# `cppls` — Powered PLS (Indahl 2005)
_Group_: **Core PLS** · _Registry tolerance_: `0.1`
## Description
CPPLS (Canonical Powered PLS, Indahl Liland & Næs 2009)
From the `pls4all.sklearn.CPPLSRegression` docstring:
> Canonical Powered PLS (Indahl, Liland & Næs 2009).
> **Registry note** — R `pls::cppls 2.9.0` Canonical Powered PLS (lower=upper=0.5 default reduces to NIPALS PLS1 with X-only deflation for q=1). pls4all default matches; SIMPLS column-σ^γ variant available via cfg.solver = SIMPLS.
### Parameters
| Name | Type | Default | Notes |
|------|------|---------|-------|
| `n_components` | `int` | `2` | Number of latent components extracted (k). |
| `gamma` | `float` | `0.5` | Covariance/correlation mixing exponent (0 → covariance-maximizing PLS, 1 → correlation-maximizing). |
| `solver` | `—` | `None` | Inner algorithm: 'nipals', 'simpls', 'svd', 'kernel', 'orthogonal-scores', 'power', 'randomized-svd', 'wide-kernel'. |
## Explanations
### Bibliographic source
Indahl, U. G. (2005). *A twist to partial least squares regression*. Journal of Chemometrics 19(1), 32–44.
### Mathematical principle
Powered PLS introduces a single hyperparameter $\gamma \in [0, 1]$ that morphs the loading-weight definition between PCA ($\gamma=0$) and PLS ($\gamma=1$). Concretely, the loading weight is $\mathbf{w} \propto \operatorname{sign}(\mathbf{X}^{\top}\mathbf{y}) \odot |\mathbf{X}^{\top}\mathbf{y}|^{\gamma}$ where $\odot$ is the element-wise product.
When $\mathbf{X}$ carries weakly informative columns alongside strongly informative ones, raising the covariance to a power $\gamma < 1$ tempers the influence of the dominant columns and produces a more uniform weighting, which empirically improves CV-RMSE on spectra with sharp absorption peaks dominating the covariance. The implementation reduces to standard SIMPLS at $\gamma=1$.
**Important nomenclature caveat:** R's `pls::cppls` (Liland 2009) implements *Canonical Powered PLS*, a completely different algorithm that orthogonalises blocks of $\mathbf{Y}$ before powering. The pls4all implementation matches Indahl 2005, **not** Liland 2009. The benchmark widens the parity tolerance for this method to surface the divergence as a drift rather than treat it as a bug.
### Implementation
`n4m_cppls_fit` (MethodResult entry point). No widely installable Python reference; R `pls::cppls 2.8.5` is the closest installable analogue but implements Liland 2009 rather than Indahl 2005.
MATLAB header (`bindings/matlab/+pls4all/CpplsRegression.m`):
```text
pls4all.CpplsRegression — Canonical Powered PLS (Indahl 2005).
```
### Usage
Direct `n4m` Python helper:
```python
import n4m
res = n4m.cppls(X, y, n_components=4, gamma=0.5)
y_hat = res["predictions"]
coef = res["coefficients"]
```
Reusable sklearn-style wrapper:
```python
from n4m.sklearn import NativeCPPLSRegressor
model = NativeCPPLSRegressor(n_components=4, gamma=0.5).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_cppls_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 cppls_fit
with pls4all.Context() as ctx, pls4all.Config() as cfg:
res = cppls_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 CPPLSRegression
mdl = CPPLSRegression(n_components=2, gamma=0.5, solver=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("cppls", X, y,
n_components = 4L, params = list(gamma = 0.5))
# res is a named list with MethodResult arrays/scalars.
# selected_indices / top_k_intervals are 1-based.
```
:::
:::{tab-item} R · `mdatools` compat
:sync: r-mdatools
:class-label: lang-r
```r
library(pls4all)
# Drop-in for `mdatools::pls(x, y, ncomp, method = "cppls")`.
fit <- pls_mdatools(X, y, ncomp = 4L, method = "cppls",
center = TRUE, scale = FALSE)
yhat <- predict(fit, newdata = X_test, ncomp = 4L)
```
:::
:::{tab-item} MATLAB · pls4all (MEX)
:sync: matlab-mex
:class-label: lang-matlab
```matlab
res = pls4all.cppls(X, y, 4);
% see header of bindings/matlab/+pls4all/cppls.m for full
% parameter surface:
% res = cppls(X, Y, n_components, gamma)
yhat = predict(res, Xtest);
```
:::
:::{tab-item} MATLAB · pls4all (classdef)
:sync: matlab-classdef
:class-label: lang-matlab
```matlab
mdl = pls4all.fit("cppls", X, y, "NumComponents", 4);
yhat = predict(mdl, Xtest);
```
:::
::::
**Registry parity references** 📐
:::{card}
:class-card: external-refs
- 📐 **`ref.r_pls`** (R · r) — `pls` 2.9.0 · qualitative (rmse_rel ≤ 1e-01) — R `pls::cppls` Canonical Powered PLS (Indahl, Liland & Næs 2009); default lower=upper=0.5 reduces to NIPALS PLS1 with X-only deflation for q=1.
