# `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_estimators_pls_qda_fit`. Reference: composite PLSRegression + sklearn `QuadraticDiscriminantAnalysis` on the scores. R roxygen note (`methods_extra.R::pls_qda_fit`): > PLS-QDA (Quadratic Discriminant Analysis on PLS scores). > @param y_labels Integer vector. Class labels. > @param n_components Integer. Number of latent components. > @param X Numeric matrix of predictors (rows = samples, cols = features). > @export MATLAB header (`bindings/matlab/+pls4all/pls_qda.m`): ```text 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** ::::{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_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); ``` ::: :::{tab-item} Python · pls4all (raw) :sync: python-raw :class-label: lang-python ```python 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"), … ``` ::: :::{tab-item} Python · pls4all.sklearn :sync: python-sklearn :class-label: lang-python ```python from pls4all.sklearn import PLSQDAClassifier mdl = PLSQDAClassifier(n_components=2) 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("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. ``` ::: :::{tab-item} R · pls4all (raw fn) :sync: r-raw :class-label: lang-r ```r library(pls4all) res <- pls_qda_fit(X, y_labels, n_components) yhat <- pls4all_predict(res, X_test) ``` ::: :::{tab-item} MATLAB · pls4all (MEX) :sync: matlab-mex :class-label: lang-matlab ```matlab 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); ``` ::: :::{tab-item} MATLAB · pls4all (classdef) :sync: matlab-classdef :class-label: lang-matlab _No idiomatic classdef wrapper — invoke `pls4all.fit("pls_qda", X, y, …)` directly from the unified MEX factory._ ::: :::: **Registry parity references** 📐 :::{card} :class-card: external-refs - 📐 **`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`](../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-06`). 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
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
::: :::{tab-item} 3 threads :sync: threads-3
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
::: :::{tab-item} 10 threads :sync: threads-10
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](../benchmarks/overview.md) · [methods index](index.md) · [interactive dashboard](../landing/dashboard.md)