# `robust_pls` — Robust PLS (Partial Robust M-regression)
_Group_: **Robust / weighted** · _Registry tolerance_: `0.1`
## Description
Robust PLS (Partial Robust M-regression, Serneels 2005)
From the `pls4all.sklearn.RobustPLSRegression` docstring:
> Robust PLS via Huber IRLS over weighted SIMPLS.
> **Registry note** — R `chemometrics::prm` (Serneels et al. 2005) — Partial Robust M-regression. pls4all defaults to PRM matching the R algorithm bit-for-bit (median centering, Fair weights on leverage + residual, univariate SIMPLS inner kernel, intercept = median(y - X@b)). The legacy Huber-IRLS over weighted SIMPLS path is reachable via ``cfg.robust_pls_legacy = 1``.
### Parameters
| Name | Type | Default | Notes |
|------|------|---------|-------|
| `n_components` | `int` | `2` | Number of latent components extracted (k). |
| `huber_k` | `float` | `1.345` | Huber threshold (in residual-stdev units) controlling IRLS reweighting; smaller = more robust. |
| `max_irls_iter` | `int` | `5` | Maximum IRLS reweighting iterations. |
## Explanations
### Bibliographic source
Serneels, S., Croux, C., Filzmoser, P. & Van Espen, P. J. (2005). *Partial Robust M-Regression*. Chemometrics and Intelligent Laboratory Systems 79(1–2), 55–64.
### Mathematical principle
Robust PLS performs a sequence of weighted PLS fits where the weights $w_i$ are reduced for samples with large current residuals — the iteratively-reweighted least-squares (IRLS) recipe.
At each iteration, the weight of sample $i$ is set from Huber's $\psi$-function applied to the standardised residual: $w_i = \psi(r_i / s) / r_i$ where $\psi(z) = z$ for $|z| \le k$ and $\psi(z) = k\,\operatorname{sign}(z)$ otherwise. $k = 1.345$ gives 95 % asymptotic efficiency at the Gaussian. The robust scale $s$ is typically the MAD of the residuals.
Convergence is rapid: 3–5 iterations typically suffice. Robust PLS down-weights — rather than removes — outliers, which is desirable when outlier-ness is a continuous concept (mild spectral artefacts) rather than binary (broken samples).
Compared to median-PLS variants, the M-regression form preserves the analytic structure of SIMPLS and offers smooth weighting; it also generalises to leverage-based weights (PRM with x-weights).
### Implementation
`n4m_robust_pls_fit`. Reference: CRAN `chemometrics::prm` (Serneels et al. authors). The exact weight schedule and scale estimator differ from `prm` so RMSE-rel parity is widened to ~2.0 to flag presence rather than enforce exact agreement.
MATLAB header (`bindings/matlab/+pls4all/RobustPlsRegression.m`):
```text
pls4all.RobustPlsRegression Robust PLS via Huber IRLS.
```
### Usage
Direct `n4m` Python helper:
```python
import n4m
res = n4m.robust_pls(
X,
y,
n_components=4,
huber_k=1.345,
max_irls_iter=5,
)
y_hat = res["predictions"]
coef = res["coefficients"]
```
The `n4m.sklearn.NativeRobustPLSRegressor` wrapper replays predictions from the
returned coefficients plus reconstructed intercept.
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_robust_pls_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 robust_pls_fit
with pls4all.Context() as ctx, pls4all.Config() as cfg:
res = robust_pls_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 RobustPLSRegression
mdl = RobustPLSRegression(n_components=2, huber_k=1.345, max_irls_iter=20)
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("robust_pls", X, y,
n_components = 4L, params = list(huber_k = 4.0, max_irls_iter = 30L))
# 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.robust_pls(X, y, 4);
% see header of bindings/matlab/+pls4all/robust_pls.m for full
% parameter surface:
% res = robust_pls(X, Y, n_components, huber_k, max_irls_iter)
yhat = predict(res, Xtest);
```
:::
:::{tab-item} MATLAB · pls4all (classdef)
:sync: matlab-classdef
:class-label: lang-matlab
```matlab
mdl = pls4all.fit("robust_pls", X, y, "NumComponents", 4);
yhat = predict(mdl, Xtest);
```
:::
::::
**Registry parity references** 📐
:::{card}
:class-card: external-refs
- 📐 **`ref.r_chemometrics`** (R · r) — `chemometrics` 0.7.x · qualitative (rmse_rel ≤ 1e-01) — R `chemometrics::prm` (Partial Robust M-regression). pls4all uses Huber IRLS over weighted SIMPLS; this is an M-estimator variant from the same family. Predictions diverge by O(0.5).
