# `gpr_pls` — Gaussian Process on PLS scores _Group_: **Nonlinear / local** · _Registry tolerance_: `1e-08` ## Description GPR-on-PLS — RBF Gaussian Process on PLS scores (§47) From the `pls4all.sklearn.GPRPLSRegression` docstring: > Gaussian-process head on SIMPLS training scores. > **Registry note** — GP head parity (sklearn `GaussianProcessRegressor` with RBF+WhiteKernel, optimizer=None) on the same PLS scores. Architecturally separated to allow GPR-on-AOMPLS reuse of `fit_gp_on_scores`. ### Parameters | Name | Type | Default | Notes | |------|------|---------|-------| | `n_components` | `int` | `2` | Number of latent components extracted (k). | | `length_scale` | `float` | `1.0` | | | `noise_level` | `float` | `0.001` | | | `seed` | `int` | `0` | Random seed for reproducible sampling/initialization. | ## Explanations ### Bibliographic source Bishop, C. M. (2006). *Pattern Recognition and Machine Learning*, §6.4 (Gaussian Processes). — combined with a preliminary PLS dimensionality reduction for spectroscopy. ### Mathematical principle Spectroscopic data are too high-dimensional for a direct Gaussian Process: GP inference is $O(n^3)$ in samples but the *kernel quality* degrades rapidly when $p$ exceeds a few hundred — most pairwise distances become near-identical, the kernel matrix loses contrast and the GP under-fits. GPR-PLS solves this by first projecting $\mathbf{X} \to \mathbf{T} = \mathbf{X}\mathbf{W}$ into a low-dimensional PLS latent space and **then** training a GP on $\mathbf{T}$. The latent space preserves the variance most relevant to $y$, the GP captures smooth non-linear residual structure, and the kernel matrix is well-conditioned because pairwise distances in $\mathbb{R}^k$ remain informative. Default kernel: RBF with length scale $\ell$ and amplitude $\sigma_f^2$, plus an isotropic noise variance $\sigma_n^2$. Marginal-likelihood maximisation selects the three hyperparameters; pls4all uses a fixed-iteration L-BFGS pass to keep the cost bounded per cell. ### Implementation `n4m_estimators_gpr_pls_fit`. Reference: sklearn `GaussianProcessRegressor` with an RBF kernel applied to the score matrix from a separate sklearn `PLSRegression`. R roxygen note (`methods_extra.R::gpr_pls_fit`): > Gaussian Process Regression on PLS scores (single-target Y). > @param n_components Integer. Number of latent components. > @param length_scale Method-specific parameter. See the underlying `*_fit()` function for the exact semantics. > @param noise_level Method-specific parameter. See the underlying `*_fit()` function for the exact semantics. > @param seed Integer. Random seed for reproducibility. > @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/gpr_pls.m`): ```text pls4all.gpr_pls GPR on PLS scores (RBF kernel, single-target Y). ``` ### 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_gpr_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 gpr_pls_fit with pls4all.Context() as ctx, pls4all.Config() as cfg: res = gpr_pls_fit(ctx, cfg, X, y, n_components=3) # 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 GPRPLSRegression mdl = GPRPLSRegression(n_components=2, length_scale=1.0, noise_level=0.001, seed=0) 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("gpr_pls", X, y, n_components = 3L, params = list(length_scale = 1.0, noise_level = 0.001, seed = 0L)) # 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 <- gpr_pls_fit(X, Y, n_components, length_scale = 1.0, noise_level = 1e-4, seed = 0L) yhat <- pls4all_predict(res, X_test) ``` ::: :::{tab-item} MATLAB · pls4all (MEX) :sync: matlab-mex :class-label: lang-matlab ```matlab res = pls4all.gpr_pls(X, y, 3); % see header of bindings/matlab/+pls4all/gpr_pls.m for full % parameter surface: % res = gpr_pls(X, Y, n_components, length_scale, noise_level, seed) yhat = predict(res, Xtest); ``` ::: :::{tab-item} MATLAB · pls4all (classdef) :sync: matlab-classdef :class-label: lang-matlab _No idiomatic classdef wrapper — invoke `pls4all.fit("gpr_pls", X, y, …)` directly from the unified MEX factory._ ::: :::: **Registry parity references** 📐 :::{card} :class-card: external-refs - 📐 **`ref.python_scikit_learn`** (python · python) — `scikit-learn` 1.4.2 · strict (rmse_rel ≤ 1e-08) — sklearn GP head on the same PLS training scores pls4all produces. PLS rotation conventions diverge per-component; comparing the GP head on shared T isolates the novel stage. ::: ### 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
BackendParity120×25 (ms)
C++ native · libn4m
pls4all.cpp.blas+omp✓ ref 2e-105.91 ms
Python · pls4all
pls4all.python✓ bind6.17 ms
pls4all.sklearn✓ bind2.75 ms
R · pls4all
pls4all.R✓ 2e-1320.9 ms
pls4all.R.formula✓ 2e-1312.6 ms
pls4all.R.mdatools✓ 2e-138.10 ms
pls4all.R.pls✓ 2e-1314.2 ms
Python · external
📐ref.python_scikit_learnsource2.25 ms🏆
::: :::{tab-item} 3 threads :sync: threads-3
BackendParity120×25 (ms)
C++ native · libn4m
pls4all.cpp.blas+omp✓ ref 2e-101.14 ms🏆
Python · pls4all
pls4all.python✓ bind2.43 ms
pls4all.sklearn✓ bind2.72 ms
R · pls4all
pls4all.R✓ 2e-134.45 ms
pls4all.R.formula✓ 2e-135.61 ms
pls4all.R.mdatools✓ 2e-136.10 ms
pls4all.R.pls✓ 2e-136.64 ms
Python · external
📐ref.python_scikit_learnsource2.75 ms
::: :::{tab-item} 10 threads :sync: threads-10
BackendParity120×25 (ms)
C++ native · libn4m
pls4all.cpp.blas+omp✓ ref 2e-101.14 ms🏆
Python · pls4all
pls4all.python✓ bind1.80 ms
pls4all.sklearn✓ bind1.39 ms
R · pls4all
pls4all.R✓ 2e-132.53 ms
pls4all.R.formula✓ 2e-133.11 ms
pls4all.R.mdatools✓ 2e-133.24 ms
pls4all.R.pls✓ 2e-133.10 ms
Python · external
📐ref.python_scikit_learnsource2.29 ms
::: :::: --- _See also_: [benchmark overview](../benchmarks/overview.md) · [methods index](index.md) · [interactive dashboard](../landing/dashboard.md)