# `spa_select` — SPA — Successive Projections Algorithm _Group_: **Variable selector** · _Registry tolerance_: `1e-06` ## Description SPA Successive Projections (§18 Phase 5e) From the `pls4all.sklearn.SPASelector` docstring: > Successive Projections Algorithm selector (Araujo 2001). > **Registry note** — R `plsVarSel::spa_pls` randomization-based SPA (Forina et al.) — paired Wilcoxon variable importance, p < 0.05 survivor set (fallback to top-`ncomp` p-values). Default `_spa_select_pls4all` path mirrors the same R call with seed=11, giving bit-exact mask parity. The deterministic top-k Araújo projection kernel is opt-in via `legacy=True`. ### Parameters | Name | Type | Default | Notes | |------|------|---------|-------| | `top_k` | `int` | `None` | Number of features to retain. | | `n_components` | `int` | `2` | Number of latent components extracted (k). | ## Explanations ### Bibliographic source Araújo, M. C. U., Saldanha, T. C. B., Galvão, R. K. H., Yoneyama, T., Chame, H. C. & Visani, V. (2001). *The successive projections algorithm for variable selection in spectroscopic multicomponent analysis*. Chemometrics and Intelligent Laboratory Systems 57(2), 65–73. ### Mathematical principle SPA is a greedy forward selector that seeks **minimally collinear** features. Starting from the feature with largest coefficient (or user-supplied seed), at each step it adds the feature whose direction is **maximally orthogonal** to all previously-selected features. Formally, at step $m$ it picks $j^{\star} = \arg\max_j \| \mathbf{P}_{S_{m-1}}^{\perp} \mathbf{x}_j \|_2$, where $\mathbf{P}_{S}^{\perp}$ projects onto the orthogonal complement of the span of the already-selected features. This produces a feature subset that is well-conditioned for downstream regression: the inverse $(\mathbf{X}_S^{\top}\mathbf{X}_S)^{-1}$ has small condition number by construction. SPA is therefore particularly effective when followed by MLR (or PLS with very few components) on the selected subset. Stops at a user-specified top-$k$. Computational cost: $O(k\, n\, p)$. ### Implementation `n4m_feature_selection_spa_select`. Reference: R `plsVarSel`. R roxygen note (`selectors.R::spa_select`): > SPA — Successive Projections Algorithm. MATLAB header (`bindings/matlab/+pls4all/spa_select.m`): ```text pls4all.spa_select Successive Projections Algorithm. res = pls4all.spa_select(X, Y, K, top_k) Output struct fields: selected_indices : 1 × top_k row vector of 1-based feature indices. ``` ### 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_feature_selection_spa_select(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 spa_select_fit with pls4all.Context() as ctx, pls4all.Config() as cfg: res = spa_select_fit(ctx, cfg, X, y, n_components=4) # 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 SPASelector mdl = SPASelector(top_k, 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("spa_select", X, y, n_components = 4L, params = list(top_k = 10L)) # 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 <- spa_select(X, Y, n_components, top_k) yhat <- pls4all_predict(res, X_test) ``` ::: :::{tab-item} MATLAB · pls4all (MEX) :sync: matlab-mex :class-label: lang-matlab ```matlab res = pls4all.spa_select(X, y, 4); % see header of bindings/matlab/+pls4all/spa_select.m for full % parameter surface: % res = spa_select(X, Y, n_components, top_k) yhat = predict(res, Xtest); ``` ::: :::{tab-item} MATLAB · pls4all (classdef) :sync: matlab-classdef :class-label: lang-matlab _No idiomatic classdef wrapper — invoke `pls4all.fit("spa_select", X, y, …)` directly from the unified MEX factory._ ::: :::: **Registry parity references** 📐 :::{card} :class-card: external-refs - 📐 **`ref.r_plsvarsel`** (R · r) — `plsVarSel` 0.10.0 · strict (rmse_rel ≤ 1e-06) — R `plsVarSel::spa_pls` — Successive Projections Algorithm. ::: ### 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
BackendParity200×40 (ms)
C++ native · libn4m
pls4all.cpp.blas+omp✓ J 1.00499.6 ms
Python · pls4all
pls4all.python✓ J 1.00515.0 ms
pls4all.sklearn⇄ J 0.121.97 ms🏆
R · pls4all
pls4all.R⇄ J 0.124.72 ms
pls4all.R.formula⇄ J 0.126.23 ms
pls4all.R.mdatools⇄ J 0.125.86 ms
pls4all.R.pls⇄ J 0.125.75 ms
R · external
📐ref.r_plsvarselsource104.0 ms
::: :::{tab-item} 3 threads :sync: threads-3
BackendParity200×40 (ms)
C++ native · libn4m
pls4all.cpp.blas+omp✓ J 1.00819.9 ms
Python · pls4all
pls4all.python✓ J 1.00530.6 ms
pls4all.sklearn⇄ J 0.121.90 ms🏆
R · pls4all
pls4all.R⇄ J 0.125.33 ms
pls4all.R.formula⇄ J 0.125.81 ms
pls4all.R.mdatools⇄ J 0.1210.9 ms
pls4all.R.pls⇄ J 0.126.22 ms
R · external
📐ref.r_plsvarselsource164.5 ms
::: :::{tab-item} 10 threads :sync: threads-10
BackendParity200×40 (ms)
C++ native · libn4m
pls4all.cpp.blas+omp✓ J 1.00557.0 ms
Python · pls4all
pls4all.python✓ J 1.00514.9 ms
pls4all.sklearn⇄ J 0.121.89 ms🏆
R · pls4all
pls4all.R⇄ J 0.124.70 ms
pls4all.R.formula⇄ J 0.125.57 ms
pls4all.R.mdatools⇄ J 0.125.52 ms
pls4all.R.pls⇄ J 0.125.17 ms
R · external
📐ref.r_plsvarselsource179.5 ms
::: :::: --- _See also_: [benchmark overview](../benchmarks/overview.md) · [methods index](index.md) · [interactive dashboard](../landing/dashboard.md)