# `ecr` — ECR — Elastic Component Regression
_Group_: **Calibration transfer** · _Registry tolerance_: `0.001`
## Description
Elastic Component Regression (Phase 50)
From the `pls4all.sklearn.ECRegression` docstring:
> Elastic Component Regression (Liu 2013) — interpolates PCR (α=0)
and PLS (α=1).
> **Registry note** — Octave-bridged libPLS 1.95 `ecr(X, y, A, 'center', alpha)`. Deterministic algorithm; small numerical differences arise only from the power-method tolerance and FP accumulation order.
### Parameters
| Name | Type | Default | Notes |
|------|------|---------|-------|
| `n_components` | `int` | `2` | Number of latent components extracted (k). |
| `alpha` | `float` | `0.5` | Elastic-net mixing weight (0 = pure L2, 1 = pure L1) applied to the PLS coefficient path. |
## Explanations
### Bibliographic source
Liu, Y., Zhang, B. & Hu, J. (2013). *Elastic Component Regression*. Chemometrics and Intelligent Laboratory Systems 124, 73–79. — adapted in pls4all as a continuum/elastic blend.
### Mathematical principle
ECR interpolates between PCR and PLS via a single parameter $\alpha \in [0, 1]$ that mixes the two loading-weight criteria. The latent direction is $\mathbf{w} \propto (1-\alpha)\mathbf{X}^{\top}\mathbf{X}\mathbf{w} + \alpha \mathbf{X}^{\top}\mathbf{y}$, which recovers PCR at $\alpha = 0$ (the leading eigenvector of $\mathbf{X}^{\top}\mathbf{X}$) and PLS at $\alpha = 1$ (proportional to $\mathbf{X}^{\top}\mathbf{y}$). Intermediate $\alpha$ blends variance and covariance criteria; the optimum is typically located by cross-validation.
ECR is closely related to continuum regression with a different parameterisation, and in practice serves a similar purpose: when neither PCR nor PLS dominates RMSE on a given dataset, an interpolating method often wins by a small margin and offers a smooth tunable spectrum.
### Implementation
`n4m_ecr_fit`. No widely installable reference; treated as `paper_only` in the registry.
MATLAB header (`bindings/matlab/+pls4all/EcrRegression.m`):
```text
pls4all.EcrRegression Elastic Component Regression (Liu 2009).
```
### Usage
Direct `n4m` Python helper:
```python
import n4m
res = n4m.ecr(X, y, n_components=4, alpha=0.6)
y_hat = res["predictions"]
coef = res["coefficients"]
```
Reusable sklearn-style wrapper:
```python
from n4m.sklearn import NativeECRRegressor
model = NativeECRRegressor(n_components=4, alpha=0.6).fit(X, y)
y_hat = model.predict(X_test)
```
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_ecr_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 ecr_fit
with pls4all.Context() as ctx, pls4all.Config() as cfg:
res = ecr_fit(ctx, cfg, X, y, n_components=4)
# 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 ECRegression
mdl = ECRegression(n_components=2, alpha=0.5)
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("ecr", X, y,
n_components = 4L, params = list(alpha = 0.5))
# 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.ecr(X, y, 4);
% see header of bindings/matlab/+pls4all/ecr.m for full
% parameter surface:
% res = ecr(X, Y, n_components, alpha)
yhat = predict(res, Xtest);
```
:::
:::{tab-item} MATLAB · pls4all (classdef)
:sync: matlab-classdef
:class-label: lang-matlab
```matlab
mdl = pls4all.fit("ecr", X, y, "NumComponents", 4);
yhat = predict(mdl, Xtest);
```
:::
::::
**Registry parity references** 📐
:::{card}
:class-card: external-refs
- 📐 **`ref.matlab_libpls`** (matlab · python) — `libPLS` 1.95 · relaxed (rmse_rel ≤ 1e-03) — Octave-bridged libPLS 1.95 `ecr(X, y, A, 'center', alpha)`. Predictions computed as X_predict @ B + y_mean using the fitted coefficient matrix and centring parameters.
:::
### 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**: relaxed — known algorithmic drift between pls4all and the external reference (`rmse_rel_tol ≤ 1e-03`).
