# `opls` — Orthogonal PLS (OPLS) _Group_: **Core PLS** · _Registry tolerance_: `1e-08` ## Description Orthogonal PLS (Trygg & Wold 2002) From the `pls4all.sklearn.OPLSRegression` docstring: > Orthogonal PLS regression (Trygg & Wold 2002). > **Registry note** — Bioconductor `ropls::opls` is the external OPLS reference; convergence and orthogonal-component conventions may differ. ### Parameters | Name | Type | Default | Notes | |------|------|---------|-------| | `n_components` | `int` | `2` | Number of latent components extracted (k). | | `solver` | `str` | `'nipals'` | Inner algorithm: 'nipals', 'simpls', 'svd', 'kernel', 'orthogonal-scores', 'power', 'randomized-svd', 'wide-kernel'. | | `center_x` | `bool` | `True` | Subtract the column mean of X before fitting. | | `scale_x` | `bool` | `True` | Standardize X columns to unit variance before fitting. | | `center_y` | `bool` | `True` | Subtract the column mean of y before fitting. | | `scale_y` | `bool` | `False` | Standardize y columns to unit variance before fitting. | | `tol` | `float` | `1e-06` | Convergence tolerance for iterative solvers (NIPALS / power-iteration). | | `max_iter` | `int` | `500` | Maximum iterations for iterative solvers. | | `store_scores` | `bool` | `False` | If True, keep the latent score matrix (`x_scores_`) after fit. | ## Explanations ### Bibliographic source Trygg, J. & Wold, S. (2002). *Orthogonal projections to latent structures (O-PLS)*. Journal of Chemometrics 16(3), 119–128. ### Mathematical principle OPLS rotates the standard PLS latent space so that a single direction captures all $\mathbf{Y}$-correlated variation while the remaining components capture $\mathbf{Y}$-orthogonal structural variation in $\mathbf{X}$. The resulting decomposition $\mathbf{X} = \mathbf{t}_p\mathbf{p}_p^{\top} + \mathbf{T}_o\mathbf{P}_o^{\top} + \mathbf{E}$ separates the **predictive component** $\mathbf{t}_p$ from the orthogonal block $\mathbf{T}_o$, which absorbs spectroscopic baselines, scatter and other nuisance factors that confound interpretation of the predictive loading. Numerically OPLS proceeds by NIPALS-deflating $\mathbf{X}$ against directions orthogonal to $\mathbf{X}^{\top}\mathbf{y}$ before each new predictive component is extracted. Predictions are identical to those of a one-component PLS on the orthogonal-corrected $\mathbf{X}$; the value is in **the interpretation of the loadings**, not in better predictions per se. OPLS shines in metabolomics and process spectroscopy where the spectra carry strong systematic but non-predictive variation; in those settings the single-vector predictive loading is far easier to relate to biology / chemistry than a multi-component PLS loading matrix. ### Implementation `Algorithm.OPLS` + `Solver.NIPALS` + `Deflation.ORTHOGONAL`. Reference: Bioconductor `ropls::opls`. Note: orthogonal-component ordering and the criterion that stops orthogonal extraction differ between implementations — exact bit parity is not expected, but RMSE-rel parity within ~1e-3 is. R roxygen note (`sklearn.R::opls`): > Formula-based OPLS regression wrapper around the n4m C ABI. MATLAB header (`bindings/matlab/+pls4all/OplsRegression.m`): ```text pls4all.OplsRegression — Orthogonal Partial Least Squares Regression model. Example: mdl = pls4all.OplsRegression(X, y, 5); yhat = predict(mdl, Xnew); ``` ### 