`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):

pls4all.RobustPlsRegression  Robust PLS via Huber IRLS.

Usage¶

Direct n4m Python helper:

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

C ABI · libn4m

/* 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);

Python · pls4all (raw)

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"), …

Python · pls4all.sklearn

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)

R · pls4all_method()

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.

MATLAB · pls4all (MEX)

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);

MATLAB · pls4all (classdef)

mdl  = pls4all.fit("robust_pls", X, y, "NumComponents", 4);
yhat = predict(mdl, Xtest);

Registry parity references 📐

📐 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. 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-08).

Rows tagged with 📐 are the canonical parity references for this method (declared in parity_timing.registry). 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.

1 thread

Backend	Parity	200×50 (ms)
C++ native · libn4m
`pls4all.cpp.blas+omp`	✓ ref 1e-14	2.07 ms
Python · pls4all
`pls4all.python`	✓ bind	1.95 ms🏆
`pls4all.sklearn`	✓ bind	2.15 ms
R · pls4all
`pls4all.R`	✓ bind	4.38 ms
`pls4all.R.formula`	✓ bind	5.55 ms
`pls4all.R.mdatools`	✓ bind	5.77 ms
`pls4all.R.pls`	✓ bind	6.62 ms
R · external
📐`ref.r_chemometrics`	source	17.9 ms

3 threads

Backend	Parity	200×50 (ms)
C++ native · libn4m
`pls4all.cpp.blas+omp`	✓ ref 1e-14	2.02 ms
Python · pls4all
`pls4all.python`	✓ bind	2.00 ms🏆
`pls4all.sklearn`	✓ bind	2.10 ms
R · pls4all
`pls4all.R`	✓ bind	5.43 ms
`pls4all.R.formula`	✓ bind	6.63 ms
`pls4all.R.mdatools`	✓ bind	5.84 ms
`pls4all.R.pls`	✓ bind	5.99 ms
R · external
📐`ref.r_chemometrics`	source	17.0 ms

10 threads

Backend	Parity	200×50 (ms)
C++ native · libn4m
`pls4all.cpp.blas+omp`	✓ ref 1e-14	2.05 ms
Python · pls4all
`pls4all.python`	✓ bind	1.98 ms🏆
`pls4all.sklearn`	✓ bind	2.25 ms
R · pls4all
`pls4all.R`	✓ bind	4.94 ms
`pls4all.R.formula`	✓ bind	7.07 ms
`pls4all.R.mdatools`	✓ bind	5.59 ms
`pls4all.R.pls`	✓ bind	5.68 ms
R · external
📐`ref.r_chemometrics`	source	17.5 ms

nirs4all-methods

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`robust_pls` — Robust PLS (Partial Robust M-regression)¶

Description¶

Parameters¶

Explanations¶

Bibliographic source¶

Mathematical principle¶

Implementation¶

Usage¶

Benchmarks¶

robust_pls — Robust PLS (Partial Robust M-regression)¶

Description¶

Parameters¶

Explanations¶

Bibliographic source¶

Mathematical principle¶

Implementation¶

Usage¶

Benchmarks¶

`robust_pls` — Robust PLS (Partial Robust M-regression)¶