# `mb_pls` — Multi-block PLS (Westerhuis 1998)
_Group_: **Multi-block / cross-modal** · _Registry tolerance_: `2.0`
## Description
MB-PLS — Multi-block PLS (§17 Phase 4)
From the `pls4all.sklearn.MBPLSRegression` docstring:
> Multi-block PLS (Westerhuis 1998).
> **Registry note** — In-tree `nirs4all.operators.models.sklearn.mbpls.MBPLS` is the sanctioned external reference (the mbpls PyPI package is broken against sklearn 1.8). pls4all's MB-PLS default now mirrors nirs4all NIPALS multi-block (standardize=False); the legacy block-balanced SIMPLS path is opt-in via cfg.scale_x=True.
### Parameters
| Name | Type | Default | Notes |
|------|------|---------|-------|
| `n_components` | `int` | `2` | Number of latent components extracted (k). |
| `block_sizes` | `—` | `None` | Sequence of contiguous block widths defining the X-block partition (columns of X). |
| `n_blocks` | `int` | `3` | registry benchmark cell value |
## Explanations
### Bibliographic source
Westerhuis, J. A., Kourti, T. & MacGregor, J. F. (1998). *Analysis of multiblock and hierarchical PCA and PLS models*. Journal of Chemometrics 12(5), 301–321.
### Mathematical principle
When predictors come from several distinct sources — NIR, MIR, Raman, process tags, lab assays — concatenating them into one wide matrix lets the block with the most variance dominate. Multi-block PLS instead **block-scales** each $\mathbf{X}_b$ so blocks contribute proportionally to their information content rather than their dimensionality.
Formally, each block is centred and autoscaled, then scaled by $1 / \sqrt{p_b}$ so its total variance is unit-normalised. PLS then runs on the concatenated $[\mathbf{X}_1, \ldots, \mathbf{X}_B]$ with optional per-block weights. Block-level *importance* statistics (block-VIP, block-RMSE) are recovered from the loadings by restriction to each block's columns.
Compared to plain concatenation, MB-PLS gives interpretable per-block contributions and is the standard approach in process spectroscopy.
### Implementation
`n4m_mb_pls_fit` — requires a `block_sizes` integer vector summing to $p$. The C ABI materialises the intercept directly (no separate $\bar{\mathbf{y}}$ key) because the block scaling changes the centring semantics. Reference: sanctioned git-pinned port `nirs4all.operators.models.sklearn.mbpls`.
MATLAB header (`bindings/matlab/+pls4all/MbPlsRegression.m`):
```text
pls4all.MbPlsRegression — Multi-block PLS.
predict uses the stored intercept directly (coefficients are already
in original X scale + intercept folds in y_mean - x_mean @ coef).
```
### 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_mb_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 mb_pls_fit
with pls4all.Context() as ctx, pls4all.Config() as cfg:
res = mb_pls_fit(ctx, cfg, X, y, n_components=3)
# 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 MBPLSRegression
mdl = MBPLSRegression(n_components=2, 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("mb_pls", X, y,
n_components = 3L, params = list(n_blocks = 3L))
# 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.mb_pls(X, y, 3);
% see header of bindings/matlab/+pls4all/mb_pls.m for full
% parameter surface:
% res = mb_pls(X, Y, n_components, block_sizes)
yhat = predict(res, Xtest);
```
:::
:::{tab-item} MATLAB · pls4all (classdef)
:sync: matlab-classdef
:class-label: lang-matlab
```matlab
mdl = pls4all.fit("mb_pls", X, y, "NumComponents", 3);
yhat = predict(mdl, Xtest);
```
:::
::::
**Registry parity references** 📐
:::{card}
:class-card: external-refs
- 📐 **`nirs4all`** (python · python) — `nirs4all` in-tree · qualitative (rmse_rel ≤ 2e+00) — In-tree Python MB-PLS (sanctioned external reference). The mbpls PyPI package is broken against sklearn 1.8 (uses the deprecated `force_all_finite` kwarg). nirs4all's implementation is a clean re-derivation of Westerhuis 1998.
:::
### 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**: qualitative — shape/smoke comparison only. The external library and pls4all do not produce numerically equivalent output for this method (see the MethodSpec notes); the `rmse_rel_tol ≤ 2e+00` budget is set wide on purpose. Treat ~ shape as *“we ran both, both finished”*, not as numerical agreement.
