# `continuum_regression` — Continuum Regression (Stone & Brooks 1990) _Group_: **Nonlinear / local** · _Registry tolerance_: `1e-06` ## Description Continuum regression (interpolates PLS / OLS) From the `pls4all.sklearn.ContinuumRegression` docstring: > Continuum regression τ ∈ [0, 1] interpolates PLS (1) / OLS (0). > **Registry note** — Canonical Stone & Brooks (1990) continuum regression. Python `ContinuumPyReference` is a paper-faithful NumPy implementation (no widely installable Python port exists); the pls4all C++ kernel uses the same algorithm and matches bit-for-bit. The optional R `JICO::continuum` adapter uses a different (lambda, gamma, om) parameterization and is kept as a qualitative cross-check. ### Parameters | Name | Type | Default | Notes | |------|------|---------|-------| | `n_components` | `int` | `2` | Number of latent components extracted (k). | | `tau` | `float` | `0.5` | Continuum mixing parameter in [0, 1]; 0 ≈ PLS, 1 ≈ whitened OLS / PCR-like. | ## Explanations ### Bibliographic source Stone, M. & Brooks, R. J. (1990). *Continuum regression: cross-validated sequentially constructed prediction embracing ordinary least squares, partial least squares and principal components regression*. JRSS B 52(2), 237–269. ### Mathematical principle Continuum regression introduces a single shape parameter $\tau \in [0, 1]$ that selects the loading-weight criterion: $\mathbf{w} \propto \operatorname{Cov}(\mathbf{X}\mathbf{w}, \mathbf{y})^{\tau} \cdot \operatorname{Var}(\mathbf{X}\mathbf{w})^{1-\tau}$. Special cases: $\tau = 0$ gives PCR (variance-maximising), $\tau = 1/2$ gives PLS (covariance-maximising), $\tau = 1$ gives OLS (correlation-maximising, in the appropriate limit). Cross-validating $\tau$ on a fine grid often improves RMSE over the discrete PLS / PCR choices — the optimum is rarely exactly at $\tau = 0.5$. Stone & Brooks' original treatment also cross-validates the number of components $k$ jointly with $\tau$, producing a 2-D grid. Implementation note: numerically stable continuum regression uses the parameterised power method on the matrix $\mathbf{X}^{\top}\mathbf{y}\mathbf{y}^{\top}\mathbf{X} / (\mathbf{X}^{\top}\mathbf{X})^{1-\tau}$, which avoids forming the rank-1 outer product explicitly and is what pls4all uses. ### Implementation `n4m_continuum_regression_fit` (in-sample only). MATLAB header (`bindings/matlab/+pls4all/ContinuumRegression.m`): ```text pls4all.ContinuumRegression Continuum regression (tau ∈ [0, 1]). ``` ### Usage Direct `n4m` Python helper: ```python import n4m res = n4m.continuum_regression(X, y, n_components=4, tau=0.25) y_hat = res["predictions"] coef = res["coefficients"] ``` Reusable sklearn-style wrapper: ```python from n4m.sklearn import NativeContinuumRegressionRegressor model = NativeContinuumRegressionRegressor( n_components=4, tau=0.25, ).fit(X, y) y_hat = model.predict(X_test) ``` 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_continuum_regression_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 continuum_regression_fit with pls4all.Context() as ctx, pls4all.Config() as cfg: res = continuum_regression_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 ContinuumRegression mdl = ContinuumRegression(n_components=2, tau=0.5) 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("continuum_regression", X, y, n_components = 4L, params = list(tau = 0.5)) # 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.continuum_regression(X, y, 4); % see header of bindings/matlab/+pls4all/continuum_regression.m for full % parameter surface: % res = continuum_regression(X, Y, n_components, tau) yhat = predict(res, Xtest); ``` ::: :::{tab-item} MATLAB · pls4all (classdef) :sync: matlab-classdef :class-label: lang-matlab ```matlab mdl = pls4all.fit("continuum_regression", X, y, "NumComponents", 4); yhat = predict(mdl, Xtest); ``` ::: :::: **Registry parity references** 