# `cppls` — Powered PLS (Indahl 2005)
_Group_: **Core PLS** · _Registry tolerance_: `1e-08`
## 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 matches at ~8e-16. SIMPLS column-σ^γ variant 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_estimators_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.
R roxygen note (`sklearn_methods.R::cppls`):
> Canonical Powered PLS — formula entry point.
MATLAB header (`bindings/matlab/+pls4all/CpplsRegression.m`):
```text
pls4all.CpplsRegression — Canonical Powered PLS (Indahl 2005).
```
### 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_estimators_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 · pls4all (raw fn)
:sync: r-raw
:class-label: lang-r
```r
library(pls4all)
res <- cppls_fit(X, Y, n_components, gamma = 0.5)
yhat <- pls4all_predict(res, X_test)
```
:::
:::{tab-item} R · pls4all (formula+S3)
:sync: r-formula
:class-label: lang-r
```r
library(pls4all)
fit <- cppls(y ~ ., data = train, ncomp = 4L)
yhat <- predict(fit, newdata = test)
summary(fit)
```
:::
:::{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 · strict (rmse_rel ≤ 1e-08) — 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 · ⇄ cross-check = documented by-design selector/RNG/model, noncanonical API/facade convention, or secondary oracle · ✗ divergent · ⚠ error · — not run. The fastest backend per column is marked 🏆.
**Reference gate**: strict — numeric equivalence (`rmse_rel_tol ≤ 1e-08`).
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 | 200×50 (ms) |
|---|
| C++ native · libn4m |
|---|
pls4all.cpp.blas+omp | ✓ ref 8e-16 | 1.88 ms |
| Python · pls4all |
|---|
pls4all.python | ✓ bind | 1.87 ms🏆 |
pls4all.sklearn | ✓ 4e-15 | 1.97 ms |
| R · pls4all |
|---|
pls4all.R | ✓ bind | 4.57 ms |
pls4all.R.formula | ✓ bind | 5.65 ms |
pls4all.R.mdatools | ⇄ +3e-02 | 6.07 ms |
pls4all.R.pls | ⇄ +3e-02 | 10.6 ms |
| R · external |
|---|
📐ref.r_pls | source | 9.91 ms |
:::
:::{tab-item} 3 threads
:sync: threads-3
| Backend | Parity | 200×50 (ms) |
|---|
| C++ native · libn4m |
|---|
pls4all.cpp.blas+omp | ✓ ref 8e-16 | 1.78 ms🏆 |
| Python · pls4all |
|---|
pls4all.python | ✓ bind | 1.79 ms |
pls4all.sklearn | ✓ 4e-15 | 1.97 ms |
| R · pls4all |
|---|
pls4all.R | ✓ bind | 4.38 ms |
pls4all.R.formula | ✓ bind | 5.49 ms |
pls4all.R.mdatools | ⇄ +3e-02 | 6.44 ms |
pls4all.R.pls | ⇄ +3e-02 | 10.6 ms |
| R · external |
|---|
📐ref.r_pls | source | 9.85 ms |
:::
:::{tab-item} 10 threads
:sync: threads-10
| Backend | Parity | 200×50 (ms) |
|---|
| C++ native · libn4m |
|---|
pls4all.cpp.blas+omp | ✓ ref 8e-16 | 1.69 ms🏆 |
| Python · pls4all |
|---|
pls4all.python | ✓ bind | 1.80 ms |
pls4all.sklearn | ✓ 4e-15 | 1.95 ms |
| R · pls4all |
|---|
pls4all.R | ✓ bind | 4.57 ms |
pls4all.R.formula | ✓ bind | 5.38 ms |
pls4all.R.mdatools | ⇄ +3e-02 | 6.59 ms |
pls4all.R.pls | ⇄ +3e-02 | 11.1 ms |
| R · external |
|---|
📐ref.r_pls | source | 10.5 ms |
:::
::::
---
_See also_: [benchmark overview](../benchmarks/overview.md) · [methods index](index.md) · [interactive dashboard](../landing/dashboard.md)