# `lw_pls` — Locally-Weighted PLS (LW-PLS)
_Group_: **Nonlinear / local** · _Registry tolerance_: `5.0`
## Description
LW-PLS — Locally-weighted PLS (§17 Phase 4)
From the `pls4all.sklearn.LWPLSRegression` docstring:
> Locally-weighted PLS (Næs & Centner 1998).
> **Registry note** — In-tree `nirs4all.operators.models.sklearn.lwpls.LWPLS` is the sanctioned external reference. pls4all defaults to the Gaussian-weighted local PLS that matches nirs4all bit-for-bit (max_abs < 1e-13); the legacy k-NN cutoff variant is opt-in via cfg.solver = SIMPLS.
### Parameters
| Name | Type | Default | Notes |
|------|------|---------|-------|
| `n_components` | `int` | `2` | Number of latent components extracted (k). |
| `n_neighbors` | `int` | `30` | Number of training neighbours used for each local prediction (LW-PLS). |
## Explanations
### Bibliographic source
Centner, V. & Massart, D. L. (1998). *Optimisation in locally weighted regression*. Analytical Chemistry 70(19), 4206–4211.
### Mathematical principle
Instead of fitting a single global PLS, LW-PLS refits a *per-prediction-point* local PLS using only the $k$-nearest calibration samples (in $\mathbf{X}$-space distance). This adapts the model to the local geometry around each query point and is effective on calibration sets that span heterogeneous regimes (e.g. a single instrument calibrated across several product classes).
The neighbourhood weight typically combines distance (Gaussian or tricube kernel on the Euclidean / Mahalanobis distance) with the inverse residual variance from a preliminary global fit. The local PLS uses few components (typically 2–4) because the neighbourhood is small.
Prediction cost is $O(n)$ for the neighbour search plus $O(k_{\mathrm{nn}} \cdot p \cdot k_{\mathrm{pls}})$ for the local fit, per query. KD-tree / ball-tree indices accelerate the neighbour search; pls4all uses an exhaustive scan because $p \gg n$ defeats most spatial indices for NIR data anyway.
### Implementation
`n4m_lw_pls_fit`. Reference: sanctioned git-pinned port `nirs4all.operators.models.sklearn.lwpls`.
MATLAB header (`bindings/matlab/+pls4all/lw_pls.m`):
```text
pls4all.lw_pls Locally-weighted PLS (Næs & Centner 1998).
```
### 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_lw_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 lw_pls_fit
with pls4all.Context() as ctx, pls4all.Config() as cfg:
res = lw_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 LWPLSRegression
mdl = LWPLSRegression(n_components=2, n_neighbors=30)
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("lw_pls", X, y,
n_components = 3L, params = list(n_neighbors = 30L))
# 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.lw_pls(X, y, 3);
% see header of bindings/matlab/+pls4all/lw_pls.m for full
% parameter surface:
% res = lw_pls(X, Y, n_components, n_neighbors)
yhat = predict(res, Xtest);
```
:::
:::{tab-item} MATLAB · pls4all (classdef)
:sync: matlab-classdef
:class-label: lang-matlab
_No idiomatic classdef wrapper — invoke `pls4all.fit("lw_pls", X, y, …)` directly from the unified MEX factory._
:::
::::
**Registry parity references** 📐
:::{card}
:class-card: external-refs
- 📐 **`nirs4all`** (python · python) — `nirs4all` in-tree · qualitative (rmse_rel ≤ 5e+00) — In-tree Python LW-PLS (sanctioned external reference). Locally-weighted PLS (Naes 1990 / Centner 1998). pls4all's default solver (NIPALS) implements the same Gaussian-weighted local PLS as the nirs4all reference, deriving the kernel bandwidth `lambda = max(1.0, 0.5 * n_neighbors)`. The legacy k-NN cutoff variant remains available via cfg.solver = SIMPLS.
