# `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`):
```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_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} 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 · ✗ 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
| Backend | Parity | 50×250 (ms) | 100×50 (ms) | 100×500 (ms) | 100×2500 (ms) | 200×30 (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 | ≈ +5e-16 | 3.90 ms | 2.73 ms | 13.3 ms🏆 | 60.8 ms🏆 | 2.15 ms | 4.00 ms | 9.06 ms | 61.0 ms🏆 | 356.3 ms🏆 | 31.3 ms | 326.9 ms🏆 | 2.6 s🏆 | 127.1 ms🏆 | 2.0 s |
pls4all.cpp.blas+omp | ≈ +5e-16 | 4.16 ms | 1.98 ms🏆 | 13.9 ms | 61.0 ms | 3.03 ms | 3.98 ms🏆 | 6.68 ms | 62.9 ms | 360.7 ms | 33.3 ms | 332.9 ms | 2.6 s | 135.1 ms | 1.8 s🏆 |
pls4all.cpp.omp | ≈ +5e-16 | 5.92 ms | 2.17 ms | 16.1 ms | 66.9 ms | 2.14 ms | 4.11 ms | 7.86 ms | 68.2 ms | 536.7 ms | 30.8 ms🏆 | 383.1 ms | 2.8 s | 128.7 ms | 2.2 s |
pls4all.cpp.ref | ≈ +5e-16 | 4.71 ms | 2.31 ms | 13.8 ms | 68.8 ms | 2.01 ms🏆 | 6.01 ms | 6.48 ms🏆 | 72.5 ms | 496.9 ms | 31.2 ms | 405.3 ms | 3.2 s | 127.2 ms | 2.1 s |
| Python · pls4all |
|---|
pls4all.python | ✓ bind | 3.51 ms🏆 | — | — | — | 2.17 ms | 4.76 ms | — | — | — | — | — | — | — | — |
pls4all.sklearn | ✗ +2e+00 | 3.97 ms | — | — | — | 1.68 ms | 3.36 ms | — | — | — | — | — | — | — | — |
| R · pls4all |
|---|
pls4all.R | ✗ +2e+00 | 12.7 ms | — | — | — | 4.82 ms | 13.3 ms | — | — | — | — | — | — | — | — |
pls4all.R.formula | ✗ +2e+00 | 25.6 ms | — | — | — | 5.58 ms | 10.9 ms | — | — | — | — | — | — | — | — |
pls4all.R.mdatools | ✗ +2e+00 | 22.5 ms | — | — | — | 5.74 ms | 10.5 ms | — | — | — | — | — | — | — | — |
pls4all.R.pls | ✗ +2e+00 | 26.6 ms | — | — | — | 5.96 ms | 12.6 ms | — | — | — | — | — | — | — | — |
| MATLAB · pls4all |
|---|
pls4all.matlab | ✗ +2e+00 | 4.46 ms | — | — | — | 2.24 ms | 4.56 ms | — | — | — | — | — | — | — | — |
pls4all.matlab.classdef | ✗ +2e+00 | 5.15 ms | — | — | — | 2.56 ms | 4.35 ms | — | — | — | — | — | — | — | — |
| Python · external |
|---|
📐ref.python_scikit_learn | source | 4.74 ms | — | — | — | 2.39 ms | 4.19 ms | — | — | — | — | — | — | — | — |
:::
:::{tab-item} 3 threads
:sync: threads-3
| Backend | Parity | 50×250 (ms) | 100×50 (ms) | 100×500 (ms) | 100×2500 (ms) | 200×30 (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 | ✓ ref 5e-16 | — | — | — | — | 2.12 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.blas+omp | ✓ ref 5e-16 | — | — | — | — | 3.06 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.omp | ✓ ref 5e-16 | — | — | — | — | 2.32 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.ref | ✓ ref 5e-16 | — | — | — | — | 2.11 ms | — | — | — | — | — | — | — | — | — |
| Python · pls4all |
|---|
pls4all.python | ✓ 7e-16 | — | — | — | — | 2.03 ms🏆 | — | — | — | — | — | — | — | — | — |
pls4all.sklearn | ✗ +2e+00 | — | — | — | — | 1.60 ms | — | — | — | — | — | — | — | — | — |
| R · pls4all |
|---|
pls4all.R | ✗ +3e-03 | — | — | — | — | 4.21 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.formula | ✗ +3e-03 | — | — | — | — | 5.81 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.mdatools | ✗ +3e-03 | — | — | — | — | 5.08 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.pls | ✗ +3e-03 | — | — | — | — | 4.95 ms | — | — | — | — | — | — | — | — | — |
| MATLAB · pls4all |
|---|
pls4all.matlab | ✗ +2e+00 | — | — | — | — | 2.06 ms | — | — | — | — | — | — | — | — | — |
pls4all.matlab.classdef | ✗ +2e+00 | — | — | — | — | 2.84 ms | — | — | — | — | — | — | — | — | — |
| Python · external |
|---|
📐ref.python_scikit_learn | source | — | — | — | — | 3.81 ms | — | — | — | — | — | — | — | — | — |
:::
:::{tab-item} 10 threads
:sync: threads-10
| Backend | Parity | 50×250 (ms) | 100×50 (ms) | 100×500 (ms) | 100×2500 (ms) | 200×30 (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 | ✓ ref 5e-16 | — | — | — | — | 1.75 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.blas+omp | ✓ ref 5e-16 | — | — | — | — | 1.71 ms🏆 | — | — | — | — | — | — | — | — | — |
pls4all.cpp.omp | ✓ ref 5e-16 | — | — | — | — | 1.73 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.ref | ✓ ref 5e-16 | — | — | — | — | 1.76 ms | — | — | — | — | — | — | — | — | — |
| Python · pls4all |
|---|
pls4all.python | ✓ 7e-16 | — | — | — | — | 1.71 ms | — | — | — | — | — | — | — | — | — |
pls4all.sklearn | ✗ +2e+00 | — | — | — | — | 1.40 ms | — | — | — | — | — | — | — | — | — |
| R · pls4all |
|---|
pls4all.R | ✗ +3e-03 | — | — | — | — | 3.20 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.formula | ✗ +3e-03 | — | — | — | — | 3.81 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.mdatools | ✗ +3e-03 | — | — | — | — | 3.82 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.pls | ✗ +3e-03 | — | — | — | — | 3.68 ms | — | — | — | — | — | — | — | — | — |
| MATLAB · pls4all |
|---|
pls4all.matlab | ✗ +2e+00 | — | — | — | — | 1.90 ms | — | — | — | — | — | — | — | — | — |
pls4all.matlab.classdef | ✗ +2e+00 | — | — | — | — | 2.21 ms | — | — | — | — | — | — | — | — | — |
| Python · external |
|---|
📐ref.python_scikit_learn | source | — | — | — | — | 2.23 ms | — | — | — | — | — | — | — | — | — |
:::
::::
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_See also_: [benchmark overview](../benchmarks/overview.md) · [methods index](index.md) · [interactive dashboard](../landing/dashboard.md)