
pls4all.python | ✓ bind | 2.14 ms |
pls4all.sklearn | ⇄ +9e-01 | 1.13 ms🏆 |
| R · pls4all |
|---|
pls4all.R | ⇄ +9e-01 | 2.80 ms |
pls4all.R.formula | ⇄ +9e-01 | 5.64 ms |
pls4all.R.mdatools | ⇄ +9e-01 | 4.37 ms |
pls4all.R.pls | ⇄ +9e-01 | 4.60 ms |
| R · external |
|---|
📐ref.r_pls | source | 11.5 ms |
:::
::::
---
_See also_: [benchmark overview](../benchmarks/overview.md) · [methods index](index.md) · [interactive dashboard](../landing/dashboard.md)