# `recursive_pls` — Recursive (moving-window) PLS
_Group_: **Core PLS** · _Registry tolerance_: `0.1`
## Description
Recursive (moving-window) PLS
From the `pls4all.sklearn.RecursivePLSRegression` docstring:
> Moving-window recursive PLS.
### Parameters
| Name | Type | Default | Notes |
|------|------|---------|-------|
| `n_components` | `int` | `2` | Number of latent components extracted (k). |
| `window_size` | `int` | `50` | Length of the moving window for recursive / interval-random-frog models. |
## Explanations
### Bibliographic source
Helland, K., Berntsen, H. E., Borgen, O. S. & Martens, H. (1992). *Recursive algorithm for partial least squares regression*. Chemometrics and Intelligent Laboratory Systems 14(1–3), 129–137.
### Mathematical principle
Process-analytical instruments produce streams of spectra under drifting conditions (changing humidity, instrument warm-up, fouling). Recursive PLS maintains a fitted model that **adapts as new samples arrive** by re-fitting on a sliding window of the most recent $w$ samples.
At time step $t$, the model is fit on $\{(\mathbf{x}_{t-w+1}, y_{t-w+1}), \ldots, (\mathbf{x}_t, y_t)\}$ and applied to incoming $\mathbf{x}_{t+1}$. Computational cost is $O(wpk)$ per step. The window width $w$ controls a stability/adaptability trade-off: short windows track drift aggressively but are noisier; long windows are stable but lag.
More sophisticated recursive variants (Qin 1998) use exponential forgetting factors instead of a hard window. pls4all's variant uses the hard-window form for deterministic parity with R `pls` rolling refits.
### Implementation
`n4m_recursive_pls_run` (returns predictions only — no global coefficient export, since the model changes per step). The Python sklearn wrapper is an in-sample-only estimator.
MATLAB header (`bindings/matlab/+pls4all/RecursivePlsRegression.m`):
```text
pls4all.RecursivePlsRegression Moving-window recursive PLS.
In-sample only: result holds in-window predictions; no global coefficient
matrix. `predict(X)` returns the stored predictions for the training X
(length-preserved; warmup samples are 0).
```
### 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_recursive_pls_run(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 recursive_pls_run
with pls4all.Context() as ctx, pls4all.Config() as cfg:
res = recursive_pls_run(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 RecursivePLSRegression
mdl = RecursivePLSRegression(n_components=2, window_size=50)
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("recursive_pls", X, y,
n_components = 3L, params = list(window_size = 60L))
# 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.recursive_pls(X, y, 3);
% see header of bindings/matlab/+pls4all/recursive_pls.m for full
% parameter surface:
% res = recursive_pls(X, Y, n_components, window_size)
yhat = predict(res, Xtest);
```
:::
:::{tab-item} MATLAB · pls4all (classdef)
:sync: matlab-classdef
:class-label: lang-matlab
```matlab
mdl = pls4all.fit("recursive_pls", X, y, "NumComponents", 3);
yhat = predict(mdl, Xtest);
```
:::
::::
**Registry parity references** 📐
:::{card}
:class-card: external-refs
- 📐 **`ref.python_scikit_learn`** (python · python) — `scikit-learn` 1.4.2 · qualitative (rmse_rel ≤ 1e-01) — Moving-window refit using sklearn PLSRegression (NIPALS).
- 📐 **`ref.r_pls`** (R · r) — `pls` 2.8.5 · qualitative (rmse_rel ≤ 1e-01) — Moving-window refit using R `pls::plsr` (SIMPLS by default).
:::
### 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 ≤ 1e-01` 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) | 200×50 (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 | — | 1.81 ms🏆 | 22.0 ms | 117.5 ms | 3.04 ms🏆 | 5.04 ms | 15.5 ms🏆 | 140.6 ms🏆 | 837.9 ms | 100.5 ms | 700.0 ms | 5.8 s | 244.1 ms🏆 | 2.4 s🏆 |
pls4all.cpp.blas+omp | ⚠ +2e-15 | — | 3.42 ms | 18.7 ms🏆 | 116.0 ms🏆 | 3.12 ms | 4.92 ms | 16.5 ms | 158.4 ms | 752.2 ms🏆 | 92.4 ms🏆 | 662.4 ms🏆 | 5.3 s🏆 | 250.1 ms | 2.4 s |
pls4all.cpp.omp | ⚠ +2e-15 | — | 2.99 ms | 26.9 ms | 132.2 ms | 3.60 ms | 5.99 ms | 19.2 ms | 202.5 ms | 996.6 ms | 103.0 ms | 926.0 ms | 6.8 s | 324.9 ms | 3.4 s |
pls4all.cpp.ref | ⚠ +2e-15 | — | 3.55 ms | 26.3 ms | 135.6 ms | 3.73 ms | 5.83 ms | 18.6 ms | 214.1 ms | 1.1 s | 115.9 ms | 1.1 s | 7.3 s | 335.6 ms | 3.5 s |
| Python · pls4all |
|---|
pls4all.python | ⚠ | — | — | — | — | 3.24 ms | 4.89 ms | — | — | — | — | — | — | — | — |