:::
### 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 | ≈ +3e-02 | 2.41 ms | 2.04 ms | 10.1 ms | 53.1 ms🏆 | 1.88 ms | 2.51 ms | 4.83 ms🏆 | 57.1 ms | 302.4 ms | 26.3 ms🏆 | 349.0 ms | 1.8 s | 114.1 ms | 1.4 s |
pls4all.cpp.blas+omp | ≈ +3e-02 | 2.68 ms | 1.00 ms🏆 | 12.0 ms | 55.3 ms | 1.89 ms | 2.67 ms | 5.12 ms | 56.3 ms | 292.8 ms🏆 | 28.0 ms | 337.5 ms🏆 | 1.8 s🏆 | 104.0 ms🏆 | 1.4 s🏆 |
pls4all.cpp.omp | ≈ +3e-02 | 2.72 ms | 1.06 ms | 9.88 ms🏆 | 54.2 ms | 1.99 ms | 2.70 ms | 5.23 ms | 58.2 ms | 294.7 ms | 26.4 ms | 350.1 ms | 1.8 s | 116.3 ms | 1.4 s |
pls4all.cpp.ref | ≈ +3e-02 | 2.44 ms | 1.08 ms | 11.2 ms | 53.6 ms | 1.85 ms🏆 | 2.48 ms🏆 | 6.29 ms | 55.8 ms🏆 | 307.6 ms | 27.4 ms | 354.9 ms | 1.8 s | 111.3 ms | 1.4 s |
| Python · pls4all |
|---|
pls4all.python | ✓ bind | 2.39 ms🏆 | — | — | — | 2.04 ms | 2.56 ms | — | — | — | — | — | — | — | — |
pls4all.sklearn | ✓ 4e-15 | 2.82 ms | — | — | — | 2.47 ms | 3.22 ms | — | — | — | — | — | — | — | — |
| R · pls4all |
|---|
pls4all.R | ✗ +3e-02 | 11.4 ms | — | — | — | 6.83 ms | 8.08 ms | — | — | — | — | — | — | — | — |
pls4all.R.formula | ✗ +3e-02 | 19.2 ms | — | — | — | 10.0 ms | 8.18 ms | — | — | — | — | — | — | — | — |
pls4all.R.mdatools | ✗ +3e-02 | 19.0 ms | — | — | — | 10.7 ms | 9.35 ms | — | — | — | — | — | — | — | — |
pls4all.R.pls | ✗ +3e-02 | 41.4 ms | — | — | — | 18.0 ms | 17.3 ms | — | — | — | — | — | — | — | — |
| MATLAB · pls4all |
|---|
pls4all.matlab | ✗ +9e+00 | 4.48 ms | — | — | — | 3.19 ms | 5.41 ms | — | — | — | — | — | — | — | — |
pls4all.matlab.classdef | ✗ +9e+00 | 4.63 ms | — | — | — | 3.75 ms | 5.33 ms | — | — | — | — | — | — | — | — |
| R · external |
|---|
📐ref.r_pls | source | 31.8 ms | — | — | — | 14.6 ms | 15.4 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 | ~ shape 3e-02 | — | — | — | — | 1.92 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.blas+omp | ~ shape 3e-02 | — | — | — | — | 2.12 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.omp | ~ shape 3e-02 | — | — | — | — | 1.97 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.ref | ~ shape 3e-02 | — | — | — | — | 1.90 ms🏆 | — | — | — | — | — | — | — | — | — |
| Python · pls4all |
|---|
pls4all.python | ✓ bind | — | — | — | — | 1.97 ms | — | — | — | — | — | — | — | — | — |
pls4all.sklearn | ✓ 4e-15 | — | — | — | — | 2.24 ms | — | — | — | — | — | — | — | — | — |
| R · pls4all |
|---|
pls4all.R | ✗ +3e-02 | — | — | — | — | 6.54 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.formula | ✗ +3e-02 | — | — | — | — | 9.89 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.mdatools | ✗ +3e-02 | — | — | — | — | 9.26 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.pls | ✗ +3e-02 | — | — | — | — | 14.6 ms | — | — | — | — | — | — | — | — | — |
| MATLAB · pls4all |
|---|
pls4all.matlab | ✗ +9e+00 | — | — | — | — | 2.94 ms | — | — | — | — | — | — | — | — | — |
pls4all.matlab.classdef | ✗ +9e+00 | — | — | — | — | 5.77 ms | — | — | — | — | — | — | — | — | — |
| R · external |
|---|
📐ref.r_pls | source | — | — | — | — | 13.6 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 3e-02 | — | — | — | — | 1.75 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.blas+omp | ~ shape 3e-02 | — | — | — | — | 1.75 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.omp | ~ shape 3e-02 | — | — | — | — | 1.74 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.ref | ~ shape 3e-02 | — | — | — | — | 1.74 ms🏆 | — | — | — | — | — | — | — | — | — |
| Python · pls4all |
|---|
pls4all.python | ✓ bind | — | — | — | — | 1.76 ms | — | — | — | — | — | — | — | — | — |
pls4all.sklearn | ✓ 4e-15 | — | — | — | — | 1.93 ms | — | — | — | — | — | — | — | — | — |
| R · pls4all |
|---|
pls4all.R | ✗ +3e-02 | — | — | — | — | 5.27 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.formula | ✗ +3e-02 | — | — | — | — | 5.97 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.mdatools | ✗ +3e-02 | — | — | — | — | 6.83 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.pls | ✗ +3e-02 | — | — | — | — | 11.8 ms | — | — | — | — | — | — | — | — | — |
| MATLAB · pls4all |
|---|
pls4all.matlab | ✗ +9e+00 | — | — | — | — | 2.89 ms | — | — | — | — | — | — | — | — | — |
pls4all.matlab.classdef | ✗ +9e+00 | — | — | — | — | 3.23 ms | — | — | — | — | — | — | — | — | — |
| R · external |
|---|
📐ref.r_pls | source | — | — | — | — | 11.6 ms | — | — | — | — | — | — | — | — | — |
:::
::::
---
_See also_: [benchmark overview](../benchmarks/overview.md) · [methods index](index.md) · [interactive dashboard](../landing/dashboard.md)