# `on_pls` — OnPLS (Orthogonal N-block PLS) _Group_: **Multi-block / cross-modal** · _Registry tolerance_: `1e-06` ## Description OnPLS — Orthogonal multi-block decomposition (§18) > **Registry note** — Python `OnPLS` (tomlof/OnPLS, vendored in bindings/python/vendor/OnPLS). Canonical Löfstedt & Trygg 2011. Both impls predict the joint-component reconstruction of block 0. _No tunable parameters declared at the binding level._ ## Explanations ### Bibliographic source Löfstedt, T. & Trygg, J. (2011). *OnPLS — a novel multiblock method for the modelling of predictive and orthogonal variation*. Journal of Chemometrics 25(8), 441–455. ### Mathematical principle OnPLS generalises OPLS to multiple blocks: it decomposes the joint structure into a globally predictive component shared by all blocks plus block-unique orthogonal components per block. This separates 'integrated' biology / chemistry information from block-specific noise. The procedure iteratively refines a joint component by alternating projections and orthogonalisations across blocks. Compared to SO-PLS — which is asymmetric in block order — OnPLS is **symmetric**: no causal directionality is implied between blocks. This is the right choice when blocks are observation modalities of the same underlying process (e.g. transcriptomics + metabolomics + proteomics on the same biological samples). The CRAN `OnPLS` package was archived in 2024 so the Python implementation is vendored at `bindings/python/vendor/OnPLS/` to remove the dependency. ### Implementation `n4m_estimators_on_pls_fit` — requires `n_joint`, `n_unique_per_block`, `block_sizes`. The CRAN OnPLS package is archived; pls4all carries an in-tree vendored port for the parity reference. R roxygen note (`methods_extra.R::on_pls_fit`): > OnPLS — Orthogonal multi-block PLS (joint + unique loadings). > @param n_joint Method-specific parameter. See the underlying `*_fit()` function for the exact semantics. > @param n_unique_per_block Method-specific parameter. See the underlying `*_fit()` function for the exact semantics. > @param block_sizes Integer vector. Per-block feature counts for multi-block PLS. > @param X Numeric matrix of predictors (rows = samples, cols = features). > @param Y Numeric matrix or vector of responses, with one row per sample. > @export MATLAB header (`bindings/matlab/+pls4all/on_pls.m`): ```text pls4all.on_pls Orthogonal multi-block PLS (joint + unique loadings). n_components arg is unused by on_pls (it has its own block structure), but the dispatcher still requires it; we pass n_joint. ``` ### 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_on_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 on_pls_fit with pls4all.Context() as ctx, pls4all.Config() as cfg: res = on_pls_fit(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 on_pls result = on_pls(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("on_pls", X, y, n_components = 2L) # 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 <- on_pls_fit(X, Y, n_joint, n_unique_per_block, block_sizes) yhat <- pls4all_predict(res, X_test) ``` ::: :::{tab-item} MATLAB · pls4all (MEX) :sync: matlab-mex :class-label: lang-matlab ```matlab res = pls4all.on_pls(X, y, 2); % see header of bindings/matlab/+pls4all/on_pls.m for full % parameter surface: % res = on_pls(X, Y, n_joint, n_unique_per_block, block_sizes) yhat = predict(res, Xtest); ``` ::: :::{tab-item} MATLAB · pls4all (classdef) :sync: matlab-classdef :class-label: lang-matlab _No idiomatic classdef wrapper — invoke `pls4all.fit("on_pls", X, y, …)` directly from the unified MEX factory._ ::: :::: **Registry parity references** 📐 :::{card} :class-card: external-refs - 📐 **`ref.python_onpls`** (python · python) — `OnPLS` github tomlof/OnPLS · strict (rmse_rel ≤ 1e-06) — Python `OnPLS` (Löfstedt & Trygg 2011). Vendored from GitHub because R `multiblock 0.8.10` lacks `onpls`. Both impls return the joint-component reconstruction X̂_0. ::: ### 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
BackendParity200×30 (ms)
C++ native · libn4m
pls4all.cpp.blas+omp✓ ref 2e-116.20 ms
Python · pls4all
pls4all.python✓ bind3.02 ms
pls4all.sklearn✓ bind2.69 ms🏆
Python · external
📐ref.python_onplssource12.2 ms
::: :::{tab-item} 3 threads :sync: threads-3
BackendParity200×30 (ms)
C++ native · libn4m
pls4all.cpp.blas+omp✓ ref 2e-111.81 ms🏆
Python · pls4all
pls4all.python✓ bind1.91 ms
pls4all.sklearn✓ bind1.84 ms
Python · external
📐ref.python_onplssource9.46 ms
::: :::{tab-item} 10 threads :sync: threads-10
BackendParity200×30 (ms)
C++ native · libn4m
pls4all.cpp.blas+omp✓ ref 2e-111.91 ms🏆
Python · pls4all
pls4all.python✓ bind2.00 ms
pls4all.sklearn✓ bind16.0 ms
Python · external
📐ref.python_onplssource13.4 ms
::: :::: --- _See also_: [benchmark overview](../benchmarks/overview.md) · [methods index](index.md) · [interactive dashboard](../landing/dashboard.md)