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) | 200×60 (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 | ≈ +2e-15 | 3.12 ms | 1.20 ms | 10.1 ms🏆 | 53.1 ms | 2.27 ms🏆 | 3.04 ms | 4.85 ms | 51.9 ms | 271.9 ms | 24.7 ms | 345.6 ms🏆 | 1.8 s🏆 | 110.2 ms🏆 | 1.6 s |
pls4all.cpp.blas+omp | ≈ +2e-15 | 2.91 ms | 1.10 ms | 10.8 ms | 51.0 ms | 2.31 ms | 2.79 ms🏆 | 6.73 ms | 51.0 ms🏆 | 268.2 ms | 26.3 ms | 356.3 ms | 1.8 s | 142.7 ms | 1.4 s🏆 |
pls4all.cpp.omp | ≈ +2e-15 | 2.91 ms | 1.67 ms | 11.5 ms | 49.6 ms🏆 | 2.43 ms | 3.00 ms | 5.21 ms | 51.9 ms | 269.3 ms | 23.6 ms🏆 | 362.4 ms | 1.9 s | 127.6 ms | 1.5 s |
pls4all.cpp.ref | ≈ +2e-15 | 3.35 ms | 1.03 ms🏆 | 11.1 ms | 52.8 ms | 2.41 ms | 2.88 ms | 4.52 ms🏆 | 53.9 ms | 260.5 ms🏆 | 27.7 ms | 362.0 ms | 1.9 s | 122.6 ms | 1.5 s |
| Python · pls4all |
|---|
pls4all.python | ✓ bind | 2.78 ms🏆 | — | — | — | 2.46 ms | 3.04 ms | — | — | — | — | — | — | — | — |
pls4all.sklearn | ✓ 4e-15 | 4.94 ms | — | — | — | 3.85 ms | 4.83 ms | — | — | — | — | — | — | — | — |
| R · pls4all |
|---|
pls4all.R | ✗ +1e-01 | 13.9 ms | — | — | — | 8.28 ms | 14.4 ms | — | — | — | — | — | — | — | — |
pls4all.R.formula | ✗ +1e-01 | 24.0 ms | — | — | — | 11.6 ms | 13.9 ms | — | — | — | — | — | — | — | — |
pls4all.R.mdatools | ✗ +1e-01 | 20.9 ms | — | — | — | 12.4 ms | 10.8 ms | — | — | — | — | — | — | — | — |
pls4all.R.pls | ✗ +1e-01 | 24.8 ms | — | — | — | 12.7 ms | 12.4 ms | — | — | — | — | — | — | — | — |
| MATLAB · pls4all |
|---|
pls4all.matlab | ✗ +9e+00 | 5.08 ms | — | — | — | 3.64 ms | 4.16 ms | — | — | — | — | — | — | — | — |
pls4all.matlab.classdef | ✗ +9e+00 | 4.72 ms | — | — | — | 4.78 ms | 5.06 ms | — | — | — | — | — | — | — | — |
| Python · external |
|---|
📐nirs4all | source | 3.83 ms | — | — | — | 2.98 ms | 3.88 ms | — | — | — | — | — | — | — | — |
:::
:::{tab-item} 3 threads
:sync: threads-3
| Backend | Parity | 50×250 (ms) | 100×50 (ms) | 100×500 (ms) | 100×2500 (ms) | 200×60 (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 | ~ shape 7e-16 | — | — | — | — | 2.33 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.blas+omp | ~ shape 7e-16 | — | — | — | — | 2.30 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.omp | ~ shape 7e-16 | — | — | — | — | 2.14 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.ref | ~ shape 7e-16 | — | — | — | — | 3.49 ms | — | — | — | — | — | — | — | — | — |
| Python · pls4all |
|---|
pls4all.python | ✓ bind | — | — | — | — | 2.10 ms🏆 | — | — | — | — | — | — | — | — | — |
pls4all.sklearn | ✓ 4e-15 | — | — | — | — | 2.67 ms | — | — | — | — | — | — | — | — | — |
| R · pls4all |
|---|
pls4all.R | ✓ 7e-15 | — | — | — | — | 7.58 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.formula | ✓ 7e-15 | — | — | — | — | 10.0 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.mdatools | ✓ 7e-15 | — | — | — | — | 9.24 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.pls | ✓ 7e-15 | — | — | — | — | 10.1 ms | — | — | — | — | — | — | — | — | — |
| MATLAB · pls4all |
|---|
pls4all.matlab | ✗ +9e+00 | — | — | — | — | 5.86 ms | — | — | — | — | — | — | — | — | — |
pls4all.matlab.classdef | ✗ +9e+00 | — | — | — | — | 4.25 ms | — | — | — | — | — | — | — | — | — |
| Python · external |
|---|
📐nirs4all | source | — | — | — | — | 3.10 ms | — | — | — | — | — | — | — | — | — |
:::
:::{tab-item} 10 threads
:sync: threads-10
| Backend | Parity | 50×250 (ms) | 100×50 (ms) | 100×500 (ms) | 100×2500 (ms) | 200×60 (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 | ~ shape 7e-16 | — | — | — | — | 2.08 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.blas+omp | ~ shape 7e-16 | — | — | — | — | 2.06 ms🏆 | — | — | — | — | — | — | — | — | — |
pls4all.cpp.omp | ~ shape 7e-16 | — | — | — | — | 2.10 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.ref | ~ shape 7e-16 | — | — | — | — | 2.08 ms | — | — | — | — | — | — | — | — | — |
| Python · pls4all |
|---|
pls4all.python | ✓ bind | — | — | — | — | 2.98 ms | — | — | — | — | — | — | — | — | — |
pls4all.sklearn | ✓ 4e-15 | — | — | — | — | 2.24 ms | — | — | — | — | — | — | — | — | — |
| R · pls4all |
|---|
pls4all.R | ✓ 7e-15 | — | — | — | — | 5.81 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.formula | ✓ 7e-15 | — | — | — | — | 7.09 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.mdatools | ✓ 7e-15 | — | — | — | — | 7.28 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.pls | ✓ 7e-15 | — | — | — | — | 7.09 ms | — | — | — | — | — | — | — | — | — |
| MATLAB · pls4all |
|---|
pls4all.matlab | ✗ +9e+00 | — | — | — | — | 3.29 ms | — | — | — | — | — | — | — | — | — |
pls4all.matlab.classdef | ✗ +9e+00 | — | — | — | — | 3.69 ms | — | — | — | — | — | — | — | — | — |
| Python · external |
|---|
📐nirs4all | source | — | — | — | — | 2.59 ms | — | — | — | — | — | — | — | — | — |
:::
::::
---
_See also_: [benchmark overview](../benchmarks/overview.md) · [methods index](index.md) · [interactive dashboard](../landing/dashboard.md)