:::
### 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-01` 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) | 200×50 (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-13 | 2.78 ms🏆 | 1.24 ms | 10.4 ms🏆 | 67.6 ms | 2.10 ms🏆 | 2.90 ms | 5.30 ms | 60.6 ms | 331.0 ms | 30.2 ms | 370.8 ms🏆 | 2.0 s | 128.8 ms | 1.8 s🏆 |
pls4all.cpp.blas+omp | ≈ +4e-13 | 2.94 ms | 1.22 ms | 11.7 ms | 65.0 ms🏆 | 2.21 ms | 2.75 ms🏆 | 5.52 ms | 60.6 ms | 320.0 ms | 27.6 ms🏆 | 378.8 ms | 1.9 s🏆 | 125.5 ms | 1.8 s |
pls4all.cpp.omp | ≈ +4e-13 | 2.94 ms | 1.21 ms🏆 | 11.6 ms | 65.0 ms | 2.14 ms | 2.88 ms | 5.09 ms🏆 | 57.7 ms🏆 | 341.6 ms | 32.7 ms | 381.1 ms | 2.0 s | 122.6 ms | 1.8 s |
pls4all.cpp.ref | ≈ +4e-13 | 2.92 ms | 1.46 ms | 10.9 ms | 69.5 ms | 2.15 ms | 2.82 ms | 5.27 ms | 62.2 ms | 316.5 ms🏆 | 30.0 ms | 376.4 ms | 2.0 s | 114.5 ms🏆 | 1.8 s |
| Python · pls4all |
|---|
pls4all.python | ✓ bind | 3.05 ms | — | — | — | 2.28 ms | 2.88 ms | — | — | — | — | — | — | — | — |
pls4all.sklearn | ✓ bind | 3.57 ms | — | — | — | 2.66 ms | 3.76 ms | — | — | — | — | — | — | — | — |
| R · pls4all |
|---|
pls4all.R | ✗ +4e+00 | 12.4 ms | — | — | — | 8.97 ms | 13.7 ms | — | — | — | — | — | — | — | — |
pls4all.R.formula | ✗ +4e+00 | 21.0 ms | — | — | — | 10.1 ms | 12.1 ms | — | — | — | — | — | — | — | — |
pls4all.R.mdatools | ✗ +4e+00 | 20.5 ms | — | — | — | 10.3 ms | 13.1 ms | — | — | — | — | — | — | — | — |
pls4all.R.pls | ✗ +4e+00 | 22.2 ms | — | — | — | 12.2 ms | 11.0 ms | — | — | — | — | — | — | — | — |
| MATLAB · pls4all |
|---|
pls4all.matlab | ✗ +9e+00 | 4.67 ms | — | — | — | 3.38 ms | 5.55 ms | — | — | — | — | — | — | — | — |
pls4all.matlab.classdef | ✗ +9e+00 | 5.54 ms | — | — | — | 4.27 ms | 5.09 ms | — | — | — | — | — | — | — | — |
| R · external |
|---|
📐ref.r_chemometrics | source | 39.2 ms | — | — | — | 25.6 ms | 22.9 ms | — | — | — | — | — | — | — | — |
:::
:::{tab-item} 3 threads
:sync: threads-3
| Backend | Parity | 50×250 (ms) | 100×50 (ms) | 100×500 (ms) | 100×2500 (ms) | 200×50 (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 1e-14 | — | — | — | — | 3.29 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.blas+omp | ~ shape 1e-14 | — | — | — | — | 3.29 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.omp | ~ shape 1e-14 | — | — | — | — | 3.31 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.ref | ~ shape 1e-14 | — | — | — | — | 2.18 ms🏆 | — | — | — | — | — | — | — | — | — |
| Python · pls4all |
|---|
pls4all.python | ✓ bind | — | — | — | — | 2.27 ms | — | — | — | — | — | — | — | — | — |
pls4all.sklearn | ✓ bind | — | — | — | — | 2.46 ms | — | — | — | — | — | — | — | — | — |
| R · pls4all |
|---|
pls4all.R | ✗ +2e-01 | — | — | — | — | 8.00 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.formula | ✗ +2e-01 | — | — | — | — | 9.90 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.mdatools | ✗ +2e-01 | — | — | — | — | 9.35 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.pls | ✗ +2e-01 | — | — | — | — | 10.2 ms | — | — | — | — | — | — | — | — | — |
| MATLAB · pls4all |
|---|
pls4all.matlab | ✗ +9e+00 | — | — | — | — | 4.22 ms | — | — | — | — | — | — | — | — | — |
pls4all.matlab.classdef | ✗ +9e+00 | — | — | — | — | 4.93 ms | — | — | — | — | — | — | — | — | — |
| R · external |
|---|
📐ref.r_chemometrics | source | — | — | — | — | 24.3 ms | — | — | — | — | — | — | — | — | — |
:::
:::{tab-item} 10 threads
:sync: threads-10
| Backend | Parity | 50×250 (ms) | 100×50 (ms) | 100×500 (ms) | 100×2500 (ms) | 200×50 (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 1e-14 | — | — | — | — | 2.00 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.blas+omp | ~ shape 1e-14 | — | — | — | — | 2.01 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.omp | ~ shape 1e-14 | — | — | — | — | 2.03 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.ref | ~ shape 1e-14 | — | — | — | — | 1.97 ms🏆 | — | — | — | — | — | — | — | — | — |
| Python · pls4all |
|---|
pls4all.python | ✓ bind | — | — | — | — | 2.09 ms | — | — | — | — | — | — | — | — | — |
pls4all.sklearn | ✓ bind | — | — | — | — | 2.14 ms | — | — | — | — | — | — | — | — | — |
| R · pls4all |
|---|
pls4all.R | ✗ +2e-01 | — | — | — | — | 6.87 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.formula | ✗ +2e-01 | — | — | — | — | 7.85 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.mdatools | ✗ +2e-01 | — | — | — | — | 7.29 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.pls | ✗ +2e-01 | — | — | — | — | 7.34 ms | — | — | — | — | — | — | — | — | — |
| MATLAB · pls4all |
|---|
pls4all.matlab | ✗ +9e+00 | — | — | — | — | 3.07 ms | — | — | — | — | — | — | — | — | — |
pls4all.matlab.classdef | ✗ +9e+00 | — | — | — | — | 3.50 ms | — | — | — | — | — | — | — | — | — |
| R · external |
|---|
📐ref.r_chemometrics | source | — | — | — | — | 18.6 ms | — | — | — | — | — | — | — | — | — |
:::
::::
---
_See also_: [benchmark overview](../benchmarks/overview.md) · [methods index](index.md) · [interactive dashboard](../landing/dashboard.md)