# `pls_lda` — PLS-LDA _Group_: **Classification & GLM** · _Registry tolerance_: `1e-08` ## Description PLS-LDA — LDA on PLS scores (§17 Phase 4) From the `pls4all.sklearn.PLSLDAClassifier` docstring: > PLS-LDA on PLS scores (in-sample only). > **Registry note** — sklearn `PLSRegression(scale=False) -> LinearDiscriminantAnalysis` is the canonical reference. The in-kernel pooled-covariance LDA head reproduces sklearn's multiclass `decision_function` at ~1e-15. `legacy=True` opts into the historical SIMPLS+scaled variant (not parity-equivalent). ### 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 Barker, M. & Rayens, W. (2003). *Partial least squares for discrimination*. Journal of Chemometrics 17(3), 166–173. ### Mathematical principle PLS-LDA is a two-stage classifier: first project $\mathbf{X}$ into the PLS latent space using one-hot encoded class labels as $\mathbf{Y}$, then fit Linear Discriminant Analysis on the resulting scores $\mathbf{T} = \mathbf{X}\mathbf{W}$. LDA in the latent space is well-conditioned (the score matrix has $k \ll p$ columns by construction), and the PLS projection has already aligned the latent axes with the class separation direction. This is more robust than applying LDA directly to high-dimensional $\mathbf{X}$, where the within-class covariance is singular. The decision boundary is **linear in the latent space** (and therefore also linear in the original feature space via $\mathbf{W}$). For non-linear class boundaries use PLS-QDA or PLS-logistic. ### Implementation `n4m_estimators_pls_lda_fit`. The reference is composite (sklearn `PLSRegression` + sklearn `LinearDiscriminantAnalysis`); no library exposes a single PLS-LDA call. R roxygen note (`methods_extra.R::pls_lda_fit`): > PLS-LDA — Linear Discriminant Analysis on PLS scores. > @param y_labels Integer vector. Class labels. > @param n_components Integer. Number of latent components. > @param n_classes Integer >= 2. Number of class labels. > @param X Numeric matrix of predictors (rows = samples, cols = features). > @export MATLAB header (`bindings/matlab/+pls4all/pls_lda.m`): ```text pls4all.pls_lda Linear Discriminant Analysis on PLS scores. ``` ### 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_lda_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_lda_fit with pls4all.Context() as ctx, pls4all.Config() as cfg: res = pls_lda_fit(ctx, cfg, X, y, n_components=3, 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 PLSLDAClassifier mdl = PLSLDAClassifier(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_lda", X, y, n_components = 3L, 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_lda_fit(X, y_labels, n_components, n_classes = NULL) yhat <- pls4all_predict(res, X_test) ``` ::: :::{tab-item} MATLAB · pls4all (MEX) :sync: matlab-mex :class-label: lang-matlab ```matlab res = pls4all.pls_lda(X, y, 3); % see header of bindings/matlab/+pls4all/pls_lda.m for full % parameter surface: % res = pls_lda(X, y_labels, n_components, n_classes) yhat = predict(res, Xtest); ``` ::: :::{tab-item} MATLAB · pls4all (classdef) :sync: matlab-classdef :class-label: lang-matlab _No idiomatic classdef wrapper — invoke `pls4all.fit("pls_lda", 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-08) — sklearn `PLSRegression -> LinearDiscriminantAnalysis` pipeline. pls4all's PLS-LDA uses a single SIMPLS pass with an internal LDA head; sklearn fits PLS on dummy-encoded targets and feeds the scores into LDA — both are LDA on PLS scores but the latent bases diverge. We compare class boundaries via one-hot decision scores. ::: ### 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
BackendParity200×40 (ms)
C++ native · libn4m
pls4all.cpp.blas+omp✓ ref 3e-163.14 ms🏆
Python · pls4all
pls4all.python✓ bind7.48 ms
Python · external
📐ref.python_scikit_learnsource7.32 ms
::: :::{tab-item} 3 threads :sync: threads-3
BackendParity200×40 (ms)
C++ native · libn4m
pls4all.cpp.blas+omp✓ ref 3e-166.86 ms
Python · pls4all
pls4all.python✓ bind4.05 ms🏆
Python · external
📐ref.python_scikit_learnsource4.54 ms
::: :::{tab-item} 10 threads :sync: threads-10
BackendParity200×40 (ms)
C++ native · libn4m
pls4all.cpp.blas+omp✓ ref 3e-161.95 ms🏆
Python · pls4all
pls4all.python✓ bind2.67 ms
Python · external
📐ref.python_scikit_learnsource3.59 ms
::: :::: --- _See also_: [benchmark overview](../benchmarks/overview.md) · [methods index](index.md) · [interactive dashboard](../landing/dashboard.md)