# `so_pls` — Sequential and Orthogonalised PLS (SO-PLS)
_Group_: **Multi-block / cross-modal** · _Registry tolerance_: `1e-06`
## Description
SO-PLS — Sequential & Orthogonalized multi-block PLS (§17)
From the `pls4all.sklearn.SOPLSRegression` docstring:
> Sequential & Orthogonalised multi-block PLS (Næs et al. 2011).
> **Registry note** — R `multiblock::sopls 0.8.10` (Næs et al. 2011) canonical SO-PLS via fit$fitted at the full (k1,..,kB) slice. pls4all's NIPALS-based SO-PLS matches to ~1e-13.
### Parameters
| Name | Type | Default | Notes |
|------|------|---------|-------|
| `n_components_per_block` | `—` | `None` | Per-block latent-component budget (one int per block). |
| `block_sizes` | `—` | `None` | Sequence of contiguous block widths defining the X-block partition (columns of X). |
## Explanations
### Bibliographic source
Næs, T., Tomic, O., Mevik, B.-H. & Martens, H. (2011). *Path modelling by sequential PLS regression*. Journal of Chemometrics 25(1), 28–40.
### Mathematical principle
SO-PLS extends MB-PLS by sequentially processing blocks in a user-specified order, orthogonalising each subsequent block against the scores extracted from the previous ones. Concretely: fit PLS on block 1, extract scores $\mathbf{T}_1$; orthogonalise block 2 to $\mathbf{T}_1$, fit PLS on the residual; and so on.
This makes each block's contribution **additive and interpretable** — the unique variance in block $b$ that is predictive of $y$ given everything in blocks $1, \ldots, b-1$. Compared to MB-PLS (which fits all blocks simultaneously), SO-PLS encodes a domain hypothesis about which block is causally upstream of which.
Tuning: a per-block $k_b$ (number of components) plus the block order. The order matters; permuting blocks changes the attribution. Cross-validating the per-block $k_b$ is the standard procedure.
### Implementation
`n4m_so_pls_fit` — requires `n_components_per_block` and `block_sizes`. Reference: CRAN `multiblock 0.8.10`.
MATLAB header (`bindings/matlab/+pls4all/so_pls.m`):
```text
pls4all.so_pls Sequential & Orthogonalised multi-block PLS (Næs 2011).
X: n × sum(block_sizes) concatenated multi-block matrix.
n_components_per_block: int32 vector of length n_blocks.
```
### 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_so_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 so_pls_fit
with pls4all.Context() as ctx, pls4all.Config() as cfg:
res = so_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 SOPLSRegression
mdl = SOPLSRegression(n_components_per_block=None, block_sizes=None)
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("so_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.so_pls(X, y, 2);
% see header of bindings/matlab/+pls4all/so_pls.m for full
% parameter surface:
% res = so_pls(X, Y, n_components_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("so_pls", X, y, …)` directly from the unified MEX factory._
:::
::::
**Registry parity references** 📐
:::{card}
:class-card: external-refs
- 📐 **`ref.r_multiblock`** (R · r) — `multiblock` 0.8.10 · strict (rmse_rel ≤ 1e-06) — R `multiblock::sopls 0.8.10` (Næs et al. 2011) canonical SO-PLS — in-sample fitted values via fit$fitted at the full (k1,..,kB) slice match pls4all's canonical NIPALS SO-PLS to ~1e-15 in centered space.
:::
### 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 | ≈ +1e-09 | 3.79 ms | 2.04 ms | 20.9 ms | 88.9 ms🏆 | 1.66 ms🏆 | 3.58 ms | 8.60 ms🏆 | 108.1 ms | 652.6 ms🏆 | 43.7 ms🏆 | 530.2 ms | 8.7 s | 233.9 ms | 3.1 s |
pls4all.cpp.blas+omp | ≈ +1e-09 | 3.81 ms | 2.46 ms | 19.8 ms | 93.4 ms | 1.78 ms | 3.62 ms | 8.77 ms | 103.4 ms | 690.4 ms | 44.2 ms | 522.9 ms | 9.3 s | 206.2 ms🏆 | 2.8 s🏆 |
pls4all.cpp.omp | ≈ +1e-09 | 4.10 ms | 1.88 ms🏆 | 20.3 ms | 94.5 ms | 1.72 ms | 3.53 ms | 8.89 ms | 99.7 ms | 672.2 ms | 45.4 ms | 504.7 ms | 9.2 s | 216.3 ms | 3.1 s |
pls4all.cpp.ref | ≈ +1e-09 | 3.48 ms🏆 | 3.27 ms | 19.2 ms🏆 | 93.8 ms | 1.74 ms | 3.52 ms🏆 | 8.80 ms | 97.2 ms🏆 | 699.0 ms | 44.9 ms | 477.7 ms🏆 | 8.6 s🏆 | 218.3 ms | 3.0 s |
