# `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_spa_select`. Reference: R `plsVarSel`.
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_spa_select_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 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} 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 · ✗ 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) | 200×40 (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 | ≈ | 492.2 ms | 3.3 s | 3.9 s | 7.0 s | 625.8 ms | 604.1 ms | 5.4 s | 8.1 s | 22.1 s | 34.0 s | 52.0 s | 216.9 s | — | 646.6 s |
pls4all.cpp.blas+omp | ≈ | 485.1 ms | 3.4 s | 3.5 s🏆 | 7.1 s | 626.2 ms | 583.1 ms | 5.0 s🏆 | 8.5 s | 23.0 s | 34.2 s | 52.2 s | 221.5 s | — | 629.9 s |
pls4all.cpp.omp | ≈ | 477.6 ms | 3.3 s🏆 | 3.8 s | 7.2 s | 618.8 ms | 576.0 ms | 5.3 s | 8.1 s | 25.8 s | 29.2 s🏆 | 52.7 s | 214.0 s🏆 | — | 614.2 s🏆 |
pls4all.cpp.ref | ≈ | 516.2 ms | 3.6 s | 3.9 s | 6.6 s🏆 | 615.7 ms | 586.9 ms | 5.2 s | 7.7 s🏆 | 21.6 s🏆 | 35.0 s | 51.1 s🏆 | 217.6 s | — | 638.9 s |
| Python · pls4all |
|---|
pls4all.python | ✓ bind | 507.1 ms | — | — | — | 611.1 ms | 570.7 ms | — | — | — | — | — | — | — | — |
pls4all.sklearn | ✗ +1e+00 | 3.51 ms | — | — | — | 2.90 ms | 3.42 ms | — | — | — | — | — | — | — | — |
| R · pls4all |
|---|
pls4all.R | ✗ +1e+00 | 11.7 ms | — | — | — | 5.82 ms | 11.4 ms | — | — | — | — | — | — | — | — |
pls4all.R.formula | ✗ +1e+00 | 19.6 ms | — | — | — | 8.28 ms | 9.52 ms | — | — | — | — | — | — | — | — |
pls4all.R.mdatools | ✗ +1e+00 | 19.2 ms | — | — | — | 7.76 ms | 9.18 ms | — | — | — | — | — | — | — | — |
pls4all.R.pls | ✗ +1e+00 | 22.8 ms | — | — | — | 8.37 ms | 10.1 ms | — | — | — | — | — | — | — | — |
| MATLAB · pls4all |
|---|
pls4all.matlab | ✗ +1e+00 | 5.06 ms | — | — | — | 3.02 ms | 4.77 ms | — | — | — | — | — | — | — | — |
pls4all.matlab.classdef | ✗ +1e+00 | 7.52 ms | — | — | — | 3.43 ms | 5.26 ms | — | — | — | — | — | — | — | — |
| R · external |
|---|
📐ref.r_plsvarsel | source | 42.4 ms🏆 | — | — | — | 126.8 ms🏆 | 131.9 ms🏆 | — | — | — | — | — | — | — | — |
:::
:::{tab-item} 3 threads
:sync: threads-3
| Backend | Parity | 50×250 (ms) | 100×50 (ms) | 100×500 (ms) | 100×2500 (ms) | 200×40 (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 | — | — | — | — | 568.0 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.blas+omp | ✓ ref | — | — | — | — | 578.1 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.omp | ✓ ref | — | — | — | — | 576.5 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.ref | ✓ ref | — | — | — | — | 560.0 ms | — | — | — | — | — | — | — | — | — |
| Python · pls4all |
|---|
pls4all.python | ✓ bind | — | — | — | — | 569.6 ms | — | — | — | — | — | — | — | — | — |
pls4all.sklearn | ✗ +1e+00 | — | — | — | — | 2.11 ms | — | — | — | — | — | — | — | — | — |
| R · pls4all |
|---|
pls4all.R | ✗ +1e+00 | — | — | — | — | 5.98 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.formula | ✗ +1e+00 | — | — | — | — | 6.56 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.mdatools | ✗ +1e+00 | — | — | — | — | 6.81 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.pls | ✗ +1e+00 | — | — | — | — | 7.02 ms | — | — | — | — | — | — | — | — | — |
| MATLAB · pls4all |
|---|
pls4all.matlab | ✗ +1e+00 | — | — | — | — | 2.95 ms | — | — | — | — | — | — | — | — | — |
pls4all.matlab.classdef | ✗ +1e+00 | — | — | — | — | 4.43 ms | — | — | — | — | — | — | — | — | — |
| R · external |
|---|
📐ref.r_plsvarsel | source | — | — | — | — | 98.8 ms🏆 | — | — | — | — | — | — | — | — | — |
:::
:::{tab-item} 10 threads
:sync: threads-10
| Backend | Parity | 50×250 (ms) | 100×50 (ms) | 100×500 (ms) | 100×2500 (ms) | 200×40 (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 | — | — | — | — | 501.7 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.blas+omp | ✓ ref | — | — | — | — | 495.7 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.omp | ✓ ref | — | — | — | — | 493.4 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.ref | ✓ ref | — | — | — | — | 494.6 ms | — | — | — | — | — | — | — | — | — |
| Python · pls4all |
|---|
pls4all.python | ✓ bind | — | — | — | — | 492.6 ms | — | — | — | — | — | — | — | — | — |
pls4all.sklearn | ✗ +1e+00 | — | — | — | — | 1.82 ms | — | — | — | — | — | — | — | — | — |
| R · pls4all |
|---|
pls4all.R | ✗ +1e+00 | — | — | — | — | 4.24 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.formula | ✗ +1e+00 | — | — | — | — | 5.16 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.mdatools | ✗ +1e+00 | — | — | — | — | 5.20 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.pls | ✗ +1e+00 | — | — | — | — | 5.20 ms | — | — | — | — | — | — | — | — | — |
| MATLAB · pls4all |
|---|
pls4all.matlab | ✗ +1e+00 | — | — | — | — | 2.61 ms | — | — | — | — | — | — | — | — | — |
pls4all.matlab.classdef | ✗ +1e+00 | — | — | — | — | 2.94 ms | — | — | — | — | — | — | — | — | — |
| R · external |
|---|
📐ref.r_plsvarsel | source | — | — | — | — | 88.5 ms🏆 | — | — | — | — | — | — | — | — | — |
:::
::::
---
_See also_: [benchmark overview](../benchmarks/overview.md) · [methods index](index.md) · [interactive dashboard](../landing/dashboard.md)