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 (Model.fit path) */ n4m_context_t* ctx = n4m_context_create(); n4m_config_t* cfg = n4m_config_create(); n4m_config_set_algorithm(cfg, N4M_ALGORITHM_PLS_REGRESSION); n4m_config_set_solver (cfg, N4M_SOLVER_SIMPLS); n4m_config_set_n_components(cfg, 4); n4m_model_t* mdl = NULL; n4m_model_fit(ctx, cfg, &x_view, &y_view, &mdl); n4m_model_predict(ctx, mdl, &x_test_view, &y_hat_view); n4m_model_destroy(mdl); 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 import Algorithm, Solver with pls4all.Context() as ctx, pls4all.Config() as cfg: cfg.algorithm = Algorithm.PLS_REGRESSION cfg.solver = Solver.SIMPLS cfg.n_components = 4 with pls4all.Model.fit(ctx, cfg, X, y) as mdl: y_hat = mdl.predict(X_test) ``` ::: :::{tab-item} Python · pls4all.sklearn :sync: python-sklearn :class-label: lang-python ```python from pls4all.sklearn import OPLSRegression mdl = OPLSRegression(n_components=2, solver='nipals', center_x=True, scale_x=True, center_y=True, scale_y=False, tol=1e-06, max_iter=500, store_scores=False) 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("opls", X, y, n_components = 4L) # res is a named list with MethodResult arrays/scalars. # selected_indices / top_k_intervals are 1-based. ``` ::: :::{tab-item} R · pls4all (formula+S3) :sync: r-formula :class-label: lang-r ```r library(pls4all) fit <- opls(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.opls(X, y, 4); % see header of bindings/matlab/+pls4all/opls.m for full % parameter surface: % [coefs, x_mean, y_mean, predictions] = opls(X, Y, n_components) yhat = predict(res, Xtest); ``` ::: :::{tab-item} MATLAB · pls4all (classdef) :sync: matlab-classdef :class-label: lang-matlab ```matlab mdl = pls4all.fit("opls", X, y, "NumComponents", 4); yhat = predict(mdl, Xtest); ``` ::: :::: **Registry parity references** 📐 :::{card} :class-card: external-refs - 📐 **`ref.r_ropls`** (R · r) — `ropls` Bioc · strict (rmse_rel ≤ 1e-08) — Bioconductor `ropls::opls` — OPLS reference. Permutations and plotting are disabled in benchmark timing; ropls still requires crossvalI >= 1 for a finite Q2 path. ::: ### 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 5e-151.87 ms🏆
Python · pls4all
pls4all.python✓ bind1.88 ms
pls4all.sklearn✓ bind2.12 ms
R · pls4all
pls4all.R✓ 2e-144.35 ms
pls4all.R.formula✓ 2e-145.13 ms
pls4all.R.mdatools✓ 2e-145.23 ms
pls4all.R.pls✓ 2e-145.40 ms
R · external
📐ref.r_roplssource15.4 ms
::: :::{tab-item} 3 threads :sync: threads-3
BackendParity200×50 (ms)
C++ native · libn4m
pls4all.cpp.blas+omp✓ ref 5e-151.96 ms
Python · pls4all
pls4all.python✓ bind1.79 ms🏆
pls4all.sklearn✓ bind2.14 ms
R · pls4all
pls4all.R✓ 2e-146.22 ms
pls4all.R.formula✓ 2e-145.57 ms
pls4all.R.mdatools✓ 2e-147.12 ms
pls4all.R.pls✓ 2e-147.88 ms
R · external
📐ref.r_roplssource16.0 ms
::: :::{tab-item} 10 threads :sync: threads-10
BackendParity200×50 (ms)
C++ native · libn4m
pls4all.cpp.blas+omp✓ ref 5e-151.73 ms🏆
Python · pls4all
pls4all.python✓ bind1.81 ms
pls4all.sklearn✓ bind1.94 ms
R · pls4all
pls4all.R✓ 2e-144.17 ms
pls4all.R.formula✓ 2e-145.09 ms
pls4all.R.mdatools✓ 2e-145.39 ms
pls4all.R.pls✓ 2e-145.17 ms
R · external
📐ref.r_roplssource16.0 ms
::: :::: --- _See also_: [benchmark overview](../benchmarks/overview.md) · [methods index](index.md) · [interactive dashboard](../landing/dashboard.md)