| Python · pls4all |
|---|
pls4all.python | ✓ bind | 3.64 ms | — | — | — | 1.78 ms | 3.97 ms | — | — | — | — | — | — | — | — |
pls4all.sklearn | ✓ bind | 3.93 ms | — | — | — | 2.10 ms | 4.13 ms | — | — | — | — | — | — | — | — |
| R · pls4all |
|---|
pls4all.R | ✗ +2e+01 | 11.8 ms | — | — | — | 13.2 ms | 9.95 ms | — | — | — | — | — | — | — | — |
pls4all.R.formula | ✗ +2e+01 | 20.4 ms | — | — | — | 13.5 ms | 9.20 ms | — | — | — | — | — | — | — | — |
pls4all.R.mdatools | ✗ +2e+01 | 18.5 ms | — | — | — | 12.4 ms | 11.6 ms | — | — | — | — | — | — | — | — |
pls4all.R.pls | ✗ +2e+01 | 18.3 ms | — | — | — | 14.1 ms | 9.86 ms | — | — | — | — | — | — | — | — |
| MATLAB · pls4all |
|---|
pls4all.matlab | ✗ +9e+00 | 5.58 ms | — | — | — | 2.72 ms | 5.47 ms | — | — | — | — | — | — | — | — |
pls4all.matlab.classdef | ✗ +9e+00 | 9.05 ms | — | — | — | 3.08 ms | 6.92 ms | — | — | — | — | — | — | — | — |
| R · external |
|---|
📐ref.r_multiblock | source | 1.0 s | — | — | — | 1.4 s | 1.1 s | — | — | — | — | — | — | — | — |
:::
:::{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 7e-10 | — | — | — | — | 1.67 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.blas+omp | ✓ ref 7e-10 | — | — | — | — | 1.71 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.omp | ✓ ref 7e-10 | — | — | — | — | 1.75 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.ref | ✓ ref 7e-10 | — | — | — | — | 1.80 ms | — | — | — | — | — | — | — | — | — |
| Python · pls4all |
|---|
pls4all.python | ✓ bind | — | — | — | — | 1.65 ms🏆 | — | — | — | — | — | — | — | — | — |
pls4all.sklearn | ✓ bind | — | — | — | — | 2.11 ms | — | — | — | — | — | — | — | — | — |
| R · pls4all |
|---|
pls4all.R | ✗ +2e+01 | — | — | — | — | 11.5 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.formula | ✗ +2e+01 | — | — | — | — | 12.1 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.mdatools | ✗ +2e+01 | — | — | — | — | 13.7 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.pls | ✗ +2e+01 | — | — | — | — | 12.9 ms | — | — | — | — | — | — | — | — | — |
| MATLAB · pls4all |
|---|
pls4all.matlab | ✗ +9e+00 | — | — | — | — | 2.57 ms | — | — | — | — | — | — | — | — | — |
pls4all.matlab.classdef | ✗ +9e+00 | — | — | — | — | 4.55 ms | — | — | — | — | — | — | — | — | — |
| R · external |
|---|
📐ref.r_multiblock | source | — | — | — | — | 1.3 s | — | — | — | — | — | — | — | — | — |
:::
:::{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 7e-10 | — | — | — | — | 1.61 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.blas+omp | ✓ ref 7e-10 | — | — | — | — | 1.59 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.omp | ✓ ref 7e-10 | — | — | — | — | 1.56 ms🏆 | — | — | — | — | — | — | — | — | — |
pls4all.cpp.ref | ✓ ref 7e-10 | — | — | — | — | 1.58 ms | — | — | — | — | — | — | — | — | — |
| Python · pls4all |
|---|
pls4all.python | ✓ bind | — | — | — | — | 1.64 ms | — | — | — | — | — | — | — | — | — |
pls4all.sklearn | ✓ bind | — | — | — | — | 1.69 ms | — | — | — | — | — | — | — | — | — |
| R · pls4all |
|---|
pls4all.R | ✗ +2e+01 | — | — | — | — | 10.2 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.formula | ✗ +2e+01 | — | — | — | — | 10.9 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.mdatools | ✗ +2e+01 | — | — | — | — | 10.9 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.pls | ✗ +2e+01 | — | — | — | — | 10.8 ms | — | — | — | — | — | — | — | — | — |
| MATLAB · pls4all |
|---|
pls4all.matlab | ✗ +9e+00 | — | — | — | — | 2.42 ms | — | — | — | — | — | — | — | — | — |
pls4all.matlab.classdef | ✗ +9e+00 | — | — | — | — | 2.62 ms | — | — | — | — | — | — | — | — | — |
| R · external |
|---|
📐ref.r_multiblock | source | — | — | — | — | 1.0 s | — | — | — | — | — | — | — | — | — |
:::
::::
---
_See also_: [benchmark overview](../benchmarks/overview.md) · [methods index](index.md) · [interactive dashboard](../landing/dashboard.md)