# `so_pls` — Sequential and Orthogonalised PLS (SO-PLS) _Group_: **Multi-block / cross-modal** · _Registry tolerance_: `1e-06` ## Description SO-PLS — Sequential & Orthogonalized multi-block PLS (§17) From the `pls4all.sklearn.SOPLSRegression` docstring: > Sequential & Orthogonalised multi-block PLS (Næs et al. 2011). > **Registry note** — R `multiblock::sopls 0.8.10` (Næs et al. 2011) canonical SO-PLS via fit$fitted at the full (k1,..,kB) slice. pls4all's NIPALS-based SO-PLS matches to ~1e-13. ### Parameters | Name | Type | Default | Notes | |------|------|---------|-------| | `n_components_per_block` | `—` | `None` | Per-block latent-component budget (one int per block). | | `block_sizes` | `—` | `None` | Sequence of contiguous block widths defining the X-block partition (columns of X). | ## Explanations ### Bibliographic source Næs, T., Tomic, O., Mevik, B.-H. & Martens, H. (2011). *Path modelling by sequential PLS regression*. Journal of Chemometrics 25(1), 28–40. ### Mathematical principle SO-PLS extends MB-PLS by sequentially processing blocks in a user-specified order, orthogonalising each subsequent block against the scores extracted from the previous ones. Concretely: fit PLS on block 1, extract scores $\mathbf{T}_1$; orthogonalise block 2 to $\mathbf{T}_1$, fit PLS on the residual; and so on. This makes each block's contribution **additive and interpretable** — the unique variance in block $b$ that is predictive of $y$ given everything in blocks $1, \ldots, b-1$. Compared to MB-PLS (which fits all blocks simultaneously), SO-PLS encodes a domain hypothesis about which block is causally upstream of which. Tuning: a per-block $k_b$ (number of components) plus the block order. The order matters; permuting blocks changes the attribution. Cross-validating the per-block $k_b$ is the standard procedure. ### Implementation `n4m_estimators_so_pls_fit` — requires `n_components_per_block` and `block_sizes`. Reference: CRAN `multiblock 0.8.10`. R roxygen note (`methods_extra.R::so_pls_fit`): > Sequential & Orthogonalised multi-block PLS (Næs et al. 2011). > `block_sizes` integer vector summing to ncol(X); > `n_components_per_block` integer vector of same length. > @param block_sizes Integer vector. Per-block feature counts for multi-block PLS. > @param n_components_per_block Method-specific parameter. See the underlying `*_fit()` function for the exact semantics. > @param X Numeric matrix of predictors (rows = samples, cols = features). > @param Y Numeric matrix or vector of responses, with one row per sample. > @export MATLAB header (`bindings/matlab/+pls4all/so_pls.m`): ```text pls4all.so_pls Sequential & Orthogonalised multi-block PLS (Næs 2011). X: n × sum(block_sizes) concatenated multi-block matrix. n_components_per_block: int32 vector of length n_blocks. ``` ### 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_so_pls_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 so_pls_fit with pls4all.Context() as ctx, pls4all.Config() as cfg: res = so_pls_fit(ctx, cfg, X, y) # 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 SOPLSRegression mdl = SOPLSRegression(n_components_per_block=None, block_sizes=None) 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("so_pls", X, y, n_components = 2L) # 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 <- so_pls_fit(X, Y, block_sizes, n_components_per_block) yhat <- pls4all_predict(res, X_test) ``` ::: :::{tab-item} MATLAB · pls4all (MEX) :sync: matlab-mex :class-label: lang-matlab ```matlab res = pls4all.so_pls(X, y, 2); % see header of bindings/matlab/+pls4all/so_pls.m for full % parameter surface: % res = so_pls(X, Y, n_components_per_block, block_sizes) yhat = predict(res, Xtest); ``` ::: :::{tab-item} MATLAB · pls4all (classdef) :sync: matlab-classdef :class-label: lang-matlab _No idiomatic classdef wrapper — invoke `pls4all.fit("so_pls", X, y, …)` directly from the unified MEX factory._ ::: :::: **Registry parity references** 📐 :::{card} :class-card: external-refs - 📐 **`ref.r_multiblock`** (R · r) — `multiblock` 0.8.10 · strict (rmse_rel ≤ 1e-06) — R `multiblock::sopls 0.8.10` (Næs et al. 2011) canonical SO-PLS — in-sample fitted values via fit$fitted at the full (k1,..,kB) slice match pls4all's canonical NIPALS SO-PLS to ~1e-15 in centered space. ::: ### 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 7e-101.62 ms🏆
Python · pls4all
pls4all.python✓ bind1.67 ms
pls4all.sklearn✓ bind1.77 ms
R · pls4all
pls4all.R✓ bind3.26 ms
pls4all.R.formula✓ bind3.99 ms
pls4all.R.mdatools✓ bind4.00 ms
pls4all.R.pls✓ bind4.11 ms
R · external
📐ref.r_multiblocksource981.7 ms
::: :::{tab-item} 3 threads :sync: threads-3
BackendParity200×30 (ms)
C++ native · libn4m
pls4all.cpp.blas+omp✓ ref 7e-101.66 ms🏆
Python · pls4all
pls4all.python✓ bind1.69 ms
pls4all.sklearn✓ bind2.44 ms
R · pls4all
pls4all.R✓ bind22.4 ms
pls4all.R.formula✓ bind31.5 ms
pls4all.R.mdatools✓ bind31.4 ms
pls4all.R.pls✓ bind34.4 ms
R · external
📐ref.r_multiblocksource1.5 s
::: :::{tab-item} 10 threads :sync: threads-10
BackendParity200×30 (ms)
C++ native · libn4m
pls4all.cpp.blas+omp✓ ref 7e-107.16 ms🏆
Python · pls4all
pls4all.python✓ bind11.3 ms
pls4all.sklearn✓ bind11.9 ms
R · pls4all
pls4all.R✓ bind11.3 ms
pls4all.R.formula✓ bind25.9 ms
pls4all.R.mdatools✓ bind10.3 ms
pls4all.R.pls✓ bind9.15 ms
R · external
📐ref.r_multiblocksource4.0 s
::: :::: --- _See also_: [benchmark overview](../benchmarks/overview.md) · [methods index](index.md) · [interactive dashboard](../landing/dashboard.md)