pls_qda — PLS-QDA

Group: Classification & GLM · Registry tolerance: 1e-06

Description

PLS-QDA (§5) — quadratic discriminant on PLS scores

From the pls4all.sklearn.PLSQDAClassifier docstring:

PLS-QDA on PLS scores (in-sample only).

Registry note — sklearn PLSRegression(scale=False) -> QuadraticDiscriminantAnalysis(reg_param=0.0) pipeline. pls4all’s default now mirrors this convention: NIPALS PLS scores via the C kernel, then sklearn-style QDA predict_proba in Python. The legacy single-pass C++ kernel (SIMPLS + identity-covariance log-posterior) is opt-in via legacy=True.

Parameters

Name

Type

Default

Notes

n_components

int

2

Number of latent components extracted (k).

n_classes

int

3

registry benchmark cell value

Explanations

Bibliographic source

Pérez-Enciso, M. & Tenenhaus, M. (2003). Prediction of clinical outcome with microarray data: a partial least squares discriminant analysis (PLS-DA) approach. Human Genetics 112(5–6), 581–592.

Mathematical principle

Replace LDA with QDA in the second stage of PLS-LDA: instead of assuming a shared covariance across classes, fit a per-class covariance \(\boldsymbol{\Sigma}_c\) on the latent scores. The resulting decision rule \(\hat{c}(\mathbf{x}) = \arg\min_c (\mathbf{t}(\mathbf{x}) - \boldsymbol{\mu}_c)^{\top} \boldsymbol{\Sigma}_c^{-1} (\mathbf{t}(\mathbf{x}) - \boldsymbol{\mu}_c) + \log|\boldsymbol{\Sigma}_c|\) is quadratic in the latent scores.

QDA needs at least \(k + 1\) samples per class to estimate \(\boldsymbol{\Sigma}_c\) stably, but otherwise gives more flexible decision boundaries than LDA. Worth trying whenever the LDA boundary visibly under-fits in a 2-D latent score plot.

Class probabilities follow from the Mahalanobis distance via the Bayes rule with uniform priors (or user-supplied priors).

Implementation

n4m_pls_qda_fit. Reference: composite PLSRegression + sklearn QuadraticDiscriminantAnalysis on the scores.

MATLAB header (bindings/matlab/+pls4all/pls_qda.m):

pls4all.pls_qda  Quadratic Discriminant Analysis on PLS scores.
 y_labels: integer class IDs in {0, …, n_classes-1}.

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

/* C ABI — libn4m */
n4m_context_t* ctx = n4m_context_create();
n4m_config_t*  cfg = n4m_config_create();
n4m_method_result_t* res = NULL;
n4m_pls_qda_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);

import pls4all
from pls4all._methods import pls_qda_fit
with pls4all.Context() as ctx, pls4all.Config() as cfg:
    res = pls_qda_fit(ctx, cfg, X, y, n_components=4, y_labels=y_labels)
# then: res.matrix("predictions"), res.matrix("coefficients"),
# res.vector("mask"), res.scalar("intercept"), …

from pls4all.sklearn import PLSQDAClassifier
mdl = PLSQDAClassifier(n_components=2)
mdl.fit(X, y)
y_hat = mdl.predict(X_test)

library(pls4all)
# Unified low-level dispatcher (May 2026 R cleanup):
res <- pls4all_method("pls_qda", X, y,
                      n_components = 4L, params = list(n_classes = 3L))
# res is a named list with MethodResult arrays/scalars.
# selected_indices / top_k_intervals are 1-based.

res = pls4all.pls_qda(X, y, 4);
% see header of bindings/matlab/+pls4all/pls_qda.m for full
% parameter surface:
%   res = pls_qda(X, y_labels, n_components)
yhat = predict(res, Xtest);

No idiomatic classdef wrapper — invoke pls4all.fit("pls_qda", X, y, …) directly from the unified MEX factory.

Registry parity references 📐

  • 📐 ref.python_scikit_learn (python · python) — scikit-learn 1.8.0 · strict (rmse_rel ≤ 1e-06) — sklearn PLSRegression(scale=False) -> QuadraticDiscriminantAnalysis(reg_param=0.0) pipeline. pls4all’s default PLS-QDA reuses the same convention: NIPALS PLS scores from the C kernel, then sklearn-style QDA in Python. Bit-for-bit parity (max_abs < 1e-6).

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-06).

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.

BackendParity200×30 (ms)
C++ native · libn4m
pls4all.cpp.blas+omp✓ ref 5e-162.65 ms
Python · pls4all
pls4all.python✓ bind2.12 ms
pls4all.sklearn⇄ +2e+002.11 ms🏆
R · pls4all
pls4all.R⇄ +2e+004.06 ms
pls4all.R.formula⇄ +2e+004.67 ms
pls4all.R.mdatools⇄ +2e+005.46 ms
pls4all.R.pls⇄ +2e+004.99 ms
Python · external
📐ref.python_scikit_learnsource2.36 ms
BackendParity200×30 (ms)
C++ native · libn4m
pls4all.cpp.blas+omp✓ ref 5e-161.94 ms
Python · pls4all
pls4all.python✓ bind1.92 ms
pls4all.sklearn⇄ +2e+001.64 ms🏆
R · pls4all
pls4all.R⇄ +2e+008.84 ms
pls4all.R.formula⇄ +2e+0018.1 ms
pls4all.R.mdatools⇄ +2e+0018.2 ms
pls4all.R.pls⇄ +2e+0013.5 ms
Python · external
📐ref.python_scikit_learnsource6.90 ms
BackendParity200×30 (ms)
C++ native · libn4m
pls4all.cpp.blas+omp✓ ref 5e-165.99 ms
Python · pls4all
pls4all.python✓ bind4.32 ms
pls4all.sklearn⇄ +2e+001.49 ms🏆
R · pls4all
pls4all.R⇄ +2e+003.90 ms
pls4all.R.formula⇄ +2e+005.11 ms
pls4all.R.mdatools⇄ +2e+005.10 ms
pls4all.R.pls⇄ +2e+005.08 ms
Python · external
📐ref.python_scikit_learnsource2.39 ms

See also: benchmark overview · methods index · interactive dashboard