# `robust_pls` — Robust PLS (Partial Robust M-regression) _Group_: **Robust / weighted** · _Registry tolerance_: `1e-08` ## 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` | `20` | 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_estimators_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. R roxygen note (`sklearn_extra.R::robust_pls`): > Robust PLS — formula entry point. MATLAB header (`bindings/matlab/+pls4all/RobustPlsRegression.m`): ```text pls4all.RobustPlsRegression Robust PLS via Huber IRLS. ``` ### 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_estimators_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} R · pls4all (raw fn) :sync: r-raw :class-label: lang-r ```r library(pls4all) res <- robust_pls_fit(X, Y, n_components, huber_k = 1.345, max_irls_iter = 20L) yhat <- pls4all_predict(res, X_test) ``` ::: :::{tab-item} R · pls4all (formula+S3) :sync: r-formula :class-label: lang-r ```r library(pls4all) fit <- robust_pls(y ~ ., data = train, ncomp = 4L) yhat <- predict(fit, newdata = test) summary(fit) ``` ::: :::{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 · strict (rmse_rel ≤ 1e-08) — 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  ·  ⇄ 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`](../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
BackendParity200×50 (ms)
C++ native · libn4m
pls4all.cpp.blas+omp✓ ref 1e-142.07 ms
Python · pls4all
pls4all.python✓ bind1.95 ms🏆
pls4all.sklearn✓ bind2.15 ms
R · pls4all
pls4all.R✓ bind4.38 ms
pls4all.R.formula✓ bind5.55 ms
pls4all.R.mdatools✓ bind5.77 ms
pls4all.R.pls✓ bind6.62 ms
R · external
📐ref.r_chemometricssource17.9 ms
::: :::{tab-item} 3 threads :sync: threads-3
BackendParity200×50 (ms)
C++ native · libn4m
pls4all.cpp.blas+omp✓ ref 1e-142.02 ms
Python · pls4all
pls4all.python✓ bind2.00 ms🏆
pls4all.sklearn✓ bind2.10 ms
R · pls4all
pls4all.R✓ bind5.43 ms
pls4all.R.formula✓ bind6.63 ms
pls4all.R.mdatools✓ bind5.84 ms
pls4all.R.pls✓ bind5.99 ms
R · external
📐ref.r_chemometricssource17.0 ms
::: :::{tab-item} 10 threads :sync: threads-10
BackendParity200×50 (ms)
C++ native · libn4m
pls4all.cpp.blas+omp✓ ref 1e-142.05 ms
Python · pls4all
pls4all.python✓ bind1.98 ms🏆
pls4all.sklearn✓ bind2.25 ms
R · pls4all
pls4all.R✓ bind4.94 ms
pls4all.R.formula✓ bind7.07 ms
pls4all.R.mdatools✓ bind5.59 ms
pls4all.R.pls✓ bind5.68 ms
R · external
📐ref.r_chemometricssource17.5 ms
::: :::: --- _See also_: [benchmark overview](../benchmarks/overview.md) · [methods index](index.md) · [interactive dashboard](../landing/dashboard.md)