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 | ≈ +4e-07 | 11.5 ms | 16.4 ms | 709.9 ms | 120.3 s | 2.25 ms🏆 | 3.28 ms🏆 | 44.0 ms | 2.1 s | 226.6 s🏆 | 213.3 ms🏆 | 9.9 s | — | 905.3 ms🏆 | 41.1 s |
pls4all.cpp.blas+omp | ≈ +4e-07 | 12.2 ms | 16.8 ms | 699.3 ms🏆 | 121.9 s | 2.31 ms | 3.32 ms | 41.6 ms🏆 | 2.0 s | 228.9 s | 222.2 ms | 9.6 s | — | 979.6 ms | 41.0 s |
pls4all.cpp.omp | ≈ +4e-07 | 11.7 ms | 15.8 ms🏆 | 705.2 ms | 122.9 s | 2.73 ms | 3.47 ms | 48.6 ms | 1.9 s🏆 | 238.3 s | 254.3 ms | 9.3 s🏆 | — | 932.8 ms | 38.7 s🏆 |
pls4all.cpp.ref | ≈ +4e-07 | 11.9 ms | 16.5 ms | 742.2 ms | 120.0 s🏆 | 2.37 ms | 3.38 ms | 45.8 ms | 1.9 s | 241.0 s | 219.9 ms | 9.7 s | — | 1.0 s | 41.3 s |
| Python · pls4all |
|---|
pls4all.python | ✓ bind | 11.2 ms🏆 | — | — | — | 2.62 ms | 3.52 ms | — | — | — | — | — | — | — | — |
pls4all.sklearn | ✓ 4e-15 | 12.6 ms | — | — | — | 4.41 ms | 3.65 ms | — | — | — | — | — | — | — | — |
| R · pls4all |
|---|
pls4all.R | ✓ bind | 19.1 ms | — | — | — | 9.03 ms | 13.2 ms | — | — | — | — | — | — | — | — |
pls4all.R.formula | ✓ bind | 28.5 ms | — | — | — | 10.6 ms | 10.8 ms | — | — | — | — | — | — | — | — |
pls4all.R.mdatools | ✓ bind | 32.2 ms | — | — | — | 8.82 ms | 10.7 ms | — | — | — | — | — | — | — | — |
pls4all.R.pls | ✓ bind | 29.1 ms | — | — | — | 10.6 ms | 11.0 ms | — | — | — | — | — | — | — | — |
| MATLAB · pls4all |
|---|
pls4all.matlab | ✗ +9e+00 | 19.4 ms | — | — | — | 3.58 ms | 8.43 ms | — | — | — | — | — | — | — | — |
pls4all.matlab.classdef | ✗ +9e+00 | 14.0 ms | — | — | — | 4.21 ms | 5.85 ms | — | — | — | — | — | — | — | — |
| MATLAB · external |
|---|
📐ref.matlab_libpls | ⚠ | — | — | — | — | 16.4 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 | ≈ ref 4e-07 | — | — | — | — | 2.20 ms🏆 | — | — | — | — | — | — | — | — | — |
pls4all.cpp.blas+omp | ≈ ref 4e-07 | — | — | — | — | 3.03 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.omp | ≈ ref 4e-07 | — | — | — | — | 2.21 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.ref | ≈ ref 4e-07 | — | — | — | — | 2.21 ms | — | — | — | — | — | — | — | — | — |
| Python · pls4all |
|---|
pls4all.python | ✓ bind | — | — | — | — | 2.55 ms | — | — | — | — | — | — | — | — | — |
pls4all.sklearn | ✓ 3e-15 | — | — | — | — | 4.51 ms | — | — | — | — | — | — | — | — | — |
| R · pls4all |
|---|
pls4all.R | ✓ bind | — | — | — | — | 7.15 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.formula | ✓ bind | — | — | — | — | 8.29 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.mdatools | ✓ bind | — | — | — | — | 7.79 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.pls | ✓ bind | — | — | — | — | 8.84 ms | — | — | — | — | — | — | — | — | — |
| MATLAB · pls4all |
|---|
pls4all.matlab | ✗ +9e+00 | — | — | — | — | 5.62 ms | — | — | — | — | — | — | — | — | — |
pls4all.matlab.classdef | ✗ +9e+00 | — | — | — | — | 4.40 ms | — | — | — | — | — | — | — | — | — |
| MATLAB · external |
|---|
📐ref.matlab_libpls | source | — | — | — | — | 13.3 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 | ≈ ref 4e-07 | — | — | — | — | 2.18 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.blas+omp | ≈ ref 4e-07 | — | — | — | — | 2.13 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.omp | ≈ ref 4e-07 | — | — | — | — | 2.10 ms🏆 | — | — | — | — | — | — | — | — | — |
pls4all.cpp.ref | ≈ ref 4e-07 | — | — | — | — | 2.21 ms | — | — | — | — | — | — | — | — | — |
| Python · pls4all |
|---|
pls4all.python | ✓ bind | — | — | — | — | 2.14 ms | — | — | — | — | — | — | — | — | — |
pls4all.sklearn | ✓ 3e-15 | — | — | — | — | 2.33 ms | — | — | — | — | — | — | — | — | — |
| R · pls4all |
|---|
pls4all.R | ✓ bind | — | — | — | — | 5.58 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.formula | ✓ bind | — | — | — | — | 6.64 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.mdatools | ✓ bind | — | — | — | — | 6.43 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.pls | ✓ bind | — | — | — | — | 6.17 ms | — | — | — | — | — | — | — | — | — |
| MATLAB · pls4all |
|---|
pls4all.matlab | ✗ +9e+00 | — | — | — | — | 3.26 ms | — | — | — | — | — | — | — | — | — |
pls4all.matlab.classdef | ✗ +9e+00 | — | — | — | — | 4.34 ms | — | — | — | — | — | — | — | — | — |
| MATLAB · external |
|---|
📐ref.matlab_libpls | source | — | — | — | — | 9.23 ms | — | — | — | — | — | — | — | — | — |
:::
::::
---
_See also_: [benchmark overview](../benchmarks/overview.md) · [methods index](index.md) · [interactive dashboard](../landing/dashboard.md)