pls4all.sklearn | ⚠ | — | — | — | — | 5.12 ms | 4.85 ms🏆 | — | — | — | — | — | — | — | — |
| R · pls4all |
|---|
pls4all.R | ✓ 9e-15 | — | — | — | — | 8.73 ms | 14.1 ms | — | — | — | — | — | — | — | — |
pls4all.R.formula | ✓ 9e-15 | — | — | — | — | 10.6 ms | 11.0 ms | — | — | — | — | — | — | — | — |
pls4all.R.mdatools | ✓ 9e-15 | — | — | — | — | 9.76 ms | 12.1 ms | — | — | — | — | — | — | — | — |
pls4all.R.pls | ✓ 9e-15 | — | — | — | — | 11.3 ms | 12.6 ms | — | — | — | — | — | — | — | — |
| MATLAB · pls4all |
|---|
pls4all.matlab | ✗ +1e+01 | — | — | — | — | 4.35 ms | 6.37 ms | — | — | — | — | — | — | — | — |
pls4all.matlab.classdef | ✗ +1e+01 | — | — | — | — | 5.16 ms | 7.59 ms | — | — | — | — | — | — | — | — |
| Python · external |
|---|
📐ref.python_scikit_learn | source | 2.50 ms🏆 | — | — | — | 70.0 ms | 93.0 ms | — | — | — | — | — | — | — | — |
| R · external |
|---|
📐ref.r_pls | ⚠ +2e-14 | — | — | — | — | 216.4 ms | 206.2 ms | — | — | — | — | — | — | — | — |
:::
:::{tab-item} 3 threads
:sync: threads-3
| Backend | Parity | 50×250 (ms) | 100×50 (ms) | 100×500 (ms) | 100×2500 (ms) | 200×50 (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 9e-16 | — | — | — | — | 3.08 ms🏆 | — | — | — | — | — | — | — | — | — |
pls4all.cpp.blas+omp | ~ shape 9e-16 | — | — | — | — | 3.36 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.omp | ~ shape 1e-15 | — | — | — | — | 5.70 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.ref | ~ shape 1e-15 | — | — | — | — | 4.10 ms | — | — | — | — | — | — | — | — | — |
| Python · pls4all |
|---|
pls4all.python | ✓ bind | — | — | — | — | 3.22 ms | — | — | — | — | — | — | — | — | — |
pls4all.sklearn | ✓ bind | — | — | — | — | 3.54 ms | — | — | — | — | — | — | — | — | — |
| R · pls4all |
|---|
pls4all.R | ✓ 4e-15 | — | — | — | — | 8.26 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.formula | ✓ 4e-15 | — | — | — | — | 9.96 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.mdatools | ✓ 4e-15 | — | — | — | — | 9.54 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.pls | ✓ 4e-15 | — | — | — | — | 9.47 ms | — | — | — | — | — | — | — | — | — |
| MATLAB · pls4all |
|---|
pls4all.matlab | ✗ +1e+01 | — | — | — | — | 4.51 ms | — | — | — | — | — | — | — | — | — |
pls4all.matlab.classdef | ✗ +1e+01 | — | — | — | — | 4.76 ms | — | — | — | — | — | — | — | — | — |
| Python · external |
|---|
📐ref.python_scikit_learn | source | — | — | — | — | 60.4 ms | — | — | — | — | — | — | — | — | — |
| R · external |
|---|
📐ref.r_pls | ~ shape 4e-15 | — | — | — | — | 195.9 ms | — | — | — | — | — | — | — | — | — |
:::
:::{tab-item} 10 threads
:sync: threads-10
| Backend | Parity | 50×250 (ms) | 100×50 (ms) | 100×500 (ms) | 100×2500 (ms) | 200×50 (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 9e-16 | — | — | — | — | 2.80 ms🏆 | — | — | — | — | — | — | — | — | — |
pls4all.cpp.blas+omp | ~ shape 9e-16 | — | — | — | — | 2.84 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.omp | ~ shape 1e-15 | — | — | — | — | 3.37 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.ref | ~ shape 1e-15 | — | — | — | — | 3.45 ms | — | — | — | — | — | — | — | — | — |
| Python · pls4all |
|---|
pls4all.python | ✓ 4e-15 | — | — | — | — | 2.91 ms | — | — | — | — | — | — | — | — | — |
pls4all.sklearn | ✓ 4e-15 | — | — | — | — | 2.98 ms | — | — | — | — | — | — | — | — | — |
| R · pls4all |
|---|
pls4all.R | ✓ bind | — | — | — | — | 6.38 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.formula | ✓ bind | — | — | — | — | 7.69 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.mdatools | ✓ bind | — | — | — | — | 7.80 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.pls | ✓ bind | — | — | — | — | 7.58 ms | — | — | — | — | — | — | — | — | — |
| MATLAB · pls4all |
|---|
pls4all.matlab | ✗ +1e+01 | — | — | — | — | 4.05 ms | — | — | — | — | — | — | — | — | — |
pls4all.matlab.classdef | ✗ +1e+01 | — | — | — | — | 4.35 ms | — | — | — | — | — | — | — | — | — |
| Python · external |
|---|
📐ref.python_scikit_learn | source | — | — | — | — | 52.4 ms | — | — | — | — | — | — | — | — | — |
| R · external |
|---|
📐ref.r_pls | ~ shape 4e-15 | — | — | — | — | 147.7 ms | — | — | — | — | — | — | — | — | — |
:::
::::
---
_See also_: [benchmark overview](../benchmarks/overview.md) · [methods index](index.md) · [interactive dashboard](../landing/dashboard.md)