`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):

pls4all.CpplsRegression — Canonical Powered PLS (Indahl 2005).

Usage¶

Direct n4m Python helper:

import n4m

res = n4m.cppls(X, y, n_components=4, gamma=0.5)
y_hat = res["predictions"]
coef = res["coefficients"]

Reusable sklearn-style wrapper:

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

C ABI · libn4m

/* 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);

Python · pls4all (raw)

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"), …

Python · pls4all.sklearn

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)

R · pls4all_method()

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.

R · mdatools compat

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)

MATLAB · pls4all (MEX)

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);

MATLAB · pls4all (classdef)

mdl  = pls4all.fit("cppls", X, y, "NumComponents", 4);
yhat = predict(mdl, Xtest);

Registry parity references 📐

📐 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. 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). 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.

1 thread

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

3 threads

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

10 threads

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

nirs4all-methods

Navigation

`cppls` — Powered PLS (Indahl 2005)¶

Description¶

Parameters¶

Explanations¶

Bibliographic source¶

Mathematical principle¶

Implementation¶

Usage¶

Benchmarks¶

cppls — Powered PLS (Indahl 2005)¶

Description¶

Parameters¶

Explanations¶

Bibliographic source¶

Mathematical principle¶

Implementation¶

Usage¶

Benchmarks¶

`cppls` — Powered PLS (Indahl 2005)¶