# `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_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.
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_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} 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 · ✗ 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
| Backend | Parity | 50×250 (ms) | 100×50 (ms) | 100×500 (ms) | 100×2500 (ms) | 200×30 (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 | ≈ +9e-11 | 34.1 ms | 4.03 ms | 210.5 ms | 19.9 s | 2.09 ms | 4.20 ms | 10.8 ms🏆 | 479.1 ms | 29.6 s | 65.4 ms | 2.8 s | 170.7 s | 351.9 ms🏆 | 16.9 s |
pls4all.cpp.blas+omp | ≈ +9e-11 | 33.4 ms | 3.70 ms | 246.0 ms | 15.6 s🏆 | 1.94 ms | 4.35 ms | 12.5 ms | 543.0 ms | 29.9 s | 62.1 ms | 3.1 s | 204.1 s | 376.9 ms | 16.7 s |
pls4all.cpp.omp | ≈ +9e-11 | 34.2 ms | 3.72 ms | 224.5 ms | 18.3 s | 1.92 ms🏆 | 3.97 ms🏆 | 11.3 ms | 477.2 ms | 29.6 s🏆 | 61.9 ms🏆 | 2.7 s🏆 | 163.2 s | 358.2 ms | 16.5 s🏆 |
pls4all.cpp.ref | ≈ +9e-11 | 33.3 ms🏆 | 3.25 ms🏆 | 209.5 ms🏆 | 20.4 s | 2.12 ms | 6.68 ms | 11.2 ms | 441.3 ms🏆 | 30.0 s | 71.9 ms | 2.9 s | 157.5 s🏆 | 358.8 ms | 18.5 s |
| Python · pls4all |
|---|
pls4all.python | ✓ bind | 33.3 ms | — | — | — | 2.01 ms | 4.20 ms | — | — | — | — | — | — | — | — |
pls4all.sklearn | ⚠ | — | — | — | — | 2.13 ms | — | — | — | — | — | — | — | — | — |
| MATLAB · pls4all |
|---|
pls4all.matlab | ≈ | 40.7 ms | — | — | — | 2.91 ms | 6.91 ms | — | — | — | — | — | — | — | — |
pls4all.matlab.classdef | ≈ | 37.8 ms | — | — | — | 5.46 ms | 9.07 ms | — | — | — | — | — | — | — | — |
| Python · external |
|---|
📐ref.python_onpls | source | 48.0 ms | — | — | — | 11.7 ms | 19.2 ms | — | — | — | — | — | — | — | — |
:::
:::{tab-item} 3 threads
:sync: threads-3
| Backend | Parity | 50×250 (ms) | 100×50 (ms) | 100×500 (ms) | 100×2500 (ms) | 200×30 (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 | ✓ ref 2e-11 | — | — | — | — | 2.11 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.blas+omp | ✓ ref 2e-11 | — | — | — | — | 3.83 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.omp | ✓ ref 2e-11 | — | — | — | — | 1.99 ms🏆 | — | — | — | — | — | — | — | — | — |
pls4all.cpp.ref | ✓ ref 2e-11 | — | — | — | — | 2.12 ms | — | — | — | — | — | — | — | — | — |
| Python · pls4all |
|---|
pls4all.python | ✓ bind | — | — | — | — | 4.07 ms | — | — | — | — | — | — | — | — | — |
pls4all.sklearn | ✓ bind | — | — | — | — | 2.26 ms | — | — | — | — | — | — | — | — | — |
| MATLAB · pls4all |
|---|
pls4all.matlab | ≈ | — | — | — | — | 2.98 ms | — | — | — | — | — | — | — | — | — |
pls4all.matlab.classdef | ≈ | — | — | — | — | 5.12 ms | — | — | — | — | — | — | — | — | — |
| Python · external |
|---|
📐ref.python_onpls | source | — | — | — | — | 11.6 ms | — | — | — | — | — | — | — | — | — |
:::
:::{tab-item} 10 threads
:sync: threads-10
| Backend | Parity | 50×250 (ms) | 100×50 (ms) | 100×500 (ms) | 100×2500 (ms) | 200×30 (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 | ✓ ref 2e-11 | — | — | — | — | 2.03 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.blas+omp | ✓ ref 2e-11 | — | — | — | — | 2.10 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.omp | ✓ ref 2e-11 | — | — | — | — | 5.00 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.ref | ✓ ref 2e-11 | — | — | — | — | 2.00 ms🏆 | — | — | — | — | — | — | — | — | — |
| Python · pls4all |
|---|
pls4all.python | ✓ bind | — | — | — | — | 4.63 ms | — | — | — | — | — | — | — | — | — |
pls4all.sklearn | ✓ bind | — | — | — | — | 2.21 ms | — | — | — | — | — | — | — | — | — |
| MATLAB · pls4all |
|---|
pls4all.matlab | ≈ | — | — | — | — | 3.47 ms | — | — | — | — | — | — | — | — | — |
pls4all.matlab.classdef | ≈ | — | — | — | — | 3.39 ms | — | — | — | — | — | — | — | — | — |
| Python · external |
|---|
📐ref.python_onpls | source | — | — | — | — | 10.2 ms | — | — | — | — | — | — | — | — | — |
:::
::::
---
_See also_: [benchmark overview](../benchmarks/overview.md) · [methods index](index.md) · [interactive dashboard](../landing/dashboard.md)