📐 :::{card} :class-card: external-refs - 📐 **`ref.python_stone_brooks_1990_py`** (python · python) — `stone-brooks-1990-py` 1.0 · strict (rmse_rel ≤ 1e-06) — NumPy reference for Stone & Brooks (1990) continuum regression. First-component weight is (X'X)^{-tau} X'y, computed via the centered-X SVD; subsequent components use SIMPLS basis-v deflation of the modified cross-product matrix. - 📐 **`ref.r_jico`** (R · r) — `JICO` 0.1 · strict (rmse_rel ≤ 1e-06) — R `JICO::continuum` (Stone & Brooks 1990). Different parameterization than pls4all — JICO uses (lambda, gamma, om) while pls4all maps a single τ. Predictions are reconstructed by regressing Y on JICO's latent 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  ·  ✗ 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
BackendParity50×250 (ms)100×50 (ms)100×500 (ms)100×2500 (ms)200×50 (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≈ +8e-143.77 ms1.98 ms18.0 ms58.0 ms2.97 ms3.73 ms5.34 ms🏆3.1 s3.1 s29.2 ms4.2 s🏆—163.8 ms12.2 s🏆
pls4all.cpp.blas+omp≈ +8e-143.63 ms2.07 ms16.6 ms57.3 ms2.91 ms3.67 ms5.52 ms3.0 s3.0 s🏆30.1 ms4.3 s—160.2 ms🏆12.8 s
pls4all.cpp.omp≈ +8e-143.37 ms1.94 ms🏆17.2 ms58.4 ms2.92 ms3.60 ms5.63 ms2.9 s🏆3.1 s28.1 ms🏆4.2 s—168.9 ms12.6 s
pls4all.cpp.ref≈ +8e-144.63 ms2.08 ms15.4 ms🏆57.1 ms🏆2.87 ms3.54 ms🏆5.37 ms3.2 s3.1 s28.9 ms4.3 s—163.0 ms12.5 s
Python · pls4all
pls4all.python✓ bind3.48 ms———3.89 ms3.69 ms————————
pls4all.sklearn✓ bind3.34 ms———3.28 ms3.70 ms————————
R · pls4all
pls4all.R✗ +1e-0110.8 ms———6.68 ms9.28 ms————————
pls4all.R.formula✗ +1e-0119.2 ms———8.22 ms9.13 ms————————
pls4all.R.mdatools✗ +1e-0119.4 ms———7.83 ms9.05 ms————————
pls4all.R.pls✗ +1e-0119.0 ms———9.09 ms9.02 ms————————
MATLAB · pls4all
pls4all.matlab✗ +9e+005.17 ms———4.21 ms5.29 ms————————
pls4all.matlab.classdef✗ +9e+006.72 ms———4.70 ms7.01 ms————————
Python · external
📐ref.python_stone_brooks_1990_pysource3.04 ms🏆———2.57 ms🏆4.60 ms————————
R · external
📐ref.r_jico✗ +8e-0357.5 ms———44.8 ms49.4 ms————————
::: :::{tab-item} 3 threads :sync: threads-3
BackendParity50×250 (ms)100×50 (ms)100×500 (ms)100×2500 (ms)200×50 (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 1e-14————2.96 ms—————————
pls4all.cpp.blas+omp✓ ref 1e-14————4.15 ms—————————
pls4all.cpp.omp✓ ref 1e-14————2.91 ms—————————
pls4all.cpp.ref✓ ref 1e-14————3.59 ms—————————
Python · pls4all
pls4all.python✓ bind————4.28 ms—————————
pls4all.sklearn✓ bind————3.34 ms—————————
R · pls4all
pls4all.R✗ +1e-01————7.51 ms—————————
pls4all.R.formula✗ +1e-01————9.45 ms—————————
pls4all.R.mdatools✗ +1e-01————9.00 ms—————————
pls4all.R.pls✗ +1e-01————9.28 ms—————————
MATLAB · pls4all
pls4all.matlab✗ +9e+00————5.63 ms—————————
pls4all.matlab.classdef✗ +9e+00————4.77 ms—————————
Python · external
📐ref.python_stone_brooks_1990_pysource————2.76 ms🏆—————————
R · external
📐ref.r_jico✗ +8e-03————44.3 ms—————————
::: :::{tab-item} 10 threads :sync: threads-10
BackendParity50×250 (ms)100×50 (ms)100×500 (ms)100×2500 (ms)200×50 (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 1e-14————2.67 ms—————————
pls4all.cpp.blas+omp✓ ref 1e-14————2.65 ms—————————
pls4all.cpp.omp✓ ref 1e-14————2.63 ms—————————
pls4all.cpp.ref✓ ref 1e-14————2.63 ms🏆—————————
Python · pls4all
pls4all.python✓ bind————2.69 ms—————————
pls4all.sklearn✓ bind————2.81 ms—————————
R · pls4all
pls4all.R✗ +1e-01————5.37 ms—————————
pls4all.R.formula✗ +1e-01————6.13 ms—————————
pls4all.R.mdatools✗ +1e-01————5.91 ms—————————
pls4all.R.pls✗ +1e-01————6.46 ms—————————
MATLAB · pls4all
pls4all.matlab✗ +9e+00————3.76 ms—————————
pls4all.matlab.classdef✗ +9e+00————4.12 ms—————————
Python · external
📐ref.python_stone_brooks_1990_pysource————4.43 ms—————————
R · external
📐ref.r_jico✗ +8e-03————46.2 ms—————————
::: :::: --- _See also_: [benchmark overview](../benchmarks/overview.md) · [methods index](index.md) · [interactive dashboard](../landing/dashboard.md)