:::
### 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 ≤ 5e+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) | 200×40 (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 | ≈ +7e-16 | 8.28 ms | 4.44 ms🏆 | 53.9 ms | 296.3 ms | 10.7 ms | 26.0 ms | 123.6 ms | 1.8 s🏆 | 19.0 s🏆 | 4.0 s | 212.9 s | — | 107.8 s | — |
pls4all.cpp.blas+omp | ≈ +7e-16 | 6.56 ms🏆 | 6.21 ms | 55.4 ms | 313.2 ms | 10.2 ms | 28.2 ms | 123.7 ms | 1.8 s | 19.1 s | 4.0 s🏆 | 215.5 s | — | 109.8 s | — |
pls4all.cpp.omp | ≈ +7e-16 | 8.37 ms | 6.03 ms | 52.0 ms🏆 | 291.6 ms🏆 | 9.62 ms | 26.3 ms | 118.5 ms🏆 | 1.8 s | 19.3 s | 4.0 s | 216.6 s | — | 109.0 s | — |
pls4all.cpp.ref | ≈ +7e-16 | 8.20 ms | 4.45 ms | 54.9 ms | 304.9 ms | 9.60 ms🏆 | 28.1 ms | 127.9 ms | 1.8 s | 19.1 s | 4.1 s | 211.3 s🏆 | — | 106.1 s🏆 | — |
| Python · pls4all |
|---|
pls4all.python | ✓ bind | 6.60 ms | — | — | — | 10.4 ms | 25.6 ms🏆 | — | — | — | — | — | — | — | — |
pls4all.sklearn | ✗ +1e+00 | 6.27 ms | — | — | — | 4.73 ms | 9.17 ms | — | — | — | — | — | — | — | — |
| R · pls4all |
|---|
pls4all.R | ✗ +4e-01 | 17.5 ms | — | — | — | 14.0 ms | 17.8 ms | — | — | — | — | — | — | — | — |
pls4all.R.formula | ✗ +4e-01 | 35.0 ms | — | — | — | 14.2 ms | 16.6 ms | — | — | — | — | — | — | — | — |
pls4all.R.mdatools | ✗ +4e-01 | 25.4 ms | — | — | — | 14.1 ms | 15.1 ms | — | — | — | — | — | — | — | — |
pls4all.R.pls | ✗ +4e-01 | 22.1 ms | — | — | — | 16.9 ms | 15.9 ms | — | — | — | — | — | — | — | — |
| MATLAB · pls4all |
|---|
pls4all.matlab | ✗ +1e+01 | 6.44 ms | — | — | — | 5.55 ms | 9.16 ms | — | — | — | — | — | — | — | — |
pls4all.matlab.classdef | ✗ +1e+01 | 8.67 ms | — | — | — | 5.76 ms | 10.8 ms | — | — | — | — | — | — | — | — |
| Python · external |
|---|
📐nirs4all | source | 11.2 ms | — | — | — | 18.2 ms | 58.6 ms | — | — | — | — | — | — | — | — |
:::
:::{tab-item} 3 threads
:sync: threads-3
| Backend | Parity | 50×250 (ms) | 100×50 (ms) | 100×500 (ms) | 100×2500 (ms) | 200×40 (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 | — | — | — | — | 10.7 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.blas+omp | ~ shape 7e-16 | — | — | — | — | 9.79 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.omp | ~ shape 7e-16 | — | — | — | — | 9.95 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.ref | ~ shape 7e-16 | — | — | — | — | 11.1 ms | — | — | — | — | — | — | — | — | — |
| Python · pls4all |
|---|
pls4all.python | ✓ bind | — | — | — | — | 9.57 ms🏆 | — | — | — | — | — | — | — | — | — |
pls4all.sklearn | ✗ +1e+00 | — | — | — | — | 4.44 ms | — | — | — | — | — | — | — | — | — |
| R · pls4all |
|---|
pls4all.R | ✓ 3e-15 | — | — | — | — | 12.9 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.formula | ✓ 3e-15 | — | — | — | — | 14.0 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.mdatools | ✓ 3e-15 | — | — | — | — | 14.5 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.pls | ✓ 3e-15 | — | — | — | — | 14.7 ms | — | — | — | — | — | — | — | — | — |
| MATLAB · pls4all |
|---|
pls4all.matlab | ✗ +1e+01 | — | — | — | — | 5.18 ms | — | — | — | — | — | — | — | — | — |
pls4all.matlab.classdef | ✗ +1e+01 | — | — | — | — | 6.15 ms | — | — | — | — | — | — | — | — | — |
| Python · external |
|---|
📐nirs4all | source | — | — | — | — | 20.4 ms | — | — | — | — | — | — | — | — | — |
:::
:::{tab-item} 10 threads
:sync: threads-10
| Backend | Parity | 50×250 (ms) | 100×50 (ms) | 100×500 (ms) | 100×2500 (ms) | 200×40 (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 | — | — | — | — | 8.99 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.blas+omp | ~ shape 7e-16 | — | — | — | — | 8.89 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.omp | ~ shape 7e-16 | — | — | — | — | 8.85 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.ref | ~ shape 7e-16 | — | — | — | — | 8.78 ms🏆 | — | — | — | — | — | — | — | — | — |
| Python · pls4all |
|---|
pls4all.python | ✓ bind | — | — | — | — | 8.99 ms | — | — | — | — | — | — | — | — | — |
pls4all.sklearn | ✗ +1e+00 | — | — | — | — | 3.92 ms | — | — | — | — | — | — | — | — | — |
| R · pls4all |
|---|
pls4all.R | ✓ 3e-15 | — | — | — | — | 11.1 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.formula | ✓ 3e-15 | — | — | — | — | 12.3 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.mdatools | ✓ 3e-15 | — | — | — | — | 12.2 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.pls | ✓ 3e-15 | — | — | — | — | 12.1 ms | — | — | — | — | — | — | — | — | — |
| MATLAB · pls4all |
|---|
pls4all.matlab | ✗ +1e+01 | — | — | — | — | 4.71 ms | — | — | — | — | — | — | — | — | — |
pls4all.matlab.classdef | ✗ +1e+01 | — | — | — | — | 5.07 ms | — | — | — | — | — | — | — | — | — |
| Python · external |
|---|
📐nirs4all | source | — | — | — | — | 16.9 ms | — | — | — | — | — | — | — | — | — |
:::
::::
---
_See also_: [benchmark overview](../benchmarks/overview.md) · [methods index](index.md) · [interactive dashboard](../landing/dashboard.md)